--- /srv/rebuilderd/tmp/rebuilderd7B3uqT/inputs/macaulay2-common_1.25.11+ds-1_all.deb +++ /srv/rebuilderd/tmp/rebuilderd7B3uqT/out/macaulay2-common_1.25.11+ds-1_all.deb ├── file list │ @@ -1,3 +1,3 @@ │ -rw-r--r-- 0 0 0 4 2025-11-14 16:08:07.000000 debian-binary │ --rw-r--r-- 0 0 0 540356 2025-11-14 16:08:07.000000 control.tar.xz │ --rw-r--r-- 0 0 0 31303512 2025-11-14 16:08:07.000000 data.tar.xz │ +-rw-r--r-- 0 0 0 540404 2025-11-14 16:08:07.000000 control.tar.xz │ +-rw-r--r-- 0 0 0 31302720 2025-11-14 16:08:07.000000 data.tar.xz ├── control.tar.xz │ ├── control.tar │ │ ├── ./control │ │ │ @@ -1,13 +1,13 @@ │ │ │ Package: macaulay2-common │ │ │ Source: macaulay2 │ │ │ Version: 1.25.11+ds-1 │ │ │ Architecture: all │ │ │ Maintainer: Debian Math Team │ │ │ -Installed-Size: 305296 │ │ │ +Installed-Size: 305292 │ │ │ Depends: fonts-katex (>= 0.16.10+~cs6.1.0), libjs-bootsidemenu (>= 1.0.0), libjs-bootstrap5 (>= 5.3.8+dfsg), libjs-d3 (>= 3.5.17), libjs-jquery (>= 3.7.1+dfsg+~3.5.33), 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), node-fortawesome-fontawesome-free (>= 6.7.2+ds1) │ │ │ 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 │ │ │ @@ -3724,18 +3724,18 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 76682 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/BeginningMacaulay2/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4226 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/BeginningMacaulay2/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 2909 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/BeginningMacaulay2/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Benchmark/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Benchmark/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 2927 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Benchmark/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Benchmark/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 423 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Benchmark/example-output/_run__Benchmarks.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 433 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Benchmark/example-output/_run__Benchmarks.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Benchmark/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 29 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Benchmark/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 5574 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Benchmark/html/_run__Benchmarks.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 5584 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Benchmark/html/_run__Benchmarks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5233 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Benchmark/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4242 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Benchmark/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 2912 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Benchmark/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/BernsteinSato/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/BernsteinSato/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 289851 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/BernsteinSato/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/BernsteinSato/example-output/ │ │ │ @@ -4316,22 +4316,22 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5291 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Browse/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4374 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Browse/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3057 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Browse/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Bruns/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Bruns/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 19549 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Bruns/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Bruns/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 4572 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Bruns/example-output/_bruns.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4571 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Bruns/example-output/_bruns.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1717 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Bruns/example-output/_bruns__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2579 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Bruns/example-output/_elementary.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1678 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Bruns/example-output/_evans__Griffith.out │ │ │ -rw-r--r-- 0 root (0) root (0) 570 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Bruns/example-output/_is__Syzygy.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Bruns/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 49 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Bruns/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 14041 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Bruns/html/_bruns.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 14040 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Bruns/html/_bruns.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8831 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Bruns/html/_bruns__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10497 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Bruns/html/_elementary.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8129 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Bruns/html/_evans__Griffith.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6572 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Bruns/html/_is__Syzygy.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9591 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Bruns/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6809 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Bruns/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4075 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Bruns/html/toc.html │ │ │ @@ -4434,15 +4434,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1448 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_is__Quasi__Isomorphism.out │ │ │ -rw-r--r-- 0 root (0) root (0) 771 2025-11-14 16:08:07.000000 ./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 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_koszul__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1962 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_minimize_lp__Chain__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 694 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_nonzero__Max.out │ │ │ -rw-r--r-- 0 root (0) root (0) 684 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_prepend__Zero__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 899 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_remove__Zero__Trailing__Terms.out │ │ │ --rw-r--r-- 0 root (0) root (0) 3452 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_resolution__Of__Chain__Complex.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3450 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_resolution__Of__Chain__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 541 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_resolution_lp__Chain__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2570 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_scarf__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 537 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_substitute_lp__Chain__Complex_cm__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 672 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_taylor.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1333 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_taylor__Resolution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1351 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_trivial__Homological__Truncation.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/ │ │ │ @@ -4465,15 +4465,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5473 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_koszul__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7220 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_koszul__Complex_lp..._cm__Length__Limit_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10096 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_minimize_lp__Chain__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6612 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_nonzero__Max.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6519 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_nonzero__Min.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5900 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_prepend__Zero__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6665 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_remove__Zero__Trailing__Terms.html │ │ │ --rw-r--r-- 0 root (0) root (0) 12264 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_resolution__Of__Chain__Complex.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 12262 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_resolution__Of__Chain__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7373 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_resolution__Of__Chain__Complex_lp..._cm__Length__Limit_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9135 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_resolution_lp__Chain__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10433 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_scarf__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5755 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_substitute_lp__Chain__Complex_cm__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5714 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_taylor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6750 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_taylor__Resolution.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7255 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_taylor__Resolution_lp..._cm__Length__Limit_eq_gt..._rp.html │ │ │ @@ -4502,49 +4502,49 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 12258 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ChainComplexOperations/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8418 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ChainComplexOperations/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4775 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ChainComplexOperations/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 123136 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 4377 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___C__S__M.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1501 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Check__Smooth.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4376 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___C__S__M.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1502 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Check__Smooth.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3419 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Check__Toric__Variety__Valid.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3442 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Chern.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2404 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Chow__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 265 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Class__In__Chow__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 775 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Class__In__Toric__Chow__Ring.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2016 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Comp__Method.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4379 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Euler.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2014 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Comp__Method.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4377 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Euler.out │ │ │ -rw-r--r-- 0 root (0) root (0) 342 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Euler__Affine.out │ │ │ --rw-r--r-- 0 root (0) root (0) 760 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Inds__Of__Smooth.out │ │ │ --rw-r--r-- 0 root (0) root (0) 667 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Input__Is__Smooth.out │ │ │ --rw-r--r-- 0 root (0) root (0) 624 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Method.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 759 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Inds__Of__Smooth.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 665 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Input__Is__Smooth.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 621 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Method.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1049 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Multi__Proj__Coord__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6662 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Output.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3327 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Segre.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1666 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Toric__Chow__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 570 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/_is__Multi__Homogeneous.out │ │ │ -rw-r--r-- 0 root (0) root (0) 802 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/_probabilistic_spalgorithm.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 665 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 55 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 23694 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___C__S__M.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6375 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Check__Smooth.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 23693 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___C__S__M.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6376 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Check__Smooth.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11239 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Check__Toric__Variety__Valid.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18179 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Chern.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9577 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Chow__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6409 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Class__In__Chow__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7339 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Class__In__Toric__Chow__Ring.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10384 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Comp__Method.html │ │ │ --rw-r--r-- 0 root (0) root (0) 18562 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Euler.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10382 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Comp__Method.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 18560 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Euler.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5955 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Euler__Affine.html │ │ │ --rw-r--r-- 0 root (0) root (0) 5903 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Inds__Of__Smooth.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6013 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Input__Is__Smooth.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6466 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Method.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 5902 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Inds__Of__Smooth.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6011 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Input__Is__Smooth.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6463 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Method.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8377 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Multi__Proj__Coord__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16398 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Output.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16710 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Segre.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8690 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Toric__Chow__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4910 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/_bertini__Check.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5128 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/_configuring_sp__Bertini.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7468 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/_is__Multi__Homogeneous.html │ │ │ @@ -4559,15 +4559,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 2624 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/___Chordal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 304 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/___Ring__Element_sp_pc_sp__Chordal__Net.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1273 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/___Ring__Map_sp__Chordal__Net.out │ │ │ -rw-r--r-- 0 root (0) root (0) 178 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_adjacent__Minors__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2151 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_chordal__Elim.out │ │ │ -rw-r--r-- 0 root (0) root (0) 969 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_chordal__Graph.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1787 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_chordal__Net.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1202 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_chordal__Net_lp__Hash__Table_cm__Hash__Table_cm__Elim__Tree_cm__Digraph_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1196 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_chordal__Net_lp__Hash__Table_cm__Hash__Table_cm__Elim__Tree_cm__Digraph_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1633 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_chordal__Tria.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2559 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_chordal_spnetworks_spexamples.out │ │ │ -rw-r--r-- 0 root (0) root (0) 429 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_chromatic__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 237 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_codim__Count.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1454 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_components_lp__Chordal__Net_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 414 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_constraint__Graph.out │ │ │ -rw-r--r-- 0 root (0) root (0) 726 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_digraph_lp__Chordal__Net_rp.out │ │ │ @@ -4600,15 +4600,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 3907 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Chordal/html/___Get__Table.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7598 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Chordal/html/___Ring__Element_sp_pc_sp__Chordal__Net.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7248 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Chordal/html/___Ring__Map_sp__Chordal__Net.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5071 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Chordal/html/_adjacent__Minors__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10905 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Chordal/html/_chordal__Elim.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7288 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Chordal/html/_chordal__Graph.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9374 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Chordal/html/_chordal__Net.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8344 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Chordal/html/_chordal__Net_lp__Hash__Table_cm__Hash__Table_cm__Elim__Tree_cm__Digraph_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8338 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Chordal/html/_chordal__Net_lp__Hash__Table_cm__Hash__Table_cm__Elim__Tree_cm__Digraph_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8861 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Chordal/html/_chordal__Tria.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10852 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Chordal/html/_chordal_spnetworks_spexamples.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5434 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Chordal/html/_chromatic__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7090 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Chordal/html/_codim__Count.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8141 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Chordal/html/_components_lp__Chordal__Net_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5721 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Chordal/html/_constraint__Graph.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7695 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Chordal/html/_digraph_lp__Chordal__Net_rp.html │ │ │ @@ -4801,23 +4801,23 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 36467 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 26490 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15240 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CohomCalg/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CohomCalg/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 13418 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CohomCalg/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CohomCalg/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 13261 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CohomCalg/example-output/___Cohom__Calg.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 13260 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CohomCalg/example-output/___Cohom__Calg.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1045 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CohomCalg/example-output/_cohom__Calg.out │ │ │ -rw-r--r-- 0 root (0) root (0) 982 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CohomCalg/example-output/_cohom__Calg_lp__Normal__Toric__Variety_rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 97 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 3840 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/___Silent.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10103 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/_cohom__Calg.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8434 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/_cohom__Calg_lp__Normal__Toric__Variety_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 25594 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/index.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 25593 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5794 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3603 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CoincidentRootLoci/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CoincidentRootLoci/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 107816 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CoincidentRootLoci/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CoincidentRootLoci/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 1103 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CoincidentRootLoci/example-output/___Coincident__Root__Locus_sp_st_sp__Coincident__Root__Locus.out │ │ │ @@ -4900,15 +4900,15 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 231378 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 650 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___B__G__G__L.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1959 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___B__Ranks.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3164 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Complete__Intersection__Resolutions.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4598 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Eisenbud__Shamash.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4595 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Eisenbud__Shamash.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4116 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Eisenbud__Shamash__Total.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2743 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Ext__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1023 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Ext__Module__Data.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2102 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___S2.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1754 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Shamash.out │ │ │ -rw-r--r-- 0 root (0) root (0) 761 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Tate__Resolution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1322 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_complexity.out │ │ │ @@ -4932,24 +4932,24 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1851 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_make__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 857 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_make__T.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1336 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_matrix__Factorization.out │ │ │ -rw-r--r-- 0 root (0) root (0) 10478 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_new__Ext.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1284 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_odd__Ext__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 498 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_regularity__Sequence.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1314 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_splittings.out │ │ │ --rw-r--r-- 0 root (0) root (0) 379 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_sum__Two__Monomials.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 377 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_sum__Two__Monomials.out │ │ │ -rw-r--r-- 0 root (0) root (0) 450 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_two__Monomials.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 50 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5986 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___A__Ranks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4935 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Augmentation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6307 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___B__G__G__L.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9094 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___B__Ranks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5966 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Check.html │ │ │ --rw-r--r-- 0 root (0) root (0) 15259 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Eisenbud__Shamash.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 15256 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Eisenbud__Shamash.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14302 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Eisenbud__Shamash__Total.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11218 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Ext__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10327 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Ext__Module__Data.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5284 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Grading.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5531 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Hom__With__Components.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5130 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Layered.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4742 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Lift.html │ │ │ @@ -4994,15 +4994,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 10902 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_module__As__Ext.html │ │ │ -rw-r--r-- 0 root (0) root (0) 23042 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_new__Ext.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8736 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_odd__Ext__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5888 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_psi__Maps.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7031 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_regularity__Sequence.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7383 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_splittings.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5224 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_stable__Hom.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6048 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_sum__Two__Monomials.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6046 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_sum__Two__Monomials.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5573 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_tensor__With__Components.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4869 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_to__Array.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6365 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_two__Monomials.html │ │ │ -rw-r--r-- 0 root (0) root (0) 56031 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 35373 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15036 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Complexes/ │ │ │ @@ -5343,29 +5343,29 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 9518 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ConformalBlocks/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 50933 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 36624 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/___Cosmological_spcorrelator_spfor_spthe_sp2-site_spchain.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1233 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/___Gauss_sq_sphypergeometric_spfunction.out │ │ │ --rw-r--r-- 0 root (0) root (0) 3314 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/___Massless_spone-loop_sptriangle_sp__Feynman_spdiagram.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3313 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/___Massless_spone-loop_sptriangle_sp__Feynman_spdiagram.out │ │ │ -rw-r--r-- 0 root (0) root (0) 240 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/_base__Fraction__Field.out │ │ │ -rw-r--r-- 0 root (0) root (0) 424 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/_connection__Matrices.out │ │ │ -rw-r--r-- 0 root (0) root (0) 240 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/_connection__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 494 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/_gauge__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 369 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/_gauge__Transform.out │ │ │ -rw-r--r-- 0 root (0) root (0) 299 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/_is__Epsilon__Factorized.out │ │ │ -rw-r--r-- 0 root (0) root (0) 263 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/_is__Integrable.out │ │ │ -rw-r--r-- 0 root (0) root (0) 202 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/_normal__Form.out │ │ │ -rw-r--r-- 0 root (0) root (0) 244 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/_standard__Monomials.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 56 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 46480 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/___Cosmological_spcorrelator_spfor_spthe_sp2-site_spchain.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7850 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/___Gauss_sq_sphypergeometric_spfunction.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9213 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/___Massless_spone-loop_sptriangle_sp__Feynman_spdiagram.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9212 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/___Massless_spone-loop_sptriangle_sp__Feynman_spdiagram.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5611 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/_base__Fraction__Field.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6834 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/_connection__Matrices.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6189 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/_connection__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7537 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/_gauge__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7266 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/_gauge__Transform.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7099 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/_is__Epsilon__Factorized.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7183 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/_is__Integrable.html │ │ │ @@ -5587,82 +5587,82 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 16572 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/CpMackeyFunctors/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Cremona/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Cremona/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 239171 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Cremona/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 2308 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Chern__Schwartz__Mac__Pherson.out │ │ │ -rw-r--r-- 0 root (0) root (0) 859 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Codim__Bs__Inv.out │ │ │ --rw-r--r-- 0 root (0) root (0) 19792 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Cremona.out │ │ │ --rw-r--r-- 0 root (0) root (0) 526 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Euler__Characteristic.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 19785 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Cremona.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 527 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Euler__Characteristic.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1795 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Rational__Map_sp!.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2550 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Rational__Map_sp^_st_st_sp__Ideal.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2551 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Rational__Map_sp^_st_st_sp__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2314 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Rational__Map_sp_eq_eq_sp__Rational__Map.out │ │ │ 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./usr/share/doc/Macaulay2/Cremona/example-output/___Segre__Class.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 6050 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_abstract__Rational__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 42548 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_approximate__Inverse__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1530 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_coefficients_lp__Rational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 33238 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_degree__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1233 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_describe_lp__Rational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 515 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_exceptional__Locus.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1047 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2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_inverse__Map_lp..._cm__Verbose_eq_gt..._rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 46183 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_inverse_lp__Rational__Map_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1574 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_is__Birational.out │ │ │ --rw-r--r-- 0 root (0) root (0) 3447 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_is__Dominant.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 46184 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_inverse_lp__Rational__Map_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1573 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_is__Birational.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3446 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_is__Dominant.out │ │ │ -rw-r--r-- 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./usr/share/doc/Macaulay2/Cremona/example-output/_point_lp__Quotient__Ring_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1460 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_point_lp__Quotient__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4713 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_projective__Degrees.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2351 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_quadro__Quadric__Cremona__Transformation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3884 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_rational__Map.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2828 2025-11-14 16:08:07.000000 ./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) 2824 2025-11-14 16:08:07.000000 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./usr/share/doc/Macaulay2/Cremona/html/index.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 66132 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Cremona/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 37101 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Cremona/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16187 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Cremona/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Cyclotomic/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Cyclotomic/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 6101 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Cyclotomic/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Cyclotomic/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 455 2025-11-14 16:08:07.000000 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./usr/share/doc/Macaulay2/DGAlgebras/html/_zeroth__Homology.html │ │ │ -rw-r--r-- 0 root (0) root (0) 33333 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 35984 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 22250 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/DecomposableSparseSystems/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/DecomposableSparseSystems/dump/ │ │ │ @@ -6145,15 +6145,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 488 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_isolated__Vertices.out │ │ │ -rw-r--r-- 0 root (0) root (0) 544 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_line__Graph.out │ │ │ 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./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_random__Hyper__Graph.out │ │ │ -rw-r--r-- 0 root (0) root (0) 808 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_random__Uniform__Hyper__Graph.out │ │ │ -rw-r--r-- 0 root (0) root (0) 269 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_ring_lp__Hyper__Graph_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 414 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_simplicial__Complex__To__Hyper__Graph.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1666 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_smallest__Cycle__Size.out │ │ │ -rw-r--r-- 0 root (0) root (0) 899 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_spanning__Tree.out │ │ │ -rw-r--r-- 0 root (0) root (0) 847 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_vertex__Cover__Number.out │ │ │ -rw-r--r-- 0 root (0) root (0) 735 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_vertex__Covers.out │ │ │ @@ -6225,15 +6225,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6818 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/html/_isolated__Vertices.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6255 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/html/_line__Graph.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7368 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/html/_neighbors.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10222 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/html/_num__Connected__Components.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9019 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/html/_num__Connected__Graph__Components.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6689 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/html/_num__Triangles.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6786 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/html/_random__Graph.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9372 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/html/_random__Hyper__Graph.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9688 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/html/_random__Hyper__Graph.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6313 2025-11-14 16:08:07.000000 ./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) 5988 2025-11-14 16:08:07.000000 ./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) 7405 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/html/_random__Uniform__Hyper__Graph.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5913 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/html/_ring_lp__Hyper__Graph_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6754 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/html/_simplicial__Complex__To__Hyper__Graph.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8160 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/html/_smallest__Cycle__Size.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6428 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/html/_spanning__Tree.html │ │ │ @@ -6255,23 +6255,23 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 9316 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EigenSolver/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4942 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EigenSolver/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3069 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EigenSolver/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 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./usr/share/doc/Macaulay2/Elimination/example-output/_discriminant_lp__Ring__Element_cm__Ring__Element_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 891 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Elimination/example-output/_eliminate.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 9030 2025-11-14 16:08:07.000000 ./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 2025-11-14 16:08:07.000000 ./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 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Elimination/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 24 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Elimination/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 7359 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Elimination/html/_discriminant_lp__Ring__Element_cm__Ring__Element_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7859 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Elimination/html/_eliminate.html │ │ │ --rw-r--r-- 0 root (0) root (0) 16452 2025-11-14 16:08:07.000000 ./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) 7360 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Elimination/html/_discriminant_lp__Ring__Element_cm__Ring__Element_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7858 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Elimination/html/_eliminate.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 16453 2025-11-14 16:08:07.000000 ./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) 15904 2025-11-14 16:08:07.000000 ./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) 7141 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Elimination/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5367 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Elimination/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3467 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Elimination/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EliminationMatrices/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EliminationMatrices/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 98399 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EliminationMatrices/dump/rawdocumentation.dump │ │ │ @@ -6460,35 +6460,35 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6990 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EngineTests/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 10834 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 367 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/example-output/_lines__Hypersurface.out │ │ │ -rw-r--r-- 0 root (0) root (0) 193 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/example-output/_multiple__Cover.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2171 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/example-output/_rational__Curve.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2172 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/example-output/_rational__Curve.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 48 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5203 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/_lines__Hypersurface.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5211 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/_multiple__Cover.html │ │ │ --rw-r--r-- 0 root (0) root (0) 11547 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/_rational__Curve.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 11548 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/_rational__Curve.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7109 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5529 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3452 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EquivariantGB/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EquivariantGB/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 52732 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EquivariantGB/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 235 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/___Equivariant__G__B.out │ │ │ -rw-r--r-- 0 root (0) 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./usr/share/doc/Macaulay2/EquivariantGB/example-output/_merge__P__Q.out │ │ │ -rw-r--r-- 0 root (0) root (0) 161 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_min_lp__Priority__Queue_rp.out │ │ │ @@ -6503,15 +6503,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 4953 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/___Shift.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3844 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/___Symmetrize.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9077 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_build__E__Monomial__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10316 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_build__E__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6607 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_build__E__Ring_lp__Ring_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5895 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_delete__Min.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6472 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_egb.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9002 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_egb__Toric.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9000 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_egb__Toric.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6615 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_egb_lp..._cm__Algorithm_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6126 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_exponent__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7150 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_inc__Orbit.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6020 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_insert_lp__Priority__Queue_cm__Thing_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5369 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_length_lp__Priority__Queue_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6478 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_merge__P__Q.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5290 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_min_lp__Priority__Queue_rp.html │ │ │ @@ -6752,70 +6752,70 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 10138 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FGLM/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4568 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FGLM/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 2981 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FGLM/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 142955 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 29586 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/___Fast__Minors__Strategy__Tutorial.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 29590 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/___Fast__Minors__Strategy__Tutorial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1035 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/___Point__Options.out │ │ │ -rw-r--r-- 0 root (0) root (0) 14279 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/___Regular__In__Codimension__Tutorial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1054 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/___Strategy__Default.out │ │ │ -rw-r--r-- 0 root (0) root (0) 337 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_choose__Good__Minors.out │ │ │ -rw-r--r-- 0 root (0) root (0) 246 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_choose__Random__Submatrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 307 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_choose__Submatrix__Largest__Degree.out │ │ │ -rw-r--r-- 0 root (0) root (0) 308 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_choose__Submatrix__Smallest__Degree.out │ │ │ -rw-r--r-- 0 root (0) root (0) 533 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_get__Submatrix__Of__Rank.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1786 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_is__Codim__At__Least.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1787 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_is__Codim__At__Least.out │ │ │ -rw-r--r-- 0 root (0) root (0) 275 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_is__Rank__At__Least.out │ │ │ --rw-r--r-- 0 root (0) root (0) 436 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_proj__Dim.out │ │ │ --rw-r--r-- 0 root (0) root (0) 424 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_recursive__Minors.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 435 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_proj__Dim.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 423 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_recursive__Minors.out │ │ │ -rw-r--r-- 0 root (0) root (0) 25045 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_regular__In__Codimension.out │ │ │ -rw-r--r-- 0 root (0) root (0) 273 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_reorder__Polynomial__Ring.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 586 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 32 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6134 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Det__Strategy.html │ │ │ --rw-r--r-- 0 root (0) root (0) 50410 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Fast__Minors__Strategy__Tutorial.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 50414 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Fast__Minors__Strategy__Tutorial.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5373 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Max__Minors.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4415 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Min__Dimension.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4447 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Modulus.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6808 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Point__Options.html │ │ │ -rw-r--r-- 0 root (0) root (0) 27490 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Regular__In__Codimension__Tutorial.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14641 2025-11-14 16:08:07.000000 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16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_is__Codim__At__Least.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6128 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_is__Dim__At__Most.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9506 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_is__Rank__At__Least.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5364 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_is__Rank__At__Least_lp..._cm__Threads_eq_gt..._rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10205 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_proj__Dim.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7912 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_recursive__Minors.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10204 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_proj__Dim.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7911 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_recursive__Minors.html │ │ │ -rw-r--r-- 0 root (0) root (0) 43512 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_regular__In__Codimension.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6458 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_reorder__Polynomial__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 24783 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 26704 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7533 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FastMinors/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 25936 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 1849 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/___Fitting_spideals_spof_spfinite_spmodules.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1848 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/___Fitting_spideals_spof_spfinite_spmodules.out │ │ │ -rw-r--r-- 0 root (0) root (0) 334 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/_affine__Part.out │ │ │ -rw-r--r-- 0 root (0) root (0) 694 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/_co1__Fitting.out │ │ │ -rw-r--r-- 0 root (0) root (0) 848 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/_gauss__Col.out │ │ │ -rw-r--r-- 0 root (0) root (0) 365 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/_gotzmann__Test.out │ │ │ -rw-r--r-- 0 root (0) root (0) 610 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/_next__Degree.out │ │ │ -rw-r--r-- 0 root (0) root (0) 197 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/_quot__Scheme.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 32 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 10957 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/___Fitting_spideals_spof_spfinite_spmodules.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10956 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/___Fitting_spideals_spof_spfinite_spmodules.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6153 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/_affine__Part.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7448 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/_co1__Fitting.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7377 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/_gauss__Col.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8186 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/_gotzmann__Test.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7246 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/_next__Degree.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5849 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/_quot__Scheme.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9538 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/index.html │ │ │ @@ -7037,16 +7037,16 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 105269 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 338 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/___Bounds.out │ │ │ -rw-r--r-- 0 root (0) root (0) 318 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/___Frobenius__Thresholds.out │ │ │ -rw-r--r-- 0 root (0) root (0) 793 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/___Guess__Strategy.out │ │ │ -rw-r--r-- 0 root (0) root (0) 866 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_compare__F__P__T.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4032 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_fpt.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2460 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_frobenius__Nu.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4033 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_fpt.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2456 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_frobenius__Nu.out │ │ │ -rw-r--r-- 0 root (0) root (0) 760 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_is__F__Jumping__Exponent.out │ │ │ -rw-r--r-- 0 root (0) root (0) 552 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_is__F__P__T.out │ │ │ -rw-r--r-- 0 root (0) root (0) 828 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_is__Simple__Normal__Crossing.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 617 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 12 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5582 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Bounds.html │ │ │ @@ -7057,16 +7057,16 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 4963 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Global__Frobenius__Root.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10580 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Guess__Strategy.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4665 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Return__List.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4769 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Search.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4735 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Standard__Power.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5945 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Use__Special__Algorithms.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14756 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_compare__F__P__T.html │ │ │ --rw-r--r-- 0 root (0) root (0) 25998 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_fpt.html │ │ │ --rw-r--r-- 0 root (0) root (0) 24543 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_frobenius__Nu.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 25999 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_fpt.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 24539 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_frobenius__Nu.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13295 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_is__F__Jumping__Exponent.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12519 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_is__F__P__T.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9887 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_is__Simple__Normal__Crossing.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19600 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20028 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7092 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/FunctionFieldDesingularization/ │ │ │ @@ -8273,49 +8273,49 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 50235 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11279 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5699 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/GroebnerWalk/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/GroebnerWalk/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 18071 2025-11-14 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2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/GroebnerWalk/example-output/_set__Walk__Trace.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/GroebnerWalk/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 36 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/GroebnerWalk/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6615 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/GroebnerWalk/html/_get__Walk__Trace.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8245 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/GroebnerWalk/html/_groebner__Walk.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13559 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/GroebnerWalk/html/_groebner__Walk_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7909 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/GroebnerWalk/html/_set__Walk__Trace.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10304 2025-11-14 16:08:07.000000 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2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_primary__Invariants_lp..._cm__Dade_eq_gt..._rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10764 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_primary__Invariants_lp..._cm__Degree__Vector_eq_gt..._rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10914 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_primary__Invariants_lp..._cm__Degree__Vector_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6147 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_rank_lp__Diagonal__Action_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7132 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_relations_lp__Finite__Group__Action_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8243 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_reynolds__Operator.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5298 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_ring_lp__Group__Action_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8361 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_schreier__Graph.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8886 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_secondary__Invariants.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11430 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_secondary__Invariants_lp..._cm__Print__Degree__Polynomial_eq_gt..._rp.html │ │ │ @@ -8934,15 +8934,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 78221 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Jets/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Jets/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 2738 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Jets/example-output/___Example_sp1.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2643 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Jets/example-output/___Example_sp2.out │ │ │ -rw-r--r-- 0 root (0) root (0) 9513 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Jets/example-output/___Example_sp3.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2465 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Jets/example-output/___Example_sp4.out │ │ │ -rw-r--r-- 0 root (0) root (0) 556 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Jets/example-output/___J__J.out │ │ │ --rw-r--r-- 0 root (0) root (0) 5997 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Jets/example-output/___Storing_sp__Computations.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 5996 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Jets/example-output/___Storing_sp__Computations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 634 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Jets/example-output/_jets__Projection.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1511 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Jets/example-output/_jets__Radical.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1210 2025-11-14 16:08:07.000000 ./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 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Jets/example-output/_jets_lp__Z__Z_cm__Graph_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2492 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Jets/example-output/_jets_lp__Z__Z_cm__Ideal_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1463 2025-11-14 16:08:07.000000 ./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 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Jets/example-output/_jets_lp__Z__Z_cm__Quotient__Ring_rp.out │ │ │ @@ -8955,15 +8955,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 70 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Jets/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 9118 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Jets/html/___Example_sp1.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9407 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Jets/html/___Example_sp2.html │ │ │ -rw-r--r-- 0 root (0) root (0) 17870 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Jets/html/___Example_sp3.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10712 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Jets/html/___Example_sp4.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5311 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Jets/html/___J__J.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4217 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Jets/html/___Saturate.html │ │ │ --rw-r--r-- 0 root (0) root (0) 16218 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Jets/html/___Storing_sp__Computations.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 16217 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Jets/html/___Storing_sp__Computations.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5102 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Jets/html/_jet.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6255 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Jets/html/_jets.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4619 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Jets/html/_jets__Base.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4534 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Jets/html/_jets__Info.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4884 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Jets/html/_jets__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4662 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Jets/html/_jets__Max__Order.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7171 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Jets/html/_jets__Projection.html │ │ │ @@ -8984,25 +8984,25 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 9828 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Jets/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7299 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Jets/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 101880 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 1949 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_all__Gradings.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2784 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_analyze__Strand.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2786 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_analyze__Strand.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5835 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_canonical__Homotopies.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1036 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2257 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Table.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2261 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Table.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3425 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Tables.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1002 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Det.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1003 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Det.out │ │ │ -rw-r--r-- 0 root (0) root (0) 268 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_compute__Bound.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6823 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_correspondence__Scroll.out │ │ │ -rw-r--r-- 0 root (0) root (0) 804 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_cox__Matrices.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1634 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_degenerate__K3.out │ │ │ --rw-r--r-- 0 root (0) root (0) 6875 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_degenerate__K3__Betti__Tables.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 6872 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_degenerate__K3__Betti__Tables.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2293 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_homotopy__Ranks.out │ │ │ -rw-r--r-- 0 root (0) root (0) 720 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_irrelevant__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 551 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_product__Of__Projective__Spaces.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1031 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_relative__Equations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1075 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_relative__Resolution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1638 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_relative__Resolution__Twists.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2221 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_resonance__Det.out │ │ │ @@ -9011,26 +9011,26 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 3037 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_schreyer__Name.out │ │ │ -rw-r--r-- 0 root (0) root (0) 813 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_small__Diagonal.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 30 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4913 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/___Fine__Grading.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4489 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/___Scrolls.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7927 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_all__Gradings.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9535 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_analyze__Strand.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9537 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_analyze__Strand.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6565 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_canonical__Carpet.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13299 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_canonical__Homotopies.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11993 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9787 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Table.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9791 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Table.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10621 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Tables.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6851 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Det.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6852 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Det.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6736 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_compute__Bound.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18189 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_correspondence__Scroll.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7392 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_cox__Matrices.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9428 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_degenerate__K3.html │ │ │ --rw-r--r-- 0 root (0) root (0) 14901 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_degenerate__K3__Betti__Tables.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 14898 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_degenerate__K3__Betti__Tables.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5106 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_gorenstein__Double.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8478 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_hankel__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8111 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_homotopy__Ranks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6177 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_irrelevant__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8020 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_product__Of__Projective__Spaces.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7118 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_relative__Equations.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7204 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_relative__Resolution.html │ │ │ @@ -9163,27 +9163,27 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1849 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LAYOUT.gz │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LLLBases/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LLLBases/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 33504 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LLLBases/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LLLBases/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 2858 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LLLBases/example-output/___L__L__L.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2411 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LLLBases/example-output/___L__L__L_lp..._cm__Change__Matrix_eq_gt..._rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1436 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LLLBases/example-output/___L__L__L_lp..._cm__Strategy_eq_gt..._rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1432 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LLLBases/example-output/___L__L__L_lp..._cm__Strategy_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 476 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LLLBases/example-output/_gcd__L__L__L.out │ │ │ -rw-r--r-- 0 root (0) root (0) 678 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LLLBases/example-output/_is__L__L__L.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LLLBases/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 28 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LLLBases/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4088 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___B__K__Z.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4034 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Cohen__Engine.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4148 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Cohen__Top__Level.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3951 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Givens.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3444 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Hermite.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10841 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___L__L__L.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9865 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___L__L__L_lp..._cm__Change__Matrix_eq_gt..._rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 22128 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___L__L__L_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 22124 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___L__L__L_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3996 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___N__T__L.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3797 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Real__F__P.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3817 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Real__Q__P.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3999 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Real__Q__P1.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3833 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Real__R__R.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3866 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Real__X__D.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4470 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Threshold.html │ │ │ @@ -9199,15 +9199,15 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 70564 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 604 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/___Working_spwith_splattice_sppolytopes.out │ │ │ -rw-r--r-- 0 root (0) root (0) 265 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_adjoint__Polytope.out │ │ │ -rw-r--r-- 0 root (0) root (0) 329 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_ambient__Halfspaces.out │ │ │ --rw-r--r-- 0 root (0) root (0) 596 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_are__Isomorphic.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 595 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_are__Isomorphic.out │ │ │ -rw-r--r-- 0 root (0) root (0) 684 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_cayley.out │ │ │ -rw-r--r-- 0 root (0) root (0) 85 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_codegree.out │ │ │ -rw-r--r-- 0 root (0) root (0) 281 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_degree__Of__Jet__Separation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 312 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_epsilon__Bounds.out │ │ │ -rw-r--r-- 0 root (0) root (0) 304 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_gauss__Fiber.out │ │ │ -rw-r--r-- 0 root (0) root (0) 373 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_gauss__Image.out │ │ │ -rw-r--r-- 0 root (0) root (0) 350 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_gaussk__Fiber.out │ │ │ @@ -9224,15 +9224,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 235 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_toric__Div.out │ │ │ -rw-r--r-- 0 root (0) root (0) 167 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_torus__Embedding.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 17 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5005 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/___Working_spwith_splattice_sppolytopes.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5659 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_adjoint__Polytope.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5749 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_ambient__Halfspaces.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7712 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_are__Isomorphic.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7711 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_are__Isomorphic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9462 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_cayley.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4957 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_codegree.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6907 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_degree__Of__Jet__Separation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6587 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_epsilon__Bounds.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6730 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_gauss__Fiber.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6764 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_gauss__Image.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7032 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_gaussk__Fiber.html │ │ │ @@ -9480,15 +9480,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 231 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_comp__Min.out │ │ │ -rw-r--r-- 0 root (0) root (0) 283 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_diagonal__Multidegrees.out │ │ │ -rw-r--r-- 0 root (0) root (0) 191 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_find__Mins.out │ │ │ -rw-r--r-- 0 root (0) root (0) 701 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_find__Region.out │ │ │ -rw-r--r-- 0 root (0) root (0) 754 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_irrelevant__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1046 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_is__Linear__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 904 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_is__Quasi__Linear.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1149 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_linear__Truncations__Bound.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1150 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_linear__Truncations__Bound.out │ │ │ -rw-r--r-- 0 root (0) root (0) 976 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_multigraded__Polynomial__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1807 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_partial__Regularities.out │ │ │ -rw-r--r-- 0 root (0) root (0) 577 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_regularity__Bound.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1734 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_support__Of__Tor.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 604 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 61 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/.Headline │ │ │ @@ -9496,29 +9496,29 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6901 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_diagonal__Multidegrees.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5897 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_find__Mins.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9399 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_find__Region.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7158 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_irrelevant__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8139 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_is__Linear__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9029 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_is__Quasi__Linear.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8199 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_linear__Truncations.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8380 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_linear__Truncations__Bound.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8381 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_linear__Truncations__Bound.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10456 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_multigraded__Polynomial__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8861 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_partial__Regularities.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7847 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_regularity__Bound.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8621 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_support__Of__Tor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 25856 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15787 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6141 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LocalRings/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LocalRings/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 68684 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LocalRings/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LocalRings/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 746 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LocalRings/example-output/___Local__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1030 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LocalRings/example-output/___Local__Rings.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1024 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LocalRings/example-output/_hilbert__Samuel__Function.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1022 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LocalRings/example-output/_hilbert__Samuel__Function.out │ │ │ -rw-r--r-- 0 root (0) root (0) 316 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LocalRings/example-output/_is__Well__Defined_lp__Local__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1328 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LocalRings/example-output/_lift__Up.out │ │ │ -rw-r--r-- 0 root (0) root (0) 372 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LocalRings/example-output/_local__Complement.out │ │ │ -rw-r--r-- 0 root (0) root (0) 412 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LocalRings/example-output/_local__Mingens.out │ │ │ -rw-r--r-- 0 root (0) root (0) 574 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LocalRings/example-output/_local__Modulo.out │ │ │ -rw-r--r-- 0 root (0) root (0) 451 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LocalRings/example-output/_local__Prune.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2748 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LocalRings/example-output/_local__Resolution.out │ │ │ @@ -9533,15 +9533,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 4116 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LocalRings/html/_char_lp__Local__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4083 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LocalRings/html/_coefficient__Ring_lp__Local__Ring_rp.html │ │ │ 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./usr/share/doc/Macaulay2/LocalRings/html/_hilbert__Samuel__Function.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4116 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LocalRings/html/_is__Commutative_lp__Local__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5391 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LocalRings/html/_is__Well__Defined_lp__Local__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10432 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LocalRings/html/_lift__Up.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7171 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LocalRings/html/_local__Complement.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7041 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LocalRings/html/_local__Mingens.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7847 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LocalRings/html/_local__Modulo.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7139 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/LocalRings/html/_local__Prune.html │ │ │ @@ -9712,15 +9712,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1999 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Expression.out │ │ │ -rw-r--r-- 0 root (0) root (0) 788 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Ext^__Z__Z_lp__Matrix_cm__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1449 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Ext^__Z__Z_lp__Module_cm__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 548 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Flat__Monoid.out │ │ │ -rw-r--r-- 0 root (0) root (0) 100 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Function__Closure.out │ │ │ -rw-r--r-- 0 root (0) root (0) 340 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Function_sp_at_at_sp__Function.out │ │ │ -rw-r--r-- 0 root (0) root (0) 836 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Function_sp_us_sp__Thing.out │ │ │ --rw-r--r-- 0 root (0) root (0) 414 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__Cstats.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 415 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__Cstats.out │ │ │ -rw-r--r-- 0 root (0) root (0) 611 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__F.out │ │ │ -rw-r--r-- 0 root (0) root (0) 194 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__Lex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 561 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__Rev__Lex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 175 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Gamma.out │ │ │ -rw-r--r-- 0 root (0) root (0) 230 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Global__Assign__Hook.out │ │ │ -rw-r--r-- 0 root (0) root (0) 371 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Global__Release__Hook.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6869 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Grassmannian.out │ │ │ @@ -9821,15 +9821,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 238 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Ring__Family_sp_us_st.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1401 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Ring__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 428 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Ring__Map_sp__Ring__Element.out │ │ │ -rw-r--r-- 0 root (0) root (0) 399 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Ring_sp^_sp__List.out │ │ │ -rw-r--r-- 0 root (0) root (0) 217 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Ring_sp^_sp__Z__Z.out │ │ │ -rw-r--r-- 0 root (0) root (0) 755 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Ring_sp_sl_sp__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1854 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___S__V__D.out │ │ │ --rw-r--r-- 0 root (0) root (0) 349 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___S__V__D_lp..._cm__Divide__Conquer_eq_gt..._rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 348 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___S__V__D_lp..._cm__Divide__Conquer_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5127 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Schreyer_sporders.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1024 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Schubert.out │ │ │ -rw-r--r-- 0 root (0) root (0) 232 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Self__Initializing__Type.out │ │ │ -rw-r--r-- 0 root (0) root (0) 272 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Sequence.out │ │ │ -rw-r--r-- 0 root (0) root (0) 950 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Set.out │ │ │ -rw-r--r-- 0 root (0) root (0) 205 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Set_sp-_sp__Set.out │ │ │ -rw-r--r-- 0 root (0) root (0) 157 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Set_sp_sh_qu_sp__Thing.out │ │ │ @@ -9989,23 +9989,23 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 202 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_between.out │ │ │ -rw-r--r-- 0 root (0) root (0) 332 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_binomial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 609 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_block__Matrix__Form.out │ │ │ -rw-r--r-- 0 root (0) root (0) 223 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_borel_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 725 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_break.out │ │ │ -rw-r--r-- 0 root (0) root (0) 755 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cache.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1307 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_caching_spcomputation_spresults.out │ │ │ --rw-r--r-- 0 root (0) root (0) 591 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cancel__Task_lp__Task_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 592 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cancel__Task_lp__Task_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1803 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_capture.out │ │ │ -rw-r--r-- 0 root (0) root (0) 77 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_ceiling_lp__Number_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 128 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_center__String.out │ │ │ -rw-r--r-- 0 root (0) root (0) 946 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_change__Base.out │ │ │ -rw-r--r-- 0 root (0) root (0) 259 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_change__Directory.out │ │ │ -rw-r--r-- 0 root (0) root (0) 227 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_char.out │ │ │ -rw-r--r-- 0 root (0) root (0) 196 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_characters.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1522 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_check.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1523 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_check.out │ │ │ -rw-r--r-- 0 root (0) root (0) 303 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_class.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1307 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_clean.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2015 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_code.out │ │ │ -rw-r--r-- 0 root (0) root (0) 543 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_codim_lp__Ideal_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 241 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_codim_lp__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 164 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_codim_lp__Monomial__Ideal_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 267 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_codim_lp__Quotient__Ring_rp.out │ │ │ @@ -10021,22 +10021,22 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 288 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_column__Swap.out │ │ │ -rw-r--r-- 0 root (0) root (0) 175 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_columnate.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1095 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_combine.out │ │ │ -rw-r--r-- 0 root (0) root (0) 198 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_command__Interpreter.out │ │ │ -rw-r--r-- 0 root (0) root (0) 149 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_comments.out │ │ │ -rw-r--r-- 0 root (0) root (0) 331 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_common__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 465 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_commonest.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1535 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_communicating_spwith_spprograms.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1547 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_communicating_spwith_spprograms.out │ │ │ -rw-r--r-- 0 root (0) root (0) 225 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_comodule.out │ │ │ -rw-r--r-- 0 root (0) root (0) 372 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compact__Matrix__Form.out │ │ │ -rw-r--r-- 0 root (0) root (0) 251 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compare__Exchange.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1085 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compose.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2477 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compositions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 272 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compress.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4286 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_computing_sp__Groebner_spbases.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4288 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_computing_sp__Groebner_spbases.out │ │ │ -rw-r--r-- 0 root (0) root (0) 640 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_computing_spsyzygies.out │ │ │ -rw-r--r-- 0 root (0) root (0) 103 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_concatenate.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1767 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_concatenating_spmatrices.out │ │ │ -rw-r--r-- 0 root (0) root (0) 392 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_conditional_spexecution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 148 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_conjugate.out │ │ │ -rw-r--r-- 0 root (0) root (0) 478 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_conjugate_lp__Partition_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 517 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_constructing_spmaps_spbetween_spmodules.out │ │ │ @@ -10047,15 +10047,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 740 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_copy__File_lp__String_cm__String_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 112 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cos.out │ │ │ -rw-r--r-- 0 root (0) root (0) 113 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cosh.out │ │ │ -rw-r--r-- 0 root (0) root (0) 114 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cot.out │ │ │ -rw-r--r-- 0 root (0) root (0) 116 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_coth.out │ │ │ -rw-r--r-- 0 root (0) root (0) 297 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cover__Map_lp__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 521 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cover_lp__Module_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 314 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cpu__Time.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 313 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cpu__Time.out │ │ │ -rw-r--r-- 0 root (0) root (0) 243 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_create__Task.out │ │ │ -rw-r--r-- 0 root (0) root (0) 483 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_creating_span_spideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 451 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_creating_spand_spwriting_spfiles.out │ │ │ -rw-r--r-- 0 root (0) root (0) 115 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_csc.out │ │ │ -rw-r--r-- 0 root (0) root (0) 116 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_csch.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1090 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_current.out │ │ │ -rw-r--r-- 0 root (0) root (0) 84 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_current__Column__Number.out │ │ │ @@ -10116,15 +10116,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1720 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_eagon__Northcott.out │ │ │ -rw-r--r-- 0 root (0) root (0) 532 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_eigenvalues.out │ │ │ -rw-r--r-- 0 root (0) root (0) 717 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_eigenvectors.out │ │ │ -rw-r--r-- 0 root (0) root (0) 111 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_eint.out │ │ │ -rw-r--r-- 0 root (0) root (0) 103 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elapsed__Time.out │ │ │ -rw-r--r-- 0 root (0) root (0) 162 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elapsed__Timing.out │ │ │ -rw-r--r-- 0 root (0) root (0) 366 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elements.out │ │ │ --rw-r--r-- 0 root (0) root (0) 21209 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elimination_spof_spvariables.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 21213 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elimination_spof_spvariables.out │ │ │ -rw-r--r-- 0 root (0) root (0) 782 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_end.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3683 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_end__Package.out │ │ │ -rw-r--r-- 0 root (0) root (0) 487 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_entries.out │ │ │ -rw-r--r-- 0 root (0) root (0) 185 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_entries_lp__Vector_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 605 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_equality_spand_spcontainment.out │ │ │ -rw-r--r-- 0 root (0) root (0) 111 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_erf.out │ │ │ -rw-r--r-- 0 root (0) root (0) 114 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_erfc.out │ │ │ @@ -10265,15 +10265,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1260 2025-11-14 16:08:07.000000 ./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 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_induced__Map_lp__Module_cm__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1035 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_inheritance.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1187 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_inputting_spa_spmatrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 592 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_insert.out │ │ │ -rw-r--r-- 0 root (0) root (0) 686 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_installing_spassignment_spmethods.out │ │ │ -rw-r--r-- 0 root (0) root (0) 936 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_installing_spaugmented_spassignment_spmethods.out │ │ │ --rw-r--r-- 0 root (0) root (0) 932 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_instances.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 933 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_instances.out │ │ │ -rw-r--r-- 0 root (0) root (0) 316 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_integers_spmodulo_spa_spprime.out │ │ │ -rw-r--r-- 0 root (0) root (0) 315 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_integrate.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1162 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersect.out │ │ │ -rw-r--r-- 0 root (0) root (0) 997 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersect_lp__Ideal_cm__Ideal_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 189 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersect_lp__Set_cm__Set_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 190 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersection_spof_spideals.out │ │ │ -rw-r--r-- 0 root (0) root (0) 188 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_inverse__Erf.out │ │ │ @@ -10405,30 +10405,30 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 769 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 424 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Mutable__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 125 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Ring__Element_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 345 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Ring__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 319 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Ring_cm__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 228 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Vector_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 560 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max.out │ │ │ --rw-r--r-- 0 root (0) root (0) 82 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max__Allowable__Threads.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 83 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max__Allowable__Threads.out │ │ │ -rw-r--r-- 0 root (0) root (0) 83 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max__Exponent.out │ │ │ -rw-r--r-- 0 root (0) root (0) 311 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max__Position.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1637 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_memoize.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1636 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_memoize.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1730 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_merge.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2673 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_method.out │ │ │ -rw-r--r-- 0 root (0) root (0) 970 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_method__Options_lp__Function_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6748 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_methods.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2721 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_methods_spfor_spnormal_spforms_spand_spremainder.out │ │ │ -rw-r--r-- 0 root (0) root (0) 551 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_min.out │ │ │ -rw-r--r-- 0 root (0) root (0) 84 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_min__Exponent.out │ │ │ -rw-r--r-- 0 root (0) root (0) 310 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_min__Position.out │ │ │ -rw-r--r-- 0 root (0) root (0) 925 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_mingens_lp__Groebner__Basis_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2181 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_mingens_lp__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 737 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_mingle.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1801 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimal__Betti.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1805 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimal__Betti.out │ │ │ -rw-r--r-- 0 root (0) root (0) 951 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimal__Presentation_lp__Ideal_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 607 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimal__Presentation_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 643 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimal__Presentation_lp__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1229 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimal__Presentation_lp__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 160 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimize__Filename.out │ │ │ -rw-r--r-- 0 root (0) root (0) 504 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minors_lp__Z__Z_cm__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 318 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_mkdir.out │ │ │ @@ -10519,15 +10519,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 232 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_peek_sq.out │ │ │ -rw-r--r-- 0 root (0) root (0) 267 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_permanents.out │ │ │ -rw-r--r-- 0 root (0) root (0) 874 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_permutations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 495 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pfaffian.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2202 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pfaffians.out │ │ │ -rw-r--r-- 0 root (0) root (0) 301 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pi.out │ │ │ -rw-r--r-- 0 root (0) root (0) 304 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pivots_lp__Matrix_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 13709 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_poincare.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 13700 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_poincare.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2331 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_polarize.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4055 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_polynomial_springs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 624 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_position.out │ │ │ -rw-r--r-- 0 root (0) root (0) 475 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_positions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 274 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_powermod.out │ │ │ -rw-r--r-- 0 root (0) root (0) 173 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_precision.out │ │ │ -rw-r--r-- 0 root (0) root (0) 657 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_prefix__Path.out │ │ │ @@ -10561,15 +10561,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 3534 2025-11-14 16:08:07.000000 ./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 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_quotient__Remainder.out │ │ │ -rw-r--r-- 0 root (0) root (0) 332 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_quotient__Remainder_lp__Ring__Element_cm__Ring__Element_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 825 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_quotient__Remainder_sq.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3975 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_quotient_lp__Matrix_cm__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1476 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_quotient_springs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1405 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_quotient_sq.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1250 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random__K__Rational__Point.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1251 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random__K__Rational__Point.out │ │ │ -rw-r--r-- 0 root (0) root (0) 669 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random__Mutable__Matrix_lp__Z__Z_cm__Z__Z_cm__R__R_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 399 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random__Subset.out │ │ │ -rw-r--r-- 0 root (0) root (0) 843 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random_lp__List_cm__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 138 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random_lp__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1109 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random_lp__Module_cm__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 232 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random_lp__Q__Q_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 869 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random_lp__Type_rp.out │ │ │ @@ -10656,15 +10656,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 418 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_show__User__Structure.out │ │ │ -rw-r--r-- 0 root (0) root (0) 175 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_sign.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2452 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_simple_sp__Groebner_spbasis_spcomputations_spover_spvarious_springs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 100 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_sin.out │ │ │ -rw-r--r-- 0 root (0) root (0) 933 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_singular__Locus.out │ │ │ -rw-r--r-- 0 root (0) root (0) 294 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_size2.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3173 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_smith__Normal__Form_lp__Matrix_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4507 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_solve.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4505 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_solve.out │ │ │ -rw-r--r-- 0 root (0) root (0) 891 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_some__Terms.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1085 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_sort__Columns.out │ │ │ -rw-r--r-- 0 root (0) root (0) 495 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_sort_lp__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 582 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_sort_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 278 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_source_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 265 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_source_lp__Ring__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 564 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_specifying_sptypical_spvalues.out │ │ │ @@ -10779,15 +10779,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 188 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_value_lp__Symbol_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 201 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_values.out │ │ │ -rw-r--r-- 0 root (0) root (0) 256 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_variables.out │ │ │ -rw-r--r-- 0 root (0) root (0) 770 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_vars_lp__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 153 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_vars_lp__Monoid_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 759 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_vars_lp__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1063 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_vector.out │ │ │ --rw-r--r-- 0 root (0) root (0) 12154 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_version.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 12160 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_version.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1121 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_viewing_spthe_spsymbols_spdefined_spso_spfar.out │ │ │ -rw-r--r-- 0 root (0) root (0) 467 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_wedge__Product_lp__Z__Z_cm__Z__Z_cm__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 610 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_weight__Range.out │ │ │ -rw-r--r-- 0 root (0) root (0) 349 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_what_spa_spclass_spis.out │ │ │ -rw-r--r-- 0 root (0) root (0) 708 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_what_spis_spa_sp__Groebner_spbasis_qu.out │ │ │ -rw-r--r-- 0 root (0) root (0) 754 2025-11-14 16:08:07.000000 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│ │ @@ -11587,15 +11587,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 12840 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_changes_cm_sp1.8.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6879 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_changes_cm_sp1.9.1.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3856 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_changes_cm_sp1.9.2.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10388 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_changes_cm_sp1.9.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6248 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_changes_spto_sp__Macaulay2_cm_spby_spversion.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5138 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_char.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5341 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_characters.html │ │ │ --rw-r--r-- 0 root (0) root (0) 13370 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_check.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 13371 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_check.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4076 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_chi.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6177 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_class.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8574 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_clean.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4359 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_clear__All.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3748 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_clear__Echo.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4197 2025-11-14 16:08:07.000000 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-rw-r--r-- 0 root (0) root (0) 4697 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_complete__Graph__Init.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5190 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_compute__Mixed__Volume.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7814 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_create__Seed__Pair.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7626 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_dynamic__Flower__Solve.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7624 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_dynamic__Flower__Solve.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5009 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_flower__Graph__Augment.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4670 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_flower__Graph__Init.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4889 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_get__Track__Time.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6199 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_homotopy__Graph.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4606 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_indices_lp__Point__Array_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4847 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_is__Member_lp__Abstract__Point_cm__Point__Array_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4736 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_length_lp__Point__Array_rp.html │ │ │ @@ -13763,24 +13763,24 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 39612 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Msolve/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 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root (0) 10503 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Msolve/html/_msolve__R__U__R.html │ │ │ --rw-r--r-- 0 root (0) root (0) 14361 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Msolve/html/_msolve__Real__Solutions.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 14378 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Msolve/html/_msolve__Real__Solutions.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7806 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Msolve/html/_msolve__Saturate.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18069 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Msolve/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11442 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Msolve/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4504 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Msolve/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiGradedRationalMap/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiGradedRationalMap/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 47286 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiGradedRationalMap/dump/rawdocumentation.dump │ │ │ @@ -13886,27 +13886,27 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 35822 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 369 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/___N__P.out │ │ │ -rw-r--r-- 0 root (0) root (0) 656 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_gr__Gr.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1249 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_hilbert__Sequence.out │ │ │ -rw-r--r-- 0 root (0) root (0) 413 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_j__Mult.out │ │ │ --rw-r--r-- 0 root (0) root (0) 260 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_mon__Analytic__Spread.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 262 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_mon__Analytic__Spread.out │ │ │ -rw-r--r-- 0 root (0) root (0) 573 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_mon__Reduction.out │ │ │ -rw-r--r-- 0 root (0) root (0) 596 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_monj__Mult.out │ │ │ -rw-r--r-- 0 root (0) root (0) 805 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_multiplicity__Sequence.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1239 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_print__Hilbert__Sequence.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 594 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 47 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6190 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/___N__P.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6706 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_gr__Gr.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8885 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_hilbert__Sequence.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6040 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_j__Mult.html │ │ │ --rw-r--r-- 0 root (0) root (0) 5704 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_mon__Analytic__Spread.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 5706 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_mon__Analytic__Spread.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6831 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_mon__Reduction.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6478 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_monj__Mult.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9585 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_multiplicity__Sequence.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6921 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_print__Hilbert__Sequence.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15265 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9576 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4371 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/toc.html │ │ │ @@ -13959,15 +13959,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5192 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiplierIdealsDim2/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 379736 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 279 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Embedded__Projective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1954 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Embedded__Projective__Variety_sp!.out │ │ │ --rw-r--r-- 0 root (0) root (0) 20227 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Embedded__Projective__Variety_sp_eq_eq_eq_gt_sp__Embedded__Projective__Variety.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 20228 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Embedded__Projective__Variety_sp_eq_eq_eq_gt_sp__Embedded__Projective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 670 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Embedded__Projective__Variety_sp_pl_pl_sp__Embedded__Projective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1106 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Fano_lp__Z__Z_cm__Embedded__Projective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 374 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___G__G.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5148 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___G__G_lp__Z__Z_cm__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1057 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Hom_lp__Multiprojective__Variety_cm__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 329 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp^_sp__Z__Z.out │ │ │ -rw-r--r-- 0 root (0) root (0) 951 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_bs_bs_sp__Multiprojective__Variety.out │ │ │ @@ -13975,15 +13975,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 507 2025-11-14 16:08:07.000000 ./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 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_pc_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 399 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_pl_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 447 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_st_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 663 2025-11-14 16:08:07.000000 ./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 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_st_st_sp__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1341 2025-11-14 16:08:07.000000 ./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 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp__Multiprojective__Variety.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1275 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1200 2025-11-14 16:08:07.000000 ./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 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp_st_sp__Multirational__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1848 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp_st_st_sp__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 772 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp_vb_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 803 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp_vb_sp__Multirational__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 763 2025-11-14 16:08:07.000000 ./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 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp_vb_vb_sp__Multirational__Map.out │ │ │ @@ -14006,45 +14006,45 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 671 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_decompose_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 324 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degree__Sequence.out │ │ │ -rw-r--r-- 0 root (0) root (0) 158 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degree_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1006 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degree_lp__Multirational__Map_cm__Option_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 428 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degree_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 426 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degrees_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 722 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_describe_lp__Multiprojective__Variety_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2263 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_describe_lp__Multirational__Map_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2265 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_describe_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 155 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_dim_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 482 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_dual_lp__Embedded__Projective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 809 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_entries_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 239 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_euler_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2030 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_factor_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1459 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_fiber__Product.out │ │ │ -rw-r--r-- 0 root (0) root (0) 673 2025-11-14 16:08:07.000000 ./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) 1857 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_graph_lp__Multirational__Map_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1856 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_graph_lp__Multirational__Map_rp.out │ │ │ 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│ │ -rw-r--r-- 0 root (0) root (0) 534 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_multidegree_lp__Multirational__Map_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 662 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_multidegree_lp__Z__Z_cm__Multirational__Map_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 664 2025-11-14 16:08:07.000000 ./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 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_multirational__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1182 2025-11-14 16:08:07.000000 ./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 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_multirational__Map_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 363 2025-11-14 16:08:07.000000 ./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 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_parametrize_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 490 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_permute_lp__Multiprojective__Variety_cm__List_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1237 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_point_lp__Multiprojective__Variety_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1236 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_point_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 811 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_projection__Maps.out │ │ │ -rw-r--r-- 0 root (0) root (0) 605 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_projection__Maps_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2190 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_projections.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2823 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_projective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 370 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_projective__Variety_lp__List_cm__List_cm__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 343 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_projective__Variety_lp__List_cm__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1055 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_projective__Variety_lp__Multidimensional__Matrix_rp.out │ │ │ @@ -14052,15 +14052,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 2154 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_random_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 582 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_rational__Map_lp__Multiprojective__Variety_cm__Tally_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 440 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_ring_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 328 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_schubert__Cycle.out │ │ │ -rw-r--r-- 0 root (0) root (0) 151 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_sectional__Genus.out │ │ │ -rw-r--r-- 0 root (0) root (0) 468 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_segre__Embedding.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6206 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_segre_lp__Multiprojective__Variety_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1051 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_segre_lp__Multirational__Map_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1050 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_segre_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 288 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_shape_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 813 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_shortcuts.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3241 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_show_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 512 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_singular__Locus_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 497 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_super_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 352 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_support_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 590 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_tangent__Cone_lp__Embedded__Projective__Variety_cm__Embedded__Projective__Variety_rp.out │ │ │ @@ -14074,15 +14074,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 631 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_⋂.out │ │ │ -rw-r--r-- 0 root (0) root (0) 443 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_⋃.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 641 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 50 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 14090 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Embedded__Projective__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8119 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Embedded__Projective__Variety_sp!.html │ │ │ --rw-r--r-- 0 root (0) root (0) 29042 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Embedded__Projective__Variety_sp_eq_eq_eq_gt_sp__Embedded__Projective__Variety.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 29043 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Embedded__Projective__Variety_sp_eq_eq_eq_gt_sp__Embedded__Projective__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6724 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Embedded__Projective__Variety_sp_pl_pl_sp__Embedded__Projective__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9902 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Fano_lp__Z__Z_cm__Embedded__Projective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6913 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___G__G.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11424 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___G__G_lp__Z__Z_cm__Multirational__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6539 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Grassmannian__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9178 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Hom_lp__Multiprojective__Variety_cm__Multiprojective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 24980 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multiprojective__Variety.html │ │ │ @@ -14093,15 +14093,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6488 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multiprojective__Variety_sp_pc_sp__Multiprojective__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7160 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multiprojective__Variety_sp_pl_sp__Multiprojective__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6880 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multiprojective__Variety_sp_st_sp__Multiprojective__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6950 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multiprojective__Variety_sp_st_st_sp__Multiprojective__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7756 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multiprojective__Variety_sp_st_st_sp__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 23929 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8185 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map_sp^_st_st_sp__Multiprojective__Variety.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8286 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map_sp__Multiprojective__Variety.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8284 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map_sp__Multiprojective__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5781 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map_sp_eq_eq_sp__Multirational__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5662 2025-11-14 16:08:07.000000 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./usr/share/doc/Macaulay2/Nauty/html/___Comparison_spof_sp__Graph6_spand_sp__Sparse6_spformats.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5714 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Nauty/html/___Example_co_sp__Checking_spfor_spisomorphic_spgraphs.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8522 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Nauty/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8520 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Nauty/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10316 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Nauty/html/_add__Edges.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6991 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Nauty/html/_are__Isomorphic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14195 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Nauty/html/_build__Graph__Filter.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7805 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Nauty/html/_count__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8103 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Nauty/html/_filter__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13715 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Nauty/html/_generate__Bipartite__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14088 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Nauty/html/_generate__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9658 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Nauty/html/_generate__Random__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8443 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Nauty/html/_generate__Random__Regular__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6502 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Nauty/html/_graph6__To__Sparse6.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8740 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Nauty/html/_graph__Complement.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8742 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Nauty/html/_graph__Complement.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8943 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Nauty/html/_graph__To__String.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5998 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Nauty/html/_is__Planar.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8315 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Nauty/html/_neighborhood__Complements.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7139 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Nauty/html/_new__Edges.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6373 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Nauty/html/_only__Planar.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7363 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Nauty/html/_relabel__Bipartite.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8940 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Nauty/html/_relabel__Graph.html │ │ │ @@ -14623,15 +14623,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 8104 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Nauty/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NautyGraphs/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NautyGraphs/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 131317 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NautyGraphs/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 375 2025-11-14 16:08:07.000000 ./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 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/___Example_co_sp__Checking_spfor_spisomorphic_spgraphs.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1406 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/___Example_co_sp__Generating_spand_spfiltering_spgraphs.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1410 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/___Example_co_sp__Generating_spand_spfiltering_spgraphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 826 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_add__Edges.out │ │ │ -rw-r--r-- 0 root (0) root (0) 388 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_are__Isomorphic.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1498 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_build__Graph__Filter.out │ │ │ -rw-r--r-- 0 root (0) root (0) 184 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_generate__Graphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 326 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_generate__Random__Graphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 124 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_generate__Random__Regular__Graphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 498 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_graph__Complement.out │ │ │ @@ -14645,15 +14645,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 755 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_remove__Edges.out │ │ │ -rw-r--r-- 0 root (0) root (0) 232 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_remove__Isomorphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 203 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_string__To__Graph.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 32 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6195 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/___Comparison_spof_sp__Graph6_spand_sp__Sparse6_spformats.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5706 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/___Example_co_sp__Checking_spfor_spisomorphic_spgraphs.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8548 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8552 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9550 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_add__Edges.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7633 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_are__Isomorphic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13153 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_build__Graph__Filter.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7800 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_count__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8080 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_filter__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9132 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_generate__Bipartite__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9062 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_generate__Graphs.html │ │ │ @@ -14830,15 +14830,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 8147 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NonminimalComplexes/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5466 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NonminimalComplexes/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3915 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NonminimalComplexes/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 608995 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 3511 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Chow_spring.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3515 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Chow_spring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3270 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___H__H^__Z__Z_lp__Normal__Toric__Variety_cm__Coherent__Sheaf_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1905 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Normal__Toric__Variety_sp^_sp__Array.out │ │ │ -rw-r--r-- 0 root (0) root (0) 908 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Normal__Toric__Variety_sp^_st_st_sp__Z__Z.out │ │ │ -rw-r--r-- 0 root (0) root (0) 840 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Normal__Toric__Variety_sp_st_st_sp__Normal__Toric__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1922 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Normal__Toric__Variety_sp_us_sp__Array.out │ │ │ -rw-r--r-- 0 root (0) root (0) 708 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Normal__Toric__Variety_sp_us_sp__Z__Z.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1977 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___O__O_sp__Toric__Divisor.out │ │ │ @@ -14902,20 +14902,20 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1271 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_lattice__Points_lp__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1229 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_make__Simplicial_lp__Normal__Toric__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1568 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_make__Smooth_lp__Normal__Toric__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 601 2025-11-14 16:08:07.000000 ./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 2025-11-14 16:08:07.000000 ./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) 912 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_matrix_lp__Toric__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 543 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_max_lp__Normal__Toric__Variety_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1562 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_monomials_lp__Toric__Divisor_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1560 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_monomials_lp__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2228 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_nef__Generators_lp__Normal__Toric__Variety_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1001 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Fan_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1002 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Fan_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3171 2025-11-14 16:08:07.000000 ./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 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Matrix_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1965 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Polyhedron_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1964 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Polyhedron_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 704 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1599 2025-11-14 16:08:07.000000 ./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 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_orbits_lp__Normal__Toric__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1605 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_picard__Group_lp__Normal__Toric__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1337 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_picard__Group_lp__Toric__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1807 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_polytope_lp__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2219 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_projective_spspace.out │ │ │ @@ -14940,15 +14940,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 881 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_vector_lp__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1765 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_vertices_lp__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1650 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_weighted__Projective__Space_lp__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 452 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_weil__Divisor__Group_lp__Normal__Toric__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 985 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_weil__Divisor__Group_lp__Toric__Map_rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 68 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 12849 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Chow_spring.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 12853 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Chow_spring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11949 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___H__H^__Z__Z_lp__Normal__Toric__Variety_cm__Coherent__Sheaf_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 27601 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Normal__Toric__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14079 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Normal__Toric__Variety_sp^_sp__Array.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9111 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Normal__Toric__Variety_sp^_st_st_sp__Z__Z.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9410 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Normal__Toric__Variety_sp_st_st_sp__Normal__Toric__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13026 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Normal__Toric__Variety_sp_us_sp__Array.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8752 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Normal__Toric__Variety_sp_us_sp__Z__Z.html │ │ │ @@ -15017,20 +15017,20 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 10561 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_make__Simplicial_lp__Normal__Toric__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13649 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_make__Smooth_lp__Normal__Toric__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9577 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_making_spnormal_sptoric_spvarieties.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11442 2025-11-14 16:08:07.000000 ./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) 12498 2025-11-14 16:08:07.000000 ./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) 11093 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_matrix_lp__Toric__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9104 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_max_lp__Normal__Toric__Variety_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10407 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_monomials_lp__Toric__Divisor_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10405 2025-11-14 16:08:07.000000 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│ │ -rw-r--r-- 0 root (0) root (0) 7658 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/___Chain__Complex_sp_us_sp__Z__Z.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6367 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/___Chain__Complex_sp_us_sp__Z__Z_sp_eq_sp__Thing.html │ │ │ --rw-r--r-- 0 root (0) root (0) 18648 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/___Fast__Nonminimal.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 18647 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/___Fast__Nonminimal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13919 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/___Graded__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13078 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/___Graded__Module__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5641 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/___Graded__Module__Map_sp_vb_sp__Graded__Module__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5731 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/___Graded__Module__Map_sp_vb_vb_sp__Graded__Module__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4108 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/___Graded__Module_sp__Array.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4152 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/___Graded__Module_sp_st_st_sp__Chain__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5700 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/___Graded__Module_sp_st_st_sp__Graded__Module.html │ │ │ @@ -16255,22 +16255,22 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6360 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/OpenMath/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4999 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/OpenMath/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3143 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/OpenMath/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 84707 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 17050 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Checking_spthe_spcodimension_spand_spirreducible_spdecomposition_spof_spthe_sp__I__G_spideal.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 17049 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Checking_spthe_spcodimension_spand_spirreducible_spdecomposition_spof_spthe_sp__I__G_spideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1080 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Example_sp4.1_co_spunique_spgraph_spon_sp8_spvertices_spwith_spexotic_spsolutions_spand_spno_spinduced_spcycle_spof_splength_spat_spleast_sp5.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2542 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Example_sp4.2_co_spa_sp__K5_spand_sppentagon_spglued_spalong_span_spedge.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2543 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Example_sp4.2_co_spa_sp__K5_spand_sppentagon_spglued_spalong_span_spedge.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2081 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Example_sp4.3_co_spexamples_spof_spgluing_sptwo_spcycles_spalong_span_spedge.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5401 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Example_sp4.4_co_sp__The_spsquare_spwithin_spa_spsquare.out │ │ │ -rw-r--r-- 0 root (0) root (0) 997 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Generation_spof_spall_sp__S__C__T_sp_lpsimple_cm_spconnected_cm_sp2-connected_rp_spgraphs_spon_spsmall_spnumbers_spof_spvertices.out │ │ │ -rw-r--r-- 0 root (0) root (0) 16187 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Oscillators.out │ │ │ --rw-r--r-- 0 root (0) root (0) 3811 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___S__C__T_spgraphs_spwith_spexotic_spsolutions.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3813 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___S__C__T_spgraphs_spwith_spexotic_spsolutions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1682 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_all__Unique__Principal__Minors.out │ │ │ -rw-r--r-- 0 root (0) root (0) 10260 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_find__Real__Solutions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 151 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_get__Angles.out │ │ │ -rw-r--r-- 0 root (0) root (0) 429 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_get__Linearly__Stable__Solutions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2330 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_identify__Stability.out │ │ │ -rw-r--r-- 0 root (0) root (0) 459 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_is__Stable__Solution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 10664 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_osc__Jacobian.out │ │ │ @@ -16278,22 +16278,22 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 625 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_osc__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 23259 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_osc__System.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2051 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_show__Exotic__Solutions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 241 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_standard__Sols.out │ │ │ -rw-r--r-- 0 root (0) root (0) 236 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_vertex__Spanning__Polynomial.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 33 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 25409 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Checking_spthe_spcodimension_spand_spirreducible_spdecomposition_spof_spthe_sp__I__G_spideal.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 25408 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Checking_spthe_spcodimension_spand_spirreducible_spdecomposition_spof_spthe_sp__I__G_spideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6234 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Example_sp4.1_co_spunique_spgraph_spon_sp8_spvertices_spwith_spexotic_spsolutions_spand_spno_spinduced_spcycle_spof_splength_spat_spleast_sp5.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7735 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Example_sp4.2_co_spa_sp__K5_spand_sppentagon_spglued_spalong_span_spedge.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7736 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Example_sp4.2_co_spa_sp__K5_spand_sppentagon_spglued_spalong_span_spedge.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7644 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Example_sp4.3_co_spexamples_spof_spgluing_sptwo_spcycles_spalong_span_spedge.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10645 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Example_sp4.4_co_sp__The_spsquare_spwithin_spa_spsquare.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6691 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Generation_spof_spall_sp__S__C__T_sp_lpsimple_cm_spconnected_cm_sp2-connected_rp_spgraphs_spon_spsmall_spnumbers_spof_spvertices.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3563 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Harrington-__Schenck-__Stillman.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8190 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___S__C__T_spgraphs_spwith_spexotic_spsolutions.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8192 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___S__C__T_spgraphs_spwith_spexotic_spsolutions.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8807 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_all__Unique__Principal__Minors.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19710 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_find__Real__Solutions.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6864 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_get__Angles.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6738 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_get__Linearly__Stable__Solutions.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10573 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_identify__Stability.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7493 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_is__Stable__Solution.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20003 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_osc__Jacobian.html │ │ │ @@ -16525,15 +16525,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 12104 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Parsing/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9154 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Parsing/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7244 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Parsing/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PathSignatures/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PathSignatures/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 136324 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PathSignatures/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 2842 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___A_spfamily_spof_sppaths_spon_spa_spcone.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2841 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___A_spfamily_spof_sppaths_spon_spa_spcone.out │ │ │ -rw-r--r-- 0 root (0) root (0) 456 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___Array_sp_us_sp__N__C__Polynomial__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 863 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___C__Axis__Tensor.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1364 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___C__Mon__Tensor.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2634 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___Path.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1530 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___Path__Signatures.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1392 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___Polynomial_sppaths_spof_spdegree_spm.out │ │ │ -rw-r--r-- 0 root (0) root (0) 11000 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___The_spuniversal_spvariety_spand_sptoric_spcoordinates.out │ │ │ @@ -16559,15 +16559,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 204 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/_tensor__Log.out │ │ │ -rw-r--r-- 0 root (0) root (0) 886 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/_tensor__Parametrization.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2023 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/_word__Algebra.out │ │ │ -rw-r--r-- 0 root (0) root (0) 387 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/_word__Format.out │ │ │ -rw-r--r-- 0 root (0) root (0) 215 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/_word__String.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 38 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 11402 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___A_spfamily_spof_sppaths_spon_spa_spcone.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 11401 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___A_spfamily_spof_sppaths_spon_spa_spcone.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7073 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___Array_sp_us_sp__N__C__Polynomial__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7959 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___C__Axis__Tensor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8285 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___C__Mon__Tensor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3941 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___Computing_sp__Path_sp__Varieties.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7451 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___N__C__Polynomial__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 21351 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___N__C__Ring__Element.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15444 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___Path.html │ │ │ @@ -16601,15 +16601,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 30145 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19861 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9130 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 138923 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 4679 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/___Lab__Book__Protocol.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4680 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/___Lab__Book__Protocol.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1213 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_centers.out │ │ │ -rw-r--r-- 0 root (0) root (0) 11148 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_ci__Module__To__Clifford__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1604 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_ci__Module__To__Matrix__Factorization.out │ │ │ -rw-r--r-- 0 root (0) root (0) 795 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_clifford__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1590 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_clifford__Module__To__C__I__Resolution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 889 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_clifford__Module__To__Matrix__Factorization.out │ │ │ -rw-r--r-- 0 root (0) root (0) 864 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_clifford__Operators.out │ │ │ @@ -16638,15 +16638,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1764 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_tensor__Product.out │ │ │ -rw-r--r-- 0 root (0) root (0) 613 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_translate__Isotropic__Subspace.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1547 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_vector__Bundle__On__E.out │ │ │ -rw-r--r-- 0 root (0) root (0) 513 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_y__Action.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 47 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 7197 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/___Clifford__Module.html │ │ │ --rw-r--r-- 0 root (0) root (0) 13648 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/___Lab__Book__Protocol.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 13649 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/___Lab__Book__Protocol.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5002 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/___Random__Nice__Pencil.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7348 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/___Vector__Bundle__On__E.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4978 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/_base__Ring_spof_spa_sp__Random__Nice__Pencil.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10358 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/_centers.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19728 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/_ci__Module__To__Clifford__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10916 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/_ci__Module__To__Matrix__Factorization.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8843 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/_clifford__Module.html │ │ │ @@ -17405,15 +17405,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 2469 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Example_co_sp__Hibi_spideals.out │ │ │ -rw-r--r-- 0 root (0) root (0) 358 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Example_co_sp__Intersection_splattices.out │ │ │ -rw-r--r-- 0 root (0) root (0) 433 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Example_co_sp__L__C__M-lattices.out │ │ │ -rw-r--r-- 0 root (0) root (0) 283 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Poset.out │ │ │ -rw-r--r-- 0 root (0) root (0) 131 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Poset_sp_us_sp__List.out │ │ │ -rw-r--r-- 0 root (0) root (0) 123 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Poset_sp_us_sp__Z__Z.out │ │ │ -rw-r--r-- 0 root (0) root (0) 150 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Poset_sp_us_st.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2467 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Precompute.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2466 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Precompute.out │ │ │ -rw-r--r-- 0 root (0) root (0) 308 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_adjoin__Max.out │ │ │ -rw-r--r-- 0 root (0) root (0) 306 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_adjoin__Min.out │ │ │ -rw-r--r-- 0 root (0) root (0) 668 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_all__Relations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 315 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_antichains.out │ │ │ -rw-r--r-- 0 root (0) root (0) 320 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_are__Isomorphic.out │ │ │ -rw-r--r-- 0 root (0) root (0) 157 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_atoms.out │ │ │ -rw-r--r-- 0 root (0) root (0) 253 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_boolean__Lattice.out │ │ │ @@ -17532,15 +17532,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 9486 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Posets/html/___Example_co_sp__Hibi_spideals.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7037 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Posets/html/___Example_co_sp__Intersection_splattices.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6041 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Posets/html/___Example_co_sp__L__C__M-lattices.html │ │ │ -rw-r--r-- 0 root (0) root (0) 38795 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Posets/html/___Poset.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5707 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Posets/html/___Poset_sp_us_sp__List.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5830 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Posets/html/___Poset_sp_us_sp__Z__Z.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5493 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Posets/html/___Poset_sp_us_st.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8667 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Posets/html/___Precompute.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8666 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Posets/html/___Precompute.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6042 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Posets/html/_adjoin__Max.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6073 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Posets/html/_adjoin__Min.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6593 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Posets/html/_all__Relations.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6851 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Posets/html/_antichains.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7024 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Posets/html/_are__Isomorphic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5632 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Posets/html/_atoms.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5773 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Posets/html/_augment__Poset.html │ │ │ @@ -17695,30 +17695,30 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 81139 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 2696 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/___Primary__Decomposition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3736 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_associated__Primes.out │ │ │ -rw-r--r-- 0 root (0) root (0) 249 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_associated_spprimes.out │ │ │ -rw-r--r-- 0 root (0) root (0) 411 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_irreducible__Decomposition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 280 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_is__Primary.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2222 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_kernel__Of__Localization.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2223 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_kernel__Of__Localization.out │ │ │ -rw-r--r-- 0 root (0) root (0) 627 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_localize.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1734 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_primary__Decomposition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 7407 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_primary__Decomposition_lp__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 765 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_primary_spdecomposition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1389 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_reg__Seq__In__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 423 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_remove__Lowest__Dimension.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2249 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_strategies_spfor_spcomputing_spprimary_spdecomposition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 242 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_top__Components.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 52 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 17690 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_associated__Primes.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4670 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_associated_spprimes.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5762 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_irreducible__Decomposition.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8232 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_is__Primary.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8732 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_kernel__Of__Localization.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8733 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_kernel__Of__Localization.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10172 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_localize.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8209 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_primary__Component.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11813 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_primary__Decomposition.html │ │ │ -rw-r--r-- 0 root (0) root (0) 21071 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_primary__Decomposition_lp__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5735 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_primary_spdecomposition.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9885 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_reg__Seq__In__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7321 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_remove__Lowest__Dimension.html │ │ │ @@ -18143,15 +18143,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1962 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Example_sp__Type_sp[300b].out │ │ │ -rw-r--r-- 0 root (0) root (0) 2633 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Example_sp__Type_sp[300c].out │ │ │ -rw-r--r-- 0 root (0) root (0) 2039 2025-11-14 16:08:07.000000 ./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 2025-11-14 16:08:07.000000 ./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 2025-11-14 16:08:07.000000 ./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 2025-11-14 16:08:07.000000 ./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 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Half_spcanonical_spdegree_sp20.out │ │ │ --rw-r--r-- 0 root (0) root (0) 21833 2025-11-14 16:08:07.000000 ./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) 22546 2025-11-14 16:08:07.000000 ./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 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Noether-__Lefschetz_spexamples.out │ │ │ -rw-r--r-- 0 root (0) root (0) 7051 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Pfaffians_spon_spquadrics.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1233 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Singularities_spof_splifting_spof_sptype_sp[300b].out │ │ │ -rw-r--r-- 0 root (0) root (0) 4091 2025-11-14 16:08:07.000000 ./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 2025-11-14 16:08:07.000000 ./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 2025-11-14 16:08:07.000000 ./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 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Type_sp[331]_cm_sp__C__Y_spof_spdegree_sp17_spvia_splinkage.out │ │ │ @@ -18185,15 +18185,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 9061 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Example_sp__Type_sp[300b].html │ │ │ -rw-r--r-- 0 root (0) root (0) 9848 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Example_sp__Type_sp[300c].html │ │ │ -rw-r--r-- 0 root (0) root (0) 6828 2025-11-14 16:08:07.000000 ./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) 16172 2025-11-14 16:08:07.000000 ./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) 19856 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Finding_spthe_sp__Betti_spstratum_spof_spa_spgiven_spquartic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9268 2025-11-14 16:08:07.000000 ./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) 8693 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Half_spcanonical_spdegree_sp20.html │ │ │ --rw-r--r-- 0 root (0) root (0) 47481 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 48194 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10984 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Noether-__Lefschetz_spexamples.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4123 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Normalize.html │ │ │ 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./usr/share/doc/Macaulay2/RInterface/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 31191 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RInterface/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7199 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RInterface/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomCanonicalCurves/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomCanonicalCurves/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 4410 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomCanonicalCurves/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomCanonicalCurves/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 416 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomCanonicalCurves/example-output/_canonical__Curve.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 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./usr/share/doc/Macaulay2/RandomCanonicalCurves/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomComplexes/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomComplexes/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 27181 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomComplexes/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 2678 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_disturb.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1358 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_histogram.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1315 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_maximal__Entry.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1522 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_normalize.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5055 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_random__Chain__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 510 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_random__Simplicial__Complex.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1161 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_test__Time__For__L__L__Lon__Syzygies.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1159 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_test__Time__For__L__L__Lon__Syzygies.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 44 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 3937 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/___Discrete.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4295 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/___With__L__L__L.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4151 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/___Zero__Mean.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9992 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_disturb.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7241 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_histogram.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7257 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_maximal__Entry.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6973 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_normalize.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14308 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_random__Chain__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6662 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_random__Simplicial__Complex.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8072 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_test__Time__For__L__L__Lon__Syzygies.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8070 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_test__Time__For__L__L__Lon__Syzygies.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11390 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9528 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4800 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomCurves/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomCurves/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 1330 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomCurves/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomCurves/html/ │ │ │ @@ -18396,33 +18396,33 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6319 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4166 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 85866 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 604 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/___Finding_sp__Extreme_sp__Examples.out │ │ │ --rw-r--r-- 0 root (0) root (0) 466 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/___Random__Ideals.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 447 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/___Random__Ideals.out │ │ │ -rw-r--r-- 0 root (0) root (0) 481 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_ideal__Chain__From__Shelling.out │ │ │ -rw-r--r-- 0 root (0) root (0) 308 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_ideal__From__Shelling.out │ │ │ -rw-r--r-- 0 root (0) root (0) 188 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_is__Shelling.out │ │ │ -rw-r--r-- 0 root (0) root (0) 250 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Addition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 894 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Binomial__Edge__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 491 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Binomial__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 564 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Edge__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 479 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Elements__From__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 461 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Ideal.out │ │ │ --rw-r--r-- 0 root (0) root (0) 285 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Monomial.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 289 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Monomial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 421 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Monomial__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 359 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Pure__Binomial__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 249 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Shellable__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 733 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Shellable__Ideal__Chain.out │ │ │ -rw-r--r-- 0 root (0) root (0) 752 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Shelling.out │ │ │ -rw-r--r-- 0 root (0) root (0) 426 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Sparse__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 491 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Square__Free__Monomial__Ideal.out │ │ │ --rw-r--r-- 0 root (0) root (0) 8929 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Square__Free__Step.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 8825 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Square__Free__Step.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1615 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Toric__Edge__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 233 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_reg__Seq.out │ │ │ -rw-r--r-- 0 root (0) root (0) 350 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_square__Free.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 39 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5095 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/___Alexander__Probability.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10300 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/___Finding_sp__Extreme_sp__Examples.html │ │ │ @@ -18431,28 +18431,28 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5789 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_is__Shelling.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7774 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Addition.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7286 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Binomial__Edge__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7700 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Binomial__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6565 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Edge__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7793 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Elements__From__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7658 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Ideal.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6272 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Monomial.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6276 2025-11-14 16:08:07.000000 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16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/index.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 25638 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 17515 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8052 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomMonomialIdeals/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomMonomialIdeals/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 151393 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomMonomialIdeals/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomMonomialIdeals/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 319 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomMonomialIdeals/example-output/___C__M__Stats.out │ │ │ @@ -18580,15 +18580,15 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomPoints/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomPoints/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 96387 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomPoints/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 356 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/___Max__Coordinates__To__Replace.out │ │ │ -rw-r--r-- 0 root (0) root (0) 268 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/___Point__Check__Attempts.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2427 2025-11-14 16:08:07.000000 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./usr/share/doc/Macaulay2/RandomPoints/html/___Max__Coordinates__To__Replace.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4903 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/___Num__Threads__To__Use.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6021 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/___Point__Check__Attempts.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9659 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/___Replacement.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8823 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/_dim__Via__Bezout.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8822 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/_dim__Via__Bezout.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13400 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/_extend__Ideal__By__Non__Zero__Minor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10917 2025-11-14 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./usr/share/doc/Macaulay2/RationalPoints2/html/_zeros.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19956 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RationalPoints2/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15620 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RationalPoints2/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4519 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/RationalPoints2/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ReactionNetworks/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ReactionNetworks/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 93198 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ReactionNetworks/dump/rawdocumentation.dump │ │ │ @@ -18883,15 +18883,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 551 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_is__Reduction.out │ │ │ -rw-r--r-- 0 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./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_rees__Ideal.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3964 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_rees__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 417 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_special__Fiber.out │ │ │ -rw-r--r-- 0 root (0) root (0) 933 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_special__Fiber__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1090 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_symmetric__Kernel.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2884 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_symmetric__Kernel_lp..._cm__Variable_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6573 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_versal__Embedding.out │ │ │ -rw-r--r-- 0 root (0) root 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./usr/share/doc/Macaulay2/ReesAlgebra/html/_rees__Ideal_lp..._cm__Pair__Limit_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15730 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/html/_rees__Ideal_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10380 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/html/_special__Fiber.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11700 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/html/_special__Fiber__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5952 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/html/_symmetric__Algebra__Ideal.html │ │ │ @@ -19089,80 +19089,80 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Resultants/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Resultants/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 114689 2025-11-14 16:08:07.000000 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./usr/share/doc/Macaulay2/SLnEquivariantMatrices/example-output/_sl2__Equivariant__Constant__Rank__Matrix.out │ │ │ @@ -19490,30 +19490,30 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 65102 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SRdeformations/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 45881 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SRdeformations/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 22045 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SRdeformations/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SVDComplexes/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SVDComplexes/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 44888 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SVDComplexes/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 2825 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/___S__V__D__Complex.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2826 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/___S__V__D__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 11853 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/___S__V__D__Complexes.out │ │ │ --rw-r--r-- 0 root (0) root (0) 3437 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/___S__V__D__Homology.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3436 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/___S__V__D__Homology.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5542 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_are__Pseudo__Inverses.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2088 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_common__Entries.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6984 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_euclidean__Distance.out │ │ │ -rw-r--r-- 0 root (0) root (0) 7308 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_laplacians.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1301 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_numeric__Rank.out │ │ │ -rw-r--r-- 0 root (0) root (0) 9529 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_project__To__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 9254 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_pseudo__Inverse.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 84 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 3737 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/___Laplacian.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3751 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/___Projection.html │ │ │ --rw-r--r-- 0 root (0) root (0) 11975 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/___S__V__D__Complex.html │ │ │ --rw-r--r-- 0 root (0) root (0) 13012 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/___S__V__D__Homology.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 11976 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/___S__V__D__Complex.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 13011 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/___S__V__D__Homology.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15236 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_are__Pseudo__Inverses.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3541 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_check__S__V__D__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10080 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_common__Entries.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14702 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_euclidean__Distance.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14335 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_laplacians.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3882 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_new__Chain__Complex__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9100 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_numeric__Rank.html │ │ │ @@ -19540,24 +19540,24 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Saturation/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 52965 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Saturation/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 859 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_annihilator.out │ │ │ -rw-r--r-- 0 root (0) root (0) 846 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_ideal_spquotients_spand_spsaturation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 259 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_is__Supported__In__Zero__Locus.out │ │ │ -rw-r--r-- 0 root (0) root (0) 807 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_module_spquotients_cm_spsaturation_cm_spand_spannihilator.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1836 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_quotient_lp..._cm__Strategy_eq_gt..._rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1835 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_quotient_lp..._cm__Strategy_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2582 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_quotient_lp__Module_cm__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 837 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_saturate.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Saturation/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 69 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Saturation/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 8747 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Saturation/html/_annihilator.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6560 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Saturation/html/_ideal_spquotients_spand_spsaturation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6876 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Saturation/html/_is__Supported__In__Zero__Locus.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7263 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Saturation/html/_module_spquotients_cm_spsaturation_cm_spand_spannihilator.html │ │ │ --rw-r--r-- 0 root (0) root (0) 17356 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Saturation/html/_quotient_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 17355 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Saturation/html/_quotient_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15598 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Saturation/html/_quotient_lp__Module_cm__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11944 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Saturation/html/_saturate.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12655 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Saturation/html/_saturate_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13853 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Saturation/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11866 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Saturation/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4273 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Saturation/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Schubert2/ │ │ │ @@ -19591,15 +19591,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1688 2025-11-14 16:08:07.000000 ./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 2025-11-14 16:08:07.000000 ./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 2025-11-14 16:08:07.000000 ./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 2025-11-14 16:08:07.000000 ./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 2025-11-14 16:08:07.000000 ./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 2025-11-14 16:08:07.000000 ./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 2025-11-14 16:08:07.000000 ./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 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Lines_spon_sphypersurfaces.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1583 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Lines_spon_sphypersurfaces.out │ │ │ -rw-r--r-- 0 root (0) root (0) 256 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___O__O_sp_us_sp__Abstract__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1332 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___O__O_sp_us_sp__Ring__Element.out │ │ │ -rw-r--r-- 0 root (0) root (0) 913 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Quotient__Bundles.out │ │ │ -rw-r--r-- 0 root (0) root (0) 869 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Riemann-__Roch_spon_spa_spcurve.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1741 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Riemann-__Roch_spon_spa_spsurface.out │ │ │ -rw-r--r-- 0 root (0) root (0) 7382 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Riemann-__Roch_spwithout_spdenominators.out │ │ │ -rw-r--r-- 0 root (0) root (0) 227 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Ring_sp_us_sp__Chern__Class__Variable.out │ │ │ @@ -19711,15 +19711,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5605 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Examples_spfrom_sp__Schubert_cm_sptranslated.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8983 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Flag__Bundle.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6370 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Hom_lp__Abstract__Sheaf_cm__Abstract__Sheaf_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5551 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Ideas_spfor_spfuture_spdevelopment.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6351 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Incidence__Correspondence.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4163 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Intersection__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4202 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Isotropic.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8064 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Lines_spon_sphypersurfaces.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8060 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Lines_spon_sphypersurfaces.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4976 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___O__O_sp_us_sp__Abstract__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6353 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___O__O_sp_us_sp__Ring__Element.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4277 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Pull__Back.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6529 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Quotient__Bundles.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5154 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Riemann-__Roch_spon_spa_spcurve.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7875 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Riemann-__Roch_spon_spa_spsurface.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11386 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Riemann-__Roch_spwithout_spdenominators.html │ │ │ @@ -19990,29 +19990,29 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SegreClasses/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SegreClasses/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 48932 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SegreClasses/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 505 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_chow__Class.out │ │ │ -rw-r--r-- 0 root (0) root (0) 690 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_contained__In__Singular__Locus.out │ │ │ -rw-r--r-- 0 root (0) root (0) 895 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_intersection__Product.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1526 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_is__Component__Contained.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1525 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_is__Component__Contained.out │ │ │ -rw-r--r-- 0 root (0) root (0) 435 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_is__Multi__Hom.out │ │ │ -rw-r--r-- 0 root (0) root (0) 277 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_make__Chow__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 345 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_make__Product__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 453 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_multiplicity.out │ │ │ -rw-r--r-- 0 root (0) root (0) 881 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_projective__Degree.out │ │ │ -rw-r--r-- 0 root (0) root (0) 932 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_projective__Degrees.out │ │ │ -rw-r--r-- 0 root (0) root (0) 876 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_segre.out │ │ │ -rw-r--r-- 0 root (0) root (0) 720 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_segre__Dim__X.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 97 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 7697 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_chow__Class.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8412 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_contained__In__Singular__Locus.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9833 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_intersection__Product.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9715 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_is__Component__Contained.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9714 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_is__Component__Contained.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6577 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_is__Multi__Hom.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5950 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_make__Chow__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6488 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_make__Product__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7807 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_multiplicity.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9364 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_projective__Degree.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9107 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_projective__Degrees.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8878 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_segre.html │ │ │ @@ -20542,18 +20542,18 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1122 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/___Multidimensional__Matrix_sp_st_sp__Multidimensional__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 646 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/___Multidimensional__Matrix_sp_us_sp__Sequence.out │ │ │ -rw-r--r-- 0 root (0) root (0) 597 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/___Ring__Element_sp_st_sp__Multidimensional__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 478 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/___Sparse__Discriminant_sp__Thing.out │ │ │ -rw-r--r-- 0 root (0) root (0) 930 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/___Sparse__Resultant_sp__Thing.out │ │ │ -rw-r--r-- 0 root (0) root (0) 326 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_char_lp__Sparse__Discriminant_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 385 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_char_lp__Sparse__Resultant_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1023 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_degree__Determinant.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1024 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_degree__Determinant.out │ │ │ -rw-r--r-- 0 root (0) root (0) 814 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_dense__Discriminant.out │ │ │ -rw-r--r-- 0 root (0) root (0) 929 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_dense__Resultant.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1363 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_determinant_lp__Multidimensional__Matrix_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1362 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_determinant_lp__Multidimensional__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 973 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_dim_lp__Multidimensional__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 494 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_entries_lp__Multidimensional__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 561 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_exponents__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 394 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_exponents_lp__Sparse__Discriminant_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 471 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_exponents_lp__Sparse__Resultant_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2043 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_flattening.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1250 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_generic__Laurent__Polynomials.out │ │ │ @@ -20567,15 +20567,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 634 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_random__Multidimensional__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2126 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_rank_lp__Multidimensional__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1772 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_reverse__Shape.out │ │ │ -rw-r--r-- 0 root (0) root (0) 537 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_ring_lp__Multidimensional__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 457 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_shape.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2104 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_sort__Shape.out │ │ │ -rw-r--r-- 0 root (0) root (0) 12400 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_sparse__Discriminant.out │ │ │ --rw-r--r-- 0 root (0) root (0) 55254 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_sparse__Resultant.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 55250 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_sparse__Resultant.out │ │ │ -rw-r--r-- 0 root (0) root (0) 911 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_sylvester__Matrix_lp__Multidimensional__Matrix_rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 611 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 35 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 10142 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/___Multidimensional__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6947 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/___Multidimensional__Matrix_sp-_sp__Multidimensional__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6345 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/___Multidimensional__Matrix_sp_eq_eq_sp__Multidimensional__Matrix.html │ │ │ @@ -20585,18 +20585,18 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6195 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/___Ring__Element_sp_st_sp__Multidimensional__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5440 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/___Sparse__Discriminant.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5997 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/___Sparse__Discriminant_sp__Thing.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5468 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/___Sparse__Resultant.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6526 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/___Sparse__Resultant_sp__Thing.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5198 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_char_lp__Sparse__Discriminant_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5264 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_char_lp__Sparse__Resultant_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6901 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_degree__Determinant.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6902 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_degree__Determinant.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8190 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_dense__Discriminant.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8266 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_dense__Resultant.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8706 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_determinant_lp__Multidimensional__Matrix_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8705 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_determinant_lp__Multidimensional__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6122 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_dim_lp__Multidimensional__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5711 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_entries_lp__Multidimensional__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7003 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_exponents__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5551 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_exponents_lp__Sparse__Discriminant_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5569 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_exponents_lp__Sparse__Resultant_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8550 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_flattening.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7760 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_generic__Laurent__Polynomials.html │ │ │ @@ -20610,15 +20610,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6579 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_random__Multidimensional__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8294 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_rank_lp__Multidimensional__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7610 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_reverse__Shape.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5628 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_ring_lp__Multidimensional__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5589 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_shape.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7887 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_sort__Shape.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20153 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_sparse__Discriminant.html │ │ │ --rw-r--r-- 0 root (0) root (0) 65535 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_sparse__Resultant.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 65531 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_sparse__Resultant.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7079 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_sylvester__Matrix_lp__Multidimensional__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 21933 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20968 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11034 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpechtModule/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpechtModule/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 178042 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpechtModule/dump/rawdocumentation.dump │ │ │ @@ -20762,43 +20762,43 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1325 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_ambient__Fivefold.out │ │ │ -rw-r--r-- 0 root (0) root (0) 954 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__Castelnuovo__Surface.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1005 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__K3surface_lp__Special__Cubic__Fourfold_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1191 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__K3surface_lp__Special__Gushel__Mukai__Fourfold_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 614 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_beauville__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 351 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_clean_lp__Hodge__Special__Fourfold_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1173 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_detect__Congruence_lp__Special__Cubic__Fourfold_cm__Z__Z_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1553 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_detect__Congruence_lp__Special__Gushel__Mukai__Fourfold_cm__Z__Z_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1555 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_detect__Congruence_lp__Special__Gushel__Mukai__Fourfold_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 298 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_discriminant_lp__Special__Cubic__Fourfold_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 287 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_discriminant_lp__Special__Gushel__Mukai__Fourfold_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 286 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_discriminant_lp__Special__Gushel__Mukai__Fourfold_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 672 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_fano__Fourfold.out │ │ │ -rw-r--r-- 0 root (0) root (0) 599 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_from__Ordinary__To__Gushel.out │ │ │ -rw-r--r-- 0 root (0) root (0) 153 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_is__Admissible.out │ │ │ -rw-r--r-- 0 root (0) root (0) 164 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_is__Admissible__G__M.out │ │ │ -rw-r--r-- 0 root (0) root (0) 716 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_map_lp__Congruence__Of__Curves_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1356 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_mirror__Fourfold.out │ │ │ -rw-r--r-- 0 root (0) root (0) 357 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_normal__Sheaf.out │ │ │ --rw-r--r-- 0 root (0) root (0) 814 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 813 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count.out │ │ │ -rw-r--r-- 0 root (0) root (0) 786 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count_lp__Special__Cubic__Fourfold_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 973 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count_lp__Special__Gushel__Mukai__Fourfold_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 653 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parametrize__Fano__Fourfold.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1972 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parametrize_lp__Hodge__Special__Fourfold_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 6387 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Cubic__Fourfold.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 6386 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Cubic__Fourfold.out │ │ │ -rw-r--r-- 0 root (0) root (0) 324 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Cubic__Fourfold_lp__Embedded__Projective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 393 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Cubic__Fourfold_lp__String_cm__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1499 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Fourfold.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1520 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Gushel__Mukai__Fourfold.out │ │ │ -rw-r--r-- 0 root (0) root (0) 762 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Gushel__Mukai__Fourfold_lp__Array_cm__Array_cm__String_cm__Thing_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 469 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Gushel__Mukai__Fourfold_lp__Embedded__Projective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 530 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Gushel__Mukai__Fourfold_lp__String_cm__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 413 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_surface.out │ │ │ -rw-r--r-- 0 root (0) root (0) 663 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_surface_lp__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2427 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_surface_lp__Multiprojective__Variety_cm__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 770 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_to__External__String_lp__Hodge__Special__Fourfold_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5325 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_to__Grass.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4324 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_to__Grass_lp__Embedded__Projective__Variety_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4322 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_to__Grass_lp__Embedded__Projective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 531 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_unirational__Parametrization.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 610 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 23 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6372 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/___Congruence__Of__Curves.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6988 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/___Congruence__Of__Curves_sp__Embedded__Projective__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7943 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/___G__Mtables.html │ │ │ @@ -20814,47 +20814,47 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 10359 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_associated__K3surface_lp__Special__Cubic__Fourfold_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10574 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_associated__K3surface_lp__Special__Gushel__Mukai__Fourfold_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6841 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_beauville__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5897 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_check_lp__Z__Z_cm__Congruence__Of__Curves_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5533 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_clean_lp__Hodge__Special__Fourfold_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6192 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_detect__Congruence.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8687 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_detect__Congruence_lp__Special__Cubic__Fourfold_cm__Z__Z_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9572 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_detect__Congruence_lp__Special__Gushel__Mukai__Fourfold_cm__Z__Z_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9574 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_detect__Congruence_lp__Special__Gushel__Mukai__Fourfold_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6654 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_discriminant_lp__Special__Cubic__Fourfold_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6746 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_discriminant_lp__Special__Gushel__Mukai__Fourfold_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6745 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_discriminant_lp__Special__Gushel__Mukai__Fourfold_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6642 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_fano__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5909 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_from__Ordinary__To__Gushel.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5585 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_is__Admissible.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5729 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_is__Admissible__G__M.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5440 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_is__Member_lp__Embedded__Projective__Variety_cm__Congruence__Of__Curves_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7955 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_map_lp__Congruence__Of__Curves_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5595 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_map_lp__Special__Cubic__Fourfold_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5686 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_map_lp__Special__Gushel__Mukai__Fourfold_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9405 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_mirror__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6394 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_normal__Sheaf.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8483 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8482 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7955 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count_lp__Special__Cubic__Fourfold_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8182 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count_lp__Special__Gushel__Mukai__Fourfold_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8186 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parametrize__Fano__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8747 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parametrize_lp__Hodge__Special__Fourfold_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 15461 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Cubic__Fourfold.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 15460 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Cubic__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6948 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Cubic__Fourfold_lp__Embedded__Projective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7039 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Cubic__Fourfold_lp__String_cm__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10491 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6591 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Fourfold_lp__String_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12074 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Gushel__Mukai__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11528 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Gushel__Mukai__Fourfold_lp__Array_cm__Array_cm__String_cm__Thing_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7954 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Gushel__Mukai__Fourfold_lp__Embedded__Projective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7745 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Gushel__Mukai__Fourfold_lp__String_cm__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7348 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_surface.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7730 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_surface_lp__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9302 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_surface_lp__Multiprojective__Variety_cm__Multiprojective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6144 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_to__External__String_lp__Hodge__Special__Fourfold_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11933 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_to__Grass.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10746 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_to__Grass_lp__Embedded__Projective__Variety_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10744 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_to__Grass_lp__Embedded__Projective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5400 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_trisecant__Flop.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7020 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_unirational__Parametrization.html │ │ │ -rw-r--r-- 0 root (0) root (0) 37413 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 33496 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14735 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpectralSequences/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/SpectralSequences/dump/ │ │ │ @@ -21790,23 +21790,23 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 606 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_descend__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 116 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_floor__Log.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1009 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1518 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Power.out │ │ │ -rw-r--r-- 0 root (0) root (0) 227 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Preimage.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1328 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Root.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1117 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Trace__On__Canonical__Module.out │ │ │ --rw-r--r-- 0 root (0) root (0) 488 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__Cohen__Macaulay.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1819 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Injective.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 487 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__Cohen__Macaulay.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1818 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Injective.out │ │ │ -rw-r--r-- 0 root (0) root (0) 637 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Pure.out │ │ │ -rw-r--r-- 0 root (0) root (0) 517 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Rational.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1492 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Regular.out │ │ │ -rw-r--r-- 0 root (0) root (0) 187 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_multiplicative__Order.out │ │ │ -rw-r--r-- 0 root (0) root (0) 416 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_parameter__Test__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 216 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Element.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1514 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Ideal.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1511 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2756 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Module.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 590 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 40 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4326 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/___Ascent__Count.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4385 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/___Assume__C__M.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5130 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/___Assume__Domain.html │ │ │ @@ -21841,23 +21841,23 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 9334 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_descend__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5225 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_floor__Log.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10611 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12408 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Power.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5237 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Preimage.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16745 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Root.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8554 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Trace__On__Canonical__Module.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8216 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__Cohen__Macaulay.html │ │ │ --rw-r--r-- 0 root (0) root (0) 15806 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Injective.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8215 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__Cohen__Macaulay.html │ │ │ +-rw-r--r-- 0 root (0) 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./usr/share/doc/Macaulay2/TestIdeals/html/_test__Ideal.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 14991 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 17708 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 37577 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 34814 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12484 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Text/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Text/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 136729 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Text/dump/rawdocumentation.dump │ │ │ @@ -22057,30 +22057,30 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 37607 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ThinSincereQuivers/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 27702 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ThinSincereQuivers/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11513 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ThinSincereQuivers/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ThreadedGB/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ThreadedGB/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 24388 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ThreadedGB/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 1510 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/___Minimal.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1467 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/___Minimal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6776 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/___Threaded__G__B.out │ │ │ --rw-r--r-- 0 root (0) root (0) 615 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_matrix_lp__Lineage__Table_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1159 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_minimize_lp__Lineage__Table_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1155 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_reduce.out │ │ │ --rw-r--r-- 0 root (0) root (0) 5165 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_tgb.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 610 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_matrix_lp__Lineage__Table_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1034 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_minimize_lp__Lineage__Table_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1143 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_reduce.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2533 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_tgb.out │ │ │ -rw-r--r-- 0 root (0) root (0) 328 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_tgb_lp..._cm__Verbose_eq_gt..._rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 605 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 77 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6425 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/___Lineage__Table.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7097 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/___Minimal.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6670 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_matrix_lp__Lineage__Table_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7101 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_minimize_lp__Lineage__Table_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7158 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_reduce.html │ │ │ --rw-r--r-- 0 root (0) root (0) 15868 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_tgb.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7054 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/___Minimal.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6665 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_matrix_lp__Lineage__Table_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6976 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_minimize_lp__Lineage__Table_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7146 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_reduce.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 13236 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_tgb.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7430 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_tgb_lp..._cm__Verbose_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 21626 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6806 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4627 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Topcom/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Topcom/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 66598 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Topcom/dump/rawdocumentation.dump │ │ │ @@ -22374,17 +22374,17 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 17380 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TriangularSets/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11888 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TriangularSets/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7217 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/TriangularSets/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 31886 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 2274 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/___Triangulations.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2275 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/___Triangulations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 18048 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/_all__Triangulations_lp__Matrix_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 75224 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/_generate__Triangulations.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 75242 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/_generate__Triangulations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 714 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/_is__Regular__Triangulation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1654 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/_triangulation.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 42 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4460 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/html/___Chirotope.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3951 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/html/___Cone__Index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7308 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/html/___Triangulation.html │ │ │ @@ -22392,29 +22392,29 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 30386 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_all__Triangulations_lp__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3970 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_bistellar__Flip.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3884 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_chirotope.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3827 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_delaunay__Subdivision.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3823 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_delaunay__Weights.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3867 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_fine__Star__Triangulation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3781 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_flips.html │ │ │ --rw-r--r-- 0 root (0) root (0) 86004 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_generate__Triangulations.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 86022 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_generate__Triangulations.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3911 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_gkz__Vector.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3873 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_is__Fine.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9223 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_is__Regular__Triangulation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3883 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_is__Star.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3814 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_naive__Chirotope.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4176 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_naive__Is__Triangulation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3807 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_neighbors.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3950 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_regular__Fine__Star__Triangulation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3970 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_regular__Fine__Triangulation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4144 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_regular__Triangulation__Weights.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9719 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_triangulation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3727 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_vectors.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3837 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_volume__Vector.html │ │ │ --rw-r--r-- 0 root (0) root (0) 24698 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/html/index.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 24699 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10341 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6323 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triangulations/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triplets/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triplets/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 49146 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triplets/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triplets/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 191 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/Triplets/example-output/___Betti1_lp__Triplet_rp.out │ │ │ @@ -23205,15 +23205,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 11114 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/VectorGraphics/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9177 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/VectorGraphics/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/VersalDeformations/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/VersalDeformations/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 226382 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/VersalDeformations/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 1050 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/___Def__Param.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1216 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/___Smart__Lift.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1214 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/___Smart__Lift.out │ │ │ -rw-r--r-- 0 root (0) root (0) 416 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/_check__Comparison__Theorem.out │ │ │ -rw-r--r-- 0 root (0) root (0) 406 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/_check__Tangent__Space.out │ │ │ -rw-r--r-- 0 root (0) root (0) 862 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/_correct__Deformation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 433 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/_cotangent__Cohomology1.out │ │ │ -rw-r--r-- 0 root (0) root (0) 774 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/_cotangent__Cohomology2.out │ │ │ -rw-r--r-- 0 root (0) root (0) 562 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/_ext__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3316 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/_families.out │ │ │ @@ -23242,15 +23242,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5713 2025-11-14 16:08:07.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Correction__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7708 2025-11-14 16:08:07.000000 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16:08:07.000000 ./usr/share/info/SchurComplexes.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 4868 2025-11-14 16:08:07.000000 ./usr/share/info/SchurFunctors.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 25155 2025-11-14 16:08:07.000000 ./usr/share/info/SchurRings.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 7965 2025-11-14 16:08:07.000000 ./usr/share/info/SchurVeronese.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 2654 2025-11-14 16:08:07.000000 ./usr/share/info/SectionRing.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 7698 2025-11-14 16:08:07.000000 ./usr/share/info/SegreClasses.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 7702 2025-11-14 16:08:07.000000 ./usr/share/info/SegreClasses.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 7469 2025-11-14 16:08:07.000000 ./usr/share/info/SemidefiniteProgramming.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 8348 2025-11-14 16:08:07.000000 ./usr/share/info/Seminormalization.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 2626 2025-11-14 16:08:07.000000 ./usr/share/info/Serialization.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 8316 2025-11-14 16:08:07.000000 ./usr/share/info/SimpleDoc.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 8317 2025-11-14 16:08:07.000000 ./usr/share/info/SimpleDoc.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 76323 2025-11-14 16:08:07.000000 ./usr/share/info/SimplicialComplexes.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 7232 2025-11-14 16:08:07.000000 ./usr/share/info/SimplicialDecomposability.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 2966 2025-11-14 16:08:07.000000 ./usr/share/info/SimplicialPosets.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 37164 2025-11-14 16:08:07.000000 ./usr/share/info/SlackIdeals.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 13525 2025-11-14 16:08:07.000000 ./usr/share/info/SpaceCurves.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 27520 2025-11-14 16:08:07.000000 ./usr/share/info/SparseResultants.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 37569 2025-11-14 16:08:07.000000 ./usr/share/info/SpechtModule.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 32630 2025-11-14 16:08:07.000000 ./usr/share/info/SpecialFanoFourfolds.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 27511 2025-11-14 16:08:07.000000 ./usr/share/info/SparseResultants.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 37567 2025-11-14 16:08:07.000000 ./usr/share/info/SpechtModule.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 32633 2025-11-14 16:08:07.000000 ./usr/share/info/SpecialFanoFourfolds.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 291857 2025-11-14 16:08:07.000000 ./usr/share/info/SpectralSequences.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 12678 2025-11-14 16:08:07.000000 ./usr/share/info/StatGraphs.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 12684 2025-11-14 16:08:07.000000 ./usr/share/info/StatGraphs.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 2631 2025-11-14 16:08:07.000000 ./usr/share/info/StatePolytope.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 7815 2025-11-14 16:08:07.000000 ./usr/share/info/StronglyStableIdeals.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 1897 2025-11-14 16:08:07.000000 ./usr/share/info/Style.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 1895 2025-11-14 16:08:07.000000 ./usr/share/info/Style.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 29769 2025-11-14 16:08:07.000000 ./usr/share/info/SubalgebraBases.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 15726 2025-11-14 16:08:07.000000 ./usr/share/info/SumsOfSquares.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 5366 2025-11-14 16:08:07.000000 ./usr/share/info/SuperLinearAlgebra.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 2753 2025-11-14 16:08:07.000000 ./usr/share/info/SwitchingFields.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 14147 2025-11-14 16:08:07.000000 ./usr/share/info/SymbolicPowers.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 14145 2025-11-14 16:08:07.000000 ./usr/share/info/SymbolicPowers.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 2157 2025-11-14 16:08:07.000000 ./usr/share/info/SymmetricPolynomials.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 14151 2025-11-14 16:08:07.000000 ./usr/share/info/TSpreadIdeals.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 17554 2025-11-14 16:08:07.000000 ./usr/share/info/Tableaux.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 1559 2025-11-14 16:08:07.000000 ./usr/share/info/TangentCone.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 47114 2025-11-14 16:08:07.000000 ./usr/share/info/TateOnProducts.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 22604 2025-11-14 16:08:07.000000 ./usr/share/info/TensorComplexes.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 2389 2025-11-14 16:08:07.000000 ./usr/share/info/TerraciniLoci.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 31169 2025-11-14 16:08:07.000000 ./usr/share/info/TestIdeals.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 31192 2025-11-14 16:08:07.000000 ./usr/share/info/TestIdeals.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 15058 2025-11-14 16:08:07.000000 ./usr/share/info/Text.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 22883 2025-11-14 16:08:07.000000 ./usr/share/info/ThinSincereQuivers.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 8487 2025-11-14 16:08:07.000000 ./usr/share/info/ThreadedGB.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 8192 2025-11-14 16:08:07.000000 ./usr/share/info/ThreadedGB.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 13901 2025-11-14 16:08:07.000000 ./usr/share/info/Topcom.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 7599 2025-11-14 16:08:07.000000 ./usr/share/info/TorAlgebra.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 7449 2025-11-14 16:08:07.000000 ./usr/share/info/ToricHigherDirectImages.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 4014 2025-11-14 16:08:07.000000 ./usr/share/info/ToricInvariants.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 4015 2025-11-14 16:08:07.000000 ./usr/share/info/ToricInvariants.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 6581 2025-11-14 16:08:07.000000 ./usr/share/info/ToricTopology.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 31435 2025-11-14 16:08:07.000000 ./usr/share/info/ToricVectorBundles.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 9584 2025-11-14 16:08:07.000000 ./usr/share/info/TriangularSets.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 13679 2025-11-14 16:08:07.000000 ./usr/share/info/Triangulations.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 7188 2025-11-14 16:08:07.000000 ./usr/share/info/Triplets.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 11010 2025-11-14 16:08:07.000000 ./usr/share/info/Tropical.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 10698 2025-11-14 16:08:07.000000 ./usr/share/info/TropicalToric.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 5772 2025-11-14 16:08:07.000000 ./usr/share/info/Truncations.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 8693 2025-11-14 16:08:07.000000 ./usr/share/info/Units.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 4687 2025-11-14 16:08:07.000000 ./usr/share/info/VNumber.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 8877 2025-11-14 16:08:07.000000 ./usr/share/info/Valuations.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 44204 2025-11-14 16:08:07.000000 ./usr/share/info/Varieties.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 19378 2025-11-14 16:08:07.000000 ./usr/share/info/VectorFields.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 51845 2025-11-14 16:08:07.000000 ./usr/share/info/VectorGraphics.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 41367 2025-11-14 16:08:07.000000 ./usr/share/info/VersalDeformations.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 41366 2025-11-14 16:08:07.000000 ./usr/share/info/VersalDeformations.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 12693 2025-11-14 16:08:07.000000 ./usr/share/info/VirtualResolutions.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 10443 2025-11-14 16:08:07.000000 ./usr/share/info/Visualize.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 38070 2025-11-14 16:08:07.000000 ./usr/share/info/WeilDivisors.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 38075 2025-11-14 16:08:07.000000 ./usr/share/info/WeilDivisors.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 10700 2025-11-14 16:08:07.000000 ./usr/share/info/WeylAlgebras.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 33226 2025-11-14 16:08:07.000000 ./usr/share/info/WeylGroups.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 14775 2025-11-14 16:08:07.000000 ./usr/share/info/WhitneyStratifications.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 33211 2025-11-14 16:08:07.000000 ./usr/share/info/WeylGroups.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 14772 2025-11-14 16:08:07.000000 ./usr/share/info/WhitneyStratifications.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 8865 2025-11-14 16:08:07.000000 ./usr/share/info/XML.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 49484 2025-11-14 16:08:07.000000 ./usr/share/info/gfanInterface.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 49483 2025-11-14 16:08:07.000000 ./usr/share/info/gfanInterface.info.gz │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/lintian/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/lintian/overrides/ │ │ │ -rw-r--r-- 0 root (0) root (0) 11345 2025-11-14 15:57:07.000000 ./usr/share/lintian/overrides/macaulay2-common │ │ │ lrwxrwxrwx 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/Macaulay2/Style/katex/contrib/auto-render.min.js -> ../../../../javascript/katex/contrib/auto-render.js │ │ │ lrwxrwxrwx 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./usr/share/Macaulay2/Style/katex/contrib/copy-tex.min.js -> ../../../../javascript/katex/contrib/copy-tex.js │ │ │ lrwxrwxrwx 0 root (0) root (0) 0 2025-11-14 16:08:07.000000 ./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 2025-11-14 16:08:07.000000 ./usr/share/Macaulay2/Style/katex/fonts/KaTeX_AMS-Regular.ttf -> ../../../../fonts/truetype/katex/KaTeX_AMS-Regular.ttf │ │ ├── ./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) │ │ │ - -- 2.01805s elapsed │ │ │ + -- 1.40564s 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) │ │ │ - -- 2.36135s elapsed │ │ │ + -- 1.81312s 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 │ │ │ @@ -90,28 +90,28 @@ │ │ │ 1 │ │ │ o2 : R-module, quotient of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i3 : elapsedTime burkeResolution(M, 7, Check => false)
│ │ │ - -- 2.01805s elapsed
│ │ │ + -- 1.40564s 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)
│ │ │ - -- 2.36135s elapsed
│ │ │ + -- 1.81312s 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 {} │ │ │ │ @@ -23,24 +23,24 @@ │ │ │ │ i2 : M = coker vars R │ │ │ │ │ │ │ │ o2 = cokernel | a b c | │ │ │ │ │ │ │ │ 1 │ │ │ │ o2 : R-module, quotient of R │ │ │ │ i3 : elapsedTime burkeResolution(M, 7, Check => false) │ │ │ │ - -- 2.01805s elapsed │ │ │ │ + -- 1.40564s 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) │ │ │ │ - -- 2.36135s elapsed │ │ │ │ + -- 1.81312s 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/AdjunctionForSurfaces/example-output/_adjoint__Matrix.out │ │ │ @@ -49,15 +49,15 @@ │ │ │ o8 : BettiTally │ │ │ │ │ │ i9 : c=codim I │ │ │ │ │ │ o9 = 4 │ │ │ │ │ │ i10 : elapsedTime fI=res I │ │ │ - -- .0216671s elapsed │ │ │ + -- .0274596s 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) │ │ │ - -- .585455s elapsed │ │ │ + -- .560889s 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; │ │ │ - -- 5.79625s elapsed │ │ │ + -- 5.02254s 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) │ │ │ - -- .0135763s elapsed │ │ │ + -- .0196097s 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) │ │ │ - -- .0549513s elapsed │ │ │ + -- .0720603s elapsed │ │ │ │ │ │ 0 1 │ │ │ o16 = total: 1 12 │ │ │ 0: 1 . │ │ │ 1: . 12 │ │ │ │ │ │ o16 : BettiTally │ │ │ │ │ │ i17 : I'== I │ │ │ │ │ │ o17 = true │ │ │ │ │ │ i18 : elapsedTime basePts=primaryDecomposition ideal H; │ │ │ - -- 2.1225s elapsed │ │ │ + -- 1.5809s 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 │ │ │ @@ -149,15 +149,15 @@ │ │ │ │ │ │ o9 = 4 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i10 : elapsedTime fI=res I
│ │ │ - -- .0216671s elapsed
│ │ │ + -- .0274596s elapsed
│ │ │  
│ │ │          1       14       33       28       8
│ │ │  o10 = Pn  <-- Pn   <-- Pn   <-- Pn   <-- Pn  <-- 0
│ │ │                                                    
│ │ │        0       1        2        3        4       5
│ │ │  
│ │ │  o10 : ChainComplex
│ │ │ ├── html2text {} │ │ │ │ @@ -54,15 +54,15 @@ │ │ │ │ 2: . 12 │ │ │ │ │ │ │ │ o8 : BettiTally │ │ │ │ i9 : c=codim I │ │ │ │ │ │ │ │ o9 = 4 │ │ │ │ i10 : elapsedTime fI=res I │ │ │ │ - -- .0216671s elapsed │ │ │ │ + -- .0274596s 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 │ │ │ @@ -217,15 +217,15 @@ │ │ │ │ │ │ o14 : RingMap P2 <-- Pn │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i15 : elapsedTime betti(I'=trim ker phi)
│ │ │ - -- .585455s elapsed
│ │ │ + -- .560889s elapsed
│ │ │  
│ │ │               0  1
│ │ │  o15 = total: 1 11
│ │ │            0: 1  .
│ │ │            1: .  3
│ │ │            2: .  8
│ │ │  
│ │ │ @@ -238,15 +238,15 @@
│ │ │  
│ │ │  o16 = true
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i17 : elapsedTime basePts=primaryDecomposition ideal H;
│ │ │ - -- 5.79625s elapsed
│ │ │ + -- 5.02254s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i18 : tally apply(basePts,c->(dim c, degree c, betti c))
│ │ │  
│ │ │                            0 1
│ │ │ ├── html2text {}
│ │ │ │ @@ -110,28 +110,28 @@
│ │ │ │            6: . 7
│ │ │ │  
│ │ │ │  o13 : BettiTally
│ │ │ │  i14 : phi=map(P2,Pn,H);
│ │ │ │  
│ │ │ │  o14 : RingMap P2 <-- Pn
│ │ │ │  i15 : elapsedTime betti(I'=trim ker phi)
│ │ │ │ - -- .585455s elapsed
│ │ │ │ + -- .560889s 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;
│ │ │ │ - -- 5.79625s elapsed
│ │ │ │ + -- 5.02254s 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
│ │ │ @@ -193,15 +193,15 @@
│ │ │  
│ │ │  o13 : BettiTally
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i14 : elapsedTime sub(I,H)
│ │ │ - -- .0135763s elapsed
│ │ │ + -- .0196097s elapsed
│ │ │  
│ │ │  o14 = ideal (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
│ │ │  
│ │ │  o14 : Ideal of P2
│ │ │ │ │ │ │ │ │ │ │ │ @@ -210,15 +210,15 @@ │ │ │ │ │ │ o15 : RingMap P2 <-- Pn │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i16 : elapsedTime betti(I'=trim ker phi)
│ │ │ - -- .0549513s elapsed
│ │ │ + -- .0720603s elapsed
│ │ │  
│ │ │               0  1
│ │ │  o16 = total: 1 12
│ │ │            0: 1  .
│ │ │            1: . 12
│ │ │  
│ │ │  o16 : BettiTally
│ │ │ @@ -230,15 +230,15 @@ │ │ │ │ │ │ o17 = true │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i18 : elapsedTime basePts=primaryDecomposition ideal H;
│ │ │ - -- 2.1225s elapsed
│ │ │ + -- 1.5809s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i19 : tally apply(basePts,c->(dim c, degree c, betti c))
│ │ │  
│ │ │                             0  1
│ │ │ ├── html2text {}
│ │ │ │ @@ -82,36 +82,36 @@
│ │ │ │            0: 1 .
│ │ │ │            1: . .
│ │ │ │            2: . .
│ │ │ │            3: . 8
│ │ │ │  
│ │ │ │  o13 : BettiTally
│ │ │ │  i14 : elapsedTime sub(I,H)
│ │ │ │ - -- .0135763s elapsed
│ │ │ │ + -- .0196097s 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)
│ │ │ │ - -- .0549513s elapsed
│ │ │ │ + -- .0720603s elapsed
│ │ │ │  
│ │ │ │               0  1
│ │ │ │  o16 = total: 1 12
│ │ │ │            0: 1  .
│ │ │ │            1: . 12
│ │ │ │  
│ │ │ │  o16 : BettiTally
│ │ │ │  i17 : I'== I
│ │ │ │  
│ │ │ │  o17 = true
│ │ │ │  i18 : elapsedTime basePts=primaryDecomposition ideal H;
│ │ │ │ - -- 2.1225s elapsed
│ │ │ │ + -- 1.5809s 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/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 0.447739s (cpu); 0.367036s (thread); 0s (gc)
│ │ │ + -- used 0.545654s (cpu); 0.447216s (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 0.480185s (cpu); 0.403507s (thread); 0s (gc)
│ │ │ + -- used 0.575256s (cpu); 0.483894s (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
│ │ │ @@ -253,15 +253,15 @@
│ │ │                4      4
│ │ │  o13 : Matrix A  <-- A
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i14 : time betti (F = pureResolution(M,{0,2,4}))
│ │ │ - -- used 0.447739s (cpu); 0.367036s (thread); 0s (gc)
│ │ │ + -- used 0.545654s (cpu); 0.447216s (thread); 0s (gc)
│ │ │  
│ │ │               0 1 2
│ │ │  o14 = total: 3 6 3
│ │ │            0: 3 . .
│ │ │            1: . 6 .
│ │ │            2: . . 3
│ │ │  
│ │ │ @@ -272,15 +272,15 @@
│ │ │          

│ │ │            
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 = Complex with 6 actors
│ │ │ │  
│ │ │ │  o19 : ActionOnComplex
│ │ │ │  i20 : elapsedTime a1 = character A1
│ │ │ │ - -- .750459s elapsed
│ │ │ │ + -- .825978s 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
│ │ │ │ - -- 34.0105s elapsed
│ │ │ │ + -- 29.3868s 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)
│ │ │ │ - -- 14.0834s elapsed
│ │ │ │ + -- 12.4792s elapsed
│ │ │ │  
│ │ │ │  o32 = Character over R
│ │ │ │  
│ │ │ │        (0, {21}) => | 1 1 1 1 1 1 |
│ │ │ │  
│ │ │ │  o32 : Character
│ │ │ │  i33 : b/T
│ │ ├── ./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 0.232837s (cpu); 0.179806s (thread); 0s (gc)
│ │ │ + -- used 0.29943s (cpu); 0.224798s (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
│ │ │ @@ -380,15 +380,15 @@
│ │ │  
│ │ │  o22 : BettiTally
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │              
│ │ │            
│ │ │          

│ │ │                
i15 : time betti (F = pureResolution(11,4,{0,2,4}))
│ │ │ - -- used 0.480185s (cpu); 0.403507s (thread); 0s (gc)
│ │ │ + -- used 0.575256s (cpu); 0.483894s (thread); 0s (gc)
│ │ │  
│ │ │               0 1 2
│ │ │  o15 = total: 3 6 3
│ │ │            0: 3 . .
│ │ │            1: . 6 .
│ │ │            2: . . 3
│ │ │ ├── html2text {}
│ │ │ │ @@ -161,30 +161,30 @@
│ │ │ │        | -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 0.447739s (cpu); 0.367036s (thread); 0s (gc)
│ │ │ │ + -- used 0.545654s (cpu); 0.447216s (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 0.480185s (cpu); 0.403507s (thread); 0s (gc)
│ │ │ │ + -- used 0.575256s (cpu); 0.483894s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               0 1 2
│ │ │ │  o15 = total: 3 6 3
│ │ │ │            0: 3 . .
│ │ │ │            1: . 6 .
│ │ │ │            2: . . 3
│ │ ├── ./usr/share/doc/Macaulay2/Benchmark/example-output/_run__Benchmarks.out
│ │ │ @@ -1,10 +1,10 @@
│ │ │  -- -*- M2-comint -*- hash: 1330545576567
│ │ │  
│ │ │  i1 : runBenchmarks "res39"
│ │ │ --- beginning computation Fri Nov 14 17:31:11 UTC 2025
│ │ │ --- Linux sbuild 6.12.48+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.48-1 (2025-09-20) x86_64 GNU/Linux
│ │ │ --- AMD EPYC 7702P 64-Core Processor  AuthenticAMD  cpu MHz 1996.250  
│ │ │ +-- beginning computation Thu Dec 18 16:03:29 UTC 2025
│ │ │ +-- Linux sbuild 6.12.57+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.57-1 (2025-11-05) x86_64 GNU/Linux
│ │ │ +-- Intel Xeon Processor (Skylake, IBRS)  GenuineIntel  cpu MHz 2099.998  
│ │ │  -- Macaulay2 1.25.11, compiled with gcc 15.2.0
│ │ │ --- res39: res of a generic 3 by 9 matrix over ZZ/101: .115525 seconds
│ │ │ +-- res39: res of a generic 3 by 9 matrix over ZZ/101: .164204 seconds
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Benchmark/html/_run__Benchmarks.html
│ │ │ @@ -75,19 +75,19 @@
│ │ │          

│ │ │            
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 Fri Nov 14 17:31:11 UTC 2025
│ │ │ --- Linux sbuild 6.12.48+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.48-1 (2025-09-20) x86_64 GNU/Linux
│ │ │ --- AMD EPYC 7702P 64-Core Processor  AuthenticAMD  cpu MHz 1996.250  
│ │ │ +-- beginning computation Thu Dec 18 16:03:29 UTC 2025
│ │ │ +-- Linux sbuild 6.12.57+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.57-1 (2025-11-05) x86_64 GNU/Linux
│ │ │ +-- Intel Xeon Processor (Skylake, IBRS)  GenuineIntel  cpu MHz 2099.998  
│ │ │  -- Macaulay2 1.25.11, compiled with gcc 15.2.0
│ │ │ --- res39: res of a generic 3 by 9 matrix over ZZ/101: .115525 seconds

│ │ │ +-- res39: res of a generic 3 by 9 matrix over ZZ/101: .164204 seconds
│ │ │              

│ │ │        
│ │ │        

│ │ │          

│ │ │            
For the programmer

│ │ │ ├── html2text {}
│ │ │ │ @@ -23,18 +23,18 @@
│ │ │ │  "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 Fri Nov 14 17:31:11 UTC 2025
│ │ │ │ --- Linux sbuild 6.12.48+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.48-1
│ │ │ │ -(2025-09-20) x86_64 GNU/Linux
│ │ │ │ --- AMD EPYC 7702P 64-Core Processor  AuthenticAMD  cpu MHz 1996.250
│ │ │ │ +-- beginning computation Thu Dec 18 16:03:29 UTC 2025
│ │ │ │ +-- Linux sbuild 6.12.57+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ │ +6.12.57-1 (2025-11-05) x86_64 GNU/Linux
│ │ │ │ +-- Intel Xeon Processor (Skylake, IBRS)  GenuineIntel  cpu MHz 2099.998
│ │ │ │  -- Macaulay2 1.25.11, compiled with gcc 15.2.0
│ │ │ │ --- res39: res of a generic 3 by 9 matrix over ZZ/101: .115525 seconds
│ │ │ │ +-- res39: res of a generic 3 by 9 matrix over ZZ/101: .164204 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.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/Benchmark.m2:297:0.
│ │ ├── ./usr/share/doc/Macaulay2/Bertini/dump/rawdocumentation.dump
│ │ │ @@ -515,15 +515,15 @@
│ │ │  Pi4uLikiLCJCZXJ0aW5pIn0sIlJhbmRvbUNvbXBsZXgifSxUVHsiID0+ICJ9LFRUeyIuLi4ifSwi
│ │ │  LCAiLFNQQU57ImRlZmF1bHQgdmFsdWUgIiwie30ifSwiLCAiLFNQQU57fX0sU1BBTntUTzJ7bmV3
│ │ │  IERvY3VtZW50VGFnIGZyb20ge1tiZXJ0aW5pVXNlckhvbW90b3B5LFJhbmRvbVJlYWxdLCJiZXJ0
│ │ │  aW5pVXNlckhvbW90b3B5KC4uLixSYW5kb21SZWFsPT4uLi4pIiwiQmVydGluaSJ9LCJSYW5kb21S
│ │ │  ZWFsIn0sVFR7IiA9PiAifSxUVHsiLi4uIn0sIiwgIixTUEFOeyJkZWZhdWx0IHZhbHVlICIsInt9
│ │ │  In0sIiwgIixTUEFOe319LFNQQU57VE8ye25ldyBEb2N1bWVudFRhZyBmcm9tIHsiVG9wRGlyZWN0
│ │ │  b3J5IiwiVG9wRGlyZWN0b3J5IiwiQmVydGluaSJ9LCJUb3BEaXJlY3RvcnkifSxUVHsiID0+ICJ9
│ │ │ -LFRUeyIuLi4ifSwiLCAiLFNQQU57ImRlZmF1bHQgdmFsdWUgIiwiXCIvdG1wL00yLTI4Njg3LTAv
│ │ │ +LFRUeyIuLi4ifSwiLCAiLFNQQU57ImRlZmF1bHQgdmFsdWUgIiwiXCIvdG1wL00yLTQwOTk4LTAv
│ │ │  MFwiIn0sIiwgIixTUEFOeyJPcHRpb24gdG8gY2hhbmdlIGRpcmVjdG9yeSBmb3IgZmlsZSBzdG9y
│ │ │  YWdlLiJ9fSxTUEFOe1RPMntuZXcgRG9jdW1lbnRUYWcgZnJvbSB7W2JlcnRpbmlVc2VySG9tb3Rv
│ │ │  cHksVmVyYm9zZV0sImJlcnRpbmlVc2VySG9tb3RvcHkoLi4uLFZlcmJvc2U9Pi4uLikiLCJCZXJ0
│ │ │  aW5pIn0sIlZlcmJvc2UifSxUVHsiID0+ICJ9LFRUeyIuLi4ifSwiLCAiLFNQQU57ImRlZmF1bHQg
│ │ │  dmFsdWUgIiwiZmFsc2UifSwiLCAiLFNQQU57Ik9wdGlvbiB0byBzaWxlbmNlIGFkZGl0aW9uYWwg
│ │ │  b3V0cHV0In19fSwgc3ltYm9sIERvY3VtZW50VGFnID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsi
│ │ │  YmVydGluaVVzZXJIb21vdG9weSIsImJlcnRpbmlVc2VySG9tb3RvcHkiLCJCZXJ0aW5pIn0sIEtl
│ │ │ @@ -1100,15 +1100,15 @@
│ │ │  ZXJ0aW5pUGFyYW1ldGVySG9tb3RvcHkoLi4uLFJhbmRvbVJlYWw9Pi4uLikiLCJCZXJ0aW5pIn0s
│ │ │  IlJhbmRvbVJlYWwifSxUVHsiID0+ICJ9LFRUeyIuLi4ifSwiLCAiLFNQQU57ImRlZmF1bHQgdmFs
│ │ │  dWUgIiwie30ifSwiLCAiLFNQQU57ImFuIG9wdGlvbiB3aGljaCBkZXNpZ25hdGVzIHN5bWJvbHMv
│ │ │  c3RyaW5ncy92YXJpYWJsZXMgdGhhdCB3aWxsIGJlIHNldCB0byBiZSBhIHJhbmRvbSByZWFsIG51
│ │ │  bWJlciBvciByYW5kb20gY29tcGxleCBudW1iZXIifX0sU1BBTntUTzJ7bmV3IERvY3VtZW50VGFn
│ │ │  IGZyb20geyJUb3BEaXJlY3RvcnkiLCJUb3BEaXJlY3RvcnkiLCJCZXJ0aW5pIn0sIlRvcERpcmVj
│ │ │  dG9yeSJ9LFRUeyIgPT4gIn0sVFR7Ii4uLiJ9LCIsICIsU1BBTnsiZGVmYXVsdCB2YWx1ZSAiLCJc
│ │ │ -Ii90bXAvTTItMjg2ODctMC8wXCIifSwiLCAiLFNQQU57Ik9wdGlvbiB0byBjaGFuZ2UgZGlyZWN0
│ │ │ +Ii90bXAvTTItNDA5OTgtMC8wXCIifSwiLCAiLFNQQU57Ik9wdGlvbiB0byBjaGFuZ2UgZGlyZWN0
│ │ │  b3J5IGZvciBmaWxlIHN0b3JhZ2UuIn19LFNQQU57VE8ye25ldyBEb2N1bWVudFRhZyBmcm9tIHtb
│ │ │  YmVydGluaVBhcmFtZXRlckhvbW90b3B5LFZlcmJvc2VdLCJiZXJ0aW5pUGFyYW1ldGVySG9tb3Rv
│ │ │  cHkoLi4uLFZlcmJvc2U9Pi4uLikiLCJCZXJ0aW5pIn0sIlZlcmJvc2UifSxUVHsiID0+ICJ9LFRU
│ │ │  eyIuLi4ifSwiLCAiLFNQQU57ImRlZmF1bHQgdmFsdWUgIiwiZmFsc2UifSwiLCAiLFNQQU57Ik9w
│ │ │  dGlvbiB0byBzaWxlbmNlIGFkZGl0aW9uYWwgb3V0cHV0In19fSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsiYmVydGluaVBhcmFtZXRlckhvbW90b3B5IiwiYmVy
│ │ │  dGluaVBhcmFtZXRlckhvbW90b3B5IiwiQmVydGluaSJ9LCBLZXkgPT4gYmVydGluaVBhcmFtZXRl
│ │ │ @@ -2449,15 +2449,15 @@
│ │ │  YWw9Pi4uLikiLCJCZXJ0aW5pIn0sIlJhbmRvbVJlYWwifSxUVHsiID0+ICJ9LFRUeyIuLi4ifSwi
│ │ │  LCAiLFNQQU57ImRlZmF1bHQgdmFsdWUgIiwie30ifSwiLCAiLFNQQU57ImFuIG9wdGlvbiB3aGlj
│ │ │  aCBkZXNpZ25hdGVzIHN5bWJvbHMvc3RyaW5ncy92YXJpYWJsZXMgdGhhdCB3aWxsIGJlIHNldCB0
│ │ │  byBiZSBhIHJhbmRvbSByZWFsIG51bWJlciBvciByYW5kb20gY29tcGxleCBudW1iZXIifX0sU1BB
│ │ │  TntUTzJ7bmV3IERvY3VtZW50VGFnIGZyb20ge1tiZXJ0aW5pWmVyb0RpbVNvbHZlLFRvcERpcmVj
│ │ │  dG9yeV0sImJlcnRpbmlaZXJvRGltU29sdmUoLi4uLFRvcERpcmVjdG9yeT0+Li4uKSIsIkJlcnRp
│ │ │  bmkifSwiVG9wRGlyZWN0b3J5In0sVFR7IiA9PiAifSxUVHsiLi4uIn0sIiwgIixTUEFOeyJkZWZh
│ │ │ -dWx0IHZhbHVlICIsIlwiL3RtcC9NMi0yODY4Ny0wLzBcIiJ9LCIsICIsU1BBTnsiT3B0aW9uIHRv
│ │ │ +dWx0IHZhbHVlICIsIlwiL3RtcC9NMi00MDk5OC0wLzBcIiJ9LCIsICIsU1BBTnsiT3B0aW9uIHRv
│ │ │  IGNoYW5nZSBkaXJlY3RvcnkgZm9yIGZpbGUgc3RvcmFnZS4ifX0sU1BBTntUTzJ7bmV3IERvY3Vt
│ │ │  ZW50VGFnIGZyb20ge1tiZXJ0aW5pWmVyb0RpbVNvbHZlLFVzZVJlZ2VuZXJhdGlvbl0sImJlcnRp
│ │ │  bmlaZXJvRGltU29sdmUoLi4uLFVzZVJlZ2VuZXJhdGlvbj0+Li4uKSIsIkJlcnRpbmkifSwiVXNl
│ │ │  UmVnZW5lcmF0aW9uIn0sVFR7IiA9PiAifSxUVHsiLi4uIn0sIiwgIixTUEFOeyJkZWZhdWx0IHZh
│ │ │  bHVlICIsIi0xIn0sIiwgIixTUEFOe319LFNQQU57VE8ye25ldyBEb2N1bWVudFRhZyBmcm9tIHtb
│ │ │  YmVydGluaVplcm9EaW1Tb2x2ZSxWZXJib3NlXSwiYmVydGluaVplcm9EaW1Tb2x2ZSguLi4sVmVy
│ │ │  Ym9zZT0+Li4uKSIsIkJlcnRpbmkifSwiVmVyYm9zZSJ9LFRUeyIgPT4gIn0sVFR7Ii4uLiJ9LCIs
│ │ ├── ./usr/share/doc/Macaulay2/Bertini/html/_bertini__Parameter__Homotopy.html
│ │ │ @@ -72,15 +72,15 @@
│ │ │              
  • HomVariableGroup => ..., default value {}, an option to group variables and use multihomogeneous homotopies
    • 
│ │ │              
  • M2Precision (missing documentation)
│ │ │   => ..., default value 53, 
    • 
│ │ │              
  • 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/M2-28687-0/0", Option to change directory for file storage.
    • 
│ │ │ +            
  • TopDirectory => ..., default value "/tmp/M2-40998-0/0", Option to change directory for file storage.
    • 
│ │ │              
  • Verbose => ..., default value false, Option to silence additional output
    • 
│ │ │            
│ │ │          
│ │ │          
  • Outputs:          
      
│ │ │              
    • S, a list, a list whose entries are lists of solutions for each target system
      • 
│ │ │            
    
│ │ │          
    • 
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,15 +26,15 @@
│ │ │ │              "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/M2-28687-0/0", Option to
│ │ │ │ +          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-40998-0/0", 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
│ │ │ @@ -77,15 +77,15 @@
│ │ │   => ..., default value 53, 
│ │ │              
  • OutputStyle (missing documentation)
│ │ │   => ..., default value "OutPoints", 
    • 
│ │ │              
  • RandomComplex (missing documentation)
│ │ │   => ..., default value {}, 
    • 
│ │ │              
  • RandomReal (missing documentation)
│ │ │   => ..., default value {}, 
    • 
│ │ │ -            
  • TopDirectory => ..., default value "/tmp/M2-28687-0/0", Option to change directory for file storage.
    • 
│ │ │ +            
  • TopDirectory => ..., default value "/tmp/M2-40998-0/0", Option to change directory for file storage.
    • 
│ │ │              
  • Verbose => ..., default value false, Option to silence additional output
    • 
│ │ │            
│ │ │          
│ │ │          
  • Outputs:          
      
│ │ │              
    • S0, a list, a list of solutions to the target system
      • 
│ │ │            
    
│ │ │          
    • 
│ │ │ ├── html2text {}
│ │ │ │ @@ -21,15 +21,15 @@
│ │ │ │              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/M2-28687-0/0", Option to
│ │ │ │ +          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-40998-0/0", 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
│ │ │ @@ -79,15 +79,15 @@
│ │ │   => ..., default value "main_data", 
│ │ │              
  • NameSolutionsFile (missing documentation)
│ │ │   => ..., default value "raw_solutions", 
    • 
│ │ │              
  • 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/M2-28687-0/0", Option to change directory for file storage.
    • 
│ │ │ +            
  • TopDirectory => ..., default value "/tmp/M2-40998-0/0", Option to change directory for file storage.
    • 
│ │ │              
  • UseRegeneration (missing documentation)
│ │ │   => ..., default value -1, 
    • 
│ │ │              
  • Verbose => ..., default value false, Option to silence additional output
    • 
│ │ │            
│ │ │          
│ │ │          
  • Outputs:          
      
│ │ │              
    • S, a list, a list of points that are contained in the variety of F
      • 
│ │ │ ├── html2text {}
│ │ │ │ @@ -32,15 +32,15 @@
│ │ │ │              "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/M2-28687-0/0", Option to
│ │ │ │ +          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-40998-0/0", 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/example-output/___Betti__Characters_sp__Example_sp1.out
│ │ │ @@ -76,15 +76,15 @@
│ │ │  i8 : A = action(RI,S7)
│ │ │  
│ │ │  o8 = Complex with 15 actors
│ │ │  
│ │ │  o8 : ActionOnComplex
│ │ │  
│ │ │  i9 : elapsedTime c = character A
│ │ │ - -- .470785s elapsed
│ │ │ + -- .420119s 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 = Complex with 11 actors
│ │ │  
│ │ │  o6 : ActionOnComplex
│ │ │  
│ │ │  i7 : elapsedTime c=character A
│ │ │ - -- .391999s elapsed
│ │ │ + -- .469453s 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 = Complex with 6 actors
│ │ │  
│ │ │  o19 : ActionOnComplex
│ │ │  
│ │ │  i20 : elapsedTime a1 = character A1
│ │ │ - -- .750459s elapsed
│ │ │ + -- .825978s 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
│ │ │ - -- 34.0105s elapsed
│ │ │ + -- 29.3868s 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)
│ │ │ - -- 14.0834s elapsed
│ │ │ + -- 12.4792s 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
│ │ │ @@ -162,15 +162,15 @@
│ │ │  
│ │ │  o8 : ActionOnComplex
    • 
│ │ │              
    
│ │ │                
    i9 : elapsedTime c = character A
│ │ │ - -- .470785s elapsed
│ │ │ + -- .420119s 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 = Complex with 15 actors
│ │ │ │  
│ │ │ │  o8 : ActionOnComplex
│ │ │ │  i9 : elapsedTime c = character A
│ │ │ │ - -- .470785s elapsed
│ │ │ │ + -- .420119s 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
│ │ │ @@ -180,15 +180,15 @@
│ │ │  
│ │ │  o6 : ActionOnComplex
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime c=character A
│ │ │ - -- .391999s elapsed
│ │ │ + -- .469453s 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 = Complex with 11 actors
│ │ │ │  
│ │ │ │  o6 : ActionOnComplex
│ │ │ │  i7 : elapsedTime c=character A
│ │ │ │ - -- .391999s elapsed
│ │ │ │ + -- .469453s 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
│ │ │ @@ -310,30 +310,30 @@
│ │ │  
│ │ │  o19 : ActionOnComplex
    
│ │ │              
    
│ │ │                
    i20 : elapsedTime a1 = character A1
│ │ │ - -- .750459s elapsed
│ │ │ + -- .825978s 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
│ │ │ - -- 34.0105s elapsed
│ │ │ + -- 29.3868s 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 |
│ │ │ @@ -467,15 +467,15 @@
│ │ │  
│ │ │  o31 : ActionOnGradedModule
    
│ │ │              
    
│ │ │                
    i32 : elapsedTime b = character(B,21)
│ │ │ - -- 14.0834s elapsed
│ │ │ + -- 12.4792s elapsed
│ │ │  
│ │ │  o32 = Character over R
│ │ │         
│ │ │        (0, {21}) => | 1 1 1 1 1 1 |
│ │ │  
│ │ │  o32 : Character
    
│ │ │              
    
│ │ │                
    i23 : time j=bruns F.dd_3;
│ │ │ - -- used 0.232837s (cpu); 0.179806s (thread); 0s (gc)
│ │ │ + -- used 0.29943s (cpu); 0.224798s (thread); 0s (gc)
│ │ │  
│ │ │  o23 : Ideal of S
    
│ │ │              
    
│ │ │                
    i24 : betti res j
│ │ │ ├── html2text {}
│ │ │ │ @@ -230,15 +230,15 @@
│ │ │ │  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 0.232837s (cpu); 0.179806s (thread); 0s (gc)
│ │ │ │ + -- used 0.29943s (cpu); 0.224798s (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/example-output/_cell__Complex_lp__Ring_cm__Simplicial__Complex_rp.out
│ │ │ @@ -24,15 +24,15 @@
│ │ │  
│ │ │  o7 : CellComplex
│ │ │  
│ │ │  i8 : applyValues(cells C, l -> apply(l,cellLabel))
│ │ │  
│ │ │                        5   4    3 2   2 3     4   5
│ │ │  o8 = HashTable{0 => {x , x y, x y , x y , x*y , x }                                       }
│ │ │ -                      5 3   5 4   4 2   4 4   5    3 3   5 2   2 4   5 3   5 4   5    5 2
│ │ │ -               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   5 3   5 4   5 2   5 4   5 3   5 4   5 2
│ │ │ +                      2 4   5 3   5 4   5    5 2   5 3   5 4   4 2   4 4   5    3 3   5 2
│ │ │ +               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/_hull__Complex.out
│ │ │ @@ -19,16 +19,16 @@
│ │ │                             
│ │ │       -1     0      1      2
│ │ │  
│ │ │  o4 : Complex
│ │ │  
│ │ │  i5 : cells(1,H)/cellLabel
│ │ │  
│ │ │ -       4 5   2        2    4 4    5 3    5 4   3 5
│ │ │ -o5 = {x y , x y*z, x*y z, x y z, x y z, x y , x y z}
│ │ │ +       5 3    5 4   3 5    4 5   2        2    4 4
│ │ │ +o5 = {x y z, x y , x y z, x y , x y*z, x*y z, x y z}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : cells(2,H)/cellLabel
│ │ │  
│ │ │         5 4    4 5
│ │ │  o6 = {x y z, x y z}
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/html/_cell__Complex_lp__Ring_cm__Simplicial__Complex_rp.html
│ │ │ @@ -129,17 +129,17 @@
│ │ │            
    
│ │ │                
    i8 : applyValues(cells C, l -> apply(l,cellLabel))
│ │ │  
│ │ │                        5   4    3 2   2 3     4   5
│ │ │  o8 = HashTable{0 => {x , x y, x y , x y , x*y , x }                                       }
│ │ │ -                      5 3   5 4   4 2   4 4   5    3 3   5 2   2 4   5 3   5 4   5    5 2
│ │ │ -               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   5 3   5 4   5 2   5 4   5 3   5 4   5 2
│ │ │ +                      2 4   5 3   5 4   5    5 2   5 3   5 4   4 2   4 4   5    3 3   5 2
│ │ │ +               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 {}
│ │ │ │ @@ -41,19 +41,19 @@
│ │ │ │  
│ │ │ │  o7 : CellComplex
│ │ │ │  i8 : applyValues(cells C, l -> apply(l,cellLabel))
│ │ │ │  
│ │ │ │                        5   4    3 2   2 3     4   5
│ │ │ │  o8 = HashTable{0 => {x , x y, x y , x y , x*y , x }
│ │ │ │  }
│ │ │ │ -                      5 3   5 4   4 2   4 4   5    3 3   5 2   2 4   5 3   5 4
│ │ │ │ -5    5 2
│ │ │ │ -               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   5 3   5 4   5 2   5 4   5 3   5 4   5 2
│ │ │ │ +                      2 4   5 3   5 4   5    5 2   5 3   5 4   4 2   4 4   5
│ │ │ │ +3 3   5 2
│ │ │ │ +               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:: **********
│ │ │ │      * _c_e_l_l_C_o_m_p_l_e_x_(_R_i_n_g_,_S_i_m_p_l_i_c_i_a_l_C_o_m_p_l_e_x_) -- Creates a cell complex from a
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/html/_hull__Complex.html
│ │ │ @@ -109,16 +109,16 @@
│ │ │  o4 : Complex
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : cells(1,H)/cellLabel
│ │ │  
│ │ │ -       4 5   2        2    4 4    5 3    5 4   3 5
│ │ │ -o5 = {x y , x y*z, x*y z, x y z, x y z, x y , x y z}
│ │ │ +       5 3    5 4   3 5    4 5   2        2    4 4
│ │ │ +o5 = {x y z, x y , x y z, x y , x y*z, x*y z, x y z}
│ │ │  
│ │ │  o5 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : cells(2,H)/cellLabel
│ │ │ ├── html2text {}
│ │ │ │ @@ -39,16 +39,16 @@
│ │ │ │  o4 = S  <-- S  <-- S  <-- S
│ │ │ │  
│ │ │ │       -1     0      1      2
│ │ │ │  
│ │ │ │  o4 : Complex
│ │ │ │  i5 : cells(1,H)/cellLabel
│ │ │ │  
│ │ │ │ -       4 5   2        2    4 4    5 3    5 4   3 5
│ │ │ │ -o5 = {x y , x y*z, x*y z, x y z, x y z, x y , x y z}
│ │ │ │ +       5 3    5 4   3 5    4 5   2        2    4 4
│ │ │ │ +o5 = {x y z, x y , x y z, x y , x y*z, x*y z, x y z}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : cells(2,H)/cellLabel
│ │ │ │  
│ │ │ │         5 4    4 5
│ │ │ │  o6 = {x y z, x y z}
│ │ ├── ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_minimize_lp__Chain__Complex_rp.out
│ │ │ @@ -63,15 +63,15 @@
│ │ │  o11 : ChainComplex
│ │ │  
│ │ │  i12 : isMinimalChainComplex E
│ │ │  
│ │ │  o12 = false
│ │ │  
│ │ │  i13 : time m = minimize (E[1]);
│ │ │ - -- used 0.273702s (cpu); 0.213569s (thread); 0s (gc)
│ │ │ + -- used 0.354472s (cpu); 0.270071s (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 0.0947774s (cpu); 0.0947771s (thread); 0s (gc)
│ │ │ + -- used 0.104671s (cpu); 0.10467s (thread); 0s (gc)
│ │ │  
│ │ │  i9 : time n = cartanEilenbergResolution C;
│ │ │ - -- used 0.215242s (cpu); 0.17428s (thread); 0s (gc)
│ │ │ + -- used 0.252815s (cpu); 0.167776s (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_lp__Chain__Complex_rp.html
│ │ │ @@ -181,15 +181,15 @@
│ │ │          
    
│ │ │            
    Now we minimize the result. The free summand we added to the end maps to zero, and thus is part of the minimization.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ @@ -301,15 +301,15 @@
│ │ │  
│ │ │  o21 : A
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -160,15 +160,15 @@
│ │ │ │  
│ │ │ │                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 0.66005s (cpu); 0.386905s (thread); 0s (gc)
│ │ │ │ + -- used 1.26396s (cpu); 0.448208s (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)
│ │ │ │ @@ -199,15 +199,15 @@
│ │ │ │  
│ │ │ │          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 0.0915819s (cpu); 0.0600642s (thread); 0s (gc)
│ │ │ │ + -- used 0.105346s (cpu); 0.0734029s (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
│ │ │ @@ -72,15 +72,15 @@
│ │ │  
│ │ │  o2 : NormalToricVariety
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i13 : time m = minimize (E[1]);
│ │ │ - -- used 0.273702s (cpu); 0.213569s (thread); 0s (gc)
    
│ │ │ + -- used 0.354472s (cpu); 0.270071s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i14 : isQuasiIsomorphism m
│ │ │  
│ │ │  o14 = true
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -81,15 +81,15 @@
│ │ │ │  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 0.273702s (cpu); 0.213569s (thread); 0s (gc)
│ │ │ │ + -- used 0.354472s (cpu); 0.270071s (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
│ │ │ @@ -129,21 +129,21 @@
│ │ │                  		
│ │ │                
    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 0.0947774s (cpu); 0.0947771s (thread); 0s (gc)
    
│ │ │ + -- used 0.104671s (cpu); 0.10467s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i9 : time n = cartanEilenbergResolution C;
│ │ │ - -- used 0.215242s (cpu); 0.17428s (thread); 0s (gc)
    
│ │ │ + -- used 0.252815s (cpu); 0.167776s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i10 : betti source m
│ │ │  
│ │ │               0  1  2   3   4   5   6   7
│ │ │ ├── html2text {}
│ │ │ │ @@ -49,17 +49,17 @@
│ │ │ │  
│ │ │ │  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 0.0947774s (cpu); 0.0947771s (thread); 0s (gc)
│ │ │ │ + -- used 0.104671s (cpu); 0.10467s (thread); 0s (gc)
│ │ │ │  i9 : time n = cartanEilenbergResolution C;
│ │ │ │ - -- used 0.215242s (cpu); 0.17428s (thread); 0s (gc)
│ │ │ │ + -- used 0.252815s (cpu); 0.167776s (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/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 0.66005s (cpu); 0.386905s (thread); 0s (gc)
│ │ │ + -- used 1.26396s (cpu); 0.448208s (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 0.0915819s (cpu); 0.0600642s (thread); 0s (gc)
│ │ │ + -- used 0.105346s (cpu); 0.0734029s (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 0.23277s (cpu); 0.15461s (thread); 0s (gc)
│ │ │ + -- used 0.26581s (cpu); 0.164203s (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 0.359124s (cpu); 0.291528s (thread); 0s (gc)
│ │ │ + -- used 0.414944s (cpu); 0.311642s (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 0.628951s (cpu); 0.306279s (thread); 0s (gc)
│ │ │ + -- used 1.26655s (cpu); 0.427004s (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 2.27529s (cpu); 1.87843s (thread); 0s (gc)
│ │ │ + -- used 2.50376s (cpu); 2.18259s (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 0.277885s (cpu); 0.20403s (thread); 0s (gc)
│ │ │ + -- used 0.367023s (cpu); 0.246429s (thread); 0s (gc)
│ │ │  
│ │ │          3     2
│ │ │  o10 = 3h  + 5h
│ │ │          1     1
│ │ │  
│ │ │        ZZ[h ]
│ │ │            1
│ │ │  o10 : ------
│ │ │           4
│ │ │          h
│ │ │           1
│ │ │  
│ │ │  i11 : time CSM(K,CompMethod=>PnResidual)
│ │ │ - -- used 0.0819154s (cpu); 0.0819232s (thread); 0s (gc)
│ │ │ + -- used 0.100399s (cpu); 0.100409s (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 0.0786223s (cpu); 0.0359069s (thread); 0s (gc)
│ │ │ + -- used 0.092511s (cpu); 0.0610662s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 4
│ │ │  
│ │ │  i5 : time Euler I
│ │ │ - -- used 0.256966s (cpu); 0.152636s (thread); 0s (gc)
│ │ │ + -- used 0.313028s (cpu); 0.176694s (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 0.133671s (cpu); 0.0748297s (thread); 0s (gc)
│ │ │ + -- used 0.230227s (cpu); 0.0931576s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 2
│ │ │  
│ │ │  i11 : time Euler(J,Method=>DirectCompleteInt,IndsOfSmooth=>{0,1})
│ │ │ - -- used 0.224953s (cpu); 0.102967s (thread); 0s (gc)
│ │ │ + -- used 0.282088s (cpu); 0.11223s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = 2
│ │ │  
│ │ │  i12 : R=MultiProjCoordRing({2,2})
│ │ │  
│ │ │  o12 = R
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Euler__Affine.out
│ │ │ @@ -13,12 +13,12 @@
│ │ │              2    2    2
│ │ │  o3 = ideal(x  + x  + x  - 1)
│ │ │              1    2    3
│ │ │  
│ │ │  o3 : Ideal of R
│ │ │  
│ │ │  i4 : time EulerAffine I
│ │ │ - -- used 0.0603683s (cpu); 0.0550831s (thread); 0s (gc)
│ │ │ + -- used 0.0904061s (cpu); 0.0672509s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 2
│ │ │  
│ │ │  i5 :
│ │ ├── ./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 2.26857s (cpu); 1.16196s (thread); 0s (gc)
│ │ │ + -- used 6.79101s (cpu); 1.46886s (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 2.85984s (cpu); 1.23776s (thread); 0s (gc)
│ │ │ + -- used 6.60161s (cpu); 1.5278s (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 0.658368s (cpu); 0.396912s (thread); 0s (gc)
│ │ │ + -- used 1.18058s (cpu); 0.513952s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o3 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │  o3 : ------
│ │ │          5
│ │ │         h
│ │ │          1
│ │ │  
│ │ │  i4 : time CSM(I,InputIsSmooth=>true)
│ │ │ - -- used 0.0464211s (cpu); 0.0327346s (thread); 0s (gc)
│ │ │ + -- used 0.0992696s (cpu); 0.0409482s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o4 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │  o4 : ------
│ │ │          5
│ │ │         h
│ │ │          1
│ │ │  
│ │ │  i5 : time Chern I
│ │ │ - -- used 0.0380228s (cpu); 0.0341599s (thread); 0s (gc)
│ │ │ + -- used 0.0721917s (cpu); 0.043178s (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 1.61391s (cpu); 0.920252s (thread); 0s (gc)
│ │ │ + -- used 4.2163s (cpu); 1.48308s (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 0.483112s (cpu); 0.214874s (thread); 0s (gc)
│ │ │ + -- used 0.846411s (cpu); 0.27438s (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
│ │ │ @@ -234,15 +234,15 @@
│ │ │  
│ │ │  o14 : Ideal of R
    
│ │ │              
    
│ │ │                
    i15 : time csmK=CSM(A,K)
│ │ │ - -- used 0.66005s (cpu); 0.386905s (thread); 0s (gc)
│ │ │ + -- used 1.26396s (cpu); 0.448208s (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 0.0915819s (cpu); 0.0600642s (thread); 0s (gc)
│ │ │ + -- used 0.105346s (cpu); 0.0734029s (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 0.23277s (cpu); 0.15461s (thread); 0s (gc)
│ │ │ + -- used 0.26581s (cpu); 0.164203s (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
│ │ │ @@ -88,15 +88,15 @@
│ │ │       (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 0.359124s (cpu); 0.291528s (thread); 0s (gc)
│ │ │ + -- used 0.414944s (cpu); 0.311642s (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 0.23277s (cpu); 0.15461s (thread); 0s (gc)
│ │ │ │ + -- used 0.26581s (cpu); 0.164203s (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 0.359124s (cpu); 0.291528s (thread); 0s (gc)
│ │ │ │ + -- used 0.414944s (cpu); 0.311642s (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
│ │ │ @@ -92,15 +92,15 @@
│ │ │  
│ │ │  o4 : Ideal of R
    
│ │ │              
    
│ │ │                
    i5 : time CSM(I,CompMethod=>ProjectiveDegree)
│ │ │ - -- used 0.628951s (cpu); 0.306279s (thread); 0s (gc)
│ │ │ + -- used 1.26655s (cpu); 0.427004s (thread); 0s (gc)
│ │ │  
│ │ │         5      4      3      2
│ │ │  o5 = 6h  + 14h  + 14h  + 10h
│ │ │         1      1      1      1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ @@ -109,15 +109,15 @@
│ │ │         h
│ │ │          1
    
│ │ │              
    
│ │ │                
    i6 : time CSM(I,CompMethod=>PnResidual)
│ │ │ - -- used 2.27529s (cpu); 1.87843s (thread); 0s (gc)
│ │ │ + -- used 2.50376s (cpu); 2.18259s (thread); 0s (gc)
│ │ │  
│ │ │         5      4      3      2
│ │ │  o6 = 6H  + 14H  + 14H  + 10H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o6 : -----
│ │ │          6
│ │ │ @@ -142,15 +142,15 @@
│ │ │  
│ │ │  o9 : Ideal of S
    
│ │ │              
    
│ │ │                
    i10 : time CSM(K,CompMethod=>ProjectiveDegree)
│ │ │ - -- used 0.277885s (cpu); 0.20403s (thread); 0s (gc)
│ │ │ + -- used 0.367023s (cpu); 0.246429s (thread); 0s (gc)
│ │ │  
│ │ │          3     2
│ │ │  o10 = 3h  + 5h
│ │ │          1     1
│ │ │  
│ │ │        ZZ[h ]
│ │ │            1
│ │ │ @@ -159,15 +159,15 @@
│ │ │          h
│ │ │           1
    
│ │ │              
    
│ │ │                
    i11 : time CSM(K,CompMethod=>PnResidual)
│ │ │ - -- used 0.0819154s (cpu); 0.0819232s (thread); 0s (gc)
│ │ │ + -- used 0.100399s (cpu); 0.100409s (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 0.628951s (cpu); 0.306279s (thread); 0s (gc)
│ │ │ │ + -- used 1.26655s (cpu); 0.427004s (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 2.27529s (cpu); 1.87843s (thread); 0s (gc)
│ │ │ │ + -- used 2.50376s (cpu); 2.18259s (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 0.277885s (cpu); 0.20403s (thread); 0s (gc)
│ │ │ │ + -- used 0.367023s (cpu); 0.246429s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          3     2
│ │ │ │  o10 = 3h  + 5h
│ │ │ │          1     1
│ │ │ │  
│ │ │ │        ZZ[h ]
│ │ │ │            1
│ │ │ │  o10 : ------
│ │ │ │           4
│ │ │ │          h
│ │ │ │           1
│ │ │ │  i11 : time CSM(K,CompMethod=>PnResidual)
│ │ │ │ - -- used 0.0819154s (cpu); 0.0819232s (thread); 0s (gc)
│ │ │ │ + -- used 0.100399s (cpu); 0.100409s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          3     2
│ │ │ │  o11 = 3H  + 5H
│ │ │ │  
│ │ │ │        ZZ[H]
│ │ │ │  o11 : -----
│ │ │ │           4
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Euler.html
│ │ │ @@ -125,23 +125,23 @@
│ │ │  
│ │ │  o3 : Ideal of R
    
│ │ │              
    
│ │ │                
    i4 : time Euler(I,InputIsSmooth=>true)
│ │ │ - -- used 0.0786223s (cpu); 0.0359069s (thread); 0s (gc)
│ │ │ + -- used 0.092511s (cpu); 0.0610662s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 4
    
│ │ │              
    
│ │ │                
    i5 : time Euler I
│ │ │ - -- used 0.256966s (cpu); 0.152636s (thread); 0s (gc)
│ │ │ + -- used 0.313028s (cpu); 0.176694s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 4
    
│ │ │              
    
│ │ │                
    i6 : EulerIHash=Euler(I,Output=>HashForm);
    
│ │ │ @@ -189,23 +189,23 @@
│ │ │          
    
│ │ │            
    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 0.133671s (cpu); 0.0748297s (thread); 0s (gc)
│ │ │ + -- used 0.230227s (cpu); 0.0931576s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 2
    
│ │ │              
    
│ │ │                
    i11 : time Euler(J,Method=>DirectCompleteInt,IndsOfSmooth=>{0,1})
│ │ │ - -- used 0.224953s (cpu); 0.102967s (thread); 0s (gc)
│ │ │ + -- used 0.282088s (cpu); 0.11223s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = 2
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Now consider an example in \PP^2 \times \PP^2.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -74,19 +74,19 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │               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 0.0786223s (cpu); 0.0359069s (thread); 0s (gc)
│ │ │ │ + -- used 0.092511s (cpu); 0.0610662s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 4
│ │ │ │  i5 : time Euler I
│ │ │ │ - -- used 0.256966s (cpu); 0.152636s (thread); 0s (gc)
│ │ │ │ + -- used 0.313028s (cpu); 0.176694s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 4
│ │ │ │  i6 : EulerIHash=Euler(I,Output=>HashForm);
│ │ │ │  i7 : A=ring EulerIHash#"CSM"
│ │ │ │  
│ │ │ │  o7 = A
│ │ │ │  
│ │ │ │ @@ -114,19 +114,19 @@
│ │ │ │  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 0.133671s (cpu); 0.0748297s (thread); 0s (gc)
│ │ │ │ + -- used 0.230227s (cpu); 0.0931576s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = 2
│ │ │ │  i11 : time Euler(J,Method=>DirectCompleteInt,IndsOfSmooth=>{0,1})
│ │ │ │ - -- used 0.224953s (cpu); 0.102967s (thread); 0s (gc)
│ │ │ │ + -- used 0.282088s (cpu); 0.11223s (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/___Euler__Affine.html
│ │ │ @@ -95,15 +95,15 @@
│ │ │  
│ │ │  o3 : Ideal of R
│ │ │              
    
│ │ │                
    i4 : time EulerAffine I
│ │ │ - -- used 0.0603683s (cpu); 0.0550831s (thread); 0s (gc)
│ │ │ + -- used 0.0904061s (cpu); 0.0672509s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 2
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Observe that the algorithm is a probabilistic algorithm and may give a wrong answer with a small but nonzero probability. Read more under probabilistic algorithm.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,15 +23,15 @@
│ │ │ │  
│ │ │ │              2    2    2
│ │ │ │  o3 = ideal(x  + x  + x  - 1)
│ │ │ │              1    2    3
│ │ │ │  
│ │ │ │  o3 : Ideal of R
│ │ │ │  i4 : time EulerAffine I
│ │ │ │ - -- used 0.0603683s (cpu); 0.0550831s (thread); 0s (gc)
│ │ │ │ + -- used 0.0904061s (cpu); 0.0672509s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 2
│ │ │ │  Observe that the algorithm is a probabilistic algorithm and may give a wrong
│ │ │ │  answer with a small but nonzero probability. Read more under _p_r_o_b_a_b_i_l_i_s_t_i_c
│ │ │ │  _a_l_g_o_r_i_t_h_m.
│ │ │ │  ********** WWaayyss ttoo uussee EEuulleerrAAffffiinnee:: **********
│ │ │ │      * EulerAffine(Ideal)
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Inds__Of__Smooth.html
│ │ │ @@ -70,15 +70,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time CSM(I,Method=>DirectCompletInt)
│ │ │ - -- used 2.26857s (cpu); 1.16196s (thread); 0s (gc)
│ │ │ + -- used 6.79101s (cpu); 1.46886s (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
│ │ │ @@ -87,15 +87,15 @@
│ │ │        (h , h )
│ │ │          1   2
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time CSM(I,Method=>DirectCompletInt,IndsOfSmooth=>{1,2})
│ │ │ - -- used 2.85984s (cpu); 1.23776s (thread); 0s (gc)
│ │ │ + -- used 6.60161s (cpu); 1.5278s (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 2.26857s (cpu); 1.16196s (thread); 0s (gc)
│ │ │ │ + -- used 6.79101s (cpu); 1.46886s (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 2.85984s (cpu); 1.23776s (thread); 0s (gc)
│ │ │ │ + -- used 6.60161s (cpu); 1.5278s (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
│ │ │ @@ -66,15 +66,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time CSM I
│ │ │ - -- used 0.658368s (cpu); 0.396912s (thread); 0s (gc)
│ │ │ + -- used 1.18058s (cpu); 0.513952s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o3 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ @@ -83,15 +83,15 @@
│ │ │         h
│ │ │          1
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time CSM(I,InputIsSmooth=>true)
│ │ │ - -- used 0.0464211s (cpu); 0.0327346s (thread); 0s (gc)
│ │ │ + -- used 0.0992696s (cpu); 0.0409482s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o4 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ @@ -105,15 +105,15 @@
│ │ │          
    
│ │ │            
    Note that one could, equivalently, use the command Chern instead in this case.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i5 : time Chern I
│ │ │ - -- used 0.0380228s (cpu); 0.0341599s (thread); 0s (gc)
│ │ │ + -- used 0.0721917s (cpu); 0.043178s (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 0.658368s (cpu); 0.396912s (thread); 0s (gc)
│ │ │ │ + -- used 1.18058s (cpu); 0.513952s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         3
│ │ │ │  o3 = 4h
│ │ │ │         1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ │ │  o3 : ------
│ │ │ │          5
│ │ │ │         h
│ │ │ │          1
│ │ │ │  i4 : time CSM(I,InputIsSmooth=>true)
│ │ │ │ - -- used 0.0464211s (cpu); 0.0327346s (thread); 0s (gc)
│ │ │ │ + -- used 0.0992696s (cpu); 0.0409482s (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 0.0380228s (cpu); 0.0341599s (thread); 0s (gc)
│ │ │ │ + -- used 0.0721917s (cpu); 0.043178s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         3
│ │ │ │  o5 = 4h
│ │ │ │         1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Method.html
│ │ │ @@ -70,15 +70,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
    
│ │ │              
    
│ │ │                
    i3 : time CSM I
│ │ │ - -- used 1.61391s (cpu); 0.920252s (thread); 0s (gc)
│ │ │ + -- used 4.2163s (cpu); 1.48308s (thread); 0s (gc)
│ │ │  
│ │ │          5      4     3
│ │ │  o3 = 12h  + 10h  + 6h
│ │ │          1      1     1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ @@ -87,15 +87,15 @@
│ │ │         h
│ │ │          1
    
│ │ │              
    
│ │ │                
    i4 : time CSM(I,Method=>DirectCompleteInt)
│ │ │ - -- used 0.483112s (cpu); 0.214874s (thread); 0s (gc)
│ │ │ + -- used 0.846411s (cpu); 0.27438s (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 1.61391s (cpu); 0.920252s (thread); 0s (gc)
│ │ │ │ + -- used 4.2163s (cpu); 1.48308s (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 0.483112s (cpu); 0.214874s (thread); 0s (gc)
│ │ │ │ + -- used 0.846411s (cpu); 0.27438s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          5      4     3
│ │ │ │  o4 = 12h  + 10h  + 6h
│ │ │ │          1      1     1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ ├── ./usr/share/doc/Macaulay2/Chordal/example-output/_chordal__Net_lp__Hash__Table_cm__Hash__Table_cm__Elim__Tree_cm__Digraph_rp.out
│ │ │ @@ -16,32 +16,32 @@
│ │ │  
│ │ │  o2 : Digraph
│ │ │  
│ │ │  i3 : G = chordalGraph digraph hashTable{a=>{b,c},b=>{c},c=>{d},d=>{}};
│ │ │  
│ │ │  i4 : tree = elimTree G
│ │ │  
│ │ │ -o4 = ElimTree{a => b   }
│ │ │ +o4 = ElimTree{a => c}
│ │ │                b => c
│ │ │                c => d
│ │ │ -              d => null
│ │ │ +              d => b
│ │ │  
│ │ │  o4 : ElimTree
│ │ │  
│ │ │  i5 : rnk = hashTable{"a0"=>a, "a1"=>a, "b0"=>b, "b1"=>b, "b2"=>b,
│ │ │                       "c0"=>c, "d0"=>d, "c1"=>c, "d1"=>d};
│ │ │  
│ │ │  i6 : eqs = hashTable{"a0" => ({a},{}), "a1" => ({},{}),
│ │ │                       "b0" => ({b},{}), "b1" => ({},{}), "b2" => ({b},{}),
│ │ │                       "c0" => ({c},{}), "c1" => ({},{}),
│ │ │                       "d0" => ({},{}), "d1" => ({d},{}) };
│ │ │  
│ │ │  i7 : chordalNet(eqs,rnk,tree,DG)
│ │ │  
│ │ │ -o7 = ChordalNet{ d => { , d}    }
│ │ │ -                 b => {b,  , b}
│ │ │ -                 a => {a,  }
│ │ │ +o7 = ChordalNet{ a => {a,  }    }
│ │ │                   c => { , c}
│ │ │ +                 d => { , d}
│ │ │ +                 b => {b,  , b}
│ │ │  
│ │ │  o7 : ChordalNet
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/Chordal/html/_chordal__Net_lp__Hash__Table_cm__Hash__Table_cm__Elim__Tree_cm__Digraph_rp.html
│ │ │ @@ -102,18 +102,18 @@
│ │ │                
    i3 : G = chordalGraph digraph hashTable{a=>{b,c},b=>{c},c=>{d},d=>{}};
    
│ │ │              
    
│ │ │                
    i4 : tree = elimTree G
│ │ │  
│ │ │ -o4 = ElimTree{a => b   }
│ │ │ +o4 = ElimTree{a => c}
│ │ │                b => c
│ │ │                c => d
│ │ │ -              d => null
│ │ │ +              d => b
│ │ │  
│ │ │  o4 : ElimTree
    
│ │ │              
    
│ │ │                
    i5 : rnk = hashTable{"a0"=>a, "a1"=>a, "b0"=>b, "b1"=>b, "b2"=>b,
│ │ │ @@ -128,18 +128,18 @@
│ │ │                       "d0" => ({},{}), "d1" => ({d},{}) };
    
│ │ │              
    
│ │ │                
    i7 : chordalNet(eqs,rnk,tree,DG)
│ │ │  
│ │ │ -o7 = ChordalNet{ d => { , d}    }
│ │ │ -                 b => {b,  , b}
│ │ │ -                 a => {a,  }
│ │ │ +o7 = ChordalNet{ a => {a,  }    }
│ │ │                   c => { , c}
│ │ │ +                 d => { , d}
│ │ │ +                 b => {b,  , b}
│ │ │  
│ │ │  o7 : ChordalNet
    
│ │ │              
    
│ │ │          
    
│ │ │        
│ │ │ ├── html2text {}
│ │ │ │ @@ -32,32 +32,32 @@
│ │ │ │               d0 => {}
│ │ │ │               d1 => {}
│ │ │ │  
│ │ │ │  o2 : Digraph
│ │ │ │  i3 : G = chordalGraph digraph hashTable{a=>{b,c},b=>{c},c=>{d},d=>{}};
│ │ │ │  i4 : tree = elimTree G
│ │ │ │  
│ │ │ │ -o4 = ElimTree{a => b   }
│ │ │ │ +o4 = ElimTree{a => c}
│ │ │ │                b => c
│ │ │ │                c => d
│ │ │ │ -              d => null
│ │ │ │ +              d => b
│ │ │ │  
│ │ │ │  o4 : ElimTree
│ │ │ │  i5 : rnk = hashTable{"a0"=>a, "a1"=>a, "b0"=>b, "b1"=>b, "b2"=>b,
│ │ │ │                       "c0"=>c, "d0"=>d, "c1"=>c, "d1"=>d};
│ │ │ │  i6 : eqs = hashTable{"a0" => ({a},{}), "a1" => ({},{}),
│ │ │ │                       "b0" => ({b},{}), "b1" => ({},{}), "b2" => ({b},{}),
│ │ │ │                       "c0" => ({c},{}), "c1" => ({},{}),
│ │ │ │                       "d0" => ({},{}), "d1" => ({d},{}) };
│ │ │ │  i7 : chordalNet(eqs,rnk,tree,DG)
│ │ │ │  
│ │ │ │ -o7 = ChordalNet{ d => { , d}    }
│ │ │ │ -                 b => {b,  , b}
│ │ │ │ -                 a => {a,  }
│ │ │ │ +o7 = ChordalNet{ a => {a,  }    }
│ │ │ │                   c => { , c}
│ │ │ │ +                 d => { , d}
│ │ │ │ +                 b => {b,  , b}
│ │ │ │  
│ │ │ │  o7 : ChordalNet
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _d_i_s_p_l_a_y_N_e_t -- displays a chordal network using Graphivz
│ │ │ │      * _d_i_g_r_a_p_h_(_C_h_o_r_d_a_l_N_e_t_) -- digraph associated to a chordal network
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _c_h_o_r_d_a_l_N_e_t_(_H_a_s_h_T_a_b_l_e_,_H_a_s_h_T_a_b_l_e_,_E_l_i_m_T_r_e_e_,_D_i_g_r_a_p_h_) -- construct chordal
│ │ ├── ./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)
│ │ │ - -- 2.97386s elapsed
│ │ │ + -- 3.15825s 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)
│ │ │ - -- .390976s elapsed
│ │ │ + -- .523507s 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))
│ │ │ - -- 10.3301s elapsed
│ │ │ + -- 9.90547s 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)
│ │ │ - -- .344562s elapsed
│ │ │ + -- .515779s 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))
│ │ │ - -- .556898s elapsed
│ │ │ - -- .556932s elapsed
│ │ │ + -- .47572s elapsed
│ │ │ + -- .475746s elapsed
│ │ │  
│ │ │  o28 = {0, 0, 0, 0, 0}
│ │ │  
│ │ │  o28 : List
│ │ │  
│ │ │  i29 : assert(cohomvec1 == cohomvec2)
│ │ ├── ./usr/share/doc/Macaulay2/CohomCalg/html/index.html
│ │ │ @@ -309,15 +309,15 @@
│ │ │  
│ │ │  o19 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i20 : elapsedTime hvecs = cohomCalg(X, D2)
│ │ │ - -- 2.97386s elapsed
│ │ │ + -- 3.15825s 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,
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -399,25 +399,25 @@
│ │ │  
│ │ │  o22 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i23 : elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 + X_8)
│ │ │ - -- .390976s elapsed
│ │ │ + -- .523507s 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))
│ │ │ - -- 10.3301s elapsed
│ │ │ + -- 9.90547s elapsed
│ │ │  
│ │ │  o24 = {1, 0, 0, 0, 0}
│ │ │  
│ │ │  o24 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -433,26 +433,26 @@
│ │ │  
│ │ │  o26 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i27 : elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 - X_8)
│ │ │ - -- .344562s elapsed
│ │ │ + -- .515779s 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))
│ │ │ - -- .556898s elapsed
│ │ │ - -- .556932s elapsed
│ │ │ + -- .47572s elapsed
│ │ │ + -- .475746s elapsed
│ │ │  
│ │ │  o28 = {0, 0, 0, 0, 0}
│ │ │  
│ │ │  o28 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── 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)
│ │ │ │ - -- 2.97386s elapsed
│ │ │ │ + -- 3.15825s 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)
│ │ │ │ - -- .390976s elapsed
│ │ │ │ + -- .523507s 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))
│ │ │ │ - -- 10.3301s elapsed
│ │ │ │ + -- 9.90547s 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)
│ │ │ │ - -- .344562s elapsed
│ │ │ │ + -- .515779s 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))
│ │ │ │ - -- .556898s elapsed
│ │ │ │ - -- .556932s elapsed
│ │ │ │ + -- .47572s elapsed
│ │ │ │ + -- .475746s 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/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 6.82125s (cpu); 5.19893s (thread); 0s (gc)
│ │ │ + -- used 8.02946s (cpu); 5.9218s (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 0.184033s (cpu); 0.107007s (thread); 0s (gc)
│ │ │ + -- used 0.211706s (cpu); 0.118398s (thread); 0s (gc)
│ │ │  
│ │ │          1       6       18       38       66
│ │ │  o20 = R1  <-- R1  <-- R1   <-- R1   <-- R1
│ │ │                                           
│ │ │        0       1       2        3        4
│ │ │  
│ │ │  o20 : Complex
│ │ │  
│ │ │  i21 : GG = time EisenbudShamash(ff,F,4)
│ │ │ - -- used 0.953554s (cpu); 0.724234s (thread); 0s (gc)
│ │ │ + -- used 1.14279s (cpu); 0.864033s (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 : Complex
│ │ │  
│ │ │  i22 : GG = time EisenbudShamash(R1,F[2],4)
│ │ │ - -- used 0.945085s (cpu); 0.740101s (thread); 0s (gc)
│ │ │ + -- used 1.15248s (cpu); 0.885315s (thread); 0s (gc)
│ │ │  
│ │ │          1       6       18       38       66
│ │ │  o22 = R1  <-- R1  <-- R1   <-- R1   <-- R1
│ │ │                                           
│ │ │        -2      -1      0        1        2
│ │ │  
│ │ │  o22 : Complex
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_sum__Two__Monomials.out
│ │ │ @@ -2,21 +2,21 @@
│ │ │  
│ │ │  i1 : setRandomSeed 0
│ │ │   -- setting random seed to 0
│ │ │  
│ │ │  o1 = 0
│ │ │  
│ │ │  i2 : sumTwoMonomials(2,3)
│ │ │ - -- used 0.404052s (cpu); 0.333488s (thread); 0s (gc)
│ │ │ + -- used 0.713384s (cpu); 0.46515s (thread); 0s (gc)
│ │ │  2
│ │ │  Tally{{{2, 2}, {1, 2}} => 3}
│ │ │  
│ │ │ - -- used 0.200262s (cpu); 0.124311s (thread); 0s (gc)
│ │ │ + -- used 0.300038s (cpu); 0.156447s (thread); 0s (gc)
│ │ │  3
│ │ │  Tally{{{2, 2}, {1, 2}} => 1}
│ │ │  
│ │ │ - -- used 3.607e-06s (cpu); 3.135e-06s (thread); 0s (gc)
│ │ │ + -- used 5.58e-06s (cpu); 3.112e-06s (thread); 0s (gc)
│ │ │  4
│ │ │  Tally{}
│ │ │  
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_two__Monomials.out
│ │ │ @@ -2,23 +2,23 @@
│ │ │  
│ │ │  i1 : setRandomSeed 0
│ │ │   -- setting random seed to 0
│ │ │  
│ │ │  o1 = 0
│ │ │  
│ │ │  i2 : twoMonomials(2,3)
│ │ │ - -- used 0.829311s (cpu); 0.613928s (thread); 0s (gc)
│ │ │ + -- used 1.15822s (cpu); 0.735085s (thread); 0s (gc)
│ │ │  2
│ │ │  Tally{{{1, 1}} => 2        }
│ │ │        {{2, 2}, {1, 2}} => 4
│ │ │  
│ │ │ - -- used 0.41129s (cpu); 0.337464s (thread); 0s (gc)
│ │ │ + -- used 0.616518s (cpu); 0.450456s (thread); 0s (gc)
│ │ │  3
│ │ │  Tally{{{2, 2}, {1, 2}} => 2}
│ │ │        {{3, 3}, {2, 3}} => 1
│ │ │  
│ │ │ - -- used 0.193175s (cpu); 0.136002s (thread); 0s (gc)
│ │ │ + -- used 0.251225s (cpu); 0.155998s (thread); 0s (gc)
│ │ │  4
│ │ │  Tally{{{2, 2}, {1, 2}} => 1}
│ │ │  
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Eisenbud__Shamash.html
│ │ │ @@ -131,15 +131,15 @@
│ │ │  
│ │ │  o6 = 10
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : time G = EisenbudShamash(ff,F,len)
│ │ │ - -- used 6.82125s (cpu); 5.19893s (thread); 0s (gc)
│ │ │ + -- used 8.02946s (cpu); 5.9218s (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 /
│ │ │                                                                                                                                                                                
│ │ │ @@ -295,28 +295,28 @@
│ │ │  
│ │ │  o19 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i20 : FF = time Shamash(R1,F,4)
│ │ │ - -- used 0.184033s (cpu); 0.107007s (thread); 0s (gc)
│ │ │ + -- used 0.211706s (cpu); 0.118398s (thread); 0s (gc)
│ │ │  
│ │ │          1       6       18       38       66
│ │ │  o20 = R1  <-- R1  <-- R1   <-- R1   <-- R1
│ │ │                                           
│ │ │        0       1       2        3        4
│ │ │  
│ │ │  o20 : Complex
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i21 : GG = time EisenbudShamash(ff,F,4)
│ │ │ - -- used 0.953554s (cpu); 0.724234s (thread); 0s (gc)
│ │ │ + -- used 1.14279s (cpu); 0.864033s (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
│ │ │ @@ -328,15 +328,15 @@
│ │ │          
    
│ │ │            
    The function also deals correctly with complexes F where min F is not 0:
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i22 : GG = time EisenbudShamash(R1,F[2],4)
│ │ │ - -- used 0.945085s (cpu); 0.740101s (thread); 0s (gc)
│ │ │ + -- used 1.15248s (cpu); 0.885315s (thread); 0s (gc)
│ │ │  
│ │ │          1       6       18       38       66
│ │ │  o22 = R1  <-- R1  <-- R1   <-- R1   <-- R1
│ │ │                                           
│ │ │        -2      -1      0        1        2
│ │ │  
│ │ │  o22 : Complex
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -49,15 +49,15 @@
│ │ │ │  o5 = R
│ │ │ │  
│ │ │ │  o5 : QuotientRing
│ │ │ │  i6 : len = 10
│ │ │ │  
│ │ │ │  o6 = 10
│ │ │ │  i7 : time G = EisenbudShamash(ff,F,len)
│ │ │ │ - -- used 6.82125s (cpu); 5.19893s (thread); 0s (gc)
│ │ │ │ + -- used 8.02946s (cpu); 5.9218s (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 = |--------|  <-- |--------|  <-- |--------|   <-- |--------|   <-- |-------
│ │ │ │  -|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |-
│ │ │ │  -------|   <-- |--------|
│ │ │ │ @@ -165,36 +165,36 @@
│ │ │ │  o18 : Matrix R  <-- R
│ │ │ │  i19 : R1 = R/ideal ff
│ │ │ │  
│ │ │ │  o19 = R1
│ │ │ │  
│ │ │ │  o19 : QuotientRing
│ │ │ │  i20 : FF = time Shamash(R1,F,4)
│ │ │ │ - -- used 0.184033s (cpu); 0.107007s (thread); 0s (gc)
│ │ │ │ + -- used 0.211706s (cpu); 0.118398s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          1       6       18       38       66
│ │ │ │  o20 = R1  <-- R1  <-- R1   <-- R1   <-- R1
│ │ │ │  
│ │ │ │        0       1       2        3        4
│ │ │ │  
│ │ │ │  o20 : Complex
│ │ │ │  i21 : GG = time EisenbudShamash(ff,F,4)
│ │ │ │ - -- used 0.953554s (cpu); 0.724234s (thread); 0s (gc)
│ │ │ │ + -- used 1.14279s (cpu); 0.864033s (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 : Complex
│ │ │ │  The function also deals correctly with complexes F where min F is not 0:
│ │ │ │  i22 : GG = time EisenbudShamash(R1,F[2],4)
│ │ │ │ - -- used 0.945085s (cpu); 0.740101s (thread); 0s (gc)
│ │ │ │ + -- used 1.15248s (cpu); 0.885315s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          1       6       18       38       66
│ │ │ │  o22 = R1  <-- R1  <-- R1   <-- R1   <-- R1
│ │ │ │  
│ │ │ │        -2      -1      0        1        2
│ │ │ │  
│ │ │ │  o22 : Complex
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_sum__Two__Monomials.html
│ │ │ @@ -79,23 +79,23 @@
│ │ │  
│ │ │  o1 = 0
│ │ │              
    
│ │ │                
    i2 : sumTwoMonomials(2,3)
│ │ │ - -- used 0.404052s (cpu); 0.333488s (thread); 0s (gc)
│ │ │ + -- used 0.713384s (cpu); 0.46515s (thread); 0s (gc)
│ │ │  2
│ │ │  Tally{{{2, 2}, {1, 2}} => 3}
│ │ │  
│ │ │ - -- used 0.200262s (cpu); 0.124311s (thread); 0s (gc)
│ │ │ + -- used 0.300038s (cpu); 0.156447s (thread); 0s (gc)
│ │ │  3
│ │ │  Tally{{{2, 2}, {1, 2}} => 1}
│ │ │  
│ │ │ - -- used 3.607e-06s (cpu); 3.135e-06s (thread); 0s (gc)
│ │ │ + -- used 5.58e-06s (cpu); 3.112e-06s (thread); 0s (gc)
│ │ │  4
│ │ │  Tally{}
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,23 +18,23 @@
│ │ │ │  appropriate syzygy M of M0 = R/(m1+m2) where m1 and m2 are monomials of the
│ │ │ │  same degree.
│ │ │ │  i1 : setRandomSeed 0
│ │ │ │   -- setting random seed to 0
│ │ │ │  
│ │ │ │  o1 = 0
│ │ │ │  i2 : sumTwoMonomials(2,3)
│ │ │ │ - -- used 0.404052s (cpu); 0.333488s (thread); 0s (gc)
│ │ │ │ + -- used 0.713384s (cpu); 0.46515s (thread); 0s (gc)
│ │ │ │  2
│ │ │ │  Tally{{{2, 2}, {1, 2}} => 3}
│ │ │ │  
│ │ │ │ - -- used 0.200262s (cpu); 0.124311s (thread); 0s (gc)
│ │ │ │ + -- used 0.300038s (cpu); 0.156447s (thread); 0s (gc)
│ │ │ │  3
│ │ │ │  Tally{{{2, 2}, {1, 2}} => 1}
│ │ │ │  
│ │ │ │ - -- used 3.607e-06s (cpu); 3.135e-06s (thread); 0s (gc)
│ │ │ │ + -- used 5.58e-06s (cpu); 3.112e-06s (thread); 0s (gc)
│ │ │ │  4
│ │ │ │  Tally{}
│ │ │ │  ********** 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 ssuummTTwwooMMoonnoommiiaallss:: **********
│ │ │ │      * sumTwoMonomials(ZZ,ZZ)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_two__Monomials.html
│ │ │ @@ -83,25 +83,25 @@
│ │ │  
│ │ │  o1 = 0
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : twoMonomials(2,3)
│ │ │ - -- used 0.829311s (cpu); 0.613928s (thread); 0s (gc)
│ │ │ + -- used 1.15822s (cpu); 0.735085s (thread); 0s (gc)
│ │ │  2
│ │ │  Tally{{{1, 1}} => 2        }
│ │ │        {{2, 2}, {1, 2}} => 4
│ │ │  
│ │ │ - -- used 0.41129s (cpu); 0.337464s (thread); 0s (gc)
│ │ │ + -- used 0.616518s (cpu); 0.450456s (thread); 0s (gc)
│ │ │  3
│ │ │  Tally{{{2, 2}, {1, 2}} => 2}
│ │ │        {{3, 3}, {2, 3}} => 1
│ │ │  
│ │ │ - -- used 0.193175s (cpu); 0.136002s (thread); 0s (gc)
│ │ │ + -- used 0.251225s (cpu); 0.155998s (thread); 0s (gc)
│ │ │  4
│ │ │  Tally{{{2, 2}, {1, 2}} => 1}
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,25 +20,25 @@
│ │ │ │  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
│ │ │ │   -- setting random seed to 0
│ │ │ │  
│ │ │ │  o1 = 0
│ │ │ │  i2 : twoMonomials(2,3)
│ │ │ │ - -- used 0.829311s (cpu); 0.613928s (thread); 0s (gc)
│ │ │ │ + -- used 1.15822s (cpu); 0.735085s (thread); 0s (gc)
│ │ │ │  2
│ │ │ │  Tally{{{1, 1}} => 2        }
│ │ │ │        {{2, 2}, {1, 2}} => 4
│ │ │ │  
│ │ │ │ - -- used 0.41129s (cpu); 0.337464s (thread); 0s (gc)
│ │ │ │ + -- used 0.616518s (cpu); 0.450456s (thread); 0s (gc)
│ │ │ │  3
│ │ │ │  Tally{{{2, 2}, {1, 2}} => 2}
│ │ │ │        {{3, 3}, {2, 3}} => 1
│ │ │ │  
│ │ │ │ - -- used 0.193175s (cpu); 0.136002s (thread); 0s (gc)
│ │ │ │ + -- used 0.251225s (cpu); 0.155998s (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/ConnectionMatrices/example-output/___Cosmological_spcorrelator_spfor_spthe_sp2-site_spchain.out
│ │ │ @@ -27,18 +27,18 @@
│ │ │       - ϵ*z*dy + 2ϵ  - ϵ, x*dx + y*dy + z*dz - 2ϵ)
│ │ │  
│ │ │  o7 : Ideal of D
│ │ │  
│ │ │  i8 : assert(holonomicRank I == 4)
│ │ │  
│ │ │  i9 : elapsedTime A = connectionMatrices I;
│ │ │ - -- 2.86586s elapsed
│ │ │ + -- 2.43094s elapsed
│ │ │  
│ │ │  i10 : elapsedTime assert isIntegrable A
│ │ │ - -- 5.93547s elapsed
│ │ │ + -- 4.22789s elapsed
│ │ │  
│ │ │  i11 : netList A
│ │ │  
│ │ │        +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  o11 = || 2ϵ/x                                                                                                                                                                                                                                                                                                                                                                                                                                    -y/x                                                                                                                                                                                                                                                                                                                                                                                                                                                                   -z/x                                                                                                                                                                                                                                                                                                                                                                                                                                                  0                                                                                                                             |                                                                                                                                                                                                                                                                                                                                     |
│ │ │        || (4x2y2ϵ^2+4xy2zϵ^2-2x2z2ϵ^2-2y2z2ϵ^2-4xz3ϵ^2+x3zϵ-3xy2zϵ+2xz3ϵ)/(2x4y2+2x3y3+x4yz+2x3y2z+x2y3z-x4z2-x3yz2-x2y2z2-xy3z2-x3z3-2x2yz3-xy2z3)                                                                                                                                                                                                                                                                                               (2x3y2ϵ-2x2y3ϵ+2x3yzϵ-2xy3zϵ-x3z2ϵ+x2yz2ϵ-xy2z2ϵ+y3z2ϵ-2x3yz+2xy3z)/(2x4y2+2x3y3+x4yz+2x3y2z+x2y3z-x4z2-x3yz2-x2y2z2-xy3z2-x3z3-2x2yz3-xy2z3)                                                                                                                                                                                                                                                                                                                          (-2x2y2zϵ-x3z2ϵ-3xy2z2ϵ+x2z3ϵ+y2z3ϵ+4xz4ϵ+2xy2z2-2xz4)/(2x4y2+2x3y3+x4yz+2x3y2z+x2y3z-x4z2-x3yz2-x2y2z2-xy3z2-x3z3-2x2yz3-xy2z3)                                                                                                                                                                                                                                                                                                                      (-xyz+xz2+yz2-z3)/(2x2y+2xy2-x2z-2xyz-y2z)                                                                                    |                                                                                                                                                                                                                                                                                                                                     |
│ │ │        || (-2xyz2ϵ^2-2y2z2ϵ^2-4yz3ϵ^2+2x2y2ϵ+x2yzϵ+xy2zϵ+2y2z2ϵ+2yz3ϵ)/(2x3y2z+x3yz2+x2y2z2-x3z3-xy2z3-x2z4-xyz4)                                                                                                                                                                                                                                                                                                                                 (x2yz2ϵ+2xy2z2ϵ+y3z2ϵ+2xyz3ϵ+2y2z3ϵ-2x2y3-x2y2z-xy3z-x2yz2-y3z2-xyz3-y2z3)/(2x3y2z+x3yz2+x2y2z2-x3z3-xy2z3-x2z4-xyz4)                                                                                                                                                                                                                                                                                                                                                  (2x2y2ϵ+x2yzϵ+xy2zϵ-2x2z2ϵ+xyz2ϵ+y2z2ϵ-2xz3ϵ+2yz3ϵ-2x2y2-x2yz-xy2z+x2z2-y2z2+xz3-yz3)/(2x3y2+x3yz+x2y2z-x3z2-xy2z2-x2z3-xyz3)                                                                                                                                                                                                                                                                                                                         (-yz+z2)/(2xy-xz-yz)                                                                                                          |                                                                                                                                                                                                                                                                                                                                     |
│ │ │ @@ -56,24 +56,24 @@
│ │ │        +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  
│ │ │  i12 : F = baseFractionField D;
│ │ │  
│ │ │  i13 : B = {1_D,dx,dy,dx*dy};
│ │ │  
│ │ │  i14 : elapsedTime g = gaugeMatrix(I, B);
│ │ │ - -- .758247s elapsed
│ │ │ + -- .542844s elapsed
│ │ │  
│ │ │                4      4
│ │ │  o14 : Matrix F  <-- F
│ │ │  
│ │ │  i15 : elapsedTime A1 = gaugeTransform(g, A);
│ │ │ - -- 1.51935s elapsed
│ │ │ + -- 1.19323s elapsed
│ │ │  
│ │ │  i16 : elapsedTime assert isIntegrable A1
│ │ │ - -- .837147s elapsed
│ │ │ + -- .901501s elapsed
│ │ │  
│ │ │  i17 : netList A1
│ │ │  
│ │ │        +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  o17 = || 0                            1                      0                                         0                                                      |                                                                         |
│ │ │        || (-2ϵ^2+ϵ)/(x2-z2)            (3xϵ+zϵ-2x)/(x2-z2)    (yϵ+zϵ)/(x2-z2)                           (-y-z)/(x-z)                                           |                                                                         |
│ │ │        || 0                            0                      0                                         1                                                      |                                                                         |
│ │ │ @@ -96,18 +96,18 @@
│ │ │                {0, 0, ϵ*(y^2-z^2), ϵ*(x+y)*(y+z)},
│ │ │                {0, 0, 0, -(x+y)*(x+z)*(y+z)}});
│ │ │  
│ │ │                4      4
│ │ │  o18 : Matrix F  <-- F
│ │ │  
│ │ │  i19 : elapsedTime A2 = gaugeTransform(changeEps, A1);
│ │ │ - -- .427948s elapsed
│ │ │ + -- .324602s elapsed
│ │ │  
│ │ │  i20 : elapsedTime assert isIntegrable A2
│ │ │ - -- .715487s elapsed
│ │ │ + -- .638904s elapsed
│ │ │  
│ │ │  i21 : netList A2
│ │ │  
│ │ │        +-------------------------------------------------------------------------------------------+
│ │ │  o21 = || ϵ/(x+z) 2zϵ/(x2-z2) 0       0                      |                                     |
│ │ │        || 0       ϵ/(x-z)     0       ϵ/(x+y)                |                                     |
│ │ │        || 0       0           ϵ/(x+z) (-yϵ+zϵ)/(x2+xy+xz+yz) |                                     |
│ │ ├── ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/___Massless_spone-loop_sptriangle_sp__Feynman_spdiagram.out
│ │ │ @@ -16,18 +16,18 @@
│ │ │  
│ │ │                     2
│ │ │  o6 = {1, dx, dy, dy }
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : elapsedTime A = connectionMatrices I;
│ │ │ - -- .226112s elapsed
│ │ │ + -- .18511s elapsed
│ │ │  
│ │ │  i8 : elapsedTime assert isIntegrable A
│ │ │ - -- .152683s elapsed
│ │ │ + -- .170753s elapsed
│ │ │  
│ │ │  i9 : netList A
│ │ │  
│ │ │       +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  o9 = || 0                                                       1                                               0                                                                     0                                                                      ||
│ │ │       || 0                                                       -1/x                                            1/x                                                                   y/x                                                                    ||
│ │ │       || -1/2xy                                                  -1/y                                            (-x-3y+1)/2xy                                                         (-x-y+1)/2x                                                            ||
│ │ ├── ./usr/share/doc/Macaulay2/ConnectionMatrices/html/___Cosmological_spcorrelator_spfor_spthe_sp2-site_spchain.html
│ │ │ @@ -118,21 +118,21 @@
│ │ │          
    
│ │ │            
    Then, we compute the system in connection form and verify that it meets the integrability conditions.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -82,15 +82,15 @@
│ │ │ │                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 0.15576s (cpu); 0.155761s (thread); 0s (gc)
│ │ │ │ + -- used 0.174302s (cpu); 0.174303s (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
│ │ │ @@ -134,59 +134,59 @@
│ │ │                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
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
    
│ │ │                
    i9 : elapsedTime A = connectionMatrices I;
│ │ │ - -- 2.86586s elapsed
    
│ │ │ + -- 2.43094s elapsed
│ │ │              
    
│ │ │                
    i10 : elapsedTime assert isIntegrable A
│ │ │ - -- 5.93547s elapsed
    
│ │ │ + -- 4.22789s elapsed
│ │ │              
    
│ │ │                
    i11 : netList A
│ │ │  
│ │ │        +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ @@ -167,30 +167,30 @@
│ │ │              
    		
│ │ │                
    i13 : B = {1_D,dx,dy,dx*dy};
    
│ │ │              
    
│ │ │                
    i14 : elapsedTime g = gaugeMatrix(I, B);
│ │ │ - -- .758247s elapsed
│ │ │ + -- .542844s elapsed
│ │ │  
│ │ │                4      4
│ │ │  o14 : Matrix F  <-- F
    
│ │ │              
    
│ │ │                
    i15 : elapsedTime A1 = gaugeTransform(g, A);
│ │ │ - -- 1.51935s elapsed
    
│ │ │ + -- 1.19323s elapsed
│ │ │              
    
│ │ │                
    i16 : elapsedTime assert isIntegrable A1
│ │ │ - -- .837147s elapsed
    
│ │ │ + -- .901501s elapsed
│ │ │              
    
│ │ │                
    i17 : netList A1
│ │ │  
│ │ │        +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ @@ -227,21 +227,21 @@
│ │ │                4      4
│ │ │  o18 : Matrix F  <-- F
    
│ │ │              
    
│ │ │                
    i19 : elapsedTime A2 = gaugeTransform(changeEps, A1);
│ │ │ - -- .427948s elapsed
    
│ │ │ + -- .324602s elapsed
│ │ │              
    
│ │ │                
    i20 : elapsedTime assert isIntegrable A2
│ │ │ - -- .715487s elapsed
    
│ │ │ + -- .638904s elapsed
│ │ │              
    
│ │ │                
    i21 : netList A2
│ │ │  
│ │ │        +-------------------------------------------------------------------------------------------+
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,17 +41,17 @@
│ │ │ │  
│ │ │ │  o7 : Ideal of D
│ │ │ │  First, we check that the system has finite holonomic rank using _h_o_l_o_n_o_m_i_c_R_a_n_k.
│ │ │ │  i8 : assert(holonomicRank I == 4)
│ │ │ │  Then, we compute the system in connection form and verify that it meets the
│ │ │ │  integrability conditions.
│ │ │ │  i9 : elapsedTime A = connectionMatrices I;
│ │ │ │ - -- 2.86586s elapsed
│ │ │ │ + -- 2.43094s elapsed
│ │ │ │  i10 : elapsedTime assert isIntegrable A
│ │ │ │ - -- 5.93547s elapsed
│ │ │ │ + -- 4.22789s elapsed
│ │ │ │  i11 : netList A
│ │ │ │  
│ │ │ │        +----------------------------------------------------------------------------------------------------------
│ │ │ │  -----------------------------------------------------------------------------------------------------------------
│ │ │ │  -----------------------------------------------------------------------------------------------------------------
│ │ │ │  -----------------------------------------------------------------------------------------------------------------
│ │ │ │  -----------------------------------------------------------------------------------------------------------------
│ │ │ │ @@ -227,22 +227,22 @@
│ │ │ │  -----------------------------------------------------------------------------------+
│ │ │ │  Next, we use _g_a_u_g_e_ _m_a_t_r_i_x for changing base to a base given by suitable set of
│ │ │ │  standard monomials, and compute the _g_a_u_g_e_ _t_r_a_n_s_f_o_r_m with respect to this gauge
│ │ │ │  matrix.
│ │ │ │  i12 : F = baseFractionField D;
│ │ │ │  i13 : B = {1_D,dx,dy,dx*dy};
│ │ │ │  i14 : elapsedTime g = gaugeMatrix(I, B);
│ │ │ │ - -- .758247s elapsed
│ │ │ │ + -- .542844s elapsed
│ │ │ │  
│ │ │ │                4      4
│ │ │ │  o14 : Matrix F  <-- F
│ │ │ │  i15 : elapsedTime A1 = gaugeTransform(g, A);
│ │ │ │ - -- 1.51935s elapsed
│ │ │ │ + -- 1.19323s elapsed
│ │ │ │  i16 : elapsedTime assert isIntegrable A1
│ │ │ │ - -- .837147s elapsed
│ │ │ │ + -- .901501s elapsed
│ │ │ │  i17 : netList A1
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  --------------------------------------------------------------------------+
│ │ │ │  o17 = || 0                            1                      0
│ │ │ │  0                                                      |
│ │ │ │ @@ -300,17 +300,17 @@
│ │ │ │                {0, ϵ*(x^2-z^2), 0, ϵ*(x+y)*(x+z)},
│ │ │ │                {0, 0, ϵ*(y^2-z^2), ϵ*(x+y)*(y+z)},
│ │ │ │                {0, 0, 0, -(x+y)*(x+z)*(y+z)}});
│ │ │ │  
│ │ │ │                4      4
│ │ │ │  o18 : Matrix F  <-- F
│ │ │ │  i19 : elapsedTime A2 = gaugeTransform(changeEps, A1);
│ │ │ │ - -- .427948s elapsed
│ │ │ │ + -- .324602s elapsed
│ │ │ │  i20 : elapsedTime assert isIntegrable A2
│ │ │ │ - -- .715487s elapsed
│ │ │ │ + -- .638904s elapsed
│ │ │ │  i21 : netList A2
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------------+
│ │ │ │  o21 = || ϵ/(x+z) 2zϵ/(x2-z2) 0       0                      |
│ │ │ │  |
│ │ │ │        || 0       ϵ/(x-z)     0       ϵ/(x+y)                |
│ │ ├── ./usr/share/doc/Macaulay2/ConnectionMatrices/html/___Massless_spone-loop_sptriangle_sp__Feynman_spdiagram.html
│ │ │ @@ -100,21 +100,21 @@
│ │ │          
    
│ │ │            
    Finally, we can compute the connection matrices.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : elapsedTime A = connectionMatrices I;
│ │ │ - -- .226112s elapsed
    
│ │ │ + -- .18511s elapsed
│ │ │              
    
│ │ │                
    i8 : elapsedTime assert isIntegrable A
│ │ │ - -- .152683s elapsed
    
│ │ │ + -- .170753s elapsed
│ │ │              
    
│ │ │                
    i9 : netList A
│ │ │  
│ │ │       +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,17 +20,17 @@
│ │ │ │  
│ │ │ │                     2
│ │ │ │  o6 = {1, dx, dy, dy }
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  Finally, we can compute the connection matrices.
│ │ │ │  i7 : elapsedTime A = connectionMatrices I;
│ │ │ │ - -- .226112s elapsed
│ │ │ │ + -- .18511s elapsed
│ │ │ │  i8 : elapsedTime assert isIntegrable A
│ │ │ │ - -- .152683s elapsed
│ │ │ │ + -- .170753s elapsed
│ │ │ │  i9 : netList A
│ │ │ │  
│ │ │ │       +-------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----------------+
│ │ │ │  o9 = || 0                                                       1
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/___Chern__Schwartz__Mac__Pherson.out
│ │ │ @@ -13,27 +13,27 @@
│ │ │  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 2.22491s (cpu); 1.26478s (thread); 0s (gc)
│ │ │ + -- used 2.53235s (cpu); 1.21957s (thread); 0s (gc)
│ │ │  
│ │ │         4     3     2
│ │ │  o3 = 3H  + 5H  + 3H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o3 : -----
│ │ │          5
│ │ │         H
│ │ │  
│ │ │  i4 : time ChernSchwartzMacPherson(C,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 1.33783s (cpu); 0.951359s (thread); 0s (gc)
│ │ │ + -- used 1.72145s (cpu); 1.07375s (thread); 0s (gc)
│ │ │  
│ │ │         4     3     2
│ │ │  o4 = 3H  + 5H  + 3H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o4 : -----
│ │ │          5
│ │ │ @@ -62,27 +62,27 @@
│ │ │          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 0.32008s (cpu); 0.18723s (thread); 0s (gc)
│ │ │ + -- used 0.393117s (cpu); 0.19787s (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)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.107679s (cpu); 0.0382531s (thread); 0s (gc)
│ │ │ + -- used 0.192302s (cpu); 0.0468254s (thread); 0s (gc)
│ │ │  
│ │ │           9      8      7      6      5      4     3
│ │ │  o10 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o10 : -----
│ │ │          10
│ │ ├── ./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 0.00444651s (cpu); 0.00444181s (thread); 0s (gc)
│ │ │ + -- used 0.00563649s (cpu); 0.00563363s (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 0.139982s (cpu); 0.0709074s (thread); 0s (gc)
│ │ │ + -- used 0.183936s (cpu); 0.0848219s (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 0.0273929s (cpu); 0.0273949s (thread); 0s (gc)
│ │ │ + -- used 0.0372504s (cpu); 0.0372589s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
│ │ │  
│ │ │  i5 : time projectiveDegrees phi
│ │ │ - -- used 0.660336s (cpu); 0.467597s (thread); 0s (gc)
│ │ │ + -- used 0.655005s (cpu); 0.4886s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {1, 3, 9, 17, 21, 15, 5}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : time projectiveDegrees(phi,NumDegrees=>0)
│ │ │ - -- used 0.061525s (cpu); 0.0615308s (thread); 0s (gc)
│ │ │ + -- used 0.0713962s (cpu); 0.0714033s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = {5}
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : time phi = toMap(phi,Dominant=>J)
│ │ │ - -- used 0.00224979s (cpu); 0.00225137s (thread); 0s (gc)
│ │ │ + -- used 0.00261791s (cpu); 0.0026234s (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 0.464618s (cpu); 0.385971s (thread); 0s (gc)
│ │ │ + -- used 0.425047s (cpu); 0.424849s (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 0.00957768s (cpu); 0.00958237s (thread); 0s (gc)
│ │ │ + -- used 0.0109158s (cpu); 0.0109172s (thread); 0s (gc)
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 : time degreeMap psi
│ │ │ - -- used 0.36608s (cpu); 0.250674s (thread); 0s (gc)
│ │ │ + -- used 0.531179s (cpu); 0.272942s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 1
│ │ │  
│ │ │  i11 : time projectiveDegrees psi
│ │ │ - -- used 5.12293s (cpu); 4.38974s (thread); 0s (gc)
│ │ │ + -- used 5.68627s (cpu); 5.2618s (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 0.00214853s (cpu); 0.00214909s (thread); 0s (gc)
│ │ │ + -- used 0.00258076s (cpu); 0.00258318s (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 0.159869s (cpu); 0.0819959s (thread); 0s (gc)
│ │ │ + -- used 0.179519s (cpu); 0.0970863s (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 0.494847s (cpu); 0.414351s (thread); 0s (gc)
│ │ │ + -- used 0.464234s (cpu); 0.463995s (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 0.343614s (cpu); 0.269496s (thread); 0s (gc)
│ │ │ + -- used 0.456608s (cpu); 0.345678s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = {5, 15, 21, 17, 9, 3, 1}
│ │ │  
│ │ │  o15 : List
│ │ │  
│ │ │  i16 : time degrees phi
│ │ │ - -- used 0.017066s (cpu); 0.0167813s (thread); 0s (gc)
│ │ │ + -- used 0.0438542s (cpu); 0.0238686s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = {1, 3, 9, 17, 21, 15, 5}
│ │ │  
│ │ │  o16 : List
│ │ │  
│ │ │  i17 : time describe phi
│ │ │ - -- used 0.00287351s (cpu); 0.00287412s (thread); 0s (gc)
│ │ │ + -- used 0.00408934s (cpu); 0.00409592s (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 0.00949367s (cpu); 0.00949464s (thread); 0s (gc)
│ │ │ + -- used 0.0114385s (cpu); 0.0114461s (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 0.00910182s (cpu); 0.00910265s (thread); 0s (gc)
│ │ │ + -- used 0.0116988s (cpu); 0.0117077s (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 1.2811s (cpu); 0.962879s (thread); 0s (gc)
│ │ │ + -- used 1.2095s (cpu); 1.00842s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = {904, 508, 268, 130, 56, 20, 5}
│ │ │  
│ │ │  o21 : List
│ │ │  
│ │ │  i22 : time degree f
│ │ │ - -- used 1.588e-05s (cpu); 1.5509e-05s (thread); 0s (gc)
│ │ │ + -- used 1.5338e-05s (cpu); 1.4902e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = 1
│ │ │  
│ │ │  i23 : time describe f
│ │ │ - -- used 0.00176246s (cpu); 0.00176335s (thread); 0s (gc)
│ │ │ + -- used 0.00169313s (cpu); 0.00169834s (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 0.32363s (cpu); 0.192112s (thread); 0s (gc)
│ │ │ + -- used 0.365833s (cpu); 0.213574s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 10
│ │ │  
│ │ │  i3 : time EulerCharacteristic(I,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0112392s (cpu); 0.0109549s (thread); 0s (gc)
│ │ │ + -- used 0.0243867s (cpu); 0.0125919s (thread); 0s (gc)
│ │ │  
│ │ │  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 0.0555729s (cpu); 0.0551512s (thread); 0s (gc)
│ │ │ + -- used 0.0985162s (cpu); 0.0672745s (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 0.0359549s (cpu); 0.0356875s (thread); 0s (gc)
│ │ │ + -- used 0.0625898s (cpu); 0.0460574s (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 0.15576s (cpu); 0.155761s (thread); 0s (gc)
│ │ │ + -- used 0.174302s (cpu); 0.174303s (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
│ │ │ @@ -47,50 +47,50 @@
│ │ │                                                                            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 0.798226s (cpu); 0.509602s (thread); 0s (gc)
│ │ │ + -- used 0.836263s (cpu); 0.55014s (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 0.549131s (cpu); 0.33326s (thread); 0s (gc)
│ │ │ + -- used 0.635674s (cpu); 0.360676s (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)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0212624s (cpu); 0.0207391s (thread); 0s (gc)
│ │ │ + -- used 0.0422507s (cpu); 0.0255555s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5       4      3
│ │ │  o6 = 3240H  - 1188H  + 396H  - 114H  + 24H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o6 : -----
│ │ │          8
│ │ │         H
│ │ │  
│ │ │  i7 : time SegreClass(lift(X,P7),Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0974975s (cpu); 0.0969659s (thread); 0s (gc)
│ │ │ + -- used 0.135886s (cpu); 0.124334s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5      4      3
│ │ │  o7 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o7 : -----
│ │ │          8
│ │ │ @@ -98,22 +98,22 @@
│ │ │  
│ │ │  i8 : o4 == o6 and o5 == o7
│ │ │  
│ │ │  o8 = true
│ │ │  
│ │ │  i9 : use ZZ/100003[x_0..x_6]
│ │ │  
│ │ │ -o9     ZZ
│ │ │ -= ------[x ..x ]
│ │ │ -  100003  0   6
│ │ │ +       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 0.197315s (cpu); 0.0983952s (thread); 0s (gc)
│ │ │ + -- used 0.259917s (cpu); 0.125368s (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 0.169509s (cpu); 0.169514s (thread); 0s (gc)
│ │ │ + -- used 0.212058s (cpu); 0.211896s (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 0.518764s (cpu); 0.315789s (thread); 0s (gc)
│ │ │ + -- used 0.433541s (cpu); 0.291859s (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 1.31381s (cpu); 0.860817s (thread); 0s (gc)
│ │ │ + -- used 1.84444s (cpu); 1.03393s (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 0.000416542s (cpu); 0.000408857s (thread); 0s (gc)
│ │ │ + -- used 0.000409484s (cpu); 0.000402295s (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 0.32875s (cpu); 0.185077s (thread); 0s (gc)
│ │ │ + -- used 0.384646s (cpu); 0.218147s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 2
│ │ │  
│ │ │  i6 : time rationalMap psi
│ │ │ - -- used 0.510248s (cpu); 0.373012s (thread); 0s (gc)
│ │ │ + -- used 0.507728s (cpu); 0.415065s (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 0.151084s (cpu); 0.0733367s (thread); 0s (gc)
│ │ │ + -- used 0.179979s (cpu); 0.0992287s (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 3.78487s (cpu); 1.95126s (thread); 0s (gc)
│ │ │ + -- used 4.37345s (cpu); 2.29473s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3
│ │ │  
│ │ │  i16 : time T2 = T * T
│ │ │ - -- used 2.8112e-05s (cpu); 2.7842e-05s (thread); 0s (gc)
│ │ │ + -- used 3.6436e-05s (cpu); 3.3449e-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 6.44263s (cpu); 3.37969s (thread); 0s (gc)
│ │ │ + -- used 7.41221s (cpu); 3.66446s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = 1
│ │ │  
│ │ │  i18 : p = apply(3,i->random(ZZ/65521))|{1}
│ │ │  
│ │ │  o18 = {-6648, -23396, -12311, 1}
│ │ │  
│ │ │ @@ -169,15 +169,15 @@
│ │ │  i20 : T q
│ │ │  
│ │ │  o20 = {-6648, -23396, -12311, 1}
│ │ │  
│ │ │  o20 : List
│ │ │  
│ │ │  i21 : time f = rationalMap T
│ │ │ - -- used 5.32932s (cpu); 2.87316s (thread); 0s (gc)
│ │ │ + -- used 6.07165s (cpu); 3.07132s (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
│ │ │ @@ -54,15 +54,15 @@
│ │ │  -- approximateInverseMap: step 4 of 10
│ │ │  -- approximateInverseMap: step 5 of 10
│ │ │  -- approximateInverseMap: step 6 of 10
│ │ │  -- approximateInverseMap: step 7 of 10
│ │ │  -- approximateInverseMap: step 8 of 10
│ │ │  -- approximateInverseMap: step 9 of 10
│ │ │  -- approximateInverseMap: step 10 of 10
│ │ │ - -- used 0.238191s (cpu); 0.186212s (thread); 0s (gc)
│ │ │ + -- used 0.31681s (cpu); 0.235711s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │                    ZZ
│ │ │       source: Proj(--[t , t , t , t , t , t , t , t , t ])
│ │ │                    97  0   1   2   3   4   5   6   7   8
│ │ │                                  ZZ
│ │ │       target: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x ]) defined by
│ │ │ @@ -109,15 +109,15 @@
│ │ │  
│ │ │  i4 : assert(phi * psi == 1 and psi * phi == 1)
│ │ │  
│ │ │  i5 : time psi' = approximateInverseMap(phi,CodimBsInv=>5);
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ - -- used 0.218651s (cpu); 0.155471s (thread); 0s (gc)
│ │ │ + -- used 0.276259s (cpu); 0.197603s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : RationalMap (quadratic rational map from PP^8 to hypersurface in PP^9)
│ │ │  
│ │ │  i6 : assert(psi == psi')
│ │ │  
│ │ │  i7 : phi = rationalMap map(P8,ZZ/97[x_0..x_11]/ideal(x_1*x_3-8*x_2*x_3+25*x_3^2-25*x_2*x_4-22*x_3*x_4+x_0*x_5+13*x_2*x_5+41*x_3*x_5-x_0*x_6+12*x_2*x_6+25*x_1*x_7+25*x_3*x_7+23*x_5*x_7-3*x_6*x_7+2*x_0*x_8+11*x_1*x_8-37*x_3*x_8-23*x_4*x_8-33*x_6*x_8+8*x_0*x_9+10*x_1*x_9-25*x_2*x_9-9*x_3*x_9+3*x_4*x_9+24*x_5*x_9-27*x_6*x_9-5*x_0*x_10+28*x_1*x_10+37*x_2*x_10+9*x_4*x_10+27*x_6*x_10-25*x_0*x_11+9*x_2*x_11+27*x_4*x_11-27*x_5*x_11,x_2^2+17*x_2*x_3-14*x_3^2-13*x_2*x_4+34*x_3*x_4+44*x_0*x_5-30*x_2*x_5+27*x_3*x_5+31*x_2*x_6-36*x_3*x_6-x_0*x_7+13*x_1*x_7+8*x_3*x_7+9*x_5*x_7+46*x_6*x_7+41*x_0*x_8-7*x_1*x_8-34*x_3*x_8-9*x_4*x_8-46*x_6*x_8-17*x_0*x_9+32*x_1*x_9-8*x_2*x_9-35*x_3*x_9-46*x_4*x_9+26*x_5*x_9+17*x_6*x_9+15*x_0*x_10+35*x_1*x_10+34*x_2*x_10+20*x_4*x_10+14*x_0*x_11+36*x_1*x_11+35*x_2*x_11-17*x_4*x_11,x_1*x_2-40*x_2*x_3+28*x_3^2-x_0*x_4+5*x_2*x_4-16*x_3*x_4+5*x_0*x_5-36*x_2*x_5+37*x_3*x_5+48*x_2*x_6-5*x_1*x_7-5*x_3*x_7+x_5*x_7+20*x_6*x_7+10*x_0*x_8+34*x_1*x_8+41*x_3*x_8-x_4*x_8+x_6*x_8+40*x_0*x_9-32*x_1*x_9+5*x_2*x_9-11*x_3*x_9-20*x_4*x_9+45*x_5*x_9-14*x_6*x_9-25*x_0*x_10+45*x_1*x_10-41*x_2*x_10-46*x_4*x_10+8*x_6*x_10-28*x_0*x_11+11*x_2*x_11+14*x_4*x_11-8*x_5*x_11),{t_4^2+t_0*t_5+t_1*t_5+35*t_2*t_5+10*t_3*t_5+25*t_4*t_5-5*t_5^2-14*t_0*t_6-14*t_1*t_6-5*t_2*t_6-13*t_4*t_6+37*t_5*t_6+22*t_6^2-31*t_3*t_7+26*t_4*t_7+12*t_5*t_7-45*t_6*t_7-46*t_3*t_8+37*t_4*t_8+28*t_5*t_8+33*t_6*t_8,t_3*t_4+4*t_0*t_5+39*t_1*t_5-40*t_2*t_5+40*t_3*t_5+26*t_4*t_5-20*t_5^2+41*t_0*t_6+36*t_1*t_6-22*t_2*t_6+36*t_4*t_6-30*t_5*t_6-13*t_6^2-25*t_3*t_7+5*t_4*t_7-35*t_5*t_7+10*t_6*t_7+11*t_3*t_8+46*t_4*t_8+29*t_5*t_8+28*t_6*t_8,t_2*t_4-5*t_0*t_5-40*t_1*t_5+12*t_2*t_5+47*t_3*t_5+37*t_4*t_5+25*t_5^2-27*t_0*t_6-22*t_1*t_6+27*t_2*t_6-23*t_4*t_6+5*t_5*t_6-13*t_6^2-39*t_3*t_7-29*t_4*t_7+9*t_5*t_7+39*t_6*t_7+36*t_3*t_8+13*t_4*t_8+26*t_5*t_8+37*t_6*t_8,t_0*t_4-t_0*t_5-8*t_1*t_5-35*t_2*t_5-10*t_3*t_5-33*t_4*t_5+5*t_5^2+15*t_0*t_6+15*t_1*t_6+5*t_2*t_6+15*t_4*t_6-38*t_5*t_6-22*t_6^2+31*t_3*t_7-25*t_4*t_7-19*t_5*t_7+47*t_6*t_7+46*t_3*t_8-36*t_4*t_8-35*t_5*t_8-31*t_6*t_8,t_2*t_3-t_0*t_5-t_1*t_5-35*t_2*t_5-10*t_3*t_5-33*t_4*t_5+5*t_5^2+14*t_0*t_6+14*t_1*t_6+5*t_2*t_6+14*t_4*t_6-31*t_5*t_6-24*t_6^2+32*t_3*t_7-25*t_4*t_7-19*t_5*t_7+47*t_6*t_7+46*t_3*t_8-36*t_4*t_8-35*t_5*t_8-31*t_6*t_8,t_1*t_3-7*t_1*t_5+t_1*t_6+t_4*t_6-7*t_5*t_6+2*t_6^2-t_3*t_7,t_0*t_3-46*t_0*t_5-39*t_1*t_5-43*t_2*t_5-41*t_3*t_5-26*t_4*t_5-28*t_5^2-35*t_0*t_6-36*t_1*t_6+20*t_2*t_6-36*t_4*t_6+9*t_5*t_6+15*t_6^2+26*t_3*t_7-5*t_4*t_7+35*t_5*t_7-10*t_6*t_7-10*t_3*t_8-46*t_4*t_8+47*t_5*t_8-25*t_6*t_8,t_2^2-46*t_1*t_4-33*t_0*t_5-45*t_1*t_5-39*t_2*t_5-39*t_3*t_5-46*t_4*t_5-29*t_5^2-48*t_0*t_6-38*t_1*t_6-30*t_2*t_6+19*t_4*t_6-44*t_5*t_6-47*t_6^2-36*t_0*t_7-46*t_1*t_7+t_2*t_7-44*t_3*t_7+48*t_4*t_7-14*t_5*t_7+4*t_6*t_7-36*t_0*t_8-46*t_1*t_8+47*t_2*t_8-34*t_3*t_8-24*t_4*t_8-12*t_5*t_8-47*t_6*t_8+47*t_7*t_8,t_1*t_2+6*t_1*t_5+5*t_0*t_6-2*t_1*t_6-t_4*t_6-t_5*t_6+5*t_0*t_7+t_1*t_7-2*t_2*t_7-7*t_5*t_7+2*t_6*t_7-2*t_1*t_8+3*t_7*t_8,t_0*t_2+t_1*t_4+5*t_0*t_5+32*t_1*t_5-20*t_2*t_5-47*t_3*t_5-37*t_4*t_5-25*t_5^2+19*t_0*t_6+22*t_1*t_6-25*t_2*t_6+25*t_4*t_6-5*t_5*t_6+13*t_6^2+5*t_0*t_7+t_1*t_7+39*t_3*t_7+28*t_4*t_7-9*t_5*t_7-39*t_6*t_7+4*t_0*t_8+t_1*t_8-36*t_3*t_8-14*t_4*t_8-26*t_5*t_8-37*t_6*t_8,t_0*t_1-39*t_1*t_4+40*t_1*t_5-37*t_0*t_6-39*t_1*t_6+19*t_4*t_6-39*t_5*t_6-38*t_0*t_7+39*t_1*t_7+19*t_2*t_7+18*t_5*t_7-19*t_6*t_7+19*t_1*t_8+20*t_7*t_8,t_0^2+12*t_1*t_4+20*t_0*t_5+27*t_1*t_5-8*t_2*t_5+37*t_3*t_5+28*t_4*t_5+30*t_5^2-46*t_0*t_6+24*t_1*t_6-40*t_2*t_6+25*t_4*t_6+16*t_5*t_6-35*t_6^2+29*t_0*t_7+12*t_1*t_7-35*t_2*t_7-8*t_3*t_7-18*t_4*t_7+42*t_5*t_7-12*t_6*t_7-6*t_0*t_8+12*t_1*t_8-15*t_3*t_8+9*t_4*t_8+20*t_5*t_8-30*t_6*t_8+4*t_7*t_8})
│ │ │  
│ │ │ @@ -192,15 +192,15 @@
│ │ │  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)
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ - -- used 2.12103s (cpu); 1.6772s (thread); 0s (gc)
│ │ │ + -- used 2.16367s (cpu); 1.81526s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = -- rational map --
│ │ │                                  ZZ
│ │ │       source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
│ │ │                                  97  0   1   2   3   4   5   6   7   8   9   10   11
│ │ │               {
│ │ │                                  2
│ │ │ @@ -258,15 +258,15 @@
│ │ │  
│ │ │  i10 : -- in this case we can remedy enabling the option Certify
│ │ │        time psi = approximateInverseMap(phi,CodimBsInv=>4,Certify=>true)
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │  Certify: output certified!
│ │ │ - -- used 3.22893s (cpu); 2.56331s (thread); 0s (gc)
│ │ │ + -- used 3.07071s (cpu); 2.65312s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = -- rational map --
│ │ │                                   ZZ
│ │ │        source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
│ │ │                                   97  0   1   2   3   4   5   6   7   8   9   10   11
│ │ │                {
│ │ │                                   2
│ │ ├── ./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 0.0457634s (cpu); 0.0457644s (thread); 0s (gc)
│ │ │ + -- used 0.0538012s (cpu); 0.0538001s (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 1.25058s (cpu); 0.717537s (thread); 0s (gc)
│ │ │ + -- used 1.45373s (cpu); 0.777154s (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 0.000650229s (cpu); 0.000643546s (thread); 0s (gc)
│ │ │ + -- used 0.000819429s (cpu); 0.000814807s (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 0.0163279s (cpu); 0.0160383s (thread); 0s (gc)
│ │ │ + -- used 0.0340261s (cpu); 0.0200554s (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 0.0333595s (cpu); 0.0330733s (thread); 0s (gc)
│ │ │ + -- used 0.0492661s (cpu); 0.0375183s (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 0.00349994s (cpu); 0.00349466s (thread); 0s (gc)
│ │ │ + -- used 0.00430652s (cpu); 0.00430037s (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 0.0958263s (cpu); 0.0958318s (thread); 0s (gc)
│ │ │ + -- used 0.111955s (cpu); 0.111956s (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 0.160001s (cpu); 0.105062s (thread); 0s (gc)
│ │ │ + -- used 0.192974s (cpu); 0.111259s (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 0.352191s (cpu); 0.207925s (thread); 0s (gc)
│ │ │ + -- used 0.376922s (cpu); 0.212254s (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
│ │ │                           290304 0    3888000  0 1    2939328000  0 1    163296000 0 1   20250 1    228614400  0 2    41150592000  0 1 2    41150592000  0 1 2    3888000 1 2     3572100  0 2    10287648000  0 1 2    342921600 1 2    114307200  0 2    63504000  1 2    25200 2     76204800  0 3    42336000  0 1 3    428652000  0 1 3    212625 1 3     5334336000  0 2 3    9601804800  0 1 2 3    489888000  1 2 3    222264000  0 2 3    12002256000 1 2 3    66679200  2 3    666792000  0 3     666792000  0 1 3    47628000 1 3    1333584000  0 2 3    444528000  1 2 3    777924000  2 3    55566000  0 3    105840 1 3    3472875 2 3    11025 3    4665600  0 4    2939328000  0 1 4     4898880000  0 1 4    29160000  1 4     41150592000  0 2 4    20575296000  0 1 2 4    4898880000  1 2 4    20575296000  0 2 4    1371686400  1 2 4    95256000  2 4     40824000  0 3 4     8573040000  0 1 3 4    11664000  1 3 4     24004512000  0 2 3 4    34292160000  1 2 3 4    12002256000  2 3 4     333396000  0 3 4    5292000  1 3 4    1333584000  2 3 4    3969000  3 4    6804000  0 4    272160000  0 1 4    58320000  1 4    190512000  0 2 4    4898880000 1 2 4    190512000 2 4    476280000  0 3 4    204120000  1 3 4    2857680000  2 3 4    23814000  3 4    30618000 0 4    46656 1 4   12757500 2 4    51030000  3 4   30375 4
│ │ │                       }
│ │ │  
│ │ │  o2 : RationalMap (rational map from PP^4 to PP^4)
│ │ │  
│ │ │  i3 : time inverse phi
│ │ │ - -- used 0.061068s (cpu); 0.0610645s (thread); 0s (gc)
│ │ │ + -- used 0.0695033s (cpu); 0.0695188s (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 0.0179228s (cpu); 0.0179213s (thread); 0s (gc)
│ │ │ + -- used 0.0219934s (cpu); 0.021953s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 : time isBirational(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0136015s (cpu); 0.0132332s (thread); 0s (gc)
│ │ │ + -- used 0.0275705s (cpu); 0.0153734s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = true
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_is__Dominant.out
│ │ │ @@ -4,15 +4,15 @@
│ │ │  
│ │ │  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)
│ │ │  Certify: output certified!
│ │ │ - -- used 2.50039s (cpu); 1.97646s (thread); 0s (gc)
│ │ │ + -- used 2.5324s (cpu); 2.21007s (thread); 0s (gc)
│ │ │  
│ │ │  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_5-24*x_5^2+21*x_0*x_6+x_1*x_6+46*x_3*x_6+27*x_4*x_6+5*x_5*x_6+35*x_6^2+20*x_0*x_7-23*x_1*x_7+8*x_2*x_7-22*x_3*x_7+20*x_4*x_7-15*x_5*x_7,x_2*x_5+47*x_5^2-40*x_0*x_6+37*x_1*x_6-25*x_2*x_6-22*x_3*x_6-8*x_4*x_6+27*x_5*x_6+15*x_6^2-23*x_0*x_7-42*x_1*x_7+27*x_2*x_7+35*x_3*x_7+39*x_4*x_7+24*x_5*x_7,x_1*x_5+15*x_5^2+49*x_0*x_6+8*x_1*x_6-31*x_2*x_6+9*x_3*x_6+38*x_4*x_6-36*x_5*x_6-30*x_6^2-33*x_0*x_7+26*x_1*x_7+32*x_2*x_7+27*x_3*x_7+6*x_4*x_7+36*x_5*x_7,x_0*x_5+30*x_5^2-11*x_0*x_6-38*x_1*x_6+13*x_2*x_6-32*x_3*x_6-30*x_4*x_6+4*x_5*x_6-28*x_6^2-30*x_0*x_7-6*x_1*x_7-45*x_2*x_7+34*x_3*x_7+20*x_4*x_7+48*x_5*x_7,x_3*x_4+46*x_5^2-37*x_0*x_6+27*x_1*x_6+33*x_2*x_6+8*x_3*x_6-32*x_4*x_6+42*x_5*x_6-34*x_6^2-37*x_0*x_7-28*x_1*x_7+10*x_2*x_7-27*x_3*x_7-42*x_4*x_7-8*x_5*x_7,x_2*x_4-25*x_5^2-4*x_0*x_6+2*x_1*x_6-31*x_2*x_6-5*x_3*x_6+16*x_4*x_6-24*x_5*x_6+31*x_6^2-30*x_0*x_7+32*x_1*x_7+12*x_2*x_7-40*x_3*x_7+3*x_4*x_7-28*x_5*x_7,x_0*x_4+15*x_5^2+48*x_0*x_6-50*x_1*x_6+46*x_2*x_6-48*x_3*x_6-23*x_4*x_6-28*x_5*x_6+39*x_6^2+38*x_1*x_7-5*x_3*x_7+5*x_4*x_7-34*x_5*x_7,x_3^2-31*x_5^2+41*x_0*x_6-30*x_1*x_6-4*x_2*x_6+43*x_3*x_6+23*x_4*x_6+7*x_5*x_6+31*x_6^2-19*x_0*x_7+25*x_1*x_7-49*x_2*x_7-16*x_3*x_7-45*x_4*x_7+25*x_5*x_7,x_2*x_3+13*x_5^2-45*x_0*x_6-22*x_1*x_6+33*x_2*x_6-26*x_3*x_6-21*x_4*x_6+34*x_5*x_6-21*x_6^2-47*x_0*x_7-10*x_1*x_7+29*x_2*x_7-46*x_3*x_7-x_4*x_7+20*x_5*x_7,x_1*x_3+22*x_5^2+4*x_0*x_6+3*x_1*x_6+45*x_2*x_6+37*x_3*x_6+17*x_4*x_6+36*x_5*x_6-2*x_6^2-31*x_0*x_7+3*x_1*x_7-12*x_2*x_7+19*x_3*x_7+28*x_4*x_7+30*x_5*x_7,x_0*x_3-47*x_5^2-43*x_0*x_6+6*x_1*x_6-40*x_2*x_6+21*x_3*x_6+26*x_4*x_6-5*x_5*x_6-5*x_6^2+4*x_0*x_7-15*x_1*x_7+18*x_2*x_7-31*x_3*x_7+50*x_4*x_7-46*x_5*x_7,x_2^2+4*x_5^2+31*x_0*x_6+41*x_1*x_6+31*x_2*x_6+28*x_3*x_6+42*x_4*x_6-28*x_5*x_6-4*x_6^2-7*x_0*x_7+15*x_1*x_7-9*x_2*x_7+31*x_3*x_7+3*x_4*x_7+7*x_5*x_7,x_1*x_2-46*x_5^2-6*x_0*x_6-50*x_1*x_6+32*x_2*x_6-10*x_3*x_6+42*x_4*x_6+33*x_5*x_6+18*x_6^2-9*x_0*x_7-20*x_1*x_7+45*x_2*x_7-9*x_3*x_7+10*x_4*x_7-8*x_5*x_7,x_0*x_2-9*x_5^2+34*x_0*x_6-45*x_1*x_6+19*x_2*x_6+24*x_3*x_6+23*x_4*x_6-37*x_5*x_6-44*x_6^2+24*x_0*x_7-33*x_2*x_7+41*x_3*x_7-40*x_4*x_7+4*x_5*x_7,x_1^2+x_1*x_4+x_4^2-28*x_5^2-33*x_0*x_6-17*x_1*x_6+11*x_3*x_6+20*x_4*x_6+25*x_5*x_6-21*x_6^2-22*x_0*x_7+24*x_1*x_7-14*x_2*x_7+5*x_3*x_7-39*x_4*x_7-18*x_5*x_7,x_0*x_1-47*x_5^2-5*x_0*x_6-9*x_1*x_6-45*x_2*x_6+48*x_3*x_6+45*x_4*x_6-29*x_5*x_6+3*x_6^2+29*x_0*x_7+40*x_1*x_7+46*x_2*x_7+27*x_3*x_7-36*x_4*x_7-39*x_5*x_7,x_0^2-31*x_5^2+36*x_0*x_6-30*x_1*x_6-10*x_2*x_6+42*x_3*x_6+9*x_4*x_6+34*x_5*x_6-6*x_6^2+48*x_0*x_7-47*x_1*x_7-19*x_2*x_7+25*x_3*x_7+28*x_4*x_7+34*x_5*x_7);
│ │ │ @@ -21,12 +21,12 @@
│ │ │  
│ │ │  i6 : phi = rationalMap(C,3,2);
│ │ │  
│ │ │  o6 : RationalMap (cubic rational map from PP^7 to PP^7)
│ │ │  
│ │ │  i7 : time isDominant(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 3.85067s (cpu); 2.52355s (thread); 0s (gc)
│ │ │ + -- used 3.72788s (cpu); 2.75047s (thread); 0s (gc)
│ │ │  
│ │ │  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 0.0172216s (cpu); 0.0172186s (thread); 0s (gc)
│ │ │ + -- used 0.0214311s (cpu); 0.021431s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = ideal ()
│ │ │  
│ │ │  o2 : Ideal of QQ[y ..y  ]
│ │ │                    0   11
│ │ │  
│ │ │  i3 : time kernel(phi,2)
│ │ │ - -- used 0.848508s (cpu); 0.44358s (thread); 0s (gc)
│ │ │ + -- used 1.1128s (cpu); 0.513164s (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 0.0052255s (cpu); 0.00522062s (thread); 0s (gc)
│ │ │ + -- used 0.00599909s (cpu); 0.00599741s (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 0.556987s (cpu); 0.397597s (thread); 0s (gc)
│ │ │ + -- used 0.57734s (cpu); 0.426018s (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 0.44597s (cpu); 0.215856s (thread); 0s (gc)
│ │ │ + -- used 0.51082s (cpu); 0.235994s (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 0.2051s (cpu); 0.130409s (thread); 0s (gc)
│ │ │ + -- used 0.224866s (cpu); 0.138715s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_projective__Degrees.out
│ │ │ @@ -8,15 +8,15 @@
│ │ │                       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)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0148041s (cpu); 0.0144893s (thread); 0s (gc)
│ │ │ + -- used 0.0809715s (cpu); 0.0244069s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = {1, 2, 4, 4, 2}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : psi=inverseMap(toMap(phi,Dominant=>infinity))
│ │ │  
│ │ │ @@ -30,15 +30,15 @@
│ │ │                           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)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0112491s (cpu); 0.0109601s (thread); 0s (gc)
│ │ │ + -- used 0.0360115s (cpu); 0.0148336s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {2, 4, 4, 2, 1}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : -- Cremona transformation of P^6 defined by the quadrics through a rational octic surface
│ │ │       phi = map specialCremonaTransformation(7,ZZ/300007)
│ │ │ @@ -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 5.0365e-05s (cpu); 4.5084e-05s (thread); 0s (gc)
│ │ │ + -- used 6.3683e-05s (cpu); 5.4318e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {1, 2, 4, 8, 8, 4, 1}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : time projectiveDegrees(phi,NumDegrees=>1)
│ │ │ - -- used 2.0038e-05s (cpu); 1.9887e-05s (thread); 0s (gc)
│ │ │ + -- used 2.9195e-05s (cpu); 2.8031e-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 0.0958343s (cpu); 0.0957853s (thread); 0s (gc)
│ │ │ + -- used 0.1119s (cpu); 0.111898s (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 0.0288893s (cpu); 0.0288899s (thread); 0s (gc)
│ │ │ + -- used 0.0361834s (cpu); 0.0361831s (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 1.28365s (cpu); 0.786418s (thread); 0s (gc)
│ │ │ + -- used 1.79185s (cpu); 0.715829s (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 1.47583s (cpu); 1.14802s (thread); 0s (gc)
│ │ │ + -- used 1.54095s (cpu); 1.17508s (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 0.0942055s (cpu); 0.0942043s (thread); 0s (gc)
│ │ │ + -- used 0.0952488s (cpu); 0.0952487s (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 0.0185862s (cpu); 0.0185866s (thread); 0s (gc)
│ │ │ + -- used 0.0203254s (cpu); 0.0203252s (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 0.0735881s (cpu); 0.0735702s (thread); 0s (gc)
│ │ │ + -- used 0.0816657s (cpu); 0.0814841s (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 0.109311s (cpu); 0.029848s (thread); 0s (gc)
│ │ │ + -- used 0.11511s (cpu); 0.0358792s (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 0.0222424s (cpu); 0.0222414s (thread); 0s (gc)
│ │ │ + -- used 0.0453571s (cpu); 0.0453573s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : RationalMap (cubic birational map from PP^3 to hypersurface in PP^4)
│ │ │  
│ │ │  i5 : time describe phi'
│ │ │ - -- used 0.00518727s (cpu); 0.00518762s (thread); 0s (gc)
│ │ │ + -- used 0.0106974s (cpu); 0.010701s (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 0.00429438s (cpu); 0.00429504s (thread); 0s (gc)
│ │ │ + -- used 0.00870284s (cpu); 0.00870936s (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
│ │ │ @@ -97,30 +97,30 @@
│ │ │  o2 : Ideal of GF 78125[x ..x ]
│ │ │                          0   4
    
│ │ │              
    
│ │ │                
    i3 : time ChernSchwartzMacPherson C
│ │ │ - -- used 2.22491s (cpu); 1.26478s (thread); 0s (gc)
│ │ │ + -- used 2.53235s (cpu); 1.21957s (thread); 0s (gc)
│ │ │  
│ │ │         4     3     2
│ │ │  o3 = 3H  + 5H  + 3H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o3 : -----
│ │ │          5
│ │ │         H
    
│ │ │              
    
│ │ │                
    i4 : time ChernSchwartzMacPherson(C,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 1.33783s (cpu); 0.951359s (thread); 0s (gc)
│ │ │ + -- used 1.72145s (cpu); 1.07375s (thread); 0s (gc)
│ │ │  
│ │ │         4     3     2
│ │ │  o4 = 3H  + 5H  + 3H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o4 : -----
│ │ │          5
│ │ │ @@ -167,30 +167,30 @@
│ │ │  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 0.32008s (cpu); 0.18723s (thread); 0s (gc)
│ │ │ + -- used 0.393117s (cpu); 0.19787s (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)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.107679s (cpu); 0.0382531s (thread); 0s (gc)
│ │ │ + -- used 0.192302s (cpu); 0.0468254s (thread); 0s (gc)
│ │ │  
│ │ │           9      8      7      6      5      4     3
│ │ │  o10 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o10 : -----
│ │ │          10
│ │ │ ├── html2text {}
│ │ │ │ @@ -39,26 +39,26 @@
│ │ │ │                 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 2.22491s (cpu); 1.26478s (thread); 0s (gc)
│ │ │ │ + -- used 2.53235s (cpu); 1.21957s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         4     3     2
│ │ │ │  o3 = 3H  + 5H  + 3H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o3 : -----
│ │ │ │          5
│ │ │ │         H
│ │ │ │  i4 : time ChernSchwartzMacPherson(C,Certify=>true)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 1.33783s (cpu); 0.951359s (thread); 0s (gc)
│ │ │ │ + -- used 1.72145s (cpu); 1.07375s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         4     3     2
│ │ │ │  o4 = 3H  + 5H  + 3H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o4 : -----
│ │ │ │          5
│ │ │ │ @@ -88,26 +88,26 @@
│ │ │ │          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 0.32008s (cpu); 0.18723s (thread); 0s (gc)
│ │ │ │ + -- used 0.393117s (cpu); 0.19787s (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)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 0.107679s (cpu); 0.0382531s (thread); 0s (gc)
│ │ │ │ + -- used 0.192302s (cpu); 0.0468254s (thread); 0s (gc)
│ │ │ │  
│ │ │ │           9      8      7      6      5      4     3
│ │ │ │  o10 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │ │  
│ │ │ │        ZZ[H]
│ │ │ │  o10 : -----
│ │ │ │          10
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/___Euler__Characteristic.html
│ │ │ @@ -85,24 +85,24 @@
│ │ │  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 0.32363s (cpu); 0.192112s (thread); 0s (gc)
│ │ │ + -- used 0.365833s (cpu); 0.213574s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 10
    
│ │ │              
    
│ │ │                
    i3 : time EulerCharacteristic(I,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0112392s (cpu); 0.0109549s (thread); 0s (gc)
│ │ │ + -- used 0.0243867s (cpu); 0.0125919s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = 10
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -31,20 +31,20 @@
│ │ │ │  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 0.32363s (cpu); 0.192112s (thread); 0s (gc)
│ │ │ │ + -- used 0.365833s (cpu); 0.213574s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = 10
│ │ │ │  i3 : time EulerCharacteristic(I,Certify=>true)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 0.0112392s (cpu); 0.0109549s (thread); 0s (gc)
│ │ │ │ + -- used 0.0243867s (cpu); 0.0125919s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  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
│ │ │ │        (smooth) projective variety
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/___Rational__Map_sp!.html
│ │ │ @@ -86,15 +86,15 @@
│ │ │       target variety: PP^5
│ │ │       coefficient ring: QQ
    
│ │ │              
    
│ │ │                
    i4 : time phi! ;
│ │ │ - -- used 0.0555729s (cpu); 0.0551512s (thread); 0s (gc)
│ │ │ + -- used 0.0985162s (cpu); 0.0672745s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : RationalMap (Cremona transformation of PP^5 of type (2,2))
    
│ │ │              
    
│ │ │                
    i5 : describe phi
│ │ │ @@ -127,15 +127,15 @@
│ │ │       target variety: PP^5
│ │ │       coefficient ring: QQ
    
│ │ │              
    
│ │ │                
    i9 : time phi! ;
│ │ │ - -- used 0.0359549s (cpu); 0.0356875s (thread); 0s (gc)
│ │ │ + -- used 0.0625898s (cpu); 0.0460574s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : RationalMap (quadratic rational map from PP^4 to PP^5)
    
│ │ │              
    
│ │ │                
    i10 : describe phi
│ │ │ ├── html2text {}
│ │ │ │ @@ -21,15 +21,15 @@
│ │ │ │  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 0.0555729s (cpu); 0.0551512s (thread); 0s (gc)
│ │ │ │ + -- used 0.0985162s (cpu); 0.0672745s (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
│ │ │ │ @@ -47,15 +47,15 @@
│ │ │ │  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 0.0359549s (cpu); 0.0356875s (thread); 0s (gc)
│ │ │ │ + -- used 0.0625898s (cpu); 0.0460574s (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
│ │ │ @@ -153,15 +153,15 @@
│ │ │  
│ │ │  o5 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v]
    
│ │ │              
    
│ │ │                
    i6 : time phi^** q
│ │ │ - -- used 0.15576s (cpu); 0.155761s (thread); 0s (gc)
│ │ │ + -- used 0.174302s (cpu); 0.174303s (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]
    
│ │ │              
    
│ │ │                
    i4 : time SegreClass X
│ │ │ - -- used 0.798226s (cpu); 0.509602s (thread); 0s (gc)
│ │ │ + -- used 0.836263s (cpu); 0.55014s (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 0.549131s (cpu); 0.33326s (thread); 0s (gc)
│ │ │ + -- used 0.635674s (cpu); 0.360676s (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)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0212624s (cpu); 0.0207391s (thread); 0s (gc)
│ │ │ + -- used 0.0422507s (cpu); 0.0255555s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5       4      3
│ │ │  o6 = 3240H  - 1188H  + 396H  - 114H  + 24H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o6 : -----
│ │ │          8
│ │ │         H
    
│ │ │              
    
│ │ │                
    i7 : time SegreClass(lift(X,P7),Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0974975s (cpu); 0.0969659s (thread); 0s (gc)
│ │ │ + -- used 0.135886s (cpu); 0.124334s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5      4      3
│ │ │  o7 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o7 : -----
│ │ │          8
│ │ │ @@ -203,25 +203,25 @@
│ │ │          
    
│ │ │          
    The method also accepts as input a ring map phi representing a rational map $\Phi:X\dashrightarrow Y$ between projective varieties. In this case, the method returns the push-forward to the Chow ring of the ambient projective space of $X$ of the Segre class of the base locus of $\Phi$ in $X$, i.e., it basically computes SegreClass ideal matrix phi. In the next example, we compute the Segre class of the base locus of a birational map $\mathbb{G}(1,4)\subset\mathbb{P}^9 \dashrightarrow \mathbb{P}^6$.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │          
    
│ │ │                
    i9 : use ZZ/100003[x_0..x_6]
│ │ │  
│ │ │ -o9     ZZ
│ │ │ -= ------[x ..x ]
│ │ │ -  100003  0   6
│ │ │ +       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 0.197315s (cpu); 0.0983952s (thread); 0s (gc)
│ │ │ + -- used 0.259917s (cpu); 0.125368s (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
│ │ │ @@ -233,15 +233,15 @@
│ │ │                (y y  - y y  + y y , 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 0.169509s (cpu); 0.169514s (thread); 0s (gc)
│ │ │ + -- used 0.212058s (cpu); 0.211896s (thread); 0s (gc)
│ │ │  
│ │ │           9      8      7      6     5
│ │ │  o11 = 23H  - 42H  + 36H  - 22H  + 9H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o11 : -----
│ │ │          10
│ │ │ @@ -267,30 +267,30 @@
│ │ │                   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 0.518764s (cpu); 0.315789s (thread); 0s (gc)
│ │ │ + -- used 0.433541s (cpu); 0.291859s (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 1.31381s (cpu); 0.860817s (thread); 0s (gc)
│ │ │ + -- used 1.84444s (cpu); 1.03393s (thread); 0s (gc)
│ │ │  
│ │ │             9       8       7      6     5
│ │ │  o14 = 2764H  - 984H  + 294H  - 67H  + 9H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o14 : -----
│ │ │          10
│ │ │ ├── html2text {}
│ │ │ │ @@ -81,47 +81,47 @@
│ │ │ │                 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 0.798226s (cpu); 0.509602s (thread); 0s (gc)
│ │ │ │ + -- used 0.836263s (cpu); 0.55014s (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 0.549131s (cpu); 0.33326s (thread); 0s (gc)
│ │ │ │ + -- used 0.635674s (cpu); 0.360676s (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)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 0.0212624s (cpu); 0.0207391s (thread); 0s (gc)
│ │ │ │ + -- used 0.0422507s (cpu); 0.0255555s (thread); 0s (gc)
│ │ │ │  
│ │ │ │            7        6       5       4      3
│ │ │ │  o6 = 3240H  - 1188H  + 396H  - 114H  + 24H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o6 : -----
│ │ │ │          8
│ │ │ │         H
│ │ │ │  i7 : time SegreClass(lift(X,P7),Certify=>true)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 0.0974975s (cpu); 0.0969659s (thread); 0s (gc)
│ │ │ │ + -- used 0.135886s (cpu); 0.124334s (thread); 0s (gc)
│ │ │ │  
│ │ │ │            7        6       5      4      3
│ │ │ │  o7 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o7 : -----
│ │ │ │          8
│ │ │ │ @@ -134,22 +134,22 @@
│ │ │ │  method returns the push-forward to the Chow ring of the ambient projective
│ │ │ │  space of $X$ of the Segre class of the base locus of $\Phi$ in $X$, i.e., it
│ │ │ │  basically computes SegreClass ideal matrix phi. In the next example, we compute
│ │ │ │  the Segre class of the base locus of a birational map $\mathbb{G}
│ │ │ │  (1,4)\subset\mathbb{P}^9 \dashrightarrow \mathbb{P}^6$.
│ │ │ │  i9 : use ZZ/100003[x_0..x_6]
│ │ │ │  
│ │ │ │ -o9     ZZ
│ │ │ │ -= ------[x ..x ]
│ │ │ │ -  100003  0   6
│ │ │ │ +       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 0.197315s (cpu); 0.0983952s (thread); 0s (gc)
│ │ │ │ + -- used 0.259917s (cpu); 0.125368s (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
│ │ │ │ @@ -169,15 +169,15 @@
│ │ │ │  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 0.169509s (cpu); 0.169514s (thread); 0s (gc)
│ │ │ │ + -- used 0.212058s (cpu); 0.211896s (thread); 0s (gc)
│ │ │ │  
│ │ │ │           9      8      7      6     5
│ │ │ │  o11 = 23H  - 42H  + 36H  - 22H  + 9H
│ │ │ │  
│ │ │ │        ZZ[H]
│ │ │ │  o11 : -----
│ │ │ │          10
│ │ │ │ @@ -198,26 +198,26 @@
│ │ │ │  ------------------------------------
│ │ │ │                 (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 0.518764s (cpu); 0.315789s (thread); 0s (gc)
│ │ │ │ + -- used 0.433541s (cpu); 0.291859s (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 1.31381s (cpu); 0.860817s (thread); 0s (gc)
│ │ │ │ + -- used 1.84444s (cpu); 1.03393s (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
│ │ │ @@ -101,15 +101,15 @@
│ │ │  
│ │ │  o3 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i4 : time psi = abstractRationalMap(P4,P5,f)
│ │ │ - -- used 0.000416542s (cpu); 0.000408857s (thread); 0s (gc)
│ │ │ + -- used 0.000409484s (cpu); 0.000402295s (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
│ │ │ @@ -119,23 +119,23 @@
│ │ │            
    
│ │ │          
    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 0.32875s (cpu); 0.185077s (thread); 0s (gc)
│ │ │ + -- used 0.384646s (cpu); 0.218147s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 2
    
│ │ │              
    
│ │ │                
    i6 : time rationalMap psi
│ │ │ - -- used 0.510248s (cpu); 0.373012s (thread); 0s (gc)
│ │ │ + -- used 0.507728s (cpu); 0.415065s (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: {
│ │ │ @@ -233,15 +233,15 @@
│ │ │  o13 : Ideal of -----[x ..x ]
│ │ │                 65521  0   3
    
│ │ │              
    
│ │ │                
    i14 : time T = abstractRationalMap(I,"OADP")
│ │ │ - -- used 0.151084s (cpu); 0.0733367s (thread); 0s (gc)
│ │ │ + -- used 0.179979s (cpu); 0.0992287s (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 ])
│ │ │ @@ -253,26 +253,26 @@
│ │ │            
    
│ │ │          
    The degree of the forms defining the abstract map T can be obtained by the following command:
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i15 : time projectiveDegrees(T,2)
│ │ │ - -- used 3.78487s (cpu); 1.95126s (thread); 0s (gc)
│ │ │ + -- used 4.37345s (cpu); 2.29473s (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 2.8112e-05s (cpu); 2.7842e-05s (thread); 0s (gc)
│ │ │ + -- used 3.6436e-05s (cpu); 3.3449e-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 ])
│ │ │ @@ -281,15 +281,15 @@
│ │ │  
│ │ │  o16 : AbstractRationalMap (rational map from PP^3 to PP^3)
    
│ │ │              
    
│ │ │                
    i17 : time projectiveDegrees(T2,2)
│ │ │ - -- used 6.44263s (cpu); 3.37969s (thread); 0s (gc)
│ │ │ + -- used 7.41221s (cpu); 3.66446s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = 1
    
│ │ │              
    
│ │ │          
    We verify that the composition of T with itself leaves a random point fixed:
    
│ │ │          
│ │ │ @@ -322,15 +322,15 @@
│ │ │            
│ │ │          
    
│ │ │          
    We now compute the concrete rational map corresponding to T:
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i21 : time f = rationalMap T
│ │ │ - -- used 5.32932s (cpu); 2.87316s (thread); 0s (gc)
│ │ │ + -- used 6.07165s (cpu); 3.07132s (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 {}
│ │ │ │ @@ -35,32 +35,32 @@
│ │ │ │  i3 : P5 := QQ[u_0..u_5]
│ │ │ │  
│ │ │ │  o3 = QQ[u ..u ]
│ │ │ │           0   5
│ │ │ │  
│ │ │ │  o3 : PolynomialRing
│ │ │ │  i4 : time psi = abstractRationalMap(P4,P5,f)
│ │ │ │ - -- used 0.000416542s (cpu); 0.000408857s (thread); 0s (gc)
│ │ │ │ + -- used 0.000409484s (cpu); 0.000402295s (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 0.32875s (cpu); 0.185077s (thread); 0s (gc)
│ │ │ │ + -- used 0.384646s (cpu); 0.218147s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 2
│ │ │ │  i6 : time rationalMap psi
│ │ │ │ - -- used 0.510248s (cpu); 0.373012s (thread); 0s (gc)
│ │ │ │ + -- used 0.507728s (cpu); 0.415065s (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: {
│ │ │ │ @@ -139,48 +139,48 @@
│ │ │ │  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 0.151084s (cpu); 0.0733367s (thread); 0s (gc)
│ │ │ │ + -- used 0.179979s (cpu); 0.0992287s (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 3.78487s (cpu); 1.95126s (thread); 0s (gc)
│ │ │ │ + -- used 4.37345s (cpu); 2.29473s (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 2.8112e-05s (cpu); 2.7842e-05s (thread); 0s (gc)
│ │ │ │ + -- used 3.6436e-05s (cpu); 3.3449e-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 6.44263s (cpu); 3.37969s (thread); 0s (gc)
│ │ │ │ + -- used 7.41221s (cpu); 3.66446s (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 = {-6648, -23396, -12311, 1}
│ │ │ │  
│ │ │ │ @@ -193,15 +193,15 @@
│ │ │ │  i20 : T q
│ │ │ │  
│ │ │ │  o20 = {-6648, -23396, -12311, 1}
│ │ │ │  
│ │ │ │  o20 : List
│ │ │ │  We now compute the concrete rational map corresponding to T:
│ │ │ │  i21 : time f = rationalMap T
│ │ │ │ - -- used 5.32932s (cpu); 2.87316s (thread); 0s (gc)
│ │ │ │ + -- used 6.07165s (cpu); 3.07132s (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
│ │ │ @@ -139,15 +139,15 @@
│ │ │  -- approximateInverseMap: step 4 of 10
│ │ │  -- approximateInverseMap: step 5 of 10
│ │ │  -- approximateInverseMap: step 6 of 10
│ │ │  -- approximateInverseMap: step 7 of 10
│ │ │  -- approximateInverseMap: step 8 of 10
│ │ │  -- approximateInverseMap: step 9 of 10
│ │ │  -- approximateInverseMap: step 10 of 10
│ │ │ - -- used 0.238191s (cpu); 0.186212s (thread); 0s (gc)
│ │ │ + -- used 0.31681s (cpu); 0.235711s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │                    ZZ
│ │ │       source: Proj(--[t , t , t , t , t , t , t , t , t ])
│ │ │                    97  0   1   2   3   4   5   6   7   8
│ │ │                                  ZZ
│ │ │       target: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x ]) defined by
│ │ │ @@ -200,15 +200,15 @@
│ │ │            
    
│ │ │                
    i5 : time psi' = approximateInverseMap(phi,CodimBsInv=>5);
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ - -- used 0.218651s (cpu); 0.155471s (thread); 0s (gc)
│ │ │ + -- used 0.276259s (cpu); 0.197603s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : RationalMap (quadratic rational map from PP^8 to hypersurface in PP^9)
    
│ │ │              
    
│ │ │                
    i6 : assert(psi == psi')
    
│ │ │ @@ -295,15 +295,15 @@
│ │ │            
    
│ │ │                
    i8 : -- without the option 'CodimBsInv=>4', it takes about triple time 
│ │ │       time psi=approximateInverseMap(phi,CodimBsInv=>4)
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ - -- used 2.12103s (cpu); 1.6772s (thread); 0s (gc)
│ │ │ + -- used 2.16367s (cpu); 1.81526s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = -- rational map --
│ │ │                                  ZZ
│ │ │       source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
│ │ │                                  97  0   1   2   3   4   5   6   7   8   9   10   11
│ │ │               {
│ │ │                                  2
│ │ │ @@ -367,15 +367,15 @@
│ │ │              
    		
│ │ │                
    i10 : -- in this case we can remedy enabling the option Certify
│ │ │        time psi = approximateInverseMap(phi,CodimBsInv=>4,Certify=>true)
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │  Certify: output certified!
│ │ │ - -- used 3.22893s (cpu); 2.56331s (thread); 0s (gc)
│ │ │ + -- used 3.07071s (cpu); 2.65312s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = -- rational map --
│ │ │                                   ZZ
│ │ │        source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
│ │ │                                   97  0   1   2   3   4   5   6   7   8   9   10   11
│ │ │                {
│ │ │                                   2
│ │ │ ├── html2text {}
│ │ │ │ @@ -135,15 +135,15 @@
│ │ │ │  -- approximateInverseMap: step 4 of 10
│ │ │ │  -- approximateInverseMap: step 5 of 10
│ │ │ │  -- approximateInverseMap: step 6 of 10
│ │ │ │  -- approximateInverseMap: step 7 of 10
│ │ │ │  -- approximateInverseMap: step 8 of 10
│ │ │ │  -- approximateInverseMap: step 9 of 10
│ │ │ │  -- approximateInverseMap: step 10 of 10
│ │ │ │ - -- used 0.238191s (cpu); 0.186212s (thread); 0s (gc)
│ │ │ │ + -- used 0.31681s (cpu); 0.235711s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = -- rational map --
│ │ │ │                    ZZ
│ │ │ │       source: Proj(--[t , t , t , t , t , t , t , t , t ])
│ │ │ │                    97  0   1   2   3   4   5   6   7   8
│ │ │ │                                  ZZ
│ │ │ │       target: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x ])
│ │ │ │ @@ -252,15 +252,15 @@
│ │ │ │  
│ │ │ │  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);
│ │ │ │  -- approximateInverseMap: step 1 of 3
│ │ │ │  -- approximateInverseMap: step 2 of 3
│ │ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ │ - -- used 0.218651s (cpu); 0.155471s (thread); 0s (gc)
│ │ │ │ + -- used 0.276259s (cpu); 0.197603s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  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
│ │ │ │  time!).
│ │ │ │  i7 : phi = rationalMap map(P8,ZZ/97[x_0..x_11]/ideal(x_1*x_3-8*x_2*x_3+25*x_3^2-25*x_2*x_4-
│ │ │ │  22*x_3*x_4+x_0*x_5+13*x_2*x_5+41*x_3*x_5-x_0*x_6+12*x_2*x_6+25*x_1*x_7+25*x_3*x_7+23*x_5*x_7-
│ │ │ │ @@ -418,15 +418,15 @@
│ │ │ │  
│ │ │ │  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)
│ │ │ │  -- approximateInverseMap: step 1 of 3
│ │ │ │  -- approximateInverseMap: step 2 of 3
│ │ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ │ - -- used 2.12103s (cpu); 1.6772s (thread); 0s (gc)
│ │ │ │ + -- used 2.16367s (cpu); 1.81526s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = -- rational map --
│ │ │ │                                  ZZ
│ │ │ │       source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
│ │ │ │                                  97  0   1   2   3   4   5   6   7   8   9   10   11
│ │ │ │               {
│ │ │ │                                  2
│ │ │ │ @@ -526,15 +526,15 @@
│ │ │ │  o9 = false
│ │ │ │  i10 : -- in this case we can remedy enabling the option Certify
│ │ │ │        time psi = approximateInverseMap(phi,CodimBsInv=>4,Certify=>true)
│ │ │ │  -- approximateInverseMap: step 1 of 3
│ │ │ │  -- approximateInverseMap: step 2 of 3
│ │ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 3.22893s (cpu); 2.56331s (thread); 0s (gc)
│ │ │ │ + -- used 3.07071s (cpu); 2.65312s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = -- rational map --
│ │ │ │                                   ZZ
│ │ │ │        source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
│ │ │ │                                   97  0   1   2   3   4   5   6   7   8   9   10   11
│ │ │ │                {
│ │ │ │                                   2
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_degree__Map.html
│ │ │ @@ -92,15 +92,15 @@
│ │ │  
│ │ │  o4 : RingMap ringP8 <-- ringP14
    
│ │ │              
    
│ │ │                
    i5 : time degreeMap phi
│ │ │ - -- used 0.0457634s (cpu); 0.0457644s (thread); 0s (gc)
│ │ │ + -- used 0.0538012s (cpu); 0.0538001s (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 
│ │ │ @@ -113,15 +113,15 @@
│ │ │  
│ │ │  o6 : RingMap ringP8 <-- ringP8
    
│ │ │              
    
│ │ │                
    i7 : time degreeMap phi'
│ │ │ - -- used 1.25058s (cpu); 0.717537s (thread); 0s (gc)
│ │ │ + -- used 1.45373s (cpu); 0.777154s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = 14
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -266,15 +266,15 @@
│ │ │ │  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 0.0457634s (cpu); 0.0457644s (thread); 0s (gc)
│ │ │ │ + -- used 0.0538012s (cpu); 0.0538001s (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))
│ │ │ │ @@ -418,15 +418,15 @@
│ │ │ │  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 1.25058s (cpu); 0.717537s (thread); 0s (gc)
│ │ │ │ + -- used 1.45373s (cpu); 0.777154s (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
│ │ │ @@ -83,15 +83,15 @@
│ │ │  
│ │ │  o3 : RationalMap (cubic rational map from PP^6 to 6-dimensional subvariety of PP^9)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time forceImage(Phi,ideal 0_(target Phi))
│ │ │ - -- used 0.000650229s (cpu); 0.000643546s (thread); 0s (gc)
    
│ │ │ + -- used 0.000819429s (cpu); 0.000814807s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : Phi;
│ │ │  
│ │ │  o5 : RationalMap (cubic dominant rational map from PP^6 to 6-dimensional subvariety of PP^9)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,15 +19,15 @@
│ │ │ │  
│ │ │ │  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 0.000650229s (cpu); 0.000643546s (thread); 0s (gc)
│ │ │ │ + -- used 0.000819429s (cpu); 0.000814807s (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 @@
│ │ │  
│ │ │  o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in PP^5)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time (p1,p2) = graph phi;
│ │ │ - -- used 0.0163279s (cpu); 0.0160383s (thread); 0s (gc)
    
│ │ │ + -- used 0.0340261s (cpu); 0.0200554s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : p1
│ │ │  
│ │ │  o4 = -- rational map --
│ │ │ @@ -272,15 +272,15 @@
│ │ │            
│ │ │          
│ │ │          
    When the source of the rational map is a multi-projective variety, the method returns all the projections.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -98,15 +98,15 @@
│ │ │ │  
│ │ │ │                        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 0.160001s (cpu); 0.105062s (thread); 0s (gc)
│ │ │ │ + -- used 0.192974s (cpu); 0.111259s (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
│ │ │ │ @@ -216,15 +216,15 @@
│ │ │ │  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 0.352191s (cpu); 0.207925s (thread); 0s (gc)
│ │ │ │ + -- used 0.376922s (cpu); 0.212254s (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
│ │ │ @@ -104,15 +104,15 @@
│ │ │  
│ │ │  o2 : RationalMap (rational map from PP^4 to PP^4)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i9 : time g = graph p2;
│ │ │ - -- used 0.0333595s (cpu); 0.0330733s (thread); 0s (gc)
    
│ │ │ + -- used 0.0492661s (cpu); 0.0375183s (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)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -50,15 +50,15 @@
│ │ │ │                        - 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 0.0163279s (cpu); 0.0160383s (thread); 0s (gc)
│ │ │ │ + -- used 0.0340261s (cpu); 0.0200554s (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
│ │ │ │ @@ -192,15 +192,15 @@
│ │ │ │  
│ │ │ │  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 0.0333595s (cpu); 0.0330733s (thread); 0s (gc)
│ │ │ │ + -- used 0.0492661s (cpu); 0.0375183s (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
│ │ │ @@ -111,15 +111,15 @@
│ │ │  
│ │ │  o2 : RationalMap (quadratic rational map from hypersurface in PP^5 to PP^4)
│ │ │              
    
│ │ │                
    i3 : time ideal phi
│ │ │ - -- used 0.00349994s (cpu); 0.00349466s (thread); 0s (gc)
│ │ │ + -- used 0.00430652s (cpu); 0.00430037s (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 )
│ │ │ @@ -195,15 +195,15 @@
│ │ │  
│ │ │  o5 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety of PP^5 x PP^4 to PP^4)
    
│ │ │              
    
│ │ │                
    i6 : time ideal phi'
│ │ │ - -- used 0.0958263s (cpu); 0.0958318s (thread); 0s (gc)
│ │ │ + -- used 0.111955s (cpu); 0.111956s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = ideal 1
│ │ │  
│ │ │                                                                                                              QQ[x ..x , y ..y ]
│ │ │                                                                                                                  0   5   0   4
│ │ │  o6 : Ideal of --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
│ │ │                                                                                                                                                                                                       2
│ │ │ ├── html2text {}
│ │ │ │ @@ -46,15 +46,15 @@
│ │ │ │                         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 0.00349994s (cpu); 0.00349466s (thread); 0s (gc)
│ │ │ │ + -- used 0.00430652s (cpu); 0.00430037s (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 )
│ │ │ │ @@ -121,15 +121,15 @@
│ │ │ │                        y
│ │ │ │                         4
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o5 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety of
│ │ │ │  PP^5 x PP^4 to PP^4)
│ │ │ │  i6 : time ideal phi'
│ │ │ │ - -- used 0.0958263s (cpu); 0.0958318s (thread); 0s (gc)
│ │ │ │ + -- used 0.111955s (cpu); 0.111956s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = ideal 1
│ │ │ │  
│ │ │ │  
│ │ │ │  QQ[x ..x , y ..y ]
│ │ │ │  
│ │ │ │  0   5   0   4
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_inverse__Map.html
│ │ │ @@ -153,15 +153,15 @@
│ │ │  
│ │ │  o1 : RationalMap (quadratic Cremona transformation of PP^20)
    
│ │ │              
    
│ │ │                
    i2 : time psi = inverseMap phi
│ │ │ - -- used 0.160001s (cpu); 0.105062s (thread); 0s (gc)
│ │ │ + -- used 0.192974s (cpu); 0.111259s (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: {
│ │ │ @@ -251,15 +251,15 @@
│ │ │  o4 : RingMap QQ[w ..w  ] <-- QQ[w ..w  ]
│ │ │                   0   26          0   26
    
│ │ │              
    
│ │ │                
    i5 : time psi = inverseMap phi
│ │ │ - -- used 0.352191s (cpu); 0.207925s (thread); 0s (gc)
│ │ │ + -- used 0.376922s (cpu); 0.212254s (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 0.061068s (cpu); 0.0610645s (thread); 0s (gc)
│ │ │ + -- used 0.0695033s (cpu); 0.0695188s (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 {}
│ │ │ │ @@ -290,15 +290,15 @@
│ │ │ │  58320000  1 4    190512000  0 2 4    4898880000 1 2 4    190512000 2 4
│ │ │ │  476280000  0 3 4    204120000  1 3 4    2857680000  2 3 4    23814000  3 4
│ │ │ │  30618000 0 4    46656 1 4   12757500 2 4    51030000  3 4   30375 4
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o2 : RationalMap (rational map from PP^4 to PP^4)
│ │ │ │  i3 : time inverse phi
│ │ │ │ - -- used 0.061068s (cpu); 0.0610645s (thread); 0s (gc)
│ │ │ │ + -- used 0.0695033s (cpu); 0.0695188s (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
│ │ │ @@ -123,24 +123,24 @@
│ │ │  
│ │ │  o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in PP^5)
    
│ │ │              
    
│ │ │                
    i3 : time isBirational phi
│ │ │ - -- used 0.0179228s (cpu); 0.0179213s (thread); 0s (gc)
│ │ │ + -- used 0.0219934s (cpu); 0.021953s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
    
│ │ │              
    
│ │ │                
    i4 : time isBirational(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0136015s (cpu); 0.0132332s (thread); 0s (gc)
│ │ │ + -- used 0.0275705s (cpu); 0.0153734s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = true
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -58,20 +58,20 @@
│ │ │ │                        - 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 0.0179228s (cpu); 0.0179213s (thread); 0s (gc)
│ │ │ │ + -- used 0.0219934s (cpu); 0.021953s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  i4 : time isBirational(phi,Certify=>true)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 0.0136015s (cpu); 0.0132332s (thread); 0s (gc)
│ │ │ │ + -- used 0.0275705s (cpu); 0.0153734s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  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)
│ │ │ │      * isBirational(RingMap)
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_is__Dominant.html
│ │ │ @@ -86,15 +86,15 @@
│ │ │  o2 : RationalMap (rational map from PP^8 to PP^8)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time isDominant(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 2.50039s (cpu); 1.97646s (thread); 0s (gc)
│ │ │ + -- used 2.5324s (cpu); 2.21007s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : P7 = ZZ/101[x_0..x_7];
    
│ │ │ @@ -115,15 +115,15 @@
│ │ │  o6 : RationalMap (cubic rational map from PP^7 to PP^7)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : time isDominant(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 3.85067s (cpu); 2.52355s (thread); 0s (gc)
│ │ │ + -- used 3.72788s (cpu); 2.75047s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = false
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,15 +20,15 @@
│ │ │ │  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)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 2.50039s (cpu); 1.97646s (thread); 0s (gc)
│ │ │ │ + -- used 2.5324s (cpu); 2.21007s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  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-
│ │ │ │  24*x_5^2+21*x_0*x_6+x_1*x_6+46*x_3*x_6+27*x_4*x_6+5*x_5*x_6+35*x_6^2+20*x_0*x_7-
│ │ │ │ @@ -65,15 +65,15 @@
│ │ │ │  
│ │ │ │  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)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 3.85067s (cpu); 2.52355s (thread); 0s (gc)
│ │ │ │ + -- used 3.72788s (cpu); 2.75047s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  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)
│ │ │ │      * isDominant(RingMap)
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_kernel_lp__Ring__Map_cm__Z__Z_rp.html
│ │ │ @@ -90,26 +90,26 @@
│ │ │  o1 : RingMap QQ[x ..x ] <-- QQ[y ..y  ]
│ │ │                   0   8          0   11
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time kernel(phi,1)
│ │ │ - -- used 0.0172216s (cpu); 0.0172186s (thread); 0s (gc)
│ │ │ + -- used 0.0214311s (cpu); 0.021431s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = ideal ()
│ │ │  
│ │ │  o2 : Ideal of QQ[y ..y  ]
│ │ │                    0   11
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time kernel(phi,2)
│ │ │ - -- used 0.848508s (cpu); 0.44358s (thread); 0s (gc)
│ │ │ + -- used 1.1128s (cpu); 0.513164s (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 {}
│ │ │ │ @@ -69,22 +69,22 @@
│ │ │ │  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 0.0172216s (cpu); 0.0172186s (thread); 0s (gc)
│ │ │ │ + -- used 0.0214311s (cpu); 0.021431s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = ideal ()
│ │ │ │  
│ │ │ │  o2 : Ideal of QQ[y ..y  ]
│ │ │ │                    0   11
│ │ │ │  i3 : time kernel(phi,2)
│ │ │ │ - -- used 0.848508s (cpu); 0.44358s (thread); 0s (gc)
│ │ │ │ + -- used 1.1128s (cpu); 0.513164s (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
│ │ │ @@ -105,15 +105,15 @@
│ │ │  o2 : Ideal of --------[x ..x ]
│ │ │                10000019  0   9
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time parametrize L
│ │ │ - -- used 0.0052255s (cpu); 0.00522062s (thread); 0s (gc)
│ │ │ + -- used 0.00599909s (cpu); 0.00599741s (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 ])
│ │ │ @@ -201,15 +201,15 @@
│ │ │  o4 : Ideal of --------[x ..x ]
│ │ │                10000019  0   9
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time parametrize Q
│ │ │ - -- used 0.556987s (cpu); 0.397597s (thread); 0s (gc)
│ │ │ + -- used 0.57734s (cpu); 0.426018s (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 {}
│ │ │ │ @@ -40,15 +40,15 @@
│ │ │ │       - 849671x  + 3034137x )
│ │ │ │                8           9
│ │ │ │  
│ │ │ │                   ZZ
│ │ │ │  o2 : Ideal of --------[x ..x ]
│ │ │ │                10000019  0   9
│ │ │ │  i3 : time parametrize L
│ │ │ │ - -- used 0.0052255s (cpu); 0.00522062s (thread); 0s (gc)
│ │ │ │ + -- used 0.00599909s (cpu); 0.00599741s (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 ])
│ │ │ │ @@ -136,15 +136,15 @@
│ │ │ │       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 0.556987s (cpu); 0.397597s (thread); 0s (gc)
│ │ │ │ + -- used 0.57734s (cpu); 0.426018s (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
│ │ │ @@ -78,15 +78,15 @@
│ │ │  
│ │ │  o1 : RationalMap (cubic rational map from 8-dimensional subvariety of PP^11 to PP^8)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time p = point source f
│ │ │ - -- used 0.44597s (cpu); 0.215856s (thread); 0s (gc)
│ │ │ + -- used 0.51082s (cpu); 0.235994s (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  
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -100,15 +100,15 @@
│ │ │                (y y  - y y  + y y  , 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 0.2051s (cpu); 0.130409s (thread); 0s (gc)
│ │ │ + -- used 0.224866s (cpu); 0.138715s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,15 +19,15 @@
│ │ │ │  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 0.44597s (cpu); 0.215856s (thread); 0s (gc)
│ │ │ │ + -- used 0.51082s (cpu); 0.235994s (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
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -40,15 +40,15 @@
│ │ │ │  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 0.2051s (cpu); 0.130409s (thread); 0s (gc)
│ │ │ │ + -- used 0.224866s (cpu); 0.138715s (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
│ │ │ @@ -89,15 +89,15 @@
│ │ │                          0   4                 0   5
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time projectiveDegrees(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0148041s (cpu); 0.0144893s (thread); 0s (gc)
│ │ │ + -- used 0.0809715s (cpu); 0.0244069s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = {1, 2, 4, 4, 2}
│ │ │  
│ │ │  o3 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -117,15 +117,15 @@
│ │ │                2 3    1 4    0 5
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time projectiveDegrees(psi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0112491s (cpu); 0.0109601s (thread); 0s (gc)
│ │ │ + -- used 0.0360115s (cpu); 0.0148336s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {2, 4, 4, 2, 1}
│ │ │  
│ │ │  o5 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │ @@ -143,25 +143,25 @@
│ │ │  o6 : RingMap ------[x ..x ] <-- ------[x ..x ]
│ │ │               300007  0   6      300007  0   6
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : time projectiveDegrees phi
│ │ │ - -- used 5.0365e-05s (cpu); 4.5084e-05s (thread); 0s (gc)
│ │ │ + -- used 6.3683e-05s (cpu); 5.4318e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {1, 2, 4, 8, 8, 4, 1}
│ │ │  
│ │ │  o7 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : time projectiveDegrees(phi,NumDegrees=>1)
│ │ │ - -- used 2.0038e-05s (cpu); 1.9887e-05s (thread); 0s (gc)
│ │ │ + -- used 2.9195e-05s (cpu); 2.8031e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = {4, 1}
│ │ │  
│ │ │  o8 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -53,15 +53,15 @@
│ │ │ │                       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)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 0.0148041s (cpu); 0.0144893s (thread); 0s (gc)
│ │ │ │ + -- used 0.0809715s (cpu); 0.0244069s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = {1, 2, 4, 4, 2}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : psi=inverseMap(toMap(phi,Dominant=>infinity))
│ │ │ │  
│ │ │ │             GF 109561[x ..x ]
│ │ │ │ @@ -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)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 0.0112491s (cpu); 0.0109601s (thread); 0s (gc)
│ │ │ │ + -- used 0.0360115s (cpu); 0.0148336s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = {2, 4, 4, 2, 1}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : -- Cremona transformation of P^6 defined by the quadrics through a
│ │ │ │  rational octic surface
│ │ │ │       phi = map specialCremonaTransformation(7,ZZ/300007)
│ │ │ │ @@ -119,21 +119,21 @@
│ │ │ │  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 5.0365e-05s (cpu); 4.5084e-05s (thread); 0s (gc)
│ │ │ │ + -- used 6.3683e-05s (cpu); 5.4318e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = {1, 2, 4, 8, 8, 4, 1}
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  i8 : time projectiveDegrees(phi,NumDegrees=>1)
│ │ │ │ - -- used 2.0038e-05s (cpu); 1.9887e-05s (thread); 0s (gc)
│ │ │ │ + -- used 2.9195e-05s (cpu); 2.8031e-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
│ │ │ @@ -88,15 +88,15 @@
│ │ │  o2 : Ideal of -----[x ..x ]
│ │ │                33331  0   6
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time phi = rationalMap(V,3,2)
│ │ │ - -- used 0.0958343s (cpu); 0.0957853s (thread); 0s (gc)
│ │ │ + -- used 0.1119s (cpu); 0.111898s (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 {}
│ │ │ │ @@ -34,15 +34,15 @@
│ │ │ │  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 0.0958343s (cpu); 0.0957853s (thread); 0s (gc)
│ │ │ │ + -- used 0.1119s (cpu); 0.111898s (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
│ │ │ @@ -111,15 +111,15 @@
│ │ │              
│ │ │                
    i5 : D = new Tally from {H => 2,C => 1};
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time phi = rationalMap D
│ │ │ - -- used 0.0288893s (cpu); 0.0288899s (thread); 0s (gc)
│ │ │ + -- used 0.0361834s (cpu); 0.0361831s (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
│ │ │ @@ -219,15 +219,15 @@
│ │ │  
│ │ │  o6 : RationalMap (cubic rational map from surface in PP^5 to PP^20)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : time ? image(phi,"F4")
│ │ │ - -- used 1.28365s (cpu); 0.786418s (thread); 0s (gc)
│ │ │ + -- used 1.79185s (cpu); 0.715829s (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 WeilDivisors, which provides general tools for working with divisors.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -40,15 +40,15 @@
│ │ │ │  
│ │ │ │  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 0.0288893s (cpu); 0.0288899s (thread); 0s (gc)
│ │ │ │ + -- used 0.0361834s (cpu); 0.0361831s (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
│ │ │ │ @@ -169,15 +169,15 @@
│ │ │ │                                                    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 1.28365s (cpu); 0.786418s (thread); 0s (gc)
│ │ │ │ + -- used 1.79185s (cpu); 0.715829s (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 _W_e_i_l_D_i_v_i_s_o_r_s, 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 1.47583s (cpu); 1.14802s (thread); 0s (gc)
│ │ │ + -- used 1.54095s (cpu); 1.17508s (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 {}
│ │ │ │ @@ -16,15 +16,15 @@
│ │ │ │              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 1.47583s (cpu); 1.14802s (thread); 0s (gc)
│ │ │ │ + -- used 1.54095s (cpu); 1.17508s (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 0.0942055s (cpu); 0.0942043s (thread); 0s (gc)
│ │ │ + -- used 0.0952488s (cpu); 0.0952487s (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
│ │ │               {
│ │ │ @@ -138,15 +138,15 @@
│ │ │  
│ │ │  o1 : RationalMap (cubic birational map from PP^6 to 6-dimensional subvariety of PP^9)
    
│ │ │              
    
│ │ │                
    i2 : time describe oo
│ │ │ - -- used 0.0185862s (cpu); 0.0185866s (thread); 0s (gc)
│ │ │ + -- used 0.0203254s (cpu); 0.0203252s (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 {}
│ │ │ │ @@ -15,15 +15,15 @@
│ │ │ │            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 0.0942055s (cpu); 0.0942043s (thread); 0s (gc)
│ │ │ │ + -- used 0.0952488s (cpu); 0.0952487s (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
│ │ │ │ @@ -323,15 +323,15 @@
│ │ │ │  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 0.0185862s (cpu); 0.0185866s (thread); 0s (gc)
│ │ │ │ + -- used 0.0203254s (cpu); 0.0203252s (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 0.0735881s (cpu); 0.0735702s (thread); 0s (gc)
│ │ │ + -- used 0.0816657s (cpu); 0.0814841s (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
│ │ │               {
│ │ │ @@ -126,15 +126,15 @@
│ │ │  
│ │ │  o1 : RationalMap (quadratic birational map from PP^8 to hypersurface in PP^9)
    
│ │ │              
    
│ │ │                
    i2 : time describe oo
│ │ │ - -- used 0.109311s (cpu); 0.029848s (thread); 0s (gc)
│ │ │ + -- used 0.11511s (cpu); 0.0358792s (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 {}
│ │ │ │ @@ -15,15 +15,15 @@
│ │ │ │            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 0.0735881s (cpu); 0.0735702s (thread); 0s (gc)
│ │ │ │ + -- used 0.0816657s (cpu); 0.0814841s (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
│ │ │ │ @@ -78,15 +78,15 @@
│ │ │ │                                                     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 0.109311s (cpu); 0.029848s (thread); 0s (gc)
│ │ │ │ + -- used 0.11511s (cpu); 0.0358792s (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
│ │ │ @@ -88,23 +88,23 @@
│ │ │  
│ │ │  o3 = 6927
    
│ │ │              
    
│ │ │                
    i4 : time phi' = value str;
│ │ │ - -- used 0.0222424s (cpu); 0.0222414s (thread); 0s (gc)
│ │ │ + -- used 0.0453571s (cpu); 0.0453573s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : RationalMap (cubic birational map from PP^3 to hypersurface in PP^4)
    
│ │ │              
    
│ │ │                
    i5 : time describe phi'
│ │ │ - -- used 0.00518727s (cpu); 0.00518762s (thread); 0s (gc)
│ │ │ + -- used 0.0106974s (cpu); 0.010701s (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}
│ │ │ @@ -113,15 +113,15 @@
│ │ │       degree base locus: 5
│ │ │       coefficient ring: ZZ/33331
    
│ │ │              
    
│ │ │                
    i6 : time describe inverse phi'
│ │ │ - -- used 0.00429438s (cpu); 0.00429504s (thread); 0s (gc)
│ │ │ + -- used 0.00870284s (cpu); 0.00870936s (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 {}
│ │ │ │ @@ -19,32 +19,32 @@
│ │ │ │  
│ │ │ │  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 0.0222424s (cpu); 0.0222414s (thread); 0s (gc)
│ │ │ │ + -- used 0.0453571s (cpu); 0.0453573s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : RationalMap (cubic birational map from PP^3 to hypersurface in PP^4)
│ │ │ │  i5 : time describe phi'
│ │ │ │ - -- used 0.00518727s (cpu); 0.00518762s (thread); 0s (gc)
│ │ │ │ + -- used 0.0106974s (cpu); 0.010701s (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 0.00429438s (cpu); 0.00429504s (thread); 0s (gc)
│ │ │ │ + -- used 0.00870284s (cpu); 0.00870936s (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
│ │ │ @@ -58,29 +58,29 @@
│ │ │              
    		
│ │ │                
    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 0.00444651s (cpu); 0.00444181s (thread); 0s (gc)
│ │ │ + -- used 0.00563649s (cpu); 0.00563363s (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 0.139982s (cpu); 0.0709074s (thread); 0s (gc)
│ │ │ + -- used 0.183936s (cpu); 0.0848219s (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
│ │ │  
│ │ │ @@ -88,43 +88,43 @@
│ │ │  o3 : Ideal of ------[x ..x ]
│ │ │                300007  0   9
    
│ │ │              
    
│ │ │                
    i4 : time degreeMap phi
│ │ │ - -- used 0.0273929s (cpu); 0.0273949s (thread); 0s (gc)
│ │ │ + -- used 0.0372504s (cpu); 0.0372589s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
    
│ │ │              
    
│ │ │                
    i5 : time projectiveDegrees phi
│ │ │ - -- used 0.660336s (cpu); 0.467597s (thread); 0s (gc)
│ │ │ + -- used 0.655005s (cpu); 0.4886s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {1, 3, 9, 17, 21, 15, 5}
│ │ │  
│ │ │  o5 : List
    
│ │ │              
    
│ │ │                
    i6 : time projectiveDegrees(phi,NumDegrees=>0)
│ │ │ - -- used 0.061525s (cpu); 0.0615308s (thread); 0s (gc)
│ │ │ + -- used 0.0713962s (cpu); 0.0714033s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = {5}
│ │ │  
│ │ │  o6 : List
    
│ │ │              
    
│ │ │                
    i7 : time phi = toMap(phi,Dominant=>J)
│ │ │ - -- used 0.00224979s (cpu); 0.00225137s (thread); 0s (gc)
│ │ │ + -- used 0.00261791s (cpu); 0.0026234s (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
│ │ │ @@ -136,15 +136,15 @@
│ │ │               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 0.464618s (cpu); 0.385971s (thread); 0s (gc)
│ │ │ + -- used 0.425047s (cpu); 0.424849s (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
│ │ │ @@ -156,44 +156,44 @@
│ │ │               (x x  - x x  + x x , 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 0.00957768s (cpu); 0.00958237s (thread); 0s (gc)
│ │ │ + -- used 0.0109158s (cpu); 0.0109172s (thread); 0s (gc)
│ │ │  
│ │ │  o9 = true
    
│ │ │              
    
│ │ │                
    i10 : time degreeMap psi
│ │ │ - -- used 0.36608s (cpu); 0.250674s (thread); 0s (gc)
│ │ │ + -- used 0.531179s (cpu); 0.272942s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 1
    
│ │ │              
    
│ │ │                
    i11 : time projectiveDegrees psi
│ │ │ - -- used 5.12293s (cpu); 4.38974s (thread); 0s (gc)
│ │ │ + -- used 5.68627s (cpu); 5.2618s (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.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -53,15 +53,15 @@
│ │ │ │        2
│ │ │ │  o6 = y T
│ │ │ │          2
│ │ │ │  
│ │ │ │  o6 : R[T ..T ]
│ │ │ │          1   3
│ │ │ │  i7 : H = HH(KR)
│ │ │ │ -Finding easy relations           :  -- used 0.0174708s (cpu); 0.016392s
│ │ │ │ +Finding easy relations           :  -- used 0.036787s (cpu); 0.0196623s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = H
│ │ │ │  
│ │ │ │  o7 : PolynomialRing, 3 skew commutative variable(s)
│ │ │ │  i8 : homologyClass(KR,z1*z2)
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_homology__Module.html
│ │ │ @@ -129,15 +129,15 @@
│ │ │  
│ │ │  o5 : Complex
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i12 : time phi = rationalMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
│ │ │ - -- used 0.00214853s (cpu); 0.00214909s (thread); 0s (gc)
│ │ │ + -- used 0.00258076s (cpu); 0.00258318s (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 ])
│ │ │ @@ -242,15 +242,15 @@
│ │ │  
│ │ │  o12 : RationalMap (cubic rational map from PP^6 to PP^9)
    
│ │ │              
    
│ │ │                
    i13 : time phi = rationalMap(phi,Dominant=>2)
│ │ │ - -- used 0.159869s (cpu); 0.0819959s (thread); 0s (gc)
│ │ │ + -- used 0.179519s (cpu); 0.0970863s (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
│ │ │ @@ -315,15 +315,15 @@
│ │ │  
│ │ │  o13 : RationalMap (cubic rational map from PP^6 to 6-dimensional subvariety of PP^9)
    
│ │ │              
    
│ │ │                
    i14 : time phi^(-1)
│ │ │ - -- used 0.494847s (cpu); 0.414351s (thread); 0s (gc)
│ │ │ + -- used 0.464234s (cpu); 0.463995s (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 ,
│ │ │ @@ -376,49 +376,49 @@
│ │ │  
│ │ │  o14 : RationalMap (cubic birational map from 6-dimensional subvariety of PP^9 to PP^6)
    
│ │ │              
    
│ │ │                
    i15 : time degrees phi^(-1)
│ │ │ - -- used 0.343614s (cpu); 0.269496s (thread); 0s (gc)
│ │ │ + -- used 0.456608s (cpu); 0.345678s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = {5, 15, 21, 17, 9, 3, 1}
│ │ │  
│ │ │  o15 : List
    
│ │ │              
    
│ │ │                
    i16 : time degrees phi
│ │ │ - -- used 0.017066s (cpu); 0.0167813s (thread); 0s (gc)
│ │ │ + -- used 0.0438542s (cpu); 0.0238686s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = {1, 3, 9, 17, 21, 15, 5}
│ │ │  
│ │ │  o16 : List
    
│ │ │              
    
│ │ │                
    i17 : time describe phi
│ │ │ - -- used 0.00287351s (cpu); 0.00287412s (thread); 0s (gc)
│ │ │ + -- used 0.00408934s (cpu); 0.00409592s (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 0.00949367s (cpu); 0.00949464s (thread); 0s (gc)
│ │ │ + -- used 0.0114385s (cpu); 0.0114461s (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}
│ │ │ @@ -427,41 +427,41 @@
│ │ │        degree base locus: 24
│ │ │        coefficient ring: ZZ/300007
    
│ │ │              
    
│ │ │                
    i19 : time (f,g) = graph phi^-1; f;
│ │ │ - -- used 0.00910182s (cpu); 0.00910265s (thread); 0s (gc)
│ │ │ + -- used 0.0116988s (cpu); 0.0117077s (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 1.2811s (cpu); 0.962879s (thread); 0s (gc)
│ │ │ + -- used 1.2095s (cpu); 1.00842s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = {904, 508, 268, 130, 56, 20, 5}
│ │ │  
│ │ │  o21 : List
    
│ │ │              
    
│ │ │                
    i22 : time degree f
│ │ │ - -- used 1.588e-05s (cpu); 1.5509e-05s (thread); 0s (gc)
│ │ │ + -- used 1.5338e-05s (cpu); 1.4902e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = 1
    
│ │ │              
    
│ │ │                
    i23 : time describe f
│ │ │ - -- used 0.00176246s (cpu); 0.00176335s (thread); 0s (gc)
│ │ │ + -- used 0.00169313s (cpu); 0.00169834s (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 0.00444651s (cpu); 0.00444181s (thread); 0s (gc)
│ │ │ │ + -- used 0.00563649s (cpu); 0.00563363s (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 0.139982s (cpu); 0.0709074s (thread); 0s (gc)
│ │ │ │ + -- used 0.183936s (cpu); 0.0848219s (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 0.0273929s (cpu); 0.0273949s (thread); 0s (gc)
│ │ │ │ + -- used 0.0372504s (cpu); 0.0372589s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 1
│ │ │ │  i5 : time projectiveDegrees phi
│ │ │ │ - -- used 0.660336s (cpu); 0.467597s (thread); 0s (gc)
│ │ │ │ + -- used 0.655005s (cpu); 0.4886s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = {1, 3, 9, 17, 21, 15, 5}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : time projectiveDegrees(phi,NumDegrees=>0)
│ │ │ │ - -- used 0.061525s (cpu); 0.0615308s (thread); 0s (gc)
│ │ │ │ + -- used 0.0713962s (cpu); 0.0714033s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = {5}
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : time phi = toMap(phi,Dominant=>J)
│ │ │ │ - -- used 0.00224979s (cpu); 0.00225137s (thread); 0s (gc)
│ │ │ │ + -- used 0.00261791s (cpu); 0.0026234s (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 0.464618s (cpu); 0.385971s (thread); 0s (gc)
│ │ │ │ + -- used 0.425047s (cpu); 0.424849s (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 0.00957768s (cpu); 0.00958237s (thread); 0s (gc)
│ │ │ │ + -- used 0.0109158s (cpu); 0.0109172s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 = true
│ │ │ │  i10 : time degreeMap psi
│ │ │ │ - -- used 0.36608s (cpu); 0.250674s (thread); 0s (gc)
│ │ │ │ + -- used 0.531179s (cpu); 0.272942s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = 1
│ │ │ │  i11 : time projectiveDegrees psi
│ │ │ │ - -- used 5.12293s (cpu); 4.38974s (thread); 0s (gc)
│ │ │ │ + -- used 5.68627s (cpu); 5.2618s (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 0.00214853s (cpu); 0.00214909s (thread); 0s (gc)
│ │ │ │ + -- used 0.00258076s (cpu); 0.00258318s (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 0.159869s (cpu); 0.0819959s (thread); 0s (gc)
│ │ │ │ + -- used 0.179519s (cpu); 0.0970863s (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 0.494847s (cpu); 0.414351s (thread); 0s (gc)
│ │ │ │ + -- used 0.464234s (cpu); 0.463995s (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 0.343614s (cpu); 0.269496s (thread); 0s (gc)
│ │ │ │ + -- used 0.456608s (cpu); 0.345678s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = {5, 15, 21, 17, 9, 3, 1}
│ │ │ │  
│ │ │ │  o15 : List
│ │ │ │  i16 : time degrees phi
│ │ │ │ - -- used 0.017066s (cpu); 0.0167813s (thread); 0s (gc)
│ │ │ │ + -- used 0.0438542s (cpu); 0.0238686s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = {1, 3, 9, 17, 21, 15, 5}
│ │ │ │  
│ │ │ │  o16 : List
│ │ │ │  i17 : time describe phi
│ │ │ │ - -- used 0.00287351s (cpu); 0.00287412s (thread); 0s (gc)
│ │ │ │ + -- used 0.00408934s (cpu); 0.00409592s (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 0.00949367s (cpu); 0.00949464s (thread); 0s (gc)
│ │ │ │ + -- used 0.0114385s (cpu); 0.0114461s (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 0.00910182s (cpu); 0.00910265s (thread); 0s (gc)
│ │ │ │ + -- used 0.0116988s (cpu); 0.0117077s (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 1.2811s (cpu); 0.962879s (thread); 0s (gc)
│ │ │ │ + -- used 1.2095s (cpu); 1.00842s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o21 = {904, 508, 268, 130, 56, 20, 5}
│ │ │ │  
│ │ │ │  o21 : List
│ │ │ │  i22 : time degree f
│ │ │ │ - -- used 1.588e-05s (cpu); 1.5509e-05s (thread); 0s (gc)
│ │ │ │ + -- used 1.5338e-05s (cpu); 1.4902e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o22 = 1
│ │ │ │  i23 : time describe f
│ │ │ │ - -- used 0.00176246s (cpu); 0.00176335s (thread); 0s (gc)
│ │ │ │ + -- used 0.00169313s (cpu); 0.00169834s (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/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
│ │ │ -Finding easy relations           :  -- used 0.0144857s (cpu); 0.0137147s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0435945s (cpu); 0.0200431s (thread); 0s (gc)
│ │ │  
│ │ │                            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
│ │ │ -Finding easy relations           :  -- used 0.0271647s (cpu); 0.0257159s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.21148s (cpu); 0.0458148s (thread); 0s (gc)
│ │ │  
│ │ │  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)
│ │ │ -Finding easy relations           :  -- used 0.0189381s (cpu); 0.0176869s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0479131s (cpu); 0.0263491s (thread); 0s (gc)
│ │ │  
│ │ │         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 : ComplexMap
│ │ │  
│ │ │  i7 : HHg = HH g
│ │ │ -Finding easy relations           :  -- used 0.014071s (cpu); 0.0132964s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0621134s (cpu); 0.0228966s (thread); 0s (gc)
│ │ │  
│ │ │                           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
│ │ │ @@ -49,15 +49,15 @@
│ │ │                                1                                                             {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a |     3
│ │ │                                                                                      
│ │ │                                                                                     2
│ │ │  
│ │ │  o6 : Complex
│ │ │  
│ │ │  i7 : HKR = HH KR
│ │ │ -Finding easy relations           :  -- used 0.125504s (cpu); 0.0655203s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.221245s (cpu); 0.0628396s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = HKR
│ │ │  
│ │ │  o7 : PolynomialRing, 4 skew commutative variable(s)
│ │ │  
│ │ │  i8 : ideal HKR
│ │ │  
│ │ │ @@ -68,15 +68,15 @@
│ │ │  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'
│ │ │ -Finding easy relations           :  -- used 0.578089s (cpu); 0.496293s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.658573s (cpu); 0.645433s (thread); 0s (gc)
│ │ │  
│ │ │  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)
│ │ │ -Finding easy relations           :  -- used 0.0244722s (cpu); 0.0208419s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0347788s (cpu); 0.0215344s (thread); 0s (gc)
│ │ │  
│ │ │  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)
│ │ │ -Finding easy relations           :  -- used 0.0182011s (cpu); 0.0174142s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.191531s (cpu); 0.04755s (thread); 0s (gc)
│ │ │  
│ │ │  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)
│ │ │ -Finding easy relations           :  -- used 0.0854845s (cpu); 0.082415s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.290394s (cpu); 0.121929s (thread); 0s (gc)
│ │ │  
│ │ │  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)
│ │ │ -Finding easy relations           :  -- used 0.053164s (cpu); 0.0517839s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.12797s (cpu); 0.0658882s (thread); 0s (gc)
│ │ │  
│ │ │  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)
│ │ │ -Finding easy relations           :  -- used 0.018217s (cpu); 0.0174421s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0816455s (cpu); 0.0286737s (thread); 0s (gc)
│ │ │  
│ │ │  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)
│ │ │ -Finding easy relations           :  -- used 0.0174708s (cpu); 0.016392s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.036787s (cpu); 0.0196623s (thread); 0s (gc)
│ │ │  
│ │ │  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 : Complex
│ │ │  
│ │ │  i6 : HKR = HH(KR)
│ │ │ - -- used 0.217734s (cpu); 0.156685s (thread); 0s (gc)
│ │ │ + -- used 0.250617s (cpu); 0.148683s (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)
│ │ │ -Finding easy relations           :  -- used 0.540699s (cpu); 0.470132s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.749341s (cpu); 0.620158s (thread); 0s (gc)
│ │ │  
│ │ │               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)
│ │ │ -Finding easy relations           :  -- used 0.248008s (cpu); 0.24363s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.172755s (cpu); 0.158389s (thread); 0s (gc)
│ │ │  
│ │ │  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 0.47227s (cpu); 0.407795s (thread); 0s (gc)
│ │ │ + -- used 0.713061s (cpu); 0.557687s (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
│ │ │ @@ -289,15 +289,15 @@
│ │ │          
    
│ │ │            
    One can also obtain the map on homology induced by a DGAlgebra map.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i17 : HHg = HH g
│ │ │ -Finding easy relations           :  -- used 0.0144857s (cpu); 0.0137147s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0435945s (cpu); 0.0200431s (thread); 0s (gc)
│ │ │  
│ │ │                            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
│ │ │ │ -Finding easy relations           :  -- used 0.0144857s (cpu); 0.0137147s
│ │ │ │ +Finding easy relations           :  -- used 0.0435945s (cpu); 0.0200431s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │                            ZZ
│ │ │ │                           ---[a..c]
│ │ │ │              ZZ           101
│ │ │ │  o17 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │ │             101  1   2           3   1     1
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___Basic_spoperations_spon_sp__D__G_sp__Algebras.html
│ │ │ @@ -113,15 +113,15 @@
│ │ │          
    
│ │ │            
    One can compute the homology algebra of a DGAlgebra using the homology (or HH) command.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -174,15 +174,15 @@
│ │ │  
│ │ │  o9 : DGAlgebra
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── 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
│ │ │ │ -Finding easy relations           :  -- used 0.0271647s (cpu); 0.0257159s
│ │ │ │ +Finding easy relations           :  -- used 0.21148s (cpu); 0.0458148s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = HB
│ │ │ │  
│ │ │ │  o5 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  i6 : describe HB
│ │ │ │  
│ │ │ │ @@ -87,15 +87,15 @@
│ │ │ │                                 1   5
│ │ │ │                                      2
│ │ │ │        Differential => {a, b, c, d, a T }
│ │ │ │                                        1
│ │ │ │  
│ │ │ │  o9 : DGAlgebra
│ │ │ │  i10 : homologyAlgebra(C,GenDegreeLimit=>4,RelDegreeLimit=>4)
│ │ │ │ -Finding easy relations           :  -- used 0.0189381s (cpu); 0.0176869s
│ │ │ │ +Finding easy relations           :  -- used 0.0479131s (cpu); 0.0263491s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │         ZZ
│ │ │ │  o10 = ---[X ..X ]
│ │ │ │        101  1   3
│ │ │ │  
│ │ │ │  o10 : PolynomialRing, 3 skew commutative variable(s)
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___H__H_sp__D__G__Algebra__Map.html
│ │ │ @@ -144,15 +144,15 @@
│ │ │  
│ │ │  o6 : ComplexMap
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -242,15 +242,15 @@
│ │ │  
│ │ │  o15 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i5 : HB = HH B
│ │ │ -Finding easy relations           :  -- used 0.0271647s (cpu); 0.0257159s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.21148s (cpu); 0.0458148s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = HB
│ │ │  
│ │ │  o5 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i10 : homologyAlgebra(C,GenDegreeLimit=>4,RelDegreeLimit=>4)
│ │ │ -Finding easy relations           :  -- used 0.0189381s (cpu); 0.0176869s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0479131s (cpu); 0.0263491s (thread); 0s (gc)
│ │ │  
│ │ │         ZZ
│ │ │  o10 = ---[X ..X ]
│ │ │        101  1   3
│ │ │  
│ │ │  o10 : PolynomialRing, 3 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i7 : HHg = HH g
│ │ │ -Finding easy relations           :  -- used 0.014071s (cpu); 0.0132964s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0621134s (cpu); 0.0228966s (thread); 0s (gc)
│ │ │  
│ │ │                           ZZ
│ │ │                          ---[a..c]
│ │ │             ZZ           101
│ │ │  o7 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │            101  1   2           3   1     1
│ │ │                         (c, b, a )
│ │ │ ├── html2text {}
│ │ │ │ @@ -62,15 +62,15 @@
│ │ │ │       2 : R  <------------- R  : 2
│ │ │ │                 {2} | 0 |
│ │ │ │                 {2} | 0 |
│ │ │ │                 {2} | 1 |
│ │ │ │  
│ │ │ │  o6 : ComplexMap
│ │ │ │  i7 : HHg = HH g
│ │ │ │ -Finding easy relations           :  -- used 0.014071s (cpu); 0.0132964s
│ │ │ │ +Finding easy relations           :  -- used 0.0621134s (cpu); 0.0228966s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │                           ZZ
│ │ │ │                          ---[a..c]
│ │ │ │             ZZ           101
│ │ │ │  o7 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │ │            101  1   2           3   1     1
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___The_sp__Koszul_spcomplex_spas_spa_sp__D__G_sp__Algebra.html
│ │ │ @@ -138,15 +138,15 @@
│ │ │          
    
│ │ │            
    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.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -166,15 +166,15 @@
│ │ │  
│ │ │  o9 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -76,15 +76,15 @@
│ │ │ │  
│ │ │ │  o6 : Complex
│ │ │ │  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
│ │ │ │ -Finding easy relations           :  -- used 0.125504s (cpu); 0.0655203s
│ │ │ │ +Finding easy relations           :  -- used 0.221245s (cpu); 0.0628396s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = HKR
│ │ │ │  
│ │ │ │  o7 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  i8 : ideal HKR
│ │ │ │  
│ │ │ │ @@ -94,15 +94,15 @@
│ │ │ │  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'
│ │ │ │ -Finding easy relations           :  -- used 0.578089s (cpu); 0.496293s
│ │ │ │ +Finding easy relations           :  -- used 0.658573s (cpu); 0.645433s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = HKR'
│ │ │ │  
│ │ │ │  o10 : QuotientRing
│ │ │ │  i11 : numgens HKR'
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_cycles.html
│ │ │ @@ -89,15 +89,15 @@
│ │ │  
│ │ │  o3 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,15 +23,15 @@
│ │ │ │  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)
│ │ │ │ -Finding easy relations           :  -- used 0.0244722s (cpu); 0.0208419s
│ │ │ │ +Finding easy relations           :  -- used 0.0347788s (cpu); 0.0215344s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = HA
│ │ │ │  
│ │ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  i5 : numgens HA
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_homology__Algebra.html
│ │ │ @@ -103,15 +103,15 @@
│ │ │  
│ │ │  o3 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : HKR = HH KR
│ │ │ -Finding easy relations           :  -- used 0.125504s (cpu); 0.0655203s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.221245s (cpu); 0.0628396s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = HKR
│ │ │  
│ │ │  o7 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i10 : HKR' = HH koszulComplexDGA R'
│ │ │ -Finding easy relations           :  -- used 0.578089s (cpu); 0.496293s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.658573s (cpu); 0.645433s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = HKR'
│ │ │  
│ │ │  o10 : QuotientRing
    
│ │ │              
    
│ │ │                
    i4 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.0244722s (cpu); 0.0208419s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0347788s (cpu); 0.0215344s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = HA
│ │ │  
│ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i4 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.0182011s (cpu); 0.0174142s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.191531s (cpu); 0.04755s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = HA
│ │ │  
│ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │ @@ -148,15 +148,15 @@
│ │ │  
│ │ │  o7 : List
    
│ │ │              
    
│ │ │                
    i8 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.0854845s (cpu); 0.082415s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.290394s (cpu); 0.121929s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = HA
│ │ │  
│ │ │  o8 : QuotientRing
    
│ │ │              
    
│ │ │                
    i16 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.053164s (cpu); 0.0517839s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.12797s (cpu); 0.0658882s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = HA
│ │ │  
│ │ │  o16 : QuotientRing
    
│ │ │              
    
│ │ │ @@ -302,15 +302,15 @@
│ │ │  
│ │ │  o20 : DGAlgebra
    
│ │ │              
    
│ │ │                
    i21 : HB = homologyAlgebra(B,GenDegreeLimit=>7,RelDegreeLimit=>14)
│ │ │ -Finding easy relations           :  -- used 0.018217s (cpu); 0.0174421s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0816455s (cpu); 0.0286737s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = HB
│ │ │  
│ │ │  o21 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -33,16 +33,16 @@
│ │ │ │  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)
│ │ │ │ -Finding easy relations           :  -- used 0.0182011s (cpu); 0.0174142s
│ │ │ │ -(thread); 0s (gc)
│ │ │ │ +Finding easy relations           :  -- used 0.191531s (cpu); 0.04755s (thread);
│ │ │ │ +0s (gc)
│ │ │ │  
│ │ │ │  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)
│ │ │ │ -Finding easy relations           :  -- used 0.0854845s (cpu); 0.082415s
│ │ │ │ +Finding easy relations           :  -- used 0.290394s (cpu); 0.121929s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = HA
│ │ │ │  
│ │ │ │  o8 : QuotientRing
│ │ │ │  i9 : numgens HA
│ │ │ │  
│ │ │ │ @@ -122,15 +122,15 @@
│ │ │ │  o14 : DGAlgebra
│ │ │ │  i15 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │ │  
│ │ │ │  o15 = {1, 7, 7, 1}
│ │ │ │  
│ │ │ │  o15 : List
│ │ │ │  i16 : HA = homologyAlgebra(A)
│ │ │ │ -Finding easy relations           :  -- used 0.053164s (cpu); 0.0517839s
│ │ │ │ +Finding easy relations           :  -- used 0.12797s (cpu); 0.0658882s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  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
│ │ │ │ @@ -158,15 +158,15 @@
│ │ │ │  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)
│ │ │ │ -Finding easy relations           :  -- used 0.018217s (cpu); 0.0174421s
│ │ │ │ +Finding easy relations           :  -- used 0.0816455s (cpu); 0.0286737s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o21 = HB
│ │ │ │  
│ │ │ │  o21 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  ********** WWaayyss ttoo uussee hhoommoollooggyyAAllggeebbrraa:: **********
│ │ │ │      * homologyAlgebra(DGAlgebra)
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_homology__Class.html
│ │ │ @@ -135,15 +135,15 @@
│ │ │  o6 : R[T ..T ]
│ │ │          1   3
    
│ │ │              
    
│ │ │                
    i7 : H = HH(KR)
│ │ │ -Finding easy relations           :  -- used 0.0174708s (cpu); 0.016392s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.036787s (cpu); 0.0196623s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = H
│ │ │  
│ │ │  o7 : PolynomialRing, 3 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i6 : HKR = HH(KR)
│ │ │ - -- used 0.217734s (cpu); 0.156685s (thread); 0s (gc)
│ │ │ + -- used 0.250617s (cpu); 0.148683s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o6 = HKR
│ │ │  
│ │ │  o6 : QuotientRing
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -54,15 +54,15 @@
│ │ │ │        1      4      6      4      1
│ │ │ │  o5 = R  <-- R  <-- R  <-- R  <-- R
│ │ │ │  
│ │ │ │       0      1      2      3      4
│ │ │ │  
│ │ │ │  o5 : Complex
│ │ │ │  i6 : HKR = HH(KR)
│ │ │ │ - -- used 0.217734s (cpu); 0.156685s (thread); 0s (gc)
│ │ │ │ + -- used 0.250617s (cpu); 0.148683s (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
│ │ │ @@ -192,15 +192,15 @@
│ │ │          
    
│ │ │            
    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:
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -49,16 +49,16 @@
│ │ │ │        Underlying algebra => R[T ..T ]
│ │ │ │                                 1   4
│ │ │ │        Differential => {t , t , t , t }
│ │ │ │                          1   2   3   4
│ │ │ │  
│ │ │ │  o4 : DGAlgebra
│ │ │ │  i5 : H = HH(KR)
│ │ │ │ -Finding easy relations           :  -- used 0.248008s (cpu); 0.24363s (thread);
│ │ │ │ -0s (gc)
│ │ │ │ +Finding easy relations           :  -- used 0.172755s (cpu); 0.158389s
│ │ │ │ +(thread); 0s (gc)
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  o3 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,15 +27,15 @@
│ │ │ │  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 0.47227s (cpu); 0.407795s (thread); 0s (gc)
│ │ │ │ + -- used 0.713061s (cpu); 0.557687s (thread); 0s (gc)
│ │ │ │  Finding easy relations           :
│ │ │ │  o4 = HB
│ │ │ │  
│ │ │ │  o4 : QuotientRing
│ │ │ │  i5 : numgens HB
│ │ │ │  
│ │ │ │  o5 = 35
│ │ ├── ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_delete__Edges.out
│ │ │ @@ -14,15 +14,15 @@
│ │ │  
│ │ │  o3 = {{a, b}, {d, e}}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : gprime = deleteEdges (g,T)
│ │ │  
│ │ │ -o4 = HyperGraph{"edges" => {{c, d}, {b, c}, {a, e}}}
│ │ │ +o4 = HyperGraph{"edges" => {{b, c}, {a, e}, {c, d}}}
│ │ │                  "ring" => S
│ │ │                  "vertices" => {a, b, c, d, e}
│ │ │  
│ │ │  o4 : HyperGraph
│ │ │  
│ │ │  i5 : h = hyperGraph {a*b*c,c*d*e,a*e}
│ │ ├── ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_random__Hyper__Graph.out
│ │ │ @@ -3,19 +3,27 @@
│ │ │  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 }}}
│ │ │ -                              3   4   5     1   4     1   2   3   5
│ │ │ +                              1   2   5     2   3     1   3   4   5
│ │ │                  "ring" => R
│ │ │                  "vertices" => {x , x , x , x , x }
│ │ │                                  1   2   3   4   5
│ │ │  
│ │ │  o3 : HyperGraph
│ │ │  
│ │ │  i4 : randomHyperGraph(R,{3,2,4})
│ │ │  
│ │ │ +o4 = HyperGraph{"edges" => {{x , x , x }, {x , x }, {x , x , x , x }}}
│ │ │ +                              1   4   5     2   5     1   2   3   4
│ │ │ +                "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/_delete__Edges.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │  o3 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,15 +26,15 @@
│ │ │ │  i3 : T = {{a,b},{d,e}}
│ │ │ │  
│ │ │ │  o3 = {{a, b}, {d, e}}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : gprime = deleteEdges (g,T)
│ │ │ │  
│ │ │ │ -o4 = HyperGraph{"edges" => {{c, d}, {b, c}, {a, e}}}
│ │ │ │ +o4 = HyperGraph{"edges" => {{b, c}, {a, e}, {c, d}}}
│ │ │ │                  "ring" => S
│ │ │ │                  "vertices" => {a, b, c, d, e}
│ │ │ │  
│ │ │ │  o4 : HyperGraph
│ │ │ │  i5 : h = hyperGraph {a*b*c,c*d*e,a*e}
│ │ │ │  
│ │ │ │  o5 = HyperGraph{"edges" => {{a, b, c}, {a, e}, {c, d, e}}}
│ │ ├── ./usr/share/doc/Macaulay2/EdgeIdeals/html/_random__Hyper__Graph.html
│ │ │ @@ -88,25 +88,33 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,21 +25,29 @@
│ │ │ │  _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 }}}
│ │ │ │ -                              3   4   5     1   4     1   2   3   5
│ │ │ │ +                              1   2   5     2   3     1   3   4   5
│ │ │ │                  "ring" => R
│ │ │ │                  "vertices" => {x , x , x , x , x }
│ │ │ │                                  1   2   3   4   5
│ │ │ │  
│ │ │ │  o3 : HyperGraph
│ │ │ │  i4 : randomHyperGraph(R,{3,2,4})
│ │ │ │ +
│ │ │ │ +o4 = HyperGraph{"edges" => {{x , x , x }, {x , x }, {x , x , x , x }}}
│ │ │ │ +                              1   4   5     2   5     1   2   3   4
│ │ │ │ +                "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/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;
│ │ │ - -- .221824s elapsed
│ │ │ + -- .228525s elapsed
│ │ │  
│ │ │  i4 : #sols -- 156 solutions
│ │ │  
│ │ │  o4 = 156
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/EigenSolver/html/index.html
│ │ │ @@ -80,15 +80,15 @@
│ │ │  
│ │ │  o2 : Ideal of QQ[a..f]
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,22 +29,22 @@
│ │ │ │  i3 : g = x^2+c*x+d
│ │ │ │  
│ │ │ │        2
│ │ │ │  o3 = x  + x*c + d
│ │ │ │  
│ │ │ │  o3 : R
│ │ │ │  i4 : time eliminate(x,ideal(f,g))
│ │ │ │ - -- used 0.00273025s (cpu); 0.0027266s (thread); 0s (gc)
│ │ │ │ + -- used 0.00322795s (cpu); 0.00322707s (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 0.00165107s (cpu); 0.00165116s (thread); 0s (gc)
│ │ │ │ + -- used 0.00178641s (cpu); 0.00178856s (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
│ │ │ @@ -97,26 +97,26 @@
│ │ │  
│ │ │  o3 : R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── 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 0.00260364s (cpu); 0.00260073s (thread); 0s (gc)
│ │ │ │ + -- used 0.00306018s (cpu); 0.00305735s (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 0.00151871s (cpu); 0.00151973s (thread); 0s (gc)
│ │ │ │ + -- used 0.00180994s (cpu); 0.0018108s (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
│ │ │ @@ -105,15 +105,15 @@
│ │ │  
│ │ │  o3 : R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i10 : z123 = masseyTripleProduct(KR,z1,z2,z3)
│ │ │ -Finding easy relations           :  -- used 0.540699s (cpu); 0.470132s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.749341s (cpu); 0.620158s (thread); 0s (gc)
│ │ │  
│ │ │               2
│ │ │  o10 = x x y z T T T T
│ │ │         1 2 2   2 3 4 5
│ │ │  
│ │ │  o10 : R[T ..T ]
│ │ │           1   5
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -90,15 +90,15 @@
│ │ │ │  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)
│ │ │ │ -Finding easy relations           :  -- used 0.540699s (cpu); 0.470132s
│ │ │ │ +Finding easy relations           :  -- used 0.749341s (cpu); 0.620158s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │               2
│ │ │ │  o10 = x x y z T T T T
│ │ │ │         1 2 2   2 3 4 5
│ │ │ │  
│ │ │ │  o10 : R[T ..T ]
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_massey__Triple__Product_lp__D__G__Algebra_cm__Z__Z_cm__Z__Z_cm__Z__Z_rp.html
│ │ │ @@ -119,15 +119,15 @@
│ │ │  
│ │ │  o4 : DGAlgebra
│ │ │              
    
│ │ │                
    i5 : H = HH(KR)
│ │ │ -Finding easy relations           :  -- used 0.248008s (cpu); 0.24363s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.172755s (cpu); 0.158389s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = H
│ │ │  
│ │ │  o5 : QuotientRing
    
│ │ │              
    
│ │ │                
    i4 : HB = torAlgebra(R,S,GenDegreeLimit=>4,RelDegreeLimit=>8)
│ │ │ - -- used 0.47227s (cpu); 0.407795s (thread); 0s (gc)
│ │ │ + -- used 0.713061s (cpu); 0.557687s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o4 = HB
│ │ │  
│ │ │  o4 : QuotientRing
    
│ │ │              
    
│ │ │                
    i4 : gprime = deleteEdges (g,T)
│ │ │  
│ │ │ -o4 = HyperGraph{"edges" => {{c, d}, {b, c}, {a, e}}}
│ │ │ +o4 = HyperGraph{"edges" => {{b, c}, {a, e}, {c, d}}}
│ │ │                  "ring" => S
│ │ │                  "vertices" => {a, b, c, d, e}
│ │ │  
│ │ │  o4 : HyperGraph
    
│ │ │              
    
│ │ │                
    i3 : randomHyperGraph(R,{3,2,4})
│ │ │  
│ │ │  o3 = HyperGraph{"edges" => {{x , x , x }, {x , x }, {x , x , x , x }}}
│ │ │ -                              3   4   5     1   4     1   2   3   5
│ │ │ +                              1   2   5     2   3     1   3   4   5
│ │ │                  "ring" => R
│ │ │                  "vertices" => {x , x , x , x , x }
│ │ │                                  1   2   3   4   5
│ │ │  
│ │ │  o3 : HyperGraph
    
│ │ │              
    
│ │ │ -              
    i4 : randomHyperGraph(R,{3,2,4})
    
│ │ │ +              
    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
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime sols = zeroDimSolve I;
│ │ │ - -- .221824s elapsed
    
│ │ │ + -- .228525s 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;
│ │ │ │ - -- .221824s elapsed
│ │ │ │ + -- .228525s 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/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 0.00273025s (cpu); 0.0027266s (thread); 0s (gc)
│ │ │ + -- used 0.00322795s (cpu); 0.00322707s (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 0.00165107s (cpu); 0.00165116s (thread); 0s (gc)
│ │ │ + -- used 0.00178641s (cpu); 0.00178856s (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 0.00260364s (cpu); 0.00260073s (thread); 0s (gc)
│ │ │ + -- used 0.00306018s (cpu); 0.00305735s (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 0.00151871s (cpu); 0.00151973s (thread); 0s (gc)
│ │ │ + -- used 0.00180994s (cpu); 0.0018108s (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 1.58822s (cpu); 1.34056s (thread); 0s (gc)
│ │ │ + -- used 1.49944s (cpu); 1.33922s (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 0.016203s (cpu); 0.0162067s (thread); 0s (gc)
│ │ │ + -- used 0.0165153s (cpu); 0.0165178s (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 1.63414s (cpu); 1.39781s (thread); 0s (gc)
│ │ │ + -- used 1.64688s (cpu); 1.49441s (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 0.0157643s (cpu); 0.0157658s (thread); 0s (gc)
│ │ │ + -- used 0.0171041s (cpu); 0.0171048s (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
│ │ │ @@ -103,26 +103,26 @@
│ │ │  
│ │ │  o3 : R
│ │ │              
    
│ │ │                
    i4 : time eliminate(x,ideal(f,g))
│ │ │ - -- used 0.00273025s (cpu); 0.0027266s (thread); 0s (gc)
│ │ │ + -- used 0.00322795s (cpu); 0.00322707s (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 0.00165107s (cpu); 0.00165116s (thread); 0s (gc)
│ │ │ + -- used 0.00178641s (cpu); 0.00178856s (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 0.00260364s (cpu); 0.00260073s (thread); 0s (gc)
│ │ │ + -- used 0.00306018s (cpu); 0.00305735s (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 0.00151871s (cpu); 0.00151973s (thread); 0s (gc)
│ │ │ + -- used 0.00180994s (cpu); 0.0018108s (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 1.58822s (cpu); 1.34056s (thread); 0s (gc)
│ │ │ + -- used 1.49944s (cpu); 1.33922s (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  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -164,15 +164,15 @@
│ │ │  
│ │ │  o4 : Ideal of R
    
│ │ │              
    
│ │ │                
    i5 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.016203s (cpu); 0.0162067s (thread); 0s (gc)
│ │ │ + -- used 0.0165153s (cpu); 0.0165178s (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 {}
│ │ │ │ @@ -35,15 +35,15 @@
│ │ │ │  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 1.58822s (cpu); 1.34056s (thread); 0s (gc)
│ │ │ │ + -- used 1.49944s (cpu); 1.33922s (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  +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -90,15 +90,15 @@
│ │ │ │       + 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 0.016203s (cpu); 0.0162067s (thread); 0s (gc)
│ │ │ │ + -- used 0.0165153s (cpu); 0.0165178s (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
│ │ │ @@ -104,15 +104,15 @@
│ │ │  
│ │ │  o4 : R
    
│ │ │              
    
│ │ │                
    i5 : time eliminate(ideal(f,g),x)
│ │ │ - -- used 1.63414s (cpu); 1.39781s (thread); 0s (gc)
│ │ │ + -- used 1.64688s (cpu); 1.49441s (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  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -163,15 +163,15 @@
│ │ │  
│ │ │  o5 : Ideal of R
    
│ │ │              
    
│ │ │                
    i6 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.0157643s (cpu); 0.0157658s (thread); 0s (gc)
│ │ │ + -- used 0.0171041s (cpu); 0.0171048s (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 {}
│ │ │ │ @@ -30,15 +30,15 @@
│ │ │ │  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 1.63414s (cpu); 1.39781s (thread); 0s (gc)
│ │ │ │ + -- used 1.64688s (cpu); 1.49441s (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  +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -85,15 +85,15 @@
│ │ │ │       + 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 0.0157643s (cpu); 0.0157658s (thread); 0s (gc)
│ │ │ │ + -- used 0.0171041s (cpu); 0.0171048s (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/EnumerationCurves/dump/rawdocumentation.dump
│ │ │ @@ -1,8 +1,8 @@
│ │ │ -# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Fri Nov 14 16:08:08 2025
│ │ │ +# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Fri Nov 14 16:08:07 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │  #:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=17
│ │ │  RW51bWVyYXRpb25DdXJ2ZXM=
│ │ ├── ./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 0.0281527s (cpu); 0.0281536s (thread); 0s (gc)
│ │ │ + -- used 0.0299567s (cpu); 0.0299549s (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 0.319804s (cpu); 0.271811s (thread); 0s (gc)
│ │ │ + -- used 0.349122s (cpu); 0.299473s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {609250, 92288, 52812, 22428, 9728}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : time rationalCurve(3)
│ │ │ - -- used 0.213965s (cpu); 0.186717s (thread); 0s (gc)
│ │ │ + -- used 0.134486s (cpu); 0.134492s (thread); 0s (gc)
│ │ │  
│ │ │       8564575000
│ │ │  o8 = ----------
│ │ │           27
│ │ │  
│ │ │  o8 : QQ
│ │ │  
│ │ │  i9 : time for D in T list rationalCurve(3,D)
│ │ │ - -- used 5.81349s (cpu); 5.09017s (thread); 0s (gc)
│ │ │ + -- used 4.96559s (cpu); 4.41394s (thread); 0s (gc)
│ │ │  
│ │ │        8564575000  422690816           4834592  11239424
│ │ │  o9 = {----------, ---------, 6424365, -------, --------}
│ │ │            27          27                 3        27
│ │ │  
│ │ │  o9 : List
│ │ │  
│ │ │  i10 : time rationalCurve(3) - rationalCurve(1)/27
│ │ │ - -- used 0.226884s (cpu); 0.173069s (thread); 0s (gc)
│ │ │ + -- used 0.136647s (cpu); 0.136652s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 317206375
│ │ │  
│ │ │  o10 : QQ
│ │ │  
│ │ │  i11 : time for D in T list rationalCurve(3,D) - rationalCurve(1,D)/27
│ │ │ - -- used 5.49819s (cpu); 4.75893s (thread); 0s (gc)
│ │ │ + -- used 5.30572s (cpu); 4.65289s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = {317206375, 15655168, 6424326, 1611504, 416256}
│ │ │  
│ │ │  o11 : List
│ │ │  
│ │ │  i12 : time rationalCurve(4)
│ │ │ - -- used 1.61994s (cpu); 1.42301s (thread); 0s (gc)
│ │ │ + -- used 1.53272s (cpu); 1.40526s (thread); 0s (gc)
│ │ │  
│ │ │        15517926796875
│ │ │  o12 = --------------
│ │ │              64
│ │ │  
│ │ │  o12 : QQ
│ │ │  
│ │ │  i13 : time rationalCurve(4,{4,2})
│ │ │ - -- used 7.4159s (cpu); 5.81762s (thread); 0s (gc)
│ │ │ + -- used 6.90367s (cpu); 5.75767s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = 3883914084
│ │ │  
│ │ │  o13 : QQ
│ │ │  
│ │ │  i14 : time rationalCurve(4) - rationalCurve(2)/8
│ │ │ - -- used 1.69129s (cpu); 1.46867s (thread); 0s (gc)
│ │ │ + -- used 1.66353s (cpu); 1.45627s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = 242467530000
│ │ │  
│ │ │  o14 : QQ
│ │ │  
│ │ │  i15 : time rationalCurve(4,{4,2}) - rationalCurve(2,{4,2})/8
│ │ │ - -- used 7.49608s (cpu); 5.96479s (thread); 0s (gc)
│ │ │ + -- used 6.92773s (cpu); 5.66149s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3883902528
│ │ │  
│ │ │  o15 : QQ
│ │ │  
│ │ │  i16 : time rationalCurve(4,{3,3}) - rationalCurve(2,{3,3})/8
│ │ │ - -- used 8.14034s (cpu); 6.23141s (thread); 0s (gc)
│ │ │ + -- used 7.04282s (cpu); 5.87212s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 1139448384
│ │ │  
│ │ │  o16 : QQ
│ │ │  
│ │ │  i17 :
│ │ ├── ./usr/share/doc/Macaulay2/EnumerationCurves/html/_lines__Hypersurface.html
│ │ │ @@ -71,15 +71,15 @@
│ │ │            
    Computes the number of lines on a general hypersurface of degree 2n - 3 in \mathbb P^n.
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,15 +11,15 @@
│ │ │ │      * 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 0.0281527s (cpu); 0.0281536s (thread); 0s (gc)
│ │ │ │ + -- used 0.0299567s (cpu); 0.0299549s (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
│ │ │ @@ -152,15 +152,15 @@
│ │ │            
    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 0.0281527s (cpu); 0.0281536s (thread); 0s (gc)
│ │ │ + -- used 0.0299567s (cpu); 0.0299549s (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 0.319804s (cpu); 0.271811s (thread); 0s (gc)
│ │ │ + -- used 0.349122s (cpu); 0.299473s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {609250, 92288, 52812, 22428, 9728}
│ │ │  
│ │ │  o7 : List
    
│ │ │              
    
│ │ │ @@ -168,27 +168,27 @@
│ │ │            
    For rational curves of degree 3:
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ @@ -198,15 +198,15 @@
│ │ │            
    The number of rational curves of degree 3 on a general quintic threefold can be computed as follows:
    
│ │ │            
    
│ │ │          
│ │ │          
    
│ │ │                
    i8 : time rationalCurve(3)
│ │ │ - -- used 0.213965s (cpu); 0.186717s (thread); 0s (gc)
│ │ │ + -- used 0.134486s (cpu); 0.134492s (thread); 0s (gc)
│ │ │  
│ │ │       8564575000
│ │ │  o8 = ----------
│ │ │           27
│ │ │  
│ │ │  o8 : QQ
    
│ │ │              
    
│ │ │                
    i9 : time for D in T list rationalCurve(3,D)
│ │ │ - -- used 5.81349s (cpu); 5.09017s (thread); 0s (gc)
│ │ │ + -- used 4.96559s (cpu); 4.41394s (thread); 0s (gc)
│ │ │  
│ │ │        8564575000  422690816           4834592  11239424
│ │ │  o9 = {----------, ---------, 6424365, -------, --------}
│ │ │            27          27                 3        27
│ │ │  
│ │ │  o9 : List
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i10 : time rationalCurve(3) - rationalCurve(1)/27
│ │ │ - -- used 0.226884s (cpu); 0.173069s (thread); 0s (gc)
│ │ │ + -- used 0.136647s (cpu); 0.136652s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 317206375
│ │ │  
│ │ │  o10 : QQ
    
│ │ │              
    
│ │ │ @@ -214,15 +214,15 @@
│ │ │            
    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 5.49819s (cpu); 4.75893s (thread); 0s (gc)
│ │ │ + -- used 5.30572s (cpu); 4.65289s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = {317206375, 15655168, 6424326, 1611504, 416256}
│ │ │  
│ │ │  o11 : List
    
│ │ │              
    
│ │ │ @@ -230,27 +230,27 @@
│ │ │            
    For rational curves of degree 4:
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i12 : time rationalCurve(4)
│ │ │ - -- used 1.61994s (cpu); 1.42301s (thread); 0s (gc)
│ │ │ + -- used 1.53272s (cpu); 1.40526s (thread); 0s (gc)
│ │ │  
│ │ │        15517926796875
│ │ │  o12 = --------------
│ │ │              64
│ │ │  
│ │ │  o12 : QQ
    
│ │ │              
    
│ │ │                
    i13 : time rationalCurve(4,{4,2})
│ │ │ - -- used 7.4159s (cpu); 5.81762s (thread); 0s (gc)
│ │ │ + -- used 6.90367s (cpu); 5.75767s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = 3883914084
│ │ │  
│ │ │  o13 : QQ
    
│ │ │              
    
│ │ │ @@ -258,15 +258,15 @@
│ │ │            
    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 1.69129s (cpu); 1.46867s (thread); 0s (gc)
│ │ │ + -- used 1.66353s (cpu); 1.45627s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = 242467530000
│ │ │  
│ │ │  o14 : QQ
    
│ │ │              
    
│ │ │ @@ -274,25 +274,25 @@
│ │ │            
    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 7.49608s (cpu); 5.96479s (thread); 0s (gc)
│ │ │ + -- used 6.92773s (cpu); 5.66149s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3883902528
│ │ │  
│ │ │  o15 : QQ
    
│ │ │              
    
│ │ │                
    i16 : time rationalCurve(4,{3,3}) - rationalCurve(2,{3,3})/8
│ │ │ - -- used 8.14034s (cpu); 6.23141s (thread); 0s (gc)
│ │ │ + -- used 7.04282s (cpu); 5.87212s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 1139448384
│ │ │  
│ │ │  o16 : QQ
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -59,85 +59,85 @@
│ │ │ │  
│ │ │ │  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 0.319804s (cpu); 0.271811s (thread); 0s (gc)
│ │ │ │ + -- used 0.349122s (cpu); 0.299473s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = {609250, 92288, 52812, 22428, 9728}
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  For rational curves of degree 3:
│ │ │ │  i8 : time rationalCurve(3)
│ │ │ │ - -- used 0.213965s (cpu); 0.186717s (thread); 0s (gc)
│ │ │ │ + -- used 0.134486s (cpu); 0.134492s (thread); 0s (gc)
│ │ │ │  
│ │ │ │       8564575000
│ │ │ │  o8 = ----------
│ │ │ │           27
│ │ │ │  
│ │ │ │  o8 : QQ
│ │ │ │  i9 : time for D in T list rationalCurve(3,D)
│ │ │ │ - -- used 5.81349s (cpu); 5.09017s (thread); 0s (gc)
│ │ │ │ + -- used 4.96559s (cpu); 4.41394s (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 0.226884s (cpu); 0.173069s (thread); 0s (gc)
│ │ │ │ + -- used 0.136647s (cpu); 0.136652s (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 5.49819s (cpu); 4.75893s (thread); 0s (gc)
│ │ │ │ + -- used 5.30572s (cpu); 4.65289s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 = {317206375, 15655168, 6424326, 1611504, 416256}
│ │ │ │  
│ │ │ │  o11 : List
│ │ │ │  For rational curves of degree 4:
│ │ │ │  i12 : time rationalCurve(4)
│ │ │ │ - -- used 1.61994s (cpu); 1.42301s (thread); 0s (gc)
│ │ │ │ + -- used 1.53272s (cpu); 1.40526s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        15517926796875
│ │ │ │  o12 = --------------
│ │ │ │              64
│ │ │ │  
│ │ │ │  o12 : QQ
│ │ │ │  i13 : time rationalCurve(4,{4,2})
│ │ │ │ - -- used 7.4159s (cpu); 5.81762s (thread); 0s (gc)
│ │ │ │ + -- used 6.90367s (cpu); 5.75767s (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 1.69129s (cpu); 1.46867s (thread); 0s (gc)
│ │ │ │ + -- used 1.66353s (cpu); 1.45627s (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 7.49608s (cpu); 5.96479s (thread); 0s (gc)
│ │ │ │ + -- used 6.92773s (cpu); 5.66149s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = 3883902528
│ │ │ │  
│ │ │ │  o15 : QQ
│ │ │ │  i16 : time rationalCurve(4,{3,3}) - rationalCurve(2,{3,3})/8
│ │ │ │ - -- used 8.14034s (cpu); 6.23141s (thread); 0s (gc)
│ │ │ │ + -- used 7.04282s (cpu); 5.87212s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = 1139448384
│ │ │ │  
│ │ │ │  o16 : QQ
│ │ │ │  ********** WWaayyss ttoo uussee rraattiioonnaallCCuurrvvee:: **********
│ │ │ │      * rationalCurve(ZZ)
│ │ │ │      * rationalCurve(ZZ,List)
│ │ ├── ./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 .00193395 seconds
│ │ │ -     -- used .000576761 seconds
│ │ │ +     -- used .00199712 seconds
│ │ │ +     -- used .00058729 seconds
│ │ │  (9, 9)
│ │ │  new stuff found
│ │ │  4
│ │ │ -     -- used .00342678 seconds
│ │ │ -     -- used .00437517 seconds
│ │ │ +     -- used .00403989 seconds
│ │ │ +     -- used .00499019 seconds
│ │ │  (16, 26)
│ │ │  new stuff found
│ │ │  5
│ │ │ -     -- used .00818257 seconds
│ │ │ -     -- used .0263901 seconds
│ │ │ +     -- used .00821606 seconds
│ │ │ +     -- used .0263956 seconds
│ │ │  (25, 60)
│ │ │  6
│ │ │ -     -- used .0178533 seconds
│ │ │ -     -- used .204629 seconds
│ │ │ +     -- used .0199498 seconds
│ │ │ +     -- used .225103 seconds
│ │ │  (36, 120)
│ │ │  7
│ │ │ -     -- used .0386938 seconds
│ │ │ -     -- used .823841 seconds
│ │ │ +     -- used .0430355 seconds
│ │ │ +     -- used .92938 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
│ │ │ @@ -101,34 +101,34 @@
│ │ │  o3 : RingMap R <-- S
    
│ │ │              
    
│ │ │                
    i4 : G = egbToric(m, OutFile=>stdio)
│ │ │  3
│ │ │ -     -- used .00193395 seconds
│ │ │ -     -- used .000576761 seconds
│ │ │ +     -- used .00199712 seconds
│ │ │ +     -- used .00058729 seconds
│ │ │  (9, 9)
│ │ │  new stuff found
│ │ │  4
│ │ │ -     -- used .00342678 seconds
│ │ │ -     -- used .00437517 seconds
│ │ │ +     -- used .00403989 seconds
│ │ │ +     -- used .00499019 seconds
│ │ │  (16, 26)
│ │ │  new stuff found
│ │ │  5
│ │ │ -     -- used .00818257 seconds
│ │ │ -     -- used .0263901 seconds
│ │ │ +     -- used .00821606 seconds
│ │ │ +     -- used .0263956 seconds
│ │ │  (25, 60)
│ │ │  6
│ │ │ -     -- used .0178533 seconds
│ │ │ -     -- used .204629 seconds
│ │ │ +     -- used .0199498 seconds
│ │ │ +     -- used .225103 seconds
│ │ │  (36, 120)
│ │ │  7
│ │ │ -     -- used .0386938 seconds
│ │ │ -     -- used .823841 seconds
│ │ │ +     -- used .0430355 seconds
│ │ │ +     -- used .92938 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 {}
│ │ │ │ @@ -33,34 +33,34 @@
│ │ │ │                    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 .00193395 seconds
│ │ │ │ -     -- used .000576761 seconds
│ │ │ │ +     -- used .00199712 seconds
│ │ │ │ +     -- used .00058729 seconds
│ │ │ │  (9, 9)
│ │ │ │  new stuff found
│ │ │ │  4
│ │ │ │ -     -- used .00342678 seconds
│ │ │ │ -     -- used .00437517 seconds
│ │ │ │ +     -- used .00403989 seconds
│ │ │ │ +     -- used .00499019 seconds
│ │ │ │  (16, 26)
│ │ │ │  new stuff found
│ │ │ │  5
│ │ │ │ -     -- used .00818257 seconds
│ │ │ │ -     -- used .0263901 seconds
│ │ │ │ +     -- used .00821606 seconds
│ │ │ │ +     -- used .0263956 seconds
│ │ │ │  (25, 60)
│ │ │ │  6
│ │ │ │ -     -- used .0178533 seconds
│ │ │ │ -     -- used .204629 seconds
│ │ │ │ +     -- used .0199498 seconds
│ │ │ │ +     -- used .225103 seconds
│ │ │ │  (36, 120)
│ │ │ │  7
│ │ │ │ -     -- used .0386938 seconds
│ │ │ │ -     -- used .823841 seconds
│ │ │ │ +     -- used .0430355 seconds
│ │ │ │ +     -- used .92938 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/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 0.157442s (cpu); 0.120698s (thread); 0s (gc)
│ │ │ + -- used 0.227414s (cpu); 0.153104s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = 2
│ │ │  
│ │ │  i29 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallest))
│ │ │ - -- used 0.302486s (cpu); 0.208099s (thread); 0s (gc)
│ │ │ + -- used 0.356466s (cpu); 0.219612s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = 3
│ │ │  
│ │ │  i30 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallestTerm))
│ │ │ - -- used 0.495085s (cpu); 0.319904s (thread); 0s (gc)
│ │ │ + -- used 0.597058s (cpu); 0.384371s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = 1
│ │ │  
│ │ │  i31 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexLargest))
│ │ │ - -- used 0.21162s (cpu); 0.169556s (thread); 0s (gc)
│ │ │ + -- used 0.276895s (cpu); 0.204345s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = 2
│ │ │  
│ │ │  i32 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallest))
│ │ │ - -- used 0.375002s (cpu); 0.205247s (thread); 0s (gc)
│ │ │ + -- used 0.443969s (cpu); 0.222349s (thread); 0s (gc)
│ │ │  
│ │ │  o32 = 3
│ │ │  
│ │ │  i33 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallestTerm))
│ │ │ - -- used 0.338148s (cpu); 0.23363s (thread); 0s (gc)
│ │ │ + -- used 0.408512s (cpu); 0.272452s (thread); 0s (gc)
│ │ │  
│ │ │  o33 = 3
│ │ │  
│ │ │  i34 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexLargest))
│ │ │ - -- used 0.298624s (cpu); 0.191542s (thread); 0s (gc)
│ │ │ + -- used 0.378983s (cpu); 0.238874s (thread); 0s (gc)
│ │ │  
│ │ │  o34 = 3
│ │ │  
│ │ │  i35 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Points))
│ │ │ - -- used 14.9955s (cpu); 10.1647s (thread); 0s (gc)
│ │ │ + -- used 18.7614s (cpu); 11.9025s (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 0.48632s (cpu); 0.444696s (thread); 0s (gc)
│ │ │ + -- used 0.439411s (cpu); 0.366736s (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 0.495057s (cpu); 0.461267s (thread); 0s (gc)
│ │ │ + -- used 0.704502s (cpu); 0.620035s (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 0.512127s (cpu); 0.390896s (thread); 0s (gc)
│ │ │ + -- used 0.598927s (cpu); 0.44656s (thread); 0s (gc)
│ │ │  
│ │ │  o46 = 2
│ │ │  
│ │ │  i47 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefault)
│ │ │ - -- used 3.35648s (cpu); 3.0338s (thread); 0s (gc)
│ │ │ + -- used 4.1081s (cpu); 3.61824s (thread); 0s (gc)
│ │ │  
│ │ │  i48 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultNonRandom)
│ │ │ - -- used 0.728099s (cpu); 0.636148s (thread); 0s (gc)
│ │ │ + -- used 0.915872s (cpu); 0.771741s (thread); 0s (gc)
│ │ │  
│ │ │  o48 = true
│ │ │  
│ │ │  i49 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Random)
│ │ │ - -- used 2.71868s (cpu); 2.52715s (thread); 0s (gc)
│ │ │ + -- used 3.50061s (cpu); 3.27946s (thread); 0s (gc)
│ │ │  
│ │ │  i50 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallest)
│ │ │ - -- used 2.3572s (cpu); 2.00446s (thread); 0s (gc)
│ │ │ + -- used 2.74894s (cpu); 2.21119s (thread); 0s (gc)
│ │ │  
│ │ │  i51 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallestTerm)
│ │ │ - -- used 0.874751s (cpu); 0.777483s (thread); 0s (gc)
│ │ │ + -- used 0.90334s (cpu); 0.831513s (thread); 0s (gc)
│ │ │  
│ │ │  o51 = true
│ │ │  
│ │ │  i52 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallest)
│ │ │ - -- used 2.70655s (cpu); 2.28905s (thread); 0s (gc)
│ │ │ + -- used 3.12236s (cpu); 2.52697s (thread); 0s (gc)
│ │ │  
│ │ │  i53 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallestTerm)
│ │ │ - -- used 3.1076s (cpu); 2.68758s (thread); 0s (gc)
│ │ │ + -- used 3.93397s (cpu); 3.30409s (thread); 0s (gc)
│ │ │  
│ │ │  i54 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Points)
│ │ │ - -- used 9.26442s (cpu); 7.71318s (thread); 0s (gc)
│ │ │ + -- used 11.5771s (cpu); 9.41579s (thread); 0s (gc)
│ │ │  
│ │ │  o54 = true
│ │ │  
│ │ │  i55 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultWithPoints)
│ │ │ - -- used 7.30189s (cpu); 6.0694s (thread); 0s (gc)
│ │ │ + -- used 9.23837s (cpu); 7.22644s (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 0.853916s (cpu); 0.601439s (thread); 0s (gc)
│ │ │ + -- used 1.06064s (cpu); 0.682954s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = true
│ │ │  
│ │ │  i5 : time regularInCodimension(2, S/J)
│ │ │ - -- used 11.1488s (cpu); 8.3419s (thread); 0s (gc)
│ │ │ + -- used 12.5666s (cpu); 8.71825s (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 1.46811s (cpu); 1.03558s (thread); 0s (gc)
│ │ │ +regularInCodimension:  Loop completed, submatrices considered = 49, and compute -- used 1.69032s (cpu); 1.16678s (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 0.181418s (cpu); 0.129355s (thread); 0s (gc)
│ │ │ + -- used 0.215701s (cpu); 0.155977s (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 0.138676s (cpu); 0.098481s (thread); 0s (gc)
│ │ │ + -- used 0.189334s (cpu); 0.126468s (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 0.568112s (cpu); 0.444722s (thread); 0s (gc)
│ │ │ + -- used 0.737494s (cpu); 0.520379s (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 0.710929s (cpu); 0.526468s (thread); 0s (gc)
│ │ │ + -- used 0.846642s (cpu); 0.572487s (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 0.467454s (cpu); 0.323483s (thread); 0s (gc)
│ │ │ + -- used 0.537942s (cpu); 0.348682s (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)
│ │ │ - -- 1.52401s elapsed
│ │ │ + -- 1.47011s 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)
│ │ │ - -- .982994s elapsed
│ │ │ + -- .867528s 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 0.00400072s (cpu); 0.00264049s (thread); 0s (gc)
│ │ │ + -- used 0.00392733s (cpu); 0.00320358s (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 0.00245125s (cpu); 0.00248215s (thread); 0s (gc)
│ │ │ + -- used 0.00248357s (cpu); 0.00299689s (thread); 0s (gc)
│ │ │  isCodimAtLeast: Computing codim of monomials based on ideal generators.
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 : time isCodimAtLeast(5, I, PairLimit => 200, Verbose=>false)
│ │ │ - -- used 0.00231197s (cpu); 0.0023101s (thread); 0s (gc)
│ │ │ + -- used 0.00159886s (cpu); 0.00285589s (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 0.237579s (cpu); 0.142956s (thread); 0s (gc)
│ │ │ + -- used 0.30941s (cpu); 0.170497s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
│ │ │  
│ │ │  i5 : time projDim(module I, Strategy=>StrategyRandom, MinDimension => 1)
│ │ │ - -- used 0.0109946s (cpu); 0.0128064s (thread); 0s (gc)
│ │ │ + -- used 0.0140121s (cpu); 0.0152185s (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 0.493613s (cpu); 0.449433s (thread); 0s (gc)
│ │ │ + -- used 0.570662s (cpu); 0.513007s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : Ideal of R
│ │ │  
│ │ │  i4 : time I1 = minors(4, M, Strategy=>Cofactor);
│ │ │ - -- used 1.51523s (cpu); 1.31036s (thread); 0s (gc)
│ │ │ + -- used 1.53975s (cpu); 1.3979s (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 0.67198s (cpu); 0.551786s (thread); 0s (gc)
│ │ │ + -- used 0.831973s (cpu); 0.612865s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
│ │ │  
│ │ │  i9 : time regularInCodimension(2, S)
│ │ │ - -- used 6.48256s (cpu); 4.93012s (thread); 0s (gc)
│ │ │ + -- used 7.53097s (cpu); 5.47385s (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 0.020027s (cpu); 0.0198154s (thread); 0s (gc)
│ │ │ + -- used 0.0241001s (cpu); 0.021182s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = -1
│ │ │  
│ │ │  i13 : time regularInCodimension(2, R)
│ │ │ - -- used 0.192016s (cpu); 0.160849s (thread); 0s (gc)
│ │ │ + -- used 0.215181s (cpu); 0.153065s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = true
│ │ │  
│ │ │  i14 : time regularInCodimension(2, R)
│ │ │ - -- used 0.909538s (cpu); 0.600324s (thread); 0s (gc)
│ │ │ + -- used 1.06255s (cpu); 0.655467s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = true
│ │ │  
│ │ │  i15 : time regularInCodimension(2, R)
│ │ │ - -- used 1.3289s (cpu); 0.899094s (thread); 0s (gc)
│ │ │ + -- used 1.54486s (cpu); 1.00318s (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 RandomNonZero
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ -internalChooseMinor: Ch -- used 6.59871s (cpu); 5.02176s (thread); 0s (gc)
│ │ │ +internalChooseMinor: Ch -- used 7.63003s (cpu); 5.64893s (thread); 0s (gc)
│ │ │  oosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ @@ -430,15 +430,15 @@
│ │ │  internalChooseMinor: Choosing Random
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 328, and computed = 180
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 1
│ │ │  regularInCodimension:  Loop completed, submatrices considered = 328, and computed = 180.  singular locus dimension appears to be = 1
│ │ │  
│ │ │  i17 : time regularInCodimension(2, S, Verbose=>true, MaxMinors=>30)
│ │ │ - -- used 1.29022s (cpu); 0.996476s (thread); 0s (gc)
│ │ │ + -- used 1.54965s (cpu); 1.15431s (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 LexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ @@ -490,59 +490,59 @@
│ │ │  i18 : StrategyCurrent#Random = 0;
│ │ │  
│ │ │  i19 : StrategyCurrent#LexSmallest = 100;
│ │ │  
│ │ │  i20 : StrategyCurrent#LexSmallestTerm = 0;
│ │ │  
│ │ │  i21 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.31652s (cpu); 0.221889s (thread); 0s (gc)
│ │ │ + -- used 0.364553s (cpu); 0.234375s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = true
│ │ │  
│ │ │  i22 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.121412s (cpu); 0.0729141s (thread); 0s (gc)
│ │ │ + -- used 0.148619s (cpu); 0.0866179s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = true
│ │ │  
│ │ │  i23 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.36055s (cpu); 0.269016s (thread); 0s (gc)
│ │ │ + -- used 0.473111s (cpu); 0.341109s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = true
│ │ │  
│ │ │  i24 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 1.61547s (cpu); 1.20594s (thread); 0s (gc)
│ │ │ + -- used 2.16122s (cpu); 1.5762s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = true
│ │ │  
│ │ │  i25 : StrategyCurrent#LexSmallest = 0;
│ │ │  
│ │ │  i26 : StrategyCurrent#LexSmallestTerm = 100;
│ │ │  
│ │ │  i27 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 2.16839s (cpu); 1.58684s (thread); 0s (gc)
│ │ │ + -- used 2.6435s (cpu); 1.82766s (thread); 0s (gc)
│ │ │  
│ │ │  i28 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 2.24976s (cpu); 1.60689s (thread); 0s (gc)
│ │ │ + -- used 2.75088s (cpu); 1.86584s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = true
│ │ │  
│ │ │  i29 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.419528s (cpu); 0.335678s (thread); 0s (gc)
│ │ │ + -- used 0.527054s (cpu); 0.391424s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = true
│ │ │  
│ │ │  i30 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.6816s (cpu); 0.545534s (thread); 0s (gc)
│ │ │ + -- used 0.856958s (cpu); 0.645873s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = true
│ │ │  
│ │ │  i31 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 0.990173s (cpu); 0.815558s (thread); 0s (gc)
│ │ │ + -- used 1.17916s (cpu); 0.909361s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = true
│ │ │  
│ │ │  i32 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 1.68103s (cpu); 1.31918s (thread); 0s (gc)
│ │ │ + -- used 1.99076s (cpu); 1.53973s (thread); 0s (gc)
│ │ │  
│ │ │  o32 = true
│ │ │  
│ │ │  i33 :
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/___Fast__Minors__Strategy__Tutorial.html
│ │ │ @@ -620,71 +620,71 @@
│ │ │          
    
│ │ │            
    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).
    
│ │ │          
    
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│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i28 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Random))
│ │ │ - -- used 0.157442s (cpu); 0.120698s (thread); 0s (gc)
│ │ │ + -- used 0.227414s (cpu); 0.153104s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = 2
    
│ │ │              
    
│ │ │                
    i29 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallest))
│ │ │ - -- used 0.302486s (cpu); 0.208099s (thread); 0s (gc)
│ │ │ + -- used 0.356466s (cpu); 0.219612s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = 3
    
│ │ │              
    
│ │ │                
    i30 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallestTerm))
│ │ │ - -- used 0.495085s (cpu); 0.319904s (thread); 0s (gc)
│ │ │ + -- used 0.597058s (cpu); 0.384371s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = 1
    
│ │ │              
    
│ │ │                
    i31 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexLargest))
│ │ │ - -- used 0.21162s (cpu); 0.169556s (thread); 0s (gc)
│ │ │ + -- used 0.276895s (cpu); 0.204345s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = 2
    
│ │ │              
    
│ │ │                
    i32 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallest))
│ │ │ - -- used 0.375002s (cpu); 0.205247s (thread); 0s (gc)
│ │ │ + -- used 0.443969s (cpu); 0.222349s (thread); 0s (gc)
│ │ │  
│ │ │  o32 = 3
    
│ │ │              
    
│ │ │                
    i33 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallestTerm))
│ │ │ - -- used 0.338148s (cpu); 0.23363s (thread); 0s (gc)
│ │ │ + -- used 0.408512s (cpu); 0.272452s (thread); 0s (gc)
│ │ │  
│ │ │  o33 = 3
    
│ │ │              
    
│ │ │                
    i34 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexLargest))
│ │ │ - -- used 0.298624s (cpu); 0.191542s (thread); 0s (gc)
│ │ │ + -- used 0.378983s (cpu); 0.238874s (thread); 0s (gc)
│ │ │  
│ │ │  o34 = 3
    
│ │ │              
    
│ │ │                
    i35 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Points))
│ │ │ - -- used 14.9955s (cpu); 10.1647s (thread); 0s (gc)
│ │ │ + -- used 18.7614s (cpu); 11.9025s (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,
    
│ │ │ @@ -709,15 +709,15 @@
│ │ │          
    
│ │ │            
    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:
    
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│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i37 : time chooseGoodMinors(20, 6, M, J, Strategy=>StrategyDefault, Verbose=>true);
│ │ │ - -- used 0.48632s (cpu); 0.444696s (thread); 0s (gc)
│ │ │ + -- used 0.439411s (cpu); 0.366736s (thread); 0s (gc)
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ @@ -820,15 +820,15 @@
│ │ │  
│ │ │  o42 = true
    
│ │ │              
    
│ │ │                
    i43 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points, PointOptions=>ptsStratGeometric))
│ │ │ - -- used 0.495057s (cpu); 0.461267s (thread); 0s (gc)
│ │ │ + -- used 0.704502s (cpu); 0.620035s (thread); 0s (gc)
│ │ │  
│ │ │  o43 = 2
    
│ │ │              
    
│ │ │                
    i44 : ptsStratRational = ptsStratGeometric++{ExtendField=>false} --change that value
│ │ │ @@ -852,15 +852,15 @@
│ │ │  
│ │ │  o45 = false
    
│ │ │              
    
│ │ │                
    i46 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points, PointOptions=>ptsStratRational))
│ │ │ - -- used 0.512127s (cpu); 0.390896s (thread); 0s (gc)
│ │ │ + -- used 0.598927s (cpu); 0.44656s (thread); 0s (gc)
│ │ │  
│ │ │  o46 = 2
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Other options may also be passed to the RandomPoints package via the PointOptions option.
    
│ │ │ @@ -868,69 +868,69 @@
│ │ │          
    
│ │ │            
    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.
    
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│ │ │          
    
│ │ │                
    i47 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefault)
│ │ │ - -- used 3.35648s (cpu); 3.0338s (thread); 0s (gc)
    
│ │ │ + -- used 4.1081s (cpu); 3.61824s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i48 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultNonRandom)
│ │ │ - -- used 0.728099s (cpu); 0.636148s (thread); 0s (gc)
│ │ │ + -- used 0.915872s (cpu); 0.771741s (thread); 0s (gc)
│ │ │  
│ │ │  o48 = true
    
│ │ │              
    
│ │ │                
    i49 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Random)
│ │ │ - -- used 2.71868s (cpu); 2.52715s (thread); 0s (gc)
    
│ │ │ + -- used 3.50061s (cpu); 3.27946s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i50 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallest)
│ │ │ - -- used 2.3572s (cpu); 2.00446s (thread); 0s (gc)
    
│ │ │ + -- used 2.74894s (cpu); 2.21119s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i51 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallestTerm)
│ │ │ - -- used 0.874751s (cpu); 0.777483s (thread); 0s (gc)
│ │ │ + -- used 0.90334s (cpu); 0.831513s (thread); 0s (gc)
│ │ │  
│ │ │  o51 = true
    
│ │ │              
    
│ │ │                
    i52 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallest)
│ │ │ - -- used 2.70655s (cpu); 2.28905s (thread); 0s (gc)
    
│ │ │ + -- used 3.12236s (cpu); 2.52697s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i53 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallestTerm)
│ │ │ - -- used 3.1076s (cpu); 2.68758s (thread); 0s (gc)
    
│ │ │ + -- used 3.93397s (cpu); 3.30409s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i54 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Points)
│ │ │ - -- used 9.26442s (cpu); 7.71318s (thread); 0s (gc)
│ │ │ + -- used 11.5771s (cpu); 9.41579s (thread); 0s (gc)
│ │ │  
│ │ │  o54 = true
    
│ │ │              
    
│ │ │                
    i55 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultWithPoints)
│ │ │ - -- used 7.30189s (cpu); 6.0694s (thread); 0s (gc)
│ │ │ + -- used 9.23837s (cpu); 7.22644s (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 0.157442s (cpu); 0.120698s (thread); 0s (gc)
│ │ │ │ + -- used 0.227414s (cpu); 0.153104s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o28 = 2
│ │ │ │  i29 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallest))
│ │ │ │ - -- used 0.302486s (cpu); 0.208099s (thread); 0s (gc)
│ │ │ │ + -- used 0.356466s (cpu); 0.219612s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o29 = 3
│ │ │ │  i30 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallestTerm))
│ │ │ │ - -- used 0.495085s (cpu); 0.319904s (thread); 0s (gc)
│ │ │ │ + -- used 0.597058s (cpu); 0.384371s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o30 = 1
│ │ │ │  i31 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexLargest))
│ │ │ │ - -- used 0.21162s (cpu); 0.169556s (thread); 0s (gc)
│ │ │ │ + -- used 0.276895s (cpu); 0.204345s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o31 = 2
│ │ │ │  i32 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallest))
│ │ │ │ - -- used 0.375002s (cpu); 0.205247s (thread); 0s (gc)
│ │ │ │ + -- used 0.443969s (cpu); 0.222349s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o32 = 3
│ │ │ │  i33 : time dim (J + chooseGoodMinors(8, 6, M, J,
│ │ │ │  Strategy=>GRevLexSmallestTerm))
│ │ │ │ - -- used 0.338148s (cpu); 0.23363s (thread); 0s (gc)
│ │ │ │ + -- used 0.408512s (cpu); 0.272452s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o33 = 3
│ │ │ │  i34 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexLargest))
│ │ │ │ - -- used 0.298624s (cpu); 0.191542s (thread); 0s (gc)
│ │ │ │ + -- used 0.378983s (cpu); 0.238874s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o34 = 3
│ │ │ │  i35 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Points))
│ │ │ │ - -- used 14.9955s (cpu); 10.1647s (thread); 0s (gc)
│ │ │ │ + -- used 18.7614s (cpu); 11.9025s (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 0.48632s (cpu); 0.444696s (thread); 0s (gc)
│ │ │ │ + -- used 0.439411s (cpu); 0.366736s (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 0.495057s (cpu); 0.461267s (thread); 0s (gc)
│ │ │ │ + -- used 0.704502s (cpu); 0.620035s (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 0.512127s (cpu); 0.390896s (thread); 0s (gc)
│ │ │ │ + -- used 0.598927s (cpu); 0.44656s (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 3.35648s (cpu); 3.0338s (thread); 0s (gc)
│ │ │ │ + -- used 4.1081s (cpu); 3.61824s (thread); 0s (gc)
│ │ │ │  i48 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>StrategyDefaultNonRandom)
│ │ │ │ - -- used 0.728099s (cpu); 0.636148s (thread); 0s (gc)
│ │ │ │ + -- used 0.915872s (cpu); 0.771741s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o48 = true
│ │ │ │  i49 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Random)
│ │ │ │ - -- used 2.71868s (cpu); 2.52715s (thread); 0s (gc)
│ │ │ │ + -- used 3.50061s (cpu); 3.27946s (thread); 0s (gc)
│ │ │ │  i50 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>LexSmallest)
│ │ │ │ - -- used 2.3572s (cpu); 2.00446s (thread); 0s (gc)
│ │ │ │ + -- used 2.74894s (cpu); 2.21119s (thread); 0s (gc)
│ │ │ │  i51 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>LexSmallestTerm)
│ │ │ │ - -- used 0.874751s (cpu); 0.777483s (thread); 0s (gc)
│ │ │ │ + -- used 0.90334s (cpu); 0.831513s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o51 = true
│ │ │ │  i52 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>GRevLexSmallest)
│ │ │ │ - -- used 2.70655s (cpu); 2.28905s (thread); 0s (gc)
│ │ │ │ + -- used 3.12236s (cpu); 2.52697s (thread); 0s (gc)
│ │ │ │  i53 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>GRevLexSmallestTerm)
│ │ │ │ - -- used 3.1076s (cpu); 2.68758s (thread); 0s (gc)
│ │ │ │ + -- used 3.93397s (cpu); 3.30409s (thread); 0s (gc)
│ │ │ │  i54 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Points)
│ │ │ │ - -- used 9.26442s (cpu); 7.71318s (thread); 0s (gc)
│ │ │ │ + -- used 11.5771s (cpu); 9.41579s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o54 = true
│ │ │ │  i55 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>StrategyDefaultWithPoints)
│ │ │ │ - -- used 7.30189s (cpu); 6.0694s (thread); 0s (gc)
│ │ │ │ + -- used 9.23837s (cpu); 7.22644s (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
│ │ │ @@ -81,23 +81,23 @@
│ │ │          
    
│ │ │            
    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 0.853916s (cpu); 0.601439s (thread); 0s (gc)
│ │ │ + -- used 1.06064s (cpu); 0.682954s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = true
    
│ │ │              
    
│ │ │                
    i5 : time regularInCodimension(2, S/J)
│ │ │ - -- used 11.1488s (cpu); 8.3419s (thread); 0s (gc)
    
│ │ │ + -- used 12.5666s (cpu); 8.71825s (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.
    
│ │ │          
    
│ │ │          
│ │ │ @@ -172,29 +172,29 @@
│ │ │  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 1.46811s (cpu); 1.03558s (thread); 0s (gc)
│ │ │ +regularInCodimension:  Loop completed, submatrices considered = 49, and compute -- used 1.69032s (cpu); 1.16678s (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 0.181418s (cpu); 0.129355s (thread); 0s (gc)
│ │ │ + -- used 0.215701s (cpu); 0.155977s (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
│ │ │ @@ -219,15 +219,15 @@
│ │ │          
    
│ │ │            
    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 0.138676s (cpu); 0.098481s (thread); 0s (gc)
│ │ │ + -- used 0.189334s (cpu); 0.126468s (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
│ │ │ @@ -252,15 +252,15 @@
│ │ │          
    
│ │ │            
    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 0.568112s (cpu); 0.444722s (thread); 0s (gc)
│ │ │ + -- used 0.737494s (cpu); 0.520379s (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.
│ │ │ @@ -291,15 +291,15 @@
│ │ │          
    
│ │ │            
    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 0.710929s (cpu); 0.526468s (thread); 0s (gc)
│ │ │ + -- used 0.846642s (cpu); 0.572487s (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
│ │ │ @@ -348,15 +348,15 @@
│ │ │          
    
│ │ │            
    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 0.467454s (cpu); 0.323483s (thread); 0s (gc)
│ │ │ + -- used 0.537942s (cpu); 0.348682s (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 0.853916s (cpu); 0.601439s (thread); 0s (gc)
│ │ │ │ + -- used 1.06064s (cpu); 0.682954s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  i5 : time regularInCodimension(2, S/J)
│ │ │ │ - -- used 11.1488s (cpu); 8.3419s (thread); 0s (gc)
│ │ │ │ + -- used 12.5666s (cpu); 8.71825s (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 1.46811s (cpu); 1.03558s (thread); 0s (gc)
│ │ │ │ +-- used 1.69032s (cpu); 1.16678s (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 0.181418s (cpu); 0.129355s (thread); 0s (gc)
│ │ │ │ + -- used 0.215701s (cpu); 0.155977s (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 0.138676s (cpu); 0.098481s (thread); 0s (gc)
│ │ │ │ + -- used 0.189334s (cpu); 0.126468s (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 0.568112s (cpu); 0.444722s (thread); 0s (gc)
│ │ │ │ + -- used 0.737494s (cpu); 0.520379s (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 0.710929s (cpu); 0.526468s (thread); 0s (gc)
│ │ │ │ + -- used 0.846642s (cpu); 0.572487s (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 0.467454s (cpu); 0.323483s (thread); 0s (gc)
│ │ │ │ + -- used 0.537942s (cpu); 0.348682s (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
│ │ │ @@ -68,15 +68,15 @@
│ │ │              
    		
│ │ │                
    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)
│ │ │ - -- 1.52401s elapsed
│ │ │ + -- 1.47011s 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
      • 
│ │ │ @@ -122,15 +122,15 @@
│ │ │            
    • 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)
│ │ │ - -- .982994s elapsed
│ │ │ + -- .867528s elapsed
│ │ │  
│ │ │  o4 = true
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── 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)
│ │ │ │ - -- 1.52401s elapsed
│ │ │ │ + -- 1.47011s 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
│ │ │ │ @@ -135,15 +135,15 @@
│ │ │ │  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)
│ │ │ │ - -- .982994s elapsed
│ │ │ │ + -- .867528s 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.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/FastMinors.m2:1993:0.
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/_is__Codim__At__Least.html
│ │ │ @@ -114,15 +114,15 @@
│ │ │  
│ │ │  o6 : Ideal of R
    
│ │ │              
    
│ │ │                
    i7 : time isCodimAtLeast(3, J)
│ │ │ - -- used 0.00400072s (cpu); 0.00264049s (thread); 0s (gc)
│ │ │ + -- used 0.00392733s (cpu); 0.00320358s (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.
    
│ │ │ @@ -136,24 +136,24 @@
│ │ │  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 0.00245125s (cpu); 0.00248215s (thread); 0s (gc)
│ │ │ + -- used 0.00248357s (cpu); 0.00299689s (thread); 0s (gc)
│ │ │  isCodimAtLeast: Computing codim of monomials based on ideal generators.
│ │ │  
│ │ │  o9 = true
    
│ │ │              
    
│ │ │                
    i10 : time isCodimAtLeast(5, I, PairLimit => 200, Verbose=>false)
│ │ │ - -- used 0.00231197s (cpu); 0.0023101s (thread); 0s (gc)
│ │ │ + -- used 0.00159886s (cpu); 0.00285589s (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 {}
│ │ │ │ @@ -38,15 +38,15 @@
│ │ │ │               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 0.00400072s (cpu); 0.00264049s (thread); 0s (gc)
│ │ │ │ + -- used 0.00392733s (cpu); 0.00320358s (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=>
│ │ │ │ @@ -72,20 +72,20 @@
│ │ │ │  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 0.00245125s (cpu); 0.00248215s (thread); 0s (gc)
│ │ │ │ + -- used 0.00248357s (cpu); 0.00299689s (thread); 0s (gc)
│ │ │ │  isCodimAtLeast: Computing codim of monomials based on ideal generators.
│ │ │ │  
│ │ │ │  o9 = true
│ │ │ │  i10 : time isCodimAtLeast(5, I, PairLimit => 200, Verbose=>false)
│ │ │ │ - -- used 0.00231197s (cpu); 0.0023101s (thread); 0s (gc)
│ │ │ │ + -- used 0.00159886s (cpu); 0.00285589s (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
│ │ │ @@ -99,23 +99,23 @@
│ │ │  
│ │ │  o3 = 2
    
│ │ │              
    
│ │ │                
    i4 : time projDim(module I, Strategy=>StrategyRandom)
│ │ │ - -- used 0.237579s (cpu); 0.142956s (thread); 0s (gc)
│ │ │ + -- used 0.30941s (cpu); 0.170497s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
    
│ │ │              
    
│ │ │                
    i5 : time projDim(module I, Strategy=>StrategyRandom, MinDimension => 1)
│ │ │ - -- used 0.0109946s (cpu); 0.0128064s (thread); 0s (gc)
│ │ │ + -- used 0.0140121s (cpu); 0.0152185s (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 {}
│ │ │ │ @@ -44,19 +44,19 @@
│ │ │ │  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 0.237579s (cpu); 0.142956s (thread); 0s (gc)
│ │ │ │ + -- used 0.30941s (cpu); 0.170497s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 1
│ │ │ │  i5 : time projDim(module I, Strategy=>StrategyRandom, MinDimension => 1)
│ │ │ │ - -- used 0.0109946s (cpu); 0.0128064s (thread); 0s (gc)
│ │ │ │ + -- used 0.0140121s (cpu); 0.0152185s (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
│ │ │ @@ -92,23 +92,23 @@
│ │ │               6      7
│ │ │  o2 : Matrix R  <-- R
    
│ │ │              
    
│ │ │                
    i3 : time I2 = recursiveMinors(4, M, Threads=>0);
│ │ │ - -- used 0.493613s (cpu); 0.449433s (thread); 0s (gc)
│ │ │ + -- used 0.570662s (cpu); 0.513007s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : Ideal of R
    
│ │ │              
    
│ │ │                
    i4 : time I1 = minors(4, M, Strategy=>Cofactor);
│ │ │ - -- used 1.51523s (cpu); 1.31036s (thread); 0s (gc)
│ │ │ + -- used 1.53975s (cpu); 1.3979s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : Ideal of R
    
│ │ │              
    
│ │ │                
    i5 : I1 == I2
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,19 +27,19 @@
│ │ │ │  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 0.493613s (cpu); 0.449433s (thread); 0s (gc)
│ │ │ │ + -- used 0.570662s (cpu); 0.513007s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 : Ideal of R
│ │ │ │  i4 : time I1 = minors(4, M, Strategy=>Cofactor);
│ │ │ │ - -- used 1.51523s (cpu); 1.31036s (thread); 0s (gc)
│ │ │ │ + -- used 1.53975s (cpu); 1.3979s (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
│ │ │ @@ -131,23 +131,23 @@
│ │ │  
│ │ │  o7 = 3
    
│ │ │              
    
│ │ │                
    i8 : time regularInCodimension(1, S)
│ │ │ - -- used 0.67198s (cpu); 0.551786s (thread); 0s (gc)
│ │ │ + -- used 0.831973s (cpu); 0.612865s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
    
│ │ │              
    
│ │ │                
    i9 : time regularInCodimension(2, S)
│ │ │ - -- used 6.48256s (cpu); 4.93012s (thread); 0s (gc)
    
│ │ │ + -- used 7.53097s (cpu); 5.47385s (thread); 0s (gc)
│ │ │              
    
│ │ │          
    
│ │ │            
    There are numerous examples where regularInCodimension is several orders of magnitude faster that calls of dim singularLocus.
    
│ │ │          
    
│ │ │          
    
│ │ │ @@ -165,39 +165,39 @@
│ │ │  
│ │ │  o11 = 2
    
│ │ │              
    
│ │ │                
    i12 : time (dim singularLocus (R))
│ │ │ - -- used 0.020027s (cpu); 0.0198154s (thread); 0s (gc)
│ │ │ + -- used 0.0241001s (cpu); 0.021182s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = -1
    
│ │ │              
    
│ │ │                
    i13 : time regularInCodimension(2, R)
│ │ │ - -- used 0.192016s (cpu); 0.160849s (thread); 0s (gc)
│ │ │ + -- used 0.215181s (cpu); 0.153065s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = true
    
│ │ │              
    
│ │ │                
    i14 : time regularInCodimension(2, R)
│ │ │ - -- used 0.909538s (cpu); 0.600324s (thread); 0s (gc)
│ │ │ + -- used 1.06255s (cpu); 0.655467s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = true
    
│ │ │              
    
│ │ │                
    i15 : time regularInCodimension(2, R)
│ │ │ - -- used 1.3289s (cpu); 0.899094s (thread); 0s (gc)
│ │ │ + -- used 1.54486s (cpu); 1.00318s (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.
    
│ │ │ @@ -537,15 +537,15 @@
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ -internalChooseMinor: Ch -- used 6.59871s (cpu); 5.02176s (thread); 0s (gc)
│ │ │ +internalChooseMinor: Ch -- used 7.63003s (cpu); 5.64893s (thread); 0s (gc)
│ │ │  oosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ @@ -589,15 +589,15 @@
│ │ │          
    
│ │ │            
    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 1.29022s (cpu); 0.996476s (thread); 0s (gc)
│ │ │ + -- used 1.54965s (cpu); 1.15431s (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 LexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ @@ -666,39 +666,39 @@
│ │ │              
    		
│ │ │                
    i20 : StrategyCurrent#LexSmallestTerm = 0;
    
│ │ │              
    
│ │ │                
    i21 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.31652s (cpu); 0.221889s (thread); 0s (gc)
│ │ │ + -- used 0.364553s (cpu); 0.234375s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = true
    
│ │ │              
    
│ │ │                
    i22 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.121412s (cpu); 0.0729141s (thread); 0s (gc)
│ │ │ + -- used 0.148619s (cpu); 0.0866179s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = true
    
│ │ │              
    
│ │ │                
    i23 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.36055s (cpu); 0.269016s (thread); 0s (gc)
│ │ │ + -- used 0.473111s (cpu); 0.341109s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = true
    
│ │ │              
    
│ │ │                
    i24 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 1.61547s (cpu); 1.20594s (thread); 0s (gc)
│ │ │ + -- used 2.16122s (cpu); 1.5762s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = true
    
│ │ │              
    
│ │ │                
    i25 : StrategyCurrent#LexSmallest = 0;
    
│ │ │ @@ -708,53 +708,53 @@
│ │ │                  		
│ │ │                
    i26 : StrategyCurrent#LexSmallestTerm = 100;
    
│ │ │              
    
│ │ │                
    i27 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 2.16839s (cpu); 1.58684s (thread); 0s (gc)
    
│ │ │ + -- used 2.6435s (cpu); 1.82766s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i28 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 2.24976s (cpu); 1.60689s (thread); 0s (gc)
│ │ │ + -- used 2.75088s (cpu); 1.86584s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = true
    
│ │ │              
    
│ │ │                
    i29 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.419528s (cpu); 0.335678s (thread); 0s (gc)
│ │ │ + -- used 0.527054s (cpu); 0.391424s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = true
    
│ │ │              
    
│ │ │                
    i30 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.6816s (cpu); 0.545534s (thread); 0s (gc)
│ │ │ + -- used 0.856958s (cpu); 0.645873s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = true
    
│ │ │              
    
│ │ │                
    i31 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 0.990173s (cpu); 0.815558s (thread); 0s (gc)
│ │ │ + -- used 1.17916s (cpu); 0.909361s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = true
    
│ │ │              
    
│ │ │                
    i32 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 1.68103s (cpu); 1.31918s (thread); 0s (gc)
│ │ │ + -- used 1.99076s (cpu); 1.53973s (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 {}
│ │ │ │ @@ -72,19 +72,19 @@
│ │ │ │  
│ │ │ │  o5 : Ideal of T
│ │ │ │  i6 : S = T/I;
│ │ │ │  i7 : dim S
│ │ │ │  
│ │ │ │  o7 = 3
│ │ │ │  i8 : time regularInCodimension(1, S)
│ │ │ │ - -- used 0.67198s (cpu); 0.551786s (thread); 0s (gc)
│ │ │ │ + -- used 0.831973s (cpu); 0.612865s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = true
│ │ │ │  i9 : time regularInCodimension(2, S)
│ │ │ │ - -- used 6.48256s (cpu); 4.93012s (thread); 0s (gc)
│ │ │ │ + -- used 7.53097s (cpu); 5.47385s (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.
│ │ │ │ @@ -92,27 +92,27 @@
│ │ │ │  (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 0.020027s (cpu); 0.0198154s (thread); 0s (gc)
│ │ │ │ + -- used 0.0241001s (cpu); 0.021182s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o12 = -1
│ │ │ │  i13 : time regularInCodimension(2, R)
│ │ │ │ - -- used 0.192016s (cpu); 0.160849s (thread); 0s (gc)
│ │ │ │ + -- used 0.215181s (cpu); 0.153065s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o13 = true
│ │ │ │  i14 : time regularInCodimension(2, R)
│ │ │ │ - -- used 0.909538s (cpu); 0.600324s (thread); 0s (gc)
│ │ │ │ + -- used 1.06255s (cpu); 0.655467s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o14 = true
│ │ │ │  i15 : time regularInCodimension(2, R)
│ │ │ │ - -- used 1.3289s (cpu); 0.899094s (thread); 0s (gc)
│ │ │ │ + -- used 1.54486s (cpu); 1.00318s (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)
│ │ │ │ @@ -461,15 +461,15 @@
│ │ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │ -internalChooseMinor: Ch -- used 6.59871s (cpu); 5.02176s (thread); 0s (gc)
│ │ │ │ +internalChooseMinor: Ch -- used 7.63003s (cpu); 5.64893s (thread); 0s (gc)
│ │ │ │  oosing GRevLexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │ @@ -515,15 +515,15 @@
│ │ │ │  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 1.29022s (cpu); 0.996476s (thread); 0s (gc)
│ │ │ │ + -- used 1.54965s (cpu); 1.15431s (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 LexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │ │ @@ -590,51 +590,51 @@
│ │ │ │  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 0.31652s (cpu); 0.221889s (thread); 0s (gc)
│ │ │ │ + -- used 0.364553s (cpu); 0.234375s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o21 = true
│ │ │ │  i22 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 0.121412s (cpu); 0.0729141s (thread); 0s (gc)
│ │ │ │ + -- used 0.148619s (cpu); 0.0866179s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o22 = true
│ │ │ │  i23 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 0.36055s (cpu); 0.269016s (thread); 0s (gc)
│ │ │ │ + -- used 0.473111s (cpu); 0.341109s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = true
│ │ │ │  i24 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 1.61547s (cpu); 1.20594s (thread); 0s (gc)
│ │ │ │ + -- used 2.16122s (cpu); 1.5762s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o24 = true
│ │ │ │  i25 : StrategyCurrent#LexSmallest = 0;
│ │ │ │  i26 : StrategyCurrent#LexSmallestTerm = 100;
│ │ │ │  i27 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 2.16839s (cpu); 1.58684s (thread); 0s (gc)
│ │ │ │ + -- used 2.6435s (cpu); 1.82766s (thread); 0s (gc)
│ │ │ │  i28 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 2.24976s (cpu); 1.60689s (thread); 0s (gc)
│ │ │ │ + -- used 2.75088s (cpu); 1.86584s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o28 = true
│ │ │ │  i29 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 0.419528s (cpu); 0.335678s (thread); 0s (gc)
│ │ │ │ + -- used 0.527054s (cpu); 0.391424s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o29 = true
│ │ │ │  i30 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 0.6816s (cpu); 0.545534s (thread); 0s (gc)
│ │ │ │ + -- used 0.856958s (cpu); 0.645873s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o30 = true
│ │ │ │  i31 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ │ - -- used 0.990173s (cpu); 0.815558s (thread); 0s (gc)
│ │ │ │ + -- used 1.17916s (cpu); 0.909361s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o31 = true
│ │ │ │  i32 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ │ - -- used 1.68103s (cpu); 1.31918s (thread); 0s (gc)
│ │ │ │ + -- used 1.99076s (cpu); 1.53973s (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/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 0.00283257s (cpu); 0.00282918s (thread); 0s (gc)
│ │ │ + -- used 0.00331665s (cpu); 0.00331373s (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 0.00630435s (cpu); 0.00630588s (thread); 0s (gc)
│ │ │ + -- used 0.00700742s (cpu); 0.0070105s (thread); 0s (gc)
│ │ │  
│ │ │  o16 : Ideal of R
│ │ │  
│ │ │  i17 : I==J
│ │ │  
│ │ │  o17 = true
│ │ ├── ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/___Fitting_spideals_spof_spfinite_spmodules.html
│ │ │ @@ -202,26 +202,26 @@
│ │ │                8      8
│ │ │  o14 : Matrix R  <-- R
    
│ │ │              
    
│ │ │                
    i15 : time I=co1Fitting(K3)
│ │ │ - -- used 0.00283257s (cpu); 0.00282918s (thread); 0s (gc)
│ │ │ + -- used 0.00331665s (cpu); 0.00331373s (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 0.00630435s (cpu); 0.00630588s (thread); 0s (gc)
│ │ │ + -- used 0.00700742s (cpu); 0.0070105s (thread); 0s (gc)
│ │ │  
│ │ │  o16 : Ideal of R
    
│ │ │              
    
│ │ │                
    i17 : I==J
│ │ │ ├── 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 0.00283257s (cpu); 0.00282918s (thread); 0s (gc)
│ │ │ │ + -- used 0.00331665s (cpu); 0.00331373s (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 0.00630435s (cpu); 0.00630588s (thread); 0s (gc)
│ │ │ │ + -- used 0.00700742s (cpu); 0.0070105s (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/ForeignFunctions/example-output/___Foreign__Object.out
│ │ │ @@ -4,19 +4,19 @@
│ │ │  
│ │ │  o1 = 5
│ │ │  
│ │ │  o1 : ForeignObject of type int32
│ │ │  
│ │ │  i2 : peek x
│ │ │  
│ │ │ -o2 = int32{Address => 0x7ff9a1968ea0}
│ │ │ +o2 = int32{Address => 0x7efef29df6a0}
│ │ │  
│ │ │  i3 : address x
│ │ │  
│ │ │ -o3 = 0x7ff9a1968ea0
│ │ │ +o3 = 0x7efef29df6a0
│ │ │  
│ │ │  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 = {0x7ff9a19b2640, 0x7ff9a19b25e0, 0x7ff9a19b25d0}
│ │ │ +o3 = {0x7efef2a2c130, 0x7efef2a2c120, 0x7efef2a2c0f0}
│ │ │  
│ │ │  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 = {0x7ff9a19b23d0, 0x7ff9a19b23c0, 0x7ff9a19b23b0}
│ │ │ +o2 = {0x7efef2a75f50, 0x7efef2a75f40, 0x7efef2a75f20}
│ │ │  
│ │ │  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 = 0x7ff9a4235e20
│ │ │ +o1 = 0x7efef52b1700
│ │ │  
│ │ │  o1 : Pointer
│ │ │  
│ │ │  i2 : voidstar ptr
│ │ │  
│ │ │ -o2 = 0x7ff9a4235e20
│ │ │ +o2 = 0x7efef52b1700
│ │ │  
│ │ │  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 = 0x7ff9a19685d0
│ │ │ +o2 = 0x7efef29df830
│ │ │  
│ │ │  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 = 0x7ff9a19681f0
│ │ │ +o1 = 0x7efef2a2cc80
│ │ │  
│ │ │  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" => 6.95187e-310}
│ │ │ +o2 = HashTable{"bar" => 6.89861e-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 => 0x7ff9a191e520}
│ │ │ +o2 = int32{Address => 0x7efef29df7d0}
│ │ │  
│ │ │  i3 : ptr = address x
│ │ │  
│ │ │ -o3 = 0x7ff9a191e520
│ │ │ +o3 = 0x7efef29df7d0
│ │ │  
│ │ │  o3 : Pointer
│ │ │  
│ │ │  i4 : ptr + 5
│ │ │  
│ │ │ -o4 = 0x7ff9a191e525
│ │ │ +o4 = 0x7efef29df7d5
│ │ │  
│ │ │  o4 : Pointer
│ │ │  
│ │ │  i5 : ptr - 3
│ │ │  
│ │ │ -o5 = 0x7ff9a191e51d
│ │ │ +o5 = 0x7efef29df7cd
│ │ │  
│ │ │  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{0x7ff9b01dc550, mpfr}
│ │ │ +o2 = SharedLibrary{0x7eff06b71550, 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 = 0x7ff9a1968480
│ │ │ +o2 = 0x7efef29dfc10
│ │ │  
│ │ │  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 = 0x563b04170b40
│ │ │ +o1 = 0x55bffd4d6b40
│ │ │  
│ │ │  o1 : Pointer
│ │ │  
│ │ │  i2 : address int 5
│ │ │  
│ │ │ -o2 = 0x7ff9a191e630
│ │ │ +o2 = 0x7efef29dfa60
│ │ │  
│ │ │  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 = 0x7fc08c06a460
│ │ │ +o17 = 0x7f612806a460
│ │ │  
│ │ │  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 = 0x7ff9aba19160
│ │ │ +o1 = 0x7eff017b9490
│ │ │  
│ │ │  o1 : ForeignObject of type void*
│ │ │  
│ │ │  i2 : ptr = getMemory(8, Atomic => true)
│ │ │  
│ │ │ -o2 = 0x7ff9a191e0c0
│ │ │ +o2 = 0x7efef29df340
│ │ │  
│ │ │  o2 : ForeignObject of type void*
│ │ │  
│ │ │  i3 : ptr = getMemory int
│ │ │  
│ │ │ -o3 = 0x7ff9a1968fd0
│ │ │ +o3 = 0x7efef29df230
│ │ │  
│ │ │  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 0x7ff99407f900
│ │ │ -freeing memory at 0x7ff99407f920
│ │ │ -freeing memory at 0x7ff99407f8a0
│ │ │ -freeing memory at 0x7ff99407f1c0
│ │ │ -freeing memory at 0x7ff99407f1a0
│ │ │ -freeing memory at 0x7ff99407f860
│ │ │ -freeing memory at 0x7ff99407f880
│ │ │ -freeing memory at 0x7ff99407f8c0
│ │ │ -freeing memory at 0x7ff99407f8e0
│ │ │ +freeing memory at 0x7efedc07f900
│ │ │ +freeing memory at 0x7efedc07f8e0
│ │ │ +freeing memory at 0x7efedc07f8c0
│ │ │ +freeing memory at 0x7efedc07f8a0
│ │ │ +freeing memory at 0x7efedc07f880
│ │ │ +freeing memory at 0x7efedc07f860
│ │ │ +freeing memory at 0x7efedc07f1a0
│ │ │ +freeing memory at 0x7efedc07f920
│ │ │ +freeing memory at 0x7efedc07f1c0
│ │ │  
│ │ │  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 = 0x7ff9a1968860
│ │ │ +o5 = 0x7efef2a2cce0
│ │ │  
│ │ │  o5 : ForeignObject of type void*
│ │ │  
│ │ │  i6 : value x
│ │ │  
│ │ │ -o6 = 0x7ff9a1968860
│ │ │ +o6 = 0x7efef2a2cce0
│ │ │  
│ │ │  o6 : Pointer
│ │ │  
│ │ │  i7 : x = charstar "Hello, world!"
│ │ │  
│ │ │  o7 = Hello, world!
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Object.html
│ │ │ @@ -64,27 +64,27 @@
│ │ │  o1 : ForeignObject of type int32
    
│ │ │              
    
│ │ │                
    i2 : peek x
│ │ │  
│ │ │ -o2 = int32{Address => 0x7ff9a1968ea0}
    
│ │ │ +o2 = int32{Address => 0x7efef29df6a0}
│ │ │              
    
│ │ │          
    
│ │ │            
    To get this, use address.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i3 : address x
│ │ │  
│ │ │ -o3 = 0x7ff9a1968ea0
│ │ │ +o3 = 0x7efef29df6a0
│ │ │  
│ │ │  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 => 0x7ff9a1968ea0}
│ │ │ │ +o2 = int32{Address => 0x7efef29df6a0}
│ │ │ │  To get this, use _a_d_d_r_e_s_s.
│ │ │ │  i3 : address x
│ │ │ │  
│ │ │ │ -o3 = 0x7ff9a1968ea0
│ │ │ │ +o3 = 0x7efef29df6a0
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -74,15 +74,15 @@
│ │ │  o2 : ForeignObject of type char**
    
│ │ │              
    
│ │ │                
    i3 : voidstarstar {address int 0, address int 1, address int 2}
│ │ │  
│ │ │ -o3 = {0x7ff9a19b2640, 0x7ff9a19b25e0, 0x7ff9a19b25d0}
│ │ │ +o3 = {0x7efef2a2c130, 0x7efef2a2c120, 0x7efef2a2c0f0}
│ │ │  
│ │ │  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 = {0x7ff9a19b2640, 0x7ff9a19b25e0, 0x7ff9a19b25d0}
│ │ │ │ +o3 = {0x7efef2a2c130, 0x7efef2a2c120, 0x7efef2a2c0f0}
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -82,15 +82,15 @@
│ │ │  o1 : ForeignObject of type char**
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : voidstarstar {address int 0, address int 1, address int 2}
│ │ │  
│ │ │ -o2 = {0x7ff9a19b23d0, 0x7ff9a19b23c0, 0x7ff9a19b23b0}
│ │ │ +o2 = {0x7efef2a75f50, 0x7efef2a75f40, 0x7efef2a75f20}
│ │ │  
│ │ │  o2 : ForeignObject of type void**
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : int2star = foreignPointerArrayType(2 * int)
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,15 +20,15 @@
│ │ │ │  i1 : charstarstar {"foo", "bar"}
│ │ │ │  
│ │ │ │  o1 = {foo, bar}
│ │ │ │  
│ │ │ │  o1 : ForeignObject of type char**
│ │ │ │  i2 : voidstarstar {address int 0, address int 1, address int 2}
│ │ │ │  
│ │ │ │ -o2 = {0x7ff9a19b23d0, 0x7ff9a19b23c0, 0x7ff9a19b23b0}
│ │ │ │ +o2 = {0x7efef2a75f50, 0x7efef2a75f40, 0x7efef2a75f20}
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -73,24 +73,24 @@
│ │ │            
    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 = 0x7ff9a4235e20
│ │ │ +o1 = 0x7efef52b1700
│ │ │  
│ │ │  o1 : Pointer
    
│ │ │              
    
│ │ │                
    i2 : voidstar ptr
│ │ │  
│ │ │ -o2 = 0x7ff9a4235e20
│ │ │ +o2 = 0x7efef52b1700
│ │ │  
│ │ │  o2 : ForeignObject of type void*
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,20 +15,20 @@
│ │ │ │      * 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 = 0x7ff9a4235e20
│ │ │ │ +o1 = 0x7efef52b1700
│ │ │ │  
│ │ │ │  o1 : Pointer
│ │ │ │  i2 : voidstar ptr
│ │ │ │  
│ │ │ │ -o2 = 0x7ff9a4235e20
│ │ │ │ +o2 = 0x7efef52b1700
│ │ │ │  
│ │ │ │  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
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Type_sp__Pointer.html
│ │ │ @@ -82,15 +82,15 @@
│ │ │  o1 : ForeignObject of type int32
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : ptr = address x
│ │ │  
│ │ │ -o2 = 0x7ff9a19685d0
│ │ │ +o2 = 0x7efef29df830
│ │ │  
│ │ │  o2 : Pointer
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : int ptr
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,15 +18,15 @@
│ │ │ │  i1 : x = int 5
│ │ │ │  
│ │ │ │  o1 = 5
│ │ │ │  
│ │ │ │  o1 : ForeignObject of type int32
│ │ │ │  i2 : ptr = address x
│ │ │ │  
│ │ │ │ -o2 = 0x7ff9a19685d0
│ │ │ │ +o2 = 0x7efef29df830
│ │ │ │  
│ │ │ │  o2 : Pointer
│ │ │ │  i3 : int ptr
│ │ │ │  
│ │ │ │  o3 = 5
│ │ │ │  
│ │ │ │  o3 : ForeignObject of type int32
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Type_sp_st_spvoidstar.html
│ │ │ @@ -73,15 +73,15 @@
│ │ │            
    This is syntactic sugar for T value ptr (see ForeignType Pointer) for dereferencing pointers.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : ptr = voidstar address int 5
│ │ │  
│ │ │ -o1 = 0x7ff9a19681f0
│ │ │ +o1 = 0x7efef2a2cc80
│ │ │  
│ │ │  o1 : ForeignObject of type void*
    
│ │ │              
    
│ │ │                
    i2 : int * ptr
│ │ │ ├── html2text {}
│ │ │ │ @@ -14,15 +14,15 @@
│ │ │ │      * 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 = 0x7ff9a19681f0
│ │ │ │ +o1 = 0x7efef2a2cc80
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -82,15 +82,15 @@
│ │ │  o1 : ForeignUnionType
    
│ │ │              
    
│ │ │                
    i2 : myunion 27
│ │ │  
│ │ │ -o2 = HashTable{"bar" => 6.95187e-310}
│ │ │ +o2 = HashTable{"bar" => 6.89861e-310}
│ │ │                 "foo" => 27
│ │ │  
│ │ │  o2 : ForeignObject of type myunion
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,15 +20,15 @@
│ │ │ │  i1 : myunion = foreignUnionType("myunion", {"foo" => int, "bar" => double})
│ │ │ │  
│ │ │ │  o1 = myunion
│ │ │ │  
│ │ │ │  o1 : ForeignUnionType
│ │ │ │  i2 : myunion 27
│ │ │ │  
│ │ │ │ -o2 = HashTable{"bar" => 6.95187e-310}
│ │ │ │ +o2 = HashTable{"bar" => 6.89861e-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
│ │ │ @@ -64,50 +64,50 @@
│ │ │  o1 : ForeignObject of type int32
│ │ │              
    
│ │ │                
    i2 : peek x
│ │ │  
│ │ │ -o2 = int32{Address => 0x7ff9a191e520}
    
│ │ │ +o2 = int32{Address => 0x7efef29df7d0}
│ │ │              
    
│ │ │          
    
│ │ │            
    These pointers can be accessed using address.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i3 : ptr = address x
│ │ │  
│ │ │ -o3 = 0x7ff9a191e520
│ │ │ +o3 = 0x7efef29df7d0
│ │ │  
│ │ │  o3 : Pointer
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Simple arithmetic can be performed on pointers.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : ptr + 5
│ │ │  
│ │ │ -o4 = 0x7ff9a191e525
│ │ │ +o4 = 0x7efef29df7d5
│ │ │  
│ │ │  o4 : Pointer
    
│ │ │              
    
│ │ │                
    i5 : ptr - 3
│ │ │  
│ │ │ -o5 = 0x7ff9a191e51d
│ │ │ +o5 = 0x7efef29df7cd
│ │ │  
│ │ │  o5 : Pointer
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,30 +10,30 @@
│ │ │ │  i1 : x = int 20
│ │ │ │  
│ │ │ │  o1 = 20
│ │ │ │  
│ │ │ │  o1 : ForeignObject of type int32
│ │ │ │  i2 : peek x
│ │ │ │  
│ │ │ │ -o2 = int32{Address => 0x7ff9a191e520}
│ │ │ │ +o2 = int32{Address => 0x7efef29df7d0}
│ │ │ │  These pointers can be accessed using _a_d_d_r_e_s_s.
│ │ │ │  i3 : ptr = address x
│ │ │ │  
│ │ │ │ -o3 = 0x7ff9a191e520
│ │ │ │ +o3 = 0x7efef29df7d0
│ │ │ │  
│ │ │ │  o3 : Pointer
│ │ │ │  Simple arithmetic can be performed on pointers.
│ │ │ │  i4 : ptr + 5
│ │ │ │  
│ │ │ │ -o4 = 0x7ff9a191e525
│ │ │ │ +o4 = 0x7efef29df7d5
│ │ │ │  
│ │ │ │  o4 : Pointer
│ │ │ │  i5 : ptr - 3
│ │ │ │  
│ │ │ │ -o5 = 0x7ff9a191e51d
│ │ │ │ +o5 = 0x7efef29df7cd
│ │ │ │  
│ │ │ │  o5 : Pointer
│ │ │ │  ******** MMeennuu ********
│ │ │ │      * _n_u_l_l_P_o_i_n_t_e_r -- the null pointer
│ │ │ │      * _a_d_d_r_e_s_s -- pointer to type or object
│ │ │ │      * _F_o_r_e_i_g_n_T_y_p_e_ _P_o_i_n_t_e_r -- dereference a pointer
│ │ │ │  ********** FFuunnccttiioonnss aanndd mmeetthhooddss rreettuurrnniinngg aa ppooiinntteerr:: **********
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Shared__Library.html
│ │ │ @@ -64,15 +64,15 @@
│ │ │  o1 : SharedLibrary
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : peek mpfr
│ │ │  
│ │ │ -o2 = SharedLibrary{0x7ff9b01dc550, mpfr}
    
│ │ │ +o2 = SharedLibrary{0x7eff06b71550, mpfr}
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │          
    Menu
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : address int
│ │ │  
│ │ │ -o1 = 0x563b04170b40
│ │ │ +o1 = 0x55bffd4d6b40
│ │ │  
│ │ │  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 = 0x7ff9a191e630
│ │ │ +o2 = 0x7efef29dfa60
│ │ │  
│ │ │  o2 : Pointer
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,22 +11,22 @@
│ │ │ │      * 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 = 0x563b04170b40
│ │ │ │ +o1 = 0x55bffd4d6b40
│ │ │ │  
│ │ │ │  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 = 0x7ff9a191e630
│ │ │ │ +o2 = 0x7efef29dfa60
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -232,15 +232,15 @@
│ │ │  o16 : ForeignFunction
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i17 : x = malloc 8
│ │ │  
│ │ │ -o17 = 0x7fc08c06a460
│ │ │ +o17 = 0x7f612806a460
│ │ │  
│ │ │  o17 : ForeignObject of type void*
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i18 : registerFinalizer(x, free)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -95,15 +95,15 @@
│ │ │ │  i16 : free = foreignFunction("free", void, voidstar)
│ │ │ │  
│ │ │ │  o16 = free
│ │ │ │  
│ │ │ │  o16 : ForeignFunction
│ │ │ │  i17 : x = malloc 8
│ │ │ │  
│ │ │ │ -o17 = 0x7fc08c06a460
│ │ │ │ +o17 = 0x7f612806a460
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -77,43 +77,43 @@
│ │ │            
    Allocate n bytes of memory using the GC garbage collector.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : ptr = getMemory 8
│ │ │  
│ │ │ -o1 = 0x7ff9aba19160
│ │ │ +o1 = 0x7eff017b9490
│ │ │  
│ │ │  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 = 0x7ff9a191e0c0
│ │ │ +o2 = 0x7efef29df340
│ │ │  
│ │ │  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 = 0x7ff9a1968fd0
│ │ │ +o3 = 0x7efef29df230
│ │ │  
│ │ │  o3 : ForeignObject of type void*
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -14,30 +14,30 @@
│ │ │ │            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 = 0x7ff9aba19160
│ │ │ │ +o1 = 0x7eff017b9490
│ │ │ │  
│ │ │ │  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 = 0x7ff9a191e0c0
│ │ │ │ +o2 = 0x7efef29df340
│ │ │ │  
│ │ │ │  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 = 0x7ff9a1968fd0
│ │ │ │ +o3 = 0x7efef29df230
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -100,23 +100,23 @@
│ │ │              
│ │ │                
    i4 : for i to 9 do (x := malloc 8; registerFinalizer(x, finalizer))
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : collectGarbage()
│ │ │ -freeing memory at 0x7ff99407f900
│ │ │ -freeing memory at 0x7ff99407f920
│ │ │ -freeing memory at 0x7ff99407f8a0
│ │ │ -freeing memory at 0x7ff99407f1c0
│ │ │ -freeing memory at 0x7ff99407f1a0
│ │ │ -freeing memory at 0x7ff99407f860
│ │ │ -freeing memory at 0x7ff99407f880
│ │ │ -freeing memory at 0x7ff99407f8c0
│ │ │ -freeing memory at 0x7ff99407f8e0
    
│ │ │ +freeing memory at 0x7efedc07f900
│ │ │ +freeing memory at 0x7efedc07f8e0
│ │ │ +freeing memory at 0x7efedc07f8c0
│ │ │ +freeing memory at 0x7efedc07f8a0
│ │ │ +freeing memory at 0x7efedc07f880
│ │ │ +freeing memory at 0x7efedc07f860
│ │ │ +freeing memory at 0x7efedc07f1a0
│ │ │ +freeing memory at 0x7efedc07f920
│ │ │ +freeing memory at 0x7efedc07f1c0
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i5 : x = voidstar address int 5
│ │ │  
│ │ │ -o5 = 0x7ff9a1968860
│ │ │ +o5 = 0x7efef2a2cce0
│ │ │  
│ │ │  o5 : ForeignObject of type void*
    
│ │ │              
    
│ │ │                
    i6 : value x
│ │ │  
│ │ │ -o6 = 0x7ff9a1968860
│ │ │ +o6 = 0x7efef2a2cce0
│ │ │  
│ │ │  o6 : Pointer
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Foreign string objects are converted to strings.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,20 +34,20 @@
│ │ │ │  
│ │ │ │  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 = 0x7ff9a1968860
│ │ │ │ +o5 = 0x7efef2a2cce0
│ │ │ │  
│ │ │ │  o5 : ForeignObject of type void*
│ │ │ │  i6 : value x
│ │ │ │  
│ │ │ │ -o6 = 0x7ff9a1968860
│ │ │ │ +o6 = 0x7efef2a2cce0
│ │ │ │  
│ │ │ │  o6 : Pointer
│ │ │ │  Foreign string objects are converted to strings.
│ │ │ │  i7 : x = charstar "Hello, world!"
│ │ │ │  
│ │ │ │  o7 = Hello, world!
│ │ ├── ./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/M2-15954-0/0
│ │ │ +o2 = /tmp/M2-20955-0/0
│ │ │  
│ │ │  i3 : F = openOut(s)
│ │ │  
│ │ │ -o3 = /tmp/M2-15954-0/0
│ │ │ +o3 = /tmp/M2-20955-0/0
│ │ │  
│ │ │  o3 : File
│ │ │  
│ │ │  i4 : putMatrix(F,A)
│ │ │  
│ │ │  i5 : close(F)
│ │ │  
│ │ │ -o5 = /tmp/M2-15954-0/0
│ │ │ +o5 = /tmp/M2-20955-0/0
│ │ │  
│ │ │  o5 : File
│ │ │  
│ │ │  i6 : getMatrix(s)
│ │ │  
│ │ │  o6 = | 1 1 1 1 |
│ │ │       | 1 2 3 4 |
│ │ ├── ./usr/share/doc/Macaulay2/FourTiTwo/html/_put__Matrix.html
│ │ │ @@ -79,36 +79,36 @@
│ │ │  o1 : Matrix ZZ  <-- ZZ
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : s = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-15954-0/0
    
│ │ │ +o2 = /tmp/M2-20955-0/0
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : F = openOut(s)
│ │ │  
│ │ │ -o3 = /tmp/M2-15954-0/0
│ │ │ +o3 = /tmp/M2-20955-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : putMatrix(F,A)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : close(F)
│ │ │  
│ │ │ -o5 = /tmp/M2-15954-0/0
│ │ │ +o5 = /tmp/M2-20955-0/0
│ │ │  
│ │ │  o5 : File
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : getMatrix(s)
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,24 +16,24 @@
│ │ │ │  o1 = | 1 1 1 1 |
│ │ │ │       | 1 2 3 4 |
│ │ │ │  
│ │ │ │                2       4
│ │ │ │  o1 : Matrix ZZ  <-- ZZ
│ │ │ │  i2 : s = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-15954-0/0
│ │ │ │ +o2 = /tmp/M2-20955-0/0
│ │ │ │  i3 : F = openOut(s)
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-15954-0/0
│ │ │ │ +o3 = /tmp/M2-20955-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : putMatrix(F,A)
│ │ │ │  i5 : close(F)
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-15954-0/0
│ │ │ │ +o5 = /tmp/M2-20955-0/0
│ │ │ │  
│ │ │ │  o5 : File
│ │ │ │  i6 : getMatrix(s)
│ │ │ │  
│ │ │ │  o6 = | 1 1 1 1 |
│ │ │ │       | 1 2 3 4 |
│ │ ├── ./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 2.29687s (cpu); 1.3621s (thread); 0s (gc)
│ │ │ + -- used 2.5999s (cpu); 1.45585s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = {.142067, .144}
│ │ │  
│ │ │  o27 : List
│ │ │  
│ │ │  i28 : time fpt(f, DepthOfSearch => 3, Attempts => 7)
│ │ │ - -- used 1.29681s (cpu); 0.79399s (thread); 0s (gc)
│ │ │ + -- used 1.49988s (cpu); 0.878563s (thread); 0s (gc)
│ │ │  
│ │ │        1
│ │ │  o28 = -
│ │ │        7
│ │ │  
│ │ │  o28 : QQ
│ │ │  
│ │ │  i29 : time fpt(f, DepthOfSearch => 4)
│ │ │ - -- used 1.10683s (cpu); 0.719671s (thread); 0s (gc)
│ │ │ + -- used 1.20403s (cpu); 0.711664s (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 0.00400825s (cpu); 0.00419427s (thread); 0s (gc)
│ │ │ + -- used 0.00396017s (cpu); 0.0053191s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3756
│ │ │  
│ │ │  i16 : time frobeniusNu(3, f, UseSpecialAlgorithms => false)
│ │ │ - -- used 0.382087s (cpu); 0.251102s (thread); 0s (gc)
│ │ │ + -- used 0.411528s (cpu); 0.286891s (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 0.270979s (cpu); 0.184269s (thread); 0s (gc)
│ │ │ + -- used 0.455469s (cpu); 0.271824s (thread); 0s (gc)
│ │ │  
│ │ │  o19 = 499
│ │ │  
│ │ │  i20 : time frobeniusNu(4, f, ContainmentTest => StandardPower)
│ │ │ - -- used 1.51677s (cpu); 1.13247s (thread); 0s (gc)
│ │ │ + -- used 1.43651s (cpu); 1.23635s (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 1.05755s (cpu); 0.695354s (thread); 0s (gc)
│ │ │ + -- used 1.16681s (cpu); 0.722538s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = 1124
│ │ │  
│ │ │  i28 : time frobeniusNu(5, f, Search => Linear)
│ │ │ - -- used 1.51572s (cpu); 0.951915s (thread); 0s (gc)
│ │ │ + -- used 1.7171s (cpu); 1.03664s (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 1.72643s (cpu); 1.38582s (thread); 0s (gc)
│ │ │ + -- used 1.73244s (cpu); 1.4386s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = 97
│ │ │  
│ │ │  i31 : time frobeniusNu(2, M, M^2, Search => Linear) -- but linear search gets luckier
│ │ │ - -- used 0.613124s (cpu); 0.515605s (thread); 0s (gc)
│ │ │ + -- used 0.555251s (cpu); 0.493124s (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
│ │ │ @@ -363,37 +363,37 @@
│ │ │          
    
│ │ │            
    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 {}
│ │ │ │ @@ -228,29 +228,29 @@
│ │ │ │  
│ │ │ │  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 2.29687s (cpu); 1.3621s (thread); 0s (gc)
│ │ │ │ + -- used 2.5999s (cpu); 1.45585s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o27 = {.142067, .144}
│ │ │ │  
│ │ │ │  o27 : List
│ │ │ │  i28 : time fpt(f, DepthOfSearch => 3, Attempts => 7)
│ │ │ │ - -- used 1.29681s (cpu); 0.79399s (thread); 0s (gc)
│ │ │ │ + -- used 1.49988s (cpu); 0.878563s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        1
│ │ │ │  o28 = -
│ │ │ │        7
│ │ │ │  
│ │ │ │  o28 : QQ
│ │ │ │  i29 : time fpt(f, DepthOfSearch => 4)
│ │ │ │ - -- used 1.10683s (cpu); 0.719671s (thread); 0s (gc)
│ │ │ │ + -- used 1.20403s (cpu); 0.711664s (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
│ │ │ @@ -192,23 +192,23 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i27 : time numeric fpt(f, DepthOfSearch => 3, FinalAttempt => true)
│ │ │ - -- used 2.29687s (cpu); 1.3621s (thread); 0s (gc)
│ │ │ + -- used 2.5999s (cpu); 1.45585s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = {.142067, .144}
│ │ │  
│ │ │  o27 : List
    
│ │ │              
    
│ │ │                
    i28 : time fpt(f, DepthOfSearch => 3, Attempts => 7)
│ │ │ - -- used 1.29681s (cpu); 0.79399s (thread); 0s (gc)
│ │ │ + -- used 1.49988s (cpu); 0.878563s (thread); 0s (gc)
│ │ │  
│ │ │        1
│ │ │  o28 = -
│ │ │        7
│ │ │  
│ │ │  o28 : QQ
    
│ │ │              
    
│ │ │                
    i29 : time fpt(f, DepthOfSearch => 4)
│ │ │ - -- used 1.10683s (cpu); 0.719671s (thread); 0s (gc)
│ │ │ + -- used 1.20403s (cpu); 0.711664s (thread); 0s (gc)
│ │ │  
│ │ │        1
│ │ │  o29 = -
│ │ │        7
│ │ │  
│ │ │  o29 : QQ
    
│ │ │                  		
│ │ │                
    i14 : f = x^3 + y^4 + z^5; -- a diagonal polynomial
    
│ │ │              
    
│ │ │                
    i15 : time frobeniusNu(3, f)
│ │ │ - -- used 0.00400825s (cpu); 0.00419427s (thread); 0s (gc)
│ │ │ + -- used 0.00396017s (cpu); 0.0053191s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3756
    
│ │ │              
    
│ │ │                
    i16 : time frobeniusNu(3, f, UseSpecialAlgorithms => false)
│ │ │ - -- used 0.382087s (cpu); 0.251102s (thread); 0s (gc)
│ │ │ + -- used 0.411528s (cpu); 0.286891s (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.
    
│ │ │ @@ -225,23 +225,23 @@
│ │ │              
│ │ │                
    i18 : f = x^3 + y^3 + z^3 + x*y*z;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i19 : time frobeniusNu(4, f) -- ContainmentTest is set to FrobeniusRoot, by default
│ │ │ - -- used 0.270979s (cpu); 0.184269s (thread); 0s (gc)
│ │ │ + -- used 0.455469s (cpu); 0.271824s (thread); 0s (gc)
│ │ │  
│ │ │  o19 = 499
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i20 : time frobeniusNu(4, f, ContainmentTest => StandardPower)
│ │ │ - -- used 1.51677s (cpu); 1.13247s (thread); 0s (gc)
│ │ │ + -- used 1.43651s (cpu); 1.23635s (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.
    
│ │ │ @@ -287,46 +287,46 @@
│ │ │              
│ │ │                
    i26 : f = x^2*y^4 + y^2*z^7 + z^2*x^8;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i27 : time frobeniusNu(5, f) -- uses binary search (default)
│ │ │ - -- used 1.05755s (cpu); 0.695354s (thread); 0s (gc)
│ │ │ + -- used 1.16681s (cpu); 0.722538s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = 1124
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i28 : time frobeniusNu(5, f, Search => Linear)
│ │ │ - -- used 1.51572s (cpu); 0.951915s (thread); 0s (gc)
│ │ │ + -- used 1.7171s (cpu); 1.03664s (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 1.72643s (cpu); 1.38582s (thread); 0s (gc)
│ │ │ + -- used 1.73244s (cpu); 1.4386s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = 97
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i31 : time frobeniusNu(2, M, M^2, Search => Linear) -- but linear search gets luckier
│ │ │ - -- used 0.613124s (cpu); 0.515605s (thread); 0s (gc)
│ │ │ + -- used 0.555251s (cpu); 0.493124s (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 {}
│ │ │ │ @@ -106,19 +106,19 @@
│ │ │ │  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 0.00400825s (cpu); 0.00419427s (thread); 0s (gc)
│ │ │ │ + -- used 0.00396017s (cpu); 0.0053191s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = 3756
│ │ │ │  i16 : time frobeniusNu(3, f, UseSpecialAlgorithms => false)
│ │ │ │ - -- used 0.382087s (cpu); 0.251102s (thread); 0s (gc)
│ │ │ │ + -- used 0.411528s (cpu); 0.286891s (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
│ │ │ │ @@ -133,19 +133,19 @@
│ │ │ │  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 0.270979s (cpu); 0.184269s (thread); 0s (gc)
│ │ │ │ + -- used 0.455469s (cpu); 0.271824s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o19 = 499
│ │ │ │  i20 : time frobeniusNu(4, f, ContainmentTest => StandardPower)
│ │ │ │ - -- used 1.51677s (cpu); 1.13247s (thread); 0s (gc)
│ │ │ │ + -- used 1.43651s (cpu); 1.23635s (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
│ │ │ │ @@ -167,31 +167,31 @@
│ │ │ │  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 1.05755s (cpu); 0.695354s (thread); 0s (gc)
│ │ │ │ + -- used 1.16681s (cpu); 0.722538s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o27 = 1124
│ │ │ │  i28 : time frobeniusNu(5, f, Search => Linear)
│ │ │ │ - -- used 1.51572s (cpu); 0.951915s (thread); 0s (gc)
│ │ │ │ + -- used 1.7171s (cpu); 1.03664s (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 1.72643s (cpu); 1.38582s (thread); 0s (gc)
│ │ │ │ + -- used 1.73244s (cpu); 1.4386s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o30 = 97
│ │ │ │  i31 : time frobeniusNu(2, M, M^2, Search => Linear) -- but linear search gets
│ │ │ │  luckier
│ │ │ │ - -- used 0.613124s (cpu); 0.515605s (thread); 0s (gc)
│ │ │ │ + -- used 0.555251s (cpu); 0.493124s (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/GKMVarieties/example-output/_orbit__Closure.out
│ │ │ @@ -208,21 +208,21 @@
│ │ │        | 3/7 5/4 3/7  10   |
│ │ │        | 6/7 2/9 5    3/2  |
│ │ │  
│ │ │                 3       4
│ │ │  o26 : Matrix QQ  <-- QQ
│ │ │  
│ │ │  i27 : time C = orbitClosure(X,Mat)
│ │ │ - -- used 0.59381s (cpu); 0.347086s (thread); 0s (gc)
│ │ │ + -- used 1.31404s (cpu); 0.400669s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = an "equivariant K-class" on a GKM variety 
│ │ │  
│ │ │  o27 : KClass
│ │ │  
│ │ │  i28 : time C = orbitClosure(X,Mat, RREFMethod => true)
│ │ │ - -- used 1.80474s (cpu); 1.03709s (thread); 0s (gc)
│ │ │ + -- used 2.78577s (cpu); 1.01172s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = an "equivariant K-class" on a GKM variety 
│ │ │  
│ │ │  o28 : KClass
│ │ │  
│ │ │  i29 :
│ │ ├── ./usr/share/doc/Macaulay2/GKMVarieties/html/_orbit__Closure.html
│ │ │ @@ -386,25 +386,25 @@
│ │ │                 3       4
│ │ │  o26 : Matrix QQ  <-- QQ
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i27 : time C = orbitClosure(X,Mat)
│ │ │ - -- used 0.59381s (cpu); 0.347086s (thread); 0s (gc)
│ │ │ + -- used 1.31404s (cpu); 0.400669s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = an "equivariant K-class" on a GKM variety 
│ │ │  
│ │ │  o27 : KClass
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i28 : time C = orbitClosure(X,Mat, RREFMethod => true)
│ │ │ - -- used 1.80474s (cpu); 1.03709s (thread); 0s (gc)
│ │ │ + -- used 2.78577s (cpu); 1.01172s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = an "equivariant K-class" on a GKM variety 
│ │ │  
│ │ │  o28 : KClass
    
│ │ │              
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -241,21 +241,21 @@
│ │ │ │  o26 = | 7   6   3/10 10/9 |
│ │ │ │        | 3/7 5/4 3/7  10   |
│ │ │ │        | 6/7 2/9 5    3/2  |
│ │ │ │  
│ │ │ │                 3       4
│ │ │ │  o26 : Matrix QQ  <-- QQ
│ │ │ │  i27 : time C = orbitClosure(X,Mat)
│ │ │ │ - -- used 0.59381s (cpu); 0.347086s (thread); 0s (gc)
│ │ │ │ + -- used 1.31404s (cpu); 0.400669s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o27 = an "equivariant K-class" on a GKM variety
│ │ │ │  
│ │ │ │  o27 : KClass
│ │ │ │  i28 : time C = orbitClosure(X,Mat, RREFMethod => true)
│ │ │ │ - -- used 1.80474s (cpu); 1.03709s (thread); 0s (gc)
│ │ │ │ + -- used 2.78577s (cpu); 1.01172s (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/Graphs/example-output/_new__Digraph.out
│ │ │ @@ -32,12 +32,12 @@
│ │ │                                           5 => {6}
│ │ │                                           6 => {}
│ │ │  
│ │ │  o2 : SortedDigraph
│ │ │  
│ │ │  i3 : keys H
│ │ │  
│ │ │ -o3 = {map, digraph, newDigraph}
│ │ │ +o3 = {digraph, map, newDigraph}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/Graphs/html/_new__Digraph.html
│ │ │ @@ -95,15 +95,15 @@
│ │ │  o2 : SortedDigraph
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : keys H
│ │ │  
│ │ │ -o3 = {map, digraph, newDigraph}
│ │ │ +o3 = {digraph, map, newDigraph}
│ │ │  
│ │ │  o3 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -36,15 +36,15 @@
│ │ │ │                                           4 => {}
│ │ │ │                                           5 => {6}
│ │ │ │                                           6 => {}
│ │ │ │  
│ │ │ │  o2 : SortedDigraph
│ │ │ │  i3 : keys H
│ │ │ │  
│ │ │ │ -o3 = {map, digraph, newDigraph}
│ │ │ │ +o3 = {digraph, map, newDigraph}
│ │ │ │  
│ │ │ │  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/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
│ │ │ - -- 2.81604s elapsed
│ │ │ + -- 2.1341s elapsed
│ │ │  
│ │ │  o5 = GroebnerBasis[status: done; S-pairs encountered up to degree 16]
│ │ │  
│ │ │  o5 : GroebnerBasis
│ │ │  
│ │ │  i6 : elapsedTime groebnerWalk(gb I1, R2)
│ │ │ - -- 2.45746s elapsed
│ │ │ + -- 1.77611s elapsed
│ │ │  
│ │ │  o6 = GroebnerBasis[status: done; S-pairs encountered up to degree 0]
│ │ │  
│ │ │  o6 : GroebnerBasis
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/GroebnerWalk/html/index.html
│ │ │ @@ -92,30 +92,30 @@
│ │ │  
│ │ │  o4 : Ideal of R2
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : elapsedTime gb I2
│ │ │ - -- 2.81604s elapsed
│ │ │ + -- 2.1341s 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)
│ │ │ - -- 2.45746s elapsed
│ │ │ + -- 1.77611s 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
│ │ │ │ - -- 2.81604s elapsed
│ │ │ │ + -- 2.1341s 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)
│ │ │ │ - -- 2.45746s elapsed
│ │ │ │ + -- 1.77611s 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/example-output/_hadamard__Power_lp__List_cm__Z__Z_rp.out
│ │ │ @@ -6,20 +6,22 @@
│ │ │  o1 = {Point{1, 1, -}, Point{1, 0, 1}, Point{1, 2, 4}}
│ │ │                    2
│ │ │  
│ │ │  o1 : List
│ │ │  
│ │ │  i2 : hadamardPower(L,3)
│ │ │  
│ │ │ -                  1                                                    
│ │ │ -o2 = {Point{1, 0, -}, Point{1, 8, 64}, Point{1, 4, 8}, Point{1, 0, 16},
│ │ │ -                  4                                                    
│ │ │ +                  1                                                  
│ │ │ +o2 = {Point{1, 0, -}, Point{1, 0, 2}, Point{1, 2, 1}, Point{1, 0, 4},
│ │ │ +                  2                                                  
│ │ │       ------------------------------------------------------------------------
│ │ │ -                                 1                               1
│ │ │ -     Point{1, 0, 1}, Point{1, 1, -}, Point{1, 0, 2}, Point{1, 0, -}, Point{1,
│ │ │ -                                 8                               2
│ │ │ +                 1                                                    
│ │ │ +     Point{1, 0, -}, Point{1, 8, 64}, Point{1, 4, 8}, Point{1, 0, 16},
│ │ │ +                 4                                                    
│ │ │       ------------------------------------------------------------------------
│ │ │ -     2, 1}, Point{1, 0, 4}}
│ │ │ +                                 1
│ │ │ +     Point{1, 0, 1}, Point{1, 1, -}}
│ │ │ +                                 8
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Hadamard/example-output/_hadamard__Product_lp__List_cm__List_rp.out
│ │ │ @@ -2,12 +2,12 @@
│ │ │  
│ │ │  i1 : L = {point{0,1}, point{1,2}};
│ │ │  
│ │ │  i2 : M = {point{1,0}, point{2,2}};
│ │ │  
│ │ │  i3 : hadamardProduct(L,M)
│ │ │  
│ │ │ -o3 = {Point{1, 0}, Point{0, 2}, Point{2, 4}}
│ │ │ +o3 = {Point{2, 4}, Point{1, 0}, Point{0, 2}}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/Hadamard/html/_hadamard__Power_lp__List_cm__Z__Z_rp.html
│ │ │ @@ -84,23 +84,25 @@
│ │ │  o1 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : hadamardPower(L,3)
│ │ │  
│ │ │ -                  1                                                    
│ │ │ -o2 = {Point{1, 0, -}, Point{1, 8, 64}, Point{1, 4, 8}, Point{1, 0, 16},
│ │ │ -                  4                                                    
│ │ │ +                  1                                                  
│ │ │ +o2 = {Point{1, 0, -}, Point{1, 0, 2}, Point{1, 2, 1}, Point{1, 0, 4},
│ │ │ +                  2                                                  
│ │ │       ------------------------------------------------------------------------
│ │ │ -                                 1                               1
│ │ │ -     Point{1, 0, 1}, Point{1, 1, -}, Point{1, 0, 2}, Point{1, 0, -}, Point{1,
│ │ │ -                                 8                               2
│ │ │ +                 1                                                    
│ │ │ +     Point{1, 0, -}, Point{1, 8, 64}, Point{1, 4, 8}, Point{1, 0, 16},
│ │ │ +                 4                                                    
│ │ │       ------------------------------------------------------------------------
│ │ │ -     2, 1}, Point{1, 0, 4}}
│ │ │ +                                 1
│ │ │ +     Point{1, 0, 1}, Point{1, 1, -}}
│ │ │ +                                 8
│ │ │  
│ │ │  o2 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -22,22 +22,24 @@
│ │ │ │  o1 = {Point{1, 1, -}, Point{1, 0, 1}, Point{1, 2, 4}}
│ │ │ │                    2
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  i2 : hadamardPower(L,3)
│ │ │ │  
│ │ │ │                    1
│ │ │ │ -o2 = {Point{1, 0, -}, Point{1, 8, 64}, Point{1, 4, 8}, Point{1, 0, 16},
│ │ │ │ -                  4
│ │ │ │ +o2 = {Point{1, 0, -}, Point{1, 0, 2}, Point{1, 2, 1}, Point{1, 0, 4},
│ │ │ │ +                  2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -                                 1                               1
│ │ │ │ -     Point{1, 0, 1}, Point{1, 1, -}, Point{1, 0, 2}, Point{1, 0, -}, Point{1,
│ │ │ │ -                                 8                               2
│ │ │ │ +                 1
│ │ │ │ +     Point{1, 0, -}, Point{1, 8, 64}, Point{1, 4, 8}, Point{1, 0, 16},
│ │ │ │ +                 4
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     2, 1}, Point{1, 0, 4}}
│ │ │ │ +                                 1
│ │ │ │ +     Point{1, 0, 1}, Point{1, 1, -}}
│ │ │ │ +                                 8
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _h_a_d_a_m_a_r_d_P_o_w_e_r_(_L_i_s_t_,_Z_Z_) -- computes the $r$-th Hadmard powers of a set
│ │ │ │        points
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/Hadamard/html/_hadamard__Product_lp__List_cm__List_rp.html
│ │ │ @@ -83,15 +83,15 @@
│ │ │                
    i2 : M = {point{1,0}, point{2,2}};
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : hadamardProduct(L,M)
│ │ │  
│ │ │ -o3 = {Point{1, 0}, Point{0, 2}, Point{2, 4}}
│ │ │ +o3 = {Point{2, 4}, Point{1, 0}, Point{0, 2}}
│ │ │  
│ │ │  o3 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,15 +17,15 @@
│ │ │ │  Given two sets of points $L$ and $M$ returns the list of (well-defined)
│ │ │ │  entrywise multiplication of pairs of points in the cartesian product $L\times
│ │ │ │  M$.
│ │ │ │  i1 : L = {point{0,1}, point{1,2}};
│ │ │ │  i2 : M = {point{1,0}, point{2,2}};
│ │ │ │  i3 : hadamardProduct(L,M)
│ │ │ │  
│ │ │ │ -o3 = {Point{1, 0}, Point{0, 2}, Point{2, 4}}
│ │ │ │ +o3 = {Point{2, 4}, Point{1, 0}, Point{0, 2}}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _h_a_d_a_m_a_r_d_P_r_o_d_u_c_t_(_L_i_s_t_,_L_i_s_t_) -- Hadamard product of two sets of points
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/Hadamard.m2:345:0.
│ │ ├── ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_css__Lead__Term.out
│ │ │ @@ -44,19 +44,19 @@
│ │ │  o5 = {9, 1, 99999, 9999999, 3, 999}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : netList cssLeadTerm(Hbeta, w)
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │ - -- .000003256s elapsed
│ │ │ - -- .000002485s elapsed
│ │ │ - -- .000004138s elapsed
│ │ │ - -- .000002776s elapsed
│ │ │ - -- .000001283s elapsed
│ │ │ + -- .000006266s elapsed
│ │ │ + -- .000005911s elapsed
│ │ │ + -- .000006867s elapsed
│ │ │ + -- .000006032s elapsed
│ │ │ + -- .000005833s elapsed
│ │ │  
│ │ │       +----------------------------------------------------+
│ │ │       |   1 5   5 5                                        |
│ │ │       | - - - - - -                                        |
│ │ │       |   2 2   2 2                                        |
│ │ │  o6 = |x   x x   x                                         |
│ │ │       | 1   2 4   5                                        |
│ │ ├── ./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
│ │ │ - -- .000003547s elapsed
│ │ │ + -- .00000608s 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)
│ │ │ - -- .000003737s elapsed
│ │ │ + -- .000005898s 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
│ │ │ @@ -134,19 +134,19 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : netList cssLeadTerm(Hbeta, w)
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │ - -- .000003256s elapsed
│ │ │ - -- .000002485s elapsed
│ │ │ - -- .000004138s elapsed
│ │ │ - -- .000002776s elapsed
│ │ │ - -- .000001283s elapsed
│ │ │ + -- .000006266s elapsed
│ │ │ + -- .000005911s elapsed
│ │ │ + -- .000006867s elapsed
│ │ │ + -- .000006032s elapsed
│ │ │ + -- .000005833s elapsed
│ │ │  
│ │ │       +----------------------------------------------------+
│ │ │       |   1 5   5 5                                        |
│ │ │       | - - - - - -                                        |
│ │ │       |   2 2   2 2                                        |
│ │ │  o6 = |x   x x   x                                         |
│ │ │       | 1   2 4   5                                        |
│ │ │ ├── html2text {}
│ │ │ │ @@ -57,19 +57,19 @@
│ │ │ │  o5 = {9, 1, 99999, 9999999, 3, 999}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : netList cssLeadTerm(Hbeta, w)
│ │ │ │  Warning:  F4 Algorithm not available over current coefficient ring or
│ │ │ │  inhomogeneous ideal.
│ │ │ │  Converting to Naive algorithm.
│ │ │ │ - -- .000003256s elapsed
│ │ │ │ - -- .000002485s elapsed
│ │ │ │ - -- .000004138s elapsed
│ │ │ │ - -- .000002776s elapsed
│ │ │ │ - -- .000001283s elapsed
│ │ │ │ + -- .000006266s elapsed
│ │ │ │ + -- .000005911s elapsed
│ │ │ │ + -- .000006867s elapsed
│ │ │ │ + -- .000006032s elapsed
│ │ │ │ + -- .000005833s elapsed
│ │ │ │  
│ │ │ │       +----------------------------------------------------+
│ │ │ │       |   1 5   5 5                                        |
│ │ │ │       | - - - - - -                                        |
│ │ │ │       |   2 2   2 2                                        |
│ │ │ │  o6 = |x   x x   x                                         |
│ │ │ │       | 1   2 4   5                                        |
│ │ ├── ./usr/share/doc/Macaulay2/HolonomicSystems/html/_solve__Frobenius__Ideal.html
│ │ │ @@ -84,15 +84,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : solveFrobeniusIdeal I
│ │ │ - -- .000003547s elapsed
│ │ │ + -- .00000608s 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 
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -113,15 +113,15 @@
│ │ │              
│ │ │                
    i4 : W = makeWeylAlgebra(QQ[x_1..x_5]);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : solveFrobeniusIdeal(I, W)
│ │ │ - -- .000003737s elapsed
│ │ │ + -- .000005898s 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
│ │ │ │ - -- .000003547s elapsed
│ │ │ │ + -- .00000608s 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)
│ │ │ │ - -- .000003737s elapsed
│ │ │ │ + -- .000005898s 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/example-output/_bracket.out
│ │ │ @@ -85,19 +85,16 @@
│ │ │  
│ │ │  o13 = 600
│ │ │  
│ │ │  i14 : H' = select(keys H, k->H#k != 0);
│ │ │  
│ │ │  i15 : H'
│ │ │  
│ │ │ -o15 = {({T , T }, T T  + T T  - z*T   + y*T  ), ({T , T }, - T T  + y*T  ),
│ │ │ -          3   7    4 6    3 7      11      13      2   9      2 9      16  
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      ({T , T }, - T T  - T T  - z*T   + x*T  ), ({T , T }, - T T  + z*T  ),
│ │ │ -         1   9      5 6    1 9      14      17      4   7      4 7      13  
│ │ │ +o15 = {({T , T }, - T T  - T T  - z*T   + x*T  ), ({T , T }, - T T  + z*T  ),
│ │ │ +          1   9      5 6    1 9      14      17      4   7      4 7      13  
│ │ │        -----------------------------------------------------------------------
│ │ │        ({T , T  }, T T  - T T   + x*T  ), ({T , T }, - T T  + z*T  ), ({T ,
│ │ │           1   10    4 6    1 10      20      5   9      5 9      16      3 
│ │ │        -----------------------------------------------------------------------
│ │ │        T }, T T  - z*T   + x*T  ), ({T , T }, - T T  - T T  - z*T   + x*T  ),
│ │ │         7    3 7      11      12      5   6      5 6    1 9      14      17  
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -151,22 +148,25 @@
│ │ │        -----------------------------------------------------------------------
│ │ │        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  + 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 
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T }, - T T  - T T   + z*T   + z*T  )}
│ │ │ -       7      5 7    4 10      12      20
│ │ │ +      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  
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      y*T  ), ({T , T }, - T T  + y*T  )}
│ │ │ +         13      2   9      2 9      16
│ │ │  
│ │ │  o15 : List
│ │ │  
│ │ │  i16 : H#(H'_0)
│ │ │  
│ │ │ -o16 = 1
│ │ │ +o16 = -1
│ │ │  
│ │ │  o16 : S[T ..T  ]
│ │ │           1   99
│ │ │  
│ │ │  i17 : bracketMatrix(A,1,2)
│ │ │  
│ │ │  o17 = | 0    -T_8 -T_6 -T_7 -T_10 |
│ │ ├── ./usr/share/doc/Macaulay2/HomotopyLieAlgebra/html/_bracket.html
│ │ │ @@ -215,19 +215,16 @@
│ │ │                
    i14 : H' = select(keys H, k->H#k != 0);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i15 : H'
│ │ │  
│ │ │ -o15 = {({T , T }, T T  + T T  - z*T   + y*T  ), ({T , T }, - T T  + y*T  ),
│ │ │ -          3   7    4 6    3 7      11      13      2   9      2 9      16  
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      ({T , T }, - T T  - T T  - z*T   + x*T  ), ({T , T }, - T T  + z*T  ),
│ │ │ -         1   9      5 6    1 9      14      17      4   7      4 7      13  
│ │ │ +o15 = {({T , T }, - T T  - T T  - z*T   + x*T  ), ({T , T }, - T T  + z*T  ),
│ │ │ +          1   9      5 6    1 9      14      17      4   7      4 7      13  
│ │ │        -----------------------------------------------------------------------
│ │ │        ({T , T  }, T T  - T T   + x*T  ), ({T , T }, - T T  + z*T  ), ({T ,
│ │ │           1   10    4 6    1 10      20      5   9      5 9      16      3 
│ │ │        -----------------------------------------------------------------------
│ │ │        T }, T T  - z*T   + x*T  ), ({T , T }, - T T  - T T  - z*T   + x*T  ),
│ │ │         7    3 7      11      12      5   6      5 6    1 9      14      17  
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -281,25 +278,28 @@
│ │ │        -----------------------------------------------------------------------
│ │ │        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  + 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 
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T }, - T T  - T T   + z*T   + z*T  )}
│ │ │ -       7      5 7    4 10      12      20
│ │ │ +      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  
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      y*T  ), ({T , T }, - T T  + y*T  )}
│ │ │ +         13      2   9      2 9      16
│ │ │  
│ │ │  o15 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i16 : H#(H'_0)
│ │ │  
│ │ │ -o16 = 1
│ │ │ +o16 = -1
│ │ │  
│ │ │  o16 : S[T ..T  ]
│ │ │           1   99
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -118,19 +118,16 @@
│ │ │ │  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  + T T  - z*T   + y*T  ), ({T , T }, - T T  + y*T  ),
│ │ │ │ -          3   7    4 6    3 7      11      13      2   9      2 9      16
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      ({T , T }, - T T  - T T  - z*T   + x*T  ), ({T , T }, - T T  + z*T  ),
│ │ │ │ -         1   9      5 6    1 9      14      17      4   7      4 7      13
│ │ │ │ +o15 = {({T , T }, - T T  - T T  - z*T   + x*T  ), ({T , T }, - T T  + z*T  ),
│ │ │ │ +          1   9      5 6    1 9      14      17      4   7      4 7      13
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        ({T , T  }, T T  - T T   + x*T  ), ({T , T }, - T T  + z*T  ), ({T ,
│ │ │ │           1   10    4 6    1 10      20      5   9      5 9      16      3
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        T }, T T  - z*T   + x*T  ), ({T , T }, - T T  - T T  - z*T   + x*T  ),
│ │ │ │         7    3 7      11      12      5   6      5 6    1 9      14      17
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ @@ -184,21 +181,24 @@
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        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  + 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
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T }, - T T  - T T   + z*T   + z*T  )}
│ │ │ │ -       7      5 7    4 10      12      20
│ │ │ │ +      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
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      y*T  ), ({T , T }, - T T  + y*T  )}
│ │ │ │ +         13      2   9      2 9      16
│ │ │ │  
│ │ │ │  o15 : List
│ │ │ │  i16 : H#(H'_0)
│ │ │ │  
│ │ │ │ -o16 = 1
│ │ │ │ +o16 = -1
│ │ │ │  
│ │ │ │  o16 : S[T ..T  ]
│ │ │ │           1   99
│ │ │ │  From this we see that [T_5, T_6] sends T_37 to -1 in kk.
│ │ │ │  Another, often simpler view of the pairing is given by _b_r_a_c_k_e_t_M_a_t_r_i_x, where the
│ │ │ │  rows and columns correspond to the generators of Pi^d and Pi^e, and the entries
│ │ │ │  are the bracket products, interpreted as elements of Pi^{d+e}. Note the anti-
│ │ ├── ./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 0.69108s (cpu); 0.387871s (thread); 0s (gc)
│ │ │ + -- used 0.851368s (cpu); 0.446788s (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 0.737984s (cpu); 0.391867s (thread); 0s (gc)
│ │ │ + -- used 0.904726s (cpu); 0.437435s (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 0.869327s (cpu); 0.426331s (thread); 0s (gc)
│ │ │ + -- used 1.01434s (cpu); 0.462974s (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 0.90652s (cpu); 0.508334s (thread); 0s (gc)
│ │ │ + -- used 1.0719s (cpu); 0.516331s (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 1.56265s (cpu); 0.75218s (thread); 0s (gc)
│ │ │ + -- used 2.03172s (cpu); 0.80295s (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 0.417337s (cpu); 0.309351s (thread); 0s (gc)
│ │ │ + -- used 0.546604s (cpu); 0.375222s (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 0.0423714s (cpu); 0.0423694s (thread); 0s (gc)
│ │ │ + -- used 0.0542393s (cpu); 0.054236s (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 0.0422364s (cpu); 0.0422368s (thread); 0s (gc)
│ │ │ + -- used 0.0554137s (cpu); 0.0552763s (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 0.0595761s (cpu); 0.0595305s (thread); 0s (gc)
│ │ │ + -- used 0.0724349s (cpu); 0.0724296s (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 0.0433813s (cpu); 0.0433775s (thread); 0s (gc)
│ │ │ + -- used 0.0529208s (cpu); 0.0529194s (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 0.0586746s (cpu); 0.0586762s (thread); 0s (gc)
│ │ │ + -- used 0.0763157s (cpu); 0.0763136s (thread); 0s (gc)
│ │ │  
│ │ │  o56 = R'
│ │ │  
│ │ │  o56 : QuotientRing
│ │ │  
│ │ │  i57 : icFractions R
│ │ │  
│ │ │ @@ -632,15 +632,15 @@
│ │ │  i66 : R = S/I
│ │ │  
│ │ │  o66 = R
│ │ │  
│ │ │  o66 : QuotientRing
│ │ │  
│ │ │  i67 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.170458s (cpu); 0.0980731s (thread); 0s (gc)
│ │ │ + -- used 0.195966s (cpu); 0.106704s (thread); 0s (gc)
│ │ │  
│ │ │  o67 = R'
│ │ │  
│ │ │  o67 : QuotientRing
│ │ │  
│ │ │  i68 : icFractions R
│ │ │  
│ │ │ @@ -721,15 +721,15 @@
│ │ │  i77 : R = S/I
│ │ │  
│ │ │  o77 = R
│ │ │  
│ │ │  o77 : QuotientRing
│ │ │  
│ │ │  i78 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.356633s (cpu); 0.315307s (thread); 0s (gc)
│ │ │ + -- used 0.486707s (cpu); 0.395743s (thread); 0s (gc)
│ │ │  
│ │ │  o78 = R'
│ │ │  
│ │ │  o78 : QuotientRing
│ │ │  
│ │ │  i79 : icFractions R
│ │ │  
│ │ │ @@ -749,15 +749,15 @@
│ │ │  i81 : R = S/sub(I,S)
│ │ │  
│ │ │  o81 = R
│ │ │  
│ │ │  o81 : QuotientRing
│ │ │  
│ │ │  i82 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.502559s (cpu); 0.351078s (thread); 0s (gc)
│ │ │ + -- used 0.608707s (cpu); 0.42427s (thread); 0s (gc)
│ │ │  
│ │ │  o82 = R'
│ │ │  
│ │ │  o82 : QuotientRing
│ │ │  
│ │ │  i83 : icFractions R
│ │ │  
│ │ │ @@ -777,20 +777,20 @@
│ │ │  i85 : R = S/sub(I,S)
│ │ │  
│ │ │  o85 = R
│ │ │  
│ │ │  o85 : QuotientRing
│ │ │  
│ │ │  i86 : time R' = integralClosure (R, Strategy => RadicalCodim1, Verbosity => 1)
│ │ │ - [jacobian time .000553749 sec #minors 4]
│ │ │ + [jacobian time .000631084 sec #minors 4]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .207126 sec  #fractions 6]
│ │ │ - [step 1:   time .236813 sec  #fractions 6]
│ │ │ - -- used 0.447945s (cpu); 0.280615s (thread); 0s (gc)
│ │ │ + [step 0:   time .241032 sec  #fractions 6]
│ │ │ + [step 1:   time .27126 sec  #fractions 6]
│ │ │ + -- used 0.517056s (cpu); 0.32369s (thread); 0s (gc)
│ │ │  
│ │ │  o86 = R'
│ │ │  
│ │ │  o86 : QuotientRing
│ │ │  
│ │ │  i87 : icFractions R
│ │ │  
│ │ │ @@ -810,20 +810,20 @@
│ │ │  i89 : R = S/sub(I,S)
│ │ │  
│ │ │  o89 = R
│ │ │  
│ │ │  o89 : QuotientRing
│ │ │  
│ │ │  i90 : time R' = integralClosure (R, Strategy => Vasconcelos, Verbosity => 1)
│ │ │ - [jacobian time .000531897 sec #minors 4]
│ │ │ + [jacobian time .000623568 sec #minors 4]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .0887676 sec  #fractions 6]
│ │ │ - [step 1:   time .423615 sec  #fractions 6]
│ │ │ - -- used 0.516097s (cpu); 0.354562s (thread); 0s (gc)
│ │ │ + [step 0:   time .107939 sec  #fractions 6]
│ │ │ + [step 1:   time .485944 sec  #fractions 6]
│ │ │ + -- used 0.598233s (cpu); 0.396706s (thread); 0s (gc)
│ │ │  
│ │ │  o90 = R'
│ │ │  
│ │ │  o90 : QuotientRing
│ │ │  
│ │ │  i91 : icFractions R
│ │ │  
│ │ │ @@ -843,20 +843,20 @@
│ │ │  i93 : R = S/sub(I,S)
│ │ │  
│ │ │  o93 = R
│ │ │  
│ │ │  o93 : QuotientRing
│ │ │  
│ │ │  i94 : time R' = integralClosure (R, Strategy => {Vasconcelos, StartWithOneMinor}, Verbosity => 1)
│ │ │ - [jacobian time .000678222 sec #minors 1]
│ │ │ + [jacobian time .00075222 sec #minors 1]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .111521 sec  #fractions 6]
│ │ │ - [step 1:   time .471115 sec  #fractions 6]
│ │ │ - -- used 0.586291s (cpu); 0.418106s (thread); 0s (gc)
│ │ │ + [step 0:   time .135517 sec  #fractions 6]
│ │ │ + [step 1:   time .582358 sec  #fractions 6]
│ │ │ + -- used 0.722333s (cpu); 0.515895s (thread); 0s (gc)
│ │ │  
│ │ │  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 .000537428 sec #minors 3]
│ │ │ + [jacobian time .000573727 sec #minors 3]
│ │ │  integral closure nvars 3 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │   [step 0: 
│ │ │ -      radical (use minprimes) .00248517 seconds
│ │ │ -      idlizer1:  .00793427 seconds
│ │ │ -      idlizer2:  .00866101 seconds
│ │ │ -      minpres:   .00851812 seconds
│ │ │ -  time .0388261 sec  #fractions 4]
│ │ │ +      radical (use minprimes) .00292905 seconds
│ │ │ +      idlizer1:  .0102355 seconds
│ │ │ +      idlizer2:  .0110145 seconds
│ │ │ +      minpres:   .0107125 seconds
│ │ │ +  time .0484343 sec  #fractions 4]
│ │ │   [step 1: 
│ │ │ -      radical (use minprimes) .00220488 seconds
│ │ │ -      idlizer1:  .011116 seconds
│ │ │ -      idlizer2:  .00998087 seconds
│ │ │ -      minpres:   .0112759 seconds
│ │ │ -  time .0452055 sec  #fractions 4]
│ │ │ +      radical (use minprimes) .00286399 seconds
│ │ │ +      idlizer1:  .0142078 seconds
│ │ │ +      idlizer2:  .0126744 seconds
│ │ │ +      minpres:   .0133739 seconds
│ │ │ +  time .0564596 sec  #fractions 4]
│ │ │   [step 2: 
│ │ │ -      radical (use minprimes) .00224902 seconds
│ │ │ -      idlizer1:  .011153 seconds
│ │ │ -      idlizer2:  .0093342 seconds
│ │ │ -      minpres:   .00877394 seconds
│ │ │ -  time .0418707 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00283532 seconds
│ │ │ +      idlizer1:  .0136529 seconds
│ │ │ +      idlizer2:  .0119381 seconds
│ │ │ +      minpres:   .0109804 seconds
│ │ │ +  time .0523117 sec  #fractions 5]
│ │ │   [step 3: 
│ │ │ -      radical (use minprimes) .00225753 seconds
│ │ │ -      idlizer1:  .117394 seconds
│ │ │ -      idlizer2:  .0133807 seconds
│ │ │ -      minpres:   .0159094 seconds
│ │ │ -  time .160923 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .0028543 seconds
│ │ │ +      idlizer1:  .16805 seconds
│ │ │ +      idlizer2:  .0164072 seconds
│ │ │ +      minpres:   .0189133 seconds
│ │ │ +  time .220935 sec  #fractions 5]
│ │ │   [step 4: 
│ │ │ -      radical (use minprimes) .00245897 seconds
│ │ │ -      idlizer1:  .00891005 seconds
│ │ │ -      idlizer2:  .0164204 seconds
│ │ │ -      minpres:   .0117435 seconds
│ │ │ -  time .0517134 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .0031623 seconds
│ │ │ +      idlizer1:  .0119864 seconds
│ │ │ +      idlizer2:  .0182659 seconds
│ │ │ +      minpres:   .0142292 seconds
│ │ │ +  time .0622556 sec  #fractions 5]
│ │ │   [step 5: 
│ │ │ -      radical (use minprimes) .00239361 seconds
│ │ │ -      idlizer1:  .00820369 seconds
│ │ │ -  time .0171387 sec  #fractions 5]
│ │ │ - -- used 0.359626s (cpu); 0.322489s (thread); 0s (gc)
│ │ │ +      radical (use minprimes) .00281792 seconds
│ │ │ +      idlizer1:  .00955899 seconds
│ │ │ +  time .0203906 sec  #fractions 5]
│ │ │ + -- used 0.465258s (cpu); 0.362806s (thread); 0s (gc)
│ │ │  
│ │ │  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 1.00107s (cpu); 0.719147s (thread); 0s (gc)
│ │ │ + -- used 1.70654s (cpu); 0.934861s (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 0.619831s (cpu); 0.485488s (thread); 0s (gc)
│ │ │ + -- used 0.982116s (cpu); 0.585975s (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
│ │ │ @@ -99,15 +99,15 @@
│ │ │  
│ │ │  o3 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time R' = integralClosure R
│ │ │ - -- used 0.69108s (cpu); 0.387871s (thread); 0s (gc)
│ │ │ + -- used 0.851368s (cpu); 0.446788s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = R'
│ │ │  
│ │ │  o4 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -186,15 +186,15 @@
│ │ │  
│ │ │  o9 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.737984s (cpu); 0.391867s (thread); 0s (gc)
│ │ │ + -- used 0.904726s (cpu); 0.437435s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = R'
│ │ │  
│ │ │  o10 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -273,15 +273,15 @@
│ │ │  
│ │ │  o15 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i16 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.869327s (cpu); 0.426331s (thread); 0s (gc)
│ │ │ + -- used 1.01434s (cpu); 0.462974s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = R'
│ │ │  
│ │ │  o16 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -348,15 +348,15 @@
│ │ │  
│ │ │  o20 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i21 : time R' = integralClosure(R, Strategy => SimplifyFractions)
│ │ │ - -- used 0.90652s (cpu); 0.508334s (thread); 0s (gc)
│ │ │ + -- used 1.0719s (cpu); 0.516331s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = R'
│ │ │  
│ │ │  o21 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -423,15 +423,15 @@
│ │ │  
│ │ │  o25 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i26 : time R' = integralClosure (R, Strategy => RadicalCodim1)
│ │ │ - -- used 1.56265s (cpu); 0.75218s (thread); 0s (gc)
│ │ │ + -- used 2.03172s (cpu); 0.80295s (thread); 0s (gc)
│ │ │  
│ │ │  o26 = R'
│ │ │  
│ │ │  o26 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -498,15 +498,15 @@
│ │ │  
│ │ │  o30 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i31 : time R' = integralClosure (R, Strategy => Vasconcelos)
│ │ │ - -- used 0.417337s (cpu); 0.309351s (thread); 0s (gc)
│ │ │ + -- used 0.546604s (cpu); 0.375222s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = R'
│ │ │  
│ │ │  o31 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -573,15 +573,15 @@
│ │ │  
│ │ │  o35 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i36 : time R' = integralClosure R
│ │ │ - -- used 0.0423714s (cpu); 0.0423694s (thread); 0s (gc)
│ │ │ + -- used 0.0542393s (cpu); 0.054236s (thread); 0s (gc)
│ │ │  
│ │ │  o36 = R'
│ │ │  
│ │ │  o36 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -643,15 +643,15 @@
│ │ │  
│ │ │  o40 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i41 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.0422364s (cpu); 0.0422368s (thread); 0s (gc)
│ │ │ + -- used 0.0554137s (cpu); 0.0552763s (thread); 0s (gc)
│ │ │  
│ │ │  o41 = R'
│ │ │  
│ │ │  o41 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -695,15 +695,15 @@
│ │ │  
│ │ │  o45 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i46 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.0595761s (cpu); 0.0595305s (thread); 0s (gc)
│ │ │ + -- used 0.0724349s (cpu); 0.0724296s (thread); 0s (gc)
│ │ │  
│ │ │  o46 = R'
│ │ │  
│ │ │  o46 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -746,15 +746,15 @@
│ │ │  
│ │ │  o50 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i51 : time R' = integralClosure (R, Strategy => RadicalCodim1)
│ │ │ - -- used 0.0433813s (cpu); 0.0433775s (thread); 0s (gc)
│ │ │ + -- used 0.0529208s (cpu); 0.0529194s (thread); 0s (gc)
│ │ │  
│ │ │  o51 = R'
│ │ │  
│ │ │  o51 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -798,15 +798,15 @@
│ │ │  
│ │ │  o55 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i56 : time R' = integralClosure (R, Strategy => Vasconcelos)
│ │ │ - -- used 0.0586746s (cpu); 0.0586762s (thread); 0s (gc)
│ │ │ + -- used 0.0763157s (cpu); 0.0763136s (thread); 0s (gc)
│ │ │  
│ │ │  o56 = R'
│ │ │  
│ │ │  o56 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -932,15 +932,15 @@
│ │ │  
│ │ │  o66 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i67 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.170458s (cpu); 0.0980731s (thread); 0s (gc)
│ │ │ + -- used 0.195966s (cpu); 0.106704s (thread); 0s (gc)
│ │ │  
│ │ │  o67 = R'
│ │ │  
│ │ │  o67 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -1056,15 +1056,15 @@
│ │ │  
│ │ │  o77 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i78 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.356633s (cpu); 0.315307s (thread); 0s (gc)
│ │ │ + -- used 0.486707s (cpu); 0.395743s (thread); 0s (gc)
│ │ │  
│ │ │  o78 = R'
│ │ │  
│ │ │  o78 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -1098,15 +1098,15 @@
│ │ │  
│ │ │  o81 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i82 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.502559s (cpu); 0.351078s (thread); 0s (gc)
│ │ │ + -- used 0.608707s (cpu); 0.42427s (thread); 0s (gc)
│ │ │  
│ │ │  o82 = R'
│ │ │  
│ │ │  o82 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -1140,20 +1140,20 @@
│ │ │  
│ │ │  o85 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i86 : time R' = integralClosure (R, Strategy => RadicalCodim1, Verbosity => 1)
│ │ │ - [jacobian time .000553749 sec #minors 4]
│ │ │ + [jacobian time .000631084 sec #minors 4]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .207126 sec  #fractions 6]
│ │ │ - [step 1:   time .236813 sec  #fractions 6]
│ │ │ - -- used 0.447945s (cpu); 0.280615s (thread); 0s (gc)
│ │ │ + [step 0:   time .241032 sec  #fractions 6]
│ │ │ + [step 1:   time .27126 sec  #fractions 6]
│ │ │ + -- used 0.517056s (cpu); 0.32369s (thread); 0s (gc)
│ │ │  
│ │ │  o86 = R'
│ │ │  
│ │ │  o86 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -1187,20 +1187,20 @@
│ │ │  
│ │ │  o89 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i90 : time R' = integralClosure (R, Strategy => Vasconcelos, Verbosity => 1)
│ │ │ - [jacobian time .000531897 sec #minors 4]
│ │ │ + [jacobian time .000623568 sec #minors 4]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .0887676 sec  #fractions 6]
│ │ │ - [step 1:   time .423615 sec  #fractions 6]
│ │ │ - -- used 0.516097s (cpu); 0.354562s (thread); 0s (gc)
│ │ │ + [step 0:   time .107939 sec  #fractions 6]
│ │ │ + [step 1:   time .485944 sec  #fractions 6]
│ │ │ + -- used 0.598233s (cpu); 0.396706s (thread); 0s (gc)
│ │ │  
│ │ │  o90 = R'
│ │ │  
│ │ │  o90 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -1237,20 +1237,20 @@
│ │ │  
│ │ │  o93 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i94 : time R' = integralClosure (R, Strategy => {Vasconcelos, StartWithOneMinor}, Verbosity => 1)
│ │ │ - [jacobian time .000678222 sec #minors 1]
│ │ │ + [jacobian time .00075222 sec #minors 1]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .111521 sec  #fractions 6]
│ │ │ - [step 1:   time .471115 sec  #fractions 6]
│ │ │ - -- used 0.586291s (cpu); 0.418106s (thread); 0s (gc)
│ │ │ + [step 0:   time .135517 sec  #fractions 6]
│ │ │ + [step 1:   time .582358 sec  #fractions 6]
│ │ │ + -- used 0.722333s (cpu); 0.515895s (thread); 0s (gc)
│ │ │  
│ │ │  o94 = R'
│ │ │  
│ │ │  o94 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -48,15 +48,15 @@
│ │ │ │  o2 : Ideal of S
│ │ │ │  i3 : R = S/f
│ │ │ │  
│ │ │ │  o3 = R
│ │ │ │  
│ │ │ │  o3 : QuotientRing
│ │ │ │  i4 : time R' = integralClosure R
│ │ │ │ - -- used 0.69108s (cpu); 0.387871s (thread); 0s (gc)
│ │ │ │ + -- used 0.851368s (cpu); 0.446788s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = R'
│ │ │ │  
│ │ │ │  o4 : QuotientRing
│ │ │ │  i5 : netList (ideal R')_*
│ │ │ │  
│ │ │ │       +------------------------------------------------------------------------+
│ │ │ │ @@ -109,15 +109,15 @@
│ │ │ │  o8 : Ideal of S
│ │ │ │  i9 : R = S/f
│ │ │ │  
│ │ │ │  o9 = R
│ │ │ │  
│ │ │ │  o9 : QuotientRing
│ │ │ │  i10 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ │ - -- used 0.737984s (cpu); 0.391867s (thread); 0s (gc)
│ │ │ │ + -- used 0.904726s (cpu); 0.437435s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = R'
│ │ │ │  
│ │ │ │  o10 : QuotientRing
│ │ │ │  i11 : netList (ideal R')_*
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ @@ -199,15 +199,15 @@
│ │ │ │  o14 : Ideal of S
│ │ │ │  i15 : R = S/f
│ │ │ │  
│ │ │ │  o15 = R
│ │ │ │  
│ │ │ │  o15 : QuotientRing
│ │ │ │  i16 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ │ - -- used 0.869327s (cpu); 0.426331s (thread); 0s (gc)
│ │ │ │ + -- used 1.01434s (cpu); 0.462974s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = R'
│ │ │ │  
│ │ │ │  o16 : QuotientRing
│ │ │ │  i17 : netList (ideal R')_*
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ @@ -281,15 +281,15 @@
│ │ │ │  o19 : Ideal of S
│ │ │ │  i20 : R = S/f
│ │ │ │  
│ │ │ │  o20 = R
│ │ │ │  
│ │ │ │  o20 : QuotientRing
│ │ │ │  i21 : time R' = integralClosure(R, Strategy => SimplifyFractions)
│ │ │ │ - -- used 0.90652s (cpu); 0.508334s (thread); 0s (gc)
│ │ │ │ + -- used 1.0719s (cpu); 0.516331s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o21 = R'
│ │ │ │  
│ │ │ │  o21 : QuotientRing
│ │ │ │  i22 : netList (ideal R')_*
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ @@ -363,15 +363,15 @@
│ │ │ │  o24 : Ideal of S
│ │ │ │  i25 : R = S/f
│ │ │ │  
│ │ │ │  o25 = R
│ │ │ │  
│ │ │ │  o25 : QuotientRing
│ │ │ │  i26 : time R' = integralClosure (R, Strategy => RadicalCodim1)
│ │ │ │ - -- used 1.56265s (cpu); 0.75218s (thread); 0s (gc)
│ │ │ │ + -- used 2.03172s (cpu); 0.80295s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o26 = R'
│ │ │ │  
│ │ │ │  o26 : QuotientRing
│ │ │ │  i27 : netList (ideal R')_*
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ @@ -445,15 +445,15 @@
│ │ │ │  o29 : Ideal of S
│ │ │ │  i30 : R = S/f
│ │ │ │  
│ │ │ │  o30 = R
│ │ │ │  
│ │ │ │  o30 : QuotientRing
│ │ │ │  i31 : time R' = integralClosure (R, Strategy => Vasconcelos)
│ │ │ │ - -- used 0.417337s (cpu); 0.309351s (thread); 0s (gc)
│ │ │ │ + -- used 0.546604s (cpu); 0.375222s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o31 = R'
│ │ │ │  
│ │ │ │  o31 : QuotientRing
│ │ │ │  i32 : netList (ideal R')_*
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ @@ -527,15 +527,15 @@
│ │ │ │  o34 : Ideal of S
│ │ │ │  i35 : R = S/f
│ │ │ │  
│ │ │ │  o35 = R
│ │ │ │  
│ │ │ │  o35 : QuotientRing
│ │ │ │  i36 : time R' = integralClosure R
│ │ │ │ - -- used 0.0423714s (cpu); 0.0423694s (thread); 0s (gc)
│ │ │ │ + -- used 0.0542393s (cpu); 0.054236s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o36 = R'
│ │ │ │  
│ │ │ │  o36 : QuotientRing
│ │ │ │  i37 : netList (ideal R')_*
│ │ │ │  
│ │ │ │        +-----------+
│ │ │ │ @@ -573,15 +573,15 @@
│ │ │ │  o39 : Ideal of S
│ │ │ │  i40 : R = S/I
│ │ │ │  
│ │ │ │  o40 = R
│ │ │ │  
│ │ │ │  o40 : QuotientRing
│ │ │ │  i41 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ │ - -- used 0.0422364s (cpu); 0.0422368s (thread); 0s (gc)
│ │ │ │ + -- used 0.0554137s (cpu); 0.0552763s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o41 = R'
│ │ │ │  
│ │ │ │  o41 : QuotientRing
│ │ │ │  i42 : icFractions R
│ │ │ │  
│ │ │ │          2
│ │ │ │ @@ -603,15 +603,15 @@
│ │ │ │  o44 : Ideal of S
│ │ │ │  i45 : R = S/I
│ │ │ │  
│ │ │ │  o45 = R
│ │ │ │  
│ │ │ │  o45 : QuotientRing
│ │ │ │  i46 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ │ - -- used 0.0595761s (cpu); 0.0595305s (thread); 0s (gc)
│ │ │ │ + -- used 0.0724349s (cpu); 0.0724296s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o46 = R'
│ │ │ │  
│ │ │ │  o46 : QuotientRing
│ │ │ │  i47 : icFractions R
│ │ │ │  
│ │ │ │         b*d
│ │ │ │ @@ -632,15 +632,15 @@
│ │ │ │  o49 : Ideal of S
│ │ │ │  i50 : R = S/I
│ │ │ │  
│ │ │ │  o50 = R
│ │ │ │  
│ │ │ │  o50 : QuotientRing
│ │ │ │  i51 : time R' = integralClosure (R, Strategy => RadicalCodim1)
│ │ │ │ - -- used 0.0433813s (cpu); 0.0433775s (thread); 0s (gc)
│ │ │ │ + -- used 0.0529208s (cpu); 0.0529194s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o51 = R'
│ │ │ │  
│ │ │ │  o51 : QuotientRing
│ │ │ │  i52 : icFractions R
│ │ │ │  
│ │ │ │          2
│ │ │ │ @@ -662,15 +662,15 @@
│ │ │ │  o54 : Ideal of S
│ │ │ │  i55 : R = S/I
│ │ │ │  
│ │ │ │  o55 = R
│ │ │ │  
│ │ │ │  o55 : QuotientRing
│ │ │ │  i56 : time R' = integralClosure (R, Strategy => Vasconcelos)
│ │ │ │ - -- used 0.0586746s (cpu); 0.0586762s (thread); 0s (gc)
│ │ │ │ + -- used 0.0763157s (cpu); 0.0763136s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o56 = R'
│ │ │ │  
│ │ │ │  o56 : QuotientRing
│ │ │ │  i57 : icFractions R
│ │ │ │  
│ │ │ │         b*d
│ │ │ │ @@ -754,15 +754,15 @@
│ │ │ │  o65 : BettiTally
│ │ │ │  i66 : R = S/I
│ │ │ │  
│ │ │ │  o66 = R
│ │ │ │  
│ │ │ │  o66 : QuotientRing
│ │ │ │  i67 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ │ - -- used 0.170458s (cpu); 0.0980731s (thread); 0s (gc)
│ │ │ │ + -- used 0.195966s (cpu); 0.106704s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o67 = R'
│ │ │ │  
│ │ │ │  o67 : QuotientRing
│ │ │ │  i68 : icFractions R
│ │ │ │  
│ │ │ │          2    2
│ │ │ │ @@ -838,15 +838,15 @@
│ │ │ │  o76 : BettiTally
│ │ │ │  i77 : R = S/I
│ │ │ │  
│ │ │ │  o77 = R
│ │ │ │  
│ │ │ │  o77 : QuotientRing
│ │ │ │  i78 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ │ - -- used 0.356633s (cpu); 0.315307s (thread); 0s (gc)
│ │ │ │ + -- used 0.486707s (cpu); 0.395743s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o78 = R'
│ │ │ │  
│ │ │ │  o78 : QuotientRing
│ │ │ │  i79 : icFractions R
│ │ │ │  
│ │ │ │          2    2   2     3      2
│ │ │ │ @@ -862,15 +862,15 @@
│ │ │ │  o80 : PolynomialRing
│ │ │ │  i81 : R = S/sub(I,S)
│ │ │ │  
│ │ │ │  o81 = R
│ │ │ │  
│ │ │ │  o81 : QuotientRing
│ │ │ │  i82 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ │ - -- used 0.502559s (cpu); 0.351078s (thread); 0s (gc)
│ │ │ │ + -- used 0.608707s (cpu); 0.42427s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o82 = R'
│ │ │ │  
│ │ │ │  o82 : QuotientRing
│ │ │ │  i83 : icFractions R
│ │ │ │  
│ │ │ │          2    2   2     3      2
│ │ │ │ @@ -886,20 +886,20 @@
│ │ │ │  o84 : PolynomialRing
│ │ │ │  i85 : R = S/sub(I,S)
│ │ │ │  
│ │ │ │  o85 = R
│ │ │ │  
│ │ │ │  o85 : QuotientRing
│ │ │ │  i86 : time R' = integralClosure (R, Strategy => RadicalCodim1, Verbosity => 1)
│ │ │ │ - [jacobian time .000553749 sec #minors 4]
│ │ │ │ + [jacobian time .000631084 sec #minors 4]
│ │ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │ │  
│ │ │ │ - [step 0:   time .207126 sec  #fractions 6]
│ │ │ │ - [step 1:   time .236813 sec  #fractions 6]
│ │ │ │ - -- used 0.447945s (cpu); 0.280615s (thread); 0s (gc)
│ │ │ │ + [step 0:   time .241032 sec  #fractions 6]
│ │ │ │ + [step 1:   time .27126 sec  #fractions 6]
│ │ │ │ + -- used 0.517056s (cpu); 0.32369s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o86 = R'
│ │ │ │  
│ │ │ │  o86 : QuotientRing
│ │ │ │  i87 : icFractions R
│ │ │ │  
│ │ │ │          2    2   2     3      2
│ │ │ │ @@ -915,20 +915,20 @@
│ │ │ │  o88 : PolynomialRing
│ │ │ │  i89 : R = S/sub(I,S)
│ │ │ │  
│ │ │ │  o89 = R
│ │ │ │  
│ │ │ │  o89 : QuotientRing
│ │ │ │  i90 : time R' = integralClosure (R, Strategy => Vasconcelos, Verbosity => 1)
│ │ │ │ - [jacobian time .000531897 sec #minors 4]
│ │ │ │ + [jacobian time .000623568 sec #minors 4]
│ │ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │ │  
│ │ │ │ - [step 0:   time .0887676 sec  #fractions 6]
│ │ │ │ - [step 1:   time .423615 sec  #fractions 6]
│ │ │ │ - -- used 0.516097s (cpu); 0.354562s (thread); 0s (gc)
│ │ │ │ + [step 0:   time .107939 sec  #fractions 6]
│ │ │ │ + [step 1:   time .485944 sec  #fractions 6]
│ │ │ │ + -- used 0.598233s (cpu); 0.396706s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o90 = R'
│ │ │ │  
│ │ │ │  o90 : QuotientRing
│ │ │ │  i91 : icFractions R
│ │ │ │  
│ │ │ │          2    2   2     3      2
│ │ │ │ @@ -947,20 +947,20 @@
│ │ │ │  i93 : R = S/sub(I,S)
│ │ │ │  
│ │ │ │  o93 = R
│ │ │ │  
│ │ │ │  o93 : QuotientRing
│ │ │ │  i94 : time R' = integralClosure (R, Strategy => {Vasconcelos,
│ │ │ │  StartWithOneMinor}, Verbosity => 1)
│ │ │ │ - [jacobian time .000678222 sec #minors 1]
│ │ │ │ + [jacobian time .00075222 sec #minors 1]
│ │ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │ │  
│ │ │ │ - [step 0:   time .111521 sec  #fractions 6]
│ │ │ │ - [step 1:   time .471115 sec  #fractions 6]
│ │ │ │ - -- used 0.586291s (cpu); 0.418106s (thread); 0s (gc)
│ │ │ │ + [step 0:   time .135517 sec  #fractions 6]
│ │ │ │ + [step 1:   time .582358 sec  #fractions 6]
│ │ │ │ + -- used 0.722333s (cpu); 0.515895s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -71,52 +71,52 @@
│ │ │              
│ │ │                
    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 .000537428 sec #minors 3]
│ │ │ + [jacobian time .000573727 sec #minors 3]
│ │ │  integral closure nvars 3 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │   [step 0: 
│ │ │ -      radical (use minprimes) .00248517 seconds
│ │ │ -      idlizer1:  .00793427 seconds
│ │ │ -      idlizer2:  .00866101 seconds
│ │ │ -      minpres:   .00851812 seconds
│ │ │ -  time .0388261 sec  #fractions 4]
│ │ │ +      radical (use minprimes) .00292905 seconds
│ │ │ +      idlizer1:  .0102355 seconds
│ │ │ +      idlizer2:  .0110145 seconds
│ │ │ +      minpres:   .0107125 seconds
│ │ │ +  time .0484343 sec  #fractions 4]
│ │ │   [step 1: 
│ │ │ -      radical (use minprimes) .00220488 seconds
│ │ │ -      idlizer1:  .011116 seconds
│ │ │ -      idlizer2:  .00998087 seconds
│ │ │ -      minpres:   .0112759 seconds
│ │ │ -  time .0452055 sec  #fractions 4]
│ │ │ +      radical (use minprimes) .00286399 seconds
│ │ │ +      idlizer1:  .0142078 seconds
│ │ │ +      idlizer2:  .0126744 seconds
│ │ │ +      minpres:   .0133739 seconds
│ │ │ +  time .0564596 sec  #fractions 4]
│ │ │   [step 2: 
│ │ │ -      radical (use minprimes) .00224902 seconds
│ │ │ -      idlizer1:  .011153 seconds
│ │ │ -      idlizer2:  .0093342 seconds
│ │ │ -      minpres:   .00877394 seconds
│ │ │ -  time .0418707 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00283532 seconds
│ │ │ +      idlizer1:  .0136529 seconds
│ │ │ +      idlizer2:  .0119381 seconds
│ │ │ +      minpres:   .0109804 seconds
│ │ │ +  time .0523117 sec  #fractions 5]
│ │ │   [step 3: 
│ │ │ -      radical (use minprimes) .00225753 seconds
│ │ │ -      idlizer1:  .117394 seconds
│ │ │ -      idlizer2:  .0133807 seconds
│ │ │ -      minpres:   .0159094 seconds
│ │ │ -  time .160923 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .0028543 seconds
│ │ │ +      idlizer1:  .16805 seconds
│ │ │ +      idlizer2:  .0164072 seconds
│ │ │ +      minpres:   .0189133 seconds
│ │ │ +  time .220935 sec  #fractions 5]
│ │ │   [step 4: 
│ │ │ -      radical (use minprimes) .00245897 seconds
│ │ │ -      idlizer1:  .00891005 seconds
│ │ │ -      idlizer2:  .0164204 seconds
│ │ │ -      minpres:   .0117435 seconds
│ │ │ -  time .0517134 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .0031623 seconds
│ │ │ +      idlizer1:  .0119864 seconds
│ │ │ +      idlizer2:  .0182659 seconds
│ │ │ +      minpres:   .0142292 seconds
│ │ │ +  time .0622556 sec  #fractions 5]
│ │ │   [step 5: 
│ │ │ -      radical (use minprimes) .00239361 seconds
│ │ │ -      idlizer1:  .00820369 seconds
│ │ │ -  time .0171387 sec  #fractions 5]
│ │ │ - -- used 0.359626s (cpu); 0.322489s (thread); 0s (gc)
│ │ │ +      radical (use minprimes) .00281792 seconds
│ │ │ +      idlizer1:  .00955899 seconds
│ │ │ +  time .0203906 sec  #fractions 5]
│ │ │ + -- used 0.465258s (cpu); 0.362806s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = R'
│ │ │  
│ │ │  o2 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,52 +12,52 @@
│ │ │ │              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 .000537428 sec #minors 3]
│ │ │ │ + [jacobian time .000573727 sec #minors 3]
│ │ │ │  integral closure nvars 3 numgens 1 is S2 codim 1 codimJ 2
│ │ │ │  
│ │ │ │   [step 0:
│ │ │ │ -      radical (use minprimes) .00248517 seconds
│ │ │ │ -      idlizer1:  .00793427 seconds
│ │ │ │ -      idlizer2:  .00866101 seconds
│ │ │ │ -      minpres:   .00851812 seconds
│ │ │ │ -  time .0388261 sec  #fractions 4]
│ │ │ │ +      radical (use minprimes) .00292905 seconds
│ │ │ │ +      idlizer1:  .0102355 seconds
│ │ │ │ +      idlizer2:  .0110145 seconds
│ │ │ │ +      minpres:   .0107125 seconds
│ │ │ │ +  time .0484343 sec  #fractions 4]
│ │ │ │   [step 1:
│ │ │ │ -      radical (use minprimes) .00220488 seconds
│ │ │ │ -      idlizer1:  .011116 seconds
│ │ │ │ -      idlizer2:  .00998087 seconds
│ │ │ │ -      minpres:   .0112759 seconds
│ │ │ │ -  time .0452055 sec  #fractions 4]
│ │ │ │ +      radical (use minprimes) .00286399 seconds
│ │ │ │ +      idlizer1:  .0142078 seconds
│ │ │ │ +      idlizer2:  .0126744 seconds
│ │ │ │ +      minpres:   .0133739 seconds
│ │ │ │ +  time .0564596 sec  #fractions 4]
│ │ │ │   [step 2:
│ │ │ │ -      radical (use minprimes) .00224902 seconds
│ │ │ │ -      idlizer1:  .011153 seconds
│ │ │ │ -      idlizer2:  .0093342 seconds
│ │ │ │ -      minpres:   .00877394 seconds
│ │ │ │ -  time .0418707 sec  #fractions 5]
│ │ │ │ +      radical (use minprimes) .00283532 seconds
│ │ │ │ +      idlizer1:  .0136529 seconds
│ │ │ │ +      idlizer2:  .0119381 seconds
│ │ │ │ +      minpres:   .0109804 seconds
│ │ │ │ +  time .0523117 sec  #fractions 5]
│ │ │ │   [step 3:
│ │ │ │ -      radical (use minprimes) .00225753 seconds
│ │ │ │ -      idlizer1:  .117394 seconds
│ │ │ │ -      idlizer2:  .0133807 seconds
│ │ │ │ -      minpres:   .0159094 seconds
│ │ │ │ -  time .160923 sec  #fractions 5]
│ │ │ │ +      radical (use minprimes) .0028543 seconds
│ │ │ │ +      idlizer1:  .16805 seconds
│ │ │ │ +      idlizer2:  .0164072 seconds
│ │ │ │ +      minpres:   .0189133 seconds
│ │ │ │ +  time .220935 sec  #fractions 5]
│ │ │ │   [step 4:
│ │ │ │ -      radical (use minprimes) .00245897 seconds
│ │ │ │ -      idlizer1:  .00891005 seconds
│ │ │ │ -      idlizer2:  .0164204 seconds
│ │ │ │ -      minpres:   .0117435 seconds
│ │ │ │ -  time .0517134 sec  #fractions 5]
│ │ │ │ +      radical (use minprimes) .0031623 seconds
│ │ │ │ +      idlizer1:  .0119864 seconds
│ │ │ │ +      idlizer2:  .0182659 seconds
│ │ │ │ +      minpres:   .0142292 seconds
│ │ │ │ +  time .0622556 sec  #fractions 5]
│ │ │ │   [step 5:
│ │ │ │ -      radical (use minprimes) .00239361 seconds
│ │ │ │ -      idlizer1:  .00820369 seconds
│ │ │ │ -  time .0171387 sec  #fractions 5]
│ │ │ │ - -- used 0.359626s (cpu); 0.322489s (thread); 0s (gc)
│ │ │ │ +      radical (use minprimes) .00281792 seconds
│ │ │ │ +      idlizer1:  .00955899 seconds
│ │ │ │ +  time .0203906 sec  #fractions 5]
│ │ │ │ + -- used 0.465258s (cpu); 0.362806s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -109,29 +109,29 @@
│ │ │  
│ │ │  o3 : Ideal of S
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time integralClosure J
│ │ │ - -- used 1.00107s (cpu); 0.719147s (thread); 0s (gc)
│ │ │ + -- used 1.70654s (cpu); 0.934861s (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 0.619831s (cpu); 0.485488s (thread); 0s (gc)
│ │ │ + -- used 0.982116s (cpu); 0.585975s (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 {}
│ │ │ │ @@ -46,25 +46,25 @@
│ │ │ │  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 1.00107s (cpu); 0.719147s (thread); 0s (gc)
│ │ │ │ + -- used 1.70654s (cpu); 0.934861s (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 0.619831s (cpu); 0.485488s (thread); 0s (gc)
│ │ │ │ + -- used 0.982116s (cpu); 0.585975s (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/example-output/_equivariant__Hilbert.out
│ │ │ @@ -25,15 +25,15 @@
│ │ │  o3 : DiagonalAction
│ │ │  
│ │ │  i4 : T.cache.?equivariantHilbert
│ │ │  
│ │ │  o4 = false
│ │ │  
│ │ │  i5 : elapsedTime equivariantHilbertSeries(T, Order => 5)
│ │ │ - -- .00250269s elapsed
│ │ │ + -- .00293395s 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);
│ │ │ - -- .000440151s elapsed
│ │ │ + -- .000549551s 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 0.767543s (cpu); 0.538787s (thread); 0s (gc)
│ │ │ + -- used 0.946589s (cpu); 0.636641s (thread); 0s (gc)
│ │ │  
│ │ │ -       2   2    2   2 2
│ │ │ -o4 = {z , x  + y , x y }
│ │ │ +       2   2    2   3       3
│ │ │ +o4 = {z , x  + y , x y - x*y }
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : time P2=primaryInvariants(C4xC2,Dade=>true)
│ │ │ - -- used 0.729386s (cpu); 0.391831s (thread); 0s (gc)
│ │ │ + -- used 0.823188s (cpu); 0.458341s (thread); 0s (gc)
│ │ │  
│ │ │                     8                 7                   6 2  
│ │ │  o5 = {656100000000x  - 4738500000000x y + 10209037500000x y  -
│ │ │       ------------------------------------------------------------------------
│ │ │                     5 3                  4 4                 3 5  
│ │ │       1232156250000x y  - 14757374609375x y  + 1232156250000x y  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -90,23 +90,23 @@
│ │ │       ------------------------------------------------------------------------
│ │ │          2 6    8
│ │ │       90y z  + z }
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : time secondaryInvariants(P1,C4xC2)
│ │ │ - -- used 0.0301536s (cpu); 0.0301554s (thread); 0s (gc)
│ │ │ + -- used 0.0281491s (cpu); 0.0281509s (thread); 0s (gc)
│ │ │  
│ │ │ -          3       3
│ │ │ -o6 = {1, x y - x*y }
│ │ │ +          4    4
│ │ │ +o6 = {1, x  + y }
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : time secondaryInvariants(P2,C4xC2)
│ │ │ - -- used 2.05836s (cpu); 1.37581s (thread); 0s (gc)
│ │ │ + -- used 2.35339s (cpu); 1.53959s (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
│ │ │ @@ -14,15 +14,15 @@
│ │ │             | 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
│ │ │ - -- .809145s elapsed
│ │ │ + -- .624282s 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 }
│ │ │ @@ -32,15 +32,15 @@
│ │ │  
│ │ │  i5 : elapsedTime invariants(S4,DegreeBound=>4)
│ │ │  
│ │ │  Warning: stopping condition not met!
│ │ │  Output may not generate the entire ring of invariants.
│ │ │  Increase value of DegreeBound.
│ │ │  
│ │ │ - -- .614685s elapsed
│ │ │ + -- .515373s elapsed
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4  
│ │ │  o5 = {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 }
│ │ ├── ./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
│ │ │ - -- .650806s elapsed
│ │ │ + -- .595105s 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)
│ │ │ - -- .117342s elapsed
│ │ │ + -- .0658119s 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/example-output/_primary__Invariants.out
│ │ │ @@ -16,16 +16,16 @@
│ │ │                    | 0 0 1 |  | 1 0 0 |
│ │ │                    | 1 0 0 |  | 0 0 1 |
│ │ │  
│ │ │  o3 : FiniteGroupAction
│ │ │  
│ │ │  i4 : primaryInvariants S3
│ │ │  
│ │ │ -                  2    2    2   2       2    2     2       2      2
│ │ │ -o4 = {x + y + z, x  + y  + z , x y + x*y  + x z + y z + x*z  + y*z }
│ │ │ +                                   3    3    3
│ │ │ +o4 = {x + y + z, x*y + x*z + y*z, x  + y  + z }
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : K=GF(101)
│ │ │  
│ │ │  o5 = K
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/example-output/_primary__Invariants_lp..._cm__Degree__Vector_eq_gt..._rp.out
│ │ │ @@ -16,13 +16,16 @@
│ │ │                    | 0 0 1 |  | 1 0 0 |
│ │ │                    | 1 0 0 |  | 0 0 1 |
│ │ │  
│ │ │  o3 : FiniteGroupAction
│ │ │  
│ │ │  i4 : primaryInvariants(S3,DegreeVector=>{3,3,4})
│ │ │  
│ │ │ -       2       2    2     2       2      2          4    4    4
│ │ │ -o4 = {x y + x*y  + x z + y z + x*z  + y*z , x*y*z, x  + y  + z }
│ │ │ +       3    3    3   2       2    2     2       2      2   3       3    3   
│ │ │ +o4 = {x  + y  + z , x y + x*y  + x z + y z + x*z  + y*z , x y + x*y  + x z +
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +      3       3      3
│ │ │ +     y z + x*z  + y*z }
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_equivariant__Hilbert.html
│ │ │ @@ -92,15 +92,15 @@
│ │ │  
│ │ │  o4 = false
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : elapsedTime equivariantHilbertSeries(T, Order => 5)
│ │ │ - -- .00250269s elapsed
│ │ │ + -- .00293395s 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 
│ │ │ @@ -124,15 +124,15 @@
│ │ │  
│ │ │  o6 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : elapsedTime equivariantHilbertSeries(T, Order => 5);
│ │ │ - -- .000440151s elapsed
    
│ │ │ + -- .000549551s elapsed
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │          
    
│ │ │            
    For the programmer
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -30,15 +30,15 @@
│ │ │ │       | 0  -1 1 |
│ │ │ │  
│ │ │ │  o3 : DiagonalAction
│ │ │ │  i4 : T.cache.?equivariantHilbert
│ │ │ │  
│ │ │ │  o4 = false
│ │ │ │  i5 : elapsedTime equivariantHilbertSeries(T, Order => 5)
│ │ │ │ - -- .00250269s elapsed
│ │ │ │ + -- .00293395s 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
│ │ │ │ @@ -54,13 +54,13 @@
│ │ │ │  
│ │ │ │  o5 : ZZ[z ..z ][T]
│ │ │ │           0   1
│ │ │ │  i6 : T.cache.?equivariantHilbert
│ │ │ │  
│ │ │ │  o6 = true
│ │ │ │  i7 : elapsedTime equivariantHilbertSeries(T, Order => 5);
│ │ │ │ - -- .000440151s elapsed
│ │ │ │ + -- .000549551s 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.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/InvariantRing/AbelianGroupsDoc.m2:185:0.
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_hsop_spalgorithms.html
│ │ │ @@ -92,26 +92,26 @@
│ │ │            
│ │ │          
│ │ │          
    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 0.767543s (cpu); 0.538787s (thread); 0s (gc)
│ │ │ + -- used 0.946589s (cpu); 0.636641s (thread); 0s (gc)
│ │ │  
│ │ │ -       2   2    2   2 2
│ │ │ -o4 = {z , x  + y , x y }
│ │ │ +       2   2    2   3       3
│ │ │ +o4 = {z , x  + y , x y - x*y }
│ │ │  
│ │ │  o4 : List
    
│ │ │              
    
│ │ │                
    i5 : time P2=primaryInvariants(C4xC2,Dade=>true)
│ │ │ - -- used 0.729386s (cpu); 0.391831s (thread); 0s (gc)
│ │ │ + -- used 0.823188s (cpu); 0.458341s (thread); 0s (gc)
│ │ │  
│ │ │                     8                 7                   6 2  
│ │ │  o5 = {656100000000x  - 4738500000000x y + 10209037500000x y  -
│ │ │       ------------------------------------------------------------------------
│ │ │                     5 3                  4 4                 3 5  
│ │ │       1232156250000x y  - 14757374609375x y  + 1232156250000x y  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -168,26 +168,26 @@
│ │ │            
    
│ │ │          
    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 0.0301536s (cpu); 0.0301554s (thread); 0s (gc)
│ │ │ + -- used 0.0281491s (cpu); 0.0281509s (thread); 0s (gc)
│ │ │  
│ │ │ -          3       3
│ │ │ -o6 = {1, x y - x*y }
│ │ │ +          4    4
│ │ │ +o6 = {1, x  + y }
│ │ │  
│ │ │  o6 : List
    
│ │ │              
    
│ │ │                
    i7 : time secondaryInvariants(P2,C4xC2)
│ │ │ - -- used 2.05836s (cpu); 1.37581s (thread); 0s (gc)
│ │ │ + -- used 2.35339s (cpu); 1.53959s (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 0.767543s (cpu); 0.538787s (thread); 0s (gc)
│ │ │ │ + -- used 0.946589s (cpu); 0.636641s (thread); 0s (gc)
│ │ │ │  
│ │ │ │ -       2   2    2   2 2
│ │ │ │ -o4 = {z , x  + y , x y }
│ │ │ │ +       2   2    2   3       3
│ │ │ │ +o4 = {z , x  + y , x y - x*y }
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : time P2=primaryInvariants(C4xC2,Dade=>true)
│ │ │ │ - -- used 0.729386s (cpu); 0.391831s (thread); 0s (gc)
│ │ │ │ + -- used 0.823188s (cpu); 0.458341s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                     8                 7                   6 2
│ │ │ │  o5 = {656100000000x  - 4738500000000x y + 10209037500000x y  -
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                     5 3                  4 4                 3 5
│ │ │ │       1232156250000x y  - 14757374609375x y  + 1232156250000x y  +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -138,22 +138,22 @@
│ │ │ │  
│ │ │ │  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 0.0301536s (cpu); 0.0301554s (thread); 0s (gc)
│ │ │ │ + -- used 0.0281491s (cpu); 0.0281509s (thread); 0s (gc)
│ │ │ │  
│ │ │ │ -          3       3
│ │ │ │ -o6 = {1, x y - x*y }
│ │ │ │ +          4    4
│ │ │ │ +o6 = {1, x  + y }
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : time secondaryInvariants(P2,C4xC2)
│ │ │ │ - -- used 2.05836s (cpu); 1.37581s (thread); 0s (gc)
│ │ │ │ + -- used 2.35339s (cpu); 1.53959s (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
│ │ │ @@ -96,15 +96,15 @@
│ │ │  
│ │ │  o3 : FiniteGroupAction
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime invariants S4
│ │ │ - -- .809145s elapsed
│ │ │ + -- .624282s 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 }
│ │ │ @@ -117,15 +117,15 @@
│ │ │              
    		
│ │ │                
    i5 : elapsedTime invariants(S4,DegreeBound=>4)
│ │ │  
│ │ │  Warning: stopping condition not met!
│ │ │  Output may not generate the entire ring of invariants.
│ │ │  Increase value of DegreeBound.
│ │ │  
│ │ │ - -- .614685s elapsed
│ │ │ + -- .515373s elapsed
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4  
│ │ │  o5 = {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 }
│ │ │ ├── html2text {}
│ │ │ │ @@ -33,15 +33,15 @@
│ │ │ │  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
│ │ │ │ - -- .809145s elapsed
│ │ │ │ + -- .624282s 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 }
│ │ │ │ @@ -50,15 +50,15 @@
│ │ │ │  o4 : List
│ │ │ │  i5 : elapsedTime invariants(S4,DegreeBound=>4)
│ │ │ │  
│ │ │ │  Warning: stopping condition not met!
│ │ │ │  Output may not generate the entire ring of invariants.
│ │ │ │  Increase value of DegreeBound.
│ │ │ │  
│ │ │ │ - -- .614685s elapsed
│ │ │ │ + -- .515373s elapsed
│ │ │ │  
│ │ │ │                            2    2    2    2   3    3    3    3   4    4    4
│ │ │ │  o5 = {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 }
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_invariants_lp..._cm__Use__Linear__Algebra_eq_gt..._rp.html
│ │ │ @@ -96,15 +96,15 @@
│ │ │  
│ │ │  o3 : FiniteGroupAction
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime invariants S4
│ │ │ - -- .650806s elapsed
│ │ │ + -- .595105s 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 }
│ │ │ @@ -112,15 +112,15 @@
│ │ │  
│ │ │  o4 : List
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime invariants(S4,UseLinearAlgebra=>true)
│ │ │ - -- .117342s elapsed
│ │ │ + -- .0658119s 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 {}
│ │ │ │ @@ -35,27 +35,27 @@
│ │ │ │  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
│ │ │ │ - -- .650806s elapsed
│ │ │ │ + -- .595105s 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)
│ │ │ │ - -- .117342s elapsed
│ │ │ │ + -- .0658119s 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/_primary__Invariants.html
│ │ │ @@ -101,16 +101,16 @@
│ │ │  o3 : FiniteGroupAction
    
│ │ │              
    
│ │ │                
    i4 : primaryInvariants S3
│ │ │  
│ │ │ -                  2    2    2   2       2    2     2       2      2
│ │ │ -o4 = {x + y + z, x  + y  + z , x y + x*y  + x z + y z + x*z  + y*z }
│ │ │ +                                   3    3    3
│ │ │ +o4 = {x + y + z, x*y + x*z + y*z, x  + y  + z }
│ │ │  
│ │ │  o4 : List
    
│ │ │              
    
│ │ │          
    Below, the invariant ring QQ[x,y,z]S3 is calculated with K being the field with 101 elements.
    
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -38,16 +38,16 @@
│ │ │ │  o3 = QQ[x..z] <- {| 0 1 0 |, | 0 1 0 |}
│ │ │ │                    | 0 0 1 |  | 1 0 0 |
│ │ │ │                    | 1 0 0 |  | 0 0 1 |
│ │ │ │  
│ │ │ │  o3 : FiniteGroupAction
│ │ │ │  i4 : primaryInvariants S3
│ │ │ │  
│ │ │ │ -                  2    2    2   2       2    2     2       2      2
│ │ │ │ -o4 = {x + y + z, x  + y  + z , x y + x*y  + x z + y z + x*z  + y*z }
│ │ │ │ +                                   3    3    3
│ │ │ │ +o4 = {x + y + z, x*y + x*z + y*z, x  + y  + z }
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  Below, the invariant ring QQ[x,y,z]S3 is calculated with K being the field with
│ │ │ │  101 elements.
│ │ │ │  i5 : K=GF(101)
│ │ │ │  
│ │ │ │  o5 = K
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_primary__Invariants_lp..._cm__Degree__Vector_eq_gt..._rp.html
│ │ │ @@ -97,16 +97,19 @@
│ │ │  o3 : FiniteGroupAction
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : primaryInvariants(S3,DegreeVector=>{3,3,4})
│ │ │  
│ │ │ -       2       2    2     2       2      2          4    4    4
│ │ │ -o4 = {x y + x*y  + x z + y z + x*z  + y*z , x*y*z, x  + y  + z }
│ │ │ +       3    3    3   2       2    2     2       2      2   3       3    3   
│ │ │ +o4 = {x  + y  + z , x y + x*y  + x z + y z + x*z  + y*z , x y + x*y  + x z +
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +      3       3      3
│ │ │ +     y z + x*z  + y*z }
│ │ │  
│ │ │  o4 : List
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,16 +34,19 @@
│ │ │ │  o3 = QQ[x..z] <- {| 0 1 0 |, | 0 1 0 |}
│ │ │ │                    | 0 0 1 |  | 1 0 0 |
│ │ │ │                    | 1 0 0 |  | 0 0 1 |
│ │ │ │  
│ │ │ │  o3 : FiniteGroupAction
│ │ │ │  i4 : primaryInvariants(S3,DegreeVector=>{3,3,4})
│ │ │ │  
│ │ │ │ -       2       2    2     2       2      2          4    4    4
│ │ │ │ -o4 = {x y + x*y  + x z + y z + x*z  + y*z , x*y*z, x  + y  + z }
│ │ │ │ +       3    3    3   2       2    2     2       2      2   3       3    3
│ │ │ │ +o4 = {x  + y  + z , x y + x*y  + x z + y z + x*z  + y*z , x y + x*y  + x z +
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +      3       3      3
│ │ │ │ +     y z + x*z  + y*z }
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  Currently users can only use _p_r_i_m_a_r_y_I_n_v_a_r_i_a_n_t_s to calculate a hsop for the
│ │ │ │  invariant ring over a finite field by using the Dade algorithm. Users should
│ │ │ │  enter the finite field as a _G_a_l_o_i_s_F_i_e_l_d or a quotient field of the form _Z_Z/
│ │ │ │  p and are advised to ensure that the ground field has cardinality greater than
│ │ ├── ./usr/share/doc/Macaulay2/Isomorphism/example-output/_is__Isomorphic.out
│ │ │ @@ -156,20 +156,20 @@
│ │ │                     {-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   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    0    0    0    0    x_2  x_1  x_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    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    0    0    0    0    0    0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    x_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     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     0   0   0     x_2 x_1 x_0^2 |
│ │ │                     {-1} | 0 0 0  0 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    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    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    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    x_0  -x_2 x_1  -x_3 x_2  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     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     x_3 x_2 x_1^2 |
│ │ │  
│ │ │                                  40
│ │ │  o22 : S-module, subquotient of S
│ │ │  
│ │ │  i23 : elapsedTime isIsomorphic(T1, T2)
│ │ │ - -- 1.50438s elapsed
│ │ │ + -- 1.66978s elapsed
│ │ │  
│ │ │  o23 = true
│ │ │  
│ │ │  i24 : elapsedTime isomorphism(T1, T2)
│ │ │ - -- .000022302s elapsed
│ │ │ + -- .000026927s elapsed
│ │ │  
│ │ │  o24 = {-1} | 1      -3976  -13490 13495  -2886  2577   14757  -881   7677  
│ │ │        {-1} | -2527  -13566 2778   -6934  -14806 4619   -13099 6022   -10907
│ │ │        {-1} | -15420 5642   1489   1354   4591   11881  -5253  7296   -1098 
│ │ │        {-1} | 7909   -12428 -2260  -8465  12113  -6893  8411   4186   -9393 
│ │ │        {-1} | -9615  2934   10440  5015   8145   -5585  1360   3295   12851 
│ │ │        {-1} | -4881  -7984  12700  -10391 -10009 -14538 13207  262    -6500
│ │ ├── ./usr/share/doc/Macaulay2/Isomorphism/html/_is__Isomorphic.html
│ │ │ @@ -328,23 +328,23 @@
│ │ │                                  40
│ │ │  o22 : S-module, subquotient of S
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i23 : elapsedTime isIsomorphic(T1, T2)
│ │ │ - -- 1.50438s elapsed
│ │ │ + -- 1.66978s elapsed
│ │ │  
│ │ │  o23 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i24 : elapsedTime isomorphism(T1, T2)
│ │ │ - -- .000022302s elapsed
│ │ │ + -- .000026927s elapsed
│ │ │  
│ │ │  o24 = {-1} | 1      -3976  -13490 13495  -2886  2577   14757  -881   7677  
│ │ │        {-1} | -2527  -13566 2778   -6934  -14806 4619   -13099 6022   -10907
│ │ │        {-1} | -15420 5642   1489   1354   4591   11881  -5253  7296   -1098 
│ │ │        {-1} | 7909   -12428 -2260  -8465  12113  -6893  8411   4186   -9393 
│ │ │        {-1} | -9615  2934   10440  5015   8145   -5585  1360   3295   12851 
│ │ │        {-1} | -4881  -7984  12700  -10391 -10009 -14538 13207  262    -6500
│ │ │ ├── html2text {}
│ │ │ │ @@ -684,19 +684,19 @@
│ │ │ │  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     0   0   0     0   0   0
│ │ │ │  0   0   0     x_3 x_2 x_1^2 |
│ │ │ │  
│ │ │ │                                  40
│ │ │ │  o22 : S-module, subquotient of S
│ │ │ │  i23 : elapsedTime isIsomorphic(T1, T2)
│ │ │ │ - -- 1.50438s elapsed
│ │ │ │ + -- 1.66978s elapsed
│ │ │ │  
│ │ │ │  o23 = true
│ │ │ │  i24 : elapsedTime isomorphism(T1, T2)
│ │ │ │ - -- .000022302s elapsed
│ │ │ │ + -- .000026927s elapsed
│ │ │ │  
│ │ │ │  o24 = {-1} | 1      -3976  -13490 13495  -2886  2577   14757  -881   7677
│ │ │ │        {-1} | -2527  -13566 2778   -6934  -14806 4619   -13099 6022   -10907
│ │ │ │        {-1} | -15420 5642   1489   1354   4591   11881  -5253  7296   -1098
│ │ │ │        {-1} | 7909   -12428 -2260  -8465  12113  -6893  8411   4186   -9393
│ │ │ │        {-1} | -9615  2934   10440  5015   8145   -5585  1360   3295   12851
│ │ │ │        {-1} | -4881  -7984  12700  -10391 -10009 -14538 13207  262    -6500
│ │ ├── ./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/M2-50412-0/0.json
│ │ │ +o9 = /tmp/M2-79339-0/0.json
│ │ │  
│ │ │  i10 : jsonFile << "[1, 2, 3]" << endl << close
│ │ │  
│ │ │ -o10 = /tmp/M2-50412-0/0.json
│ │ │ +o10 = /tmp/M2-79339-0/0.json
│ │ │  
│ │ │  o10 : File
│ │ │  
│ │ │  i11 : fromJSON openIn jsonFile
│ │ │  
│ │ │  o11 = {1, 2, 3}
│ │ ├── ./usr/share/doc/Macaulay2/JSON/html/_from__J__S__O__N.html
│ │ │ @@ -167,22 +167,22 @@
│ │ │            
    The input may also be a file containing JSON data.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i9 : jsonFile = temporaryFileName() | ".json"
│ │ │  
│ │ │ -o9 = /tmp/M2-50412-0/0.json
    
│ │ │ +o9 = /tmp/M2-79339-0/0.json
│ │ │              
    
│ │ │                
    i10 : jsonFile << "[1, 2, 3]" << endl << close
│ │ │  
│ │ │ -o10 = /tmp/M2-50412-0/0.json
│ │ │ +o10 = /tmp/M2-79339-0/0.json
│ │ │  
│ │ │  o10 : File
    
│ │ │              
    
│ │ │                
    i11 : fromJSON openIn jsonFile
│ │ │ ├── html2text {}
│ │ │ │ @@ -53,18 +53,18 @@
│ │ │ │  
│ │ │ │  o8 = {1, 2, 3}
│ │ │ │  
│ │ │ │  o8 : List
│ │ │ │  The input may also be a file containing JSON data.
│ │ │ │  i9 : jsonFile = temporaryFileName() | ".json"
│ │ │ │  
│ │ │ │ -o9 = /tmp/M2-50412-0/0.json
│ │ │ │ +o9 = /tmp/M2-79339-0/0.json
│ │ │ │  i10 : jsonFile << "[1, 2, 3]" << endl << close
│ │ │ │  
│ │ │ │ -o10 = /tmp/M2-50412-0/0.json
│ │ │ │ +o10 = /tmp/M2-79339-0/0.json
│ │ │ │  
│ │ │ │  o10 : File
│ │ │ │  i11 : fromJSON openIn jsonFile
│ │ │ │  
│ │ │ │  o11 = {1, 2, 3}
│ │ │ │  
│ │ │ │  o11 : List
│ │ ├── ./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)
│ │ │ - -- .00230799s elapsed
│ │ │ + -- .00259876s 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
│ │ │ - -- .357303s elapsed
│ │ │ + -- .281794s 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)
│ │ │ - -- .00965176s elapsed
│ │ │ + -- .0107672s 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)
│ │ │ - -- .00257115s elapsed
│ │ │ + -- .00292815s 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)
│ │ │ - -- .00225703s elapsed
│ │ │ + -- .00270648s 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)
│ │ │ - -- .0121591s elapsed
│ │ │ + -- .0142956s 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)
│ │ │ - -- .000624555s elapsed
│ │ │ + -- .000794984s 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
│ │ │ @@ -87,27 +87,27 @@
│ │ │          
    
│ │ │            
    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)
│ │ │ │ - -- .00230799s elapsed
│ │ │ │ + -- .00259876s 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
│ │ │ │ - -- .357303s elapsed
│ │ │ │ + -- .281794s 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
│ │ │ @@ -117,15 +117,15 @@
│ │ │  
│ │ │  o7 = false
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -146,26 +146,26 @@
│ │ │                                 | x0^2-y0        |
│ │ │                   jetsMaxOrder => 3
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -290,15 +290,15 @@
│ │ │  
│ │ │  o23 = false
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : elapsedTime jetsRadical(2,I)
│ │ │ - -- .00230799s elapsed
│ │ │ + -- .00259876s 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
│ │ │ - -- .357303s elapsed
│ │ │ + -- .281794s 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]
    
│ │ │              
    
│ │ │                
    i8 : elapsedTime jets(3,I)
│ │ │ - -- .00965176s elapsed
│ │ │ + -- .0107672s 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)
│ │ │ - -- .00257115s elapsed
│ │ │ + -- .00292815s 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)
│ │ │ - -- .00225703s elapsed
│ │ │ + -- .00270648s elapsed
│ │ │  
│ │ │                               2                 2
│ │ │  o12 = ideal (2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │  o12 : Ideal of QQ[x0, y0][x1, y1][x2, y2]
    
│ │ │              
    
│ │ │                
    i24 : elapsedTime jets(3,f)
│ │ │ - -- .0121591s elapsed
│ │ │ + -- .0142956s 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]
│ │ │ @@ -324,15 +324,15 @@
│ │ │                                 | t0 t0^2        |
│ │ │                   jetsMaxOrder => 3
    
│ │ │              
    
│ │ │                
    i27 : elapsedTime jets(2,f)
│ │ │ - -- .000624555s elapsed
│ │ │ + -- .000794984s 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)
│ │ │ │ - -- .00965176s elapsed
│ │ │ │ + -- .0107672s 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)
│ │ │ │ - -- .00257115s elapsed
│ │ │ │ + -- .00292815s 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)
│ │ │ │ - -- .00225703s elapsed
│ │ │ │ + -- .00270648s 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)
│ │ │ │ - -- .0121591s elapsed
│ │ │ │ + -- .0142956s 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)
│ │ │ │ - -- .000624555s elapsed
│ │ │ │ + -- .000794984s 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/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
│ │ │                                                                                                                                                                                                                                                                                           
│ │ │       0                            1                             2                              3                              4                              5                              6                              7                              8                             9
│ │ │  
│ │ │  o3 : Complex
│ │ │  
│ │ │  i4 : L = analyzeStrand(F,a); #L
│ │ │ - -- .023996s elapsed
│ │ │ + -- .0280581s 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)
│ │ │ - -- .00242104s elapsed
│ │ │ - -- .00673308s elapsed
│ │ │ - -- .034067s elapsed
│ │ │ - -- .0999449s elapsed
│ │ │ - -- .00367219s elapsed
│ │ │ + -- .00297478s elapsed
│ │ │ + -- .00829941s elapsed
│ │ │ + -- .0275714s elapsed
│ │ │ + -- .0454032s elapsed
│ │ │ + -- .00423662s 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)
│ │ │ - -- .00266181s elapsed
│ │ │ - -- .00676057s elapsed
│ │ │ - -- .023025s elapsed
│ │ │ - -- .00985475s elapsed
│ │ │ - -- .0035774s elapsed
│ │ │ - -- .425089s elapsed
│ │ │ + -- .00432022s elapsed
│ │ │ + -- .00686264s elapsed
│ │ │ + -- .0328347s elapsed
│ │ │ + -- .0116131s elapsed
│ │ │ + -- .00413709s elapsed
│ │ │ + -- .483909s 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
│ │ │ - -- .22587s elapsed
│ │ │ + -- .209815s 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);
│ │ │ - -- .00455268s elapsed
│ │ │ - -- .0173468s elapsed
│ │ │ - -- .126286s elapsed
│ │ │ - -- 1.19253s elapsed
│ │ │ - -- .40692s elapsed
│ │ │ - -- .0843344s elapsed
│ │ │ - -- .006517s elapsed
│ │ │ - -- 6.20834s elapsed
│ │ │ + -- .00514119s elapsed
│ │ │ + -- .0241156s elapsed
│ │ │ + -- .12517s elapsed
│ │ │ + -- .980881s elapsed
│ │ │ + -- .451633s elapsed
│ │ │ + -- .0417908s elapsed
│ │ │ + -- .00765904s elapsed
│ │ │ + -- 5.93912s 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)
│ │ │ - -- .00258607s elapsed
│ │ │ - -- .00645513s elapsed
│ │ │ - -- .0463112s elapsed
│ │ │ - -- .011534s elapsed
│ │ │ - -- .00360799s elapsed
│ │ │ + -- .00282683s elapsed
│ │ │ + -- .00693946s elapsed
│ │ │ + -- .0295112s elapsed
│ │ │ + -- .0111707s elapsed
│ │ │ + -- .00417127s 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
│ │ │ - -- .227924s elapsed
│ │ │ + -- .219522s 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);
│ │ │ - -- .0046955s elapsed
│ │ │ - -- .0175456s elapsed
│ │ │ - -- .0976552s elapsed
│ │ │ - -- 1.36978s elapsed
│ │ │ - -- .488999s elapsed
│ │ │ - -- .0396705s elapsed
│ │ │ - -- .00677344s elapsed
│ │ │ - -- 7.12863s elapsed
│ │ │ + -- .00519211s elapsed
│ │ │ + -- .0191225s elapsed
│ │ │ + -- .11103s elapsed
│ │ │ + -- 1.09954s elapsed
│ │ │ + -- .464044s elapsed
│ │ │ + -- .0449083s elapsed
│ │ │ + -- .00771454s elapsed
│ │ │ + -- 5.99796s elapsed
│ │ │  
│ │ │  i8 : keys h
│ │ │  
│ │ │  o8 = {0, 2, 3, 5}
│ │ │  
│ │ │  o8 : List
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Det.out
│ │ │ @@ -3,82 +3,82 @@
│ │ │  i1 : a=4,b=4
│ │ │  
│ │ │  o1 = (4, 4)
│ │ │  
│ │ │  o1 : Sequence
│ │ │  
│ │ │  i2 : d=carpetDet(a,b)
│ │ │ - -- .0065779s elapsed
│ │ │ - -- .0185038s elapsed
│ │ │ + -- .00776234s elapsed
│ │ │ + -- .013589s elapsed
│ │ │  (number Of blocks, 26)
│ │ │ - -- .000299108s elapsed
│ │ │ + -- .000261781s elapsed
│ │ │  1
│ │ │ - -- .000140792s elapsed
│ │ │ + -- .000154249s elapsed
│ │ │  1
│ │ │ - -- .000145011s elapsed
│ │ │ + -- .0001552s elapsed
│ │ │  1
│ │ │ - -- .000132407s elapsed
│ │ │ + -- .000182441s elapsed
│ │ │  1
│ │ │ - -- .000131325s elapsed
│ │ │ + -- .000204565s elapsed
│ │ │  2
│ │ │ - -- .000139931s elapsed
│ │ │ + -- .000163673s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000154649s elapsed
│ │ │ + -- .000206228s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .0028279s elapsed
│ │ │ + -- .00019892s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .00015534s elapsed
│ │ │ + -- .000194086s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000137435s elapsed
│ │ │ + -- .000176726s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000132116s elapsed
│ │ │ + -- .000166712s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000129923s elapsed
│ │ │ + -- .000171777s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000126245s elapsed
│ │ │ + -- .000165045s elapsed
│ │ │  2
│ │ │ - -- .000126947s elapsed
│ │ │ + -- .000157514s elapsed
│ │ │  2
│ │ │ - -- .000129502s elapsed
│ │ │ + -- .000177225s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .00012912s elapsed
│ │ │ + -- .000159847s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000131656s elapsed
│ │ │ + -- .000188009s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000189794s elapsed
│ │ │ + -- .000185462s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000149969s elapsed
│ │ │ + -- .000195831s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000272388s elapsed
│ │ │ + -- .000179646s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000124412s elapsed
│ │ │ + -- .000170159s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000124452s elapsed
│ │ │ + -- .000151583s elapsed
│ │ │  2
│ │ │ - -- .000124672s elapsed
│ │ │ + -- .000153802s elapsed
│ │ │  1
│ │ │ - -- .000125894s elapsed
│ │ │ + -- .000177315s elapsed
│ │ │  1
│ │ │ - -- .000125735s elapsed
│ │ │ + -- .000165283s elapsed
│ │ │  1
│ │ │ - -- .000130323s elapsed
│ │ │ + -- .000157005s elapsed
│ │ │  1
│ │ │  
│ │ │  o2 = 3131031158784
│ │ │  
│ │ │  i3 : factor d
│ │ │  
│ │ │        32 6
│ │ ├── ./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
│ │ │ - -- .196956s elapsed
│ │ │ - -- .286293s elapsed
│ │ │ - -- .219371s elapsed
│ │ │ - -- .280074s elapsed
│ │ │ - -- .290248s elapsed
│ │ │ - -- .27205s elapsed
│ │ │ + -- .171072s elapsed
│ │ │ + -- .168393s elapsed
│ │ │ + -- .151337s elapsed
│ │ │ + -- .181682s elapsed
│ │ │ + -- .195996s elapsed
│ │ │ + -- .21695s 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)
│ │ │ - -- .00296643s elapsed
│ │ │ - -- .00668595s elapsed
│ │ │ - -- .0246113s elapsed
│ │ │ - -- .00899396s elapsed
│ │ │ - -- .00342295s elapsed
│ │ │ + -- .0236348s elapsed
│ │ │ + -- .00732165s elapsed
│ │ │ + -- .026486s elapsed
│ │ │ + -- .0104324s elapsed
│ │ │ + -- .00425117s 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)
│ │ │ - -- .306947s elapsed
│ │ │ + -- .248937s 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)
│ │ │ - -- .00447685s elapsed
│ │ │ - -- .0115156s elapsed
│ │ │ - -- .0436223s elapsed
│ │ │ - -- .0170221s elapsed
│ │ │ - -- .00643044s elapsed
│ │ │ + -- .00268156s elapsed
│ │ │ + -- .00678883s elapsed
│ │ │ + -- .0272227s elapsed
│ │ │ + -- .0109641s elapsed
│ │ │ + -- .0045422s 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
│ │ │ @@ -1,172 +1,172 @@
│ │ │  -- -*- M2-comint -*- hash: 1729182891690704738
│ │ │  
│ │ │  i1 : a=4
│ │ │  
│ │ │  o1 = 4
│ │ │  
│ │ │  i2 : (d1,d2)=resonanceDet(a)
│ │ │ - -- .0182126s elapsed
│ │ │ + -- .0204392s elapsed
│ │ │  (number of blocks= , 18)
│ │ │  (size of the matrices, Tally{1 => 4})
│ │ │                               2 => 6
│ │ │                               3 => 2
│ │ │                               4 => 6
│ │ │         0 1
│ │ │  total: 1 1
│ │ │      7: 1 1
│ │ │ - -- .000053781s elapsed
│ │ │ + -- .000045466s elapsed
│ │ │  (e )(-1)
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      7: 2 .
│ │ │      8: . 2
│ │ │ - -- .000088806s elapsed
│ │ │ + -- .000117751s elapsed
│ │ │      2
│ │ │  (e ) (e )(-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      7: 2 .
│ │ │      8: . .
│ │ │      9: . 2
│ │ │ - -- .000067035s elapsed
│ │ │ + -- .000086583s elapsed
│ │ │      2    2
│ │ │  (e ) (e )
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 3 3
│ │ │      7: 2 .
│ │ │      8: 1 .
│ │ │      9: . 1
│ │ │     10: . 2
│ │ │ - -- .000084928s elapsed
│ │ │ + -- .000115611s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (-3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      7: 1 .
│ │ │      8: 1 .
│ │ │      9: 2 2
│ │ │     10: . 1
│ │ │     11: . 1
│ │ │ - -- .000080921s elapsed
│ │ │ + -- .00010608s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      8: 1 .
│ │ │      9: 2 1
│ │ │     10: 1 2
│ │ │     11: . 1
│ │ │ - -- .000090478s elapsed
│ │ │ + -- .000100194s elapsed
│ │ │      2    3
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │      9: 1 1
│ │ │ - -- .000022591s elapsed
│ │ │ + -- .000031765s elapsed
│ │ │  (e )(-1)
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      9: 1 1
│ │ │     10: 1 1
│ │ │ - -- .000064409s elapsed
│ │ │ + -- .000078961s elapsed
│ │ │      2
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 2 1
│ │ │     10: 1 1
│ │ │     11: 1 2
│ │ │ - -- .000081843s elapsed
│ │ │ + -- .000108942s elapsed
│ │ │      2    2
│ │ │  (e ) (e ) (-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 1 .
│ │ │     10: 2 1
│ │ │     11: 1 2
│ │ │     12: . 1
│ │ │ - -- .000084347s elapsed
│ │ │ + -- .000103516s elapsed
│ │ │      2    3
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 1 .
│ │ │     10: 1 .
│ │ │     11: 2 2
│ │ │     12: . 1
│ │ │     13: . 1
│ │ │ - -- .000072405s elapsed
│ │ │ + -- .000099847s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 2 1
│ │ │     10: 1 1
│ │ │     11: 1 2
│ │ │ - -- .000076343s elapsed
│ │ │ + -- .000098747s elapsed
│ │ │      2    2
│ │ │  (e ) (e ) (-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 3 3
│ │ │     10: 2 .
│ │ │     11: 1 .
│ │ │     12: . 1
│ │ │     13: . 2
│ │ │ - -- .000070171s elapsed
│ │ │ + -- .000094411s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     10: 1 1
│ │ │     11: 1 1
│ │ │ - -- .00005925s elapsed
│ │ │ + -- .000082291s elapsed
│ │ │      2
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     11: 2 .
│ │ │     12: . .
│ │ │     13: . 2
│ │ │ - -- .00006973s elapsed
│ │ │ + -- .000104083s elapsed
│ │ │      2    2
│ │ │  (e ) (e )
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │     11: 1 1
│ │ │ - -- .000021691s elapsed
│ │ │ + -- .00003169s elapsed
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     12: 2 .
│ │ │     13: . 2
│ │ │ - -- .000068347s elapsed
│ │ │ + -- .000091179s elapsed
│ │ │      2
│ │ │  (e ) (e )(-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │     13: 1 1
│ │ │ - -- .000024396s elapsed
│ │ │ + -- .000032253s elapsed
│ │ │  (e )
│ │ │    1
│ │ │  
│ │ │         6      32    32
│ │ │  o2 = (3 , (e )  (e )  )
│ │ │              1     2
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_analyze__Strand.html
│ │ │ @@ -102,15 +102,15 @@
│ │ │  
│ │ │  o3 : Complex
    
│ │ │              
    
│ │ │                
    i4 : L = analyzeStrand(F,a); #L
│ │ │ - -- .023996s elapsed
│ │ │ + -- .0280581s elapsed
│ │ │  
│ │ │  o5 = 350
    
│ │ │              
    
│ │ │                
    i6 : betti F_a, betti F
│ │ │ @@ -141,19 +141,19 @@
│ │ │  
│ │ │  o9 = 14
    
│ │ │              
    
│ │ │                
    i10 : carpetBettiTable(a,b,3)
│ │ │ - -- .00242104s elapsed
│ │ │ - -- .00673308s elapsed
│ │ │ - -- .034067s elapsed
│ │ │ - -- .0999449s elapsed
│ │ │ - -- .00367219s elapsed
│ │ │ + -- .00297478s elapsed
│ │ │ + -- .00829941s elapsed
│ │ │ + -- .0275714s elapsed
│ │ │ + -- .0454032s elapsed
│ │ │ + -- .00423662s 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 {}
│ │ │ │ @@ -49,15 +49,15 @@
│ │ │ │       0                            1                             2
│ │ │ │  3                              4                              5
│ │ │ │  6                              7                              8
│ │ │ │  9
│ │ │ │  
│ │ │ │  o3 : Complex
│ │ │ │  i4 : L = analyzeStrand(F,a); #L
│ │ │ │ - -- .023996s elapsed
│ │ │ │ + -- .0280581s 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  .   .   .   .   .   .   .  . .
│ │ │ │ @@ -72,19 +72,19 @@
│ │ │ │  o7 = 2   3
│ │ │ │  
│ │ │ │  o7 : Expression of class Product
│ │ │ │  i8 : L3 = select(L,c->c%3==0); #L3
│ │ │ │  
│ │ │ │  o9 = 14
│ │ │ │  i10 : carpetBettiTable(a,b,3)
│ │ │ │ - -- .00242104s elapsed
│ │ │ │ - -- .00673308s elapsed
│ │ │ │ - -- .034067s elapsed
│ │ │ │ - -- .0999449s elapsed
│ │ │ │ - -- .00367219s elapsed
│ │ │ │ + -- .00297478s elapsed
│ │ │ │ + -- .00829941s elapsed
│ │ │ │ + -- .0275714s elapsed
│ │ │ │ + -- .0454032s elapsed
│ │ │ │ + -- .00423662s 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 : Sequence
    
│ │ │              
    
│ │ │                
    i2 : elapsedTime T=carpetBettiTable(a,b,3)
│ │ │ - -- .00266181s elapsed
│ │ │ - -- .00676057s elapsed
│ │ │ - -- .023025s elapsed
│ │ │ - -- .00985475s elapsed
│ │ │ - -- .0035774s elapsed
│ │ │ - -- .425089s elapsed
│ │ │ + -- .00432022s elapsed
│ │ │ + -- .00686264s elapsed
│ │ │ + -- .0328347s elapsed
│ │ │ + -- .0116131s elapsed
│ │ │ + -- .00413709s elapsed
│ │ │ + -- .483909s 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
│ │ │ @@ -112,15 +112,15 @@
│ │ │  o3 : Ideal of --[x ..x , y ..y ]
│ │ │                 3  0   5   0   5
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime T'=minimalBetti J
│ │ │ - -- .22587s elapsed
│ │ │ + -- .209815s 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
│ │ │ @@ -140,22 +140,22 @@
│ │ │  
│ │ │  o5 : BettiTally
    
│ │ │              
    
│ │ │                
    i6 : elapsedTime h=carpetBettiTables(6,6);
│ │ │ - -- .00455268s elapsed
│ │ │ - -- .0173468s elapsed
│ │ │ - -- .126286s elapsed
│ │ │ - -- 1.19253s elapsed
│ │ │ - -- .40692s elapsed
│ │ │ - -- .0843344s elapsed
│ │ │ - -- .006517s elapsed
│ │ │ - -- 6.20834s elapsed
    
│ │ │ + -- .00514119s elapsed
│ │ │ + -- .0241156s elapsed
│ │ │ + -- .12517s elapsed
│ │ │ + -- .980881s elapsed
│ │ │ + -- .451633s elapsed
│ │ │ + -- .0417908s elapsed
│ │ │ + -- .00765904s elapsed
│ │ │ + -- 5.93912s elapsed
│ │ │              
    
│ │ │                
    i7 : carpetBettiTable(h,7)
│ │ │  
│ │ │              0  1   2   3    4    5    6    7   8   9 10 11
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,20 +25,20 @@
│ │ │ │  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)
│ │ │ │ - -- .00266181s elapsed
│ │ │ │ - -- .00676057s elapsed
│ │ │ │ - -- .023025s elapsed
│ │ │ │ - -- .00985475s elapsed
│ │ │ │ - -- .0035774s elapsed
│ │ │ │ - -- .425089s elapsed
│ │ │ │ + -- .00432022s elapsed
│ │ │ │ + -- .00686264s elapsed
│ │ │ │ + -- .0328347s elapsed
│ │ │ │ + -- .0116131s elapsed
│ │ │ │ + -- .00413709s elapsed
│ │ │ │ + -- .483909s 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
│ │ │ │ @@ -46,15 +46,15 @@
│ │ │ │  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
│ │ │ │ - -- .22587s elapsed
│ │ │ │ + -- .209815s 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
│ │ │ │ @@ -66,22 +66,22 @@
│ │ │ │  o5 = total: . . . . . . . . . .
│ │ │ │           1: . . . . . . . . . .
│ │ │ │           2: . . . . . . . . . .
│ │ │ │           3: . . . . . . . . . .
│ │ │ │  
│ │ │ │  o5 : BettiTally
│ │ │ │  i6 : elapsedTime h=carpetBettiTables(6,6);
│ │ │ │ - -- .00455268s elapsed
│ │ │ │ - -- .0173468s elapsed
│ │ │ │ - -- .126286s elapsed
│ │ │ │ - -- 1.19253s elapsed
│ │ │ │ - -- .40692s elapsed
│ │ │ │ - -- .0843344s elapsed
│ │ │ │ - -- .006517s elapsed
│ │ │ │ - -- 6.20834s elapsed
│ │ │ │ + -- .00514119s elapsed
│ │ │ │ + -- .0241156s elapsed
│ │ │ │ + -- .12517s elapsed
│ │ │ │ + -- .980881s elapsed
│ │ │ │ + -- .451633s elapsed
│ │ │ │ + -- .0417908s elapsed
│ │ │ │ + -- .00765904s elapsed
│ │ │ │ + -- 5.93912s 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
│ │ │ @@ -80,19 +80,19 @@
│ │ │  
│ │ │  o1 : Sequence
    
│ │ │              
    
│ │ │                
    i2 : h=carpetBettiTables(a,b)
│ │ │ - -- .00258607s elapsed
│ │ │ - -- .00645513s elapsed
│ │ │ - -- .0463112s elapsed
│ │ │ - -- .011534s elapsed
│ │ │ - -- .00360799s elapsed
│ │ │ + -- .00282683s elapsed
│ │ │ + -- .00693946s elapsed
│ │ │ + -- .0295112s elapsed
│ │ │ + -- .0111707s elapsed
│ │ │ + -- .00417127s 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
│ │ │ @@ -134,15 +134,15 @@
│ │ │  o4 : Ideal of --[x ..x , y ..y ]
│ │ │                 3  0   5   0   5
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime T'=minimalBetti J
│ │ │ - -- .227924s elapsed
│ │ │ + -- .219522s 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
│ │ │ @@ -162,22 +162,22 @@
│ │ │  
│ │ │  o6 : BettiTally
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime h=carpetBettiTables(6,6);
│ │ │ - -- .0046955s elapsed
│ │ │ - -- .0175456s elapsed
│ │ │ - -- .0976552s elapsed
│ │ │ - -- 1.36978s elapsed
│ │ │ - -- .488999s elapsed
│ │ │ - -- .0396705s elapsed
│ │ │ - -- .00677344s elapsed
│ │ │ - -- 7.12863s elapsed
    
│ │ │ + -- .00519211s elapsed
│ │ │ + -- .0191225s elapsed
│ │ │ + -- .11103s elapsed
│ │ │ + -- 1.09954s elapsed
│ │ │ + -- .464044s elapsed
│ │ │ + -- .0449083s elapsed
│ │ │ + -- .00771454s elapsed
│ │ │ + -- 5.99796s elapsed
│ │ │              
    
│ │ │                
    i8 : keys h
│ │ │  
│ │ │  o8 = {0, 2, 3, 5}
│ │ │ ├── html2text {}
│ │ │ │ @@ -21,19 +21,19 @@
│ │ │ │  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)
│ │ │ │ - -- .00258607s elapsed
│ │ │ │ - -- .00645513s elapsed
│ │ │ │ - -- .0463112s elapsed
│ │ │ │ - -- .011534s elapsed
│ │ │ │ - -- .00360799s elapsed
│ │ │ │ + -- .00282683s elapsed
│ │ │ │ + -- .00693946s elapsed
│ │ │ │ + -- .0295112s elapsed
│ │ │ │ + -- .0111707s elapsed
│ │ │ │ + -- .00417127s 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
│ │ │ │ @@ -63,15 +63,15 @@
│ │ │ │  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
│ │ │ │ - -- .227924s elapsed
│ │ │ │ + -- .219522s 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
│ │ │ │ @@ -83,22 +83,22 @@
│ │ │ │  o6 = total: . . . . . . . . . .
│ │ │ │           1: . . . . . . . . . .
│ │ │ │           2: . . . . . . . . . .
│ │ │ │           3: . . . . . . . . . .
│ │ │ │  
│ │ │ │  o6 : BettiTally
│ │ │ │  i7 : elapsedTime h=carpetBettiTables(6,6);
│ │ │ │ - -- .0046955s elapsed
│ │ │ │ - -- .0175456s elapsed
│ │ │ │ - -- .0976552s elapsed
│ │ │ │ - -- 1.36978s elapsed
│ │ │ │ - -- .488999s elapsed
│ │ │ │ - -- .0396705s elapsed
│ │ │ │ - -- .00677344s elapsed
│ │ │ │ - -- 7.12863s elapsed
│ │ │ │ + -- .00519211s elapsed
│ │ │ │ + -- .0191225s elapsed
│ │ │ │ + -- .11103s elapsed
│ │ │ │ + -- 1.09954s elapsed
│ │ │ │ + -- .464044s elapsed
│ │ │ │ + -- .0449083s elapsed
│ │ │ │ + -- .00771454s elapsed
│ │ │ │ + -- 5.99796s elapsed
│ │ │ │  i8 : keys h
│ │ │ │  
│ │ │ │  o8 = {0, 2, 3, 5}
│ │ │ │  
│ │ │ │  o8 : List
│ │ │ │  i9 : carpetBettiTable(h,7)
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Det.html
│ │ │ @@ -80,82 +80,82 @@
│ │ │  
│ │ │  o1 : Sequence
    
│ │ │              
    
│ │ │                
    i2 : d=carpetDet(a,b)
│ │ │ - -- .0065779s elapsed
│ │ │ - -- .0185038s elapsed
│ │ │ + -- .00776234s elapsed
│ │ │ + -- .013589s elapsed
│ │ │  (number Of blocks, 26)
│ │ │ - -- .000299108s elapsed
│ │ │ + -- .000261781s elapsed
│ │ │  1
│ │ │ - -- .000140792s elapsed
│ │ │ + -- .000154249s elapsed
│ │ │  1
│ │ │ - -- .000145011s elapsed
│ │ │ + -- .0001552s elapsed
│ │ │  1
│ │ │ - -- .000132407s elapsed
│ │ │ + -- .000182441s elapsed
│ │ │  1
│ │ │ - -- .000131325s elapsed
│ │ │ + -- .000204565s elapsed
│ │ │  2
│ │ │ - -- .000139931s elapsed
│ │ │ + -- .000163673s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000154649s elapsed
│ │ │ + -- .000206228s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .0028279s elapsed
│ │ │ + -- .00019892s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .00015534s elapsed
│ │ │ + -- .000194086s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000137435s elapsed
│ │ │ + -- .000176726s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000132116s elapsed
│ │ │ + -- .000166712s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000129923s elapsed
│ │ │ + -- .000171777s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000126245s elapsed
│ │ │ + -- .000165045s elapsed
│ │ │  2
│ │ │ - -- .000126947s elapsed
│ │ │ + -- .000157514s elapsed
│ │ │  2
│ │ │ - -- .000129502s elapsed
│ │ │ + -- .000177225s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .00012912s elapsed
│ │ │ + -- .000159847s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000131656s elapsed
│ │ │ + -- .000188009s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000189794s elapsed
│ │ │ + -- .000185462s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000149969s elapsed
│ │ │ + -- .000195831s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000272388s elapsed
│ │ │ + -- .000179646s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000124412s elapsed
│ │ │ + -- .000170159s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000124452s elapsed
│ │ │ + -- .000151583s elapsed
│ │ │  2
│ │ │ - -- .000124672s elapsed
│ │ │ + -- .000153802s elapsed
│ │ │  1
│ │ │ - -- .000125894s elapsed
│ │ │ + -- .000177315s elapsed
│ │ │  1
│ │ │ - -- .000125735s elapsed
│ │ │ + -- .000165283s elapsed
│ │ │  1
│ │ │ - -- .000130323s elapsed
│ │ │ + -- .000157005s elapsed
│ │ │  1
│ │ │  
│ │ │  o2 = 3131031158784
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,82 +19,82 @@
│ │ │ │  determinants and return their product.
│ │ │ │  i1 : a=4,b=4
│ │ │ │  
│ │ │ │  o1 = (4, 4)
│ │ │ │  
│ │ │ │  o1 : Sequence
│ │ │ │  i2 : d=carpetDet(a,b)
│ │ │ │ - -- .0065779s elapsed
│ │ │ │ - -- .0185038s elapsed
│ │ │ │ + -- .00776234s elapsed
│ │ │ │ + -- .013589s elapsed
│ │ │ │  (number Of blocks, 26)
│ │ │ │ - -- .000299108s elapsed
│ │ │ │ + -- .000261781s elapsed
│ │ │ │  1
│ │ │ │ - -- .000140792s elapsed
│ │ │ │ + -- .000154249s elapsed
│ │ │ │  1
│ │ │ │ - -- .000145011s elapsed
│ │ │ │ + -- .0001552s elapsed
│ │ │ │  1
│ │ │ │ - -- .000132407s elapsed
│ │ │ │ + -- .000182441s elapsed
│ │ │ │  1
│ │ │ │ - -- .000131325s elapsed
│ │ │ │ + -- .000204565s elapsed
│ │ │ │  2
│ │ │ │ - -- .000139931s elapsed
│ │ │ │ + -- .000163673s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000154649s elapsed
│ │ │ │ + -- .000206228s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .0028279s elapsed
│ │ │ │ + -- .00019892s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .00015534s elapsed
│ │ │ │ + -- .000194086s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000137435s elapsed
│ │ │ │ + -- .000176726s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000132116s elapsed
│ │ │ │ + -- .000166712s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000129923s elapsed
│ │ │ │ + -- .000171777s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000126245s elapsed
│ │ │ │ + -- .000165045s elapsed
│ │ │ │  2
│ │ │ │ - -- .000126947s elapsed
│ │ │ │ + -- .000157514s elapsed
│ │ │ │  2
│ │ │ │ - -- .000129502s elapsed
│ │ │ │ + -- .000177225s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .00012912s elapsed
│ │ │ │ + -- .000159847s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000131656s elapsed
│ │ │ │ + -- .000188009s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000189794s elapsed
│ │ │ │ + -- .000185462s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000149969s elapsed
│ │ │ │ + -- .000195831s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000272388s elapsed
│ │ │ │ + -- .000179646s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000124412s elapsed
│ │ │ │ + -- .000170159s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000124452s elapsed
│ │ │ │ + -- .000151583s elapsed
│ │ │ │  2
│ │ │ │ - -- .000124672s elapsed
│ │ │ │ + -- .000153802s elapsed
│ │ │ │  1
│ │ │ │ - -- .000125894s elapsed
│ │ │ │ + -- .000177315s elapsed
│ │ │ │  1
│ │ │ │ - -- .000125735s elapsed
│ │ │ │ + -- .000165283s elapsed
│ │ │ │  1
│ │ │ │ - -- .000130323s elapsed
│ │ │ │ + -- .000157005s elapsed
│ │ │ │  1
│ │ │ │  
│ │ │ │  o2 = 3131031158784
│ │ │ │  i3 : factor d
│ │ │ │  
│ │ │ │        32 6
│ │ │ │  o3 = 2  3
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_compute__Bound.html
│ │ │ @@ -85,20 +85,20 @@
│ │ │  
│ │ │  o1 : Sequence
│ │ │              
    
│ │ │                
    i2 : computeBound 3
│ │ │ - -- .196956s elapsed
│ │ │ - -- .286293s elapsed
│ │ │ - -- .219371s elapsed
│ │ │ - -- .280074s elapsed
│ │ │ - -- .290248s elapsed
│ │ │ - -- .27205s elapsed
│ │ │ + -- .171072s elapsed
│ │ │ + -- .168393s elapsed
│ │ │ + -- .151337s elapsed
│ │ │ + -- .181682s elapsed
│ │ │ + -- .195996s elapsed
│ │ │ + -- .21695s elapsed
│ │ │  
│ │ │  o2 = 6
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,20 +25,20 @@
│ │ │ │  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
│ │ │ │ - -- .196956s elapsed
│ │ │ │ - -- .286293s elapsed
│ │ │ │ - -- .219371s elapsed
│ │ │ │ - -- .280074s elapsed
│ │ │ │ - -- .290248s elapsed
│ │ │ │ - -- .27205s elapsed
│ │ │ │ + -- .171072s elapsed
│ │ │ │ + -- .168393s elapsed
│ │ │ │ + -- .151337s elapsed
│ │ │ │ + -- .181682s elapsed
│ │ │ │ + -- .195996s elapsed
│ │ │ │ + -- .21695s 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
│ │ │ @@ -90,19 +90,19 @@
│ │ │  
│ │ │  o2 : Sequence
    
│ │ │              
    
│ │ │                
    i3 : h=degenerateK3BettiTables(a,b,e)
│ │ │ - -- .00296643s elapsed
│ │ │ - -- .00668595s elapsed
│ │ │ - -- .0246113s elapsed
│ │ │ - -- .00899396s elapsed
│ │ │ - -- .00342295s elapsed
│ │ │ + -- .0236348s elapsed
│ │ │ + -- .00732165s elapsed
│ │ │ + -- .026486s elapsed
│ │ │ + -- .0104324s elapsed
│ │ │ + -- .00425117s 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
│ │ │ @@ -136,15 +136,15 @@
│ │ │  
│ │ │  o4 : List
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime T= minimalBetti degenerateK3(a,b,e,Characteristic=>5)
│ │ │ - -- .306947s elapsed
│ │ │ + -- .248937s 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
│ │ │ @@ -178,19 +178,19 @@
│ │ │  
│ │ │  o7 : Sequence
    
│ │ │              
    
│ │ │                
    i8 : h=degenerateK3BettiTables(a,b,e)
│ │ │ - -- .00447685s elapsed
│ │ │ - -- .0115156s elapsed
│ │ │ - -- .0436223s elapsed
│ │ │ - -- .0170221s elapsed
│ │ │ - -- .00643044s elapsed
│ │ │ + -- .00268156s elapsed
│ │ │ + -- .00678883s elapsed
│ │ │ + -- .0272227s elapsed
│ │ │ + -- .0109641s elapsed
│ │ │ + -- .0045422s 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 {}
│ │ │ │ @@ -27,19 +27,19 @@
│ │ │ │  o1 : Sequence
│ │ │ │  i2 : e=(-1,5)
│ │ │ │  
│ │ │ │  o2 = (-1, 5)
│ │ │ │  
│ │ │ │  o2 : Sequence
│ │ │ │  i3 : h=degenerateK3BettiTables(a,b,e)
│ │ │ │ - -- .00296643s elapsed
│ │ │ │ - -- .00668595s elapsed
│ │ │ │ - -- .0246113s elapsed
│ │ │ │ - -- .00899396s elapsed
│ │ │ │ - -- .00342295s elapsed
│ │ │ │ + -- .0236348s elapsed
│ │ │ │ + -- .00732165s elapsed
│ │ │ │ + -- .026486s elapsed
│ │ │ │ + -- .0104324s elapsed
│ │ │ │ + -- .00425117s 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
│ │ │ │ @@ -65,15 +65,15 @@
│ │ │ │  o3 : HashTable
│ │ │ │  i4 : keys h
│ │ │ │  
│ │ │ │  o4 = {0, 2, 3, 5}
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : elapsedTime T= minimalBetti degenerateK3(a,b,e,Characteristic=>5)
│ │ │ │ - -- .306947s elapsed
│ │ │ │ + -- .248937s 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
│ │ │ │ @@ -94,19 +94,19 @@
│ │ │ │  these mistakes.
│ │ │ │  i7 : e=(-1,5^2)
│ │ │ │  
│ │ │ │  o7 = (-1, 25)
│ │ │ │  
│ │ │ │  o7 : Sequence
│ │ │ │  i8 : h=degenerateK3BettiTables(a,b,e)
│ │ │ │ - -- .00447685s elapsed
│ │ │ │ - -- .0115156s elapsed
│ │ │ │ - -- .0436223s elapsed
│ │ │ │ - -- .0170221s elapsed
│ │ │ │ - -- .00643044s elapsed
│ │ │ │ + -- .00268156s elapsed
│ │ │ │ + -- .00678883s elapsed
│ │ │ │ + -- .0272227s elapsed
│ │ │ │ + -- .0109641s elapsed
│ │ │ │ + -- .0045422s 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
│ │ │ @@ -78,172 +78,172 @@
│ │ │  
│ │ │  o1 = 4
    
│ │ │              
    
│ │ │                
    i2 : (d1,d2)=resonanceDet(a)
│ │ │ - -- .0182126s elapsed
│ │ │ + -- .0204392s elapsed
│ │ │  (number of blocks= , 18)
│ │ │  (size of the matrices, Tally{1 => 4})
│ │ │                               2 => 6
│ │ │                               3 => 2
│ │ │                               4 => 6
│ │ │         0 1
│ │ │  total: 1 1
│ │ │      7: 1 1
│ │ │ - -- .000053781s elapsed
│ │ │ + -- .000045466s elapsed
│ │ │  (e )(-1)
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      7: 2 .
│ │ │      8: . 2
│ │ │ - -- .000088806s elapsed
│ │ │ + -- .000117751s elapsed
│ │ │      2
│ │ │  (e ) (e )(-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      7: 2 .
│ │ │      8: . .
│ │ │      9: . 2
│ │ │ - -- .000067035s elapsed
│ │ │ + -- .000086583s elapsed
│ │ │      2    2
│ │ │  (e ) (e )
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 3 3
│ │ │      7: 2 .
│ │ │      8: 1 .
│ │ │      9: . 1
│ │ │     10: . 2
│ │ │ - -- .000084928s elapsed
│ │ │ + -- .000115611s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (-3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      7: 1 .
│ │ │      8: 1 .
│ │ │      9: 2 2
│ │ │     10: . 1
│ │ │     11: . 1
│ │ │ - -- .000080921s elapsed
│ │ │ + -- .00010608s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      8: 1 .
│ │ │      9: 2 1
│ │ │     10: 1 2
│ │ │     11: . 1
│ │ │ - -- .000090478s elapsed
│ │ │ + -- .000100194s elapsed
│ │ │      2    3
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │      9: 1 1
│ │ │ - -- .000022591s elapsed
│ │ │ + -- .000031765s elapsed
│ │ │  (e )(-1)
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      9: 1 1
│ │ │     10: 1 1
│ │ │ - -- .000064409s elapsed
│ │ │ + -- .000078961s elapsed
│ │ │      2
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 2 1
│ │ │     10: 1 1
│ │ │     11: 1 2
│ │ │ - -- .000081843s elapsed
│ │ │ + -- .000108942s elapsed
│ │ │      2    2
│ │ │  (e ) (e ) (-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 1 .
│ │ │     10: 2 1
│ │ │     11: 1 2
│ │ │     12: . 1
│ │ │ - -- .000084347s elapsed
│ │ │ + -- .000103516s elapsed
│ │ │      2    3
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 1 .
│ │ │     10: 1 .
│ │ │     11: 2 2
│ │ │     12: . 1
│ │ │     13: . 1
│ │ │ - -- .000072405s elapsed
│ │ │ + -- .000099847s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 2 1
│ │ │     10: 1 1
│ │ │     11: 1 2
│ │ │ - -- .000076343s elapsed
│ │ │ + -- .000098747s elapsed
│ │ │      2    2
│ │ │  (e ) (e ) (-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 3 3
│ │ │     10: 2 .
│ │ │     11: 1 .
│ │ │     12: . 1
│ │ │     13: . 2
│ │ │ - -- .000070171s elapsed
│ │ │ + -- .000094411s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     10: 1 1
│ │ │     11: 1 1
│ │ │ - -- .00005925s elapsed
│ │ │ + -- .000082291s elapsed
│ │ │      2
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     11: 2 .
│ │ │     12: . .
│ │ │     13: . 2
│ │ │ - -- .00006973s elapsed
│ │ │ + -- .000104083s elapsed
│ │ │      2    2
│ │ │  (e ) (e )
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │     11: 1 1
│ │ │ - -- .000021691s elapsed
│ │ │ + -- .00003169s elapsed
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     12: 2 .
│ │ │     13: . 2
│ │ │ - -- .000068347s elapsed
│ │ │ + -- .000091179s elapsed
│ │ │      2
│ │ │  (e ) (e )(-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │     13: 1 1
│ │ │ - -- .000024396s elapsed
│ │ │ + -- .000032253s elapsed
│ │ │  (e )
│ │ │    1
│ │ │  
│ │ │         6      32    32
│ │ │  o2 = (3 , (e )  (e )  )
│ │ │              1     2
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,172 +19,172 @@
│ │ │ │  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)
│ │ │ │ - -- .0182126s elapsed
│ │ │ │ + -- .0204392s elapsed
│ │ │ │  (number of blocks= , 18)
│ │ │ │  (size of the matrices, Tally{1 => 4})
│ │ │ │                               2 => 6
│ │ │ │                               3 => 2
│ │ │ │                               4 => 6
│ │ │ │         0 1
│ │ │ │  total: 1 1
│ │ │ │      7: 1 1
│ │ │ │ - -- .000053781s elapsed
│ │ │ │ + -- .000045466s elapsed
│ │ │ │  (e )(-1)
│ │ │ │    1
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │      7: 2 .
│ │ │ │      8: . 2
│ │ │ │ - -- .000088806s elapsed
│ │ │ │ + -- .000117751s elapsed
│ │ │ │      2
│ │ │ │  (e ) (e )(-1)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │      7: 2 .
│ │ │ │      8: . .
│ │ │ │      9: . 2
│ │ │ │ - -- .000067035s elapsed
│ │ │ │ + -- .000086583s elapsed
│ │ │ │      2    2
│ │ │ │  (e ) (e )
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 3 3
│ │ │ │      7: 2 .
│ │ │ │      8: 1 .
│ │ │ │      9: . 1
│ │ │ │     10: . 2
│ │ │ │ - -- .000084928s elapsed
│ │ │ │ + -- .000115611s elapsed
│ │ │ │      2    4
│ │ │ │  (e ) (e ) (-3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      7: 1 .
│ │ │ │      8: 1 .
│ │ │ │      9: 2 2
│ │ │ │     10: . 1
│ │ │ │     11: . 1
│ │ │ │ - -- .000080921s elapsed
│ │ │ │ + -- .00010608s elapsed
│ │ │ │      2    4
│ │ │ │  (e ) (e ) (3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      8: 1 .
│ │ │ │      9: 2 1
│ │ │ │     10: 1 2
│ │ │ │     11: . 1
│ │ │ │ - -- .000090478s elapsed
│ │ │ │ + -- .000100194s elapsed
│ │ │ │      2    3
│ │ │ │  (e ) (e ) (3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 1 1
│ │ │ │      9: 1 1
│ │ │ │ - -- .000022591s elapsed
│ │ │ │ + -- .000031765s elapsed
│ │ │ │  (e )(-1)
│ │ │ │    1
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │      9: 1 1
│ │ │ │     10: 1 1
│ │ │ │ - -- .000064409s elapsed
│ │ │ │ + -- .000078961s elapsed
│ │ │ │      2
│ │ │ │  (e )
│ │ │ │    1
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      9: 2 1
│ │ │ │     10: 1 1
│ │ │ │     11: 1 2
│ │ │ │ - -- .000081843s elapsed
│ │ │ │ + -- .000108942s elapsed
│ │ │ │      2    2
│ │ │ │  (e ) (e ) (-1)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      9: 1 .
│ │ │ │     10: 2 1
│ │ │ │     11: 1 2
│ │ │ │     12: . 1
│ │ │ │ - -- .000084347s elapsed
│ │ │ │ + -- .000103516s elapsed
│ │ │ │      2    3
│ │ │ │  (e ) (e ) (3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      9: 1 .
│ │ │ │     10: 1 .
│ │ │ │     11: 2 2
│ │ │ │     12: . 1
│ │ │ │     13: . 1
│ │ │ │ - -- .000072405s elapsed
│ │ │ │ + -- .000099847s elapsed
│ │ │ │      2    4
│ │ │ │  (e ) (e ) (3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      9: 2 1
│ │ │ │     10: 1 1
│ │ │ │     11: 1 2
│ │ │ │ - -- .000076343s elapsed
│ │ │ │ + -- .000098747s elapsed
│ │ │ │      2    2
│ │ │ │  (e ) (e ) (-1)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 3 3
│ │ │ │     10: 2 .
│ │ │ │     11: 1 .
│ │ │ │     12: . 1
│ │ │ │     13: . 2
│ │ │ │ - -- .000070171s elapsed
│ │ │ │ + -- .000094411s elapsed
│ │ │ │      2    4
│ │ │ │  (e ) (e ) (3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │     10: 1 1
│ │ │ │     11: 1 1
│ │ │ │ - -- .00005925s elapsed
│ │ │ │ + -- .000082291s elapsed
│ │ │ │      2
│ │ │ │  (e )
│ │ │ │    1
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │     11: 2 .
│ │ │ │     12: . .
│ │ │ │     13: . 2
│ │ │ │ - -- .00006973s elapsed
│ │ │ │ + -- .000104083s elapsed
│ │ │ │      2    2
│ │ │ │  (e ) (e )
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 1 1
│ │ │ │     11: 1 1
│ │ │ │ - -- .000021691s elapsed
│ │ │ │ + -- .00003169s elapsed
│ │ │ │  (e )
│ │ │ │    1
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │     12: 2 .
│ │ │ │     13: . 2
│ │ │ │ - -- .000068347s elapsed
│ │ │ │ + -- .000091179s elapsed
│ │ │ │      2
│ │ │ │  (e ) (e )(-1)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 1 1
│ │ │ │     13: 1 1
│ │ │ │ - -- .000024396s elapsed
│ │ │ │ + -- .000032253s elapsed
│ │ │ │  (e )
│ │ │ │    1
│ │ │ │  
│ │ │ │         6      32    32
│ │ │ │  o2 = (3 , (e )  (e )  )
│ │ │ │              1     2
│ │ ├── ./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 0.00931681s (cpu); 0.00931233s (thread); 0s (gc)
│ │ │ + -- used 0.0100824s (cpu); 0.010082s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o3 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i4 : time LLL(m, Strategy=>CohenEngine);
│ │ │ - -- used 0.0281993s (cpu); 0.0282009s (thread); 0s (gc)
│ │ │ + -- used 0.0321147s (cpu); 0.0321202s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o4 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i5 : time LLL(m, Strategy=>CohenTopLevel);
│ │ │ - -- used 0.109453s (cpu); 0.109456s (thread); 0s (gc)
│ │ │ + -- used 0.120087s (cpu); 0.120095s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o5 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i6 : time LLL(m, Strategy=>{Givens,RealFP});
│ │ │ - -- used 0.0121699s (cpu); 0.0121735s (thread); 0s (gc)
│ │ │ + -- used 0.0127551s (cpu); 0.0127627s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o6 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i7 : time LLL(m, Strategy=>{Givens,RealQP});
│ │ │ - -- used 0.048273s (cpu); 0.0482762s (thread); 0s (gc)
│ │ │ + -- used 0.0616423s (cpu); 0.0616611s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o7 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i8 : time LLL(m, Strategy=>{Givens,RealXD});
│ │ │ - -- used 0.0589056s (cpu); 0.058906s (thread); 0s (gc)
│ │ │ + -- used 0.0627626s (cpu); 0.06277s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o8 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i9 : time LLL(m, Strategy=>{Givens,RealRR});
│ │ │ - -- used 0.411173s (cpu); 0.411147s (thread); 0s (gc)
│ │ │ + -- used 0.35504s (cpu); 0.354912s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o9 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i10 : time LLL(m, Strategy=>{BKZ,Givens,RealQP});
│ │ │ - -- used 0.156182s (cpu); 0.156183s (thread); 0s (gc)
│ │ │ + -- used 0.151405s (cpu); 0.151411s (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
│ │ │ @@ -139,78 +139,78 @@
│ │ │                50       47
│ │ │  o2 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │                
    i3 : time LLL m;
│ │ │ - -- used 0.00931681s (cpu); 0.00931233s (thread); 0s (gc)
│ │ │ + -- used 0.0100824s (cpu); 0.010082s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o3 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │                
    i4 : time LLL(m, Strategy=>CohenEngine);
│ │ │ - -- used 0.0281993s (cpu); 0.0282009s (thread); 0s (gc)
│ │ │ + -- used 0.0321147s (cpu); 0.0321202s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o4 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │                
    i5 : time LLL(m, Strategy=>CohenTopLevel);
│ │ │ - -- used 0.109453s (cpu); 0.109456s (thread); 0s (gc)
│ │ │ + -- used 0.120087s (cpu); 0.120095s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o5 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │                
    i6 : time LLL(m, Strategy=>{Givens,RealFP});
│ │ │ - -- used 0.0121699s (cpu); 0.0121735s (thread); 0s (gc)
│ │ │ + -- used 0.0127551s (cpu); 0.0127627s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o6 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │                
    i7 : time LLL(m, Strategy=>{Givens,RealQP});
│ │ │ - -- used 0.048273s (cpu); 0.0482762s (thread); 0s (gc)
│ │ │ + -- used 0.0616423s (cpu); 0.0616611s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o7 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │                
    i8 : time LLL(m, Strategy=>{Givens,RealXD});
│ │ │ - -- used 0.0589056s (cpu); 0.058906s (thread); 0s (gc)
│ │ │ + -- used 0.0627626s (cpu); 0.06277s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o8 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │                
    i9 : time LLL(m, Strategy=>{Givens,RealRR});
│ │ │ - -- used 0.411173s (cpu); 0.411147s (thread); 0s (gc)
│ │ │ + -- used 0.35504s (cpu); 0.354912s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o9 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │                
    i10 : time LLL(m, Strategy=>{BKZ,Givens,RealQP});
│ │ │ - -- used 0.156182s (cpu); 0.156183s (thread); 0s (gc)
│ │ │ + -- used 0.151405s (cpu); 0.151411s (thread); 0s (gc)
│ │ │  
│ │ │                 50       47
│ │ │  o10 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -115,50 +115,50 @@
│ │ │ │                50       50
│ │ │ │  o1 : Matrix ZZ   <-- ZZ
│ │ │ │  i2 : m = syz m1;
│ │ │ │  
│ │ │ │                50       47
│ │ │ │  o2 : Matrix ZZ   <-- ZZ
│ │ │ │  i3 : time LLL m;
│ │ │ │ - -- used 0.00931681s (cpu); 0.00931233s (thread); 0s (gc)
│ │ │ │ + -- used 0.0100824s (cpu); 0.010082s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                50       47
│ │ │ │  o3 : Matrix ZZ   <-- ZZ
│ │ │ │  i4 : time LLL(m, Strategy=>CohenEngine);
│ │ │ │ - -- used 0.0281993s (cpu); 0.0282009s (thread); 0s (gc)
│ │ │ │ + -- used 0.0321147s (cpu); 0.0321202s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                50       47
│ │ │ │  o4 : Matrix ZZ   <-- ZZ
│ │ │ │  i5 : time LLL(m, Strategy=>CohenTopLevel);
│ │ │ │ - -- used 0.109453s (cpu); 0.109456s (thread); 0s (gc)
│ │ │ │ + -- used 0.120087s (cpu); 0.120095s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                50       47
│ │ │ │  o5 : Matrix ZZ   <-- ZZ
│ │ │ │  i6 : time LLL(m, Strategy=>{Givens,RealFP});
│ │ │ │ - -- used 0.0121699s (cpu); 0.0121735s (thread); 0s (gc)
│ │ │ │ + -- used 0.0127551s (cpu); 0.0127627s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                50       47
│ │ │ │  o6 : Matrix ZZ   <-- ZZ
│ │ │ │  i7 : time LLL(m, Strategy=>{Givens,RealQP});
│ │ │ │ - -- used 0.048273s (cpu); 0.0482762s (thread); 0s (gc)
│ │ │ │ + -- used 0.0616423s (cpu); 0.0616611s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                50       47
│ │ │ │  o7 : Matrix ZZ   <-- ZZ
│ │ │ │  i8 : time LLL(m, Strategy=>{Givens,RealXD});
│ │ │ │ - -- used 0.0589056s (cpu); 0.058906s (thread); 0s (gc)
│ │ │ │ + -- used 0.0627626s (cpu); 0.06277s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                50       47
│ │ │ │  o8 : Matrix ZZ   <-- ZZ
│ │ │ │  i9 : time LLL(m, Strategy=>{Givens,RealRR});
│ │ │ │ - -- used 0.411173s (cpu); 0.411147s (thread); 0s (gc)
│ │ │ │ + -- used 0.35504s (cpu); 0.354912s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                50       47
│ │ │ │  o9 : Matrix ZZ   <-- ZZ
│ │ │ │  i10 : time LLL(m, Strategy=>{BKZ,Givens,RealQP});
│ │ │ │ - -- used 0.156182s (cpu); 0.156183s (thread); 0s (gc)
│ │ │ │ + -- used 0.151405s (cpu); 0.151411s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                 50       47
│ │ │ │  o10 : Matrix ZZ   <-- ZZ
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  For most of the options, the columns do not need to be linearly independent.
│ │ │ │  The strategies CohenEngine and CohenTopLevel currently require the columns to
│ │ │ │  be linearly independent.
│ │ ├── ./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 0.644317s (cpu); 0.527424s (thread); 0s (gc)
│ │ │ + -- used 0.932299s (cpu); 0.547316s (thread); 0s (gc)
│ │ │  
│ │ │  i7 : time areIsomorphic(P,P,smoothTest=>false);
│ │ │ - -- used 0.455289s (cpu); 0.310632s (thread); 0s (gc)
│ │ │ + -- used 0.86894s (cpu); 0.359988s (thread); 0s (gc)
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/LatticePolytopes/html/_are__Isomorphic.html
│ │ │ @@ -120,21 +120,21 @@
│ │ │              
│ │ │                
    i5 : P = convexHull(M);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time areIsomorphic(P,P);
│ │ │ - -- used 0.644317s (cpu); 0.527424s (thread); 0s (gc)
    
│ │ │ + -- used 0.932299s (cpu); 0.547316s (thread); 0s (gc)    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : time areIsomorphic(P,P,smoothTest=>false);
│ │ │ - -- used 0.455289s (cpu); 0.310632s (thread); 0s (gc)
    
│ │ │ + -- used 0.86894s (cpu); 0.359988s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │          
│ │ │        
│ │ │        
    
│ │ │          
    
│ │ │            
    Ways to use areIsomorphic:
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,17 +35,17 @@
│ │ │ │       | 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 0.644317s (cpu); 0.527424s (thread); 0s (gc)
│ │ │ │ + -- used 0.932299s (cpu); 0.547316s (thread); 0s (gc)
│ │ │ │  i7 : time areIsomorphic(P,P,smoothTest=>false);
│ │ │ │ - -- used 0.455289s (cpu); 0.310632s (thread); 0s (gc)
│ │ │ │ + -- used 0.86894s (cpu); 0.359988s (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.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./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)
│ │ │ - -- .120994s elapsed
│ │ │ + -- .085409s 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}})
│ │ │ - -- .0122579s elapsed
│ │ │ + -- .0160746s 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)
│ │ │ - -- 3.5732s elapsed
│ │ │ + -- 2.93272s elapsed
│ │ │  
│ │ │  o6 = {{4, 3, 3}, {4, 4, 2}}
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : elapsedTime linearTruncationsBound M
│ │ │ - -- .0393284s elapsed
│ │ │ + -- .0310116s elapsed
│ │ │  
│ │ │  o7 = {{4, 3, 3}, {4, 4, 2}}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/LinearTruncations/html/_find__Region.html
│ │ │ @@ -125,25 +125,25 @@
│ │ │          
    
│ │ │            
    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)
│ │ │ - -- .120994s elapsed
│ │ │ + -- .085409s 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}})
│ │ │ - -- .0122579s elapsed
│ │ │ + -- .0160746s elapsed
│ │ │  
│ │ │  o7 = {{1, 2}, {3, 1}}
│ │ │  
│ │ │  o7 : List
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -48,22 +48,22 @@
│ │ │ │  
│ │ │ │  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)
│ │ │ │ - -- .120994s elapsed
│ │ │ │ + -- .085409s 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}})
│ │ │ │ - -- .0122579s elapsed
│ │ │ │ + -- .0160746s 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
│ │ │ @@ -123,25 +123,25 @@
│ │ │          
    
│ │ │            
    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)
│ │ │ - -- 3.5732s elapsed
│ │ │ + -- 2.93272s elapsed
│ │ │  
│ │ │  o6 = {{4, 3, 3}, {4, 4, 2}}
│ │ │  
│ │ │  o6 : List
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime linearTruncationsBound M
│ │ │ - -- .0393284s elapsed
│ │ │ + -- .0310116s elapsed
│ │ │  
│ │ │  o7 = {{4, 3, 3}, {4, 4, 2}}
│ │ │  
│ │ │  o7 : List
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -48,21 +48,21 @@
│ │ │ │  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)
│ │ │ │ - -- 3.5732s elapsed
│ │ │ │ + -- 2.93272s elapsed
│ │ │ │  
│ │ │ │  o6 = {{4, 3, 3}, {4, 4, 2}}
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : elapsedTime linearTruncationsBound M
│ │ │ │ - -- .0393284s elapsed
│ │ │ │ + -- .0310116s 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/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)
│ │ │ - -- .257637s elapsed
│ │ │ + -- .213559s 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
│ │ │ - -- .0603843s elapsed
│ │ │ + -- .01654s elapsed
│ │ │  
│ │ │  o11 = {1, 2, 3, 4, 5, 6}
│ │ │  
│ │ │  o11 : List
│ │ │  
│ │ │  i12 : elapsedTime hilbertSamuelFunction(q, N, 0, 5) -- 6(n+1) -- 0.32 seconds
│ │ │ - -- .271484s elapsed
│ │ │ + -- .277517s elapsed
│ │ │  
│ │ │  o12 = {6, 12, 18, 24, 30, 36}
│ │ │  
│ │ │  o12 : List
│ │ │  
│ │ │  i13 :
│ │ ├── ./usr/share/doc/Macaulay2/LocalRings/html/_hilbert__Samuel__Function.html
│ │ │ @@ -111,15 +111,15 @@
│ │ │                                1
│ │ │  o4 : RP-module, quotient of RP
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : elapsedTime hilbertSamuelFunction(M, 0, 6)
│ │ │ - -- .257637s elapsed
│ │ │ + -- .213559s elapsed
│ │ │  
│ │ │  o5 = {1, 3, 6, 7, 6, 3, 1}
│ │ │  
│ │ │  o5 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -163,25 +163,25 @@
│ │ │  
│ │ │  o10 : Ideal of RP
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i11 : elapsedTime hilbertSamuelFunction(N, 0, 5) -- n+1 -- 0.02 seconds
│ │ │ - -- .0603843s elapsed
│ │ │ + -- .01654s elapsed
│ │ │  
│ │ │  o11 = {1, 2, 3, 4, 5, 6}
│ │ │  
│ │ │  o11 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : elapsedTime hilbertSamuelFunction(q, N, 0, 5) -- 6(n+1) -- 0.32 seconds
│ │ │ - -- .271484s elapsed
│ │ │ + -- .277517s elapsed
│ │ │  
│ │ │  o12 = {6, 12, 18, 24, 30, 36}
│ │ │  
│ │ │  o12 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,15 +41,15 @@
│ │ │ │  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)
│ │ │ │ - -- .257637s elapsed
│ │ │ │ + -- .213559s elapsed
│ │ │ │  
│ │ │ │  o5 = {1, 3, 6, 7, 6, 3, 1}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : oo//sum
│ │ │ │  
│ │ │ │  o6 = 27
│ │ │ │ @@ -65,21 +65,21 @@
│ │ │ │  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
│ │ │ │ - -- .0603843s elapsed
│ │ │ │ + -- .01654s elapsed
│ │ │ │  
│ │ │ │  o11 = {1, 2, 3, 4, 5, 6}
│ │ │ │  
│ │ │ │  o11 : List
│ │ │ │  i12 : elapsedTime hilbertSamuelFunction(q, N, 0, 5) -- 6(n+1) -- 0.32 seconds
│ │ │ │ - -- .271484s elapsed
│ │ │ │ + -- .277517s 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/Macaulay2Doc/example-output/___Command.out
│ │ │ @@ -5,12 +5,12 @@
│ │ │  i2 : f
│ │ │  
│ │ │  o2 = 1073741824
│ │ │  
│ │ │  i3 : (c = Command "date";)
│ │ │  
│ │ │  i4 : c
│ │ │ -Fri Nov 14 17:26:26 UTC 2025
│ │ │ +Thu Dec 18 15:59:22 UTC 2025
│ │ │  
│ │ │  o4 = 0
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Database.out
│ │ │ @@ -1,16 +1,16 @@
│ │ │  -- -*- M2-comint -*- hash: 9579076464446459296
│ │ │  
│ │ │  i1 : filename = temporaryFileName () | ".dbm"
│ │ │  
│ │ │ -o1 = /tmp/M2-11641-0/0.dbm
│ │ │ +o1 = /tmp/M2-13231-0/0.dbm
│ │ │  
│ │ │  i2 : x = openDatabaseOut filename
│ │ │  
│ │ │ -o2 = /tmp/M2-11641-0/0.dbm
│ │ │ +o2 = /tmp/M2-13231-0/0.dbm
│ │ │  
│ │ │  o2 : Database
│ │ │  
│ │ │  i3 : x#"first" = "hi there"
│ │ │  
│ │ │  o3 = hi there
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__Cstats.out
│ │ │ @@ -1,19 +1,19 @@
│ │ │  -- -*- M2-comint -*- hash: 1731899428494721487
│ │ │  
│ │ │  i1 : s = GCstats()
│ │ │  
│ │ │ -o1 = HashTable{"bytesAlloc" => 42968369946        }
│ │ │ +o1 = HashTable{"bytesAlloc" => 43082233722        }
│ │ │                 "GC_free_space_divisor" => 3
│ │ │                 "GC_LARGE_ALLOC_WARN_INTERVAL" => 1
│ │ │                 "gcCpuTimeSecs" => 0
│ │ │ -               "heapSize" => 206622720
│ │ │ -               "numGCs" => 795
│ │ │ -               "numGCThreads" => 6
│ │ │ +               "heapSize" => 225443840
│ │ │ +               "numGCs" => 783
│ │ │ +               "numGCThreads" => 16
│ │ │  
│ │ │  o1 : HashTable
│ │ │  
│ │ │  i2 : s#"heapSize"
│ │ │  
│ │ │ -o2 = 206622720
│ │ │ +o2 = 225443840
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Minimal__Generators.out
│ │ │ @@ -40,20 +40,20 @@
│ │ │  o6 : PolynomialRing
│ │ │  
│ │ │  i7 : I = monomialCurveIdeal(R, {1,4,5,9});
│ │ │  
│ │ │  o7 : Ideal of R
│ │ │  
│ │ │  i8 : time J = truncate(8, I, MinimalGenerators => false);
│ │ │ - -- used 0.00897861s (cpu); 0.00897229s (thread); 0s (gc)
│ │ │ + -- used 0.00967653s (cpu); 0.00967333s (thread); 0s (gc)
│ │ │  
│ │ │  o8 : Ideal of R
│ │ │  
│ │ │  i9 : time K = truncate(8, I, MinimalGenerators => true);
│ │ │ - -- used 0.0801371s (cpu); 0.080144s (thread); 0s (gc)
│ │ │ + -- used 0.0823194s (cpu); 0.082328s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : Ideal of R
│ │ │  
│ │ │  i10 : numgens J
│ │ │  
│ │ │  o10 = 1067
│ │ ├── ./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 0.0241449s (cpu); 0.0241435s (thread); 0s (gc)
│ │ │ + -- used 0.0440342s (cpu); 0.0440332s (thread); 0s (gc)
│ │ │  
│ │ │  i3 : time SVD(M, DivideConquer=>true);
│ │ │ - -- used 0.0237108s (cpu); 0.0237156s (thread); 0s (gc)
│ │ │ + -- used 0.038972s (cpu); 0.0389862s (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 0.00188708s (cpu); 0.00187997s (thread); 0s (gc)
│ │ │ + -- used 0.00219741s (cpu); 0.00219001s (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 = .0002898248142239025
│ │ │ +o1 = .0003476298017475148
│ │ │  
│ │ │  o1 : RR (of precision 53)
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_caching_spcomputation_spresults.out
│ │ │ @@ -4,20 +4,20 @@
│ │ │  
│ │ │  i2 : R = QQ[x,y,z];
│ │ │  
│ │ │  i3 : M = coker vars R;
│ │ │  
│ │ │  i4 : elapsedTime pdim' M
│ │ │   -- computing pdim'
│ │ │ - -- .00646567s elapsed
│ │ │ + -- .00399088s elapsed
│ │ │  
│ │ │  o4 = 3
│ │ │  
│ │ │  i5 : elapsedTime pdim' M
│ │ │ - -- .000001924s elapsed
│ │ │ + -- .000002475s elapsed
│ │ │  
│ │ │  o5 = 3
│ │ │  
│ │ │  i6 : peek M.cache
│ │ │  
│ │ │  o6 = CacheTable{cache => MutableHashTable{}                                              }
│ │ │                  isHomogeneous => true
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cancel__Task_lp__Task_rp.out
│ │ │ @@ -18,29 +18,29 @@
│ │ │  
│ │ │  o4 = <>
│ │ │  
│ │ │  o4 : Task
│ │ │  
│ │ │  i5 : n
│ │ │  
│ │ │ -o5 = 711247
│ │ │ +o5 = 1095402
│ │ │  
│ │ │  i6 : sleep 1
│ │ │  
│ │ │  o6 = 0
│ │ │  
│ │ │  i7 : t
│ │ │  
│ │ │  o7 = <>
│ │ │  
│ │ │  o7 : Task
│ │ │  
│ │ │  i8 : n
│ │ │  
│ │ │ -o8 = 1453370
│ │ │ +o8 = 2223655
│ │ │  
│ │ │  i9 : isReady t
│ │ │  
│ │ │  o9 = false
│ │ │  
│ │ │  i10 : cancelTask t
│ │ │  
│ │ │ @@ -53,22 +53,22 @@
│ │ │  
│ │ │  o12 = <>
│ │ │  
│ │ │  o12 : Task
│ │ │  
│ │ │  i13 : n
│ │ │  
│ │ │ -o13 = 1453595
│ │ │ +o13 = 2223866
│ │ │  
│ │ │  i14 : sleep 1
│ │ │  
│ │ │  o14 = 0
│ │ │  
│ │ │  i15 : n
│ │ │  
│ │ │ -o15 = 1453595
│ │ │ +o15 = 2223866
│ │ │  
│ │ │  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/M2-10463-0/0
│ │ │ +o1 = /tmp/M2-10833-0/0
│ │ │  
│ │ │  i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-10463-0/0
│ │ │ +o2 = /tmp/M2-10833-0/0
│ │ │  
│ │ │  i3 : changeDirectory dir
│ │ │  
│ │ │ -o3 = /tmp/M2-10463-0/0/
│ │ │ +o3 = /tmp/M2-10833-0/0/
│ │ │  
│ │ │  i4 : currentDirectory()
│ │ │  
│ │ │ -o4 = /tmp/M2-10463-0/0/
│ │ │ +o4 = /tmp/M2-10833-0/0/
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_check.out
│ │ │ @@ -4,51 +4,51 @@
│ │ │  
│ │ │  o1 = FirstPackage
│ │ │  
│ │ │  o1 : Package
│ │ │  
│ │ │  i2 : check_1 FirstPackage
│ │ │   -- warning: reloading FirstPackage; recreate instances of types from this package
│ │ │ - -- capturing check(1, "FirstPackage")        -- .157225s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .116799s elapsed
│ │ │  
│ │ │  i3 : check FirstPackage
│ │ │ - -- capturing check(0, "FirstPackage")        -- .177874s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .230699s elapsed
│ │ │ + -- capturing check(0, "FirstPackage")        -- .120579s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .120961s elapsed
│ │ │  
│ │ │  i4 : check_1 "FirstPackage"
│ │ │ - -- capturing check(1, "FirstPackage")        -- .220889s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .127722s elapsed
│ │ │  
│ │ │  i5 : check "FirstPackage"
│ │ │ - -- capturing check(0, "FirstPackage")        -- .21817s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .226677s elapsed
│ │ │ + -- capturing check(0, "FirstPackage")        -- .126603s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .118973s elapsed
│ │ │  
│ │ │  i6 : tests(1, "FirstPackage")
│ │ │  
│ │ │  o6 = TestInput[/usr/share/Macaulay2/FirstPackage.m2:58:5-60:3]
│ │ │  
│ │ │  o6 : TestInput
│ │ │  
│ │ │  i7 : check oo
│ │ │ - -- capturing check(1, "FirstPackage")        -- .220218s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .116001s elapsed
│ │ │  
│ │ │  i8 : tests "FirstPackage"
│ │ │  
│ │ │  o8 = {0 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:54:5-56:3]}
│ │ │       {1 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:58:5-60:3]}
│ │ │  
│ │ │  o8 : NumberedVerticalList
│ │ │  
│ │ │  i9 : check oo
│ │ │ - -- capturing check(0, "FirstPackage")        -- .230323s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .223717s elapsed
│ │ │ + -- capturing check(0, "FirstPackage")        -- .123068s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .116914s elapsed
│ │ │  
│ │ │  i10 : tests "FirstPackage"
│ │ │  
│ │ │  o10 = {0 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:54:5-56:3]}
│ │ │        {1 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:58:5-60:3]}
│ │ │  
│ │ │  o10 : NumberedVerticalList
│ │ │  
│ │ │  i11 : check 1
│ │ │ - -- capturing check(1, "FirstPackage")        -- .230812s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .120197s elapsed
│ │ │  
│ │ │  i12 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_communicating_spwith_spprograms.out
│ │ │ @@ -1,26 +1,26 @@
│ │ │  -- -*- M2-comint -*- hash: 10365735446967377456
│ │ │  
│ │ │  i1 : run "uname -a"
│ │ │ -Linux sbuild 6.12.48+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.48-1 (2025-09-20) x86_64 GNU/Linux
│ │ │ +Linux sbuild 6.12.57+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.57-1 (2025-11-05) 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 6.12.48+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ -     6.12.48-1 (2025-09-20) x86_64 GNU/Linux\n"
│ │ │ +o3 = "Linux sbuild 6.12.57+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ +     6.12.57-1 (2025-11-05) x86_64 GNU/Linux\n"
│ │ │  
│ │ │  i4 : f = openInOut "!grep -E '^in'"
│ │ │  
│ │ │  o4 = !grep -E '^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 0x7f4e93a66540
│ │ │ +   -- registering gb 5 at 0x7f83eaa25540
│ │ │  
│ │ │     -- [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 0.27951s (cpu); 0.279785s (thread); 0s (gc)
│ │ │ + -- used 0.227838s (cpu); 0.225912s (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 0.00392072s (cpu); 0.00469829s (thread); 0s (gc)
│ │ │ + -- used 0.000148154s (cpu); 0.00367111s (thread); 0s (gc)
│ │ │  
│ │ │               0  1
│ │ │  o36 = total: 1 53
│ │ │            0: 1  .
│ │ │            1: .  .
│ │ │            2: .  2
│ │ │            3: .  1
│ │ ├── ./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/M2-11185-0/0/
│ │ │ +o1 = /tmp/M2-12295-0/0/
│ │ │  
│ │ │  i2 : dst = temporaryFileName() | "/"
│ │ │  
│ │ │ -o2 = /tmp/M2-11185-0/1/
│ │ │ +o2 = /tmp/M2-12295-0/1/
│ │ │  
│ │ │  i3 : makeDirectory (src|"a/")
│ │ │  
│ │ │ -o3 = /tmp/M2-11185-0/0/a/
│ │ │ +o3 = /tmp/M2-12295-0/0/a/
│ │ │  
│ │ │  i4 : makeDirectory (src|"b/")
│ │ │  
│ │ │ -o4 = /tmp/M2-11185-0/0/b/
│ │ │ +o4 = /tmp/M2-12295-0/0/b/
│ │ │  
│ │ │  i5 : makeDirectory (src|"b/c/")
│ │ │  
│ │ │ -o5 = /tmp/M2-11185-0/0/b/c/
│ │ │ +o5 = /tmp/M2-12295-0/0/b/c/
│ │ │  
│ │ │  i6 : src|"a/f" << "hi there" << close
│ │ │  
│ │ │ -o6 = /tmp/M2-11185-0/0/a/f
│ │ │ +o6 = /tmp/M2-12295-0/0/a/f
│ │ │  
│ │ │  o6 : File
│ │ │  
│ │ │  i7 : src|"a/g" << "hi there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11185-0/0/a/g
│ │ │ +o7 = /tmp/M2-12295-0/0/a/g
│ │ │  
│ │ │  o7 : File
│ │ │  
│ │ │  i8 : src|"b/c/g" << "ho there" << close
│ │ │  
│ │ │ -o8 = /tmp/M2-11185-0/0/b/c/g
│ │ │ +o8 = /tmp/M2-12295-0/0/b/c/g
│ │ │  
│ │ │  o8 : File
│ │ │  
│ │ │  i9 : stack findFiles src
│ │ │  
│ │ │ -o9 = /tmp/M2-11185-0/0/
│ │ │ -     /tmp/M2-11185-0/0/b/
│ │ │ -     /tmp/M2-11185-0/0/b/c/
│ │ │ -     /tmp/M2-11185-0/0/b/c/g
│ │ │ -     /tmp/M2-11185-0/0/a/
│ │ │ -     /tmp/M2-11185-0/0/a/g
│ │ │ -     /tmp/M2-11185-0/0/a/f
│ │ │ +o9 = /tmp/M2-12295-0/0/
│ │ │ +     /tmp/M2-12295-0/0/a/
│ │ │ +     /tmp/M2-12295-0/0/a/g
│ │ │ +     /tmp/M2-12295-0/0/a/f
│ │ │ +     /tmp/M2-12295-0/0/b/
│ │ │ +     /tmp/M2-12295-0/0/b/c/
│ │ │ +     /tmp/M2-12295-0/0/b/c/g
│ │ │  
│ │ │  i10 : copyDirectory(src,dst,Verbose=>true)
│ │ │ - -- copying: /tmp/M2-11185-0/0/b/c/g -> /tmp/M2-11185-0/1/b/c/g
│ │ │ - -- copying: /tmp/M2-11185-0/0/a/g -> /tmp/M2-11185-0/1/a/g
│ │ │ - -- copying: /tmp/M2-11185-0/0/a/f -> /tmp/M2-11185-0/1/a/f
│ │ │ + -- copying: /tmp/M2-12295-0/0/a/g -> /tmp/M2-12295-0/1/a/g
│ │ │ + -- copying: /tmp/M2-12295-0/0/a/f -> /tmp/M2-12295-0/1/a/f
│ │ │ + -- copying: /tmp/M2-12295-0/0/b/c/g -> /tmp/M2-12295-0/1/b/c/g
│ │ │  
│ │ │  i11 : copyDirectory(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-11185-0/0/b/c/g not newer than /tmp/M2-11185-0/1/b/c/g
│ │ │ - -- skipping: /tmp/M2-11185-0/0/a/g not newer than /tmp/M2-11185-0/1/a/g
│ │ │ - -- skipping: /tmp/M2-11185-0/0/a/f not newer than /tmp/M2-11185-0/1/a/f
│ │ │ + -- skipping: /tmp/M2-12295-0/0/a/g not newer than /tmp/M2-12295-0/1/a/g
│ │ │ + -- skipping: /tmp/M2-12295-0/0/a/f not newer than /tmp/M2-12295-0/1/a/f
│ │ │ + -- skipping: /tmp/M2-12295-0/0/b/c/g not newer than /tmp/M2-12295-0/1/b/c/g
│ │ │  
│ │ │  i12 : stack findFiles dst
│ │ │  
│ │ │ -o12 = /tmp/M2-11185-0/1/
│ │ │ -      /tmp/M2-11185-0/1/a/
│ │ │ -      /tmp/M2-11185-0/1/a/f
│ │ │ -      /tmp/M2-11185-0/1/a/g
│ │ │ -      /tmp/M2-11185-0/1/b/
│ │ │ -      /tmp/M2-11185-0/1/b/c/
│ │ │ -      /tmp/M2-11185-0/1/b/c/g
│ │ │ +o12 = /tmp/M2-12295-0/1/
│ │ │ +      /tmp/M2-12295-0/1/a/
│ │ │ +      /tmp/M2-12295-0/1/a/g
│ │ │ +      /tmp/M2-12295-0/1/a/f
│ │ │ +      /tmp/M2-12295-0/1/b/
│ │ │ +      /tmp/M2-12295-0/1/b/c/
│ │ │ +      /tmp/M2-12295-0/1/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/M2-10970-0/0
│ │ │ +o1 = /tmp/M2-11860-0/0
│ │ │  
│ │ │  i2 : dst = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-10970-0/1
│ │ │ +o2 = /tmp/M2-11860-0/1
│ │ │  
│ │ │  i3 : src << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-10970-0/0
│ │ │ +o3 = /tmp/M2-11860-0/0
│ │ │  
│ │ │  o3 : File
│ │ │  
│ │ │  i4 : copyFile(src,dst,Verbose=>true)
│ │ │ - -- copying: /tmp/M2-10970-0/0 -> /tmp/M2-10970-0/1
│ │ │ + -- copying: /tmp/M2-11860-0/0 -> /tmp/M2-11860-0/1
│ │ │  
│ │ │  i5 : get dst
│ │ │  
│ │ │  o5 = hi there
│ │ │  
│ │ │  i6 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-10970-0/0 not newer than /tmp/M2-10970-0/1
│ │ │ + -- skipping: /tmp/M2-11860-0/0 not newer than /tmp/M2-11860-0/1
│ │ │  
│ │ │  i7 : src << "ho there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-10970-0/0
│ │ │ +o7 = /tmp/M2-11860-0/0
│ │ │  
│ │ │  o7 : File
│ │ │  
│ │ │  i8 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-10970-0/0 not newer than /tmp/M2-10970-0/1
│ │ │ + -- skipping: /tmp/M2-11860-0/0 not newer than /tmp/M2-11860-0/1
│ │ │  
│ │ │  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 = 363.561188353
│ │ │ +o1 = 349.628677912
│ │ │  
│ │ │  o1 : RR (of precision 53)
│ │ │  
│ │ │  i2 : for i from 0 to 1000000 do 223131321321*324234324324;
│ │ │  
│ │ │  i3 : t2 = cpuTime()
│ │ │  
│ │ │ -o3 = 365.292753193
│ │ │ +o3 = 350.644170628
│ │ │  
│ │ │  o3 : RR (of precision 53)
│ │ │  
│ │ │  i4 : t2-t1
│ │ │  
│ │ │ -o4 = 1.731564839999976
│ │ │ +o4 = 1.01549271600004
│ │ │  
│ │ │  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 = 1763141264
│ │ │ +o1 = 1766073617
│ │ │  
│ │ │  i2 : currentTime() /( (365 + 97./400) * 24 * 60 * 60 )
│ │ │  
│ │ │ -o2 = 55.87172246546498
│ │ │ +o2 = 55.96464503289165
│ │ │  
│ │ │  o2 : RR (of precision 53)
│ │ │  
│ │ │  i3 : 12 * (oo - floor oo)
│ │ │  
│ │ │ -o3 = 10.46066958557981
│ │ │ +o3 = 11.57574039469984
│ │ │  
│ │ │  o3 : RR (of precision 53)
│ │ │  
│ │ │  i4 : run "date"
│ │ │ -Fri Nov 14 17:27:44 UTC 2025
│ │ │ +Thu Dec 18 16:00:17 UTC 2025
│ │ │  
│ │ │  o4 = 0
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elapsed__Time.out
│ │ │ @@ -1,8 +1,8 @@
│ │ │  -- -*- M2-comint -*- hash: 1330565958025
│ │ │  
│ │ │  i1 : elapsedTime sleep 1
│ │ │ - -- 1.00014s elapsed
│ │ │ + -- 1.00011s elapsed
│ │ │  
│ │ │  o1 = 0
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elapsed__Timing.out
│ │ │ @@ -1,14 +1,14 @@
│ │ │  -- -*- M2-comint -*- hash: 1731106803207298715
│ │ │  
│ │ │  i1 : elapsedTiming sleep 1
│ │ │  
│ │ │  o1 = 0
│ │ │ -     -- 1.00015 seconds
│ │ │ +     -- 1.00013 seconds
│ │ │  
│ │ │  o1 : Time
│ │ │  
│ │ │  i2 : peek oo
│ │ │  
│ │ │ -o2 = Time{1.00015, 0}
│ │ │ +o2 = Time{1.00013, 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 0.474296s (cpu); 0.280859s (thread); 0s (gc)
│ │ │ + -- used 0.133595s (cpu); 0.133594s (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 0.497187s (cpu); 0.266913s (thread); 0s (gc)
│ │ │ + -- used 0.370662s (cpu); 0.186441s (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 0.340207s (cpu); 0.126135s (thread); 0s (gc)
│ │ │ + -- used 0.0321744s (cpu); 0.0321782s (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 0.445785s (cpu); 0.239366s (thread); 0s (gc)
│ │ │ + -- used 0.104043s (cpu); 0.104048s (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 0.0020489s (cpu); 0.00204903s (thread); 0s (gc)
│ │ │ + -- used 0.00173352s (cpu); 0.00173146s (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 0.056264s (cpu); 0.0562726s (thread); 0s (gc)
│ │ │ + -- used 0.0388167s (cpu); 0.0388294s (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/M2-10191-0/91-rundir/
│ │ │ +             source directory => /tmp/M2-10311-0/91-rundir/
│ │ │               source file => stdio
│ │ │               test inputs => MutableList{}
│ │ │  
│ │ │  i7 : dictionaryPath
│ │ │  
│ │ │  o7 = {Foo.Dictionary, Varieties.Dictionary, Isomorphism.Dictionary,
│ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Exists.out
│ │ │ @@ -1,20 +1,20 @@
│ │ │  -- -*- M2-comint -*- hash: 7475038936570224899
│ │ │  
│ │ │  i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10558-0/0
│ │ │ +o1 = /tmp/M2-11028-0/0
│ │ │  
│ │ │  i2 : fileExists fn
│ │ │  
│ │ │  o2 = false
│ │ │  
│ │ │  i3 : fn << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-10558-0/0
│ │ │ +o3 = /tmp/M2-11028-0/0
│ │ │  
│ │ │  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/M2-12150-0/0
│ │ │ +o1 = /tmp/M2-14270-0/0
│ │ │  
│ │ │  o1 : File
│ │ │  
│ │ │  i2 : fileLength f
│ │ │  
│ │ │  o2 = 8
│ │ │  
│ │ │  i3 : close f
│ │ │  
│ │ │ -o3 = /tmp/M2-12150-0/0
│ │ │ +o3 = /tmp/M2-14270-0/0
│ │ │  
│ │ │  o3 : File
│ │ │  
│ │ │  i4 : filename = toString f
│ │ │  
│ │ │ -o4 = /tmp/M2-12150-0/0
│ │ │ +o4 = /tmp/M2-14270-0/0
│ │ │  
│ │ │  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/M2-11375-0/0
│ │ │ +o1 = /tmp/M2-12685-0/0
│ │ │  
│ │ │  i2 : f = fn << "hi there"
│ │ │  
│ │ │ -o2 = /tmp/M2-11375-0/0
│ │ │ +o2 = /tmp/M2-12685-0/0
│ │ │  
│ │ │  o2 : File
│ │ │  
│ │ │  i3 : fileMode f
│ │ │  
│ │ │  o3 = 420
│ │ │  
│ │ │  i4 : close f
│ │ │  
│ │ │ -o4 = /tmp/M2-11375-0/0
│ │ │ +o4 = /tmp/M2-12685-0/0
│ │ │  
│ │ │  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/M2-10989-0/0
│ │ │ +o1 = /tmp/M2-11899-0/0
│ │ │  
│ │ │  i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-10989-0/0
│ │ │ +o2 = /tmp/M2-11899-0/0
│ │ │  
│ │ │  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/M2-10854-0/0
│ │ │ +o1 = /tmp/M2-11624-0/0
│ │ │  
│ │ │  i2 : f = fn << "hi there"
│ │ │  
│ │ │ -o2 = /tmp/M2-10854-0/0
│ │ │ +o2 = /tmp/M2-11624-0/0
│ │ │  
│ │ │  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/M2-10854-0/0
│ │ │ +o6 = /tmp/M2-11624-0/0
│ │ │  
│ │ │  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/M2-11977-0/0
│ │ │ +o1 = /tmp/M2-13917-0/0
│ │ │  
│ │ │  i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-11977-0/0
│ │ │ +o2 = /tmp/M2-13917-0/0
│ │ │  
│ │ │  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 = 66
│ │ │ +o1 = 49
│ │ │  
│ │ │  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 0x7f499763be00
│ │ │ +   -- registering gb 0 at 0x7f60362e5e00
│ │ │  
│ │ │     -- [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 = Fri Nov 14 17:26:54 UTC 2025
│ │ │ +o4 = Thu Dec 18 15:59:41 UTC 2025
│ │ │  
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_instances.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/M2-10380-0/0
│ │ │ +o2 = /tmp/M2-10670-0/0
│ │ │  
│ │ │  i3 : fn << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-10380-0/0
│ │ │ +o3 = /tmp/M2-10670-0/0
│ │ │  
│ │ │  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)
│ │ │ - -- 4.26787s elapsed
│ │ │ + -- 5.61121s 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
│ │ │ - -- .0564874s elapsed
│ │ │ + -- .0593425s elapsed
│ │ │  
│ │ │  o23 = true
│ │ │  
│ │ │  i24 : elapsedTime isPseudoprime m3
│ │ │ - -- .000106819s elapsed
│ │ │ + -- .000124275s 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/M2-12188-0/0
│ │ │ +o1 = /tmp/M2-14348-0/0
│ │ │  
│ │ │  i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-12188-0/0
│ │ │ +o2 = /tmp/M2-14348-0/0
│ │ │  
│ │ │  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/M2-10722-0/0
│ │ │ +o1 = /tmp/M2-11352-0/0
│ │ │  
│ │ │  i2 : makeDirectory (dir|"/a/b/c")
│ │ │  
│ │ │ -o2 = /tmp/M2-10722-0/0/a/b/c
│ │ │ +o2 = /tmp/M2-11352-0/0/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 1.38174s (cpu); 0.797855s (thread); 0s (gc)
│ │ │ + -- used 0.818496s (cpu); 0.62457s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 514229
│ │ │  
│ │ │  i3 : fib = memoize fib
│ │ │  
│ │ │  o3 = fib
│ │ │  
│ │ │  o3 : FunctionClosure
│ │ │  
│ │ │  i4 : time fib 28
│ │ │ - -- used 8.4749e-05s (cpu); 8.4468e-05s (thread); 0s (gc)
│ │ │ + -- used 7.7903e-05s (cpu); 7.4132e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 514229
│ │ │  
│ │ │  i5 : time fib 28
│ │ │ - -- used 4.028e-06s (cpu); 3.647e-06s (thread); 0s (gc)
│ │ │ + -- used 3.788e-06s (cpu); 3.64e-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
│ │ │ @@ -17,20 +17,20 @@
│ │ │       {12 => (poincare, BettiTally)                                }
│ │ │       {13 => (hilbertPolynomial, ZZ, BettiTally)                   }
│ │ │       {14 => (degree, BettiTally)                                  }
│ │ │       {15 => (hilbertSeries, ZZ, BettiTally)                       }
│ │ │       {16 => (pdim, BettiTally)                                    }
│ │ │       {17 => (regularity, BettiTally)                              }
│ │ │       {18 => (mathML, BettiTally)                                  }
│ │ │ -     {19 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │ -     {20 => (dual, BettiTally)                                    }
│ │ │ -     {21 => (codim, BettiTally)                                   }
│ │ │ +     {19 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ +     {20 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │ +     {21 => (dual, BettiTally)                                    }
│ │ │       {22 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │       {23 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ -     {24 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ +     {24 => (codim, BettiTally)                                   }
│ │ │       {25 => (^, Ring, BettiTally)                                 }
│ │ │  
│ │ │  o1 : NumberedVerticalList
│ │ │  
│ │ │  i2 : methods resolution
│ │ │  
│ │ │  o2 = {0 => (resolution, Ideal) }
│ │ │ @@ -61,19 +61,19 @@
│ │ │       {2 => (++, Module, Module)      }
│ │ │  
│ │ │  o4 : NumberedVerticalList
│ │ │  
│ │ │  i5 : methods( Matrix, Matrix )
│ │ │  
│ │ │  o5 = {0 => (diff', Matrix, Matrix)                               }
│ │ │ -     {1 => (-, Matrix, Matrix)                                   }
│ │ │ -     {2 => (+, Matrix, Matrix)                                   }
│ │ │ -     {3 => (diff, Matrix, Matrix)                                }
│ │ │ -     {4 => (contract, Matrix, Matrix)                            }
│ │ │ -     {5 => (contract', Matrix, Matrix)                           }
│ │ │ +     {1 => (diff, Matrix, Matrix)                                }
│ │ │ +     {2 => (contract, Matrix, Matrix)                            }
│ │ │ +     {3 => (contract', Matrix, Matrix)                           }
│ │ │ +     {4 => (-, Matrix, Matrix)                                   }
│ │ │ +     {5 => (+, Matrix, Matrix)                                   }
│ │ │       {6 => (markedGB, Matrix, Matrix)                            }
│ │ │       {7 => (Hom, Matrix, Matrix)                                 }
│ │ │       {8 => (==, Matrix, Matrix)                                  }
│ │ │       {9 => (*, Matrix, Matrix)                                   }
│ │ │       {10 => (|, Matrix, Matrix)                                  }
│ │ │       {11 => (||, Matrix, Matrix)                                 }
│ │ │       {12 => (subquotient, Matrix, Matrix)                        }
│ │ │ @@ -84,22 +84,22 @@
│ │ │       {17 => (quotientRemainder', Matrix, Matrix)                 }
│ │ │       {18 => (quotientRemainder, Matrix, Matrix)                  }
│ │ │       {19 => (//, Matrix, Matrix)                                 }
│ │ │       {20 => (\\, Matrix, Matrix)                                 }
│ │ │       {21 => (quotient, Matrix, Matrix)                           }
│ │ │       {22 => (quotient', Matrix, Matrix)                          }
│ │ │       {23 => (remainder', Matrix, Matrix)                         }
│ │ │ -     {24 => (remainder, Matrix, Matrix)                          }
│ │ │ -     {25 => (%, Matrix, Matrix)                                  }
│ │ │ +     {24 => (%, Matrix, Matrix)                                  }
│ │ │ +     {25 => (remainder, Matrix, Matrix)                          }
│ │ │       {26 => (pushout, Matrix, Matrix)                            }
│ │ │       {27 => (solve, Matrix, Matrix)                              }
│ │ │ -     {28 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │ -     {29 => (intersect, Matrix, Matrix)                          }
│ │ │ -     {30 => (pullback, Matrix, Matrix)                           }
│ │ │ -     {31 => (tensor, Matrix, Matrix)                             }
│ │ │ +     {28 => (tensor, Matrix, Matrix)                             }
│ │ │ +     {29 => (pullback, Matrix, Matrix)                           }
│ │ │ +     {30 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │ +     {31 => (intersect, Matrix, Matrix)                          }
│ │ │       {32 => (substitute, Matrix, Matrix)                         }
│ │ │       {33 => (yonedaProduct, Matrix, Matrix)                      }
│ │ │       {34 => (isShortExactSequence, Matrix, Matrix)               }
│ │ │       {35 => (horseshoeResolution, Matrix, Matrix)                }
│ │ │       {36 => (connectingExtMap, Module, Matrix, Matrix)           }
│ │ │       {37 => (connectingExtMap, Matrix, Matrix, Module)           }
│ │ │       {38 => (connectingTorMap, Module, Matrix, Matrix)           }
│ │ ├── ./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
│ │ │ - -- 1.919s elapsed
│ │ │ + -- 2.34322s 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)
│ │ │ - -- .74946s elapsed
│ │ │ + -- .978961s 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)
│ │ │ - -- .0315312s elapsed
│ │ │ + -- .0378645s 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)
│ │ │ - -- 1.2026s elapsed
│ │ │ + -- 1.60704s 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/M2-10741-0/0/
│ │ │ +o1 = /tmp/M2-11391-0/0/
│ │ │  
│ │ │  i2 : mkdir p
│ │ │  
│ │ │  i3 : isDirectory p
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 : (fn = p | "foo") << "hi there" << close
│ │ │  
│ │ │ -o4 = /tmp/M2-10741-0/0/foo
│ │ │ +o4 = /tmp/M2-11391-0/0/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/M2-10615-0/0
│ │ │ +o1 = /tmp/M2-11145-0/0
│ │ │  
│ │ │  i2 : dst = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-10615-0/1
│ │ │ +o2 = /tmp/M2-11145-0/1
│ │ │  
│ │ │  i3 : src << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-10615-0/0
│ │ │ +o3 = /tmp/M2-11145-0/0
│ │ │  
│ │ │  o3 : File
│ │ │  
│ │ │  i4 : moveFile(src,dst,Verbose=>true)
│ │ │ ---moving: /tmp/M2-10615-0/0 -> /tmp/M2-10615-0/1
│ │ │ +--moving: /tmp/M2-11145-0/0 -> /tmp/M2-11145-0/1
│ │ │  
│ │ │  i5 : get dst
│ │ │  
│ │ │  o5 = hi there
│ │ │  
│ │ │  i6 : bak = moveFile(dst,Verbose=>true)
│ │ │ ---backup file created: /tmp/M2-10615-0/1.bak
│ │ │ +--backup file created: /tmp/M2-11145-0/1.bak
│ │ │  
│ │ │ -o6 = /tmp/M2-10615-0/1.bak
│ │ │ +o6 = /tmp/M2-11145-0/1.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
│ │ │ - -- .500135s elapsed
│ │ │ + -- .500116s elapsed
│ │ │  
│ │ │  o1 = 0
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_options_lp__Function_rp.out
│ │ │ @@ -21,35 +21,35 @@
│ │ │  
│ │ │  o3 = OptionTable{Generic => false}
│ │ │  
│ │ │  o3 : OptionTable
│ │ │  
│ │ │  i4 : methods codim
│ │ │  
│ │ │ -o4 = {0 => (codim, BettiTally)    }
│ │ │ -     {1 => (codim, Module)        }
│ │ │ -     {2 => (codim, QuotientRing)  }
│ │ │ -     {3 => (codim, Ideal)         }
│ │ │ -     {4 => (codim, MonomialIdeal) }
│ │ │ -     {5 => (codim, PolynomialRing)}
│ │ │ -     {6 => (codim, CoherentSheaf) }
│ │ │ -     {7 => (codim, Variety)       }
│ │ │ +o4 = {0 => (codim, Module)        }
│ │ │ +     {1 => (codim, QuotientRing)  }
│ │ │ +     {2 => (codim, Ideal)         }
│ │ │ +     {3 => (codim, MonomialIdeal) }
│ │ │ +     {4 => (codim, PolynomialRing)}
│ │ │ +     {5 => (codim, CoherentSheaf) }
│ │ │ +     {6 => (codim, Variety)       }
│ │ │ +     {7 => (codim, BettiTally)    }
│ │ │  
│ │ │  o4 : NumberedVerticalList
│ │ │  
│ │ │  i5 : options oo
│ │ │  
│ │ │ -o5 = {0 => (OptionTable{})                }
│ │ │ +o5 = {0 => (OptionTable{Generic => false})}
│ │ │       {1 => (OptionTable{Generic => false})}
│ │ │       {2 => (OptionTable{Generic => false})}
│ │ │       {3 => (OptionTable{Generic => false})}
│ │ │       {4 => (OptionTable{Generic => false})}
│ │ │       {5 => (OptionTable{Generic => false})}
│ │ │       {6 => (OptionTable{Generic => false})}
│ │ │ -     {7 => (OptionTable{Generic => false})}
│ │ │ +     {7 => (OptionTable{})                }
│ │ │  
│ │ │  o5 : NumberedVerticalList
│ │ │  
│ │ │  i6 : methods intersect
│ │ │  
│ │ │  o6 = {0 => (intersect, List)                           }
│ │ │       {1 => (intersect, Ideal, Ideal)                   }
│ │ ├── ./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);
│ │ │ - -- .636785s elapsed
│ │ │ + -- .70432s elapsed
│ │ │  
│ │ │  i4 : elapsedTime parallelApply(1..100, n -> sort L);
│ │ │ - -- .302911s elapsed
│ │ │ + -- .187029s elapsed
│ │ │  
│ │ │  i5 : allowableThreads
│ │ │  
│ │ │  o5 = 5
│ │ │  
│ │ │  i6 : allowableThreads = maxAllowableThreads
│ │ │  
│ │ │ -o6 = 7
│ │ │ +o6 = 17
│ │ │  
│ │ │  i7 : R = QQ[x,y,z];
│ │ │  
│ │ │  i8 : I = ideal(x^2 + 2*y^2 - y - 2*z, x^2 - 8*y^2 + 10*z - 1, x^2 - 7*y*z)
│ │ │  
│ │ │               2     2            2     2             2
│ │ │  o8 = ideal (x  + 2y  - y - 2z, x  - 8y  + 10z - 1, x  - 7y*z)
│ │ │ @@ -74,15 +74,15 @@
│ │ │  
│ │ │  o16 : Task
│ │ │  
│ │ │  i17 : schedule t';
│ │ │  
│ │ │  i18 : t'
│ │ │  
│ │ │ -o18 = <>
│ │ │ +o18 = <>
│ │ │  
│ │ │  o18 : Task
│ │ │  
│ │ │  i19 : taskResult t'
│ │ │  
│ │ │  o19 = | 980z2-18y-201z+13 35yz-4y+2z-1 10y2-y-12z+1 5x2-4y+2z-1 |
│ │ ├── ./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
│ │ │ - -- 2.04775s elapsed
│ │ │ + -- 2.38342s 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)
│ │ │ - -- 1.80304s elapsed
│ │ │ + -- 2.35357s 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)
│ │ │ - -- 5.57414s elapsed
│ │ │ + -- 2.33547s 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)
│ │ │ - -- 2.03698s elapsed
│ │ │ + -- 2.71086s 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)
│ │ │ - -- 1.946s elapsed
│ │ │ + -- 2.65371s 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");
│ │ │ - -- 4.82802s elapsed
│ │ │ + -- 4.21704s 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");
│ │ │ - -- 6.58749s elapsed
│ │ │ + -- 8.52357s 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");
│ │ │ - -- 4.12035s elapsed
│ │ │ + -- 3.78655s 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);
│ │ │ - -- 1.03876s elapsed
│ │ │ + -- 1.021s 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);
│ │ │ - -- 1.17701s elapsed
│ │ │ + -- 1.42759s 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 0.000521488s (cpu); 2.0348e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00399979s (cpu); 0s (thread); 0s (gc)
│ │ │  
│ │ │              3     6    9
│ │ │  o28 = 1 - 3T  + 3T  - T
│ │ │  
│ │ │  o28 : ZZ[T]
│ │ │  
│ │ │  i29 : time gens gb I;
│ │ │  
│ │ │ -   -- registering gb 19 at 0x7fe6adf6d380
│ │ │ +   -- registering gb 19 at 0x7f963d095540
│ │ │  
│ │ │     -- [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 0.0234308s (cpu); 0.0247955s (thread); 0s (gc)
│ │ │ +   --  -- used 0.0118597s (cpu); 0.0150154s (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 0x7fe6adf6d1c0
│ │ │ +   -- registering gb 20 at 0x7f963d095380
│ │ │  
│ │ │     -- [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 0x7fe6adf6d000
│ │ │ +   -- registering gb 21 at 0x7f963d095000
│ │ │  
│ │ │     -- [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 0.00793104s (cpu); 0.00887304s (thread); 0s (gc)
│ │ │ +   --  -- used 0.00403043s (cpu); 0.00586067s (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 0x7fe6ad908e00
│ │ │ +   -- registering gb 22 at 0x7f963d1f0e00
│ │ │  
│ │ │     -- [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 0.0839992s (cpu); 0.083356s (thread); 0s (gc)
│ │ │ +   --  -- used 0.0557631s (cpu); 0.0543421s (thread); 0s (gc)
│ │ │  
│ │ │                1      39
│ │ │  o37 : Matrix S  <-- S
│ │ │  
│ │ │  i38 : selectInSubring(1, gens gb J)
│ │ │  
│ │ │  o38 = | 188529931266160087758259645374082357642621166724936033369975727480205
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_printing_spto_spa_spfile.out
│ │ │ @@ -12,19 +12,19 @@
│ │ │  
│ │ │  o2 = stdio
│ │ │  
│ │ │  o2 : File
│ │ │  
│ │ │  i3 : fn = temporaryFileName()
│ │ │  
│ │ │ -o3 = /tmp/M2-10932-0/0
│ │ │ +o3 = /tmp/M2-11782-0/0
│ │ │  
│ │ │  i4 : fn << "hi there" << endl << close
│ │ │  
│ │ │ -o4 = /tmp/M2-10932-0/0
│ │ │ +o4 = /tmp/M2-11782-0/0
│ │ │  
│ │ │  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/M2-10932-0/0
│ │ │ +o9 = /tmp/M2-11782-0/0
│ │ │  
│ │ │  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/M2-10932-0/0
│ │ │ +o11 = /tmp/M2-11782-0/0
│ │ │  
│ │ │  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/_process__I__D.out
│ │ │ @@ -1,7 +1,7 @@
│ │ │  -- -*- M2-comint -*- hash: 1330513630563
│ │ │  
│ │ │  i1 : processID()
│ │ │  
│ │ │ -o1 = 10191
│ │ │ +o1 = 10311
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_profile.out
│ │ │ @@ -9,35 +9,35 @@
│ │ │  
│ │ │                4       5
│ │ │  o1 : Matrix ZZ  <-- ZZ
│ │ │  
│ │ │  i2 : profileSummary
│ │ │  
│ │ │  o2 = #run  %time   position                         
│ │ │ -     1     94.37   ../../m2/matrix1.m2:279:4-282:58 
│ │ │ -     1     91.83   ../../m2/matrix1.m2:281:22-281:43
│ │ │ -     1     44.29   ../../m2/matrix1.m2:193:25-193:52
│ │ │ -     1     30.52   ../../m2/matrix1.m2:114:5-156:72 
│ │ │ -     1     29.41   ../../m2/matrix1.m2:140:10-155:16
│ │ │ -     1     22      ../../m2/matrix1.m2:181:4-181:42 
│ │ │ +     1     94.74   ../../m2/matrix1.m2:279:4-282:58 
│ │ │ +     1     92.3    ../../m2/matrix1.m2:281:22-281:43
│ │ │ +     1     45.8    ../../m2/matrix1.m2:193:25-193:52
│ │ │ +     1     32.19   ../../m2/matrix1.m2:114:5-156:72 
│ │ │ +     1     31      ../../m2/matrix1.m2:140:10-155:16
│ │ │ +     1     22.36   ../../m2/matrix1.m2:45:10-49:22  
│ │ │ +     1     21.94   ../../m2/matrix1.m2:181:4-181:42 
│ │ │       1     20.71   ../../m2/set.m2:127:5-127:61     
│ │ │ -     1     20.64   ../../m2/matrix1.m2:45:10-49:22  
│ │ │ -     1     3.36    ../../m2/matrix1.m2:112:5-112:29 
│ │ │ -     1     2.46    ../../m2/matrix1.m2:141:13-141:78
│ │ │ -     1     2.17    ../../m2/matrix1.m2:96:5-109:11  
│ │ │ -     1     1.50    ../../m2/matrix1.m2:281:7-281:16 
│ │ │ -     1     1.46    ../../m2/matrix1.m2:147:20-147:64
│ │ │ -     1     1.28    ../../m2/matrix1.m2:111:5-111:91 
│ │ │ -     1     1.21    ../../m2/matrix1.m2:276:4-277:73 
│ │ │ -     1     1.18    ../../m2/matrix1.m2:182:4-184:74 
│ │ │ -     1     1.09    ../../m2/matrix1.m2:98:10-98:46  
│ │ │ -     1     .74     ../../m2/modules.m2:279:4-279:52 
│ │ │ -     20    .63     ../../m2/matrix1.m2:191:14-192:67
│ │ │ -     20    .45     ../../m2/matrix1.m2:47:43-47:71  
│ │ │ -     1     .0039s  elapsed total                    
│ │ │ +     1     3.59    ../../m2/matrix1.m2:112:5-112:29 
│ │ │ +     1     2.38    ../../m2/matrix1.m2:141:13-141:78
│ │ │ +     1     2.15    ../../m2/matrix1.m2:96:5-109:11  
│ │ │ +     1     1.46    ../../m2/matrix1.m2:281:7-281:16 
│ │ │ +     1     1.38    ../../m2/matrix1.m2:276:4-277:73 
│ │ │ +     1     1.37    ../../m2/matrix1.m2:147:20-147:64
│ │ │ +     1     1.2     ../../m2/matrix1.m2:111:5-111:91 
│ │ │ +     1     1.11    ../../m2/matrix1.m2:98:10-98:46  
│ │ │ +     1     1.04    ../../m2/matrix1.m2:182:4-184:74 
│ │ │ +     20    .95     ../../m2/matrix1.m2:191:14-192:67
│ │ │ +     20    .76     ../../m2/matrix1.m2:47:43-47:71  
│ │ │ +     1     .63     ../../m2/modules.m2:279:4-279:52 
│ │ │ +     1     .0038s  elapsed total                    
│ │ │  
│ │ │  i3 : coverageSummary
│ │ │  
│ │ │  o3 = covered lines:
│ │ │       ../../m2/lists.m2:145:24-145:32
│ │ │       ../../m2/lists.m2:145:34-145:58
│ │ │       ../../m2/matrix.m2:12:5-12:35
│ │ ├── ./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 0.177655s (cpu); 0.142413s (thread); 0s (gc)
│ │ │ + -- used 0.255888s (cpu); 0.124419s (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 0.710725s (cpu); 0.37679s (thread); 0s (gc)
│ │ │ + -- used 0.801597s (cpu); 0.319583s (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 4.02482s (cpu); 2.02411s (thread); 0s (gc)
│ │ │ + -- used 3.99909s (cpu); 2.04708s (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/M2-11565-0/0
│ │ │ +o1 = /tmp/M2-13075-0/0
│ │ │  
│ │ │  i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-11565-0/0
│ │ │ +o2 = /tmp/M2-13075-0/0
│ │ │  
│ │ │  i3 : (fn = dir | "/" | "foo") << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11565-0/0/foo
│ │ │ +o3 = /tmp/M2-13075-0/0/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/M2-11107-0/0
│ │ │ +o1 = /tmp/M2-12137-0/0
│ │ │  
│ │ │  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/M2-11107-0/0
│ │ │ +o2 = /tmp/M2-12137-0/0
│ │ │  
│ │ │  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/M2-11107-0/0
│ │ │ +o7 = /tmp/M2-12137-0/0
│ │ │  
│ │ │  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/M2-11806-0/0
│ │ │ +o1 = /tmp/M2-13556-0/0
│ │ │  
│ │ │  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/M2-10191-0/86-rundir/
│ │ │ +o1 = /tmp/M2-10311-0/86-rundir/
│ │ │  
│ │ │  i2 : p = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-11825-0/0
│ │ │ +o2 = /tmp/M2-13595-0/0
│ │ │  
│ │ │  i3 : q = temporaryFileName()
│ │ │  
│ │ │ -o3 = /tmp/M2-11825-0/1
│ │ │ +o3 = /tmp/M2-13595-0/1
│ │ │  
│ │ │  i4 : symlinkFile(p,q)
│ │ │  
│ │ │  i5 : p << close
│ │ │  
│ │ │ -o5 = /tmp/M2-11825-0/0
│ │ │ +o5 = /tmp/M2-13595-0/0
│ │ │  
│ │ │  o5 : File
│ │ │  
│ │ │  i6 : readlink q
│ │ │  
│ │ │ -o6 = /tmp/M2-11825-0/0
│ │ │ +o6 = /tmp/M2-13595-0/0
│ │ │  
│ │ │  i7 : realpath q
│ │ │  
│ │ │ -o7 = /tmp/M2-11825-0/0
│ │ │ +o7 = /tmp/M2-13595-0/0
│ │ │  
│ │ │  i8 : removeFile p
│ │ │  
│ │ │  i9 : removeFile q
│ │ │  
│ │ │  i10 : realpath ""
│ │ │  
│ │ │ -o10 = /tmp/M2-10191-0/86-rundir/
│ │ │ +o10 = /tmp/M2-10311-0/86-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)[0]: -- finalizing sequence #1 --
│ │ │ ---finalization: (2)[5]: -- finalizing sequence #6 --
│ │ │ ---finalization: (3)[3]: -- finalizing sequence #4 --
│ │ │ ---finalization: (4)[6]: -- finalizing sequence #7 --
│ │ │ ---finalization: (5)[7]: -- finalizing sequence #8 --
│ │ │ ---finalization: (7)[1]: -- finalizing sequence #2 --
│ │ │ +--finalization: (1)[1]: -- finalizing sequence #2 --
│ │ │ +--finalization: (3)[5]: -- finalizing sequence #6 --
│ │ │ +--finalization: (4)[2]: -- finalizing sequence #3 --
│ │ │ +--finalization: (5)[3]: -- finalizing sequence #4 --
│ │ │ +--finalization: (6)[0]: -- finalizing sequence #1 --
│ │ │ +--finalization: (7)[6]: -- finalizing sequence #7 --
│ │ │  --finalization: (8)[4]: -- finalizing sequence #5 --
│ │ │ ---finalization: (6)[2]: -- finalizing sequence #3 --
│ │ │ +--finalization: (2)[7]: -- finalizing sequence #8 --
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_remove__Directory.out
│ │ │ @@ -1,19 +1,19 @@
│ │ │  -- -*- M2-comint -*- hash: 8532980310097060089
│ │ │  
│ │ │  i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10779-0/0
│ │ │ +o1 = /tmp/M2-11469-0/0
│ │ │  
│ │ │  i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-10779-0/0
│ │ │ +o2 = /tmp/M2-11469-0/0
│ │ │  
│ │ │  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/M2-10283-0/0
│ │ │ +o1 = /tmp/M2-10473-0/0
│ │ │  
│ │ │  i2 : rootPath | fn
│ │ │  
│ │ │ -o2 = /tmp/M2-10283-0/0
│ │ │ +o2 = /tmp/M2-10473-0/0
│ │ │  
│ │ │  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/M2-11508-0/0
│ │ │ +o1 = /tmp/M2-12958-0/0
│ │ │  
│ │ │  i2 : rootURI | fn
│ │ │  
│ │ │ -o2 = file:///tmp/M2-11508-0/0
│ │ │ +o2 = file:///tmp/M2-12958-0/0
│ │ │  
│ │ │  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/M2-11356-0/0
│ │ │ +o5 = /tmp/M2-12646-0/0
│ │ │  
│ │ │  i6 : f << toString (p,m,M) << close
│ │ │  
│ │ │ -o6 = /tmp/M2-11356-0/0
│ │ │ +o6 = /tmp/M2-12646-0/0
│ │ │  
│ │ │  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/_serial__Number.out
│ │ │ @@ -1,15 +1,15 @@
│ │ │  -- -*- M2-comint -*- hash: 5271760183816554957
│ │ │  
│ │ │  i1 : serialNumber asdf
│ │ │  
│ │ │ -o1 = 1426273
│ │ │ +o1 = 1526273
│ │ │  
│ │ │  i2 : serialNumber foo
│ │ │  
│ │ │ -o2 = 1426275
│ │ │ +o2 = 1526275
│ │ │  
│ │ │  i3 : serialNumber ZZ
│ │ │  
│ │ │  o3 = 1000050
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_solve.out
│ │ │ @@ -189,18 +189,18 @@
│ │ │  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 0.000225452s (cpu); 0.000217157s (thread); 0s (gc)
│ │ │ + -- used 0.000210794s (cpu); 0.00020165s (thread); 0s (gc)
│ │ │  
│ │ │  i31 : time X = solve(A,B, MaximalRank=>true);
│ │ │ - -- used 0.000165831s (cpu); 0.000165941s (thread); 0s (gc)
│ │ │ + -- used 0.000105086s (cpu); 0.000105046s (thread); 0s (gc)
│ │ │  
│ │ │  i32 : norm(A*X-B)
│ │ │  
│ │ │  o32 = 5.111850690840453e-15
│ │ │  
│ │ │  o32 : RR (of precision 53)
│ │ │  
│ │ │ @@ -209,18 +209,18 @@
│ │ │  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 0.494046s (cpu); 0.307142s (thread); 0s (gc)
│ │ │ + -- used 0.141612s (cpu); 0.141359s (thread); 0s (gc)
│ │ │  
│ │ │  i39 : time X = solve(A,B, MaximalRank=>true);
│ │ │ - -- used 0.241089s (cpu); 0.24109s (thread); 0s (gc)
│ │ │ + -- used 0.14493s (cpu); 0.14494s (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/M2-11147-0/0/
│ │ │ +o1 = /tmp/M2-12217-0/0/
│ │ │  
│ │ │  i2 : dst = temporaryFileName() | "/"
│ │ │  
│ │ │ -o2 = /tmp/M2-11147-0/1/
│ │ │ +o2 = /tmp/M2-12217-0/1/
│ │ │  
│ │ │  i3 : makeDirectory (src|"a/")
│ │ │  
│ │ │ -o3 = /tmp/M2-11147-0/0/a/
│ │ │ +o3 = /tmp/M2-12217-0/0/a/
│ │ │  
│ │ │  i4 : makeDirectory (src|"b/")
│ │ │  
│ │ │ -o4 = /tmp/M2-11147-0/0/b/
│ │ │ +o4 = /tmp/M2-12217-0/0/b/
│ │ │  
│ │ │  i5 : makeDirectory (src|"b/c/")
│ │ │  
│ │ │ -o5 = /tmp/M2-11147-0/0/b/c/
│ │ │ +o5 = /tmp/M2-12217-0/0/b/c/
│ │ │  
│ │ │  i6 : src|"a/f" << "hi there" << close
│ │ │  
│ │ │ -o6 = /tmp/M2-11147-0/0/a/f
│ │ │ +o6 = /tmp/M2-12217-0/0/a/f
│ │ │  
│ │ │  o6 : File
│ │ │  
│ │ │  i7 : src|"a/g" << "hi there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11147-0/0/a/g
│ │ │ +o7 = /tmp/M2-12217-0/0/a/g
│ │ │  
│ │ │  o7 : File
│ │ │  
│ │ │  i8 : src|"b/c/g" << "ho there" << close
│ │ │  
│ │ │ -o8 = /tmp/M2-11147-0/0/b/c/g
│ │ │ +o8 = /tmp/M2-12217-0/0/b/c/g
│ │ │  
│ │ │  o8 : File
│ │ │  
│ │ │  i9 : symlinkDirectory(src,dst,Verbose=>true)
│ │ │ ---symlinking: ../../../0/b/c/g -> /tmp/M2-11147-0/1/b/c/g
│ │ │ ---symlinking: ../../0/a/g -> /tmp/M2-11147-0/1/a/g
│ │ │ ---symlinking: ../../0/a/f -> /tmp/M2-11147-0/1/a/f
│ │ │ +--symlinking: ../../0/a/g -> /tmp/M2-12217-0/1/a/g
│ │ │ +--symlinking: ../../0/a/f -> /tmp/M2-12217-0/1/a/f
│ │ │ +--symlinking: ../../../0/b/c/g -> /tmp/M2-12217-0/1/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/M2-11147-0/1/b/c/g
│ │ │ ---unsymlinking: ../../0/a/g -> /tmp/M2-11147-0/1/a/g
│ │ │ ---unsymlinking: ../../0/a/f -> /tmp/M2-11147-0/1/a/f
│ │ │ +--unsymlinking: ../../0/a/g -> /tmp/M2-12217-0/1/a/g
│ │ │ +--unsymlinking: ../../0/a/f -> /tmp/M2-12217-0/1/a/f
│ │ │ +--unsymlinking: ../../../0/b/c/g -> /tmp/M2-12217-0/1/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/M2-11204-0/0
│ │ │ +o1 = /tmp/M2-12334-0/0
│ │ │  
│ │ │  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/M2-12169-0/0.tex
│ │ │ +o1 = /tmp/M2-14309-0/0.tex
│ │ │  
│ │ │  i2 : temporaryFileName () | ".html"
│ │ │  
│ │ │ -o2 = /tmp/M2-12169-0/1.html
│ │ │ +o2 = /tmp/M2-14309-0/1.html
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_time.out
│ │ │ @@ -1,8 +1,8 @@
│ │ │  -- -*- M2-comint -*- hash: 1332435500723
│ │ │  
│ │ │  i1 : time 3^30
│ │ │ - -- used 1.4376e-05s (cpu); 7.053e-06s (thread); 0s (gc)
│ │ │ + -- used 1.7726e-05s (cpu); 5.692e-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
│ │ │ -     -- .000015599 seconds
│ │ │ +     -- .000020481 seconds
│ │ │  
│ │ │  o1 : Time
│ │ │  
│ │ │  i2 : peek oo
│ │ │  
│ │ │ -o2 = Time{.000015599, 205891132094649}
│ │ │ +o2 = Time{.000020481, 205891132094649}
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_version.out
│ │ │ @@ -34,15 +34,15 @@
│ │ │                 "memtailor version" => 1.0
│ │ │                 "mpfi version" => 1.5.4
│ │ │                 "mpfr version" => 4.2.2
│ │ │                 "mpsolve version" => 3.2.2
│ │ │                 "mysql version" => not present
│ │ │                 "normaliz version" => 3.10.5
│ │ │                 "ntl version" => 11.5.1
│ │ │ -               "operating system release" => 6.12.48+deb13-amd64
│ │ │ +               "operating system release" => 6.12.57+deb13-cloud-amd64
│ │ │                 "operating system" => Linux
│ │ │                 "packages" => Style FirstPackage Macaulay2Doc Parsing Classic Browse Benchmark Text SimpleDoc PackageTemplate Saturation PrimaryDecomposition FourierMotzkin Dmodules WeylAlgebras HolonomicSystems BernsteinSato ConnectionMatrices 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 WeilDivisors EllipticCurves HighestWeights MinimalPrimes Bertini TorAlgebra Permanents BinomialEdgeIdeals TateOnProducts LatticePolytopes FiniteFittingIdeals HigherCIOperators LieAlgebraRepresentations 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 TestIdeals FrobeniusThresholds NonPrincipalTestIdeals Seminormalization AlgebraicSplines TriangularSets Chordal Tropical SymbolicPowers Complexes OldChainComplexes 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 ExteriorExtensions Oscillators IncidenceCorrespondenceCohomology ToricHigherDirectImages Brackets IntegerProgramming GameTheory AllMarkovBases Tableaux CpMackeyFunctors JSONRPC MatrixFactorizations PathSignatures
│ │ │                 "pointer size" => 8
│ │ │                 "python version" => 3.13.9
│ │ │                 "readline version" => 8.3
│ │ │                 "scscp version" => not present
│ │ │                 "tbb version" => 2022.1
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Command.html
│ │ │ @@ -84,15 +84,15 @@
│ │ │              
│ │ │                
    i3 : (c = Command "date";)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : c
│ │ │ -Fri Nov 14 17:26:26 UTC 2025
│ │ │ +Thu Dec 18 15:59:22 UTC 2025
│ │ │  
│ │ │  o4 = 0
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── 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
│ │ │ │ -Fri Nov 14 17:26:26 UTC 2025
│ │ │ │ +Thu Dec 18 15:59:22 UTC 2025
│ │ │ │  
│ │ │ │  o4 = 0
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_u_n -- run an external command
│ │ │ │      * _A_f_t_e_r_E_v_a_l -- top level method applied after evaluation
│ │ │ │  ********** MMeetthhooddss tthhaatt uussee aa ccoommmmaanndd:: **********
│ │ │ │      * code(Command) -- see _c_o_d_e -- display source code
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Database.html
│ │ │ @@ -52,22 +52,22 @@
│ │ │        
    
│ │ │          
    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/M2-11641-0/0.dbm
    
│ │ │ +o1 = /tmp/M2-13231-0/0.dbm
│ │ │              
    
│ │ │                
    i2 : x = openDatabaseOut filename
│ │ │  
│ │ │ -o2 = /tmp/M2-11641-0/0.dbm
│ │ │ +o2 = /tmp/M2-13231-0/0.dbm
│ │ │  
│ │ │  o2 : Database
    
│ │ │              
    
│ │ │                
    i3 : x#"first" = "hi there"
│ │ │ ├── html2text {}
│ │ │ │ @@ -7,18 +7,18 @@
│ │ │ │  ************ 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/M2-11641-0/0.dbm
│ │ │ │ +o1 = /tmp/M2-13231-0/0.dbm
│ │ │ │  i2 : x = openDatabaseOut filename
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11641-0/0.dbm
│ │ │ │ +o2 = /tmp/M2-13231-0/0.dbm
│ │ │ │  
│ │ │ │  o2 : Database
│ │ │ │  i3 : x#"first" = "hi there"
│ │ │ │  
│ │ │ │  o3 = hi there
│ │ │ │  i4 : x#"first"
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___G__Cstats.html
│ │ │ @@ -53,33 +53,33 @@
│ │ │          
    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" => 42968369946        }
│ │ │ +o1 = HashTable{"bytesAlloc" => 43082233722        }
│ │ │                 "GC_free_space_divisor" => 3
│ │ │                 "GC_LARGE_ALLOC_WARN_INTERVAL" => 1
│ │ │                 "gcCpuTimeSecs" => 0
│ │ │ -               "heapSize" => 206622720
│ │ │ -               "numGCs" => 795
│ │ │ -               "numGCThreads" => 6
│ │ │ +               "heapSize" => 225443840
│ │ │ +               "numGCs" => 783
│ │ │ +               "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 = 206622720
    
│ │ │ +o2 = 225443840
│ │ │              
    
│ │ │          
    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.
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -9,28 +9,28 @@
│ │ │ │  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" => 42968369946        }
│ │ │ │ +o1 = HashTable{"bytesAlloc" => 43082233722        }
│ │ │ │                 "GC_free_space_divisor" => 3
│ │ │ │                 "GC_LARGE_ALLOC_WARN_INTERVAL" => 1
│ │ │ │                 "gcCpuTimeSecs" => 0
│ │ │ │ -               "heapSize" => 206622720
│ │ │ │ -               "numGCs" => 795
│ │ │ │ -               "numGCThreads" => 6
│ │ │ │ +               "heapSize" => 225443840
│ │ │ │ +               "numGCs" => 783
│ │ │ │ +               "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 = 206622720
│ │ │ │ +o2 = 225443840
│ │ │ │  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
│ │ │ @@ -128,23 +128,23 @@
│ │ │  
│ │ │  o7 : Ideal of R
    
│ │ │              
    
│ │ │                
    i8 : time J = truncate(8, I, MinimalGenerators => false);
│ │ │ - -- used 0.00897861s (cpu); 0.00897229s (thread); 0s (gc)
│ │ │ + -- used 0.00967653s (cpu); 0.00967333s (thread); 0s (gc)
│ │ │  
│ │ │  o8 : Ideal of R
    
│ │ │              
    
│ │ │                
    i9 : time K = truncate(8, I, MinimalGenerators => true);
│ │ │ - -- used 0.0801371s (cpu); 0.080144s (thread); 0s (gc)
│ │ │ + -- used 0.0823194s (cpu); 0.082328s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : Ideal of R
    
│ │ │              
    
│ │ │                
    i10 : numgens J
│ │ │ ├── html2text {}
│ │ │ │ @@ -46,19 +46,19 @@
│ │ │ │  o6 = R
│ │ │ │  
│ │ │ │  o6 : PolynomialRing
│ │ │ │  i7 : I = monomialCurveIdeal(R, {1,4,5,9});
│ │ │ │  
│ │ │ │  o7 : Ideal of R
│ │ │ │  i8 : time J = truncate(8, I, MinimalGenerators => false);
│ │ │ │ - -- used 0.00897861s (cpu); 0.00897229s (thread); 0s (gc)
│ │ │ │ + -- used 0.00967653s (cpu); 0.00967333s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 : Ideal of R
│ │ │ │  i9 : time K = truncate(8, I, MinimalGenerators => true);
│ │ │ │ - -- used 0.0801371s (cpu); 0.080144s (thread); 0s (gc)
│ │ │ │ + -- used 0.0823194s (cpu); 0.082328s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 : Ideal of R
│ │ │ │  i10 : numgens J
│ │ │ │  
│ │ │ │  o10 = 1067
│ │ │ │  i11 : numgens K
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___S__V__D_lp..._cm__Divide__Conquer_eq_gt..._rp.html
│ │ │ @@ -68,21 +68,21 @@
│ │ │  o1 : Matrix RR      <-- RR
│ │ │                53          53
    
│ │ │              
    
│ │ │                
    i2 : time SVD(M);
│ │ │ - -- used 0.0241449s (cpu); 0.0241435s (thread); 0s (gc)
    
│ │ │ + -- used 0.0440342s (cpu); 0.0440332s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i3 : time SVD(M, DivideConquer=>true);
│ │ │ - -- used 0.0237108s (cpu); 0.0237156s (thread); 0s (gc)
    
│ │ │ + -- used 0.038972s (cpu); 0.0389862s (thread); 0s (gc)
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    
│ │ │            
    Functions with optional argument named DivideConquer:
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,17 +11,17 @@
│ │ │ │  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 0.0241449s (cpu); 0.0241435s (thread); 0s (gc)
│ │ │ │ + -- used 0.0440342s (cpu); 0.0440332s (thread); 0s (gc)
│ │ │ │  i3 : time SVD(M, DivideConquer=>true);
│ │ │ │ - -- used 0.0237108s (cpu); 0.0237156s (thread); 0s (gc)
│ │ │ │ + -- used 0.038972s (cpu); 0.0389862s (thread); 0s (gc)
│ │ │ │  ********** FFuunnccttiioonnss wwiitthh ooppttiioonnaall aarrgguummeenntt nnaammeedd DDiivviiddeeCCoonnqquueerr:: **********
│ │ │ │      * _S_V_D_(_._._._,_D_i_v_i_d_e_C_o_n_q_u_e_r_=_>_._._._) -- whether to use the LAPACK divide and
│ │ │ │        conquer SVD algorithm
│ │ │ │  ********** 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
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_a_spfirst_sp__Macaulay2_spsession.html
│ │ │ @@ -826,15 +826,15 @@
│ │ │          
    
│ │ │            
    We may use resolution to produce a projective resolution of it, and time to report the time required.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i59 : time C = resolution M
│ │ │ - -- used 0.00188708s (cpu); 0.00187997s (thread); 0s (gc)
│ │ │ + -- used 0.00219741s (cpu); 0.00219001s (thread); 0s (gc)
│ │ │  
│ │ │         3      6      15      18      6
│ │ │  o59 = R  <-- R  <-- R   <-- R   <-- R  <-- 0
│ │ │                                              
│ │ │        0      1      2       3       4      5
│ │ │  
│ │ │  o59 : ChainComplex
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -390,15 +390,15 @@
│ │ │ │                 | 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 0.00188708s (cpu); 0.00187997s (thread); 0s (gc)
│ │ │ │ + -- used 0.00219741s (cpu); 0.00219001s (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
│ │ │ @@ -68,15 +68,15 @@
│ │ │        
    
│ │ │          
    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 = .0002898248142239025
│ │ │ +o1 = .0003476298017475148
│ │ │  
│ │ │  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.      
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,15 +12,15 @@
│ │ │ │            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 = .0002898248142239025
│ │ │ │ +o1 = .0003476298017475148
│ │ │ │  
│ │ │ │  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/_caching_spcomputation_spresults.html
│ │ │ @@ -69,23 +69,23 @@
│ │ │                
    i3 : M = coker vars R;
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime pdim' M
│ │ │   -- computing pdim'
│ │ │ - -- .00646567s elapsed
│ │ │ + -- .00399088s elapsed
│ │ │  
│ │ │  o4 = 3
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime pdim' M
│ │ │ - -- .000001924s elapsed
│ │ │ + -- .000002475s elapsed
│ │ │  
│ │ │  o5 = 3
    
│ │ │              
    
│ │ │                
    i6 : peek M.cache
│ │ │ ├── html2text {}
│ │ │ │ @@ -8,19 +8,19 @@
│ │ │ │  Here is a simple example of caching a computation in a _C_a_c_h_e_T_a_b_l_e, using the
│ │ │ │  augmented null coalescing operator _?_?_=.
│ │ │ │  i1 : pdim' = M -> M.cache.pdim' ??= ( printerr "computing pdim'"; pdim M );
│ │ │ │  i2 : R = QQ[x,y,z];
│ │ │ │  i3 : M = coker vars R;
│ │ │ │  i4 : elapsedTime pdim' M
│ │ │ │   -- computing pdim'
│ │ │ │ - -- .00646567s elapsed
│ │ │ │ + -- .00399088s elapsed
│ │ │ │  
│ │ │ │  o4 = 3
│ │ │ │  i5 : elapsedTime pdim' M
│ │ │ │ - -- .000001924s elapsed
│ │ │ │ + -- .000002475s elapsed
│ │ │ │  
│ │ │ │  o5 = 3
│ │ │ │  i6 : peek M.cache
│ │ │ │  
│ │ │ │  o6 = CacheTable{cache => MutableHashTable{}
│ │ │ │  }
│ │ │ │                  isHomogeneous => true
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_cancel__Task_lp__Task_rp.html
│ │ │ @@ -104,15 +104,15 @@
│ │ │  o4 : Task
    
│ │ │              
    
│ │ │                
    i5 : n
│ │ │  
│ │ │ -o5 = 711247
    
│ │ │ +o5 = 1095402
│ │ │              
    
│ │ │                
    i6 : sleep 1
│ │ │  
│ │ │  o6 = 0
    
│ │ │ @@ -127,15 +127,15 @@
│ │ │  o7 : Task
│ │ │              
    
│ │ │                
    i8 : n
│ │ │  
│ │ │ -o8 = 1453370
    
│ │ │ +o8 = 2223655
│ │ │              
    
│ │ │                
    i9 : isReady t
│ │ │  
│ │ │  o9 = false
    
│ │ │ @@ -163,29 +163,29 @@
│ │ │  o12 : Task
│ │ │              
    
│ │ │                
    i13 : n
│ │ │  
│ │ │ -o13 = 1453595
    
│ │ │ +o13 = 2223866
│ │ │              
    
│ │ │                
    i14 : sleep 1
│ │ │  
│ │ │  o14 = 0
    
│ │ │              
    
│ │ │                
    i15 : n
│ │ │  
│ │ │ -o15 = 1453595
    
│ │ │ +o15 = 2223866
│ │ │              
    
│ │ │                
    i16 : isReady t
│ │ │  
│ │ │  o16 = false
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,26 +28,26 @@
│ │ │ │  i4 : t
│ │ │ │  
│ │ │ │  o4 = <>
│ │ │ │  
│ │ │ │  o4 : Task
│ │ │ │  i5 : n
│ │ │ │  
│ │ │ │ -o5 = 711247
│ │ │ │ +o5 = 1095402
│ │ │ │  i6 : sleep 1
│ │ │ │  
│ │ │ │  o6 = 0
│ │ │ │  i7 : t
│ │ │ │  
│ │ │ │  o7 = <>
│ │ │ │  
│ │ │ │  o7 : Task
│ │ │ │  i8 : n
│ │ │ │  
│ │ │ │ -o8 = 1453370
│ │ │ │ +o8 = 2223655
│ │ │ │  i9 : isReady t
│ │ │ │  
│ │ │ │  o9 = false
│ │ │ │  i10 : cancelTask t
│ │ │ │  i11 : sleep 2
│ │ │ │  stdio:2:25:(3):[1]: error: interrupted
│ │ │ │  
│ │ │ │ @@ -55,21 +55,21 @@
│ │ │ │  i12 : t
│ │ │ │  
│ │ │ │  o12 = <>
│ │ │ │  
│ │ │ │  o12 : Task
│ │ │ │  i13 : n
│ │ │ │  
│ │ │ │ -o13 = 1453595
│ │ │ │ +o13 = 2223866
│ │ │ │  i14 : sleep 1
│ │ │ │  
│ │ │ │  o14 = 0
│ │ │ │  i15 : n
│ │ │ │  
│ │ │ │ -o15 = 1453595
│ │ │ │ +o15 = 2223866
│ │ │ │  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
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_change__Directory.html
│ │ │ @@ -71,36 +71,36 @@
│ │ │            
    Change the current working directory to dir.
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10463-0/0
    
│ │ │ +o1 = /tmp/M2-10833-0/0
│ │ │              
    
│ │ │                
    i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-10463-0/0
    
│ │ │ +o2 = /tmp/M2-10833-0/0
│ │ │              
    
│ │ │                
    i3 : changeDirectory dir
│ │ │  
│ │ │ -o3 = /tmp/M2-10463-0/0/
    
│ │ │ +o3 = /tmp/M2-10833-0/0/
│ │ │              
    
│ │ │                
    i4 : currentDirectory()
│ │ │  
│ │ │ -o4 = /tmp/M2-10463-0/0/
    
│ │ │ +o4 = /tmp/M2-10833-0/0/
│ │ │              
    
│ │ │          
    
│ │ │            
    If dir is omitted, then the current working directory is changed to the user's home directory.
    
│ │ │          
    
│ │ │        
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,24 +11,24 @@
│ │ │ │            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/M2-10463-0/0
│ │ │ │ +o1 = /tmp/M2-10833-0/0
│ │ │ │  i2 : makeDirectory dir
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10463-0/0
│ │ │ │ +o2 = /tmp/M2-10833-0/0
│ │ │ │  i3 : changeDirectory dir
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-10463-0/0/
│ │ │ │ +o3 = /tmp/M2-10833-0/0/
│ │ │ │  i4 : currentDirectory()
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-10463-0/0/
│ │ │ │ +o4 = /tmp/M2-10833-0/0/
│ │ │ │  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/_check.html
│ │ │ @@ -95,40 +95,40 @@
│ │ │  o1 : Package
│ │ │              
    
│ │ │                
    i2 : check_1 FirstPackage
│ │ │   -- warning: reloading FirstPackage; recreate instances of types from this package
│ │ │ - -- capturing check(1, "FirstPackage")        -- .157225s elapsed
    
│ │ │ + -- capturing check(1, "FirstPackage")        -- .116799s elapsed
│ │ │              
    
│ │ │                
    i3 : check FirstPackage
│ │ │ - -- capturing check(0, "FirstPackage")        -- .177874s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .230699s elapsed
    
│ │ │ + -- capturing check(0, "FirstPackage")        -- .120579s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .120961s elapsed
│ │ │              
    
│ │ │          
    
│ │ │            
    Alternatively, if the package is installed somewhere accessible, one can do the following.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : check_1 "FirstPackage"
│ │ │ - -- capturing check(1, "FirstPackage")        -- .220889s elapsed
    
│ │ │ + -- capturing check(1, "FirstPackage")        -- .127722s elapsed
│ │ │              
    
│ │ │                
    i5 : check "FirstPackage"
│ │ │ - -- capturing check(0, "FirstPackage")        -- .21817s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .226677s elapsed
    
│ │ │ + -- capturing check(0, "FirstPackage")        -- .126603s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .118973s elapsed
│ │ │              
    
│ │ │          
    
│ │ │            
    A TestInput object (or a list of such objects) can also be run directly.
    
│ │ │          
    
│ │ │          
│ │ │ @@ -140,15 +140,15 @@
│ │ │  
│ │ │  o6 : TestInput
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : check oo
│ │ │ - -- capturing check(1, "FirstPackage")        -- .220218s elapsed
    
│ │ │ + -- capturing check(1, "FirstPackage")        -- .116001s elapsed
│ │ │              
    
│ │ │                
    i8 : tests "FirstPackage"
│ │ │  
│ │ │  o8 = {0 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:54:5-56:3]}
│ │ │ @@ -156,16 +156,16 @@
│ │ │  
│ │ │  o8 : NumberedVerticalList
    
│ │ │              
    
│ │ │                
    i9 : check oo
│ │ │ - -- capturing check(0, "FirstPackage")        -- .230323s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .223717s elapsed
    
│ │ │ + -- capturing check(0, "FirstPackage")        -- .123068s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .116914s elapsed
│ │ │              
    
│ │ │          
    
│ │ │            
    If only an integer is passed as an argument, then the test with that index from the last call to tests is run.
    
│ │ │          
    
│ │ │          
│ │ │ @@ -178,15 +178,15 @@
│ │ │  
│ │ │  o10 : NumberedVerticalList
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i11 : check 1
│ │ │ - -- capturing check(1, "FirstPackage")        -- .230812s elapsed
    
│ │ │ + -- capturing check(1, "FirstPackage")        -- .120197s elapsed
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    Caveat
    
│ │ │          
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -42,52 +42,52 @@
│ │ │ │  
│ │ │ │  o1 = FirstPackage
│ │ │ │  
│ │ │ │  o1 : Package
│ │ │ │  i2 : check_1 FirstPackage
│ │ │ │   -- warning: reloading FirstPackage; recreate instances of types from this
│ │ │ │  package
│ │ │ │ - -- capturing check(1, "FirstPackage")        -- .157225s elapsed
│ │ │ │ + -- capturing check(1, "FirstPackage")        -- .116799s elapsed
│ │ │ │  i3 : check FirstPackage
│ │ │ │ - -- capturing check(0, "FirstPackage")        -- .177874s elapsed
│ │ │ │ - -- capturing check(1, "FirstPackage")        -- .230699s elapsed
│ │ │ │ + -- capturing check(0, "FirstPackage")        -- .120579s elapsed
│ │ │ │ + -- capturing check(1, "FirstPackage")        -- .120961s elapsed
│ │ │ │  Alternatively, if the package is installed somewhere accessible, one can do the
│ │ │ │  following.
│ │ │ │  i4 : check_1 "FirstPackage"
│ │ │ │ - -- capturing check(1, "FirstPackage")        -- .220889s elapsed
│ │ │ │ + -- capturing check(1, "FirstPackage")        -- .127722s elapsed
│ │ │ │  i5 : check "FirstPackage"
│ │ │ │ - -- capturing check(0, "FirstPackage")        -- .21817s elapsed
│ │ │ │ - -- capturing check(1, "FirstPackage")        -- .226677s elapsed
│ │ │ │ + -- capturing check(0, "FirstPackage")        -- .126603s elapsed
│ │ │ │ + -- capturing check(1, "FirstPackage")        -- .118973s elapsed
│ │ │ │  A _T_e_s_t_I_n_p_u_t object (or a list of such objects) can also be run directly.
│ │ │ │  i6 : tests(1, "FirstPackage")
│ │ │ │  
│ │ │ │  o6 = TestInput[/usr/share/Macaulay2/FirstPackage.m2:58:5-60:3]
│ │ │ │  
│ │ │ │  o6 : TestInput
│ │ │ │  i7 : check oo
│ │ │ │ - -- capturing check(1, "FirstPackage")        -- .220218s elapsed
│ │ │ │ + -- capturing check(1, "FirstPackage")        -- .116001s elapsed
│ │ │ │  i8 : tests "FirstPackage"
│ │ │ │  
│ │ │ │  o8 = {0 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:54:5-56:3]}
│ │ │ │       {1 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:58:5-60:3]}
│ │ │ │  
│ │ │ │  o8 : NumberedVerticalList
│ │ │ │  i9 : check oo
│ │ │ │ - -- capturing check(0, "FirstPackage")        -- .230323s elapsed
│ │ │ │ - -- capturing check(1, "FirstPackage")        -- .223717s elapsed
│ │ │ │ + -- capturing check(0, "FirstPackage")        -- .123068s elapsed
│ │ │ │ + -- capturing check(1, "FirstPackage")        -- .116914s elapsed
│ │ │ │  If only an integer is passed as an argument, then the test with that index from
│ │ │ │  the last call to _t_e_s_t_s is run.
│ │ │ │  i10 : tests "FirstPackage"
│ │ │ │  
│ │ │ │  o10 = {0 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:54:5-56:3]}
│ │ │ │        {1 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:58:5-60:3]}
│ │ │ │  
│ │ │ │  o10 : NumberedVerticalList
│ │ │ │  i11 : check 1
│ │ │ │ - -- capturing check(1, "FirstPackage")        -- .230812s elapsed
│ │ │ │ + -- capturing check(1, "FirstPackage")        -- .120197s elapsed
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  Currently, if the package was only partially loaded because the documentation
│ │ │ │  was obtainable from a database (see _b_e_g_i_n_D_o_c_u_m_e_n_t_a_t_i_o_n), then the package will
│ │ │ │  be reloaded, this time completely, to ensure that all tests are considered;
│ │ │ │  this may affect user objects of types declared by the package, as they may be
│ │ │ │  not usable by the new instance of the package. In a future version, either the
│ │ │ │  tests and the documentation will both be cached, or neither will.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_communicating_spwith_spprograms.html
│ │ │ @@ -50,15 +50,15 @@
│ │ │      
    
│ │ │        
    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 6.12.48+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.48-1 (2025-09-20) x86_64 GNU/Linux
│ │ │ +Linux sbuild 6.12.57+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.57-1 (2025-11-05) 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.        
│ │ │            
│ │ │ @@ -74,16 +74,16 @@
│ │ │            
│ │ │          
    
│ │ │  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 6.12.48+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ -     6.12.48-1 (2025-09-20) x86_64 GNU/Linux\n"
    
│ │ │ +o3 = "Linux sbuild 6.12.57+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ +     6.12.57-1 (2025-11-05) x86_64 GNU/Linux\n"
│ │ │              
    
│ │ │  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 grep and use it to locate the symbol names in Macaulay2 that begin with in.        
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : f = openInOut "!grep -E '^in'"
│ │ │ ├── html2text {}
│ │ │ │ @@ -5,16 +5,16 @@
│ │ │ │  _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 6.12.48+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.48-1 (2025-
│ │ │ │ -09-20) x86_64 GNU/Linux
│ │ │ │ +Linux sbuild 6.12.57+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.57-1
│ │ │ │ +(2025-11-05) 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
│ │ │ │ @@ -26,16 +26,16 @@
│ │ │ │  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 6.12.48+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ │ -     6.12.48-1 (2025-09-20) x86_64 GNU/Linux\n"
│ │ │ │ +o3 = "Linux sbuild 6.12.57+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ │ +     6.12.57-1 (2025-11-05) x86_64 GNU/Linux\n"
│ │ │ │  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 grep and use it to locate the symbol names in Macaulay2 that
│ │ │ │  begin with in.
│ │ │ │  i4 : f = openInOut "!grep -E '^in'"
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_computing_sp__Groebner_spbases.html
│ │ │ @@ -269,15 +269,15 @@
│ │ │                 1277
    
│ │ │              
    
│ │ │                
    i24 : gb I
│ │ │  
│ │ │ -   -- registering gb 5 at 0x7f4e93a66540
│ │ │ +   -- registering gb 5 at 0x7f83eaa25540
│ │ │  
│ │ │     -- [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
│ │ │ @@ -373,15 +373,15 @@
│ │ │                1      4
│ │ │  o32 : Matrix R  <-- R
    
│ │ │              
    
│ │ │                
    i33 : time betti gb f
│ │ │ - -- used 0.27951s (cpu); 0.279785s (thread); 0s (gc)
│ │ │ + -- used 0.227838s (cpu); 0.225912s (thread); 0s (gc)
│ │ │  
│ │ │               0  1
│ │ │  o33 = total: 1 53
│ │ │            0: 1  .
│ │ │            1: .  .
│ │ │            2: .  2
│ │ │            3: .  1
│ │ │ @@ -417,15 +417,15 @@
│ │ │  
│ │ │  o35 : ZZ[T]
    
│ │ │              
    
│ │ │                
    i36 : time betti gb f
│ │ │ - -- used 0.00392072s (cpu); 0.00469829s (thread); 0s (gc)
│ │ │ + -- used 0.000148154s (cpu); 0.00367111s (thread); 0s (gc)
│ │ │  
│ │ │               0  1
│ │ │  o36 = total: 1 53
│ │ │            0: 1  .
│ │ │            1: .  .
│ │ │            2: .  2
│ │ │            3: .  1
│ │ │ ├── html2text {}
│ │ │ │ @@ -140,15 +140,15 @@
│ │ │ │  o23 = ideal (x*y - z , y  - w )
│ │ │ │  
│ │ │ │                  ZZ
│ │ │ │  o23 : Ideal of ----[x..z, w]
│ │ │ │                 1277
│ │ │ │  i24 : gb I
│ │ │ │  
│ │ │ │ -   -- registering gb 5 at 0x7f4e93a66540
│ │ │ │ +   -- registering gb 5 at 0x7f83eaa25540
│ │ │ │  
│ │ │ │     -- [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
│ │ │ │ @@ -213,15 +213,15 @@
│ │ │ │  
│ │ │ │  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 0.27951s (cpu); 0.279785s (thread); 0s (gc)
│ │ │ │ + -- used 0.227838s (cpu); 0.225912s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               0  1
│ │ │ │  o33 = total: 1 53
│ │ │ │            0: 1  .
│ │ │ │            1: .  .
│ │ │ │            2: .  2
│ │ │ │            3: .  1
│ │ │ │ @@ -245,15 +245,15 @@
│ │ │ │  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 0.00392072s (cpu); 0.00469829s (thread); 0s (gc)
│ │ │ │ + -- used 0.000148154s (cpu); 0.00367111s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               0  1
│ │ │ │  o36 = total: 1 53
│ │ │ │            0: 1  .
│ │ │ │            1: .  .
│ │ │ │            2: .  2
│ │ │ │            3: .  1
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_copy__Directory_lp__String_cm__String_rp.html
│ │ │ @@ -80,112 +80,112 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── 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/M2-53603-0/0-in.ms -o /tmp/
│ │ │ │ -M2-53603-0/0-out.ms
│ │ │ │ + -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-84646-0/0-in.ms -o /tmp/
│ │ │ │ +M2-84646-0/0-out.ms
│ │ │ │  
│ │ │ │  --------------- INPUT DATA ---------------
│ │ │ │  #variables                       3
│ │ │ │  #equations                       3
│ │ │ │  #invalid equations               0
│ │ │ │  field characteristic             0
│ │ │ │  homogeneous input?               1
│ │ │ │ @@ -51,15 +51,15 @@
│ │ │ │  initial hash table size     131072 (2^17)
│ │ │ │  max pair selection             ALL
│ │ │ │  reduce gb                        1
│ │ │ │  #threads                         6
│ │ │ │  info level                       2
│ │ │ │  generate pbm files               0
│ │ │ │  ------------------------------------------
│ │ │ │ -Initial prime = 1235341781
│ │ │ │ +Initial prime = 1166555107
│ │ │ │  
│ │ │ │  Legend for f4 information
│ │ │ │  --------------------------------------------------------
│ │ │ │  deg       current degree of pairs selected in this round
│ │ │ │  sel       number of pairs selected in this round
│ │ │ │  pairs     total number of pairs in pair list
│ │ │ │  mat       matrix dimensions (# rows x # columns)
│ │ │ │ @@ -73,26 +73,26 @@
│ │ │ │  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
│ │ │ │ -0.06 | 0.11
│ │ │ │ +0.00 | 0.00
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----------------------
│ │ │ │  
│ │ │ │  ---------------- TIMINGS ----------------
│ │ │ │ -overall(elapsed)        0.11 sec
│ │ │ │ -overall(cpu)            0.22 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                  0.05 sec  43.2%
│ │ │ │ -convert                 0.06 sec  56.7%
│ │ │ │ -linear algebra          0.00 sec   0.0%
│ │ │ │ +symbolic prep.          0.00 sec   0.5%
│ │ │ │ +update                  0.00 sec  72.8%
│ │ │ │ +convert                 0.00 sec   3.2%
│ │ │ │ +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
│ │ │ │ @@ -106,18 +106,18 @@
│ │ │ │  -----------------------------------------
│ │ │ │  
│ │ │ │  
│ │ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │ │  [3]
│ │ │ │  #polynomials to lift              3
│ │ │ │  -----------------------------------------
│ │ │ │ -New prime = 1138309223
│ │ │ │ +New prime = 1152244237
│ │ │ │  
│ │ │ │  ---------------- TIMINGS ----------------
│ │ │ │ -multi-mod overall(elapsed)      0.03 sec
│ │ │ │ +multi-mod overall(elapsed)      0.00 sec
│ │ │ │  learning phase                  0.00 Gops/sec
│ │ │ │  application phase               0.00 Gops/sec
│ │ │ │  -----------------------------------------
│ │ │ │  
│ │ │ │  multi-modular steps
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----------------------
│ │ │ │ @@ -136,15 +136,15 @@
│ │ │ │  CRT     (elapsed)               0.00 sec
│ │ │ │  ratrecon(elapsed)               0.00 sec
│ │ │ │  -----------------------------------------
│ │ │ │  
│ │ │ │  
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----
│ │ │ │ -msolve overall time           0.26 sec (elapsed) /  0.60 sec (cpu)
│ │ │ │ +msolve overall time           0.01 sec (elapsed) /  0.04 sec (cpu)
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----
│ │ │ │  
│ │ │ │  o3 = | z y x |
│ │ │ │  
│ │ │ │               1      3
│ │ │ │  o3 : Matrix R  <-- R
│ │ ├── ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_components__Of__Kernel.out
│ │ │ @@ -23,19 +23,19 @@
│ │ │  o4 : RingMap S <-- R
│ │ │  
│ │ │  i5 : peek componentsOfKernel(2, F)
│ │ │  warning: computation begun over finite field. resulting polynomials may not lie in the ideal
│ │ │  computing total degree: 1
│ │ │  number of monomials = 6
│ │ │  number of distinct multidegrees = 6
│ │ │ - -- .00369113s elapsed
│ │ │ + -- .00249226s elapsed
│ │ │  computing total degree: 2
│ │ │  number of monomials = 21
│ │ │  number of distinct multidegrees = 18
│ │ │ - -- .0155544s elapsed
│ │ │ + -- .0106654s elapsed
│ │ │  
│ │ │  o5 = MutableHashTable{{0, 1, 0, 0, 1} => {}                   }
│ │ │                        {0, 1, 0, 1, 0} => {}
│ │ │                        {0, 1, 1, 0, 0} => {}
│ │ │                        {0, 2, 0, 0, 2} => {}
│ │ │                        {0, 2, 0, 1, 1} => {}
│ │ │                        {0, 2, 0, 2, 0} => {}
│ │ ├── ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_components__Of__Kernel.html
│ │ │ @@ -117,19 +117,19 @@
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,23 +20,23 @@
│ │ │ │  o1 : PolynomialRing
│ │ │ │  i2 : I = ideal"xy,yz,zx"
│ │ │ │  
│ │ │ │  o2 = ideal (x*y, y*z, x*z)
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : elapsedTime jMult I
│ │ │ │ - -- .0238525s elapsed
│ │ │ │ + -- .028425s elapsed
│ │ │ │  
│ │ │ │  o3 = 2
│ │ │ │  i4 : elapsedTime monjMult I
│ │ │ │ - -- .114646s elapsed
│ │ │ │ + -- .0934295s elapsed
│ │ │ │  
│ │ │ │  o4 = 2
│ │ │ │  i5 : elapsedTime multiplicitySequence I
│ │ │ │ - -- .137039s elapsed
│ │ │ │ + -- .152938s 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
│ │ │ @@ -89,15 +89,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : src = temporaryFileName() | "/"
│ │ │  
│ │ │ -o1 = /tmp/M2-11185-0/0/
    
│ │ │ +o1 = /tmp/M2-12295-0/0/
│ │ │              
    
│ │ │                
    i2 : dst = temporaryFileName() | "/"
│ │ │  
│ │ │ -o2 = /tmp/M2-11185-0/1/
    
│ │ │ +o2 = /tmp/M2-12295-0/1/
│ │ │              
    
│ │ │                
    i3 : makeDirectory (src|"a/")
│ │ │  
│ │ │ -o3 = /tmp/M2-11185-0/0/a/
    
│ │ │ +o3 = /tmp/M2-12295-0/0/a/
│ │ │              
    
│ │ │                
    i4 : makeDirectory (src|"b/")
│ │ │  
│ │ │ -o4 = /tmp/M2-11185-0/0/b/
    
│ │ │ +o4 = /tmp/M2-12295-0/0/b/
│ │ │              
    
│ │ │                
    i5 : makeDirectory (src|"b/c/")
│ │ │  
│ │ │ -o5 = /tmp/M2-11185-0/0/b/c/
    
│ │ │ +o5 = /tmp/M2-12295-0/0/b/c/
│ │ │              
    
│ │ │                
    i6 : src|"a/f" << "hi there" << close
│ │ │  
│ │ │ -o6 = /tmp/M2-11185-0/0/a/f
│ │ │ +o6 = /tmp/M2-12295-0/0/a/f
│ │ │  
│ │ │  o6 : File
    
│ │ │              
    
│ │ │                
    i7 : src|"a/g" << "hi there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11185-0/0/a/g
│ │ │ +o7 = /tmp/M2-12295-0/0/a/g
│ │ │  
│ │ │  o7 : File
    
│ │ │              
    
│ │ │                
    i8 : src|"b/c/g" << "ho there" << close
│ │ │  
│ │ │ -o8 = /tmp/M2-11185-0/0/b/c/g
│ │ │ +o8 = /tmp/M2-12295-0/0/b/c/g
│ │ │  
│ │ │  o8 : File
    
│ │ │              
    
│ │ │                
    i9 : stack findFiles src
│ │ │  
│ │ │ -o9 = /tmp/M2-11185-0/0/
│ │ │ -     /tmp/M2-11185-0/0/b/
│ │ │ -     /tmp/M2-11185-0/0/b/c/
│ │ │ -     /tmp/M2-11185-0/0/b/c/g
│ │ │ -     /tmp/M2-11185-0/0/a/
│ │ │ -     /tmp/M2-11185-0/0/a/g
│ │ │ -     /tmp/M2-11185-0/0/a/f
    
│ │ │ +o9 = /tmp/M2-12295-0/0/
│ │ │ +     /tmp/M2-12295-0/0/a/
│ │ │ +     /tmp/M2-12295-0/0/a/g
│ │ │ +     /tmp/M2-12295-0/0/a/f
│ │ │ +     /tmp/M2-12295-0/0/b/
│ │ │ +     /tmp/M2-12295-0/0/b/c/
│ │ │ +     /tmp/M2-12295-0/0/b/c/g
│ │ │              
    
│ │ │                
    i10 : copyDirectory(src,dst,Verbose=>true)
│ │ │ - -- copying: /tmp/M2-11185-0/0/b/c/g -> /tmp/M2-11185-0/1/b/c/g
│ │ │ - -- copying: /tmp/M2-11185-0/0/a/g -> /tmp/M2-11185-0/1/a/g
│ │ │ - -- copying: /tmp/M2-11185-0/0/a/f -> /tmp/M2-11185-0/1/a/f
    
│ │ │ + -- copying: /tmp/M2-12295-0/0/a/g -> /tmp/M2-12295-0/1/a/g
│ │ │ + -- copying: /tmp/M2-12295-0/0/a/f -> /tmp/M2-12295-0/1/a/f
│ │ │ + -- copying: /tmp/M2-12295-0/0/b/c/g -> /tmp/M2-12295-0/1/b/c/g
│ │ │              
    
│ │ │                
    i11 : copyDirectory(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-11185-0/0/b/c/g not newer than /tmp/M2-11185-0/1/b/c/g
│ │ │ - -- skipping: /tmp/M2-11185-0/0/a/g not newer than /tmp/M2-11185-0/1/a/g
│ │ │ - -- skipping: /tmp/M2-11185-0/0/a/f not newer than /tmp/M2-11185-0/1/a/f
    
│ │ │ + -- skipping: /tmp/M2-12295-0/0/a/g not newer than /tmp/M2-12295-0/1/a/g
│ │ │ + -- skipping: /tmp/M2-12295-0/0/a/f not newer than /tmp/M2-12295-0/1/a/f
│ │ │ + -- skipping: /tmp/M2-12295-0/0/b/c/g not newer than /tmp/M2-12295-0/1/b/c/g
│ │ │              
    
│ │ │                
    i12 : stack findFiles dst
│ │ │  
│ │ │ -o12 = /tmp/M2-11185-0/1/
│ │ │ -      /tmp/M2-11185-0/1/a/
│ │ │ -      /tmp/M2-11185-0/1/a/f
│ │ │ -      /tmp/M2-11185-0/1/a/g
│ │ │ -      /tmp/M2-11185-0/1/b/
│ │ │ -      /tmp/M2-11185-0/1/b/c/
│ │ │ -      /tmp/M2-11185-0/1/b/c/g
    
│ │ │ +o12 = /tmp/M2-12295-0/1/
│ │ │ +      /tmp/M2-12295-0/1/a/
│ │ │ +      /tmp/M2-12295-0/1/a/g
│ │ │ +      /tmp/M2-12295-0/1/a/f
│ │ │ +      /tmp/M2-12295-0/1/b/
│ │ │ +      /tmp/M2-12295-0/1/b/c/
│ │ │ +      /tmp/M2-12295-0/1/b/c/g
│ │ │              
    
│ │ │                
    i13 : get (dst|"b/c/g")
│ │ │  
│ │ │  o13 = ho there
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,68 +25,68 @@
│ │ │ │              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/M2-11185-0/0/
│ │ │ │ +o1 = /tmp/M2-12295-0/0/
│ │ │ │  i2 : dst = temporaryFileName() | "/"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11185-0/1/
│ │ │ │ +o2 = /tmp/M2-12295-0/1/
│ │ │ │  i3 : makeDirectory (src|"a/")
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11185-0/0/a/
│ │ │ │ +o3 = /tmp/M2-12295-0/0/a/
│ │ │ │  i4 : makeDirectory (src|"b/")
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-11185-0/0/b/
│ │ │ │ +o4 = /tmp/M2-12295-0/0/b/
│ │ │ │  i5 : makeDirectory (src|"b/c/")
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-11185-0/0/b/c/
│ │ │ │ +o5 = /tmp/M2-12295-0/0/b/c/
│ │ │ │  i6 : src|"a/f" << "hi there" << close
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-11185-0/0/a/f
│ │ │ │ +o6 = /tmp/M2-12295-0/0/a/f
│ │ │ │  
│ │ │ │  o6 : File
│ │ │ │  i7 : src|"a/g" << "hi there" << close
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-11185-0/0/a/g
│ │ │ │ +o7 = /tmp/M2-12295-0/0/a/g
│ │ │ │  
│ │ │ │  o7 : File
│ │ │ │  i8 : src|"b/c/g" << "ho there" << close
│ │ │ │  
│ │ │ │ -o8 = /tmp/M2-11185-0/0/b/c/g
│ │ │ │ +o8 = /tmp/M2-12295-0/0/b/c/g
│ │ │ │  
│ │ │ │  o8 : File
│ │ │ │  i9 : stack findFiles src
│ │ │ │  
│ │ │ │ -o9 = /tmp/M2-11185-0/0/
│ │ │ │ -     /tmp/M2-11185-0/0/b/
│ │ │ │ -     /tmp/M2-11185-0/0/b/c/
│ │ │ │ -     /tmp/M2-11185-0/0/b/c/g
│ │ │ │ -     /tmp/M2-11185-0/0/a/
│ │ │ │ -     /tmp/M2-11185-0/0/a/g
│ │ │ │ -     /tmp/M2-11185-0/0/a/f
│ │ │ │ +o9 = /tmp/M2-12295-0/0/
│ │ │ │ +     /tmp/M2-12295-0/0/a/
│ │ │ │ +     /tmp/M2-12295-0/0/a/g
│ │ │ │ +     /tmp/M2-12295-0/0/a/f
│ │ │ │ +     /tmp/M2-12295-0/0/b/
│ │ │ │ +     /tmp/M2-12295-0/0/b/c/
│ │ │ │ +     /tmp/M2-12295-0/0/b/c/g
│ │ │ │  i10 : copyDirectory(src,dst,Verbose=>true)
│ │ │ │ - -- copying: /tmp/M2-11185-0/0/b/c/g -> /tmp/M2-11185-0/1/b/c/g
│ │ │ │ - -- copying: /tmp/M2-11185-0/0/a/g -> /tmp/M2-11185-0/1/a/g
│ │ │ │ - -- copying: /tmp/M2-11185-0/0/a/f -> /tmp/M2-11185-0/1/a/f
│ │ │ │ + -- copying: /tmp/M2-12295-0/0/a/g -> /tmp/M2-12295-0/1/a/g
│ │ │ │ + -- copying: /tmp/M2-12295-0/0/a/f -> /tmp/M2-12295-0/1/a/f
│ │ │ │ + -- copying: /tmp/M2-12295-0/0/b/c/g -> /tmp/M2-12295-0/1/b/c/g
│ │ │ │  i11 : copyDirectory(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ │ - -- skipping: /tmp/M2-11185-0/0/b/c/g not newer than /tmp/M2-11185-0/1/b/c/g
│ │ │ │ - -- skipping: /tmp/M2-11185-0/0/a/g not newer than /tmp/M2-11185-0/1/a/g
│ │ │ │ - -- skipping: /tmp/M2-11185-0/0/a/f not newer than /tmp/M2-11185-0/1/a/f
│ │ │ │ + -- skipping: /tmp/M2-12295-0/0/a/g not newer than /tmp/M2-12295-0/1/a/g
│ │ │ │ + -- skipping: /tmp/M2-12295-0/0/a/f not newer than /tmp/M2-12295-0/1/a/f
│ │ │ │ + -- skipping: /tmp/M2-12295-0/0/b/c/g not newer than /tmp/M2-12295-0/1/b/c/g
│ │ │ │  i12 : stack findFiles dst
│ │ │ │  
│ │ │ │ -o12 = /tmp/M2-11185-0/1/
│ │ │ │ -      /tmp/M2-11185-0/1/a/
│ │ │ │ -      /tmp/M2-11185-0/1/a/f
│ │ │ │ -      /tmp/M2-11185-0/1/a/g
│ │ │ │ -      /tmp/M2-11185-0/1/b/
│ │ │ │ -      /tmp/M2-11185-0/1/b/c/
│ │ │ │ -      /tmp/M2-11185-0/1/b/c/g
│ │ │ │ +o12 = /tmp/M2-12295-0/1/
│ │ │ │ +      /tmp/M2-12295-0/1/a/
│ │ │ │ +      /tmp/M2-12295-0/1/a/g
│ │ │ │ +      /tmp/M2-12295-0/1/a/f
│ │ │ │ +      /tmp/M2-12295-0/1/b/
│ │ │ │ +      /tmp/M2-12295-0/1/b/c/
│ │ │ │ +      /tmp/M2-12295-0/1/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
│ │ │ @@ -78,65 +78,65 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : src = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10970-0/0
    
│ │ │ +o1 = /tmp/M2-11860-0/0
│ │ │              
    
│ │ │                
    i2 : dst = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-10970-0/1
    
│ │ │ +o2 = /tmp/M2-11860-0/1
│ │ │              
    
│ │ │                
    i3 : src << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-10970-0/0
│ │ │ +o3 = /tmp/M2-11860-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : copyFile(src,dst,Verbose=>true)
│ │ │ - -- copying: /tmp/M2-10970-0/0 -> /tmp/M2-10970-0/1
    
│ │ │ + -- copying: /tmp/M2-11860-0/0 -> /tmp/M2-11860-0/1
│ │ │              
    
│ │ │                
    i5 : get dst
│ │ │  
│ │ │  o5 = hi there
    
│ │ │              
    
│ │ │                
    i6 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-10970-0/0 not newer than /tmp/M2-10970-0/1
    
│ │ │ + -- skipping: /tmp/M2-11860-0/0 not newer than /tmp/M2-11860-0/1
│ │ │              
    
│ │ │                
    i7 : src << "ho there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-10970-0/0
│ │ │ +o7 = /tmp/M2-11860-0/0
│ │ │  
│ │ │  o7 : File
    
│ │ │              
    
│ │ │                
    i8 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-10970-0/0 not newer than /tmp/M2-10970-0/1
    
│ │ │ + -- skipping: /tmp/M2-11860-0/0 not newer than /tmp/M2-11860-0/1
│ │ │              
    
│ │ │                
    i9 : get dst
│ │ │  
│ │ │  o9 = hi there
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,37 +18,37 @@
│ │ │ │            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/M2-10970-0/0
│ │ │ │ +o1 = /tmp/M2-11860-0/0
│ │ │ │  i2 : dst = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10970-0/1
│ │ │ │ +o2 = /tmp/M2-11860-0/1
│ │ │ │  i3 : src << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-10970-0/0
│ │ │ │ +o3 = /tmp/M2-11860-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : copyFile(src,dst,Verbose=>true)
│ │ │ │ - -- copying: /tmp/M2-10970-0/0 -> /tmp/M2-10970-0/1
│ │ │ │ + -- copying: /tmp/M2-11860-0/0 -> /tmp/M2-11860-0/1
│ │ │ │  i5 : get dst
│ │ │ │  
│ │ │ │  o5 = hi there
│ │ │ │  i6 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ │ - -- skipping: /tmp/M2-10970-0/0 not newer than /tmp/M2-10970-0/1
│ │ │ │ + -- skipping: /tmp/M2-11860-0/0 not newer than /tmp/M2-11860-0/1
│ │ │ │  i7 : src << "ho there" << close
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-10970-0/0
│ │ │ │ +o7 = /tmp/M2-11860-0/0
│ │ │ │  
│ │ │ │  o7 : File
│ │ │ │  i8 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ │ - -- skipping: /tmp/M2-10970-0/0 not newer than /tmp/M2-10970-0/1
│ │ │ │ + -- skipping: /tmp/M2-11860-0/0 not newer than /tmp/M2-11860-0/1
│ │ │ │  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
│ │ │ @@ -64,38 +64,38 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : t1 = cpuTime()
│ │ │  
│ │ │ -o1 = 363.561188353
│ │ │ +o1 = 349.628677912
│ │ │  
│ │ │  o1 : RR (of precision 53)
    
│ │ │              
    
│ │ │                
    i2 : for i from 0 to 1000000 do 223131321321*324234324324;
    
│ │ │              
    
│ │ │                
    i3 : t2 = cpuTime()
│ │ │  
│ │ │ -o3 = 365.292753193
│ │ │ +o3 = 350.644170628
│ │ │  
│ │ │  o3 : RR (of precision 53)
    
│ │ │              
    
│ │ │                
    i4 : t2-t1
│ │ │  
│ │ │ -o4 = 1.731564839999976
│ │ │ +o4 = 1.01549271600004
│ │ │  
│ │ │  o4 : RR (of precision 53)
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── 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 = 363.561188353
│ │ │ │ +o1 = 349.628677912
│ │ │ │  
│ │ │ │  o1 : RR (of precision 53)
│ │ │ │  i2 : for i from 0 to 1000000 do 223131321321*324234324324;
│ │ │ │  i3 : t2 = cpuTime()
│ │ │ │  
│ │ │ │ -o3 = 365.292753193
│ │ │ │ +o3 = 350.644170628
│ │ │ │  
│ │ │ │  o3 : RR (of precision 53)
│ │ │ │  i4 : t2-t1
│ │ │ │  
│ │ │ │ -o4 = 1.731564839999976
│ │ │ │ +o4 = 1.01549271600004
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -64,48 +64,48 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : currentTime()
│ │ │  
│ │ │ -o1 = 1763141264
    
│ │ │ +o1 = 1766073617
│ │ │              
    
│ │ │          
    We can compute, roughly, how many years ago the epoch began as follows.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i2 : currentTime() /( (365 + 97./400) * 24 * 60 * 60 )
│ │ │  
│ │ │ -o2 = 55.87172246546498
│ │ │ +o2 = 55.96464503289165
│ │ │  
│ │ │  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 = 10.46066958557981
│ │ │ +o3 = 11.57574039469984
│ │ │  
│ │ │  o3 : RR (of precision 53)
    
│ │ │              
    
│ │ │          
    Compare that to the current date, available from a standard Unix command.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : run "date"
│ │ │ -Fri Nov 14 17:27:44 UTC 2025
│ │ │ +Thu Dec 18 16:00:17 UTC 2025
│ │ │  
│ │ │  o4 = 0
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -9,31 +9,31 @@
│ │ │ │              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 = 1763141264
│ │ │ │ +o1 = 1766073617
│ │ │ │  We can compute, roughly, how many years ago the epoch began as follows.
│ │ │ │  i2 : currentTime() /( (365 + 97./400) * 24 * 60 * 60 )
│ │ │ │  
│ │ │ │ -o2 = 55.87172246546498
│ │ │ │ +o2 = 55.96464503289165
│ │ │ │  
│ │ │ │  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 = 10.46066958557981
│ │ │ │ +o3 = 11.57574039469984
│ │ │ │  
│ │ │ │  o3 : RR (of precision 53)
│ │ │ │  Compare that to the current date, available from a standard Unix command.
│ │ │ │  i4 : run "date"
│ │ │ │ -Fri Nov 14 17:27:44 UTC 2025
│ │ │ │ +Thu Dec 18 16:00:17 UTC 2025
│ │ │ │  
│ │ │ │  o4 = 0
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _c_u_r_r_e_n_t_T_i_m_e is a _c_o_m_p_i_l_e_d_ _f_u_n_c_t_i_o_n.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_system.m2:1849:0.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_elapsed__Time.html
│ │ │ @@ -59,15 +59,15 @@
│ │ │        
│ │ │        
    
│ │ │          
    Description
    
│ │ │  elapsedTime e evaluates e, prints the amount of time elapsed, and returns the value of e.        
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : elapsedTime sleep 1
│ │ │ - -- 1.00014s elapsed
│ │ │ + -- 1.00011s elapsed
│ │ │  
│ │ │  o1 = 0
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -7,15 +7,15 @@
│ │ │ │  ************ eellaappsseeddTTiimmee ---- ttiimmee aa ccoommppuuttaattiioonn iinncclluuddiinngg ttiimmee eellaappsseedd ************
│ │ │ │      *   Usage:
│ │ │ │              elapsedTime e
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  elapsedTime e evaluates e, prints the amount of time elapsed, and returns the
│ │ │ │  value of e.
│ │ │ │  i1 : elapsedTime sleep 1
│ │ │ │ - -- 1.00014s elapsed
│ │ │ │ + -- 1.00011s elapsed
│ │ │ │  
│ │ │ │  o1 = 0
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _e_l_a_p_s_e_d_T_i_m_i_n_g -- time a computation using time elapsed
│ │ │ │      * _c_p_u_T_i_m_e -- seconds of cpu time used since Macaulay2 began
│ │ │ │      * _G_C_s_t_a_t_s -- information about the status of the garbage collector
│ │ │ │      * _p_a_r_a_l_l_e_l_ _p_r_o_g_r_a_m_m_i_n_g_ _w_i_t_h_ _t_h_r_e_a_d_s_ _a_n_d_ _t_a_s_k_s
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_elapsed__Timing.html
│ │ │ @@ -54,24 +54,24 @@
│ │ │  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
│ │ │ -     -- 1.00015 seconds
│ │ │ +     -- 1.00013 seconds
│ │ │  
│ │ │  o1 : Time
    
│ │ │              
    
│ │ │                
    i2 : peek oo
│ │ │  
│ │ │ -o2 = Time{1.00015, 0}
    
│ │ │ +o2 = Time{1.00013, 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
│ │ │ │ -     -- 1.00015 seconds
│ │ │ │ +     -- 1.00013 seconds
│ │ │ │  
│ │ │ │  o1 : Time
│ │ │ │  i2 : peek oo
│ │ │ │  
│ │ │ │ -o2 = Time{1.00015, 0}
│ │ │ │ +o2 = Time{1.00013, 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
│ │ │ @@ -65,15 +65,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │              
    
│ │ │                
    i3 : time leadTerm gens gb I
│ │ │ - -- used 0.474296s (cpu); 0.280859s (thread); 0s (gc)
│ │ │ + -- used 0.133595s (cpu); 0.133594s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = | x3y9 5148txy3 108729sxy2z2 sy4z 46644741sxy3z 143sy5 6sxy4
│ │ │       ------------------------------------------------------------------------
│ │ │       563515116021sx2y3 4374txy2z3 612704350498473090tx2yz3 217458ty4z2
│ │ │       ------------------------------------------------------------------------
│ │ │       267076255345488270sy3z4 5256861933965245618410txyz6
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -162,15 +162,15 @@
│ │ │  
│ │ │  o7 : Ideal of R
    
│ │ │              
    
│ │ │                
    i8 : time G = eliminate(I,{s,t})
│ │ │ - -- used 0.497187s (cpu); 0.266913s (thread); 0s (gc)
│ │ │ + -- used 0.370662s (cpu); 0.186441s (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
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -245,15 +245,15 @@
│ │ │  
│ │ │  o11 : Ideal of R1
    
│ │ │              
    
│ │ │                
    i12 : time G = eliminate(I1,{s,t})
│ │ │ - -- used 0.340207s (cpu); 0.126135s (thread); 0s (gc)
│ │ │ + -- used 0.0321744s (cpu); 0.0321782s (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  -
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -337,15 +337,15 @@
│ │ │  
│ │ │  o16 : RingMap A <-- B
    
│ │ │              
    
│ │ │                
    i17 : time G = kernel F
│ │ │ - -- used 0.445785s (cpu); 0.239366s (thread); 0s (gc)
│ │ │ + -- used 0.104043s (cpu); 0.104048s (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
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -418,26 +418,26 @@
│ │ │  
│ │ │  o19 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i20 : time f1 = resultant(I_0,I_2,s)
│ │ │ - -- used 0.0020489s (cpu); 0.00204903s (thread); 0s (gc)
│ │ │ + -- used 0.00173352s (cpu); 0.00173146s (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 0.056264s (cpu); 0.0562726s (thread); 0s (gc)
│ │ │ + -- used 0.0388167s (cpu); 0.0388294s (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 0.474296s (cpu); 0.280859s (thread); 0s (gc)
│ │ │ │ + -- used 0.133595s (cpu); 0.133594s (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 0.497187s (cpu); 0.266913s (thread); 0s (gc)
│ │ │ │ + -- used 0.370662s (cpu); 0.186441s (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 0.340207s (cpu); 0.126135s (thread); 0s (gc)
│ │ │ │ + -- used 0.0321744s (cpu); 0.0321782s (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 0.445785s (cpu); 0.239366s (thread); 0s (gc)
│ │ │ │ + -- used 0.104043s (cpu); 0.104048s (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 0.0020489s (cpu); 0.00204903s (thread); 0s (gc)
│ │ │ │ + -- used 0.00173352s (cpu); 0.00173146s (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 0.056264s (cpu); 0.0562726s (thread); 0s (gc)
│ │ │ │ + -- used 0.0388167s (cpu); 0.0388294s (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
│ │ │ @@ -154,15 +154,15 @@
│ │ │                                      Version => 0.0
│ │ │               package prefix => /usr/
│ │ │               PackageIsLoaded => true
│ │ │               pkgname => Foo
│ │ │               private dictionary => Foo#"private dictionary"
│ │ │               processed documentation => MutableHashTable{}
│ │ │               raw documentation => MutableHashTable{}
│ │ │ -             source directory => /tmp/M2-10191-0/91-rundir/
│ │ │ +             source directory => /tmp/M2-10311-0/91-rundir/
│ │ │               source file => stdio
│ │ │               test inputs => MutableList{}
    
│ │ │              
    
│ │ │                
    i7 : dictionaryPath
│ │ │ ├── html2text {}
│ │ │ │ @@ -77,15 +77,15 @@
│ │ │ │                                      Version => 0.0
│ │ │ │               package prefix => /usr/
│ │ │ │               PackageIsLoaded => true
│ │ │ │               pkgname => Foo
│ │ │ │               private dictionary => Foo#"private dictionary"
│ │ │ │               processed documentation => MutableHashTable{}
│ │ │ │               raw documentation => MutableHashTable{}
│ │ │ │ -             source directory => /tmp/M2-10191-0/91-rundir/
│ │ │ │ +             source directory => /tmp/M2-10311-0/91-rundir/
│ │ │ │               source file => stdio
│ │ │ │               test inputs => MutableList{}
│ │ │ │  i7 : dictionaryPath
│ │ │ │  
│ │ │ │  o7 = {Foo.Dictionary, Varieties.Dictionary, Isomorphism.Dictionary,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       Truncations.Dictionary, Polyhedra.Dictionary, Saturation.Dictionary,
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Exists.html
│ │ │ @@ -68,29 +68,29 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10558-0/0
    
│ │ │ +o1 = /tmp/M2-11028-0/0
│ │ │              
    
│ │ │                
    i2 : fileExists fn
│ │ │  
│ │ │  o2 = false
    
│ │ │              
    
│ │ │                
    i3 : fn << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-10558-0/0
│ │ │ +o3 = /tmp/M2-11028-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : fileExists fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,21 +10,21 @@
│ │ │ │      * 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/M2-10558-0/0
│ │ │ │ +o1 = /tmp/M2-11028-0/0
│ │ │ │  i2 : fileExists fn
│ │ │ │  
│ │ │ │  o2 = false
│ │ │ │  i3 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-10558-0/0
│ │ │ │ +o3 = /tmp/M2-11028-0/0
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -69,15 +69,15 @@
│ │ │          
    Description
    
│ │ │          
    The length of an open output file is determined from the internal count of the number of bytes written so far.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -510,27 +510,27 @@
│ │ │  o36 = 3
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : f = temporaryFileName() << "hi there"
│ │ │  
│ │ │ -o1 = /tmp/M2-12150-0/0
│ │ │ +o1 = /tmp/M2-14270-0/0
│ │ │  
│ │ │  o1 : File
    
│ │ │              
    
│ │ │                
    i2 : fileLength f
│ │ │ @@ -85,24 +85,24 @@
│ │ │  o2 = 8
    
│ │ │              
    
│ │ │                
    i3 : close f
│ │ │  
│ │ │ -o3 = /tmp/M2-12150-0/0
│ │ │ +o3 = /tmp/M2-14270-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : filename = toString f
│ │ │  
│ │ │ -o4 = /tmp/M2-12150-0/0
    
│ │ │ +o4 = /tmp/M2-14270-0/0
│ │ │              
    
│ │ │                
    i5 : fileLength filename
│ │ │  
│ │ │  o5 = 8
    
│ │ │ ├── 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/M2-12150-0/0
│ │ │ │ +o1 = /tmp/M2-14270-0/0
│ │ │ │  
│ │ │ │  o1 : File
│ │ │ │  i2 : fileLength f
│ │ │ │  
│ │ │ │  o2 = 8
│ │ │ │  i3 : close f
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-12150-0/0
│ │ │ │ +o3 = /tmp/M2-14270-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : filename = toString f
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-12150-0/0
│ │ │ │ +o4 = /tmp/M2-14270-0/0
│ │ │ │  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
│ │ │ @@ -69,22 +69,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11375-0/0
    
│ │ │ +o1 = /tmp/M2-12685-0/0
│ │ │              
    
│ │ │                
    i2 : f = fn << "hi there"
│ │ │  
│ │ │ -o2 = /tmp/M2-11375-0/0
│ │ │ +o2 = /tmp/M2-12685-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │                
    i3 : fileMode f
│ │ │ @@ -92,15 +92,15 @@
│ │ │  o3 = 420
    
│ │ │              
    
│ │ │                
    i4 : close f
│ │ │  
│ │ │ -o4 = /tmp/M2-11375-0/0
│ │ │ +o4 = /tmp/M2-12685-0/0
│ │ │  
│ │ │  o4 : File
    
│ │ │              
    
│ │ │                
    i5 : removeFile fn
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,26 +11,26 @@
│ │ │ │      * Inputs:
│ │ │ │            o f, a _f_i_l_e
│ │ │ │      * Outputs:
│ │ │ │            o the mode of the open file f
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11375-0/0
│ │ │ │ +o1 = /tmp/M2-12685-0/0
│ │ │ │  i2 : f = fn << "hi there"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11375-0/0
│ │ │ │ +o2 = /tmp/M2-12685-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : fileMode f
│ │ │ │  
│ │ │ │  o3 = 420
│ │ │ │  i4 : close f
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-11375-0/0
│ │ │ │ +o4 = /tmp/M2-12685-0/0
│ │ │ │  
│ │ │ │  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
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Mode_lp__String_rp.html
│ │ │ @@ -69,22 +69,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │              
│ │ │              
│ │ │                
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -425,15 +425,15 @@
│ │ │  
│ │ │  o33 = 1
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10989-0/0
    
│ │ │ +o1 = /tmp/M2-11899-0/0
│ │ │              
    
│ │ │                
    i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-10989-0/0
│ │ │ +o2 = /tmp/M2-11899-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │                
    i3 : fileMode fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,18 +11,18 @@
│ │ │ │      * Inputs:
│ │ │ │            o fn, a _s_t_r_i_n_g
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the mode of the file located at the filename or path fn
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-10989-0/0
│ │ │ │ +o1 = /tmp/M2-11899-0/0
│ │ │ │  i2 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10989-0/0
│ │ │ │ +o2 = /tmp/M2-11899-0/0
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -73,22 +73,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -245,23 +245,23 @@
│ │ │  o22 = 10000000000000000000000000001710000000000000000000000000097470000000000
│ │ │        00000000000000185613
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10854-0/0
    
│ │ │ +o1 = /tmp/M2-11624-0/0
│ │ │              
    
│ │ │                
    i2 : f = fn << "hi there"
│ │ │  
│ │ │ -o2 = /tmp/M2-10854-0/0
│ │ │ +o2 = /tmp/M2-11624-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │                
    i3 : m = 7 + 7*8 + 7*64
│ │ │ @@ -108,15 +108,15 @@
│ │ │  o5 = 511
    
│ │ │              
    
│ │ │                
    i6 : close f
│ │ │  
│ │ │ -o6 = /tmp/M2-10854-0/0
│ │ │ +o6 = /tmp/M2-11624-0/0
│ │ │  
│ │ │  o6 : File
    
│ │ │              
    
│ │ │                
    i7 : fileMode fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,30 +12,30 @@
│ │ │ │            o mo, an _i_n_t_e_g_e_r
│ │ │ │            o f, a _f_i_l_e
│ │ │ │      * Consequences:
│ │ │ │            o the mode of the open file f is set to mo
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-10854-0/0
│ │ │ │ +o1 = /tmp/M2-11624-0/0
│ │ │ │  i2 : f = fn << "hi there"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10854-0/0
│ │ │ │ +o2 = /tmp/M2-11624-0/0
│ │ │ │  
│ │ │ │  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/M2-10854-0/0
│ │ │ │ +o6 = /tmp/M2-11624-0/0
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -73,22 +73,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11977-0/0
    
│ │ │ +o1 = /tmp/M2-13917-0/0
│ │ │              
    
│ │ │                
    i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-11977-0/0
│ │ │ +o2 = /tmp/M2-13917-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │                
    i3 : m = fileMode fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,18 +13,18 @@
│ │ │ │            o fn, a _s_t_r_i_n_g
│ │ │ │      * Consequences:
│ │ │ │            o the mode of the file located at the filename or path fn is set to
│ │ │ │              mo
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11977-0/0
│ │ │ │ +o1 = /tmp/M2-13917-0/0
│ │ │ │  i2 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11977-0/0
│ │ │ │ +o2 = /tmp/M2-13917-0/0
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -76,15 +76,15 @@
│ │ │        
    
│ │ │          
    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 = 66
    
│ │ │ +o1 = 49
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,15 +18,15 @@
│ │ │ │              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 = 66
│ │ │ │ +o1 = 49
│ │ │ │  ********** 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.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_force__G__B_lp..._cm__Syzygy__Matrix_eq_gt..._rp.html
│ │ │ @@ -120,15 +120,15 @@
│ │ │  o6 : Matrix R  <-- R
    
│ │ │              
    
│ │ │                
    i7 : syz f
│ │ │  
│ │ │ -   -- registering gb 0 at 0x7f499763be00
│ │ │ +   -- registering gb 0 at 0x7f60362e5e00
│ │ │  
│ │ │     -- [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 0x7f499763be00
│ │ │ │ +   -- registering gb 0 at 0x7f60362e5e00
│ │ │ │  
│ │ │ │     -- [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
│ │ │ @@ -96,15 +96,15 @@
│ │ │                
    i3 : removeFile "test-file"
    
│ │ │              
    
│ │ │                
    i4 : get "!date"
│ │ │  
│ │ │ -o4 = Fri Nov 14 17:26:54 UTC 2025
    
│ │ │ +o4 = Thu Dec 18 15:59:41 UTC 2025
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,15 +25,15 @@
│ │ │ │  o1 : File
│ │ │ │  i2 : get "test-file"
│ │ │ │  
│ │ │ │  o2 = hi there
│ │ │ │  i3 : removeFile "test-file"
│ │ │ │  i4 : get "!date"
│ │ │ │  
│ │ │ │ -o4 = Fri Nov 14 17:26:54 UTC 2025
│ │ │ │ +o4 = Thu Dec 18 15:59:41 UTC 2025
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_a_d -- read from a file
│ │ │ │      * _r_e_m_o_v_e_F_i_l_e -- remove a file
│ │ │ │      * _c_l_o_s_e -- close a file
│ │ │ │      * _F_i_l_e_ _<_<_ _T_h_i_n_g -- print to a file
│ │ │ │  ********** WWaayyss ttoo uussee ggeett:: **********
│ │ │ │      * get(File)
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_instances.html
│ │ │ @@ -84,15 +84,15 @@
│ │ │                 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 {}
│ │ │ │ @@ -23,15 +23,15 @@
│ │ │ │                 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
│ │ │ @@ -75,22 +75,22 @@
│ │ │  o1 = true
    
│ │ │              
    
│ │ │                
    i2 : fn = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-10380-0/0
    
│ │ │ +o2 = /tmp/M2-10670-0/0
│ │ │              
    
│ │ │                
    i3 : fn << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-10380-0/0
│ │ │ +o3 = /tmp/M2-10670-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : isDirectory fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,18 +13,18 @@
│ │ │ │            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/M2-10380-0/0
│ │ │ │ +o2 = /tmp/M2-10670-0/0
│ │ │ │  i3 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-10380-0/0
│ │ │ │ +o3 = /tmp/M2-10670-0/0
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -211,15 +211,15 @@
│ │ │  
│ │ │  o18 = false
    
│ │ │              
    
│ │ │                
    i19 : elapsedTime facs = factor(m*m1)
│ │ │ - -- 4.26787s elapsed
│ │ │ + -- 5.61121s elapsed
│ │ │  
│ │ │  o19 = 1000000000000000000000000000057*1000000000000000000010000000083
│ │ │  
│ │ │  o19 : Expression of class Product
    
│ │ │              
    
│ │ │                
    i23 : elapsedTime isPrime m3
│ │ │ - -- .0564874s elapsed
│ │ │ + -- .0593425s elapsed
│ │ │  
│ │ │  o23 = true
    
│ │ │              
    
│ │ │                
    i24 : elapsedTime isPseudoprime m3
│ │ │ - -- .000106819s elapsed
│ │ │ + -- .000124275s elapsed
│ │ │  
│ │ │  o24 = true
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── 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)
│ │ │ │ - -- 4.26787s elapsed
│ │ │ │ + -- 5.61121s 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
│ │ │ │ - -- .0564874s elapsed
│ │ │ │ + -- .0593425s elapsed
│ │ │ │  
│ │ │ │  o23 = true
│ │ │ │  i24 : elapsedTime isPseudoprime m3
│ │ │ │ - -- .000106819s elapsed
│ │ │ │ + -- .000124275s 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
│ │ │ @@ -68,22 +68,22 @@
│ │ │        
    
│ │ │          
    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/M2-12188-0/0
    
│ │ │ +o1 = /tmp/M2-14348-0/0
│ │ │              
    
│ │ │                
    i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-12188-0/0
│ │ │ +o2 = /tmp/M2-14348-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │                
    i3 : isRegularFile fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,18 +13,18 @@
│ │ │ │            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/M2-12188-0/0
│ │ │ │ +o1 = /tmp/M2-14348-0/0
│ │ │ │  i2 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12188-0/0
│ │ │ │ +o2 = /tmp/M2-14348-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : isRegularFile fn
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  i4 : removeFile fn
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_make__Directory_lp__String_rp.html
│ │ │ @@ -76,22 +76,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,18 +13,18 @@
│ │ │ │      * 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/M2-10722-0/0
│ │ │ │ +o1 = /tmp/M2-11352-0/0
│ │ │ │  i2 : makeDirectory (dir|"/a/b/c")
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10722-0/0/a/b/c
│ │ │ │ +o2 = /tmp/M2-11352-0/0/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
│ │ │ @@ -64,15 +64,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    
│ │ │                
    i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10722-0/0
    
│ │ │ +o1 = /tmp/M2-11352-0/0
│ │ │              
    
│ │ │                
    i2 : makeDirectory (dir|"/a/b/c")
│ │ │  
│ │ │ -o2 = /tmp/M2-10722-0/0/a/b/c
    
│ │ │ +o2 = /tmp/M2-11352-0/0/a/b/c
│ │ │              
    
│ │ │                
    i3 : removeDirectory (dir|"/a/b/c")
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : maxAllowableThreads
│ │ │  
│ │ │ -o1 = 7
    
│ │ │ +o1 = 17
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -9,15 +9,15 @@
│ │ │ │      *   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.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_threads.m2:498:0.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_memoize.html
│ │ │ @@ -61,15 +61,15 @@
│ │ │  
│ │ │  o1 : FunctionClosure
    
│ │ │              
    
│ │ │                
    i2 : time fib 28
│ │ │ - -- used 1.38174s (cpu); 0.797855s (thread); 0s (gc)
│ │ │ + -- used 0.818496s (cpu); 0.62457s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 514229
    
│ │ │              
    
│ │ │                
    i3 : fib = memoize fib
│ │ │ @@ -78,23 +78,23 @@
│ │ │  
│ │ │  o3 : FunctionClosure
    
│ │ │              
    
│ │ │                
    i4 : time fib 28
│ │ │ - -- used 8.4749e-05s (cpu); 8.4468e-05s (thread); 0s (gc)
│ │ │ + -- used 7.7903e-05s (cpu); 7.4132e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 514229
    
│ │ │              
    
│ │ │                
    i5 : time fib 28
│ │ │ - -- used 4.028e-06s (cpu); 3.647e-06s (thread); 0s (gc)
│ │ │ + -- used 3.788e-06s (cpu); 3.64e-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 {}
│ │ │ │ @@ -11,28 +11,28 @@
│ │ │ │  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 1.38174s (cpu); 0.797855s (thread); 0s (gc)
│ │ │ │ + -- used 0.818496s (cpu); 0.62457s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = 514229
│ │ │ │  i3 : fib = memoize fib
│ │ │ │  
│ │ │ │  o3 = fib
│ │ │ │  
│ │ │ │  o3 : FunctionClosure
│ │ │ │  i4 : time fib 28
│ │ │ │ - -- used 8.4749e-05s (cpu); 8.4468e-05s (thread); 0s (gc)
│ │ │ │ + -- used 7.7903e-05s (cpu); 7.4132e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 514229
│ │ │ │  i5 : time fib 28
│ │ │ │ - -- used 4.028e-06s (cpu); 3.647e-06s (thread); 0s (gc)
│ │ │ │ + -- used 3.788e-06s (cpu); 3.64e-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
│ │ │ @@ -89,20 +89,20 @@
│ │ │       {12 => (poincare, BettiTally)                                }
│ │ │       {13 => (hilbertPolynomial, ZZ, BettiTally)                   }
│ │ │       {14 => (degree, BettiTally)                                  }
│ │ │       {15 => (hilbertSeries, ZZ, BettiTally)                       }
│ │ │       {16 => (pdim, BettiTally)                                    }
│ │ │       {17 => (regularity, BettiTally)                              }
│ │ │       {18 => (mathML, BettiTally)                                  }
│ │ │ -     {19 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │ -     {20 => (dual, BettiTally)                                    }
│ │ │ -     {21 => (codim, BettiTally)                                   }
│ │ │ +     {19 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ +     {20 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │ +     {21 => (dual, BettiTally)                                    }
│ │ │       {22 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │       {23 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ -     {24 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ +     {24 => (codim, BettiTally)                                   }
│ │ │       {25 => (^, Ring, BettiTally)                                 }
│ │ │  
│ │ │  o1 : NumberedVerticalList
    
│ │ │                
    
│ │ │ @@ -189,19 +189,19 @@
│ │ │            
│ │ │            
│ │ │              
│ │ │                
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                  
    i5 : methods( Matrix, Matrix )
│ │ │  
│ │ │  o5 = {0 => (diff', Matrix, Matrix)                               }
│ │ │ -     {1 => (-, Matrix, Matrix)                                   }
│ │ │ -     {2 => (+, Matrix, Matrix)                                   }
│ │ │ -     {3 => (diff, Matrix, Matrix)                                }
│ │ │ -     {4 => (contract, Matrix, Matrix)                            }
│ │ │ -     {5 => (contract', Matrix, Matrix)                           }
│ │ │ +     {1 => (diff, Matrix, Matrix)                                }
│ │ │ +     {2 => (contract, Matrix, Matrix)                            }
│ │ │ +     {3 => (contract', Matrix, Matrix)                           }
│ │ │ +     {4 => (-, Matrix, Matrix)                                   }
│ │ │ +     {5 => (+, Matrix, Matrix)                                   }
│ │ │       {6 => (markedGB, Matrix, Matrix)                            }
│ │ │       {7 => (Hom, Matrix, Matrix)                                 }
│ │ │       {8 => (==, Matrix, Matrix)                                  }
│ │ │       {9 => (*, Matrix, Matrix)                                   }
│ │ │       {10 => (|, Matrix, Matrix)                                  }
│ │ │       {11 => (||, Matrix, Matrix)                                 }
│ │ │       {12 => (subquotient, Matrix, Matrix)                        }
│ │ │ @@ -212,22 +212,22 @@
│ │ │       {17 => (quotientRemainder', Matrix, Matrix)                 }
│ │ │       {18 => (quotientRemainder, Matrix, Matrix)                  }
│ │ │       {19 => (//, Matrix, Matrix)                                 }
│ │ │       {20 => (\\, Matrix, Matrix)                                 }
│ │ │       {21 => (quotient, Matrix, Matrix)                           }
│ │ │       {22 => (quotient', Matrix, Matrix)                          }
│ │ │       {23 => (remainder', Matrix, Matrix)                         }
│ │ │ -     {24 => (remainder, Matrix, Matrix)                          }
│ │ │ -     {25 => (%, Matrix, Matrix)                                  }
│ │ │ +     {24 => (%, Matrix, Matrix)                                  }
│ │ │ +     {25 => (remainder, Matrix, Matrix)                          }
│ │ │       {26 => (pushout, Matrix, Matrix)                            }
│ │ │       {27 => (solve, Matrix, Matrix)                              }
│ │ │ -     {28 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │ -     {29 => (intersect, Matrix, Matrix)                          }
│ │ │ -     {30 => (pullback, Matrix, Matrix)                           }
│ │ │ -     {31 => (tensor, Matrix, Matrix)                             }
│ │ │ +     {28 => (tensor, Matrix, Matrix)                             }
│ │ │ +     {29 => (pullback, Matrix, Matrix)                           }
│ │ │ +     {30 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │ +     {31 => (intersect, Matrix, Matrix)                          }
│ │ │       {32 => (substitute, Matrix, Matrix)                         }
│ │ │       {33 => (yonedaProduct, Matrix, Matrix)                      }
│ │ │       {34 => (isShortExactSequence, Matrix, Matrix)               }
│ │ │       {35 => (horseshoeResolution, Matrix, Matrix)                }
│ │ │       {36 => (connectingExtMap, Module, Matrix, Matrix)           }
│ │ │       {37 => (connectingExtMap, Matrix, Matrix, Module)           }
│ │ │       {38 => (connectingTorMap, Module, Matrix, Matrix)           }
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,20 +29,20 @@
│ │ │ │       {12 => (poincare, BettiTally)                                }
│ │ │ │       {13 => (hilbertPolynomial, ZZ, BettiTally)                   }
│ │ │ │       {14 => (degree, BettiTally)                                  }
│ │ │ │       {15 => (hilbertSeries, ZZ, BettiTally)                       }
│ │ │ │       {16 => (pdim, BettiTally)                                    }
│ │ │ │       {17 => (regularity, BettiTally)                              }
│ │ │ │       {18 => (mathML, BettiTally)                                  }
│ │ │ │ -     {19 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │ │ -     {20 => (dual, BettiTally)                                    }
│ │ │ │ -     {21 => (codim, BettiTally)                                   }
│ │ │ │ +     {19 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ │ +     {20 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │ │ +     {21 => (dual, BettiTally)                                    }
│ │ │ │       {22 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ │       {23 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ │ -     {24 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ │ +     {24 => (codim, BettiTally)                                   }
│ │ │ │       {25 => (^, Ring, BettiTally)                                 }
│ │ │ │  
│ │ │ │  o1 : NumberedVerticalList
│ │ │ │  i2 : methods resolution
│ │ │ │  
│ │ │ │  o2 = {0 => (resolution, Ideal) }
│ │ │ │       {1 => (resolution, Module)}
│ │ │ │ @@ -86,19 +86,19 @@
│ │ │ │            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 => (diff', Matrix, Matrix)                               }
│ │ │ │ -     {1 => (-, Matrix, Matrix)                                   }
│ │ │ │ -     {2 => (+, Matrix, Matrix)                                   }
│ │ │ │ -     {3 => (diff, Matrix, Matrix)                                }
│ │ │ │ -     {4 => (contract, Matrix, Matrix)                            }
│ │ │ │ -     {5 => (contract', Matrix, Matrix)                           }
│ │ │ │ +     {1 => (diff, Matrix, Matrix)                                }
│ │ │ │ +     {2 => (contract, Matrix, Matrix)                            }
│ │ │ │ +     {3 => (contract', Matrix, Matrix)                           }
│ │ │ │ +     {4 => (-, Matrix, Matrix)                                   }
│ │ │ │ +     {5 => (+, Matrix, Matrix)                                   }
│ │ │ │       {6 => (markedGB, Matrix, Matrix)                            }
│ │ │ │       {7 => (Hom, Matrix, Matrix)                                 }
│ │ │ │       {8 => (==, Matrix, Matrix)                                  }
│ │ │ │       {9 => (*, Matrix, Matrix)                                   }
│ │ │ │       {10 => (|, Matrix, Matrix)                                  }
│ │ │ │       {11 => (||, Matrix, Matrix)                                 }
│ │ │ │       {12 => (subquotient, Matrix, Matrix)                        }
│ │ │ │ @@ -109,22 +109,22 @@
│ │ │ │       {17 => (quotientRemainder', Matrix, Matrix)                 }
│ │ │ │       {18 => (quotientRemainder, Matrix, Matrix)                  }
│ │ │ │       {19 => (//, Matrix, Matrix)                                 }
│ │ │ │       {20 => (\\, Matrix, Matrix)                                 }
│ │ │ │       {21 => (quotient, Matrix, Matrix)                           }
│ │ │ │       {22 => (quotient', Matrix, Matrix)                          }
│ │ │ │       {23 => (remainder', Matrix, Matrix)                         }
│ │ │ │ -     {24 => (remainder, Matrix, Matrix)                          }
│ │ │ │ -     {25 => (%, Matrix, Matrix)                                  }
│ │ │ │ +     {24 => (%, Matrix, Matrix)                                  }
│ │ │ │ +     {25 => (remainder, Matrix, Matrix)                          }
│ │ │ │       {26 => (pushout, Matrix, Matrix)                            }
│ │ │ │       {27 => (solve, Matrix, Matrix)                              }
│ │ │ │ -     {28 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │ │ -     {29 => (intersect, Matrix, Matrix)                          }
│ │ │ │ -     {30 => (pullback, Matrix, Matrix)                           }
│ │ │ │ -     {31 => (tensor, Matrix, Matrix)                             }
│ │ │ │ +     {28 => (tensor, Matrix, Matrix)                             }
│ │ │ │ +     {29 => (pullback, Matrix, Matrix)                           }
│ │ │ │ +     {30 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │ │ +     {31 => (intersect, Matrix, Matrix)                          }
│ │ │ │       {32 => (substitute, Matrix, Matrix)                         }
│ │ │ │       {33 => (yonedaProduct, Matrix, Matrix)                      }
│ │ │ │       {34 => (isShortExactSequence, Matrix, Matrix)               }
│ │ │ │       {35 => (horseshoeResolution, Matrix, Matrix)                }
│ │ │ │       {36 => (connectingExtMap, Module, Matrix, Matrix)           }
│ │ │ │       {37 => (connectingExtMap, Matrix, Matrix, Module)           }
│ │ │ │       {38 => (connectingTorMap, Module, Matrix, Matrix)           }
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_minimal__Betti.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  o2 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime C = minimalBetti I
│ │ │ - -- 1.919s elapsed
│ │ │ + -- 2.34322s 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  .
│ │ │ @@ -125,15 +125,15 @@
│ │ │  
│ │ │  o4 : Ideal of S
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime C = minimalBetti(I, DegreeLimit=>2)
│ │ │ - -- .74946s elapsed
│ │ │ + -- .978961s 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
│ │ │  
│ │ │ @@ -146,15 +146,15 @@
│ │ │  
│ │ │  o6 : Ideal of S
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime C = minimalBetti(I, DegreeLimit=>1, LengthLimit=>5)
│ │ │ - -- .0315312s elapsed
│ │ │ + -- .0378645s elapsed
│ │ │  
│ │ │              0  1   2   3  4
│ │ │  o7 = total: 1 35 140 189 84
│ │ │           0: 1  .   .   .  .
│ │ │           1: . 35 140 189 84
│ │ │  
│ │ │  o7 : BettiTally
    
│ │ │ @@ -166,15 +166,15 @@
│ │ │  
│ │ │  o8 : Ideal of S
│ │ │              
    
│ │ │                
    i9 : elapsedTime C = minimalBetti(I, LengthLimit=>5)
│ │ │ - -- 1.2026s elapsed
│ │ │ + -- 1.60704s 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 {}
│ │ │ │ @@ -43,15 +43,15 @@
│ │ │ │  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
│ │ │ │ - -- 1.919s elapsed
│ │ │ │ + -- 2.34322s 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  .
│ │ │ │ @@ -60,40 +60,40 @@
│ │ │ │  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)
│ │ │ │ - -- .74946s elapsed
│ │ │ │ + -- .978961s 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)
│ │ │ │ - -- .0315312s elapsed
│ │ │ │ + -- .0378645s 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)
│ │ │ │ - -- 1.2026s elapsed
│ │ │ │ + -- 1.60704s 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
│ │ │ @@ -72,15 +72,15 @@
│ │ │          
    Description
    
│ │ │          
    Only one directory will be made, so the components of the path p other than the last must already exist.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -91,15 +91,15 @@
│ │ │  o3 = true
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : p = temporaryFileName() | "/"
│ │ │  
│ │ │ -o1 = /tmp/M2-10741-0/0/
    
│ │ │ +o1 = /tmp/M2-11391-0/0/
│ │ │              
    
│ │ │                
    i2 : mkdir p
    
│ │ │              
    
│ │ │                
    i4 : (fn = p | "foo") << "hi there" << close
│ │ │  
│ │ │ -o4 = /tmp/M2-10741-0/0/foo
│ │ │ +o4 = /tmp/M2-11391-0/0/foo
│ │ │  
│ │ │  o4 : File
    
│ │ │              
    
│ │ │                
    i5 : get fn
│ │ │ ├── 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/M2-10741-0/0/
│ │ │ │ +o1 = /tmp/M2-11391-0/0/
│ │ │ │  i2 : mkdir p
│ │ │ │  i3 : isDirectory p
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  i4 : (fn = p | "foo") << "hi there" << close
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-10741-0/0/foo
│ │ │ │ +o4 = /tmp/M2-11391-0/0/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
│ │ │ @@ -81,52 +81,52 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,32 +20,32 @@
│ │ │ │            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/M2-10615-0/0
│ │ │ │ +o1 = /tmp/M2-11145-0/0
│ │ │ │  i2 : dst = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10615-0/1
│ │ │ │ +o2 = /tmp/M2-11145-0/1
│ │ │ │  i3 : src << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-10615-0/0
│ │ │ │ +o3 = /tmp/M2-11145-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : moveFile(src,dst,Verbose=>true)
│ │ │ │ ---moving: /tmp/M2-10615-0/0 -> /tmp/M2-10615-0/1
│ │ │ │ +--moving: /tmp/M2-11145-0/0 -> /tmp/M2-11145-0/1
│ │ │ │  i5 : get dst
│ │ │ │  
│ │ │ │  o5 = hi there
│ │ │ │  i6 : bak = moveFile(dst,Verbose=>true)
│ │ │ │ ---backup file created: /tmp/M2-10615-0/1.bak
│ │ │ │ +--backup file created: /tmp/M2-11145-0/1.bak
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-10615-0/1.bak
│ │ │ │ +o6 = /tmp/M2-11145-0/1.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
│ │ │ @@ -51,15 +51,15 @@
│ │ │        
    nanosleep -- sleep for a given number of nanoseconds
    
│ │ │        
    
│ │ │          
    Description
    
│ │ │  nanosleep n -- sleeps for n nanoseconds.        
    
│ │ │                
    i1 : src = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10615-0/0
    
│ │ │ +o1 = /tmp/M2-11145-0/0
│ │ │              
    
│ │ │                
    i2 : dst = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-10615-0/1
    
│ │ │ +o2 = /tmp/M2-11145-0/1
│ │ │              
    
│ │ │                
    i3 : src << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-10615-0/0
│ │ │ +o3 = /tmp/M2-11145-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : moveFile(src,dst,Verbose=>true)
│ │ │ ---moving: /tmp/M2-10615-0/0 -> /tmp/M2-10615-0/1
    
│ │ │ +--moving: /tmp/M2-11145-0/0 -> /tmp/M2-11145-0/1
│ │ │              
    
│ │ │                
    i5 : get dst
│ │ │  
│ │ │  o5 = hi there
    
│ │ │              
    
│ │ │                
    i6 : bak = moveFile(dst,Verbose=>true)
│ │ │ ---backup file created: /tmp/M2-10615-0/1.bak
│ │ │ +--backup file created: /tmp/M2-11145-0/1.bak
│ │ │  
│ │ │ -o6 = /tmp/M2-10615-0/1.bak
    
│ │ │ +o6 = /tmp/M2-11145-0/1.bak
│ │ │              
    
│ │ │                
    i7 : removeFile bak
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : elapsedTime nanosleep 500000000
│ │ │ - -- .500135s elapsed
│ │ │ + -- .500116s elapsed
│ │ │  
│ │ │  o1 = 0
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -4,15 +4,15 @@
│ │ │ │  [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
│ │ │ │ - -- .500135s elapsed
│ │ │ │ + -- .500116s 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/_options_lp__Function_rp.html
│ │ │ @@ -104,38 +104,38 @@
│ │ │  o3 : OptionTable
    
│ │ │              
    
│ │ │                
    i4 : methods codim
│ │ │  
│ │ │ -o4 = {0 => (codim, BettiTally)    }
│ │ │ -     {1 => (codim, Module)        }
│ │ │ -     {2 => (codim, QuotientRing)  }
│ │ │ -     {3 => (codim, Ideal)         }
│ │ │ -     {4 => (codim, MonomialIdeal) }
│ │ │ -     {5 => (codim, PolynomialRing)}
│ │ │ -     {6 => (codim, CoherentSheaf) }
│ │ │ -     {7 => (codim, Variety)       }
│ │ │ +o4 = {0 => (codim, Module)        }
│ │ │ +     {1 => (codim, QuotientRing)  }
│ │ │ +     {2 => (codim, Ideal)         }
│ │ │ +     {3 => (codim, MonomialIdeal) }
│ │ │ +     {4 => (codim, PolynomialRing)}
│ │ │ +     {5 => (codim, CoherentSheaf) }
│ │ │ +     {6 => (codim, Variety)       }
│ │ │ +     {7 => (codim, BettiTally)    }
│ │ │  
│ │ │  o4 : NumberedVerticalList
    
│ │ │              
    
│ │ │                
    i5 : options oo
│ │ │  
│ │ │ -o5 = {0 => (OptionTable{})                }
│ │ │ +o5 = {0 => (OptionTable{Generic => false})}
│ │ │       {1 => (OptionTable{Generic => false})}
│ │ │       {2 => (OptionTable{Generic => false})}
│ │ │       {3 => (OptionTable{Generic => false})}
│ │ │       {4 => (OptionTable{Generic => false})}
│ │ │       {5 => (OptionTable{Generic => false})}
│ │ │       {6 => (OptionTable{Generic => false})}
│ │ │ -     {7 => (OptionTable{Generic => false})}
│ │ │ +     {7 => (OptionTable{})                }
│ │ │  
│ │ │  o5 : NumberedVerticalList
    
│ │ │              
    
│ │ │                
    i6 : methods intersect
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,34 +35,34 @@
│ │ │ │  i3 : options(codim, Ideal)
│ │ │ │  
│ │ │ │  o3 = OptionTable{Generic => false}
│ │ │ │  
│ │ │ │  o3 : OptionTable
│ │ │ │  i4 : methods codim
│ │ │ │  
│ │ │ │ -o4 = {0 => (codim, BettiTally)    }
│ │ │ │ -     {1 => (codim, Module)        }
│ │ │ │ -     {2 => (codim, QuotientRing)  }
│ │ │ │ -     {3 => (codim, Ideal)         }
│ │ │ │ -     {4 => (codim, MonomialIdeal) }
│ │ │ │ -     {5 => (codim, PolynomialRing)}
│ │ │ │ -     {6 => (codim, CoherentSheaf) }
│ │ │ │ -     {7 => (codim, Variety)       }
│ │ │ │ +o4 = {0 => (codim, Module)        }
│ │ │ │ +     {1 => (codim, QuotientRing)  }
│ │ │ │ +     {2 => (codim, Ideal)         }
│ │ │ │ +     {3 => (codim, MonomialIdeal) }
│ │ │ │ +     {4 => (codim, PolynomialRing)}
│ │ │ │ +     {5 => (codim, CoherentSheaf) }
│ │ │ │ +     {6 => (codim, Variety)       }
│ │ │ │ +     {7 => (codim, BettiTally)    }
│ │ │ │  
│ │ │ │  o4 : NumberedVerticalList
│ │ │ │  i5 : options oo
│ │ │ │  
│ │ │ │ -o5 = {0 => (OptionTable{})                }
│ │ │ │ +o5 = {0 => (OptionTable{Generic => false})}
│ │ │ │       {1 => (OptionTable{Generic => false})}
│ │ │ │       {2 => (OptionTable{Generic => false})}
│ │ │ │       {3 => (OptionTable{Generic => false})}
│ │ │ │       {4 => (OptionTable{Generic => false})}
│ │ │ │       {5 => (OptionTable{Generic => false})}
│ │ │ │       {6 => (OptionTable{Generic => false})}
│ │ │ │ -     {7 => (OptionTable{Generic => false})}
│ │ │ │ +     {7 => (OptionTable{})                }
│ │ │ │  
│ │ │ │  o5 : NumberedVerticalList
│ │ │ │  i6 : methods intersect
│ │ │ │  
│ │ │ │  o6 = {0 => (intersect, List)                           }
│ │ │ │       {1 => (intersect, Ideal, Ideal)                   }
│ │ │ │       {2 => (intersect, RRi, RRi)                       }
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_parallel_spprogramming_spwith_spthreads_spand_sptasks.html
│ │ │ @@ -72,21 +72,21 @@
│ │ │              
    		
│ │ │                
    i2 : L = random toList (1..10000);
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime         apply(1..100, n -> sort L);
│ │ │ - -- .636785s elapsed
    
│ │ │ + -- .70432s elapsed
│ │ │              
    
│ │ │                
    i4 : elapsedTime parallelApply(1..100, n -> sort L);
│ │ │ - -- .302911s elapsed
    
│ │ │ + -- .187029s 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.
    
│ │ │ @@ -100,15 +100,15 @@
│ │ │  o5 = 5
    
│ │ │              
    
│ │ │                
    i6 : allowableThreads = maxAllowableThreads
│ │ │  
│ │ │ -o6 = 7
    
│ │ │ +o6 = 17
│ │ │              
    
│ │ │          
    
│ │ │            
    To run a function in another thread use schedule, as in the following example.
    
│ │ │          
    
│ │ │          
│ │ │ @@ -224,15 +224,15 @@
│ │ │                
    i17 : schedule t';
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i18 : t'
│ │ │  
│ │ │ -o18 = <<task, running>>
│ │ │ +o18 = <<task, created>>
│ │ │  
│ │ │  o18 : Task
    
│ │ │              
    
│ │ │                
    i19 : taskResult t'
│ │ │ ├── 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);
│ │ │ │ - -- .636785s elapsed
│ │ │ │ + -- .70432s elapsed
│ │ │ │  i4 : elapsedTime parallelApply(1..100, n -> sort L);
│ │ │ │ - -- .302911s elapsed
│ │ │ │ + -- .187029s 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 = QQ[x,y,z];
│ │ │ │  i8 : I = ideal(x^2 + 2*y^2 - y - 2*z, x^2 - 8*y^2 + 10*z - 1, x^2 - 7*y*z)
│ │ │ │  
│ │ │ │               2     2            2     2             2
│ │ │ │  o8 = ideal (x  + 2y  - y - 2z, x  - 8y  + 10z - 1, x  - 7y*z)
│ │ │ │  
│ │ │ │ @@ -92,15 +92,15 @@
│ │ │ │  o16 = <>
│ │ │ │  
│ │ │ │  o16 : Task
│ │ │ │  Start it running with _s_c_h_e_d_u_l_e.
│ │ │ │  i17 : schedule t';
│ │ │ │  i18 : t'
│ │ │ │  
│ │ │ │ -o18 = <>
│ │ │ │ +o18 = <>
│ │ │ │  
│ │ │ │  o18 : Task
│ │ │ │  i19 : taskResult t'
│ │ │ │  
│ │ │ │  o19 = | 980z2-18y-201z+13 35yz-4y+2z-1 10y2-y-12z+1 5x2-4y+2z-1 |
│ │ │ │  
│ │ │ │                1      4
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_parallelism_spin_spengine_spcomputations.html
│ │ │ @@ -137,15 +137,15 @@
│ │ │  
│ │ │  o3 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime minimalBetti I
│ │ │ - -- 2.04775s elapsed
│ │ │ + -- 2.38342s 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  .
│ │ │ @@ -160,15 +160,15 @@
│ │ │  
│ │ │  o5 : Ideal of S
    
│ │ │              
    
│ │ │                
    i6 : elapsedTime minimalBetti(I, ParallelizeByDegree => true)
│ │ │ - -- 1.80304s elapsed
│ │ │ + -- 2.35357s 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  .
│ │ │ @@ -190,15 +190,15 @@
│ │ │  
│ │ │  o8 = 1
    
│ │ │              
    
│ │ │                
    i9 : elapsedTime minimalBetti(I)
│ │ │ - -- 5.57414s elapsed
│ │ │ + -- 2.33547s 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  .
│ │ │ @@ -231,15 +231,15 @@
│ │ │  
│ │ │  o12 : Ideal of S
    
│ │ │              
    
│ │ │                
    i13 : elapsedTime freeResolution(I, Strategy => Nonminimal)
│ │ │ - -- 2.03698s elapsed
│ │ │ + -- 2.71086s 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
    
│ │ │ @@ -258,15 +258,15 @@
│ │ │  
│ │ │  o15 : Ideal of S
│ │ │              
    
│ │ │                
    i16 : elapsedTime freeResolution(I, Strategy => Nonminimal)
│ │ │ - -- 1.946s elapsed
│ │ │ + -- 2.65371s 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
    
│ │ │ @@ -299,15 +299,15 @@
│ │ │  
│ │ │  o19 : Ideal of S
│ │ │              
    
│ │ │                
    i20 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 4.82802s elapsed
│ │ │ + -- 4.21704s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o20 : Matrix S  <-- S
    
│ │ │              
    
│ │ │ @@ -322,15 +322,15 @@
│ │ │  
│ │ │  o22 : Ideal of S
│ │ │              
    
│ │ │                
    i23 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 6.58749s elapsed
│ │ │ + -- 8.52357s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o23 : Matrix S  <-- S
    
│ │ │              
    
│ │ │ @@ -345,15 +345,15 @@
│ │ │  
│ │ │  o25 : Ideal of S
│ │ │              
    
│ │ │                
    i26 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 4.12035s elapsed
│ │ │ + -- 3.78655s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o26 : Matrix S  <-- S
    
│ │ │              
    
│ │ │          
    
│ │ │ @@ -396,15 +396,15 @@
│ │ │  o30 : Ideal of ---<|a, b, c|>
│ │ │                 101
│ │ │              
    
│ │ │                
    i31 : elapsedTime NCGB(I, 22);
│ │ │ - -- 1.03876s elapsed
│ │ │ + -- 1.021s elapsed
│ │ │  
│ │ │                 ZZ            1       ZZ            148
│ │ │  o31 : Matrix (---<|a, b, c|>)  <-- (---<|a, b, c|>)
│ │ │                101                   101
    
│ │ │              
    
│ │ │                
    i34 : elapsedTime NCGB(I, 22);
│ │ │ - -- 1.17701s elapsed
│ │ │ + -- 1.42759s elapsed
│ │ │  
│ │ │                 ZZ            1       ZZ            148
│ │ │  o34 : Matrix (---<|a, b, c|>)  <-- (---<|a, b, c|>)
│ │ │                101                   101
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -93,30 +93,30 @@
│ │ │ │  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
│ │ │ │ - -- 2.04775s elapsed
│ │ │ │ + -- 2.38342s 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)
│ │ │ │ - -- 1.80304s elapsed
│ │ │ │ + -- 2.35357s 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  .
│ │ │ │ @@ -126,15 +126,15 @@
│ │ │ │  i7 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o7 : Ideal of S
│ │ │ │  i8 : numTBBThreads = 1
│ │ │ │  
│ │ │ │  o8 = 1
│ │ │ │  i9 : elapsedTime minimalBetti(I)
│ │ │ │ - -- 5.57414s elapsed
│ │ │ │ + -- 2.33547s 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  .
│ │ │ │ @@ -149,15 +149,15 @@
│ │ │ │  i11 : numTBBThreads = 0
│ │ │ │  
│ │ │ │  o11 = 0
│ │ │ │  i12 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o12 : Ideal of S
│ │ │ │  i13 : elapsedTime freeResolution(I, Strategy => Nonminimal)
│ │ │ │ - -- 2.03698s elapsed
│ │ │ │ + -- 2.71086s elapsed
│ │ │ │  
│ │ │ │         1      35      241      841      1781      2464      2294      1432
│ │ │ │  576      135      14
│ │ │ │  o13 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-
│ │ │ │  - S    <-- S    <-- S
│ │ │ │  
│ │ │ │  
│ │ │ │ @@ -168,15 +168,15 @@
│ │ │ │  i14 : numTBBThreads = 1
│ │ │ │  
│ │ │ │  o14 = 1
│ │ │ │  i15 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o15 : Ideal of S
│ │ │ │  i16 : elapsedTime freeResolution(I, Strategy => Nonminimal)
│ │ │ │ - -- 1.946s elapsed
│ │ │ │ + -- 2.65371s elapsed
│ │ │ │  
│ │ │ │         1      35      241      841      1781      2464      2294      1432
│ │ │ │  576      135      14
│ │ │ │  o16 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-
│ │ │ │  - S    <-- S    <-- S
│ │ │ │  
│ │ │ │  
│ │ │ │ @@ -195,37 +195,37 @@
│ │ │ │  o18 = S
│ │ │ │  
│ │ │ │  o18 : PolynomialRing
│ │ │ │  i19 : I = ideal random(S^1, S^{4:-5});
│ │ │ │  
│ │ │ │  o19 : Ideal of S
│ │ │ │  i20 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ │ - -- 4.82802s elapsed
│ │ │ │ + -- 4.21704s 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");
│ │ │ │ - -- 6.58749s elapsed
│ │ │ │ + -- 8.52357s 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");
│ │ │ │ - -- 4.12035s elapsed
│ │ │ │ + -- 3.78655s elapsed
│ │ │ │  
│ │ │ │                1      108
│ │ │ │  o26 : Matrix S  <-- S
│ │ │ │  For Gröbner basis computation in associative algebras, ParallelizeByDegree is
│ │ │ │  not relevant. In this case, use numTBBThreads to control the amount of
│ │ │ │  parallelism.
│ │ │ │  i27 : needsPackage "AssociativeAlgebras"
│ │ │ │ @@ -246,15 +246,15 @@
│ │ │ │                 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);
│ │ │ │ - -- 1.03876s elapsed
│ │ │ │ + -- 1.021s elapsed
│ │ │ │  
│ │ │ │                 ZZ            1       ZZ            148
│ │ │ │  o31 : Matrix (---<|a, b, c|>)  <-- (---<|a, b, c|>)
│ │ │ │                101                   101
│ │ │ │  i32 : I = ideal I_*
│ │ │ │  
│ │ │ │                 2                         2                         2
│ │ │ │ @@ -263,15 +263,15 @@
│ │ │ │                  ZZ
│ │ │ │  o32 : Ideal of ---<|a, b, c|>
│ │ │ │                 101
│ │ │ │  i33 : numTBBThreads = 1
│ │ │ │  
│ │ │ │  o33 = 1
│ │ │ │  i34 : elapsedTime NCGB(I, 22);
│ │ │ │ - -- 1.17701s elapsed
│ │ │ │ + -- 1.42759s 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
│ │ │ @@ -370,36 +370,36 @@
│ │ │  
│ │ │  o27 = 3
│ │ │              
    
│ │ │                
    i28 : time poincare I
│ │ │ - -- used 0.000521488s (cpu); 2.0348e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00399979s (cpu); 0s (thread); 0s (gc)
│ │ │  
│ │ │              3     6    9
│ │ │  o28 = 1 - 3T  + 3T  - T
│ │ │  
│ │ │  o28 : ZZ[T]
    
│ │ │              
    
│ │ │                
    i29 : time gens gb I;
│ │ │  
│ │ │ -   -- registering gb 19 at 0x7fe6adf6d380
│ │ │ +   -- registering gb 19 at 0x7f963d095540
│ │ │  
│ │ │     -- [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 0.0234308s (cpu); 0.0247955s (thread); 0s (gc)
│ │ │ +   --  -- used 0.0118597s (cpu); 0.0150154s (thread); 0s (gc)
│ │ │  
│ │ │                1      11
│ │ │  o29 : Matrix R  <-- R
    
│ │ │              
    
│ │ │          
    
│ │ │ @@ -411,15 +411,15 @@
│ │ │                
    i30 : R = QQ[a..d];
    
│ │ │              
    
│ │ │                
    i31 : I = ideal random(R^1, R^{3:-3});
│ │ │  
│ │ │ -   -- registering gb 20 at 0x7fe6adf6d1c0
│ │ │ +   -- registering gb 20 at 0x7f963d095380
│ │ │  
│ │ │     -- [gb]number of (nonminimal) gb elements = 0
│ │ │     -- number of monomials                = 0
│ │ │     -- #reduction steps = 0
│ │ │     -- #spairs done = 0
│ │ │     -- ncalls = 0
│ │ │     -- nloop = 0
│ │ │ @@ -428,24 +428,24 @@
│ │ │  o31 : Ideal of R
    
│ │ │              
    
│ │ │                
    i32 : time p = poincare I
│ │ │  
│ │ │ -   -- registering gb 21 at 0x7fe6adf6d000
│ │ │ +   -- registering gb 21 at 0x7f963d095000
│ │ │  
│ │ │     -- [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 0.00793104s (cpu); 0.00887304s (thread); 0s (gc)
│ │ │ +   --  -- used 0.00403043s (cpu); 0.00586067s (thread); 0s (gc)
│ │ │  
│ │ │              3     6    9
│ │ │  o32 = 1 - 3T  + 3T  - T
│ │ │  
│ │ │  o32 : ZZ[T]
    
│ │ │              
    
│ │ │                
    i37 : time gens gb J;
│ │ │  
│ │ │ -   -- registering gb 22 at 0x7fe6ad908e00
│ │ │ +   -- registering gb 22 at 0x7f963d1f0e00
│ │ │  
│ │ │     -- [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 0.0839992s (cpu); 0.083356s (thread); 0s (gc)
│ │ │ +   --  -- used 0.0557631s (cpu); 0.0543421s (thread); 0s (gc)
│ │ │  
│ │ │                1      39
│ │ │  o37 : Matrix S  <-- S
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -177,66 +177,66 @@
│ │ │ │  o26 = 1 - 3T  + 3T  - T
│ │ │ │  
│ │ │ │  o26 : ZZ[T]
│ │ │ │  i27 : gbTrace = 3
│ │ │ │  
│ │ │ │  o27 = 3
│ │ │ │  i28 : time poincare I
│ │ │ │ - -- used 0.000521488s (cpu); 2.0348e-05s (thread); 0s (gc)
│ │ │ │ + -- used 0.00399979s (cpu); 0s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              3     6    9
│ │ │ │  o28 = 1 - 3T  + 3T  - T
│ │ │ │  
│ │ │ │  o28 : ZZ[T]
│ │ │ │  i29 : time gens gb I;
│ │ │ │  
│ │ │ │ -   -- registering gb 19 at 0x7fe6adf6d380
│ │ │ │ +   -- registering gb 19 at 0x7f963d095540
│ │ │ │  
│ │ │ │     -- [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 0.0234308s (cpu); 0.0247955s (thread); 0s (gc)
│ │ │ │ +   --  -- used 0.0118597s (cpu); 0.0150154s (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 0x7fe6adf6d1c0
│ │ │ │ +   -- registering gb 20 at 0x7f963d095380
│ │ │ │  
│ │ │ │     -- [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 0x7fe6adf6d000
│ │ │ │ +   -- registering gb 21 at 0x7f963d095000
│ │ │ │  
│ │ │ │     -- [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 0.00793104s (cpu); 0.00887304s (thread); 0s (gc)
│ │ │ │ +   --  -- used 0.00403043s (cpu); 0.00586067s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              3     6    9
│ │ │ │  o32 = 1 - 3T  + 3T  - T
│ │ │ │  
│ │ │ │  o32 : ZZ[T]
│ │ │ │  i33 : S = QQ[a..d, MonomialOrder => Eliminate 2]
│ │ │ │  
│ │ │ │ @@ -281,30 +281,30 @@
│ │ │ │  
│ │ │ │  o35 : ZZ[T]
│ │ │ │  i36 : gbTrace = 3
│ │ │ │  
│ │ │ │  o36 = 3
│ │ │ │  i37 : time gens gb J;
│ │ │ │  
│ │ │ │ -   -- registering gb 22 at 0x7fe6ad908e00
│ │ │ │ +   -- registering gb 22 at 0x7f963d1f0e00
│ │ │ │  
│ │ │ │     -- [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 0.0839992s (cpu); 0.083356s (thread); 0s (gc)
│ │ │ │ +   --  -- used 0.0557631s (cpu); 0.0543421s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                1      39
│ │ │ │  o37 : Matrix S  <-- S
│ │ │ │  i38 : selectInSubring(1, gens gb J)
│ │ │ │  
│ │ │ │  o38 = | 188529931266160087758259645374082357642621166724936033369975727480205
│ │ │ │        -----------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_printing_spto_spa_spfile.html
│ │ │ @@ -97,22 +97,22 @@
│ │ │  o2 : File
│ │ │              
    
│ │ │                
    i3 : fn = temporaryFileName()
│ │ │  
│ │ │ -o3 = /tmp/M2-10932-0/0
    
│ │ │ +o3 = /tmp/M2-11782-0/0
│ │ │              
    
│ │ │                
    i4 : fn << "hi there" << endl << close
│ │ │  
│ │ │ -o4 = /tmp/M2-10932-0/0
│ │ │ +o4 = /tmp/M2-11782-0/0
│ │ │  
│ │ │  o4 : File
    
│ │ │              
    
│ │ │                
    i5 : get fn
│ │ │ @@ -151,15 +151,15 @@
│ │ │  o8 : File
    
│ │ │              
    
│ │ │                
    i9 : fn << f << close
│ │ │  
│ │ │ -o9 = /tmp/M2-10932-0/0
│ │ │ +o9 = /tmp/M2-11782-0/0
│ │ │  
│ │ │  o9 : File
    
│ │ │              
    
│ │ │                
    i10 : get fn
│ │ │ @@ -169,15 +169,15 @@
│ │ │        + 1
    
│ │ │              
    
│ │ │                
    i11 : fn << toExternalString f << close
│ │ │  
│ │ │ -o11 = /tmp/M2-10932-0/0
│ │ │ +o11 = /tmp/M2-11782-0/0
│ │ │  
│ │ │  o11 : File
    
│ │ │              
    
│ │ │                
    i12 : get fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -36,18 +36,18 @@
│ │ │ │  -- ho there --
│ │ │ │  
│ │ │ │  o2 = stdio
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-10932-0/0
│ │ │ │ +o3 = /tmp/M2-11782-0/0
│ │ │ │  i4 : fn << "hi there" << endl << close
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-10932-0/0
│ │ │ │ +o4 = /tmp/M2-11782-0/0
│ │ │ │  
│ │ │ │  o4 : File
│ │ │ │  i5 : get fn
│ │ │ │  
│ │ │ │  o5 = hi there
│ │ │ │  i6 : R = QQ[x]
│ │ │ │  
│ │ │ │ @@ -66,25 +66,25 @@
│ │ │ │   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/M2-10932-0/0
│ │ │ │ +o9 = /tmp/M2-11782-0/0
│ │ │ │  
│ │ │ │  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/M2-10932-0/0
│ │ │ │ +o11 = /tmp/M2-11782-0/0
│ │ │ │  
│ │ │ │  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/_process__I__D.html
│ │ │ @@ -64,15 +64,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : processID()
│ │ │  
│ │ │ -o1 = 10191
    
│ │ │ +o1 = 10311
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -8,15 +8,15 @@
│ │ │ │      *   Usage:
│ │ │ │              processID()
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the process identifier of the current Macaulay2 process
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : processID()
│ │ │ │  
│ │ │ │ -o1 = 10191
│ │ │ │ +o1 = 10311
│ │ │ │  ********** 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.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_profile.html
│ │ │ @@ -91,35 +91,35 @@
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -131,15 +131,15 @@
│ │ │  
│ │ │  o9 : Sequence
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -173,15 +173,15 @@
│ │ │  o14 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i2 : profileSummary
│ │ │  
│ │ │  o2 = #run  %time   position                         
│ │ │ -     1     94.37   ../../m2/matrix1.m2:279:4-282:58 
│ │ │ -     1     91.83   ../../m2/matrix1.m2:281:22-281:43
│ │ │ -     1     44.29   ../../m2/matrix1.m2:193:25-193:52
│ │ │ -     1     30.52   ../../m2/matrix1.m2:114:5-156:72 
│ │ │ -     1     29.41   ../../m2/matrix1.m2:140:10-155:16
│ │ │ -     1     22      ../../m2/matrix1.m2:181:4-181:42 
│ │ │ +     1     94.74   ../../m2/matrix1.m2:279:4-282:58 
│ │ │ +     1     92.3    ../../m2/matrix1.m2:281:22-281:43
│ │ │ +     1     45.8    ../../m2/matrix1.m2:193:25-193:52
│ │ │ +     1     32.19   ../../m2/matrix1.m2:114:5-156:72 
│ │ │ +     1     31      ../../m2/matrix1.m2:140:10-155:16
│ │ │ +     1     22.36   ../../m2/matrix1.m2:45:10-49:22  
│ │ │ +     1     21.94   ../../m2/matrix1.m2:181:4-181:42 
│ │ │       1     20.71   ../../m2/set.m2:127:5-127:61     
│ │ │ -     1     20.64   ../../m2/matrix1.m2:45:10-49:22  
│ │ │ -     1     3.36    ../../m2/matrix1.m2:112:5-112:29 
│ │ │ -     1     2.46    ../../m2/matrix1.m2:141:13-141:78
│ │ │ -     1     2.17    ../../m2/matrix1.m2:96:5-109:11  
│ │ │ -     1     1.50    ../../m2/matrix1.m2:281:7-281:16 
│ │ │ -     1     1.46    ../../m2/matrix1.m2:147:20-147:64
│ │ │ -     1     1.28    ../../m2/matrix1.m2:111:5-111:91 
│ │ │ -     1     1.21    ../../m2/matrix1.m2:276:4-277:73 
│ │ │ -     1     1.18    ../../m2/matrix1.m2:182:4-184:74 
│ │ │ -     1     1.09    ../../m2/matrix1.m2:98:10-98:46  
│ │ │ -     1     .74     ../../m2/modules.m2:279:4-279:52 
│ │ │ -     20    .63     ../../m2/matrix1.m2:191:14-192:67
│ │ │ -     20    .45     ../../m2/matrix1.m2:47:43-47:71  
│ │ │ -     1     .0039s  elapsed total                    
    
│ │ │ +     1     3.59    ../../m2/matrix1.m2:112:5-112:29 
│ │ │ +     1     2.38    ../../m2/matrix1.m2:141:13-141:78
│ │ │ +     1     2.15    ../../m2/matrix1.m2:96:5-109:11  
│ │ │ +     1     1.46    ../../m2/matrix1.m2:281:7-281:16 
│ │ │ +     1     1.38    ../../m2/matrix1.m2:276:4-277:73 
│ │ │ +     1     1.37    ../../m2/matrix1.m2:147:20-147:64
│ │ │ +     1     1.2     ../../m2/matrix1.m2:111:5-111:91 
│ │ │ +     1     1.11    ../../m2/matrix1.m2:98:10-98:46  
│ │ │ +     1     1.04    ../../m2/matrix1.m2:182:4-184:74 
│ │ │ +     20    .95     ../../m2/matrix1.m2:191:14-192:67
│ │ │ +     20    .76     ../../m2/matrix1.m2:47:43-47:71  
│ │ │ +     1     .63     ../../m2/modules.m2:279:4-279:52 
│ │ │ +     1     .0038s  elapsed total                    
│ │ │              
    
│ │ │                
    i3 : coverageSummary
│ │ │  
│ │ │  o3 = covered lines:
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,35 +25,35 @@
│ │ │ │                4       5
│ │ │ │  o1 : Matrix ZZ  <-- ZZ
│ │ │ │  Afterwards, running profileSummary and coverageSummary produces easy to read
│ │ │ │  tables summarizing the accumulated data so far in different ways.
│ │ │ │  i2 : profileSummary
│ │ │ │  
│ │ │ │  o2 = #run  %time   position
│ │ │ │ -     1     94.37   ../../m2/matrix1.m2:279:4-282:58
│ │ │ │ -     1     91.83   ../../m2/matrix1.m2:281:22-281:43
│ │ │ │ -     1     44.29   ../../m2/matrix1.m2:193:25-193:52
│ │ │ │ -     1     30.52   ../../m2/matrix1.m2:114:5-156:72
│ │ │ │ -     1     29.41   ../../m2/matrix1.m2:140:10-155:16
│ │ │ │ -     1     22      ../../m2/matrix1.m2:181:4-181:42
│ │ │ │ +     1     94.74   ../../m2/matrix1.m2:279:4-282:58
│ │ │ │ +     1     92.3    ../../m2/matrix1.m2:281:22-281:43
│ │ │ │ +     1     45.8    ../../m2/matrix1.m2:193:25-193:52
│ │ │ │ +     1     32.19   ../../m2/matrix1.m2:114:5-156:72
│ │ │ │ +     1     31      ../../m2/matrix1.m2:140:10-155:16
│ │ │ │ +     1     22.36   ../../m2/matrix1.m2:45:10-49:22
│ │ │ │ +     1     21.94   ../../m2/matrix1.m2:181:4-181:42
│ │ │ │       1     20.71   ../../m2/set.m2:127:5-127:61
│ │ │ │ -     1     20.64   ../../m2/matrix1.m2:45:10-49:22
│ │ │ │ -     1     3.36    ../../m2/matrix1.m2:112:5-112:29
│ │ │ │ -     1     2.46    ../../m2/matrix1.m2:141:13-141:78
│ │ │ │ -     1     2.17    ../../m2/matrix1.m2:96:5-109:11
│ │ │ │ -     1     1.50    ../../m2/matrix1.m2:281:7-281:16
│ │ │ │ -     1     1.46    ../../m2/matrix1.m2:147:20-147:64
│ │ │ │ -     1     1.28    ../../m2/matrix1.m2:111:5-111:91
│ │ │ │ -     1     1.21    ../../m2/matrix1.m2:276:4-277:73
│ │ │ │ -     1     1.18    ../../m2/matrix1.m2:182:4-184:74
│ │ │ │ -     1     1.09    ../../m2/matrix1.m2:98:10-98:46
│ │ │ │ -     1     .74     ../../m2/modules.m2:279:4-279:52
│ │ │ │ -     20    .63     ../../m2/matrix1.m2:191:14-192:67
│ │ │ │ -     20    .45     ../../m2/matrix1.m2:47:43-47:71
│ │ │ │ -     1     .0039s  elapsed total
│ │ │ │ +     1     3.59    ../../m2/matrix1.m2:112:5-112:29
│ │ │ │ +     1     2.38    ../../m2/matrix1.m2:141:13-141:78
│ │ │ │ +     1     2.15    ../../m2/matrix1.m2:96:5-109:11
│ │ │ │ +     1     1.46    ../../m2/matrix1.m2:281:7-281:16
│ │ │ │ +     1     1.38    ../../m2/matrix1.m2:276:4-277:73
│ │ │ │ +     1     1.37    ../../m2/matrix1.m2:147:20-147:64
│ │ │ │ +     1     1.2     ../../m2/matrix1.m2:111:5-111:91
│ │ │ │ +     1     1.11    ../../m2/matrix1.m2:98:10-98:46
│ │ │ │ +     1     1.04    ../../m2/matrix1.m2:182:4-184:74
│ │ │ │ +     20    .95     ../../m2/matrix1.m2:191:14-192:67
│ │ │ │ +     20    .76     ../../m2/matrix1.m2:47:43-47:71
│ │ │ │ +     1     .63     ../../m2/modules.m2:279:4-279:52
│ │ │ │ +     1     .0038s  elapsed total
│ │ │ │  i3 : coverageSummary
│ │ │ │  
│ │ │ │  o3 = covered lines:
│ │ │ │       ../../m2/lists.m2:145:24-145:32
│ │ │ │       ../../m2/lists.m2:145:34-145:58
│ │ │ │       ../../m2/matrix.m2:12:5-12:35
│ │ │ │       ../../m2/matrix.m2:13:5-13:46
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_random__K__Rational__Point.html
│ │ │ @@ -99,15 +99,15 @@
│ │ │  
│ │ │  o5 : Sequence
    
│ │ │              
    
│ │ │                
    i6 : time randomKRationalPoint(I)
│ │ │ - -- used 0.177655s (cpu); 0.142413s (thread); 0s (gc)
│ │ │ + -- used 0.255888s (cpu); 0.124419s (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 0.710725s (cpu); 0.37679s (thread); 0s (gc)
│ │ │ + -- used 0.801597s (cpu); 0.319583s (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
    
│ │ │              
    
│ │ │                
    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 4.02482s (cpu); 2.02411s (thread); 0s (gc)
│ │ │ + -- used 3.99909s (cpu); 2.04708s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 58
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │  o4 : Ideal of R
│ │ │ │  i5 : codim I, degree I
│ │ │ │  
│ │ │ │  o5 = (2, 10)
│ │ │ │  
│ │ │ │  o5 : Sequence
│ │ │ │  i6 : time randomKRationalPoint(I)
│ │ │ │ - -- used 0.177655s (cpu); 0.142413s (thread); 0s (gc)
│ │ │ │ + -- used 0.255888s (cpu); 0.124419s (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}));
│ │ │ │ @@ -45,15 +45,15 @@
│ │ │ │  o8 : Ideal of R
│ │ │ │  i9 : codim I, degree I
│ │ │ │  
│ │ │ │  o9 = (3, 10)
│ │ │ │  
│ │ │ │  o9 : Sequence
│ │ │ │  i10 : time randomKRationalPoint(I)
│ │ │ │ - -- used 0.710725s (cpu); 0.37679s (thread); 0s (gc)
│ │ │ │ + -- used 0.801597s (cpu); 0.319583s (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:
│ │ │ │ @@ -69,15 +69,15 @@
│ │ │ │  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 4.02482s (cpu); 2.02411s (thread); 0s (gc)
│ │ │ │ + -- used 3.99909s (cpu); 2.04708s (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
│ │ │ @@ -68,38 +68,38 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11565-0/0
    
│ │ │ +o1 = /tmp/M2-13075-0/0
│ │ │              
    
│ │ │                
    i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-11565-0/0
    
│ │ │ +o2 = /tmp/M2-13075-0/0
│ │ │              
    
│ │ │                
    i3 : (fn = dir | "/" | "foo") << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11565-0/0/foo
│ │ │ +o3 = /tmp/M2-13075-0/0/foo
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : readDirectory dir
│ │ │  
│ │ │ -o4 = {., .., foo}
│ │ │ +o4 = {.., ., foo}
│ │ │  
│ │ │  o4 : List
    
│ │ │              
    
│ │ │                
    i5 : removeFile fn
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,26 +10,26 @@
│ │ │ │      * 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/M2-11565-0/0
│ │ │ │ +o1 = /tmp/M2-13075-0/0
│ │ │ │  i2 : makeDirectory dir
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11565-0/0
│ │ │ │ +o2 = /tmp/M2-13075-0/0
│ │ │ │  i3 : (fn = dir | "/" | "foo") << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11565-0/0/foo
│ │ │ │ +o3 = /tmp/M2-13075-0/0/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
│ │ │ @@ -52,22 +52,22 @@
│ │ │        
    
│ │ │  Sometimes a file will contain a single expression whose value you wish to have access to.  For example, it might be a polynomial produced by another program.  The function get can be used to obtain the entire contents of a file as a single string.  We illustrate this here with a file whose name is expression.        
    
│ │ │  First we create the file by writing the desired text to it.        
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11107-0/0
    
│ │ │ +o1 = /tmp/M2-12137-0/0
│ │ │              
    
│ │ │                
    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/M2-11107-0/0
│ │ │ +o2 = /tmp/M2-12137-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │  Now we get the contents of the file, as a single string.        
│ │ │            
│ │ │ @@ -116,15 +116,15 @@
│ │ │  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/M2-11107-0/0
│ │ │ +o7 = /tmp/M2-12137-0/0
│ │ │  
│ │ │  o7 : File
    
│ │ │              
    
│ │ │  Now verify that it contains the desired text with get.        
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -8,20 +8,20 @@
│ │ │ │  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/M2-11107-0/0
│ │ │ │ +o1 = /tmp/M2-12137-0/0
│ │ │ │  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/M2-11107-0/0
│ │ │ │ +o2 = /tmp/M2-12137-0/0
│ │ │ │  
│ │ │ │  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
│ │ │ │ @@ -51,15 +51,15 @@
│ │ │ │  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/M2-11107-0/0
│ │ │ │ +o7 = /tmp/M2-12137-0/0
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -68,15 +68,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,15 +11,15 @@
│ │ │ │            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, the resolved path to a symbolic link, or null if the file
│ │ │ │              was not a symbolic link.
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : p = temporaryFileName ()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11806-0/0
│ │ │ │ +o1 = /tmp/M2-13556-0/0
│ │ │ │  i2 : symlinkFile ("foo", p)
│ │ │ │  i3 : readlink p
│ │ │ │  
│ │ │ │  o3 = foo
│ │ │ │  i4 : removeFile p
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_a_l_p_a_t_h -- convert a filename to one passing through no symbolic links
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_realpath.html
│ │ │ @@ -68,57 +68,57 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    
│ │ │                
    i1 : p = temporaryFileName ()
│ │ │  
│ │ │ -o1 = /tmp/M2-11806-0/0
    
│ │ │ +o1 = /tmp/M2-13556-0/0
│ │ │              
    
│ │ │                
    i2 : symlinkFile ("foo", p)
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -130,15 +130,15 @@
│ │ │          
    
│ │ │                
    i1 : realpath "."
│ │ │  
│ │ │ -o1 = /tmp/M2-10191-0/86-rundir/
    
│ │ │ +o1 = /tmp/M2-10311-0/86-rundir/
│ │ │              
    
│ │ │                
    i2 : p = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-11825-0/0
    
│ │ │ +o2 = /tmp/M2-13595-0/0
│ │ │              
    
│ │ │                
    i3 : q = temporaryFileName()
│ │ │  
│ │ │ -o3 = /tmp/M2-11825-0/1
    
│ │ │ +o3 = /tmp/M2-13595-0/1
│ │ │              
    
│ │ │                
    i4 : symlinkFile(p,q)
    
│ │ │              
    
│ │ │                
    i5 : p << close
│ │ │  
│ │ │ -o5 = /tmp/M2-11825-0/0
│ │ │ +o5 = /tmp/M2-13595-0/0
│ │ │  
│ │ │  o5 : File
    
│ │ │              
    
│ │ │                
    i6 : readlink q
│ │ │  
│ │ │ -o6 = /tmp/M2-11825-0/0
    
│ │ │ +o6 = /tmp/M2-13595-0/0
│ │ │              
    
│ │ │                
    i7 : realpath q
│ │ │  
│ │ │ -o7 = /tmp/M2-11825-0/0
    
│ │ │ +o7 = /tmp/M2-13595-0/0
│ │ │              
    
│ │ │                
    i8 : removeFile p
    
│ │ │              
    
│ │ │          
    The empty string is interpreted as a reference to the current directory.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i10 : realpath ""
│ │ │  
│ │ │ -o10 = /tmp/M2-10191-0/86-rundir/
    
│ │ │ +o10 = /tmp/M2-10311-0/86-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 {}
│ │ │ │ @@ -12,39 +12,39 @@
│ │ │ │            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/M2-10191-0/86-rundir/
│ │ │ │ +o1 = /tmp/M2-10311-0/86-rundir/
│ │ │ │  i2 : p = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11825-0/0
│ │ │ │ +o2 = /tmp/M2-13595-0/0
│ │ │ │  i3 : q = temporaryFileName()
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11825-0/1
│ │ │ │ +o3 = /tmp/M2-13595-0/1
│ │ │ │  i4 : symlinkFile(p,q)
│ │ │ │  i5 : p << close
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-11825-0/0
│ │ │ │ +o5 = /tmp/M2-13595-0/0
│ │ │ │  
│ │ │ │  o5 : File
│ │ │ │  i6 : readlink q
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-11825-0/0
│ │ │ │ +o6 = /tmp/M2-13595-0/0
│ │ │ │  i7 : realpath q
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-11825-0/0
│ │ │ │ +o7 = /tmp/M2-13595-0/0
│ │ │ │  i8 : removeFile p
│ │ │ │  i9 : removeFile q
│ │ │ │  The empty string is interpreted as a reference to the current directory.
│ │ │ │  i10 : realpath ""
│ │ │ │  
│ │ │ │ -o10 = /tmp/M2-10191-0/86-rundir/
│ │ │ │ +o10 = /tmp/M2-10311-0/86-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
│ │ │ @@ -76,22 +76,22 @@
│ │ │                  		
│ │ │                
    i1 : for i from 1 to 9 do (x := 0 .. 10000 ; registerFinalizer(x, "-- finalizing sequence #"|i|" --"))
    
│ │ │              
    
│ │ │                
    i2 : collectGarbage() 
│ │ │ ---finalization: (1)[0]: -- finalizing sequence #1 --
│ │ │ ---finalization: (2)[5]: -- finalizing sequence #6 --
│ │ │ ---finalization: (3)[3]: -- finalizing sequence #4 --
│ │ │ ---finalization: (4)[6]: -- finalizing sequence #7 --
│ │ │ ---finalization: (5)[7]: -- finalizing sequence #8 --
│ │ │ ---finalization: (7)[1]: -- finalizing sequence #2 --
│ │ │ +--finalization: (1)[1]: -- finalizing sequence #2 --
│ │ │ +--finalization: (3)[5]: -- finalizing sequence #6 --
│ │ │ +--finalization: (4)[2]: -- finalizing sequence #3 --
│ │ │ +--finalization: (5)[3]: -- finalizing sequence #4 --
│ │ │ +--finalization: (6)[0]: -- finalizing sequence #1 --
│ │ │ +--finalization: (7)[6]: -- finalizing sequence #7 --
│ │ │  --finalization: (8)[4]: -- finalizing sequence #5 --
│ │ │ ---finalization: (6)[2]: -- finalizing sequence #3 --
    
│ │ │ +--finalization: (2)[7]: -- finalizing sequence #8 --
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    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 {}
│ │ │ │ @@ -14,22 +14,22 @@
│ │ │ │      * Consequences:
│ │ │ │            o A finalizer is registered with the garbage collector to print a
│ │ │ │              string when that object is collected as garbage
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : for i from 1 to 9 do (x := 0 .. 10000 ; registerFinalizer(x, "-
│ │ │ │  - finalizing sequence #"|i|" --"))
│ │ │ │  i2 : collectGarbage()
│ │ │ │ ---finalization: (1)[0]: -- finalizing sequence #1 --
│ │ │ │ ---finalization: (2)[5]: -- finalizing sequence #6 --
│ │ │ │ ---finalization: (3)[3]: -- finalizing sequence #4 --
│ │ │ │ ---finalization: (4)[6]: -- finalizing sequence #7 --
│ │ │ │ ---finalization: (5)[7]: -- finalizing sequence #8 --
│ │ │ │ ---finalization: (7)[1]: -- finalizing sequence #2 --
│ │ │ │ +--finalization: (1)[1]: -- finalizing sequence #2 --
│ │ │ │ +--finalization: (3)[5]: -- finalizing sequence #6 --
│ │ │ │ +--finalization: (4)[2]: -- finalizing sequence #3 --
│ │ │ │ +--finalization: (5)[3]: -- finalizing sequence #4 --
│ │ │ │ +--finalization: (6)[0]: -- finalizing sequence #1 --
│ │ │ │ +--finalization: (7)[6]: -- finalizing sequence #7 --
│ │ │ │  --finalization: (8)[4]: -- finalizing sequence #5 --
│ │ │ │ ---finalization: (6)[2]: -- finalizing sequence #3 --
│ │ │ │ +--finalization: (2)[7]: -- finalizing sequence #8 --
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  This function should mainly be used for debugging. Having a large number of
│ │ │ │  finalizers might degrade the performance of the program. Moreover, registering
│ │ │ │  two or more objects that are members of a circular chain of pointers for
│ │ │ │  finalization will result in a memory leak, with none of the objects in the
│ │ │ │  chain being freed, even if nothing else points to any of them.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_remove__Directory.html
│ │ │ @@ -71,29 +71,29 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10779-0/0
    
│ │ │ +o1 = /tmp/M2-11469-0/0
│ │ │              
    
│ │ │                
    i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-10779-0/0
    
│ │ │ +o2 = /tmp/M2-11469-0/0
│ │ │              
    
│ │ │                
    i3 : readDirectory dir
│ │ │  
│ │ │ -o3 = {., ..}
│ │ │ +o3 = {.., .}
│ │ │  
│ │ │  o3 : List
    
│ │ │              
    
│ │ │                
    i4 : removeDirectory dir
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,21 +10,21 @@
│ │ │ │      * 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/M2-10779-0/0
│ │ │ │ +o1 = /tmp/M2-11469-0/0
│ │ │ │  i2 : makeDirectory dir
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10779-0/0
│ │ │ │ +o2 = /tmp/M2-11469-0/0
│ │ │ │  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
│ │ │ @@ -65,22 +65,22 @@
│ │ │          
    Description
    
│ │ │          
    This string may be concatenated with an absolute path to get one understandable by external programs.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10283-0/0
    
│ │ │ +o1 = /tmp/M2-10473-0/0
│ │ │              
    
│ │ │                
    i2 : rootPath | fn
│ │ │  
│ │ │ -o2 = /tmp/M2-10283-0/0
    
│ │ │ +o2 = /tmp/M2-10473-0/0
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,18 +11,18 @@
│ │ │ │            o a _s_t_r_i_n_g, the path, as seen by external programs, to the root of
│ │ │ │              the file system seen by Macaulay2
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This string may be concatenated with an absolute path to get one understandable
│ │ │ │  by external programs.
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-10283-0/0
│ │ │ │ +o1 = /tmp/M2-10473-0/0
│ │ │ │  i2 : rootPath | fn
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10283-0/0
│ │ │ │ +o2 = /tmp/M2-10473-0/0
│ │ │ │  ********** 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.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_system.m2:2025:0.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_root__U__R__I.html
│ │ │ @@ -65,22 +65,22 @@
│ │ │          
      Description
      
│ │ │          
      This string may be concatenated with an absolute path to get one understandable by an external browser.
      
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
      
│ │ │                
      i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11508-0/0
      
│ │ │ +o1 = /tmp/M2-12958-0/0
│ │ │              
      
│ │ │                
      i2 : rootURI | fn
│ │ │  
│ │ │ -o2 = file:///tmp/M2-11508-0/0
      
│ │ │ +o2 = file:///tmp/M2-12958-0/0
│ │ │              
      
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,18 +11,18 @@
│ │ │ │            o a _s_t_r_i_n_g, the path, as seen by an external browser, to the root of
│ │ │ │              the file system seen by Macaulay2
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This string may be concatenated with an absolute path to get one understandable
│ │ │ │  by an external browser.
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11508-0/0
│ │ │ │ +o1 = /tmp/M2-12958-0/0
│ │ │ │  i2 : rootURI | fn
│ │ │ │  
│ │ │ │ -o2 = file:///tmp/M2-11508-0/0
│ │ │ │ +o2 = file:///tmp/M2-12958-0/0
│ │ │ │  ********** 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.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_system.m2:2041:0.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_saving_sppolynomials_spand_spmatrices_spin_spfiles.html
│ │ │ @@ -90,22 +90,22 @@
│ │ │  o4 : R-module, submodule of R
│ │ │              
    
│ │ │                
    i5 : f = temporaryFileName()
│ │ │  
│ │ │ -o5 = /tmp/M2-11356-0/0
    
│ │ │ +o5 = /tmp/M2-12646-0/0
│ │ │              
    
│ │ │                
    i6 : f << toString (p,m,M) << close
│ │ │  
│ │ │ -o6 = /tmp/M2-11356-0/0
│ │ │ +o6 = /tmp/M2-12646-0/0
│ │ │  
│ │ │  o6 : File
    
│ │ │              
    
│ │ │                
    i7 : get f
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,18 +28,18 @@
│ │ │ │  
│ │ │ │  o4 = image | x2 x2-y2 xyz7 |
│ │ │ │  
│ │ │ │                               1
│ │ │ │  o4 : R-module, submodule of R
│ │ │ │  i5 : f = temporaryFileName()
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-11356-0/0
│ │ │ │ +o5 = /tmp/M2-12646-0/0
│ │ │ │  i6 : f << toString (p,m,M) << close
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-11356-0/0
│ │ │ │ +o6 = /tmp/M2-12646-0/0
│ │ │ │  
│ │ │ │  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/_serial__Number.html
│ │ │ @@ -68,22 +68,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : serialNumber asdf
│ │ │  
│ │ │ -o1 = 1426273
    
│ │ │ +o1 = 1526273
│ │ │              
    
│ │ │                
    i2 : serialNumber foo
│ │ │  
│ │ │ -o2 = 1426275
    
│ │ │ +o2 = 1526275
│ │ │              
    
│ │ │                
    i3 : serialNumber ZZ
│ │ │  
│ │ │  o3 = 1000050
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,18 +10,18 @@
│ │ │ │      * Inputs:
│ │ │ │            o x
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the serial number of x
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : serialNumber asdf
│ │ │ │  
│ │ │ │ -o1 = 1426273
│ │ │ │ +o1 = 1526273
│ │ │ │  i2 : serialNumber foo
│ │ │ │  
│ │ │ │ -o2 = 1426275
│ │ │ │ +o2 = 1526275
│ │ │ │  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.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_solve.html
│ │ │ @@ -366,21 +366,21 @@
│ │ │                  		
│ │ │                
    i28 : B = mutableMatrix(CC_53, N, 2); fillMatrix B;
    
│ │ │              
    
│ │ │                
    i30 : time X = solve(A,B);
│ │ │ - -- used 0.000225452s (cpu); 0.000217157s (thread); 0s (gc)
    
│ │ │ + -- used 0.000210794s (cpu); 0.00020165s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i31 : time X = solve(A,B, MaximalRank=>true);
│ │ │ - -- used 0.000165831s (cpu); 0.000165941s (thread); 0s (gc)
    
│ │ │ + -- used 0.000105086s (cpu); 0.000105046s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i32 : norm(A*X-B)
│ │ │  
│ │ │  o32 = 5.111850690840453e-15
│ │ │ @@ -411,21 +411,21 @@
│ │ │              
    		
│ │ │                
    i36 : B = mutableMatrix(CC_100, N, 2); fillMatrix B;
    
│ │ │              
    
│ │ │                
    i38 : time X = solve(A,B);
│ │ │ - -- used 0.494046s (cpu); 0.307142s (thread); 0s (gc)
    
│ │ │ + -- used 0.141612s (cpu); 0.141359s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i39 : time X = solve(A,B, MaximalRank=>true);
│ │ │ - -- used 0.241089s (cpu); 0.24109s (thread); 0s (gc)
    
│ │ │ + -- used 0.14493s (cpu); 0.14494s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i40 : norm(A*X-B)
│ │ │  
│ │ │  o40 = 1.491578274689709814082355885932e-28
│ │ │ ├── html2text {}
│ │ │ │ @@ -192,33 +192,33 @@
│ │ │ │  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 0.000225452s (cpu); 0.000217157s (thread); 0s (gc)
│ │ │ │ + -- used 0.000210794s (cpu); 0.00020165s (thread); 0s (gc)
│ │ │ │  i31 : time X = solve(A,B, MaximalRank=>true);
│ │ │ │ - -- used 0.000165831s (cpu); 0.000165941s (thread); 0s (gc)
│ │ │ │ + -- used 0.000105086s (cpu); 0.000105046s (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 0.494046s (cpu); 0.307142s (thread); 0s (gc)
│ │ │ │ + -- used 0.141612s (cpu); 0.141359s (thread); 0s (gc)
│ │ │ │  i39 : time X = solve(A,B, MaximalRank=>true);
│ │ │ │ - -- used 0.241089s (cpu); 0.24109s (thread); 0s (gc)
│ │ │ │ + -- used 0.14493s (cpu); 0.14494s (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/_symlink__Directory_lp__String_cm__String_rp.html
│ │ │ @@ -80,93 +80,93 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : src = temporaryFileName() | "/"
│ │ │  
│ │ │ -o1 = /tmp/M2-11147-0/0/
    
│ │ │ +o1 = /tmp/M2-12217-0/0/
│ │ │              
    
│ │ │                
    i2 : dst = temporaryFileName() | "/"
│ │ │  
│ │ │ -o2 = /tmp/M2-11147-0/1/
    
│ │ │ +o2 = /tmp/M2-12217-0/1/
│ │ │              
    
│ │ │                
    i3 : makeDirectory (src|"a/")
│ │ │  
│ │ │ -o3 = /tmp/M2-11147-0/0/a/
    
│ │ │ +o3 = /tmp/M2-12217-0/0/a/
│ │ │              
    
│ │ │                
    i4 : makeDirectory (src|"b/")
│ │ │  
│ │ │ -o4 = /tmp/M2-11147-0/0/b/
    
│ │ │ +o4 = /tmp/M2-12217-0/0/b/
│ │ │              
    
│ │ │                
    i5 : makeDirectory (src|"b/c/")
│ │ │  
│ │ │ -o5 = /tmp/M2-11147-0/0/b/c/
    
│ │ │ +o5 = /tmp/M2-12217-0/0/b/c/
│ │ │              
    
│ │ │                
    i6 : src|"a/f" << "hi there" << close
│ │ │  
│ │ │ -o6 = /tmp/M2-11147-0/0/a/f
│ │ │ +o6 = /tmp/M2-12217-0/0/a/f
│ │ │  
│ │ │  o6 : File
    
│ │ │              
    
│ │ │                
    i7 : src|"a/g" << "hi there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11147-0/0/a/g
│ │ │ +o7 = /tmp/M2-12217-0/0/a/g
│ │ │  
│ │ │  o7 : File
    
│ │ │              
    
│ │ │                
    i8 : src|"b/c/g" << "ho there" << close
│ │ │  
│ │ │ -o8 = /tmp/M2-11147-0/0/b/c/g
│ │ │ +o8 = /tmp/M2-12217-0/0/b/c/g
│ │ │  
│ │ │  o8 : File
    
│ │ │              
    
│ │ │                
    i9 : symlinkDirectory(src,dst,Verbose=>true)
│ │ │ ---symlinking: ../../../0/b/c/g -> /tmp/M2-11147-0/1/b/c/g
│ │ │ ---symlinking: ../../0/a/g -> /tmp/M2-11147-0/1/a/g
│ │ │ ---symlinking: ../../0/a/f -> /tmp/M2-11147-0/1/a/f
    
│ │ │ +--symlinking: ../../0/a/g -> /tmp/M2-12217-0/1/a/g
│ │ │ +--symlinking: ../../0/a/f -> /tmp/M2-12217-0/1/a/f
│ │ │ +--symlinking: ../../../0/b/c/g -> /tmp/M2-12217-0/1/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/M2-11147-0/1/b/c/g
│ │ │ ---unsymlinking: ../../0/a/g -> /tmp/M2-11147-0/1/a/g
│ │ │ ---unsymlinking: ../../0/a/f -> /tmp/M2-11147-0/1/a/f
    
│ │ │ +--unsymlinking: ../../0/a/g -> /tmp/M2-12217-0/1/a/g
│ │ │ +--unsymlinking: ../../0/a/f -> /tmp/M2-12217-0/1/a/f
│ │ │ +--unsymlinking: ../../../0/b/c/g -> /tmp/M2-12217-0/1/b/c/g
│ │ │              
    
│ │ │  Now we remove the files and directories we created.        
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i12 : rm = d -> if isDirectory d then removeDirectory d else removeFile d
│ │ │ ├── html2text {}
│ │ │ │ @@ -30,53 +30,53 @@
│ │ │ │            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/M2-11147-0/0/
│ │ │ │ +o1 = /tmp/M2-12217-0/0/
│ │ │ │  i2 : dst = temporaryFileName() | "/"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11147-0/1/
│ │ │ │ +o2 = /tmp/M2-12217-0/1/
│ │ │ │  i3 : makeDirectory (src|"a/")
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11147-0/0/a/
│ │ │ │ +o3 = /tmp/M2-12217-0/0/a/
│ │ │ │  i4 : makeDirectory (src|"b/")
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-11147-0/0/b/
│ │ │ │ +o4 = /tmp/M2-12217-0/0/b/
│ │ │ │  i5 : makeDirectory (src|"b/c/")
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-11147-0/0/b/c/
│ │ │ │ +o5 = /tmp/M2-12217-0/0/b/c/
│ │ │ │  i6 : src|"a/f" << "hi there" << close
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-11147-0/0/a/f
│ │ │ │ +o6 = /tmp/M2-12217-0/0/a/f
│ │ │ │  
│ │ │ │  o6 : File
│ │ │ │  i7 : src|"a/g" << "hi there" << close
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-11147-0/0/a/g
│ │ │ │ +o7 = /tmp/M2-12217-0/0/a/g
│ │ │ │  
│ │ │ │  o7 : File
│ │ │ │  i8 : src|"b/c/g" << "ho there" << close
│ │ │ │  
│ │ │ │ -o8 = /tmp/M2-11147-0/0/b/c/g
│ │ │ │ +o8 = /tmp/M2-12217-0/0/b/c/g
│ │ │ │  
│ │ │ │  o8 : File
│ │ │ │  i9 : symlinkDirectory(src,dst,Verbose=>true)
│ │ │ │ ---symlinking: ../../../0/b/c/g -> /tmp/M2-11147-0/1/b/c/g
│ │ │ │ ---symlinking: ../../0/a/g -> /tmp/M2-11147-0/1/a/g
│ │ │ │ ---symlinking: ../../0/a/f -> /tmp/M2-11147-0/1/a/f
│ │ │ │ +--symlinking: ../../0/a/g -> /tmp/M2-12217-0/1/a/g
│ │ │ │ +--symlinking: ../../0/a/f -> /tmp/M2-12217-0/1/a/f
│ │ │ │ +--symlinking: ../../../0/b/c/g -> /tmp/M2-12217-0/1/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/M2-11147-0/1/b/c/g
│ │ │ │ ---unsymlinking: ../../0/a/g -> /tmp/M2-11147-0/1/a/g
│ │ │ │ ---unsymlinking: ../../0/a/f -> /tmp/M2-11147-0/1/a/f
│ │ │ │ +--unsymlinking: ../../0/a/g -> /tmp/M2-12217-0/1/a/g
│ │ │ │ +--unsymlinking: ../../0/a/f -> /tmp/M2-12217-0/1/a/f
│ │ │ │ +--unsymlinking: ../../../0/b/c/g -> /tmp/M2-12217-0/1/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
│ │ │ @@ -72,15 +72,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,15 +12,15 @@
│ │ │ │            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/M2-11204-0/0
│ │ │ │ +o1 = /tmp/M2-12334-0/0
│ │ │ │  i2 : symlinkFile("qwert", fn)
│ │ │ │  i3 : fileExists fn
│ │ │ │  
│ │ │ │  o3 = false
│ │ │ │  i4 : readlink fn
│ │ │ │  
│ │ │ │  o4 = qwert
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_temporary__File__Name.html
│ │ │ @@ -64,22 +64,22 @@
│ │ │        
    
│ │ │          
    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 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11204-0/0
    
│ │ │ +o1 = /tmp/M2-12334-0/0
│ │ │              
    
│ │ │                
    i2 : symlinkFile("qwert", fn)
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : temporaryFileName () | ".tex"
│ │ │  
│ │ │ -o1 = /tmp/M2-12169-0/0.tex
    
│ │ │ +o1 = /tmp/M2-14309-0/0.tex
│ │ │              
    
│ │ │                
    i2 : temporaryFileName () | ".html"
│ │ │  
│ │ │ -o2 = /tmp/M2-12169-0/1.html
    
│ │ │ +o2 = /tmp/M2-14309-0/1.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 {}
│ │ │ │ @@ -11,18 +11,18 @@
│ │ │ │            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/M2-12169-0/0.tex
│ │ │ │ +o1 = /tmp/M2-14309-0/0.tex
│ │ │ │  i2 : temporaryFileName () | ".html"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12169-0/1.html
│ │ │ │ +o2 = /tmp/M2-14309-0/1.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
│ │ │ @@ -59,15 +59,15 @@
│ │ │        
│ │ │        
    
│ │ │          
    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 1.4376e-05s (cpu); 7.053e-06s (thread); 0s (gc)
│ │ │ + -- used 1.7726e-05s (cpu); 5.692e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = 205891132094649
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -7,15 +7,15 @@
│ │ │ │      *   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 1.4376e-05s (cpu); 7.053e-06s (thread); 0s (gc)
│ │ │ │ + -- used 1.7726e-05s (cpu); 5.692e-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
│ │ │ @@ -54,24 +54,24 @@
│ │ │  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
│ │ │ -     -- .000015599 seconds
│ │ │ +     -- .000020481 seconds
│ │ │  
│ │ │  o1 : Time
    
│ │ │              
    
│ │ │                
    i2 : peek oo
│ │ │  
│ │ │ -o2 = Time{.000015599, 205891132094649}
    
│ │ │ +o2 = Time{.000020481, 205891132094649}
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,20 +10,20 @@
│ │ │ │  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
│ │ │ │ -     -- .000015599 seconds
│ │ │ │ +     -- .000020481 seconds
│ │ │ │  
│ │ │ │  o1 : Time
│ │ │ │  i2 : peek oo
│ │ │ │  
│ │ │ │ -o2 = Time{.000015599, 205891132094649}
│ │ │ │ +o2 = Time{.000020481, 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
│ │ │ @@ -103,15 +103,15 @@
│ │ │                 "memtailor version" => 1.0
│ │ │                 "mpfi version" => 1.5.4
│ │ │                 "mpfr version" => 4.2.2
│ │ │                 "mpsolve version" => 3.2.2
│ │ │                 "mysql version" => not present
│ │ │                 "normaliz version" => 3.10.5
│ │ │                 "ntl version" => 11.5.1
│ │ │ -               "operating system release" => 6.12.48+deb13-amd64
│ │ │ +               "operating system release" => 6.12.57+deb13-cloud-amd64
│ │ │                 "operating system" => Linux
│ │ │                 "packages" => Style FirstPackage Macaulay2Doc Parsing Classic Browse Benchmark Text SimpleDoc PackageTemplate Saturation PrimaryDecomposition FourierMotzkin Dmodules WeylAlgebras HolonomicSystems BernsteinSato ConnectionMatrices 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 WeilDivisors EllipticCurves HighestWeights MinimalPrimes Bertini TorAlgebra Permanents BinomialEdgeIdeals TateOnProducts LatticePolytopes FiniteFittingIdeals HigherCIOperators LieAlgebraRepresentations 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 TestIdeals FrobeniusThresholds NonPrincipalTestIdeals Seminormalization AlgebraicSplines TriangularSets Chordal Tropical SymbolicPowers Complexes OldChainComplexes 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 ExteriorExtensions Oscillators IncidenceCorrespondenceCohomology ToricHigherDirectImages Brackets IntegerProgramming GameTheory AllMarkovBases Tableaux CpMackeyFunctors JSONRPC MatrixFactorizations PathSignatures
│ │ │                 "pointer size" => 8
│ │ │                 "python version" => 3.13.9
│ │ │                 "readline version" => 8.3
│ │ │                 "scscp version" => not present
│ │ │                 "tbb version" => 2022.1
│ │ │ ├── html2text {}
│ │ │ │ @@ -64,15 +64,15 @@
│ │ │ │                 "memtailor version" => 1.0
│ │ │ │                 "mpfi version" => 1.5.4
│ │ │ │                 "mpfr version" => 4.2.2
│ │ │ │                 "mpsolve version" => 3.2.2
│ │ │ │                 "mysql version" => not present
│ │ │ │                 "normaliz version" => 3.10.5
│ │ │ │                 "ntl version" => 11.5.1
│ │ │ │ -               "operating system release" => 6.12.48+deb13-amd64
│ │ │ │ +               "operating system release" => 6.12.57+deb13-cloud-amd64
│ │ │ │                 "operating system" => Linux
│ │ │ │                 "packages" => Style FirstPackage Macaulay2Doc Parsing Classic
│ │ │ │  Browse Benchmark Text SimpleDoc PackageTemplate Saturation PrimaryDecomposition
│ │ │ │  FourierMotzkin Dmodules WeylAlgebras HolonomicSystems BernsteinSato
│ │ │ │  ConnectionMatrices Depth Elimination GenericInitialIdeal IntegralClosure
│ │ │ │  HyperplaneArrangements LexIdeals Markov NoetherNormalization Points ReesAlgebra
│ │ │ │  Regularity SchurRings SymmetricPolynomials SchurFunctors SimplicialComplexes
│ │ ├── ./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 3.41653s (cpu); 1.67417s (thread); 0s (gc)
│ │ │ + -- used 4.10494s (cpu); 1.76791s (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
│ │ │ @@ -161,15 +161,15 @@
│ │ │          
      
│ │ │            
      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 3.41653s (cpu); 1.67417s (thread); 0s (gc)
│ │ │ + -- used 4.10494s (cpu); 1.76791s (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 3.41653s (cpu); 1.67417s (thread); 0s (gc)
│ │ │ │ + -- used 4.10494s (cpu); 1.76791s (thread); 0s (gc)
│ │ │ │  
│ │ │ │       +-------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ ├── ./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 0.077181s (cpu); 0.0286174s (thread); 0s (gc)
│ │ │ + -- used 0.112917s (cpu); 0.0361625s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = 1
│ │ │  
│ │ │  i24 : time regularity comodule schubertDeterminantalIdeal B
│ │ │ - -- used 0.0152261s (cpu); 0.0152272s (thread); 0s (gc)
│ │ │ + -- used 0.0187435s (cpu); 0.0187515s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = 1
│ │ │  
│ │ │  i25 :
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/___Investigating_spmatrix_sp__Schubert_spvarieties.out
│ │ │ @@ -178,17 +178,17 @@
│ │ │        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
│ │ │  
│ │ │  o15 : Ideal of QQ[z   ..z   ]
│ │ │                     1,1   5,5
│ │ │  
│ │ │  i16 : time schubertRegularity p
│ │ │ - -- used 0.000280436s (cpu); 0.000276258s (thread); 0s (gc)
│ │ │ + -- used 0.000301348s (cpu); 0.000292271s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 5
│ │ │  
│ │ │  i17 : time regularity comodule I
│ │ │ - -- used 0.015625s (cpu); 0.0156272s (thread); 0s (gc)
│ │ │ + -- used 0.0173602s (cpu); 0.0173665s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = 5
│ │ │  
│ │ │  i18 :
│ │ ├── ./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 0.00429046s (cpu); 0.00428737s (thread); 0s (gc)
│ │ │ + -- used 0.00581035s (cpu); 0.00580945s (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 0.00217528s (cpu); 0.00217604s (thread); 0s (gc)
│ │ │ + -- used 0.00271794s (cpu); 0.00271925s (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
│ │ │ @@ -383,23 +383,23 @@
│ │ │          
      
│ │ │            
      Additionally, this package facilitates investigating homological invariants of ASM ideals such as regularity (schubertRegularity) and codimension (schubertCodim). 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 0.077181s (cpu); 0.0286174s (thread); 0s (gc)
│ │ │ + -- used 0.112917s (cpu); 0.0361625s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = 1
      
│ │ │              
      
│ │ │                
      i24 : time regularity comodule schubertDeterminantalIdeal B
│ │ │ - -- used 0.0152261s (cpu); 0.0152272s (thread); 0s (gc)
│ │ │ + -- used 0.0187435s (cpu); 0.0187515s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = 1
      
│ │ │              
      
│ │ │          
      
│ │ │            
      Functions for investigating ASM varieties
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -244,19 +244,19 @@
│ │ │ │  ASM ideals such as regularity (_s_c_h_u_b_e_r_t_R_e_g_u_l_a_r_i_t_y) and codimension
│ │ │ │  (_s_c_h_u_b_e_r_t_C_o_d_i_m). 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 0.077181s (cpu); 0.0286174s (thread); 0s (gc)
│ │ │ │ + -- used 0.112917s (cpu); 0.0361625s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = 1
│ │ │ │  i24 : time regularity comodule schubertDeterminantalIdeal B
│ │ │ │ - -- used 0.0152261s (cpu); 0.0152272s (thread); 0s (gc)
│ │ │ │ + -- used 0.0187435s (cpu); 0.0187515s (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
│ │ │ @@ -315,23 +315,23 @@
│ │ │          
      
│ │ │            
      Finally, this package contains functions for investigating homological invariants of matrix Schubert varieties efficiently through combinatorial algorithms produced in [PSW24] via schubertRegularityschubertCodim.
      
│ │ │          
      
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
      
│ │ │                
      i16 : time schubertRegularity p
│ │ │ - -- used 0.000280436s (cpu); 0.000276258s (thread); 0s (gc)
│ │ │ + -- used 0.000301348s (cpu); 0.000292271s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 5
      
│ │ │              
      
│ │ │                
      i17 : time regularity comodule I
│ │ │ - -- used 0.015625s (cpu); 0.0156272s (thread); 0s (gc)
│ │ │ + -- used 0.0173602s (cpu); 0.0173665s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = 5
      
│ │ │              
      
│ │ │          
      
│ │ │            
      Functions for investigating matrix Schubert varieties
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -545,19 +545,19 @@
│ │ │ │  
│ │ │ │  o15 : 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 [PSW24] via _s_c_h_u_b_e_r_t_R_e_g_u_l_a_r_i_t_y_s_c_h_u_b_e_r_t_C_o_d_i_m.
│ │ │ │  i16 : time schubertRegularity p
│ │ │ │ - -- used 0.000280436s (cpu); 0.000276258s (thread); 0s (gc)
│ │ │ │ + -- used 0.000301348s (cpu); 0.000292271s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = 5
│ │ │ │  i17 : time regularity comodule I
│ │ │ │ - -- used 0.015625s (cpu); 0.0156272s (thread); 0s (gc)
│ │ │ │ + -- used 0.0173602s (cpu); 0.0173665s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o17 = 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 the
│ │ │ │        corresponding ASM or matrix Schubert variety
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/html/_grothendieck__Polynomial.html
│ │ │ @@ -80,28 +80,28 @@
│ │ │  
│ │ │  o1 : List
      
│ │ │              
      
│ │ │                
      i2 : time grothendieckPolynomial w
│ │ │ - -- used 0.00429046s (cpu); 0.00428737s (thread); 0s (gc)
│ │ │ + -- used 0.00581035s (cpu); 0.00580945s (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 0.00217528s (cpu); 0.00217604s (thread); 0s (gc)
│ │ │ + -- used 0.00271794s (cpu); 0.00271925s (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 0.00429046s (cpu); 0.00428737s (thread); 0s (gc)
│ │ │ │ + -- used 0.00581035s (cpu); 0.00580945s (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 0.00217528s (cpu); 0.00217604s (thread); 0s (gc)
│ │ │ │ + -- used 0.00271794s (cpu); 0.00271925s (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/example-output/___Matroid.out
│ │ │ @@ -51,20 +51,20 @@
│ │ │  i9 : keys R10.cache
│ │ │  
│ │ │  o9 = {groundSet, rankFunction, storedRepresentation}
│ │ │  
│ │ │  o9 : List
│ │ │  
│ │ │  i10 : time isWellDefined R10
│ │ │ - -- used 0.077174s (cpu); 0.07565s (thread); 0s (gc)
│ │ │ + -- used 0.060015s (cpu); 0.0581348s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = true
│ │ │  
│ │ │  i11 : time fVector R10
│ │ │ - -- used 0.177179s (cpu); 0.083325s (thread); 0s (gc)
│ │ │ + -- used 0.218879s (cpu); 0.0899998s (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 0.000406302s (cpu); 0.000208892s (thread); 0s (gc)
│ │ │ + -- used 0.000375344s (cpu); 0.000198276s (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);
│ │ │ - -- .111077s elapsed
│ │ │ + -- .0687175s 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 0.121357s (cpu); 0.0623868s (thread); 0s (gc)
│ │ │ + -- used 0.330752s (cpu); 0.0937671s (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 1.76467s (cpu); 1.25038s (thread); 0s (gc)
│ │ │ + -- used 2.07944s (cpu); 1.36877s (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 0.0159915s (cpu); 0.0134391s (thread); 0s (gc)
│ │ │ + -- used 0.0120011s (cpu); 0.0146825s (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 4.44315s (cpu); 2.84198s (thread); 0s (gc)
│ │ │ + -- used 4.84192s (cpu); 2.77699s (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 0.000917851s (cpu); 0.000605195s (thread); 0s (gc)
│ │ │ + -- used 0.00152239s (cpu); 0.000513019s (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
│ │ │ - -- .0133096s elapsed
│ │ │ + -- .014264s elapsed
│ │ │  
│ │ │  o6 = HashTable{0 => 1 }
│ │ │                 1 => 6
│ │ │                 2 => 15
│ │ │                 3 => 20
│ │ │                 4 => 1
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/___Matroid.html
│ │ │ @@ -148,23 +148,23 @@
│ │ │  
│ │ │  o9 : List
│ │ │              
      
│ │ │                
      i10 : time isWellDefined R10
│ │ │ - -- used 0.077174s (cpu); 0.07565s (thread); 0s (gc)
│ │ │ + -- used 0.060015s (cpu); 0.0581348s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = true
      
│ │ │              
      
│ │ │                
      i11 : time fVector R10
│ │ │ - -- used 0.177179s (cpu); 0.083325s (thread); 0s (gc)
│ │ │ + -- used 0.218879s (cpu); 0.0899998s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = HashTable{0 => 1 }
│ │ │                  1 => 10
│ │ │                  2 => 45
│ │ │                  3 => 75
│ │ │                  4 => 30
│ │ │                  5 => 1
│ │ │ @@ -182,15 +182,15 @@
│ │ │  
│ │ │  o12 : List
      
│ │ │              
      
│ │ │                
      i13 : time fVector R10
│ │ │ - -- used 0.000406302s (cpu); 0.000208892s (thread); 0s (gc)
│ │ │ + -- used 0.000375344s (cpu); 0.000198276s (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 0.077174s (cpu); 0.07565s (thread); 0s (gc)
│ │ │ │ + -- used 0.060015s (cpu); 0.0581348s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = true
│ │ │ │  i11 : time fVector R10
│ │ │ │ - -- used 0.177179s (cpu); 0.083325s (thread); 0s (gc)
│ │ │ │ + -- used 0.218879s (cpu); 0.0899998s (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 0.000406302s (cpu); 0.000208892s (thread); 0s (gc)
│ │ │ │ + -- used 0.000375344s (cpu); 0.000198276s (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
│ │ │ @@ -92,15 +92,15 @@
│ │ │  
│ │ │  o2 : Matroid
      
│ │ │              
      
│ │ │                
      i3 : elapsedTime L = allMinors(V, U25);
│ │ │ - -- .111077s elapsed
      
│ │ │ + -- .0687175s elapsed
│ │ │              
      
│ │ │                
      i4 : #L
│ │ │  
│ │ │  o4 = 64
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,15 +27,15 @@
│ │ │ │  o1 : Matroid
│ │ │ │  i2 : U25 = uniformMatroid(2,5)
│ │ │ │  
│ │ │ │  o2 = a "matroid" of rank 2 on 5 elements
│ │ │ │  
│ │ │ │  o2 : Matroid
│ │ │ │  i3 : elapsedTime L = allMinors(V, U25);
│ │ │ │ - -- .111077s elapsed
│ │ │ │ + -- .0687175s 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
│ │ │ @@ -135,15 +135,15 @@
│ │ │  
│ │ │  o6 : Matroid
│ │ │              
      
│ │ │                
      i7 : time autF7 = getIsos(F7, F7);
│ │ │ - -- used 0.121357s (cpu); 0.0623868s (thread); 0s (gc)
      
│ │ │ + -- used 0.330752s (cpu); 0.0937671s (thread); 0s (gc)
│ │ │              
      
│ │ │                
      i8 : #autF7
│ │ │  
│ │ │  o8 = 168
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -51,15 +51,15 @@
│ │ │ │  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 0.121357s (cpu); 0.0623868s (thread); 0s (gc)
│ │ │ │ + -- used 0.330752s (cpu); 0.0937671s (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
│ │ │ @@ -96,15 +96,15 @@
│ │ │  
│ │ │  o2 = false
│ │ │              
      
│ │ │                
      i3 : time hasMinor(M6, M5)
│ │ │ - -- used 1.76467s (cpu); 1.25038s (thread); 0s (gc)
│ │ │ + -- used 2.07944s (cpu); 1.36877s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
      
│ │ │              
      
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,15 +34,15 @@
│ │ │ │       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 1.76467s (cpu); 1.25038s (thread); 0s (gc)
│ │ │ │ + -- used 2.07944s (cpu); 1.36877s (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
│ │ │ @@ -118,15 +118,15 @@
│ │ │  
│ │ │  o4 : Matroid
    
│ │ │              
    
│ │ │                
    i5 : time isomorphism(M5, minorM6)
│ │ │ - -- used 0.0159915s (cpu); 0.0134391s (thread); 0s (gc)
│ │ │ + -- used 0.0120011s (cpu); 0.0146825s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = HashTable{0 => 1}
│ │ │                 1 => 0
│ │ │                 2 => 3
│ │ │                 3 => 2
│ │ │                 4 => 6
│ │ │                 5 => 5
│ │ │ @@ -164,15 +164,15 @@
│ │ │  
│ │ │  o7 : Matroid
    
│ │ │              
    
│ │ │                
    i8 : time phi = isomorphism(N,M6)
│ │ │ - -- used 4.44315s (cpu); 2.84198s (thread); 0s (gc)
│ │ │ + -- used 4.84192s (cpu); 2.77699s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = HashTable{0 => 11 }
│ │ │                 1 => 0
│ │ │                 2 => 1
│ │ │                 3 => 6
│ │ │                 4 => 9
│ │ │                 5 => 8
│ │ │ ├── html2text {}
│ │ │ │ @@ -40,15 +40,15 @@
│ │ │ │  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 0.0159915s (cpu); 0.0134391s (thread); 0s (gc)
│ │ │ │ + -- used 0.0120011s (cpu); 0.0146825s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = HashTable{0 => 1}
│ │ │ │                 1 => 0
│ │ │ │                 2 => 3
│ │ │ │                 3 => 2
│ │ │ │                 4 => 6
│ │ │ │                 5 => 5
│ │ │ │ @@ -74,15 +74,15 @@
│ │ │ │  o6 : HashTable
│ │ │ │  i7 : N = relabel M6
│ │ │ │  
│ │ │ │  o7 = a "matroid" of rank 5 on 15 elements
│ │ │ │  
│ │ │ │  o7 : Matroid
│ │ │ │  i8 : time phi = isomorphism(N,M6)
│ │ │ │ - -- used 4.44315s (cpu); 2.84198s (thread); 0s (gc)
│ │ │ │ + -- used 4.84192s (cpu); 2.77699s (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
│ │ │ @@ -137,15 +137,15 @@
│ │ │  
│ │ │  o9 = true
    
│ │ │              
    
│ │ │                
    i10 : time quickIsomorphismTest(M0, M1)
│ │ │ - -- used 0.000917851s (cpu); 0.000605195s (thread); 0s (gc)
│ │ │ + -- used 0.00152239s (cpu); 0.000513019s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = false
    
│ │ │              
    
│ │ │                
    i11 : value oo === false
│ │ │ ├── html2text {}
│ │ │ │ @@ -51,15 +51,15 @@
│ │ │ │  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 0.000917851s (cpu); 0.000605195s (thread); 0s (gc)
│ │ │ │ + -- used 0.00152239s (cpu); 0.000513019s (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
│ │ │ @@ -126,15 +126,15 @@
│ │ │  
│ │ │  o5 : List
    
│ │ │              
    
│ │ │                
    i6 : elapsedTime fVector M
│ │ │ - -- .0133096s elapsed
│ │ │ + -- .014264s elapsed
│ │ │  
│ │ │  o6 = HashTable{0 => 1 }
│ │ │                 1 => 6
│ │ │                 2 => 15
│ │ │                 3 => 20
│ │ │                 4 => 1
│ │ │ ├── html2text {}
│ │ │ │ @@ -48,15 +48,15 @@
│ │ │ │  o4 : Matrix QQ  <-- QQ
│ │ │ │  i5 : keys M.cache
│ │ │ │  
│ │ │ │  o5 = {groundSet, rankFunction, storedRepresentation}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : elapsedTime fVector M
│ │ │ │ - -- .0133096s elapsed
│ │ │ │ + -- .014264s elapsed
│ │ │ │  
│ │ │ │  o6 = HashTable{0 => 1 }
│ │ │ │                 1 => 6
│ │ │ │                 2 => 15
│ │ │ │                 3 => 20
│ │ │ │                 4 => 1
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/example-output/___Hybrid.out
│ │ │ @@ -5,16 +5,16 @@
│ │ │  i2 : R = ZZ/101[w..z];
│ │ │  
│ │ │  i3 : I = ideal(w*x^2-42*y*z, x^6+12*w*y+x^3*z, w^2-47*x^4*z-47*x*z^2);
│ │ │  
│ │ │  o3 : Ideal of R
│ │ │  
│ │ │  i4 : elapsedTime minimalPrimes(ideal I_*, Strategy => Hybrid{Linear,Birational,Factorization,DecomposeMonomials}, Verbosity => 2);
│ │ │ -  Strategy: Linear            (time .000975029)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: Birational        (time .01961)    #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: Factorization     (time .000355446)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: DecomposeMonomials(time .000024205)  #primes = 1 #prunedViaCodim = 0
│ │ │ +  Strategy: Linear            (time .0011929)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Birational        (time .0388872)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Factorization     (time .000363558)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: DecomposeMonomials(time .000018103)  #primes = 1 #prunedViaCodim = 0
│ │ │   -- Converting annotated ideals to ideals and selecting minimal primes...
│ │ │ - --  Time taken : .000744596
│ │ │ - -- .0495058s elapsed
│ │ │ + --  Time taken : .000826281
│ │ │ + -- .0287952s elapsed
│ │ │  
│ │ │  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)
│ │ │ - -- .000429381s elapsed
│ │ │ + -- .000553931s elapsed
│ │ │  
│ │ │  o6 = ideal (a, b, c)
│ │ │  
│ │ │  o6 : Ideal of R
│ │ │  
│ │ │  i7 : elapsedTime radical(ideal I_*, Unmixed => true)
│ │ │ - -- .0174113s elapsed
│ │ │ + -- .0146503s 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)
│ │ │ - -- .0676229s elapsed
│ │ │ + -- .07832s elapsed
│ │ │  
│ │ │  o7 = true
│ │ │  
│ │ │  i8 : elapsedTime radicalContainment(x_0, I, Strategy => "Kollar")
│ │ │ - -- .00169895s elapsed
│ │ │ + -- .00194255s elapsed
│ │ │  
│ │ │  o8 = true
│ │ │  
│ │ │  i9 : elapsedTime radicalContainment(x_n, I, Strategy => "Kollar")
│ │ │ - -- .00237249s elapsed
│ │ │ + -- .00170115s elapsed
│ │ │  
│ │ │  o9 = false
│ │ │  
│ │ │  i10 :
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/html/___Hybrid.html
│ │ │ @@ -72,21 +72,21 @@
│ │ │  
│ │ │  o3 : Ideal of R
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime minimalPrimes(ideal I_*, Strategy => Hybrid{Linear,Birational,Factorization,DecomposeMonomials}, Verbosity => 2);
│ │ │ -  Strategy: Linear            (time .000975029)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: Birational        (time .01961)    #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: Factorization     (time .000355446)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: DecomposeMonomials(time .000024205)  #primes = 1 #prunedViaCodim = 0
│ │ │ +  Strategy: Linear            (time .0011929)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Birational        (time .0388872)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Factorization     (time .000363558)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: DecomposeMonomials(time .000018103)  #primes = 1 #prunedViaCodim = 0
│ │ │   -- Converting annotated ideals to ideals and selecting minimal primes...
│ │ │ - --  Time taken : .000744596
│ │ │ - -- .0495058s elapsed
    
│ │ │ + --  Time taken : .000826281
│ │ │ + -- .0287952s elapsed
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,24 +11,23 @@
│ │ │ │  i1 : debug MinimalPrimes
│ │ │ │  i2 : R = ZZ/101[w..z];
│ │ │ │  i3 : I = ideal(w*x^2-42*y*z, x^6+12*w*y+x^3*z, w^2-47*x^4*z-47*x*z^2);
│ │ │ │  
│ │ │ │  o3 : Ideal of R
│ │ │ │  i4 : elapsedTime minimalPrimes(ideal I_*, Strategy => Hybrid
│ │ │ │  {Linear,Birational,Factorization,DecomposeMonomials}, Verbosity => 2);
│ │ │ │ -  Strategy: Linear            (time .000975029)  #primes = 0 #prunedViaCodim =
│ │ │ │ +  Strategy: Linear            (time .0011929)  #primes = 0 #prunedViaCodim = 0
│ │ │ │ +  Strategy: Birational        (time .0388872)  #primes = 0 #prunedViaCodim = 0
│ │ │ │ +  Strategy: Factorization     (time .000363558)  #primes = 0 #prunedViaCodim =
│ │ │ │  0
│ │ │ │ -  Strategy: Birational        (time .01961)    #primes = 0 #prunedViaCodim = 0
│ │ │ │ -  Strategy: Factorization     (time .000355446)  #primes = 0 #prunedViaCodim =
│ │ │ │ -0
│ │ │ │ -  Strategy: DecomposeMonomials(time .000024205)  #primes = 1 #prunedViaCodim =
│ │ │ │ +  Strategy: DecomposeMonomials(time .000018103)  #primes = 1 #prunedViaCodim =
│ │ │ │  0
│ │ │ │   -- Converting annotated ideals to ideals and selecting minimal primes...
│ │ │ │ - --  Time taken : .000744596
│ │ │ │ - -- .0495058s elapsed
│ │ │ │ + --  Time taken : .000826281
│ │ │ │ + -- .0287952s elapsed
│ │ │ │  ********** 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.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/html/_radical.html
│ │ │ @@ -131,25 +131,25 @@
│ │ │  
│ │ │  o5 : Ideal of R
    
│ │ │              
    
│ │ │                
    i6 : elapsedTime radical(ideal I_*, Strategy => Monomial)
│ │ │ - -- .000429381s elapsed
│ │ │ + -- .000553931s elapsed
│ │ │  
│ │ │  o6 = ideal (a, b, c)
│ │ │  
│ │ │  o6 : Ideal of R
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime radical(ideal I_*, Unmixed => true)
│ │ │ - -- .0174113s elapsed
│ │ │ + -- .0146503s elapsed
│ │ │  
│ │ │  o7 = ideal (c, b, a)
│ │ │  
│ │ │  o7 : Ideal of R
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -62,21 +62,21 @@
│ │ │ │  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)
│ │ │ │ - -- .000429381s elapsed
│ │ │ │ + -- .000553931s elapsed
│ │ │ │  
│ │ │ │  o6 = ideal (a, b, c)
│ │ │ │  
│ │ │ │  o6 : Ideal of R
│ │ │ │  i7 : elapsedTime radical(ideal I_*, Unmixed => true)
│ │ │ │ - -- .0174113s elapsed
│ │ │ │ + -- .0146503s 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
│ │ │ @@ -125,31 +125,31 @@
│ │ │  
│ │ │  o6 = true
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime radicalContainment(x_0, I)
│ │ │ - -- .0676229s elapsed
│ │ │ + -- .07832s elapsed
│ │ │  
│ │ │  o7 = true
    
│ │ │              
    
│ │ │                
    i8 : elapsedTime radicalContainment(x_0, I, Strategy => "Kollar")
│ │ │ - -- .00169895s elapsed
│ │ │ + -- .00194255s elapsed
│ │ │  
│ │ │  o8 = true
    
│ │ │              
    
│ │ │                
    i9 : elapsedTime radicalContainment(x_n, I, Strategy => "Kollar")
│ │ │ - -- .00237249s elapsed
│ │ │ + -- .00170115s elapsed
│ │ │  
│ │ │  o9 = false
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -50,23 +50,23 @@
│ │ │ │  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)
│ │ │ │ - -- .0676229s elapsed
│ │ │ │ + -- .07832s elapsed
│ │ │ │  
│ │ │ │  o7 = true
│ │ │ │  i8 : elapsedTime radicalContainment(x_0, I, Strategy => "Kollar")
│ │ │ │ - -- .00169895s elapsed
│ │ │ │ + -- .00194255s elapsed
│ │ │ │  
│ │ │ │  o8 = true
│ │ │ │  i9 : elapsedTime radicalContainment(x_n, I, Strategy => "Kollar")
│ │ │ │ - -- .00237249s elapsed
│ │ │ │ + -- .00170115s 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/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 0.14126s (cpu); 0.0850548s (thread); 0s (gc)
│ │ │ + -- used 0.356404s (cpu); 0.115557s (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 0.0901457s (cpu); 0.0329154s (thread); 0s (gc)
│ │ │ + -- used 0.108637s (cpu); 0.0293922s (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
│ │ │ @@ -173,15 +173,15 @@
│ │ │  
│ │ │  o9 : Ideal of U
    
│ │ │              
    
│ │ │                
    i10 : time multiReesIdeal J
│ │ │ - -- used 0.14126s (cpu); 0.0850548s (thread); 0s (gc)
│ │ │ + -- used 0.356404s (cpu); 0.115557s (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 )
│ │ │ @@ -190,15 +190,15 @@
│ │ │  o10 : Ideal of U[X ..X ]
│ │ │                    0   2
    
│ │ │              
    
│ │ │                
    i11 : time multiReesIdeal (J, a)
│ │ │ - -- used 0.0901457s (cpu); 0.0329154s (thread); 0s (gc)
│ │ │ + -- used 0.108637s (cpu); 0.0293922s (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 {}
│ │ │ │ @@ -79,28 +79,28 @@
│ │ │ │  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 0.14126s (cpu); 0.0850548s (thread); 0s (gc)
│ │ │ │ + -- used 0.356404s (cpu); 0.115557s (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 0.0901457s (cpu); 0.0329154s (thread); 0s (gc)
│ │ │ │ + -- used 0.108637s (cpu); 0.0293922s (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/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 0.431533s (cpu); 0.30545s (thread); 0s (gc)
│ │ │ + -- used 0.613563s (cpu); 0.416443s (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 0.654654s (cpu); 0.513161s (thread); 0s (gc)
│ │ │ + -- used 0.86658s (cpu); 0.602979s (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
│ │ │ @@ -145,15 +145,15 @@
│ │ │                               1
│ │ │  o7 : R-module, submodule of R
    
│ │ │              
    
│ │ │                
    i8 : (S,N) = time deformMCMModule N0 
│ │ │ - -- used 0.431533s (cpu); 0.30545s (thread); 0s (gc)
│ │ │ + -- used 0.613563s (cpu); 0.416443s (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                                           |
│ │ │  
│ │ │ @@ -186,15 +186,15 @@
│ │ │                               2
│ │ │  o10 : R-module, quotient of R
    
│ │ │              
    
│ │ │                
    i11 : (S',N') = time deformMCMModule N0'
│ │ │ - -- used 0.654654s (cpu); 0.513161s (thread); 0s (gc)
│ │ │ + -- used 0.86658s (cpu); 0.602979s (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 {}
│ │ │ │ @@ -70,15 +70,15 @@
│ │ │ │  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 0.431533s (cpu); 0.30545s (thread); 0s (gc)
│ │ │ │ + -- used 0.613563s (cpu); 0.416443s (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                                           |
│ │ │ │  
│ │ │ │ @@ -103,15 +103,15 @@
│ │ │ │  
│ │ │ │  o10 = cokernel | x2 y2  |
│ │ │ │                 | -y -x2 |
│ │ │ │  
│ │ │ │                               2
│ │ │ │  o10 : R-module, quotient of R
│ │ │ │  i11 : (S',N') = time deformMCMModule N0'
│ │ │ │ - -- used 0.654654s (cpu); 0.513161s (thread); 0s (gc)
│ │ │ │ + -- used 0.86658s (cpu); 0.602979s (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/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})
│ │ │ - -- .00311573s elapsed
│ │ │ - -- .00283883s elapsed
│ │ │ - -- .000324065s elapsed
│ │ │ - -- .00284892s elapsed
│ │ │ - -- .00294607s elapsed
│ │ │ - -- .000298176s elapsed
│ │ │ - -- .00286122s elapsed
│ │ │ - -- .00290687s elapsed
│ │ │ - -- .000226291s elapsed
│ │ │ - -- .00292072s elapsed
│ │ │ - -- .00297273s elapsed
│ │ │ - -- .000231462s elapsed
│ │ │ ---backup directory created: /tmp/M2-33427-0/1
│ │ │ + -- .00449025s elapsed
│ │ │ + -- .00443229s elapsed
│ │ │ + -- .000625274s elapsed
│ │ │ + -- .00433788s elapsed
│ │ │ + -- .00389609s elapsed
│ │ │ + -- .00046102s elapsed
│ │ │ + -- .00374982s elapsed
│ │ │ + -- .00369769s elapsed
│ │ │ + -- .000283168s elapsed
│ │ │ + -- .00350731s elapsed
│ │ │ + -- .0038125s elapsed
│ │ │ + -- .000507948s elapsed
│ │ │ +--backup directory created: /tmp/M2-49353-0/1
│ │ │    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
│ │ │ @@ -15,128 +15,128 @@
│ │ │  
│ │ │  i7 : dLoss = diff(varMatrix, gateMatrix{{loss}});
│ │ │  
│ │ │  i8 : G = gateSystem(paramMatrix,varMatrix,transpose dLoss);
│ │ │  
│ │ │  i9 : monodromyGroup(G,"msOptions" => {NumberOfEdges=>10})
│ │ │  
│ │ │ -o9 = {{2, 0, 11, 3, 10, 4, 14, 6, 5, 1, 13, 9, 7, 8, 16, 15, 12, 17, 18, 19,
│ │ │ +o9 = {{16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13, 20, 6, 9,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18,
│ │ │ +     4}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18, 16, 19, 15,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9, 4, 6}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16,
│ │ │ +     17, 3}, {3, 16, 14, 4, 5, 2, 7, 8, 1, 11, 0, 12, 10, 15, 13, 6, 17, 9,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     18, 9, 4, 6}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18,
│ │ │ +     18, 19, 20}, {12, 16, 14, 19, 4, 7, 6, 11, 1, 9, 0, 5, 10, 2, 13, 20, 8,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 19, 15, 17, 3}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
│ │ │ +     18, 3, 15, 17}, {16, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 0, 14,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 16, 17, 18, 19, 20}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8,
│ │ │ +     3, 13, 15, 6, 9, 4}, {0, 1, 3, 19, 12, 5, 2, 7, 15, 8, 10, 11, 17, 13,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 15, 12, 17, 18, 19, 20}, {0, 1, 2, 3, 4, 11, 6, 5, 8, 9, 10, 7, 12,
│ │ │ +     14, 20, 16, 18, 9, 4, 6}, {12, 16, 7, 17, 10, 9, 14, 4, 11, 1, 0, 6, 5,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     13, 14, 15, 16, 17, 18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10,
│ │ │ +     2, 13, 3, 8, 15, 18, 19, 20}, {12, 16, 14, 17, 4, 7, 6, 11, 1, 9, 0, 5,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {3, 1, 12, 4, 0, 11, 13, 5, 2, 16,
│ │ │ +     10, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 7, 8, 15, 14, 6, 17, 9, 18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15,
│ │ │ +     7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {1, 7, 12, 19, 16, 4, 0, 6,
│ │ │ +     11, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {4, 16, 14, 3, 5, 2, 7, 8, 1,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     2, 13, 11, 9, 8, 10, 5, 20, 14, 18, 3, 15, 17}, {1, 16, 12, 19, 11, 15,
│ │ │ +     11, 0, 12, 10, 6, 13, 15, 9, 17, 18, 19, 20}, {12, 16, 7, 17, 10, 9, 14,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     5, 17, 2, 7, 0, 3, 8, 10, 13, 20, 14, 18, 9, 4, 6}, {2, 0, 1, 3, 6, 5,
│ │ │ +     4, 11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 11, 2, 3, 10, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9, 7, 10, 4, 13, 11, 14, 8, 16, 15, 12, 17, 18, 19, 20}, {3, 16, 12, 4,
│ │ │ +     14, 6, 8, 1, 5, 9, 12, 13, 7, 15, 16, 17, 18, 19, 20}, {16, 14, 17, 18,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 11, 14, 5, 2, 1, 0, 7, 8, 15, 13, 6, 17, 9, 18, 19, 20}, {12, 1, 7,
│ │ │ +     8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13, 20, 6, 9, 4}, {2, 0, 11, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     17, 16, 9, 0, 4, 11, 13, 10, 6, 5, 2, 14, 3, 8, 15, 20, 18, 19}, {0, 1,
│ │ │ +     10, 4, 14, 6, 5, 1, 13, 9, 7, 8, 16, 15, 12, 17, 18, 19, 20}, {0, 1, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6},
│ │ │ +     19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0, 1,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {17, 16, 14, 9, 8, 7, 12, 11, 1, 2, 0, 5, 10, 3, 13, 4, 15, 6, 20, 18,
│ │ │ +     3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     19}, {1, 16, 12, 19, 11, 15, 5, 17, 2, 7, 0, 3, 8, 10, 13, 20, 14, 18,
│ │ │ +     1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18, 16, 19, 15, 17, 3},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9, 4, 6}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,
│ │ │ +     {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     18, 19, 20}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12,
│ │ │ +     20}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12, 17, 18,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     17, 18, 19, 20}, {0, 9, 2, 3, 10, 5, 14, 7, 8, 1, 4, 11, 12, 13, 6, 15,
│ │ │ +     19, 20}, {0, 1, 2, 3, 4, 11, 6, 5, 8, 9, 10, 7, 12, 13, 14, 15, 16, 17,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 17, 18, 19, 20}, {16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0,
│ │ │ +     18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 19, 13, 20, 6, 9, 4}, {2, 1, 0, 3, 4, 5, 6, 7, 13, 9, 10, 11, 16, 8,
│ │ │ +     16, 18, 9, 4, 6}, {3, 1, 12, 4, 0, 11, 13, 5, 2, 16, 10, 7, 8, 15, 14,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     14, 15, 12, 17, 18, 19, 20}, {12, 16, 14, 19, 4, 3, 6, 15, 1, 9, 0, 17,
│ │ │ +     6, 17, 9, 18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 2, 13, 20, 8, 18, 7, 11, 5}, {0, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10,
│ │ │ +     14, 20, 16, 18, 9, 4, 6}, {1, 7, 12, 19, 16, 4, 0, 6, 2, 13, 11, 9, 8,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20, 5, 13, 14, 3, 16, 15, 6, 9, 4}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10,
│ │ │ +     10, 5, 20, 14, 18, 3, 15, 17}, {1, 16, 12, 19, 11, 15, 5, 17, 2, 7, 0,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     11, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {12, 16, 7, 17, 10, 9, 14, 4,
│ │ │ +     3, 8, 10, 13, 20, 14, 18, 9, 4, 6}, {2, 0, 1, 3, 6, 5, 9, 7, 10, 4, 13,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 1, 2, 3, 9, 11, 4, 5,
│ │ │ +     11, 14, 8, 16, 15, 12, 17, 18, 19, 20}, {3, 16, 12, 4, 10, 11, 14, 5, 2,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     8, 6, 10, 7, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {16, 7, 12, 19, 9, 1,
│ │ │ +     1, 0, 7, 8, 15, 13, 6, 17, 9, 18, 19, 20}, {12, 1, 7, 17, 16, 9, 0, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     4, 10, 2, 6, 11, 14, 8, 0, 5, 20, 13, 18, 3, 15, 17}, {0, 1, 2, 3, 4, 5,
│ │ │ +     11, 13, 10, 6, 5, 2, 14, 3, 8, 15, 20, 18, 19}, {0, 1, 3, 19, 12, 11, 2,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {12, 16, 3, 19,
│ │ │ +     5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {17, 16, 14, 9, 8, 7,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 11, 14, 5, 15, 1, 0, 7, 17, 2, 13, 20, 8, 18, 9, 4, 6}, {0, 9, 2, 3,
│ │ │ +     12, 11, 1, 2, 0, 5, 10, 3, 13, 4, 15, 6, 20, 18, 19}, {1, 16, 12, 19,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 5, 14, 7, 8, 1, 4, 11, 12, 13, 6, 15, 16, 17, 18, 19, 20}, {0, 1, 2,
│ │ │ +     11, 15, 5, 17, 2, 7, 0, 3, 8, 10, 13, 20, 14, 18, 9, 4, 6}, {0, 1, 2, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {12,
│ │ │ +     4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {2, 0, 9,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     1, 3, 19, 0, 11, 13, 5, 15, 16, 10, 7, 17, 2, 14, 20, 8, 18, 9, 4, 6},
│ │ │ +     3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12, 17, 18, 19, 20}, {0, 9,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4,
│ │ │ +     2, 3, 10, 5, 14, 7, 8, 1, 4, 11, 12, 13, 6, 15, 16, 17, 18, 19, 20},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     6}, {4, 1, 2, 3, 0, 5, 13, 7, 8, 16, 10, 11, 12, 6, 14, 15, 9, 17, 18,
│ │ │ +     {16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13, 20, 6, 9,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16,
│ │ │ +     4}, {2, 1, 0, 3, 4, 5, 6, 7, 13, 9, 10, 11, 16, 8, 14, 15, 12, 17, 18,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     18, 9, 4, 6}, {0, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 13, 14, 3,
│ │ │ +     19, 20}, {12, 16, 14, 19, 4, 3, 6, 15, 1, 9, 0, 17, 10, 2, 13, 20, 8,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 15, 6, 9, 4}, {0, 14, 12, 19, 11, 15, 5, 17, 2, 7, 1, 3, 8, 13, 10,
│ │ │ +     18, 7, 11, 5}, {0, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 13, 14, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20, 16, 18, 9, 4, 6}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16,
│ │ │ +     16, 15, 6, 9, 4}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 12, 17, 20, 18, 19}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17,
│ │ │ +     18, 16, 19, 15, 17, 3}, {12, 16, 7, 17, 10, 9, 14, 4, 11, 1, 0, 6, 5, 2,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     13, 14, 20, 16, 18, 9, 4, 6}, {16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5,
│ │ │ +     13, 3, 8, 15, 20, 18, 19}, {0, 1, 2, 3, 9, 11, 4, 5, 8, 6, 10, 7, 12,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 0, 10, 19, 13, 20, 6, 9, 4}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11,
│ │ │ +     13, 14, 15, 16, 17, 18, 19, 20}, {16, 7, 12, 19, 9, 1, 4, 10, 2, 6, 11,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     12, 13, 14, 18, 16, 19, 15, 17, 3}, {3, 16, 14, 4, 5, 2, 7, 8, 1, 11, 0,
│ │ │ +     14, 8, 0, 5, 20, 13, 18, 3, 15, 17}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     12, 10, 15, 13, 6, 17, 9, 18, 19, 20}, {12, 16, 14, 19, 4, 7, 6, 11, 1,
│ │ │ +     11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {12, 16, 3, 19, 10, 11, 14, 5,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9, 0, 5, 10, 2, 13, 20, 8, 18, 3, 15, 17}, {16, 1, 7, 17, 12, 18, 2, 19,
│ │ │ +     15, 1, 0, 7, 17, 2, 13, 20, 8, 18, 9, 4, 6}, {0, 9, 2, 3, 10, 5, 14, 7,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     11, 8, 10, 20, 5, 0, 14, 3, 13, 15, 6, 9, 4}, {0, 1, 3, 19, 12, 5, 2, 7,
│ │ │ +     8, 1, 4, 11, 12, 13, 6, 15, 16, 17, 18, 19, 20}, {0, 1, 2, 3, 4, 5, 6,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 8, 10, 11, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {12, 16, 7, 17, 10, 9,
│ │ │ +     7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {12, 1, 3, 19, 0,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     14, 4, 11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 18, 19, 20}, {12, 16, 14, 17, 4,
│ │ │ +     11, 13, 5, 15, 16, 10, 7, 17, 2, 14, 20, 8, 18, 9, 4, 6}, {0, 1, 3, 19,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     7, 6, 11, 1, 9, 0, 5, 10, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 1, 3, 19,
│ │ │ +     12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {4, 1, 2,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0, 1, 2,
│ │ │ +     3, 0, 5, 13, 7, 8, 16, 10, 11, 12, 6, 14, 15, 9, 17, 18, 19, 20}, {0, 1,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {4,
│ │ │ +     3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 14, 3, 5, 2, 7, 8, 1, 11, 0, 12, 10, 6, 13, 15, 9, 17, 18, 19, 20},
│ │ │ +     1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 13, 14, 3, 16, 15, 6, 9, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {12, 16, 7, 17, 10, 9, 14, 4, 11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 20, 18,
│ │ │ +     {0, 14, 12, 19, 11, 15, 5, 17, 2, 7, 1, 3, 8, 13, 10, 20, 16, 18, 9, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     19}, {0, 11, 2, 3, 10, 4, 14, 6, 8, 1, 5, 9, 12, 13, 7, 15, 16, 17, 18,
│ │ │ +     6}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12, 17, 20,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     19, 20}, {16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13,
│ │ │ +     18, 19}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20, 6, 9, 4}}
│ │ │ +     18, 9, 4, 6}}
│ │ │  
│ │ │  o9 : List
│ │ │  
│ │ │  i10 :
│ │ ├── ./usr/share/doc/Macaulay2/MonodromySolver/html/_dynamic__Flower__Solve.html
│ │ │ @@ -96,27 +96,27 @@
│ │ │              
    		
│ │ │                
    i3 : (p0, x0) = createSeedPair polys;
    
│ │ │              
    
│ │ │                
    i4 : (L, npaths) = dynamicFlowerSolve(polys.PolyMap,p0,{x0})
│ │ │ - -- .00311573s elapsed
│ │ │ - -- .00283883s elapsed
│ │ │ - -- .000324065s elapsed
│ │ │ - -- .00284892s elapsed
│ │ │ - -- .00294607s elapsed
│ │ │ - -- .000298176s elapsed
│ │ │ - -- .00286122s elapsed
│ │ │ - -- .00290687s elapsed
│ │ │ - -- .000226291s elapsed
│ │ │ - -- .00292072s elapsed
│ │ │ - -- .00297273s elapsed
│ │ │ - -- .000231462s elapsed
│ │ │ ---backup directory created: /tmp/M2-33427-0/1
│ │ │ + -- .00449025s elapsed
│ │ │ + -- .00443229s elapsed
│ │ │ + -- .000625274s elapsed
│ │ │ + -- .00433788s elapsed
│ │ │ + -- .00389609s elapsed
│ │ │ + -- .00046102s elapsed
│ │ │ + -- .00374982s elapsed
│ │ │ + -- .00369769s elapsed
│ │ │ + -- .000283168s elapsed
│ │ │ + -- .00350731s elapsed
│ │ │ + -- .0038125s elapsed
│ │ │ + -- .000507948s elapsed
│ │ │ +--backup directory created: /tmp/M2-49353-0/1
│ │ │    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 {}
│ │ │ │ @@ -22,27 +22,27 @@
│ │ │ │            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})
│ │ │ │ - -- .00311573s elapsed
│ │ │ │ - -- .00283883s elapsed
│ │ │ │ - -- .000324065s elapsed
│ │ │ │ - -- .00284892s elapsed
│ │ │ │ - -- .00294607s elapsed
│ │ │ │ - -- .000298176s elapsed
│ │ │ │ - -- .00286122s elapsed
│ │ │ │ - -- .00290687s elapsed
│ │ │ │ - -- .000226291s elapsed
│ │ │ │ - -- .00292072s elapsed
│ │ │ │ - -- .00297273s elapsed
│ │ │ │ - -- .000231462s elapsed
│ │ │ │ ---backup directory created: /tmp/M2-33427-0/1
│ │ │ │ + -- .00449025s elapsed
│ │ │ │ + -- .00443229s elapsed
│ │ │ │ + -- .000625274s elapsed
│ │ │ │ + -- .00433788s elapsed
│ │ │ │ + -- .00389609s elapsed
│ │ │ │ + -- .00046102s elapsed
│ │ │ │ + -- .00374982s elapsed
│ │ │ │ + -- .00369769s elapsed
│ │ │ │ + -- .000283168s elapsed
│ │ │ │ + -- .00350731s elapsed
│ │ │ │ + -- .0038125s elapsed
│ │ │ │ + -- .000507948s elapsed
│ │ │ │ +--backup directory created: /tmp/M2-49353-0/1
│ │ │ │    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
│ │ │ @@ -118,131 +118,131 @@
│ │ │                
    i8 : G = gateSystem(paramMatrix,varMatrix,transpose dLoss);
    
│ │ │              
    
│ │ │                
    i9 : monodromyGroup(G,"msOptions" => {NumberOfEdges=>10})
│ │ │  
│ │ │ -o9 = {{2, 0, 11, 3, 10, 4, 14, 6, 5, 1, 13, 9, 7, 8, 16, 15, 12, 17, 18, 19,
│ │ │ +o9 = {{16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13, 20, 6, 9,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18,
│ │ │ +     4}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18, 16, 19, 15,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9, 4, 6}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16,
│ │ │ +     17, 3}, {3, 16, 14, 4, 5, 2, 7, 8, 1, 11, 0, 12, 10, 15, 13, 6, 17, 9,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     18, 9, 4, 6}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18,
│ │ │ +     18, 19, 20}, {12, 16, 14, 19, 4, 7, 6, 11, 1, 9, 0, 5, 10, 2, 13, 20, 8,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 19, 15, 17, 3}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
│ │ │ +     18, 3, 15, 17}, {16, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 0, 14,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 16, 17, 18, 19, 20}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8,
│ │ │ +     3, 13, 15, 6, 9, 4}, {0, 1, 3, 19, 12, 5, 2, 7, 15, 8, 10, 11, 17, 13,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 15, 12, 17, 18, 19, 20}, {0, 1, 2, 3, 4, 11, 6, 5, 8, 9, 10, 7, 12,
│ │ │ +     14, 20, 16, 18, 9, 4, 6}, {12, 16, 7, 17, 10, 9, 14, 4, 11, 1, 0, 6, 5,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     13, 14, 15, 16, 17, 18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10,
│ │ │ +     2, 13, 3, 8, 15, 18, 19, 20}, {12, 16, 14, 17, 4, 7, 6, 11, 1, 9, 0, 5,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {3, 1, 12, 4, 0, 11, 13, 5, 2, 16,
│ │ │ +     10, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 7, 8, 15, 14, 6, 17, 9, 18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15,
│ │ │ +     7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {1, 7, 12, 19, 16, 4, 0, 6,
│ │ │ +     11, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {4, 16, 14, 3, 5, 2, 7, 8, 1,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     2, 13, 11, 9, 8, 10, 5, 20, 14, 18, 3, 15, 17}, {1, 16, 12, 19, 11, 15,
│ │ │ +     11, 0, 12, 10, 6, 13, 15, 9, 17, 18, 19, 20}, {12, 16, 7, 17, 10, 9, 14,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     5, 17, 2, 7, 0, 3, 8, 10, 13, 20, 14, 18, 9, 4, 6}, {2, 0, 1, 3, 6, 5,
│ │ │ +     4, 11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 11, 2, 3, 10, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9, 7, 10, 4, 13, 11, 14, 8, 16, 15, 12, 17, 18, 19, 20}, {3, 16, 12, 4,
│ │ │ +     14, 6, 8, 1, 5, 9, 12, 13, 7, 15, 16, 17, 18, 19, 20}, {16, 14, 17, 18,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 11, 14, 5, 2, 1, 0, 7, 8, 15, 13, 6, 17, 9, 18, 19, 20}, {12, 1, 7,
│ │ │ +     8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13, 20, 6, 9, 4}, {2, 0, 11, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     17, 16, 9, 0, 4, 11, 13, 10, 6, 5, 2, 14, 3, 8, 15, 20, 18, 19}, {0, 1,
│ │ │ +     10, 4, 14, 6, 5, 1, 13, 9, 7, 8, 16, 15, 12, 17, 18, 19, 20}, {0, 1, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6},
│ │ │ +     19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0, 1,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {17, 16, 14, 9, 8, 7, 12, 11, 1, 2, 0, 5, 10, 3, 13, 4, 15, 6, 20, 18,
│ │ │ +     3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     19}, {1, 16, 12, 19, 11, 15, 5, 17, 2, 7, 0, 3, 8, 10, 13, 20, 14, 18,
│ │ │ +     1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18, 16, 19, 15, 17, 3},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9, 4, 6}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,
│ │ │ +     {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     18, 19, 20}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12,
│ │ │ +     20}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12, 17, 18,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     17, 18, 19, 20}, {0, 9, 2, 3, 10, 5, 14, 7, 8, 1, 4, 11, 12, 13, 6, 15,
│ │ │ +     19, 20}, {0, 1, 2, 3, 4, 11, 6, 5, 8, 9, 10, 7, 12, 13, 14, 15, 16, 17,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 17, 18, 19, 20}, {16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0,
│ │ │ +     18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 19, 13, 20, 6, 9, 4}, {2, 1, 0, 3, 4, 5, 6, 7, 13, 9, 10, 11, 16, 8,
│ │ │ +     16, 18, 9, 4, 6}, {3, 1, 12, 4, 0, 11, 13, 5, 2, 16, 10, 7, 8, 15, 14,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     14, 15, 12, 17, 18, 19, 20}, {12, 16, 14, 19, 4, 3, 6, 15, 1, 9, 0, 17,
│ │ │ +     6, 17, 9, 18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 2, 13, 20, 8, 18, 7, 11, 5}, {0, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10,
│ │ │ +     14, 20, 16, 18, 9, 4, 6}, {1, 7, 12, 19, 16, 4, 0, 6, 2, 13, 11, 9, 8,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20, 5, 13, 14, 3, 16, 15, 6, 9, 4}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10,
│ │ │ +     10, 5, 20, 14, 18, 3, 15, 17}, {1, 16, 12, 19, 11, 15, 5, 17, 2, 7, 0,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     11, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {12, 16, 7, 17, 10, 9, 14, 4,
│ │ │ +     3, 8, 10, 13, 20, 14, 18, 9, 4, 6}, {2, 0, 1, 3, 6, 5, 9, 7, 10, 4, 13,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 1, 2, 3, 9, 11, 4, 5,
│ │ │ +     11, 14, 8, 16, 15, 12, 17, 18, 19, 20}, {3, 16, 12, 4, 10, 11, 14, 5, 2,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     8, 6, 10, 7, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {16, 7, 12, 19, 9, 1,
│ │ │ +     1, 0, 7, 8, 15, 13, 6, 17, 9, 18, 19, 20}, {12, 1, 7, 17, 16, 9, 0, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     4, 10, 2, 6, 11, 14, 8, 0, 5, 20, 13, 18, 3, 15, 17}, {0, 1, 2, 3, 4, 5,
│ │ │ +     11, 13, 10, 6, 5, 2, 14, 3, 8, 15, 20, 18, 19}, {0, 1, 3, 19, 12, 11, 2,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {12, 16, 3, 19,
│ │ │ +     5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {17, 16, 14, 9, 8, 7,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 11, 14, 5, 15, 1, 0, 7, 17, 2, 13, 20, 8, 18, 9, 4, 6}, {0, 9, 2, 3,
│ │ │ +     12, 11, 1, 2, 0, 5, 10, 3, 13, 4, 15, 6, 20, 18, 19}, {1, 16, 12, 19,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 5, 14, 7, 8, 1, 4, 11, 12, 13, 6, 15, 16, 17, 18, 19, 20}, {0, 1, 2,
│ │ │ +     11, 15, 5, 17, 2, 7, 0, 3, 8, 10, 13, 20, 14, 18, 9, 4, 6}, {0, 1, 2, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {12,
│ │ │ +     4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {2, 0, 9,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     1, 3, 19, 0, 11, 13, 5, 15, 16, 10, 7, 17, 2, 14, 20, 8, 18, 9, 4, 6},
│ │ │ +     3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12, 17, 18, 19, 20}, {0, 9,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4,
│ │ │ +     2, 3, 10, 5, 14, 7, 8, 1, 4, 11, 12, 13, 6, 15, 16, 17, 18, 19, 20},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     6}, {4, 1, 2, 3, 0, 5, 13, 7, 8, 16, 10, 11, 12, 6, 14, 15, 9, 17, 18,
│ │ │ +     {16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13, 20, 6, 9,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16,
│ │ │ +     4}, {2, 1, 0, 3, 4, 5, 6, 7, 13, 9, 10, 11, 16, 8, 14, 15, 12, 17, 18,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     18, 9, 4, 6}, {0, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 13, 14, 3,
│ │ │ +     19, 20}, {12, 16, 14, 19, 4, 3, 6, 15, 1, 9, 0, 17, 10, 2, 13, 20, 8,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 15, 6, 9, 4}, {0, 14, 12, 19, 11, 15, 5, 17, 2, 7, 1, 3, 8, 13, 10,
│ │ │ +     18, 7, 11, 5}, {0, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 13, 14, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20, 16, 18, 9, 4, 6}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16,
│ │ │ +     16, 15, 6, 9, 4}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 12, 17, 20, 18, 19}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17,
│ │ │ +     18, 16, 19, 15, 17, 3}, {12, 16, 7, 17, 10, 9, 14, 4, 11, 1, 0, 6, 5, 2,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     13, 14, 20, 16, 18, 9, 4, 6}, {16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5,
│ │ │ +     13, 3, 8, 15, 20, 18, 19}, {0, 1, 2, 3, 9, 11, 4, 5, 8, 6, 10, 7, 12,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 0, 10, 19, 13, 20, 6, 9, 4}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11,
│ │ │ +     13, 14, 15, 16, 17, 18, 19, 20}, {16, 7, 12, 19, 9, 1, 4, 10, 2, 6, 11,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     12, 13, 14, 18, 16, 19, 15, 17, 3}, {3, 16, 14, 4, 5, 2, 7, 8, 1, 11, 0,
│ │ │ +     14, 8, 0, 5, 20, 13, 18, 3, 15, 17}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     12, 10, 15, 13, 6, 17, 9, 18, 19, 20}, {12, 16, 14, 19, 4, 7, 6, 11, 1,
│ │ │ +     11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {12, 16, 3, 19, 10, 11, 14, 5,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9, 0, 5, 10, 2, 13, 20, 8, 18, 3, 15, 17}, {16, 1, 7, 17, 12, 18, 2, 19,
│ │ │ +     15, 1, 0, 7, 17, 2, 13, 20, 8, 18, 9, 4, 6}, {0, 9, 2, 3, 10, 5, 14, 7,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     11, 8, 10, 20, 5, 0, 14, 3, 13, 15, 6, 9, 4}, {0, 1, 3, 19, 12, 5, 2, 7,
│ │ │ +     8, 1, 4, 11, 12, 13, 6, 15, 16, 17, 18, 19, 20}, {0, 1, 2, 3, 4, 5, 6,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 8, 10, 11, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {12, 16, 7, 17, 10, 9,
│ │ │ +     7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {12, 1, 3, 19, 0,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     14, 4, 11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 18, 19, 20}, {12, 16, 14, 17, 4,
│ │ │ +     11, 13, 5, 15, 16, 10, 7, 17, 2, 14, 20, 8, 18, 9, 4, 6}, {0, 1, 3, 19,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     7, 6, 11, 1, 9, 0, 5, 10, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 1, 3, 19,
│ │ │ +     12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {4, 1, 2,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0, 1, 2,
│ │ │ +     3, 0, 5, 13, 7, 8, 16, 10, 11, 12, 6, 14, 15, 9, 17, 18, 19, 20}, {0, 1,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {4,
│ │ │ +     3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 14, 3, 5, 2, 7, 8, 1, 11, 0, 12, 10, 6, 13, 15, 9, 17, 18, 19, 20},
│ │ │ +     1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 13, 14, 3, 16, 15, 6, 9, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {12, 16, 7, 17, 10, 9, 14, 4, 11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 20, 18,
│ │ │ +     {0, 14, 12, 19, 11, 15, 5, 17, 2, 7, 1, 3, 8, 13, 10, 20, 16, 18, 9, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     19}, {0, 11, 2, 3, 10, 4, 14, 6, 8, 1, 5, 9, 12, 13, 7, 15, 16, 17, 18,
│ │ │ +     6}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12, 17, 20,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     19, 20}, {16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13,
│ │ │ +     18, 19}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20, 6, 9, 4}}
│ │ │ +     18, 9, 4, 6}}
│ │ │  
│ │ │  o9 : List
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── 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 = {{2, 0, 11, 3, 10, 4, 14, 6, 5, 1, 13, 9, 7, 8, 16, 15, 12, 17, 18, 19,
│ │ │ │ +o9 = {{16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13, 20, 6, 9,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18,
│ │ │ │ +     4}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18, 16, 19, 15,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     9, 4, 6}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16,
│ │ │ │ +     17, 3}, {3, 16, 14, 4, 5, 2, 7, 8, 1, 11, 0, 12, 10, 15, 13, 6, 17, 9,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     18, 9, 4, 6}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18,
│ │ │ │ +     18, 19, 20}, {12, 16, 14, 19, 4, 7, 6, 11, 1, 9, 0, 5, 10, 2, 13, 20, 8,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     16, 19, 15, 17, 3}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
│ │ │ │ +     18, 3, 15, 17}, {16, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 0, 14,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     15, 16, 17, 18, 19, 20}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8,
│ │ │ │ +     3, 13, 15, 6, 9, 4}, {0, 1, 3, 19, 12, 5, 2, 7, 15, 8, 10, 11, 17, 13,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     16, 15, 12, 17, 18, 19, 20}, {0, 1, 2, 3, 4, 11, 6, 5, 8, 9, 10, 7, 12,
│ │ │ │ +     14, 20, 16, 18, 9, 4, 6}, {12, 16, 7, 17, 10, 9, 14, 4, 11, 1, 0, 6, 5,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     13, 14, 15, 16, 17, 18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10,
│ │ │ │ +     2, 13, 3, 8, 15, 18, 19, 20}, {12, 16, 14, 17, 4, 7, 6, 11, 1, 9, 0, 5,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {3, 1, 12, 4, 0, 11, 13, 5, 2, 16,
│ │ │ │ +     10, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     10, 7, 8, 15, 14, 6, 17, 9, 18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15,
│ │ │ │ +     7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {1, 7, 12, 19, 16, 4, 0, 6,
│ │ │ │ +     11, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {4, 16, 14, 3, 5, 2, 7, 8, 1,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     2, 13, 11, 9, 8, 10, 5, 20, 14, 18, 3, 15, 17}, {1, 16, 12, 19, 11, 15,
│ │ │ │ +     11, 0, 12, 10, 6, 13, 15, 9, 17, 18, 19, 20}, {12, 16, 7, 17, 10, 9, 14,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     5, 17, 2, 7, 0, 3, 8, 10, 13, 20, 14, 18, 9, 4, 6}, {2, 0, 1, 3, 6, 5,
│ │ │ │ +     4, 11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 11, 2, 3, 10, 4,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     9, 7, 10, 4, 13, 11, 14, 8, 16, 15, 12, 17, 18, 19, 20}, {3, 16, 12, 4,
│ │ │ │ +     14, 6, 8, 1, 5, 9, 12, 13, 7, 15, 16, 17, 18, 19, 20}, {16, 14, 17, 18,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     10, 11, 14, 5, 2, 1, 0, 7, 8, 15, 13, 6, 17, 9, 18, 19, 20}, {12, 1, 7,
│ │ │ │ +     8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13, 20, 6, 9, 4}, {2, 0, 11, 3,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     17, 16, 9, 0, 4, 11, 13, 10, 6, 5, 2, 14, 3, 8, 15, 20, 18, 19}, {0, 1,
│ │ │ │ +     10, 4, 14, 6, 5, 1, 13, 9, 7, 8, 16, 15, 12, 17, 18, 19, 20}, {0, 1, 3,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6},
│ │ │ │ +     19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0, 1,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {17, 16, 14, 9, 8, 7, 12, 11, 1, 2, 0, 5, 10, 3, 13, 4, 15, 6, 20, 18,
│ │ │ │ +     3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     19}, {1, 16, 12, 19, 11, 15, 5, 17, 2, 7, 0, 3, 8, 10, 13, 20, 14, 18,
│ │ │ │ +     1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18, 16, 19, 15, 17, 3},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     9, 4, 6}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,
│ │ │ │ +     {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     18, 19, 20}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12,
│ │ │ │ +     20}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12, 17, 18,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     17, 18, 19, 20}, {0, 9, 2, 3, 10, 5, 14, 7, 8, 1, 4, 11, 12, 13, 6, 15,
│ │ │ │ +     19, 20}, {0, 1, 2, 3, 4, 11, 6, 5, 8, 9, 10, 7, 12, 13, 14, 15, 16, 17,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     16, 17, 18, 19, 20}, {16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0,
│ │ │ │ +     18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     10, 19, 13, 20, 6, 9, 4}, {2, 1, 0, 3, 4, 5, 6, 7, 13, 9, 10, 11, 16, 8,
│ │ │ │ +     16, 18, 9, 4, 6}, {3, 1, 12, 4, 0, 11, 13, 5, 2, 16, 10, 7, 8, 15, 14,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     14, 15, 12, 17, 18, 19, 20}, {12, 16, 14, 19, 4, 3, 6, 15, 1, 9, 0, 17,
│ │ │ │ +     6, 17, 9, 18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     10, 2, 13, 20, 8, 18, 7, 11, 5}, {0, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10,
│ │ │ │ +     14, 20, 16, 18, 9, 4, 6}, {1, 7, 12, 19, 16, 4, 0, 6, 2, 13, 11, 9, 8,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     20, 5, 13, 14, 3, 16, 15, 6, 9, 4}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10,
│ │ │ │ +     10, 5, 20, 14, 18, 3, 15, 17}, {1, 16, 12, 19, 11, 15, 5, 17, 2, 7, 0,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     11, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {12, 16, 7, 17, 10, 9, 14, 4,
│ │ │ │ +     3, 8, 10, 13, 20, 14, 18, 9, 4, 6}, {2, 0, 1, 3, 6, 5, 9, 7, 10, 4, 13,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 1, 2, 3, 9, 11, 4, 5,
│ │ │ │ +     11, 14, 8, 16, 15, 12, 17, 18, 19, 20}, {3, 16, 12, 4, 10, 11, 14, 5, 2,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     8, 6, 10, 7, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {16, 7, 12, 19, 9, 1,
│ │ │ │ +     1, 0, 7, 8, 15, 13, 6, 17, 9, 18, 19, 20}, {12, 1, 7, 17, 16, 9, 0, 4,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     4, 10, 2, 6, 11, 14, 8, 0, 5, 20, 13, 18, 3, 15, 17}, {0, 1, 2, 3, 4, 5,
│ │ │ │ +     11, 13, 10, 6, 5, 2, 14, 3, 8, 15, 20, 18, 19}, {0, 1, 3, 19, 12, 11, 2,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {12, 16, 3, 19,
│ │ │ │ +     5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {17, 16, 14, 9, 8, 7,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     10, 11, 14, 5, 15, 1, 0, 7, 17, 2, 13, 20, 8, 18, 9, 4, 6}, {0, 9, 2, 3,
│ │ │ │ +     12, 11, 1, 2, 0, 5, 10, 3, 13, 4, 15, 6, 20, 18, 19}, {1, 16, 12, 19,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     10, 5, 14, 7, 8, 1, 4, 11, 12, 13, 6, 15, 16, 17, 18, 19, 20}, {0, 1, 2,
│ │ │ │ +     11, 15, 5, 17, 2, 7, 0, 3, 8, 10, 13, 20, 14, 18, 9, 4, 6}, {0, 1, 2, 3,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {12,
│ │ │ │ +     4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {2, 0, 9,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     1, 3, 19, 0, 11, 13, 5, 15, 16, 10, 7, 17, 2, 14, 20, 8, 18, 9, 4, 6},
│ │ │ │ +     3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12, 17, 18, 19, 20}, {0, 9,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4,
│ │ │ │ +     2, 3, 10, 5, 14, 7, 8, 1, 4, 11, 12, 13, 6, 15, 16, 17, 18, 19, 20},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     6}, {4, 1, 2, 3, 0, 5, 13, 7, 8, 16, 10, 11, 12, 6, 14, 15, 9, 17, 18,
│ │ │ │ +     {16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13, 20, 6, 9,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16,
│ │ │ │ +     4}, {2, 1, 0, 3, 4, 5, 6, 7, 13, 9, 10, 11, 16, 8, 14, 15, 12, 17, 18,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     18, 9, 4, 6}, {0, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 13, 14, 3,
│ │ │ │ +     19, 20}, {12, 16, 14, 19, 4, 3, 6, 15, 1, 9, 0, 17, 10, 2, 13, 20, 8,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     16, 15, 6, 9, 4}, {0, 14, 12, 19, 11, 15, 5, 17, 2, 7, 1, 3, 8, 13, 10,
│ │ │ │ +     18, 7, 11, 5}, {0, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 13, 14, 3,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     20, 16, 18, 9, 4, 6}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16,
│ │ │ │ +     16, 15, 6, 9, 4}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     15, 12, 17, 20, 18, 19}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17,
│ │ │ │ +     18, 16, 19, 15, 17, 3}, {12, 16, 7, 17, 10, 9, 14, 4, 11, 1, 0, 6, 5, 2,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     13, 14, 20, 16, 18, 9, 4, 6}, {16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5,
│ │ │ │ +     13, 3, 8, 15, 20, 18, 19}, {0, 1, 2, 3, 9, 11, 4, 5, 8, 6, 10, 7, 12,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     15, 0, 10, 19, 13, 20, 6, 9, 4}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11,
│ │ │ │ +     13, 14, 15, 16, 17, 18, 19, 20}, {16, 7, 12, 19, 9, 1, 4, 10, 2, 6, 11,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     12, 13, 14, 18, 16, 19, 15, 17, 3}, {3, 16, 14, 4, 5, 2, 7, 8, 1, 11, 0,
│ │ │ │ +     14, 8, 0, 5, 20, 13, 18, 3, 15, 17}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     12, 10, 15, 13, 6, 17, 9, 18, 19, 20}, {12, 16, 14, 19, 4, 7, 6, 11, 1,
│ │ │ │ +     11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {12, 16, 3, 19, 10, 11, 14, 5,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     9, 0, 5, 10, 2, 13, 20, 8, 18, 3, 15, 17}, {16, 1, 7, 17, 12, 18, 2, 19,
│ │ │ │ +     15, 1, 0, 7, 17, 2, 13, 20, 8, 18, 9, 4, 6}, {0, 9, 2, 3, 10, 5, 14, 7,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     11, 8, 10, 20, 5, 0, 14, 3, 13, 15, 6, 9, 4}, {0, 1, 3, 19, 12, 5, 2, 7,
│ │ │ │ +     8, 1, 4, 11, 12, 13, 6, 15, 16, 17, 18, 19, 20}, {0, 1, 2, 3, 4, 5, 6,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     15, 8, 10, 11, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {12, 16, 7, 17, 10, 9,
│ │ │ │ +     7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {12, 1, 3, 19, 0,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     14, 4, 11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 18, 19, 20}, {12, 16, 14, 17, 4,
│ │ │ │ +     11, 13, 5, 15, 16, 10, 7, 17, 2, 14, 20, 8, 18, 9, 4, 6}, {0, 1, 3, 19,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     7, 6, 11, 1, 9, 0, 5, 10, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 1, 3, 19,
│ │ │ │ +     12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {4, 1, 2,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0, 1, 2,
│ │ │ │ +     3, 0, 5, 13, 7, 8, 16, 10, 11, 12, 6, 14, 15, 9, 17, 18, 19, 20}, {0, 1,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {4,
│ │ │ │ +     3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     16, 14, 3, 5, 2, 7, 8, 1, 11, 0, 12, 10, 6, 13, 15, 9, 17, 18, 19, 20},
│ │ │ │ +     1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 13, 14, 3, 16, 15, 6, 9, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {12, 16, 7, 17, 10, 9, 14, 4, 11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 20, 18,
│ │ │ │ +     {0, 14, 12, 19, 11, 15, 5, 17, 2, 7, 1, 3, 8, 13, 10, 20, 16, 18, 9, 4,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     19}, {0, 11, 2, 3, 10, 4, 14, 6, 8, 1, 5, 9, 12, 13, 7, 15, 16, 17, 18,
│ │ │ │ +     6}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12, 17, 20,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     19, 20}, {16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13,
│ │ │ │ +     18, 19}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     20, 6, 9, 4}}
│ │ │ │ +     18, 9, 4, 6}}
│ │ │ │  
│ │ │ │  o9 : List
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  This is still somewhat experimental.
│ │ │ │  ********** WWaayyss ttoo uussee mmoonnooddrroommyyGGrroouupp:: **********
│ │ │ │      * monodromyGroup(System)
│ │ │ │      * monodromyGroup(System,AbstractPoint,List)
│ │ ├── ./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/M2-53603-0/0-in.ms -o /tmp/M2-53603-0/0-out.ms
│ │ │ + -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-84646-0/0-in.ms -o /tmp/M2-84646-0/0-out.ms
│ │ │  
│ │ │  --------------- INPUT DATA ---------------
│ │ │  #variables                       3
│ │ │  #equations                       3
│ │ │  #invalid equations               0
│ │ │  field characteristic             0
│ │ │  homogeneous input?               1
│ │ │ @@ -28,15 +28,15 @@
│ │ │  initial hash table size     131072 (2^17)
│ │ │  max pair selection             ALL
│ │ │  reduce gb                        1
│ │ │  #threads                         6
│ │ │  info level                       2
│ │ │  generate pbm files               0
│ │ │  ------------------------------------------
│ │ │ -Initial prime = 1235341781
│ │ │ +Initial prime = 1166555107
│ │ │  
│ │ │  Legend for f4 information
│ │ │  --------------------------------------------------------
│ │ │  deg       current degree of pairs selected in this round
│ │ │  sel       number of pairs selected in this round
│ │ │  pairs     total number of pairs in pair list
│ │ │  mat       matrix dimensions (# rows x # columns)
│ │ │ @@ -46,25 +46,25 @@
│ │ │  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         0.06 | 0.11         
│ │ │ +reduce final basis        3 x 3          33.33%        3 new       0 zero         0.00 | 0.00         
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  
│ │ │  ---------------- TIMINGS ----------------
│ │ │ -overall(elapsed)        0.11 sec
│ │ │ -overall(cpu)            0.22 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                  0.05 sec  43.2%
│ │ │ -convert                 0.06 sec  56.7%
│ │ │ -linear algebra          0.00 sec   0.0%
│ │ │ +symbolic prep.          0.00 sec   0.5%
│ │ │ +update                  0.00 sec  72.8%
│ │ │ +convert                 0.00 sec   3.2%
│ │ │ +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
│ │ │ @@ -78,18 +78,18 @@
│ │ │  -----------------------------------------
│ │ │  
│ │ │  
│ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │  [3]
│ │ │  #polynomials to lift              3
│ │ │  -----------------------------------------
│ │ │ -New prime = 1138309223
│ │ │ +New prime = 1152244237
│ │ │  
│ │ │  ---------------- TIMINGS ----------------
│ │ │ -multi-mod overall(elapsed)      0.03 sec
│ │ │ +multi-mod overall(elapsed)      0.00 sec
│ │ │  learning phase                  0.00 Gops/sec
│ │ │  application phase               0.00 Gops/sec
│ │ │  -----------------------------------------
│ │ │  
│ │ │  multi-modular steps
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  {1}{2}<100.00%> 
│ │ │ @@ -105,15 +105,15 @@
│ │ │  ---------------- TIMINGS ----------------
│ │ │  CRT     (elapsed)               0.00 sec
│ │ │  ratrecon(elapsed)               0.00 sec
│ │ │  -----------------------------------------
│ │ │  
│ │ │  
│ │ │  ------------------------------------------------------------------------------------
│ │ │ -msolve overall time           0.26 sec (elapsed) /  0.60 sec (cpu)
│ │ │ +msolve overall time           0.01 sec (elapsed) /  0.04 sec (cpu)
│ │ │  ------------------------------------------------------------------------------------
│ │ │  
│ │ │  o3 = | z y x |
│ │ │  
│ │ │               1      3
│ │ │  o3 : Matrix R  <-- R
│ │ ├── ./usr/share/doc/Macaulay2/Msolve/example-output/_msolve__Real__Solutions.out
│ │ │ @@ -11,111 +11,111 @@
│ │ │               2       2
│ │ │  o2 = ideal (x  - x, y  - 5)
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : rationalIntervalSols = msolveRealSolutions I
│ │ │  
│ │ │ -                    3394316794873318371           
│ │ │ -o3 = {{{- ---------------------------------------,
│ │ │ -          340282366920938463463374607431768211456 
│ │ │ +        18446744073709551615  18446744073709551617    41248173712355948587 
│ │ │ +o3 = {{{--------------------, --------------------}, {--------------------,
│ │ │ +        18446744073709551616  18446744073709551616    18446744073709551616 
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     10312043428088987147                  1246751016747761987           
│ │ │ +     --------------------}}, {{- ---------------------------------------,
│ │ │ +      4611686018427387904        340282366920938463463374607431768211456 
│ │ │       ------------------------------------------------------------------------
│ │ │ -               9090979498917902541              41248173712355948585 
│ │ │ +               1216753468475198861              41248173712355948587 
│ │ │       ---------------------------------------}, {--------------------,
│ │ │       340282366920938463463374607431768211456    18446744073709551616 
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20624086856177974295      18446744073709551615  18446744073709551617  
│ │ │ +     41248173712355948589      18446744073709551615  18446744073709551617  
│ │ │       --------------------}}, {{--------------------, --------------------},
│ │ │ -      9223372036854775808      18446744073709551616  18446744073709551616  
│ │ │ +     18446744073709551616      18446744073709551616  18446744073709551616  
│ │ │       ------------------------------------------------------------------------
│ │ │ -      41248173712355948587  10312043428088987147       
│ │ │ -     {--------------------, --------------------}}, {{-
│ │ │ -      18446744073709551616   4611686018427387904       
│ │ │ +        10312043428088987147    41248173712355948587       
│ │ │ +     {- --------------------, - --------------------}}, {{-
│ │ │ +         4611686018427387904    18446744073709551616       
│ │ │       ------------------------------------------------------------------------
│ │ │ -               27943880522800669215          
│ │ │ +                713863252272186849           
│ │ │       ---------------------------------------,
│ │ │ -     340282366920938463463374607431768211456 
│ │ │ +     170141183460469231731687303715884105728 
│ │ │       ------------------------------------------------------------------------
│ │ │ -               22702554030152245153               41248173712355948597   
│ │ │ +                765114892922892901                41248173712355948589   
│ │ │       ---------------------------------------}, {- --------------------, -
│ │ │       340282366920938463463374607431768211456      18446744073709551616   
│ │ │       ------------------------------------------------------------------------
│ │ │ -     41248173712355948577      9223372036854775807  9223372036854775809     
│ │ │ -     --------------------}}, {{-------------------, -------------------}, {-
│ │ │ -     18446744073709551616      9223372036854775808  9223372036854775808     
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     41248173712355948591    41248173712355948585
│ │ │ -     --------------------, - --------------------}}}
│ │ │ -     18446744073709551616    18446744073709551616
│ │ │ +     41248173712355948587
│ │ │ +     --------------------}}}
│ │ │ +     18446744073709551616
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : rationalApproxSols = msolveRealSolutions(I, QQ)
│ │ │  
│ │ │ -                 2848331352022292085            82496347424711897175      
│ │ │ -o4 = {{---------------------------------------, --------------------}, {1,
│ │ │ -       340282366920938463463374607431768211456  36893488147419103232      
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     82496347424711897175                2620663246324212031             
│ │ │ -     --------------------}, {- ---------------------------------------, -
│ │ │ -     36893488147419103232      340282366920938463463374607431768211456   
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     41248173712355948587         10312043428088987147
│ │ │ -     --------------------}, {1, - --------------------}}
│ │ │ -     18446744073709551616          4611686018427387904
│ │ │ +          82496347424711897175                 14998774136281563            
│ │ │ +o4 = {{1, --------------------}, {- ---------------------------------------,
│ │ │ +          36893488147419103232      340282366920938463463374607431768211456 
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     10312043428088987147         82496347424711897175     
│ │ │ +     --------------------}, {1, - --------------------}, {-
│ │ │ +      4611686018427387904         36893488147419103232     
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +                662611611621480797              10312043428088987147
│ │ │ +     ---------------------------------------, - --------------------}}
│ │ │ +     680564733841876926926749214863536422912     4611686018427387904
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : floatIntervalSols = msolveRealSolutions(I, RRi)
│ │ │  
│ │ │ -o5 = {{[1,1], [2.23607,2.23607]}, {[-8.59181e-22,1.1497e-21],
│ │ │ +o5 = {{[1,1], [2.23607,2.23607]}, {[-3.66387e-21,3.57572e-21],
│ │ │       ------------------------------------------------------------------------
│ │ │       [2.23607,2.23607]}, {[1,1], [-2.23607,-2.23607]},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {[-2.79363e-21,6.6574e-21], [-2.23607,-2.23607]}}
│ │ │ +     {[-4.19571e-21,2.24847e-21], [-2.23607,-2.23607]}}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : floatIntervalSols = msolveRealSolutions(I, RRi_10)
│ │ │  
│ │ │ -o6 = {{[.999999,1], [2.23607,2.23607]}, {[-2.3434e-7,3.31111e-7],
│ │ │ +o6 = {{[.999999,1], [2.23607,2.23607]}, {[-9.85169e-7,.0000010526],
│ │ │       ------------------------------------------------------------------------
│ │ │ -     [2.23606,2.23607]}, {[.999998,1], [-2.23607,-2.23606]},
│ │ │ +     [2.23606,2.23608]}, {[.999999,1], [-2.23607,-2.23607]},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {[-5.92081e-9,4.47108e-9], [-2.23607,-2.23607]}}
│ │ │ +     {[-7.05173e-7,.0000011087], [-2.23607,-2.23606]}}
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : floatApproxSols = msolveRealSolutions(I, RR)
│ │ │  
│ │ │ -o7 = {{1, -2.23607}, {1.03407e-19, -2.23607}, {1, 2.23607}, {2.40267e-22,
│ │ │ +o7 = {{1, 2.23607}, {-4.40774e-23, 2.23607}, {1, -2.23607}, {-9.7362e-22,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     2.23607}}
│ │ │ +     -2.23607}}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : floatApproxSols = msolveRealSolutions(I, RR_10)
│ │ │  
│ │ │ -o8 = {{-1.92693e-9, 2.23607}, {1, 2.23607}, {-2.2555e-8, -2.23607}, {1,
│ │ │ +o8 = {{1, 2.23607}, {3.37145e-8, 2.23607}, {1, -2.23607}, {2.01766e-7,
│ │ │       ------------------------------------------------------------------------
│ │ │       -2.23607}}
│ │ │  
│ │ │  o8 : List
│ │ │  
│ │ │  i9 : I = ideal {(x-1)*x^3, (y^2-5)^2}
│ │ │  
│ │ │               4    3   4      2
│ │ │  o9 = ideal (x  - x , y  - 10y  + 25)
│ │ │  
│ │ │  o9 : Ideal of R
│ │ │  
│ │ │  i10 : floatApproxSols = msolveRealSolutions(I, RRi)
│ │ │  
│ │ │ -o10 = {{[-3.61574e-21,5.06446e-21], [2.23607,2.23607]}, {[1,1],
│ │ │ +o10 = {{[1,1], [2.23607,2.23607]}, {[-3.66387e-21,3.57572e-21],
│ │ │        -----------------------------------------------------------------------
│ │ │ -      [2.23607,2.23607]}, {[-6.42531e-29,4.90632e-29], [-2.23607,-2.23607]},
│ │ │ +      [2.23607,2.23607]}, {[1,1], [-2.23607,-2.23607]},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {[1,1], [-2.23607,-2.23607]}}
│ │ │ +      {[-4.19571e-21,2.24847e-21], [-2.23607,-2.23607]}}
│ │ │  
│ │ │  o10 : List
│ │ │  
│ │ │  i11 :
│ │ ├── ./usr/share/doc/Macaulay2/Msolve/html/_msolve__Real__Solutions.html
│ │ │ @@ -100,110 +100,110 @@
│ │ │  o2 : Ideal of R
    
│ │ │              
    
│ │ │                
    i3 : rationalIntervalSols = msolveRealSolutions I
│ │ │  
│ │ │ -                    3394316794873318371           
│ │ │ -o3 = {{{- ---------------------------------------,
│ │ │ -          340282366920938463463374607431768211456 
│ │ │ +        18446744073709551615  18446744073709551617    41248173712355948587 
│ │ │ +o3 = {{{--------------------, --------------------}, {--------------------,
│ │ │ +        18446744073709551616  18446744073709551616    18446744073709551616 
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     10312043428088987147                  1246751016747761987           
│ │ │ +     --------------------}}, {{- ---------------------------------------,
│ │ │ +      4611686018427387904        340282366920938463463374607431768211456 
│ │ │       ------------------------------------------------------------------------
│ │ │ -               9090979498917902541              41248173712355948585 
│ │ │ +               1216753468475198861              41248173712355948587 
│ │ │       ---------------------------------------}, {--------------------,
│ │ │       340282366920938463463374607431768211456    18446744073709551616 
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20624086856177974295      18446744073709551615  18446744073709551617  
│ │ │ +     41248173712355948589      18446744073709551615  18446744073709551617  
│ │ │       --------------------}}, {{--------------------, --------------------},
│ │ │ -      9223372036854775808      18446744073709551616  18446744073709551616  
│ │ │ +     18446744073709551616      18446744073709551616  18446744073709551616  
│ │ │       ------------------------------------------------------------------------
│ │ │ -      41248173712355948587  10312043428088987147       
│ │ │ -     {--------------------, --------------------}}, {{-
│ │ │ -      18446744073709551616   4611686018427387904       
│ │ │ +        10312043428088987147    41248173712355948587       
│ │ │ +     {- --------------------, - --------------------}}, {{-
│ │ │ +         4611686018427387904    18446744073709551616       
│ │ │       ------------------------------------------------------------------------
│ │ │ -               27943880522800669215          
│ │ │ +                713863252272186849           
│ │ │       ---------------------------------------,
│ │ │ -     340282366920938463463374607431768211456 
│ │ │ +     170141183460469231731687303715884105728 
│ │ │       ------------------------------------------------------------------------
│ │ │ -               22702554030152245153               41248173712355948597   
│ │ │ +                765114892922892901                41248173712355948589   
│ │ │       ---------------------------------------}, {- --------------------, -
│ │ │       340282366920938463463374607431768211456      18446744073709551616   
│ │ │       ------------------------------------------------------------------------
│ │ │ -     41248173712355948577      9223372036854775807  9223372036854775809     
│ │ │ -     --------------------}}, {{-------------------, -------------------}, {-
│ │ │ -     18446744073709551616      9223372036854775808  9223372036854775808     
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     41248173712355948591    41248173712355948585
│ │ │ -     --------------------, - --------------------}}}
│ │ │ -     18446744073709551616    18446744073709551616
│ │ │ +     41248173712355948587
│ │ │ +     --------------------}}}
│ │ │ +     18446744073709551616
│ │ │  
│ │ │  o3 : List
    
│ │ │              
    
│ │ │                
    i4 : rationalApproxSols = msolveRealSolutions(I, QQ)
│ │ │  
│ │ │ -                 2848331352022292085            82496347424711897175      
│ │ │ -o4 = {{---------------------------------------, --------------------}, {1,
│ │ │ -       340282366920938463463374607431768211456  36893488147419103232      
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     82496347424711897175                2620663246324212031             
│ │ │ -     --------------------}, {- ---------------------------------------, -
│ │ │ -     36893488147419103232      340282366920938463463374607431768211456   
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     41248173712355948587         10312043428088987147
│ │ │ -     --------------------}, {1, - --------------------}}
│ │ │ -     18446744073709551616          4611686018427387904
│ │ │ +          82496347424711897175                 14998774136281563            
│ │ │ +o4 = {{1, --------------------}, {- ---------------------------------------,
│ │ │ +          36893488147419103232      340282366920938463463374607431768211456 
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     10312043428088987147         82496347424711897175     
│ │ │ +     --------------------}, {1, - --------------------}, {-
│ │ │ +      4611686018427387904         36893488147419103232     
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +                662611611621480797              10312043428088987147
│ │ │ +     ---------------------------------------, - --------------------}}
│ │ │ +     680564733841876926926749214863536422912     4611686018427387904
│ │ │  
│ │ │  o4 : List
    
│ │ │              
    
│ │ │                
    i5 : floatIntervalSols = msolveRealSolutions(I, RRi)
│ │ │  
│ │ │ -o5 = {{[1,1], [2.23607,2.23607]}, {[-8.59181e-22,1.1497e-21],
│ │ │ +o5 = {{[1,1], [2.23607,2.23607]}, {[-3.66387e-21,3.57572e-21],
│ │ │       ------------------------------------------------------------------------
│ │ │       [2.23607,2.23607]}, {[1,1], [-2.23607,-2.23607]},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {[-2.79363e-21,6.6574e-21], [-2.23607,-2.23607]}}
│ │ │ +     {[-4.19571e-21,2.24847e-21], [-2.23607,-2.23607]}}
│ │ │  
│ │ │  o5 : List
    
│ │ │              
    
│ │ │                
    i6 : floatIntervalSols = msolveRealSolutions(I, RRi_10)
│ │ │  
│ │ │ -o6 = {{[.999999,1], [2.23607,2.23607]}, {[-2.3434e-7,3.31111e-7],
│ │ │ +o6 = {{[.999999,1], [2.23607,2.23607]}, {[-9.85169e-7,.0000010526],
│ │ │       ------------------------------------------------------------------------
│ │ │ -     [2.23606,2.23607]}, {[.999998,1], [-2.23607,-2.23606]},
│ │ │ +     [2.23606,2.23608]}, {[.999999,1], [-2.23607,-2.23607]},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {[-5.92081e-9,4.47108e-9], [-2.23607,-2.23607]}}
│ │ │ +     {[-7.05173e-7,.0000011087], [-2.23607,-2.23606]}}
│ │ │  
│ │ │  o6 : List
    
│ │ │              
    
│ │ │                
    i7 : floatApproxSols = msolveRealSolutions(I, RR)
│ │ │  
│ │ │ -o7 = {{1, -2.23607}, {1.03407e-19, -2.23607}, {1, 2.23607}, {2.40267e-22,
│ │ │ +o7 = {{1, 2.23607}, {-4.40774e-23, 2.23607}, {1, -2.23607}, {-9.7362e-22,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     2.23607}}
│ │ │ +     -2.23607}}
│ │ │  
│ │ │  o7 : List
    
│ │ │              
    
│ │ │                
    i8 : floatApproxSols = msolveRealSolutions(I, RR_10)
│ │ │  
│ │ │ -o8 = {{-1.92693e-9, 2.23607}, {1, 2.23607}, {-2.2555e-8, -2.23607}, {1,
│ │ │ +o8 = {{1, 2.23607}, {3.37145e-8, 2.23607}, {1, -2.23607}, {2.01766e-7,
│ │ │       ------------------------------------------------------------------------
│ │ │       -2.23607}}
│ │ │  
│ │ │  o8 : List
    
│ │ │              
    
│ │ │ @@ -221,19 +221,19 @@
│ │ │  o9 : Ideal of R
    
│ │ │              
    
│ │ │                
    i10 : floatApproxSols = msolveRealSolutions(I, RRi)
│ │ │  
│ │ │ -o10 = {{[-3.61574e-21,5.06446e-21], [2.23607,2.23607]}, {[1,1],
│ │ │ +o10 = {{[1,1], [2.23607,2.23607]}, {[-3.66387e-21,3.57572e-21],
│ │ │        -----------------------------------------------------------------------
│ │ │ -      [2.23607,2.23607]}, {[-6.42531e-29,4.90632e-29], [-2.23607,-2.23607]},
│ │ │ +      [2.23607,2.23607]}, {[1,1], [-2.23607,-2.23607]},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {[1,1], [-2.23607,-2.23607]}}
│ │ │ +      {[-4.19571e-21,2.24847e-21], [-2.23607,-2.23607]}}
│ │ │  
│ │ │  o10 : List
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -43,90 +43,90 @@
│ │ │ │  
│ │ │ │               2       2
│ │ │ │  o2 = ideal (x  - x, y  - 5)
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : rationalIntervalSols = msolveRealSolutions I
│ │ │ │  
│ │ │ │ -                    3394316794873318371
│ │ │ │ -o3 = {{{- ---------------------------------------,
│ │ │ │ -          340282366920938463463374607431768211456
│ │ │ │ +        18446744073709551615  18446744073709551617    41248173712355948587
│ │ │ │ +o3 = {{{--------------------, --------------------}, {--------------------,
│ │ │ │ +        18446744073709551616  18446744073709551616    18446744073709551616
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     10312043428088987147                  1246751016747761987
│ │ │ │ +     --------------------}}, {{- ---------------------------------------,
│ │ │ │ +      4611686018427387904        340282366920938463463374607431768211456
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -               9090979498917902541              41248173712355948585
│ │ │ │ +               1216753468475198861              41248173712355948587
│ │ │ │       ---------------------------------------}, {--------------------,
│ │ │ │       340282366920938463463374607431768211456    18446744073709551616
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     20624086856177974295      18446744073709551615  18446744073709551617
│ │ │ │ +     41248173712355948589      18446744073709551615  18446744073709551617
│ │ │ │       --------------------}}, {{--------------------, --------------------},
│ │ │ │ -      9223372036854775808      18446744073709551616  18446744073709551616
│ │ │ │ +     18446744073709551616      18446744073709551616  18446744073709551616
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -      41248173712355948587  10312043428088987147
│ │ │ │ -     {--------------------, --------------------}}, {{-
│ │ │ │ -      18446744073709551616   4611686018427387904
│ │ │ │ +        10312043428088987147    41248173712355948587
│ │ │ │ +     {- --------------------, - --------------------}}, {{-
│ │ │ │ +         4611686018427387904    18446744073709551616
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -               27943880522800669215
│ │ │ │ +                713863252272186849
│ │ │ │       ---------------------------------------,
│ │ │ │ -     340282366920938463463374607431768211456
│ │ │ │ +     170141183460469231731687303715884105728
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -               22702554030152245153               41248173712355948597
│ │ │ │ +                765114892922892901                41248173712355948589
│ │ │ │       ---------------------------------------}, {- --------------------, -
│ │ │ │       340282366920938463463374607431768211456      18446744073709551616
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     41248173712355948577      9223372036854775807  9223372036854775809
│ │ │ │ -     --------------------}}, {{-------------------, -------------------}, {-
│ │ │ │ -     18446744073709551616      9223372036854775808  9223372036854775808
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     41248173712355948591    41248173712355948585
│ │ │ │ -     --------------------, - --------------------}}}
│ │ │ │ -     18446744073709551616    18446744073709551616
│ │ │ │ +     41248173712355948587
│ │ │ │ +     --------------------}}}
│ │ │ │ +     18446744073709551616
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : rationalApproxSols = msolveRealSolutions(I, QQ)
│ │ │ │  
│ │ │ │ -                 2848331352022292085            82496347424711897175
│ │ │ │ -o4 = {{---------------------------------------, --------------------}, {1,
│ │ │ │ -       340282366920938463463374607431768211456  36893488147419103232
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     82496347424711897175                2620663246324212031
│ │ │ │ -     --------------------}, {- ---------------------------------------, -
│ │ │ │ -     36893488147419103232      340282366920938463463374607431768211456
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     41248173712355948587         10312043428088987147
│ │ │ │ -     --------------------}, {1, - --------------------}}
│ │ │ │ -     18446744073709551616          4611686018427387904
│ │ │ │ +          82496347424711897175                 14998774136281563
│ │ │ │ +o4 = {{1, --------------------}, {- ---------------------------------------,
│ │ │ │ +          36893488147419103232      340282366920938463463374607431768211456
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     10312043428088987147         82496347424711897175
│ │ │ │ +     --------------------}, {1, - --------------------}, {-
│ │ │ │ +      4611686018427387904         36893488147419103232
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +                662611611621480797              10312043428088987147
│ │ │ │ +     ---------------------------------------, - --------------------}}
│ │ │ │ +     680564733841876926926749214863536422912     4611686018427387904
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : floatIntervalSols = msolveRealSolutions(I, RRi)
│ │ │ │  
│ │ │ │ -o5 = {{[1,1], [2.23607,2.23607]}, {[-8.59181e-22,1.1497e-21],
│ │ │ │ +o5 = {{[1,1], [2.23607,2.23607]}, {[-3.66387e-21,3.57572e-21],
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       [2.23607,2.23607]}, {[1,1], [-2.23607,-2.23607]},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {[-2.79363e-21,6.6574e-21], [-2.23607,-2.23607]}}
│ │ │ │ +     {[-4.19571e-21,2.24847e-21], [-2.23607,-2.23607]}}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : floatIntervalSols = msolveRealSolutions(I, RRi_10)
│ │ │ │  
│ │ │ │ -o6 = {{[.999999,1], [2.23607,2.23607]}, {[-2.3434e-7,3.31111e-7],
│ │ │ │ +o6 = {{[.999999,1], [2.23607,2.23607]}, {[-9.85169e-7,.0000010526],
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     [2.23606,2.23607]}, {[.999998,1], [-2.23607,-2.23606]},
│ │ │ │ +     [2.23606,2.23608]}, {[.999999,1], [-2.23607,-2.23607]},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {[-5.92081e-9,4.47108e-9], [-2.23607,-2.23607]}}
│ │ │ │ +     {[-7.05173e-7,.0000011087], [-2.23607,-2.23606]}}
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : floatApproxSols = msolveRealSolutions(I, RR)
│ │ │ │  
│ │ │ │ -o7 = {{1, -2.23607}, {1.03407e-19, -2.23607}, {1, 2.23607}, {2.40267e-22,
│ │ │ │ +o7 = {{1, 2.23607}, {-4.40774e-23, 2.23607}, {1, -2.23607}, {-9.7362e-22,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     2.23607}}
│ │ │ │ +     -2.23607}}
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  i8 : floatApproxSols = msolveRealSolutions(I, RR_10)
│ │ │ │  
│ │ │ │ -o8 = {{-1.92693e-9, 2.23607}, {1, 2.23607}, {-2.2555e-8, -2.23607}, {1,
│ │ │ │ +o8 = {{1, 2.23607}, {3.37145e-8, 2.23607}, {1, -2.23607}, {2.01766e-7,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       -2.23607}}
│ │ │ │  
│ │ │ │  o8 : List
│ │ │ │  Note in cases where solutions have multiplicity this is not reflected in the
│ │ │ │  output. While the solver does not return multiplicities, it reliably outputs
│ │ │ │  the verified isolating intervals for multiple solutions.
│ │ │ │ @@ -134,19 +134,19 @@
│ │ │ │  
│ │ │ │               4    3   4      2
│ │ │ │  o9 = ideal (x  - x , y  - 10y  + 25)
│ │ │ │  
│ │ │ │  o9 : Ideal of R
│ │ │ │  i10 : floatApproxSols = msolveRealSolutions(I, RRi)
│ │ │ │  
│ │ │ │ -o10 = {{[-3.61574e-21,5.06446e-21], [2.23607,2.23607]}, {[1,1],
│ │ │ │ +o10 = {{[1,1], [2.23607,2.23607]}, {[-3.66387e-21,3.57572e-21],
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      [2.23607,2.23607]}, {[-6.42531e-29,4.90632e-29], [-2.23607,-2.23607]},
│ │ │ │ +      [2.23607,2.23607]}, {[1,1], [-2.23607,-2.23607]},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {[1,1], [-2.23607,-2.23607]}}
│ │ │ │ +      {[-4.19571e-21,2.24847e-21], [-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
│ │ │ @@ -78,15 +78,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │              
    
│ │ │                
    i3 : msolveGB(I, Verbosity => 2, Threads => 6) 
│ │ │ - -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-53603-0/0-in.ms -o /tmp/M2-53603-0/0-out.ms
│ │ │ + -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-84646-0/0-in.ms -o /tmp/M2-84646-0/0-out.ms
│ │ │  
│ │ │  --------------- INPUT DATA ---------------
│ │ │  #variables                       3
│ │ │  #equations                       3
│ │ │  #invalid equations               0
│ │ │  field characteristic             0
│ │ │  homogeneous input?               1
│ │ │ @@ -97,15 +97,15 @@
│ │ │  initial hash table size     131072 (2^17)
│ │ │  max pair selection             ALL
│ │ │  reduce gb                        1
│ │ │  #threads                         6
│ │ │  info level                       2
│ │ │  generate pbm files               0
│ │ │  ------------------------------------------
│ │ │ -Initial prime = 1235341781
│ │ │ +Initial prime = 1166555107
│ │ │  
│ │ │  Legend for f4 information
│ │ │  --------------------------------------------------------
│ │ │  deg       current degree of pairs selected in this round
│ │ │  sel       number of pairs selected in this round
│ │ │  pairs     total number of pairs in pair list
│ │ │  mat       matrix dimensions (# rows x # columns)
│ │ │ @@ -115,25 +115,25 @@
│ │ │  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         0.06 | 0.11         
│ │ │ +reduce final basis        3 x 3          33.33%        3 new       0 zero         0.00 | 0.00         
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  
│ │ │  ---------------- TIMINGS ----------------
│ │ │ -overall(elapsed)        0.11 sec
│ │ │ -overall(cpu)            0.22 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                  0.05 sec  43.2%
│ │ │ -convert                 0.06 sec  56.7%
│ │ │ -linear algebra          0.00 sec   0.0%
│ │ │ +symbolic prep.          0.00 sec   0.5%
│ │ │ +update                  0.00 sec  72.8%
│ │ │ +convert                 0.00 sec   3.2%
│ │ │ +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
│ │ │ @@ -147,18 +147,18 @@
│ │ │  -----------------------------------------
│ │ │  
│ │ │  
│ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │  [3]
│ │ │  #polynomials to lift              3
│ │ │  -----------------------------------------
│ │ │ -New prime = 1138309223
│ │ │ +New prime = 1152244237
│ │ │  
│ │ │  ---------------- TIMINGS ----------------
│ │ │ -multi-mod overall(elapsed)      0.03 sec
│ │ │ +multi-mod overall(elapsed)      0.00 sec
│ │ │  learning phase                  0.00 Gops/sec
│ │ │  application phase               0.00 Gops/sec
│ │ │  -----------------------------------------
│ │ │  
│ │ │  multi-modular steps
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  {1}{2}<100.00%> 
│ │ │ @@ -174,15 +174,15 @@
│ │ │  ---------------- TIMINGS ----------------
│ │ │  CRT     (elapsed)               0.00 sec
│ │ │  ratrecon(elapsed)               0.00 sec
│ │ │  -----------------------------------------
│ │ │  
│ │ │  
│ │ │  ------------------------------------------------------------------------------------
│ │ │ -msolve overall time           0.26 sec (elapsed) /  0.60 sec (cpu)
│ │ │ +msolve overall time           0.01 sec (elapsed) /  0.04 sec (cpu)
│ │ │  ------------------------------------------------------------------------------------
│ │ │  
│ │ │  o3 = | z y x |
│ │ │  
│ │ │               1      3
│ │ │  o3 : Matrix R  <-- R
    
│ │ │              
    
│ │ │                
    i5 : peek componentsOfKernel(2, F)
│ │ │  warning: computation begun over finite field. resulting polynomials may not lie in the ideal
│ │ │  computing total degree: 1
│ │ │  number of monomials = 6
│ │ │  number of distinct multidegrees = 6
│ │ │ - -- .00369113s elapsed
│ │ │ + -- .00249226s elapsed
│ │ │  computing total degree: 2
│ │ │  number of monomials = 21
│ │ │  number of distinct multidegrees = 18
│ │ │ - -- .0155544s elapsed
│ │ │ + -- .0106654s elapsed
│ │ │  
│ │ │  o5 = MutableHashTable{{0, 1, 0, 0, 1} => {}                   }
│ │ │                        {0, 1, 0, 1, 0} => {}
│ │ │                        {0, 1, 1, 0, 0} => {}
│ │ │                        {0, 2, 0, 0, 2} => {}
│ │ │                        {0, 2, 0, 1, 1} => {}
│ │ │                        {0, 2, 0, 2, 0} => {}
│ │ │ ├── html2text {}
│ │ │ │ @@ -51,19 +51,19 @@
│ │ │ │  o4 : RingMap S <-- R
│ │ │ │  i5 : peek componentsOfKernel(2, F)
│ │ │ │  warning: computation begun over finite field. resulting polynomials may not lie
│ │ │ │  in the ideal
│ │ │ │  computing total degree: 1
│ │ │ │  number of monomials = 6
│ │ │ │  number of distinct multidegrees = 6
│ │ │ │ - -- .00369113s elapsed
│ │ │ │ + -- .00249226s elapsed
│ │ │ │  computing total degree: 2
│ │ │ │  number of monomials = 21
│ │ │ │  number of distinct multidegrees = 18
│ │ │ │ - -- .0155544s elapsed
│ │ │ │ + -- .0106654s elapsed
│ │ │ │  
│ │ │ │  o5 = MutableHashTable{{0, 1, 0, 0, 1} => {}                   }
│ │ │ │                        {0, 1, 0, 1, 0} => {}
│ │ │ │                        {0, 1, 1, 0, 0} => {}
│ │ │ │                        {0, 2, 0, 0, 2} => {}
│ │ │ │                        {0, 2, 0, 1, 1} => {}
│ │ │ │                        {0, 2, 0, 2, 0} => {}
│ │ ├── ./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
│ │ │ - -- .0238525s elapsed
│ │ │ + -- .028425s elapsed
│ │ │  
│ │ │  o3 = 2
│ │ │  
│ │ │  i4 : elapsedTime monjMult I
│ │ │ - -- .114646s elapsed
│ │ │ + -- .0934295s elapsed
│ │ │  
│ │ │  o4 = 2
│ │ │  
│ │ │  i5 : elapsedTime multiplicitySequence I
│ │ │ - -- .137039s elapsed
│ │ │ + -- .152938s 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
│ │ │ - -- .12587s elapsed
│ │ │ + -- .0886905s 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
│ │ │ - -- .150963s elapsed
│ │ │ + -- .114823s elapsed
│ │ │  
│ │ │  o3 = 192
│ │ │  
│ │ │  i4 : elapsedTime jMult I
│ │ │ - -- 1.50536s elapsed
│ │ │ + -- 1.42555s elapsed
│ │ │  
│ │ │  o4 = 192
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_j__Mult.html
│ │ │ @@ -88,31 +88,31 @@
│ │ │  
│ │ │  o2 : Ideal of R
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime jMult I
│ │ │ - -- .0238525s elapsed
│ │ │ + -- .028425s elapsed
│ │ │  
│ │ │  o3 = 2
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime monjMult I
│ │ │ - -- .114646s elapsed
│ │ │ + -- .0934295s elapsed
│ │ │  
│ │ │  o4 = 2
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime multiplicitySequence I
│ │ │ - -- .137039s elapsed
│ │ │ + -- .152938s elapsed
│ │ │  
│ │ │  o5 = HashTable{2 => 3}
│ │ │                 3 => 2
│ │ │  
│ │ │  o5 : HashTable
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime monAnalyticSpread I
│ │ │ - -- .12587s elapsed
│ │ │ + -- .0886905s elapsed
│ │ │  
│ │ │  o3 = 2
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -22,15 +22,15 @@
│ │ │ │  i2 : I = ideal"x2,xy,y3"
│ │ │ │  
│ │ │ │               2        3
│ │ │ │  o2 = ideal (x , x*y, y )
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : elapsedTime monAnalyticSpread I
│ │ │ │ - -- .12587s elapsed
│ │ │ │ + -- .0886905s 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
│ │ │ @@ -92,23 +92,23 @@
│ │ │  
│ │ │  o2 : Ideal of R
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime monjMult I
│ │ │ - -- .150963s elapsed
│ │ │ + -- .114823s elapsed
│ │ │  
│ │ │  o3 = 192
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime jMult I
│ │ │ - -- 1.50536s elapsed
│ │ │ + -- 1.42555s elapsed
│ │ │  
│ │ │  o4 = 192
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -24,19 +24,19 @@
│ │ │ │  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
│ │ │ │ - -- .150963s elapsed
│ │ │ │ + -- .114823s elapsed
│ │ │ │  
│ │ │ │  o3 = 192
│ │ │ │  i4 : elapsedTime jMult I
│ │ │ │ - -- 1.50536s elapsed
│ │ │ │ + -- 1.42555s 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/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 2.8986s (cpu); 1.72302s (thread); 0s (gc)
│ │ │ + -- used 3.61985s (cpu); 1.97822s (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 3.15417s (cpu); 1.94205s (thread); 0s (gc)
│ │ │ + -- used 3.77703s (cpu); 2.13829s (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 7.83915s (cpu); 4.80537s (thread); 0s (gc)
│ │ │ + -- used 6.64387s (cpu); 4.72889s (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 4.99846s (cpu); 3.67423s (thread); 0s (gc)
│ │ │ + -- used 4.71011s (cpu); 3.92244s (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 0.0924424s (cpu); 0.0925199s (thread); 0s (gc)
│ │ │ + -- used 0.183458s (cpu); 0.127465s (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 1.89973s (cpu); 1.35924s (thread); 0s (gc)
│ │ │ + -- used 2.25009s (cpu); 1.43156s (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 3.56256s (cpu); 2.40057s (thread); 0s (gc)
│ │ │ + -- used 4.96963s (cpu); 2.71989s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 2
│ │ │  
│ │ │  i5 : time degree(Phi,Strategy=>"0-th projective degree")
│ │ │ - -- used 0.331353s (cpu); 0.28086s (thread); 0s (gc)
│ │ │ + -- used 0.388566s (cpu); 0.30027s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 2
│ │ │  
│ │ │  i6 : time degree Phi
│ │ │ - -- used 0.326329s (cpu); 0.269225s (thread); 0s (gc)
│ │ │ + -- used 0.363562s (cpu); 0.278197s (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 0.452916s (cpu); 0.395006s (thread); 0s (gc)
│ │ │ + -- used 0.618659s (cpu); 0.401444s (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 0.00318912s (cpu); 0.000168737s (thread); 0s (gc)
│ │ │ + -- used 0.00314006s (cpu); 0.000180686s (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 0.0014133s (cpu); 0.00021309s (thread); 0s (gc)
│ │ │ + -- used 0.00280647s (cpu); 0.000241696s (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 1.36042s (cpu); 1.01983s (thread); 0s (gc)
│ │ │ + -- used 1.21715s (cpu); 1.00258s (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 0.000156814s (cpu); 0.000370705s (thread); 0s (gc)
│ │ │ + -- used 0.000122347s (cpu); 0.000446119s (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 0.111155s (cpu); 0.0496517s (thread); 0s (gc)
│ │ │ + -- used 0.0931118s (cpu); 0.037018s (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 0.0355309s (cpu); 0.0346335s (thread); 0s (gc)
│ │ │ + -- used 0.107805s (cpu); 0.0616844s (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 0.215498s (cpu); 0.148226s (thread); 0s (gc)
│ │ │ + -- used 0.305142s (cpu); 0.180013s (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 0.199139s (cpu); 0.133717s (thread); 0s (gc)
│ │ │ + -- used 0.140302s (cpu); 0.128387s (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 5.22579s (cpu); 2.68091s (thread); 0s (gc)
│ │ │ + -- used 10.4993s (cpu); 2.77588s (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 0.373003s (cpu); 0.295065s (thread); 0s (gc)
│ │ │ + -- used 0.446548s (cpu); 0.345478s (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 1.40947s (cpu); 1.02276s (thread); 0s (gc)
│ │ │ + -- used 1.31876s (cpu); 1.10132s (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 0.140994s (cpu); 0.0876716s (thread); 0s (gc)
│ │ │ + -- used 0.128258s (cpu); 0.0762614s (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 0.160203s (cpu); 0.0924297s (thread); 0s (gc)
│ │ │ + -- used 0.321423s (cpu); 0.134113s (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 0.442904s (cpu); 0.290422s (thread); 0s (gc)
│ │ │ + -- used 0.642818s (cpu); 0.35201s (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 0.00368849s (cpu); 1.1823e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00373306s (cpu); 1.1672e-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 0.32581s (cpu); 0.169415s (thread); 0s (gc)
│ │ │ + -- used 0.53311s (cpu); 0.220708s (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 1.54364s (cpu); 0.844695s (thread); 0s (gc)
│ │ │ + -- used 1.83477s (cpu); 0.953102s (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 0.352309s (cpu); 0.232452s (thread); 0s (gc)
│ │ │ + -- used 0.525903s (cpu); 0.279373s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = false
│ │ │  
│ │ │  i4 : time Psi = first graph Phi;
│ │ │ - -- used 0.161522s (cpu); 0.101427s (thread); 0s (gc)
│ │ │ + -- used 0.124235s (cpu); 0.0747201s (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 4.17216s (cpu); 3.04184s (thread); 0s (gc)
│ │ │ + -- used 3.84185s (cpu); 3.14294s (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 0.00800035s (cpu); 0.00921257s (thread); 0s (gc)
│ │ │ + -- used 0.116384s (cpu); 0.038162s (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 0.558273s (cpu); 0.439528s (thread); 0s (gc)
│ │ │ + -- used 0.533222s (cpu); 0.453712s (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 0.510118s (cpu); 0.379256s (thread); 0s (gc)
│ │ │ + -- used 0.585522s (cpu); 0.405119s (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 0.00244457s (cpu); 0.0012392s (thread); 0s (gc)
│ │ │ - -- used 0.280015s (cpu); 0.154499s (thread); 0s (gc)
│ │ │ - -- used 0.221312s (cpu); 0.167638s (thread); 0s (gc)
│ │ │ - -- used 0.215238s (cpu); 0.14879s (thread); 0s (gc)
│ │ │ - -- used 0.191957s (cpu); 0.116338s (thread); 0s (gc)
│ │ │ + -- used 0.00250687s (cpu); 0.00137215s (thread); 0s (gc)
│ │ │ + -- used 0.230082s (cpu); 0.149713s (thread); 0s (gc)
│ │ │ + -- used 0.260478s (cpu); 0.177666s (thread); 0s (gc)
│ │ │ + -- used 0.245703s (cpu); 0.159553s (thread); 0s (gc)
│ │ │ + -- used 0.219403s (cpu); 0.133707s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = {51, 28, 14, 6, 2}
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 : time assert(oo == multidegree Phi)
│ │ │ - -- used 0.160984s (cpu); 0.101194s (thread); 0s (gc)
│ │ │ + -- used 0.243937s (cpu); 0.100052s (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 0.0160578s (cpu); 0.0164613s (thread); 0s (gc)
│ │ │ + -- used 0.0779279s (cpu); 0.026532s (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 1.86005s (cpu); 1.10026s (thread); 0s (gc)
│ │ │ + -- used 1.58984s (cpu); 1.06685s (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 0.716809s (cpu); 0.551533s (thread); 0s (gc)
│ │ │ + -- used 1.10689s (cpu); 0.570786s (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 0.32872s (cpu); 0.173534s (thread); 0s (gc)
│ │ │ + -- used 0.24132s (cpu); 0.157477s (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
│ │ │ @@ -103,15 +103,15 @@
│ │ │  
│ │ │  o5 = curve in PP^8 cut out by 17 hypersurfaces of degrees 1^2 2^15 
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time f = X ===> Y;
│ │ │ - -- used 2.8986s (cpu); 1.72302s (thread); 0s (gc)
│ │ │ + -- used 3.61985s (cpu); 1.97822s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : MultirationalMap (automorphism of PP^8)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : f X
│ │ │ @@ -143,15 +143,15 @@
│ │ │  
│ │ │  o10 : ProjectiveVariety, 6-dimensional subvariety of PP^8
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i11 : time g = V ===> W;
│ │ │ - -- used 3.15417s (cpu); 1.94205s (thread); 0s (gc)
│ │ │ + -- used 3.77703s (cpu); 2.13829s (thread); 0s (gc)
│ │ │  
│ │ │  o11 : MultirationalMap (automorphism of PP^8)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : g||W
│ │ │ @@ -252,15 +252,15 @@
│ │ │  
│ │ │  o16 = 6-dimensional subvariety of PP^9 cut out by 5 hypersurfaces of degree 2
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i17 : time h = Z ===> GG_K(1,4)
│ │ │ - -- used 7.83915s (cpu); 4.80537s (thread); 0s (gc)
│ │ │ + -- used 6.64387s (cpu); 4.72889s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = h
│ │ │  
│ │ │  o17 : MultirationalMap (isomorphism from PP^9 to PP^9)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -33,15 +33,15 @@
│ │ │ │  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 2.8986s (cpu); 1.72302s (thread); 0s (gc)
│ │ │ │ + -- used 3.61985s (cpu); 1.97822s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 : MultirationalMap (automorphism of PP^8)
│ │ │ │  i7 : f X
│ │ │ │  
│ │ │ │  o7 = Y
│ │ │ │  
│ │ │ │  o7 : ProjectiveVariety, curve in PP^8
│ │ │ │ @@ -53,15 +53,15 @@
│ │ │ │  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 3.15417s (cpu); 1.94205s (thread); 0s (gc)
│ │ │ │ + -- used 3.77703s (cpu); 2.13829s (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
│ │ │ │ @@ -144,15 +144,15 @@
│ │ │ │  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 7.83915s (cpu); 4.80537s (thread); 0s (gc)
│ │ │ │ + -- used 6.64387s (cpu); 4.72889s (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
│ │ │ @@ -89,15 +89,15 @@
│ │ │  
│ │ │  o3 : ProjectiveVariety, 4-dimensional subvariety of PP^2 x PP^4
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time X = Phi^* Y;
│ │ │ - -- used 4.99846s (cpu); 3.67423s (thread); 0s (gc)
│ │ │ + -- used 4.71011s (cpu); 3.92244s (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
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,15 +26,15 @@
│ │ │ │  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 4.99846s (cpu); 3.67423s (thread); 0s (gc)
│ │ │ │ + -- used 4.71011s (cpu); 3.92244s (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
│ │ │ @@ -95,15 +95,15 @@
│ │ │  
│ │ │  o4 : ProjectiveVariety, 4-dimensional subvariety of PP^4 x PP^7
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time Phi Z;
│ │ │ - -- used 0.0924424s (cpu); 0.0925199s (thread); 0s (gc)
│ │ │ + -- used 0.183458s (cpu); 0.127465s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^7
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : dim oo, degree oo, degrees oo
│ │ │ @@ -112,15 +112,15 @@
│ │ │  
│ │ │  o6 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : time Phi (point Z + point Z + point Z)
│ │ │ - -- used 1.89973s (cpu); 1.35924s (thread); 0s (gc)
│ │ │ + -- used 2.25009s (cpu); 1.43156s (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
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,24 +25,24 @@
│ │ │ │  
│ │ │ │  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 0.0924424s (cpu); 0.0925199s (thread); 0s (gc)
│ │ │ │ + -- used 0.183458s (cpu); 0.127465s (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 1.89973s (cpu); 1.35924s (thread); 0s (gc)
│ │ │ │ + -- used 2.25009s (cpu); 1.43156s (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
│ │ │ @@ -93,31 +93,31 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       multi-degrees (0,2)^1 (1,1)^3 (2,1)^8 (4,0)^1
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time degree(Phi,Strategy=>"random point")
│ │ │ - -- used 3.56256s (cpu); 2.40057s (thread); 0s (gc)
│ │ │ + -- used 4.96963s (cpu); 2.71989s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 2
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time degree(Phi,Strategy=>"0-th projective degree")
│ │ │ - -- used 0.331353s (cpu); 0.28086s (thread); 0s (gc)
│ │ │ + -- used 0.388566s (cpu); 0.30027s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 2
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time degree Phi
│ │ │ - -- used 0.326329s (cpu); 0.269225s (thread); 0s (gc)
│ │ │ + -- used 0.363562s (cpu); 0.278197s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = 2
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    Note, as in the example above, that calculation times may vary depending on the strategy used.
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,23 +27,23 @@
│ │ │ │  
│ │ │ │  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 3.56256s (cpu); 2.40057s (thread); 0s (gc)
│ │ │ │ + -- used 4.96963s (cpu); 2.71989s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 2
│ │ │ │  i5 : time degree(Phi,Strategy=>"0-th projective degree")
│ │ │ │ - -- used 0.331353s (cpu); 0.28086s (thread); 0s (gc)
│ │ │ │ + -- used 0.388566s (cpu); 0.30027s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 2
│ │ │ │  i6 : time degree Phi
│ │ │ │ - -- used 0.326329s (cpu); 0.269225s (thread); 0s (gc)
│ │ │ │ + -- used 0.363562s (cpu); 0.278197s (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
│ │ │ @@ -81,15 +81,15 @@
│ │ │  
│ │ │  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 0.452916s (cpu); 0.395006s (thread); 0s (gc)
│ │ │ + -- used 0.618659s (cpu); 0.401444s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = 1
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,15 +18,15 @@
│ │ │ │  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 0.452916s (cpu); 0.395006s (thread); 0s (gc)
│ │ │ │ + -- used 0.618659s (cpu); 0.401444s (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
│ │ │ @@ -77,15 +77,15 @@
│ │ │  
│ │ │  o1 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 to PP^4 x PP^5)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time ? Phi
│ │ │ - -- used 0.00318912s (cpu); 0.000168737s (thread); 0s (gc)
│ │ │ + -- used 0.00314006s (cpu); 0.000180686s (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
    
│ │ │              
│ │ │ @@ -96,27 +96,27 @@
│ │ │  
│ │ │  o3 : ProjectiveVariety, 4-dimensional subvariety of PP^4 x PP^5
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time ? Phi
│ │ │ - -- used 0.0014133s (cpu); 0.00021309s (thread); 0s (gc)
│ │ │ + -- used 0.00280647s (cpu); 0.000241696s (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 1.36042s (cpu); 1.01983s (thread); 0s (gc)
│ │ │ + -- used 1.21715s (cpu); 1.00258s (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 
│ │ │ @@ -126,15 +126,15 @@
│ │ │       degree sequence (map 2/2): [(0,1), (2,0)]
│ │ │       coefficient ring: ZZ/65521
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time ? Phi
│ │ │ - -- used 0.000156814s (cpu); 0.000370705s (thread); 0s (gc)
│ │ │ + -- used 0.000122347s (cpu); 0.000446119s (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 {}
│ │ │ │ @@ -16,36 +16,36 @@
│ │ │ │  ? 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 0.00318912s (cpu); 0.000168737s (thread); 0s (gc)
│ │ │ │ + -- used 0.00314006s (cpu); 0.000180686s (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 0.0014133s (cpu); 0.00021309s (thread); 0s (gc)
│ │ │ │ + -- used 0.00280647s (cpu); 0.000241696s (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 1.36042s (cpu); 1.01983s (thread); 0s (gc)
│ │ │ │ + -- used 1.21715s (cpu); 1.00258s (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
│ │ │ │ @@ -53,15 +53,15 @@
│ │ │ │  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 0.000156814s (cpu); 0.000370705s (thread); 0s (gc)
│ │ │ │ + -- used 0.000122347s (cpu); 0.000446119s (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
│ │ │ @@ -83,15 +83,15 @@
│ │ │  
│ │ │  o1 : MultirationalMap (dominant rational map from PP^4 to hypersurface in PP^5)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time (Phi1,Phi2) = graph Phi
│ │ │ - -- used 0.111155s (cpu); 0.0496517s (thread); 0s (gc)
│ │ │ + -- used 0.0931118s (cpu); 0.037018s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = (Phi1, Phi2)
│ │ │  
│ │ │  o2 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -107,15 +107,15 @@
│ │ │  
│ │ │  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 0.0355309s (cpu); 0.0346335s (thread); 0s (gc)
│ │ │ + -- used 0.107805s (cpu); 0.0616844s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = (Phi21, Phi22)
│ │ │  
│ │ │  o5 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -131,15 +131,15 @@
│ │ │  
│ │ │  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 0.215498s (cpu); 0.148226s (thread); 0s (gc)
│ │ │ + -- used 0.305142s (cpu); 0.180013s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = (Phi211, Phi212)
│ │ │  
│ │ │  o8 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,43 +19,43 @@
│ │ │ │  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 0.111155s (cpu); 0.0496517s (thread); 0s (gc)
│ │ │ │ + -- used 0.0931118s (cpu); 0.037018s (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 0.0355309s (cpu); 0.0346335s (thread); 0s (gc)
│ │ │ │ + -- used 0.107805s (cpu); 0.0616844s (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 0.215498s (cpu); 0.148226s (thread); 0s (gc)
│ │ │ │ + -- used 0.305142s (cpu); 0.180013s (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
│ │ │ @@ -95,15 +95,15 @@
│ │ │  
│ │ │  o4 : MultirationalMap (rational map from PP^4 to PP^7 x PP^4)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time Z = image Phi;
│ │ │ - -- used 0.199139s (cpu); 0.133717s (thread); 0s (gc)
│ │ │ + -- used 0.140302s (cpu); 0.128387s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^4
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : dim Z, degree Z, degrees Z
│ │ │ @@ -115,15 +115,15 @@
│ │ │            
│ │ │          
│ │ │          
    Alternatively, the calculation can be performed using the Segre embedding as follows:
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── 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 multidegree Phi
│ │ │ │ - -- used 0.510118s (cpu); 0.379256s (thread); 0s (gc)
│ │ │ │ + -- used 0.585522s (cpu); 0.405119s (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
│ │ │ @@ -77,29 +77,29 @@
│ │ │  
│ │ │  o1 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 to PP^5)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : time Z' = projectiveVariety (map segre target Phi) image(segre Phi,"F4");
│ │ │ - -- used 5.22579s (cpu); 2.68091s (thread); 0s (gc)
│ │ │ + -- used 10.4993s (cpu); 2.77588s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^4
    
│ │ │              
    
│ │ │                
    i8 : assert(Z == Z')
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,26 +23,26 @@
│ │ │ │  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 0.199139s (cpu); 0.133717s (thread); 0s (gc)
│ │ │ │ + -- used 0.140302s (cpu); 0.128387s (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 5.22579s (cpu); 2.68091s (thread); 0s (gc)
│ │ │ │ + -- used 10.4993s (cpu); 2.77588s (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
│ │ │ @@ -83,15 +83,15 @@
│ │ │  
│ │ │  o2 : MultirationalMap (rational map from PP^6 to GG(2,4))
│ │ │              
    
│ │ │                
    i3 : time Psi = inverse2 Phi;
│ │ │ - -- used 0.373003s (cpu); 0.295065s (thread); 0s (gc)
│ │ │ + -- used 0.446548s (cpu); 0.345478s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : MultirationalMap (birational map from GG(2,4) to PP^6)
    
│ │ │              
    
│ │ │                
    i4 : assert(Phi * Psi == 1)
    
│ │ │ @@ -103,15 +103,15 @@
│ │ │  
│ │ │  o5 : MultirationalMap (rational map from PP^6 x PP^6 to GG(2,4) x GG(2,4))
│ │ │              
    
│ │ │                
    i6 : time Psi' = inverse2 Phi';
│ │ │ - -- used 1.40947s (cpu); 1.02276s (thread); 0s (gc)
│ │ │ + -- used 1.31876s (cpu); 1.10132s (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 {}
│ │ │ │ @@ -24,23 +24,23 @@
│ │ │ │  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 0.373003s (cpu); 0.295065s (thread); 0s (gc)
│ │ │ │ + -- used 0.446548s (cpu); 0.345478s (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 1.40947s (cpu); 1.02276s (thread); 0s (gc)
│ │ │ │ + -- used 1.31876s (cpu); 1.10132s (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
│ │ │ @@ -88,45 +88,45 @@
│ │ │  
│ │ │  o2 : MultirationalMap (dominant rational map from PP^4 to hypersurface in PP^5)
│ │ │              
    
│ │ │                
    i3 : time inverse Phi;
│ │ │ - -- used 0.140994s (cpu); 0.0876716s (thread); 0s (gc)
│ │ │ + -- used 0.128258s (cpu); 0.0762614s (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 0.160203s (cpu); 0.0924297s (thread); 0s (gc)
│ │ │ + -- used 0.321423s (cpu); 0.134113s (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 0.442904s (cpu); 0.290422s (thread); 0s (gc)
│ │ │ + -- used 0.642818s (cpu); 0.35201s (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 {}
│ │ │ │ @@ -24,32 +24,32 @@
│ │ │ │  
│ │ │ │  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 0.140994s (cpu); 0.0876716s (thread); 0s (gc)
│ │ │ │ + -- used 0.128258s (cpu); 0.0762614s (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 0.160203s (cpu); 0.0924297s (thread); 0s (gc)
│ │ │ │ + -- used 0.321423s (cpu); 0.134113s (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 0.442904s (cpu); 0.290422s (thread); 0s (gc)
│ │ │ │ + -- used 0.642818s (cpu); 0.35201s (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
│ │ │ @@ -83,45 +83,45 @@
│ │ │  
│ │ │  o3 : MultirationalMap (rational map from PP^3 to PP^2 x PP^2)
│ │ │              
    
│ │ │                
    i4 : time isIsomorphism Phi
│ │ │ - -- used 0.00368849s (cpu); 1.1823e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00373306s (cpu); 1.1672e-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 0.32581s (cpu); 0.169415s (thread); 0s (gc)
│ │ │ + -- used 0.53311s (cpu); 0.220708s (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 1.54364s (cpu); 0.844695s (thread); 0s (gc)
│ │ │ + -- used 1.83477s (cpu); 0.953102s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
    
│ │ │              
    
│ │ │                
    i9 : assert(o8 and (not o6) and (not o4))
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,31 +17,31 @@
│ │ │ │       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 0.00368849s (cpu); 1.1823e-05s (thread); 0s (gc)
│ │ │ │ + -- used 0.00373306s (cpu); 1.1672e-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 0.32581s (cpu); 0.169415s (thread); 0s (gc)
│ │ │ │ + -- used 0.53311s (cpu); 0.220708s (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 1.54364s (cpu); 0.844695s (thread); 0s (gc)
│ │ │ │ + -- used 1.83477s (cpu); 0.953102s (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
│ │ │ @@ -80,31 +80,31 @@
│ │ │  
│ │ │  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 0.352309s (cpu); 0.232452s (thread); 0s (gc)
│ │ │ + -- used 0.525903s (cpu); 0.279373s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = false
    
│ │ │              
    
│ │ │                
    i4 : time Psi = first graph Phi;
│ │ │ - -- used 0.161522s (cpu); 0.101427s (thread); 0s (gc)
│ │ │ + -- used 0.124235s (cpu); 0.0747201s (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 4.17216s (cpu); 3.04184s (thread); 0s (gc)
│ │ │ + -- used 3.84185s (cpu); 3.14294s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = true
    
│ │ │              
    
│ │ │                
    i6 : assert((not o3) and o5)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,24 +17,24 @@
│ │ │ │  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 0.352309s (cpu); 0.232452s (thread); 0s (gc)
│ │ │ │ + -- used 0.525903s (cpu); 0.279373s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = false
│ │ │ │  i4 : time Psi = first graph Phi;
│ │ │ │ - -- used 0.161522s (cpu); 0.101427s (thread); 0s (gc)
│ │ │ │ + -- used 0.124235s (cpu); 0.0747201s (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 4.17216s (cpu); 3.04184s (thread); 0s (gc)
│ │ │ │ + -- used 3.84185s (cpu); 3.14294s (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
│ │ │ @@ -79,30 +79,30 @@
│ │ │  
│ │ │  o2 : ProjectiveVariety, curve in PP^7
│ │ │              
    
│ │ │                
    i3 : time f = linearlyNormalEmbedding X;
│ │ │ - -- used 0.00800035s (cpu); 0.00921257s (thread); 0s (gc)
│ │ │ + -- used 0.116384s (cpu); 0.038162s (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 0.558273s (cpu); 0.439528s (thread); 0s (gc)
│ │ │ + -- used 0.533222s (cpu); 0.453712s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : MultirationalMap (birational map from Y to curve in PP^7)
    
│ │ │              
    
│ │ │                
    i6 : assert(isIsomorphism g)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,23 +13,23 @@
│ │ │ │              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 0.00800035s (cpu); 0.00921257s (thread); 0s (gc)
│ │ │ │ + -- used 0.116384s (cpu); 0.038162s (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 0.558273s (cpu); 0.439528s (thread); 0s (gc)
│ │ │ │ + -- used 0.533222s (cpu); 0.453712s (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
│ │ │ @@ -81,15 +81,15 @@
│ │ │  
│ │ │  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 0.510118s (cpu); 0.379256s (thread); 0s (gc)
│ │ │ + -- used 0.585522s (cpu); 0.405119s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = {66, 46, 31, 20}
│ │ │  
│ │ │  o3 : List
    
│ │ │              
    
│ │ │                
    i2 : for i in {4,3,2,1,0} list time multidegree(i,Phi)
│ │ │ - -- used 0.00244457s (cpu); 0.0012392s (thread); 0s (gc)
│ │ │ - -- used 0.280015s (cpu); 0.154499s (thread); 0s (gc)
│ │ │ - -- used 0.221312s (cpu); 0.167638s (thread); 0s (gc)
│ │ │ - -- used 0.215238s (cpu); 0.14879s (thread); 0s (gc)
│ │ │ - -- used 0.191957s (cpu); 0.116338s (thread); 0s (gc)
│ │ │ + -- used 0.00250687s (cpu); 0.00137215s (thread); 0s (gc)
│ │ │ + -- used 0.230082s (cpu); 0.149713s (thread); 0s (gc)
│ │ │ + -- used 0.260478s (cpu); 0.177666s (thread); 0s (gc)
│ │ │ + -- used 0.245703s (cpu); 0.159553s (thread); 0s (gc)
│ │ │ + -- used 0.219403s (cpu); 0.133707s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = {51, 28, 14, 6, 2}
│ │ │  
│ │ │  o2 : List
    
│ │ │              
    
│ │ │                
    i3 : time assert(oo == multidegree Phi)
│ │ │ - -- used 0.160984s (cpu); 0.101194s (thread); 0s (gc)
    
│ │ │ + -- used 0.243937s (cpu); 0.100052s (thread); 0s (gc)
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    References
    
│ │ │  ArXiv preprint: Computations with rational maps between multi-projective varieties.      
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,25 +17,25 @@
│ │ │ │  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 0.00244457s (cpu); 0.0012392s (thread); 0s (gc)
│ │ │ │ - -- used 0.280015s (cpu); 0.154499s (thread); 0s (gc)
│ │ │ │ - -- used 0.221312s (cpu); 0.167638s (thread); 0s (gc)
│ │ │ │ - -- used 0.215238s (cpu); 0.14879s (thread); 0s (gc)
│ │ │ │ - -- used 0.191957s (cpu); 0.116338s (thread); 0s (gc)
│ │ │ │ + -- used 0.00250687s (cpu); 0.00137215s (thread); 0s (gc)
│ │ │ │ + -- used 0.230082s (cpu); 0.149713s (thread); 0s (gc)
│ │ │ │ + -- used 0.260478s (cpu); 0.177666s (thread); 0s (gc)
│ │ │ │ + -- used 0.245703s (cpu); 0.159553s (thread); 0s (gc)
│ │ │ │ + -- used 0.219403s (cpu); 0.133707s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = {51, 28, 14, 6, 2}
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  i3 : time assert(oo == multidegree Phi)
│ │ │ │ - -- used 0.160984s (cpu); 0.101194s (thread); 0s (gc)
│ │ │ │ + -- used 0.243937s (cpu); 0.100052s (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
│ │ │ @@ -80,15 +80,15 @@
│ │ │  
│ │ │  o2 : ProjectiveVariety, 4-dimensional subvariety of PP^3 x PP^2 x PP^9    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time p := point X
│ │ │ - -- used 0.0160578s (cpu); 0.0164613s (thread); 0s (gc)
│ │ │ + -- used 0.0779279s (cpu); 0.026532s (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
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  o4 : ProjectiveVariety, hypersurface in PP^3 x PP^2 x PP^9
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time q = point Y
│ │ │ - -- used 1.86005s (cpu); 1.10026s (thread); 0s (gc)
│ │ │ + -- used 1.58984s (cpu); 1.06685s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = q
│ │ │  
│ │ │  o5 : ProjectiveVariety, a point in PP^3 x PP^2 x PP^9
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -14,25 +14,25 @@
│ │ │ │            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 0.0160578s (cpu); 0.0164613s (thread); 0s (gc)
│ │ │ │ + -- used 0.0779279s (cpu); 0.026532s (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 1.86005s (cpu); 1.10026s (thread); 0s (gc)
│ │ │ │ + -- used 1.58984s (cpu); 1.06685s (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
│ │ │ @@ -101,15 +101,15 @@
│ │ │  
│ │ │  o5 : MultirationalMap (rational map from PP^4 to hypersurface in PP^5 x PP^4 x PP^4)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time segre Phi;
│ │ │ - -- used 0.716809s (cpu); 0.551533s (thread); 0s (gc)
│ │ │ + -- used 1.10689s (cpu); 0.570786s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : RationalMap (rational map from PP^4 to PP^149)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : describe segre Phi
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │  
│ │ │ │  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 0.716809s (cpu); 0.551533s (thread); 0s (gc)
│ │ │ │ + -- used 1.10689s (cpu); 0.570786s (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
│ │ │ @@ -77,15 +77,15 @@
│ │ │  
│ │ │  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 0.32872s (cpu); 0.173534s (thread); 0s (gc)
│ │ │ + -- used 0.24132s (cpu); 0.157477s (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 {}
│ │ │ │ @@ -15,15 +15,15 @@
│ │ │ │  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 0.32872s (cpu); 0.173534s (thread); 0s (gc)
│ │ │ │ + -- used 0.24132s (cpu); 0.157477s (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/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 = (73, 77, 92, 92, 92, 93, 97, 95, 96, 96, 97, 98, 97, 94, 98, 100, 99,
│ │ │ +o9 = (64, 80, 89, 89, 96, 96, 95, 95, 98, 99, 96, 98, 94, 96, 97, 97, 97, 98,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     97, 98, 96, 99, 100, 97, 99, 99, 96, 98, 100, 97)
│ │ │ +     99, 95, 99, 100, 99, 97, 98, 96, 98, 98, 98)
│ │ │  
│ │ │  o9 : Sequence
│ │ │  
│ │ │  i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), connected))
│ │ │  
│ │ │ -o10 = (15, 8, 5, 2, 3, 2, 2, 2, 2, 5, 1, 1, 2, 1, 0, 0, 0, 2, 1, 0, 0, 0, 0,
│ │ │ +o10 = (15, 6, 6, 4, 4, 2, 0, 2, 2, 2, 3, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      1, 0, 0, 0, 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 = {DmS, Djk, DpC, DB[, DrS}
│ │ │ +o2 = {DyC, DTw, DQ_, Dok, DWW}
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │  
│ │ │  o3 = {DDO, Dx_, Dlw, Dx{, D_K}
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/example-output/_generate__Random__Regular__Graphs.out
│ │ │ @@ -1,21 +1,21 @@
│ │ │  -- -*- M2-comint -*- hash: 1729831171060067675
│ │ │  
│ │ │  i1 : R = QQ[a..e];
│ │ │  
│ │ │  i2 : generateRandomRegularGraphs(R, 3, 2)
│ │ │  
│ │ │ -o2 = {Graph{"edges" => {{b, c}, {a, d}, {b, d}, {a, e}, {c, e}}},
│ │ │ +o2 = {Graph{"edges" => {{a, b}, {b, c}, {a, d}, {c, e}, {d, e}}},
│ │ │              "ring" => R                                          
│ │ │              "vertices" => {a, b, c, d, e}                        
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Graph{"edges" => {{a, b}, {a, c}, {c, d}, {b, e}, {d, e}}},
│ │ │ +     Graph{"edges" => {{a, b}, {a, c}, {b, 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}}}}
│ │ │ +     Graph{"edges" => {{a, b}, {b, c}, {c, d}, {a, e}, {d, e}}}}
│ │ │             "ring" => R
│ │ │             "vertices" => {a, b, c, d, e}
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/example-output/_graph__Complement.out
│ │ │ @@ -13,13 +13,13 @@
│ │ │  i3 : graphComplement "Dhc"
│ │ │  
│ │ │  o3 = DUW
│ │ │  
│ │ │  i4 : G = generateBipartiteGraphs 7;
│ │ │  
│ │ │  i5 : time graphComplement G;
│ │ │ - -- used 0.00057649s (cpu); 0.000563907s (thread); 0s (gc)
│ │ │ + -- used 0.000769548s (cpu); 0.000583866s (thread); 0s (gc)
│ │ │  
│ │ │  i6 : time (graphComplement \ G);
│ │ │ - -- used 0.142673s (cpu); 0.072388s (thread); 0s (gc)
│ │ │ + -- used 0.175907s (cpu); 0.0811833s (thread); 0s (gc)
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html
│ │ │ @@ -117,28 +117,28 @@
│ │ │                
    i8 : prob = n -> log(n)/n;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, 2*(prob n)), connected))
│ │ │  
│ │ │ -o9 = (73, 77, 92, 92, 92, 93, 97, 95, 96, 96, 97, 98, 97, 94, 98, 100, 99,
│ │ │ +o9 = (64, 80, 89, 89, 96, 96, 95, 95, 98, 99, 96, 98, 94, 96, 97, 97, 97, 98,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     97, 98, 96, 99, 100, 97, 99, 99, 96, 98, 100, 97)
│ │ │ +     99, 95, 99, 100, 99, 97, 98, 96, 98, 98, 98)
│ │ │  
│ │ │  o9 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), connected))
│ │ │  
│ │ │ -o10 = (15, 8, 5, 2, 3, 2, 2, 2, 2, 5, 1, 1, 2, 1, 0, 0, 0, 2, 1, 0, 0, 0, 0,
│ │ │ +o10 = (15, 6, 6, 4, 4, 2, 0, 2, 2, 2, 3, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      1, 0, 0, 0, 0, 0)
│ │ │ +      0, 0, 0, 0, 0, 0)
│ │ │  
│ │ │  o10 : Sequence
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── 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 = (73, 77, 92, 92, 92, 93, 97, 95, 96, 96, 97, 98, 97, 94, 98, 100, 99,
│ │ │ │ +o9 = (64, 80, 89, 89, 96, 96, 95, 95, 98, 99, 96, 98, 94, 96, 97, 97, 97, 98,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     97, 98, 96, 99, 100, 97, 99, 99, 96, 98, 100, 97)
│ │ │ │ +     99, 95, 99, 100, 99, 97, 98, 96, 98, 98, 98)
│ │ │ │  
│ │ │ │  o9 : Sequence
│ │ │ │  i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2),
│ │ │ │  connected))
│ │ │ │  
│ │ │ │ -o10 = (15, 8, 5, 2, 3, 2, 2, 2, 2, 5, 1, 1, 2, 1, 0, 0, 0, 2, 1, 0, 0, 0, 0,
│ │ │ │ +o10 = (15, 6, 6, 4, 4, 2, 0, 2, 2, 2, 3, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      1, 0, 0, 0, 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
│ │ │ @@ -100,15 +100,15 @@
│ │ │  o1 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : generateRandomGraphs(5, 5)
│ │ │  
│ │ │ -o2 = {DmS, Djk, DpC, DB[, DrS}
│ │ │ +o2 = {DyC, DTw, DQ_, Dok, DWW}
│ │ │  
│ │ │  o2 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │ ├── html2text {}
│ │ │ │ @@ -37,15 +37,15 @@
│ │ │ │  i1 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │ │  
│ │ │ │  o1 = {DDO, Dx_, Dlw, Dx{, D_K}
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  i2 : generateRandomGraphs(5, 5)
│ │ │ │  
│ │ │ │ -o2 = {DmS, Djk, DpC, DB[, DrS}
│ │ │ │ +o2 = {DyC, DTw, DQ_, Dok, DWW}
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -87,23 +87,23 @@
│ │ │                
    i1 : R = QQ[a..e];
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : generateRandomRegularGraphs(R, 3, 2)
│ │ │  
│ │ │ -o2 = {Graph{"edges" => {{b, c}, {a, d}, {b, d}, {a, e}, {c, e}}},
│ │ │ +o2 = {Graph{"edges" => {{a, b}, {b, c}, {a, d}, {c, e}, {d, e}}},
│ │ │              "ring" => R                                          
│ │ │              "vertices" => {a, b, c, d, e}                        
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Graph{"edges" => {{a, b}, {a, c}, {c, d}, {b, e}, {d, e}}},
│ │ │ +     Graph{"edges" => {{a, b}, {a, c}, {b, 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}}}}
│ │ │ +     Graph{"edges" => {{a, b}, {b, c}, {c, d}, {a, e}, {d, e}}}}
│ │ │             "ring" => R
│ │ │             "vertices" => {a, b, c, d, e}
│ │ │  
│ │ │  o2 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -24,23 +24,23 @@
│ │ │ │  vertices with a given regularity. Note that some graphs may be isomorphic.
│ │ │ │  If a _P_o_l_y_n_o_m_i_a_l_R_i_n_g $R$ is supplied instead, then the number of vertices is the
│ │ │ │  number of generators. Moreover, the nauty-based strings are automatically
│ │ │ │  converted to instances of the class _G_r_a_p_h in $R$.
│ │ │ │  i1 : R = QQ[a..e];
│ │ │ │  i2 : generateRandomRegularGraphs(R, 3, 2)
│ │ │ │  
│ │ │ │ -o2 = {Graph{"edges" => {{b, c}, {a, d}, {b, d}, {a, e}, {c, e}}},
│ │ │ │ +o2 = {Graph{"edges" => {{a, b}, {b, c}, {a, d}, {c, e}, {d, e}}},
│ │ │ │              "ring" => R
│ │ │ │              "vertices" => {a, b, c, d, e}
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     Graph{"edges" => {{a, b}, {a, c}, {c, d}, {b, e}, {d, e}}},
│ │ │ │ +     Graph{"edges" => {{a, b}, {a, c}, {b, 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}}}}
│ │ │ │ +     Graph{"edges" => {{a, b}, {b, c}, {c, d}, {a, e}, {d, e}}}}
│ │ │ │             "ring" => R
│ │ │ │             "vertices" => {a, b, c, d, e}
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  The number of vertices $n$ must be positive as nauty cannot handle graphs with
│ │ │ │  zero vertices.
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/html/_graph__Complement.html
│ │ │ @@ -116,21 +116,21 @@
│ │ │              
│ │ │                
    i4 : G = generateBipartiteGraphs 7;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time graphComplement G;
│ │ │ - -- used 0.00057649s (cpu); 0.000563907s (thread); 0s (gc)
    
│ │ │ + -- used 0.000769548s (cpu); 0.000583866s (thread); 0s (gc)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time (graphComplement \ G);
│ │ │ - -- used 0.142673s (cpu); 0.072388s (thread); 0s (gc)
    
│ │ │ + -- used 0.175907s (cpu); 0.0811833s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,17 +41,17 @@
│ │ │ │  
│ │ │ │  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 0.00057649s (cpu); 0.000563907s (thread); 0s (gc)
│ │ │ │ + -- used 0.000769548s (cpu); 0.000583866s (thread); 0s (gc)
│ │ │ │  i6 : time (graphComplement \ G);
│ │ │ │ - -- used 0.142673s (cpu); 0.072388s (thread); 0s (gc)
│ │ │ │ + -- used 0.175907s (cpu); 0.0811833s (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/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 = (76, 79, 91, 92, 91, 96, 96, 89, 96, 94, 96, 99, 98, 96, 98, 99, 100,
│ │ │ +o9 = (74, 85, 89, 90, 98, 94, 93, 93, 97, 97, 100, 100, 97, 100, 98, 99, 96,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     97, 99, 95, 97, 99, 100, 96, 99, 99, 97, 98, 100)
│ │ │ +     99, 97, 98, 98, 100, 99, 98, 100, 96, 100, 100, 98)
│ │ │  
│ │ │  o9 : Sequence
│ │ │  
│ │ │  i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), connected))
│ │ │  
│ │ │ -o10 = (19, 7, 3, 6, 5, 4, 2, 1, 4, 0, 0, 2, 0, 0, 0, 0, 2, 1, 2, 0, 0, 1, 0,
│ │ │ +o10 = (14, 4, 8, 3, 5, 0, 4, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      0, 0, 0, 0, 0, 1)
│ │ │ +      0, 0, 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 = {Dd?, D?_, DXk, DIS, DVc}
│ │ │ +o2 = {DLG, Dfs, DVs, DlC, D\s}
│ │ │  
│ │ │  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 = {DLo, DqK, D[S}
│ │ │ +o1 = {DkK, DpS, Dhc}
│ │ │  
│ │ │  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 0.000660357s (cpu); 0.000530855s (thread); 0s (gc)
│ │ │ + -- used 0.000694753s (cpu); 0.000568642s (thread); 0s (gc)
│ │ │  
│ │ │  i5 : time (graphComplement \ G);
│ │ │ - -- used 0.144771s (cpu); 0.0960107s (thread); 0s (gc)
│ │ │ + -- used 0.173964s (cpu); 0.0830496s (thread); 0s (gc)
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html
│ │ │ @@ -117,28 +117,28 @@
│ │ │                
      i8 : prob = n -> log(n)/n;
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i9 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, 2*(prob n)), connected))
│ │ │  
│ │ │ -o9 = (76, 79, 91, 92, 91, 96, 96, 89, 96, 94, 96, 99, 98, 96, 98, 99, 100,
│ │ │ +o9 = (74, 85, 89, 90, 98, 94, 93, 93, 97, 97, 100, 100, 97, 100, 98, 99, 96,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     97, 99, 95, 97, 99, 100, 96, 99, 99, 97, 98, 100)
│ │ │ +     99, 97, 98, 98, 100, 99, 98, 100, 96, 100, 100, 98)
│ │ │  
│ │ │  o9 : Sequence
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), connected))
│ │ │  
│ │ │ -o10 = (19, 7, 3, 6, 5, 4, 2, 1, 4, 0, 0, 2, 0, 0, 0, 0, 2, 1, 2, 0, 0, 1, 0,
│ │ │ +o10 = (14, 4, 8, 3, 5, 0, 4, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      0, 0, 0, 0, 0, 1)
│ │ │ +      0, 0, 0, 0, 0, 0)
│ │ │  
│ │ │  o10 : Sequence
      
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── 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 = (76, 79, 91, 92, 91, 96, 96, 89, 96, 94, 96, 99, 98, 96, 98, 99, 100,
│ │ │ │ +o9 = (74, 85, 89, 90, 98, 94, 93, 93, 97, 97, 100, 100, 97, 100, 98, 99, 96,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     97, 99, 95, 97, 99, 100, 96, 99, 99, 97, 98, 100)
│ │ │ │ +     99, 97, 98, 98, 100, 99, 98, 100, 96, 100, 100, 98)
│ │ │ │  
│ │ │ │  o9 : Sequence
│ │ │ │  i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2),
│ │ │ │  connected))
│ │ │ │  
│ │ │ │ -o10 = (19, 7, 3, 6, 5, 4, 2, 1, 4, 0, 0, 2, 0, 0, 0, 0, 2, 1, 2, 0, 0, 1, 0,
│ │ │ │ +o10 = (14, 4, 8, 3, 5, 0, 4, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      0, 0, 0, 0, 0, 1)
│ │ │ │ +      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/NautyGraphs/html/_generate__Random__Graphs.html
│ │ │ @@ -93,15 +93,15 @@
│ │ │  o1 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : generateRandomGraphs(5, 5)
│ │ │  
│ │ │ -o2 = {Dd?, D?_, DXk, DIS, DVc}
│ │ │ +o2 = {DLG, Dfs, DVs, DlC, D\s}
│ │ │  
│ │ │  o2 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │ ├── html2text {}
│ │ │ │ @@ -30,15 +30,15 @@
│ │ │ │  i1 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │ │  
│ │ │ │  o1 = {DDO, Dx_, Dlw, Dx{, D_K}
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  i2 : generateRandomGraphs(5, 5)
│ │ │ │  
│ │ │ │ -o2 = {Dd?, D?_, DXk, DIS, DVc}
│ │ │ │ +o2 = {DLG, Dfs, DVs, DlC, D\s}
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -77,15 +77,15 @@
│ │ │            
    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 = {DLo, DqK, D[S}
│ │ │ +o1 = {DkK, DpS, Dhc}
│ │ │  
│ │ │  o1 : List
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,15 +18,15 @@
│ │ │ │      * 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 = {DLo, DqK, D[S}
│ │ │ │ +o1 = {DkK, DpS, Dhc}
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -110,21 +110,21 @@
│ │ │              
│ │ │                
    i3 : G = generateBipartiteGraphs 7;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time graphComplement G;
│ │ │ - -- used 0.000660357s (cpu); 0.000530855s (thread); 0s (gc)
    
│ │ │ + -- used 0.000694753s (cpu); 0.000568642s (thread); 0s (gc)    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time (graphComplement \ G);
│ │ │ - -- used 0.144771s (cpu); 0.0960107s (thread); 0s (gc)
    
│ │ │ + -- used 0.173964s (cpu); 0.0830496s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │          
    
│ │ │            
    Ways to use graphComplement:
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -38,17 +38,17 @@
│ │ │ │  
│ │ │ │  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 0.000660357s (cpu); 0.000530855s (thread); 0s (gc)
│ │ │ │ + -- used 0.000694753s (cpu); 0.000568642s (thread); 0s (gc)
│ │ │ │  i5 : time (graphComplement \ G);
│ │ │ │ - -- used 0.144771s (cpu); 0.0960107s (thread); 0s (gc)
│ │ │ │ + -- used 0.173964s (cpu); 0.0830496s (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/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")
│ │ │ - -- .171815s elapsed
│ │ │ + -- .108199s 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
│ │ │ @@ -120,15 +120,15 @@
│ │ │  
│ │ │  o5 = true
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : elapsedTime noetherianOperators(Q, Strategy => "PunctualQuot")
│ │ │ - -- .171815s elapsed
│ │ │ + -- .108199s 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")
│ │ │ │ - -- .171815s elapsed
│ │ │ │ + -- .108199s 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/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 0.197856s (cpu); 0.199371s (thread); 0s (gc)
│ │ │ + -- used 0.263958s (cpu); 0.263215s (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 0.00244851s (cpu); 0.00123145s (thread); 0s (gc)
│ │ │ - -- used 2.3725e-05s (cpu); 8.1352e-05s (thread); 0s (gc)
│ │ │ - -- used 9.157e-06s (cpu); 7.0322e-05s (thread); 0s (gc)
│ │ │ - -- used 8.777e-06s (cpu); 7.5351e-05s (thread); 0s (gc)
│ │ │ - -- used 5.0966e-05s (cpu); 8.8336e-05s (thread); 0s (gc)
│ │ │ - -- used 1.0339e-05s (cpu); 7.1384e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00400116s (cpu); 0.0017862s (thread); 0s (gc)
│ │ │ + -- used 2.8325e-05s (cpu); 0.000137574s (thread); 0s (gc)
│ │ │ + -- used 1.2727e-05s (cpu); 0.000113617s (thread); 0s (gc)
│ │ │ + -- used 1.14e-05s (cpu); 0.000106904s (thread); 0s (gc)
│ │ │ + -- used 1.1421e-05s (cpu); 0.000107042s (thread); 0s (gc)
│ │ │ + -- used 1.1216e-05s (cpu); 0.000106693s (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
│ │ │ @@ -1,29 +1,29 @@
│ │ │  -- -*- M2-comint -*- hash: 16408385764843695632
│ │ │  
│ │ │  i1 : assert all (5, d -> isWellDefined toricProjectiveSpace (d+1))
│ │ │  
│ │ │  i2 : setRandomSeed (currentTime ());
│ │ │ - -- setting random seed to 1763141963
│ │ │ + -- setting random seed to 1766074209
│ │ │  
│ │ │  i3 : a = sort apply (3, i -> random (7))
│ │ │  
│ │ │ -o3 = {0, 3, 4}
│ │ │ +o3 = {1, 3, 5}
│ │ │  
│ │ │  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, 1, 6, 7}
│ │ │ +o7 = {6, 6, 7, 7, 9}
│ │ │  
│ │ │  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
│ │ │ - -- .0576176s elapsed
│ │ │ + -- .0386635s 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))
│ │ │ - -- .00125706s elapsed
│ │ │ + -- .00137045s 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
│ │ │ - -- .0350012s elapsed
│ │ │ + -- .033201s 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))
│ │ │ - -- .000940674s elapsed
│ │ │ + -- .00117243s 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
│ │ │ - -- 40.9805s elapsed
│ │ │ + -- 30.0784s elapsed
│ │ │  
│ │ │  o11 = 7909
│ │ │  
│ │ │  i12 : elapsedTime assert (m3 === #first entries basis (degree D3, ring variety D3))
│ │ │ - -- .0316693s elapsed
│ │ │ + -- .0307763s 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 2.6249e-05s (cpu); 2.0469e-05s (thread); 0s (gc)
│ │ │ + -- used 4.5716e-05s (cpu); 3.6209e-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 0.043392s (cpu); 0.0433981s (thread); 0s (gc)
│ │ │ + -- used 0.0602178s (cpu); 0.0602276s (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 0.0223801s (cpu); 0.0223778s (thread); 0s (gc)
│ │ │ + -- used 0.0371457s (cpu); 0.0371442s (thread); 0s (gc)
│ │ │  
│ │ │  i20 : X2 = time normalToricVariety vertMatrix;
│ │ │ - -- used 0.00239766s (cpu); 0.00239836s (thread); 0s (gc)
│ │ │ + -- used 0.0034646s (cpu); 0.00347125s (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
│ │ │ @@ -207,15 +207,15 @@
│ │ │          
    
│ │ │            
    We end with a slightly larger example.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -254,20 +254,20 @@
│ │ │  
│ │ │  o18 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i14 : Y = time smoothFanoToricVariety(5,100);
│ │ │ - -- used 0.197856s (cpu); 0.199371s (thread); 0s (gc)
    
│ │ │ + -- used 0.263958s (cpu); 0.263215s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i15 : A2 = intersectionRing Y;
    
│ │ │              
    
│ │ │                
    i19 : for i to dim Y list time hilbertFunction (i, A2)
│ │ │ - -- used 0.00244851s (cpu); 0.00123145s (thread); 0s (gc)
│ │ │ - -- used 2.3725e-05s (cpu); 8.1352e-05s (thread); 0s (gc)
│ │ │ - -- used 9.157e-06s (cpu); 7.0322e-05s (thread); 0s (gc)
│ │ │ - -- used 8.777e-06s (cpu); 7.5351e-05s (thread); 0s (gc)
│ │ │ - -- used 5.0966e-05s (cpu); 8.8336e-05s (thread); 0s (gc)
│ │ │ - -- used 1.0339e-05s (cpu); 7.1384e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00400116s (cpu); 0.0017862s (thread); 0s (gc)
│ │ │ + -- used 2.8325e-05s (cpu); 0.000137574s (thread); 0s (gc)
│ │ │ + -- used 1.2727e-05s (cpu); 0.000113617s (thread); 0s (gc)
│ │ │ + -- used 1.14e-05s (cpu); 0.000106904s (thread); 0s (gc)
│ │ │ + -- used 1.1421e-05s (cpu); 0.000107042s (thread); 0s (gc)
│ │ │ + -- used 1.1216e-05s (cpu); 0.000106693s (thread); 0s (gc)
│ │ │  
│ │ │  o19 = {1, 6, 13, 13, 6, 1}
│ │ │  
│ │ │  o19 : List
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -96,15 +96,15 @@
│ │ │ │  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 0.197856s (cpu); 0.199371s (thread); 0s (gc)
│ │ │ │ + -- used 0.263958s (cpu); 0.263215s (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
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ @@ -129,20 +129,20 @@
│ │ │ │        (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 0.00244851s (cpu); 0.00123145s (thread); 0s (gc)
│ │ │ │ - -- used 2.3725e-05s (cpu); 8.1352e-05s (thread); 0s (gc)
│ │ │ │ - -- used 9.157e-06s (cpu); 7.0322e-05s (thread); 0s (gc)
│ │ │ │ - -- used 8.777e-06s (cpu); 7.5351e-05s (thread); 0s (gc)
│ │ │ │ - -- used 5.0966e-05s (cpu); 8.8336e-05s (thread); 0s (gc)
│ │ │ │ - -- used 1.0339e-05s (cpu); 7.1384e-05s (thread); 0s (gc)
│ │ │ │ + -- used 0.00400116s (cpu); 0.0017862s (thread); 0s (gc)
│ │ │ │ + -- used 2.8325e-05s (cpu); 0.000137574s (thread); 0s (gc)
│ │ │ │ + -- used 1.2727e-05s (cpu); 0.000113617s (thread); 0s (gc)
│ │ │ │ + -- used 1.14e-05s (cpu); 0.000106904s (thread); 0s (gc)
│ │ │ │ + -- used 1.1421e-05s (cpu); 0.000107042s (thread); 0s (gc)
│ │ │ │ + -- used 1.1216e-05s (cpu); 0.000106693s (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
│ │ │ @@ -93,22 +93,22 @@
│ │ │          
    
│ │ │            
    The second examples show that a randomly selected Kleinschmidt toric variety and a weighted projective space are also well-defined.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i2 : setRandomSeed (currentTime ());
│ │ │ - -- setting random seed to 1763141963
    
│ │ │ + -- setting random seed to 1766074209
│ │ │              
    
│ │ │                
    i3 : a = sort apply (3, i -> random (7))
│ │ │  
│ │ │ -o3 = {0, 3, 4}
│ │ │ +o3 = {1, 3, 5}
│ │ │  
│ │ │  o3 : List
    
│ │ │              
    
│ │ │                
    i4 : assert isWellDefined kleinschmidt (4,a)
    
│ │ │ @@ -126,15 +126,15 @@
│ │ │                
    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, 1, 6, 7}
│ │ │ +o7 = {6, 6, 7, 7, 9}
│ │ │  
│ │ │  o7 : List
    
│ │ │              
    
│ │ │                
    i8 : assert isWellDefined weightedProjectiveSpace q
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,27 +28,27 @@
│ │ │ │      * the intersection of the cones associated to two elements of coneList is a
│ │ │ │        face of each cone.
│ │ │ │  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 ());
│ │ │ │ - -- setting random seed to 1763141963
│ │ │ │ + -- setting random seed to 1766074209
│ │ │ │  i3 : a = sort apply (3, i -> random (7))
│ │ │ │  
│ │ │ │ -o3 = {0, 3, 4}
│ │ │ │ +o3 = {1, 3, 5}
│ │ │ │  
│ │ │ │  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, 1, 6, 7}
│ │ │ │ +o7 = {6, 6, 7, 7, 9}
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -96,15 +96,15 @@
│ │ │  
│ │ │  o2 : ToricDivisor on PP2
│ │ │              
    
│ │ │                
    i3 : M1 = elapsedTime monomials D1
│ │ │ - -- .0576176s elapsed
│ │ │ + -- .0386635s 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 }
│ │ │ @@ -112,15 +112,15 @@
│ │ │  
│ │ │  o3 : List
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime assert (set M1 === set first entries basis(degree D1, ring variety D1))
│ │ │ - -- .00125706s elapsed
    
│ │ │ + -- .00137045s elapsed
│ │ │              
    
│ │ │          
    
│ │ │            
    Toric varieties of Picard-rank 2 are slightly more interesting.
    
│ │ │          
    
│ │ │          
│ │ │ @@ -138,27 +138,27 @@
│ │ │  
│ │ │  o6 : ToricDivisor on FF2
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -171,23 +171,23 @@
│ │ │  
│ │ │  o10 : ToricDivisor on X
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : M2 = elapsedTime monomials D2
│ │ │ - -- .0350012s elapsed
│ │ │ + -- .033201s 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))
│ │ │ - -- .000940674s elapsed
    
│ │ │ + -- .00117243s elapsed
│ │ │              
    
│ │ │                
    i9 : X = kleinschmidt (5, {1,2,3});
    
│ │ │              
    
│ │ │                
    i11 : m3 = elapsedTime # monomials D3
│ │ │ - -- 40.9805s elapsed
│ │ │ + -- 30.0784s elapsed
│ │ │  
│ │ │  o11 = 7909
    
│ │ │              
    
│ │ │                
    i12 : elapsedTime assert (m3 === #first entries basis (degree D3, ring variety D3))
│ │ │ - -- .0316693s elapsed
    
│ │ │ + -- .0307763s elapsed
│ │ │              
    
│ │ │          
    
│ │ │            
    By exploiting latticePoints, this method function avoids using the basis function.
    
│ │ │          
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,61 +27,61 @@
│ │ │ │  i2 : D1 = 5*PP2_0
│ │ │ │  
│ │ │ │  o2 = 5*PP2
│ │ │ │            0
│ │ │ │  
│ │ │ │  o2 : ToricDivisor on PP2
│ │ │ │  i3 : M1 = elapsedTime monomials D1
│ │ │ │ - -- .0576176s elapsed
│ │ │ │ + -- .0386635s 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))
│ │ │ │ - -- .00125706s elapsed
│ │ │ │ + -- .00137045s 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
│ │ │ │ - -- .0350012s elapsed
│ │ │ │ + -- .033201s 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))
│ │ │ │ - -- .000940674s elapsed
│ │ │ │ + -- .00117243s 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
│ │ │ │ - -- 40.9805s elapsed
│ │ │ │ + -- 30.0784s elapsed
│ │ │ │  
│ │ │ │  o11 = 7909
│ │ │ │  i12 : elapsedTime assert (m3 === #first entries basis (degree D3, ring variety
│ │ │ │  D3))
│ │ │ │ - -- .0316693s elapsed
│ │ │ │ + -- .0307763s 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
│ │ │ @@ -125,25 +125,25 @@
│ │ │          
    
│ │ │            
    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.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -48,22 +48,22 @@
│ │ │ │  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 2.6249e-05s (cpu); 2.0469e-05s (thread); 0s (gc)
│ │ │ │ + -- used 4.5716e-05s (cpu); 3.6209e-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 0.043392s (cpu); 0.0433981s (thread); 0s (gc)
│ │ │ │ + -- used 0.0602178s (cpu); 0.0602276s (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
│ │ │ @@ -233,21 +233,21 @@
│ │ │                 2       3
│ │ │  o18 : Matrix ZZ  <-- ZZ
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -102,17 +102,17 @@
│ │ │ │  
│ │ │ │  o18 = | 0 1 0 |
│ │ │ │        | 0 0 1 |
│ │ │ │  
│ │ │ │                 2       3
│ │ │ │  o18 : Matrix ZZ  <-- ZZ
│ │ │ │  i19 : X1 = time normalToricVariety convexHull (vertMatrix);
│ │ │ │ - -- used 0.0223801s (cpu); 0.0223778s (thread); 0s (gc)
│ │ │ │ + -- used 0.0371457s (cpu); 0.0371442s (thread); 0s (gc)
│ │ │ │  i20 : X2 = time normalToricVariety vertMatrix;
│ │ │ │ - -- used 0.00239766s (cpu); 0.00239836s (thread); 0s (gc)
│ │ │ │ + -- used 0.0034646s (cpu); 0.00347125s (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/NumericalImplicitization/example-output/___Convert__To__Cone.out
│ │ │ @@ -21,21 +21,21 @@
│ │ │  
│ │ │  i4 : (numericalHilbertFunction(F, I, 3, Verbose => false)).hilbertFunctionValue == 0
│ │ │  
│ │ │  o4 = true
│ │ │  
│ │ │  i5 : T = numericalHilbertFunction(F, I, 3, ConvertToCone => true)
│ │ │  Sampling image points ...
│ │ │ -     -- used .0790187 seconds
│ │ │ +     -- used .0978784 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .00605006 seconds
│ │ │ +     -- used .00661361 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00292018 seconds
│ │ │ +     -- used .00322982 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000456115 seconds
│ │ │ +     -- used .000464308 seconds
│ │ │  
│ │ │  o5 = a "numerical interpolation table", indicating
│ │ │       the space of degree 3 forms in the ideal of the image has dimension 3
│ │ │  
│ │ │  o5 : NumericalInterpolationTable
│ │ │  
│ │ │  i6 : extractImageEquations(T, AttemptZZ => true)
│ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_extract__Image__Equations.out
│ │ │ @@ -11,21 +11,21 @@
│ │ │  o2 = | s3 s2t st2 t3 |
│ │ │  
│ │ │               1      4
│ │ │  o2 : Matrix R  <-- R
│ │ │  
│ │ │  i3 : extractImageEquations(F, ideal 0_R, 2, AttemptZZ => true)
│ │ │  Sampling image points ...
│ │ │ -     -- used .00365705 seconds
│ │ │ +     -- used .00454359 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .00270941 seconds
│ │ │ +     -- used .0033195 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00107503 seconds
│ │ │ +     -- used .00123816 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000286057 seconds
│ │ │ +     -- used .0003305 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 ])
│ │ │                 53  0   3           53  0   3
│ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Hilbert__Function.out
│ │ │ @@ -11,40 +11,40 @@
│ │ │  o2 = | s3 s2t st2 t3 |
│ │ │  
│ │ │               1      4
│ │ │  o2 : Matrix R  <-- R
│ │ │  
│ │ │  i3 : numericalHilbertFunction(F, ideal 0_R, 4)
│ │ │  Sampling image points ...
│ │ │ -     -- used .012393 seconds
│ │ │ +     -- used .0146129 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .0114083 seconds
│ │ │ +     -- used .0129944 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00726179 seconds
│ │ │ +     -- used .00766644 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000761889 seconds
│ │ │ +     -- used .000825249 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 .00312588 seconds
│ │ │ +     -- used .0033447 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00846492 seconds
│ │ │ +     -- used .00904545 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000905679 seconds
│ │ │ +     -- used .00110696 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
│ │ │ @@ -20,12 +20,12 @@
│ │ │  
│ │ │  i8 : F = sum(1..14, i -> basis(4, R, Variables=>toList(a_(i,1)..a_(i,5))));
│ │ │  
│ │ │               1      70
│ │ │  o8 : Matrix R  <-- R
│ │ │  
│ │ │  i9 : time numericalImageDim(F, ideal 0_R)
│ │ │ - -- used 0.066304s (cpu); 0.0663008s (thread); 0s (gc)
│ │ │ + -- used 0.0740739s (cpu); 0.0740672s (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)
│ │ │ - -- .619309s elapsed
│ │ │ + -- .47259s elapsed
│ │ │  
│ │ │  o7 = p
│ │ │  
│ │ │  o7 : Point
│ │ │  
│ │ │  i8 : matrix pack(5, p#Coordinates)
│ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/___Convert__To__Cone.html
│ │ │ @@ -100,21 +100,21 @@
│ │ │  o4 = true
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -33,21 +33,21 @@
│ │ │ │  o3 : Ideal of R
│ │ │ │  i4 : (numericalHilbertFunction(F, I, 3, Verbose => false)).hilbertFunctionValue
│ │ │ │  == 0
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  i5 : T = numericalHilbertFunction(F, I, 3, ConvertToCone => true)
│ │ │ │  Sampling image points ...
│ │ │ │ -     -- used .0790187 seconds
│ │ │ │ +     -- used .0978784 seconds
│ │ │ │  Creating interpolation matrix ...
│ │ │ │ -     -- used .00605006 seconds
│ │ │ │ +     -- used .00661361 seconds
│ │ │ │  Performing normalization preconditioning ...
│ │ │ │ -     -- used .00292018 seconds
│ │ │ │ +     -- used .00322982 seconds
│ │ │ │  Computing numerical kernel ...
│ │ │ │ -     -- used .000456115 seconds
│ │ │ │ +     -- used .000464308 seconds
│ │ │ │  
│ │ │ │  o5 = a "numerical interpolation table", indicating
│ │ │ │       the space of degree 3 forms in the ideal of the image has dimension 3
│ │ │ │  
│ │ │ │  o5 : NumericalInterpolationTable
│ │ │ │  i6 : extractImageEquations(T, AttemptZZ => true)
│ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_extract__Image__Equations.html
│ │ │ @@ -102,21 +102,21 @@
│ │ │  o2 : Matrix R  <-- R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -38,21 +38,21 @@
│ │ │ │  
│ │ │ │  o2 = | s3 s2t st2 t3 |
│ │ │ │  
│ │ │ │               1      4
│ │ │ │  o2 : Matrix R  <-- R
│ │ │ │  i3 : extractImageEquations(F, ideal 0_R, 2, AttemptZZ => true)
│ │ │ │  Sampling image points ...
│ │ │ │ -     -- used .00365705 seconds
│ │ │ │ +     -- used .00454359 seconds
│ │ │ │  Creating interpolation matrix ...
│ │ │ │ -     -- used .00270941 seconds
│ │ │ │ +     -- used .0033195 seconds
│ │ │ │  Performing normalization preconditioning ...
│ │ │ │ -     -- used .00107503 seconds
│ │ │ │ +     -- used .00123816 seconds
│ │ │ │  Computing numerical kernel ...
│ │ │ │ -     -- used .000286057 seconds
│ │ │ │ +     -- used .0003305 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 ])
│ │ │ │                 53  0   3           53  0   3
│ │ │ │  Here is how to do the same computation symbolically.
│ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Hilbert__Function.html
│ │ │ @@ -107,21 +107,21 @@
│ │ │  o2 : Matrix R  <-- R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -147,19 +147,19 @@
│ │ │                
    i6 : S = numericalImageSample(F, ideal 0_R, 60);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -57,39 +57,39 @@
│ │ │ │  
│ │ │ │  o2 = | s3 s2t st2 t3 |
│ │ │ │  
│ │ │ │               1      4
│ │ │ │  o2 : Matrix R  <-- R
│ │ │ │  i3 : numericalHilbertFunction(F, ideal 0_R, 4)
│ │ │ │  Sampling image points ...
│ │ │ │ -     -- used .012393 seconds
│ │ │ │ +     -- used .0146129 seconds
│ │ │ │  Creating interpolation matrix ...
│ │ │ │ -     -- used .0114083 seconds
│ │ │ │ +     -- used .0129944 seconds
│ │ │ │  Performing normalization preconditioning ...
│ │ │ │ -     -- used .00726179 seconds
│ │ │ │ +     -- used .00766644 seconds
│ │ │ │  Computing numerical kernel ...
│ │ │ │ -     -- used .000761889 seconds
│ │ │ │ +     -- used .000825249 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 .00312588 seconds
│ │ │ │ +     -- used .0033447 seconds
│ │ │ │  Performing normalization preconditioning ...
│ │ │ │ -     -- used .00846492 seconds
│ │ │ │ +     -- used .00904545 seconds
│ │ │ │  Computing numerical kernel ...
│ │ │ │ -     -- used .000905679 seconds
│ │ │ │ +     -- used .00110696 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
│ │ │ @@ -140,15 +140,15 @@
│ │ │               1      70
│ │ │  o8 : Matrix R  <-- R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i6 : X1 = time normalToricVariety ({{-1,-1},{1,0},{0,1}}, {{0,1},{1,2},{0,2}})
│ │ │ - -- used 2.6249e-05s (cpu); 2.0469e-05s (thread); 0s (gc)
│ │ │ + -- used 4.5716e-05s (cpu); 3.6209e-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 0.043392s (cpu); 0.0433981s (thread); 0s (gc)
    
│ │ │ + -- used 0.0602178s (cpu); 0.0602276s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i8 : assert (sort rays X1 == sort rays X2 and max X1 == max X2)
    
│ │ │              
    
│ │ │                
    i19 : X1 = time normalToricVariety convexHull (vertMatrix);
│ │ │ - -- used 0.0223801s (cpu); 0.0223778s (thread); 0s (gc)
    
│ │ │ + -- used 0.0371457s (cpu); 0.0371442s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i20 : X2 = time normalToricVariety vertMatrix;
│ │ │ - -- used 0.00239766s (cpu); 0.00239836s (thread); 0s (gc)
    
│ │ │ + -- used 0.0034646s (cpu); 0.00347125s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i21 : assert (set rays X2 === set rays X1 and max X1 === max X2)
    
│ │ │              
    
│ │ │                
    i5 : T = numericalHilbertFunction(F, I, 3, ConvertToCone => true)
│ │ │  Sampling image points ...
│ │ │ -     -- used .0790187 seconds
│ │ │ +     -- used .0978784 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .00605006 seconds
│ │ │ +     -- used .00661361 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00292018 seconds
│ │ │ +     -- used .00322982 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000456115 seconds
│ │ │ +     -- used .000464308 seconds
│ │ │  
│ │ │  o5 = a "numerical interpolation table", indicating
│ │ │       the space of degree 3 forms in the ideal of the image has dimension 3
│ │ │  
│ │ │  o5 : NumericalInterpolationTable
    
│ │ │              
    
│ │ │                
    i3 : extractImageEquations(F, ideal 0_R, 2, AttemptZZ => true)
│ │ │  Sampling image points ...
│ │ │ -     -- used .00365705 seconds
│ │ │ +     -- used .00454359 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .00270941 seconds
│ │ │ +     -- used .0033195 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00107503 seconds
│ │ │ +     -- used .00123816 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000286057 seconds
│ │ │ +     -- used .0003305 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 ])
│ │ │                 53  0   3           53  0   3
    
│ │ │              
    
│ │ │                
    i3 : numericalHilbertFunction(F, ideal 0_R, 4)
│ │ │  Sampling image points ...
│ │ │ -     -- used .012393 seconds
│ │ │ +     -- used .0146129 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .0114083 seconds
│ │ │ +     -- used .0129944 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00726179 seconds
│ │ │ +     -- used .00766644 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000761889 seconds
│ │ │ +     -- used .000825249 seconds
│ │ │  
│ │ │  o3 = a "numerical interpolation table", indicating
│ │ │       the space of degree 4 forms in the ideal of the image has dimension 22
│ │ │  
│ │ │  o3 : NumericalInterpolationTable
    
│ │ │              
    
│ │ │                
    i7 : numericalHilbertFunction(F, ideal 0_R, S, 2, UseSLP => true)
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .00312588 seconds
│ │ │ +     -- used .0033447 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00846492 seconds
│ │ │ +     -- used .00904545 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000905679 seconds
│ │ │ +     -- used .00110696 seconds
│ │ │  
│ │ │  o7 = a "numerical interpolation table", indicating
│ │ │       the space of degree 2 forms in the ideal of the image has dimension 1
│ │ │  
│ │ │  o7 : NumericalInterpolationTable
    
│ │ │              
    
│ │ │                
    i9 : time numericalImageDim(F, ideal 0_R)
│ │ │ - -- used 0.066304s (cpu); 0.0663008s (thread); 0s (gc)
│ │ │ + -- used 0.0740739s (cpu); 0.0740672s (thread); 0s (gc)
│ │ │  
│ │ │  o9 = 69
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -43,15 +43,15 @@
│ │ │ │  201-222. We numerically verify this below.
│ │ │ │  i7 : R = CC[a_(1,1)..a_(14,5)];
│ │ │ │  i8 : F = sum(1..14, i -> basis(4, R, Variables=>toList(a_(i,1)..a_(i,5))));
│ │ │ │  
│ │ │ │               1      70
│ │ │ │  o8 : Matrix R  <-- R
│ │ │ │  i9 : time numericalImageDim(F, ideal 0_R)
│ │ │ │ - -- used 0.066304s (cpu); 0.0663008s (thread); 0s (gc)
│ │ │ │ + -- used 0.0740739s (cpu); 0.0740672s (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
│ │ │ @@ -132,15 +132,15 @@
│ │ │  
│ │ │  o6 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : elapsedTime p = realPoint(I, Iterations => 100)
│ │ │ - -- .619309s elapsed
│ │ │ + -- .47259s elapsed
│ │ │  
│ │ │  o7 = p
│ │ │  
│ │ │  o7 : Point
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -49,15 +49,15 @@
│ │ │ │  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)
│ │ │ │ - -- .619309s elapsed
│ │ │ │ + -- .47259s 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
│ │ │ @@ -1,8 +1,8 @@
│ │ │ -# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Fri Nov 14 16:08:08 2025
│ │ │ +# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Fri Nov 14 16:08:07 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │  #:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=35
│ │ │  bnVtZXJpY2FsS2VybmVsKC4uLixUb2xlcmFuY2U9Pi4uLik=
│ │ ├── ./usr/share/doc/Macaulay2/NumericalSchubertCalculus/example-output/_set__Verbose__Level.out
│ │ │ @@ -52,92 +52,92 @@
│ │ │  
│ │ │  i4 : assert all(S,s->checkIncidenceSolution(s,SchPblm))
│ │ │  
│ │ │  i5 : setVerboseLevel 1; 
│ │ │  
│ │ │  i6 : S = solveSchubertProblem(SchPblm,2,4)
│ │ │  -- playCheckers
│ │ │ --- cpu time = .00707217
│ │ │ +-- cpu time = .0120019
│ │ │  -- making a recursive call to resolveNode
│ │ │  -- playCheckers
│ │ │ --- cpu time = .00399967
│ │ │ +-- cpu time = .00406253
│ │ │  -- making a recursive call to resolveNode
│ │ │  -- playCheckers
│ │ │  -- cpu time = 0
│ │ │  resolveNode reached node of no remaining conditions
│ │ │ --- time to make equations: .00399831
│ │ │ +-- time to make equations: .00816898
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .00653127 sec. for [{0, 1, 2, 3}, {0, infinity, 2, infinity}]
│ │ │ --- time of performing one checker move: .0159992
│ │ │ --- time of performing one checker move: 0
│ │ │ --- time of performing one checker move: 0
│ │ │ --- time to make equations: .00799998
│ │ │ + -- trackHomotopy time = .00826948 sec. for [{0, 1, 2, 3}, {0, infinity, 2, infinity}]
│ │ │ +-- time of performing one checker move: .0239408
│ │ │ +-- time of performing one checker move: .118184
│ │ │ +-- time of performing one checker move: .00398542
│ │ │ +-- time to make equations: .00397353
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .00612084 sec. for [{1, 2, 3, 0}, {1, infinity, infinity, 2}]
│ │ │ --- time of performing one checker move: .0160045
│ │ │ --- time to make equations: .00515796
│ │ │ -Setup time: 1
│ │ │ + -- trackHomotopy time = .00743253 sec. for [{1, 2, 3, 0}, {1, infinity, infinity, 2}]
│ │ │ +-- time of performing one checker move: .0199985
│ │ │ +-- time to make equations: .00410781
│ │ │ +Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .0870607 sec. for [{1, 3, 2, 0}, {1, infinity, infinity, 2}]
│ │ │ --- time of performing one checker move: .111595
│ │ │ --- time to make equations: .00399871
│ │ │ + -- trackHomotopy time = .0277843 sec. for [{1, 3, 2, 0}, {1, infinity, infinity, 2}]
│ │ │ +-- time of performing one checker move: .130759
│ │ │ +-- time to make equations: .00799933
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .00592173 sec. for [{2, 3, 1, 0}, {2, infinity, infinity, 1}]
│ │ │ --- time of performing one checker move: .015998
│ │ │ --- time to make equations: .0854627
│ │ │ + -- trackHomotopy time = .00734998 sec. for [{2, 3, 1, 0}, {2, infinity, infinity, 1}]
│ │ │ +-- time of performing one checker move: .020074
│ │ │ +-- time to make equations: .109524
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .0112502 sec. for [{0, 1, 2, 3}, {infinity, 1, 2, infinity}]
│ │ │ --- time of performing one checker move: .0974532
│ │ │ --- time to make equations: .012
│ │ │ + -- trackHomotopy time = .00901353 sec. for [{0, 1, 2, 3}, {infinity, 1, 2, infinity}]
│ │ │ +-- time of performing one checker move: .129491
│ │ │ +-- time to make equations: .116336
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .0691376 sec. for [{0, 1, 3, 2}, {infinity, 1, infinity, 2}]
│ │ │ --- time of performing one checker move: .109427
│ │ │ --- time of performing one checker move: 0
│ │ │ --- time of performing one checker move: .0039993
│ │ │ --- time to make equations: .012002
│ │ │ + -- trackHomotopy time = .00895515 sec. for [{0, 1, 3, 2}, {infinity, 1, infinity, 2}]
│ │ │ +-- time of performing one checker move: .128425
│ │ │ +-- time of performing one checker move: .0039221
│ │ │ +-- time of performing one checker move: .00404729
│ │ │ +-- time to make equations: .0159434
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .0829076 sec. for [{1, 3, 2, 0}, {infinity, 3, infinity, 1}]
│ │ │ --- time of performing one checker move: .111029
│ │ │ + -- trackHomotopy time = .0287004 sec. for [{1, 3, 2, 0}, {infinity, 3, infinity, 1}]
│ │ │ +-- time of performing one checker move: .129882
│ │ │  -- making a recursive call to resolveNode
│ │ │  -- playCheckers
│ │ │ --- cpu time = .00799904
│ │ │ +-- cpu time = .00403028
│ │ │  -- making a recursive call to resolveNode
│ │ │  -- playCheckers
│ │ │  -- cpu time = 0
│ │ │  resolveNode reached node of no remaining conditions
│ │ │ --- time to make equations: .00399999
│ │ │ +-- time to make equations: .109975
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .00633084 sec. for [{0, 1, 2, 3}, {0, infinity, 2, infinity}]
│ │ │ --- time of performing one checker move: .0989545
│ │ │ + -- trackHomotopy time = .00755955 sec. for [{0, 1, 2, 3}, {0, infinity, 2, infinity}]
│ │ │ +-- time of performing one checker move: .126017
│ │ │  -- time of performing one checker move: 0
│ │ │ --- time to make equations: .00399939
│ │ │ +-- time to make equations: .00799392
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .00612709 sec. for [{0, 2, 3, 1}, {0, infinity, infinity, 2}]
│ │ │ --- time of performing one checker move: .0199873
│ │ │ --- time of performing one checker move: .0917858
│ │ │ --- time of performing one checker move: .00398251
│ │ │ + -- trackHomotopy time = .00774006 sec. for [{0, 2, 3, 1}, {0, infinity, infinity, 2}]
│ │ │ +-- time of performing one checker move: .0239359
│ │ │ +-- time of performing one checker move: 0
│ │ │ +-- time of performing one checker move: 0
│ │ │ +-- time of performing one checker move: .00400311
│ │ │  -- time of performing one checker move: 0
│ │ │ --- time of performing one checker move: .00400001
│ │ │  -- time of performing one checker move: 0
│ │ │ --- time of performing one checker move: .00399726
│ │ │ +-- time of performing one checker move: .00400024
│ │ │  -- time of performing one checker move: 0
│ │ │ --- time to make equations: .0119992
│ │ │ +-- time to make equations: .119348
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .0924931 sec. for [{1, 3, 2, 0}, {1, infinity, infinity, 3}]
│ │ │ --- time of performing one checker move: .111872
│ │ │ --- time of performing one checker move: .00399915
│ │ │ + -- trackHomotopy time = .00924347 sec. for [{1, 3, 2, 0}, {1, infinity, infinity, 3}]
│ │ │ +-- time of performing one checker move: .13532
│ │ │ +-- time of performing one checker move: .00401344
│ │ │  
│ │ │  o6 = {| -1.65573-.600637ii .0201935+.0437095ii   |, | -.154703+.175591ii 
│ │ │        | -1.23037-1.66989ii -.0308057-.00120618ii |  | -.801221-.0354303ii
│ │ │        | 1.35971-.743988ii  -.0713133-.049047ii   |  | .325581-2.08048ii  
│ │ │        | -.397038-1.8974ii  .0102261-.024397ii    |  | -.475895-.209388ii 
│ │ │       ------------------------------------------------------------------------
│ │ │       .0376857+.0683239ii   |}
│ │ ├── ./usr/share/doc/Macaulay2/NumericalSchubertCalculus/html/_set__Verbose__Level.html
│ │ │ @@ -147,92 +147,92 @@
│ │ │                
    i5 : setVerboseLevel 1; 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : S = solveSchubertProblem(SchPblm,2,4)
│ │ │  -- playCheckers
│ │ │ --- cpu time = .00707217
│ │ │ +-- cpu time = .0120019
│ │ │  -- making a recursive call to resolveNode
│ │ │  -- playCheckers
│ │ │ --- cpu time = .00399967
│ │ │ +-- cpu time = .00406253
│ │ │  -- making a recursive call to resolveNode
│ │ │  -- playCheckers
│ │ │  -- cpu time = 0
│ │ │  resolveNode reached node of no remaining conditions
│ │ │ --- time to make equations: .00399831
│ │ │ +-- time to make equations: .00816898
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .00653127 sec. for [{0, 1, 2, 3}, {0, infinity, 2, infinity}]
│ │ │ --- time of performing one checker move: .0159992
│ │ │ --- time of performing one checker move: 0
│ │ │ --- time of performing one checker move: 0
│ │ │ --- time to make equations: .00799998
│ │ │ + -- trackHomotopy time = .00826948 sec. for [{0, 1, 2, 3}, {0, infinity, 2, infinity}]
│ │ │ +-- time of performing one checker move: .0239408
│ │ │ +-- time of performing one checker move: .118184
│ │ │ +-- time of performing one checker move: .00398542
│ │ │ +-- time to make equations: .00397353
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .00612084 sec. for [{1, 2, 3, 0}, {1, infinity, infinity, 2}]
│ │ │ --- time of performing one checker move: .0160045
│ │ │ --- time to make equations: .00515796
│ │ │ -Setup time: 1
│ │ │ + -- trackHomotopy time = .00743253 sec. for [{1, 2, 3, 0}, {1, infinity, infinity, 2}]
│ │ │ +-- time of performing one checker move: .0199985
│ │ │ +-- time to make equations: .00410781
│ │ │ +Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .0870607 sec. for [{1, 3, 2, 0}, {1, infinity, infinity, 2}]
│ │ │ --- time of performing one checker move: .111595
│ │ │ --- time to make equations: .00399871
│ │ │ + -- trackHomotopy time = .0277843 sec. for [{1, 3, 2, 0}, {1, infinity, infinity, 2}]
│ │ │ +-- time of performing one checker move: .130759
│ │ │ +-- time to make equations: .00799933
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .00592173 sec. for [{2, 3, 1, 0}, {2, infinity, infinity, 1}]
│ │ │ --- time of performing one checker move: .015998
│ │ │ --- time to make equations: .0854627
│ │ │ + -- trackHomotopy time = .00734998 sec. for [{2, 3, 1, 0}, {2, infinity, infinity, 1}]
│ │ │ +-- time of performing one checker move: .020074
│ │ │ +-- time to make equations: .109524
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .0112502 sec. for [{0, 1, 2, 3}, {infinity, 1, 2, infinity}]
│ │ │ --- time of performing one checker move: .0974532
│ │ │ --- time to make equations: .012
│ │ │ + -- trackHomotopy time = .00901353 sec. for [{0, 1, 2, 3}, {infinity, 1, 2, infinity}]
│ │ │ +-- time of performing one checker move: .129491
│ │ │ +-- time to make equations: .116336
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .0691376 sec. for [{0, 1, 3, 2}, {infinity, 1, infinity, 2}]
│ │ │ --- time of performing one checker move: .109427
│ │ │ --- time of performing one checker move: 0
│ │ │ --- time of performing one checker move: .0039993
│ │ │ --- time to make equations: .012002
│ │ │ + -- trackHomotopy time = .00895515 sec. for [{0, 1, 3, 2}, {infinity, 1, infinity, 2}]
│ │ │ +-- time of performing one checker move: .128425
│ │ │ +-- time of performing one checker move: .0039221
│ │ │ +-- time of performing one checker move: .00404729
│ │ │ +-- time to make equations: .0159434
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .0829076 sec. for [{1, 3, 2, 0}, {infinity, 3, infinity, 1}]
│ │ │ --- time of performing one checker move: .111029
│ │ │ + -- trackHomotopy time = .0287004 sec. for [{1, 3, 2, 0}, {infinity, 3, infinity, 1}]
│ │ │ +-- time of performing one checker move: .129882
│ │ │  -- making a recursive call to resolveNode
│ │ │  -- playCheckers
│ │ │ --- cpu time = .00799904
│ │ │ +-- cpu time = .00403028
│ │ │  -- making a recursive call to resolveNode
│ │ │  -- playCheckers
│ │ │  -- cpu time = 0
│ │ │  resolveNode reached node of no remaining conditions
│ │ │ --- time to make equations: .00399999
│ │ │ +-- time to make equations: .109975
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .00633084 sec. for [{0, 1, 2, 3}, {0, infinity, 2, infinity}]
│ │ │ --- time of performing one checker move: .0989545
│ │ │ + -- trackHomotopy time = .00755955 sec. for [{0, 1, 2, 3}, {0, infinity, 2, infinity}]
│ │ │ +-- time of performing one checker move: .126017
│ │ │  -- time of performing one checker move: 0
│ │ │ --- time to make equations: .00399939
│ │ │ +-- time to make equations: .00799392
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .00612709 sec. for [{0, 2, 3, 1}, {0, infinity, infinity, 2}]
│ │ │ --- time of performing one checker move: .0199873
│ │ │ --- time of performing one checker move: .0917858
│ │ │ --- time of performing one checker move: .00398251
│ │ │ + -- trackHomotopy time = .00774006 sec. for [{0, 2, 3, 1}, {0, infinity, infinity, 2}]
│ │ │ +-- time of performing one checker move: .0239359
│ │ │ +-- time of performing one checker move: 0
│ │ │ +-- time of performing one checker move: 0
│ │ │ +-- time of performing one checker move: .00400311
│ │ │  -- time of performing one checker move: 0
│ │ │ --- time of performing one checker move: .00400001
│ │ │  -- time of performing one checker move: 0
│ │ │ --- time of performing one checker move: .00399726
│ │ │ +-- time of performing one checker move: .00400024
│ │ │  -- time of performing one checker move: 0
│ │ │ --- time to make equations: .0119992
│ │ │ +-- time to make equations: .119348
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .0924931 sec. for [{1, 3, 2, 0}, {1, infinity, infinity, 3}]
│ │ │ --- time of performing one checker move: .111872
│ │ │ --- time of performing one checker move: .00399915
│ │ │ + -- trackHomotopy time = .00924347 sec. for [{1, 3, 2, 0}, {1, infinity, infinity, 3}]
│ │ │ +-- time of performing one checker move: .13532
│ │ │ +-- time of performing one checker move: .00401344
│ │ │  
│ │ │  o6 = {| -1.65573-.600637ii .0201935+.0437095ii   |, | -.154703+.175591ii 
│ │ │        | -1.23037-1.66989ii -.0308057-.00120618ii |  | -.801221-.0354303ii
│ │ │        | 1.35971-.743988ii  -.0713133-.049047ii   |  | .325581-2.08048ii  
│ │ │        | -.397038-1.8974ii  .0102261-.024397ii    |  | -.475895-.209388ii 
│ │ │       ------------------------------------------------------------------------
│ │ │       .0376857+.0683239ii   |}
│ │ │ ├── html2text {}
│ │ │ │ @@ -65,102 +65,102 @@
│ │ │ │       -.0336427+.0141017ii  |
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : assert all(S,s->checkIncidenceSolution(s,SchPblm))
│ │ │ │  i5 : setVerboseLevel 1;
│ │ │ │  i6 : S = solveSchubertProblem(SchPblm,2,4)
│ │ │ │  -- playCheckers
│ │ │ │ --- cpu time = .00707217
│ │ │ │ +-- cpu time = .0120019
│ │ │ │  -- making a recursive call to resolveNode
│ │ │ │  -- playCheckers
│ │ │ │ --- cpu time = .00399967
│ │ │ │ +-- cpu time = .00406253
│ │ │ │  -- making a recursive call to resolveNode
│ │ │ │  -- playCheckers
│ │ │ │  -- cpu time = 0
│ │ │ │  resolveNode reached node of no remaining conditions
│ │ │ │ --- time to make equations: .00399831
│ │ │ │ +-- time to make equations: .00816898
│ │ │ │  Setup time: 0
│ │ │ │  Computing time:0
│ │ │ │ - -- trackHomotopy time = .00653127 sec. for [{0, 1, 2, 3}, {0, infinity, 2,
│ │ │ │ + -- trackHomotopy time = .00826948 sec. for [{0, 1, 2, 3}, {0, infinity, 2,
│ │ │ │  infinity}]
│ │ │ │ --- time of performing one checker move: .0159992
│ │ │ │ --- time of performing one checker move: 0
│ │ │ │ --- time of performing one checker move: 0
│ │ │ │ --- time to make equations: .00799998
│ │ │ │ +-- time of performing one checker move: .0239408
│ │ │ │ +-- time of performing one checker move: .118184
│ │ │ │ +-- time of performing one checker move: .00398542
│ │ │ │ +-- time to make equations: .00397353
│ │ │ │  Setup time: 0
│ │ │ │  Computing time:0
│ │ │ │ - -- trackHomotopy time = .00612084 sec. for [{1, 2, 3, 0}, {1, infinity,
│ │ │ │ + -- trackHomotopy time = .00743253 sec. for [{1, 2, 3, 0}, {1, infinity,
│ │ │ │  infinity, 2}]
│ │ │ │ --- time of performing one checker move: .0160045
│ │ │ │ --- time to make equations: .00515796
│ │ │ │ -Setup time: 1
│ │ │ │ +-- time of performing one checker move: .0199985
│ │ │ │ +-- time to make equations: .00410781
│ │ │ │ +Setup time: 0
│ │ │ │  Computing time:0
│ │ │ │ - -- trackHomotopy time = .0870607 sec. for [{1, 3, 2, 0}, {1, infinity,
│ │ │ │ + -- trackHomotopy time = .0277843 sec. for [{1, 3, 2, 0}, {1, infinity,
│ │ │ │  infinity, 2}]
│ │ │ │ --- time of performing one checker move: .111595
│ │ │ │ --- time to make equations: .00399871
│ │ │ │ +-- time of performing one checker move: .130759
│ │ │ │ +-- time to make equations: .00799933
│ │ │ │  Setup time: 0
│ │ │ │  Computing time:0
│ │ │ │ - -- trackHomotopy time = .00592173 sec. for [{2, 3, 1, 0}, {2, infinity,
│ │ │ │ + -- trackHomotopy time = .00734998 sec. for [{2, 3, 1, 0}, {2, infinity,
│ │ │ │  infinity, 1}]
│ │ │ │ --- time of performing one checker move: .015998
│ │ │ │ --- time to make equations: .0854627
│ │ │ │ +-- time of performing one checker move: .020074
│ │ │ │ +-- time to make equations: .109524
│ │ │ │  Setup time: 0
│ │ │ │  Computing time:0
│ │ │ │ - -- trackHomotopy time = .0112502 sec. for [{0, 1, 2, 3}, {infinity, 1, 2,
│ │ │ │ + -- trackHomotopy time = .00901353 sec. for [{0, 1, 2, 3}, {infinity, 1, 2,
│ │ │ │  infinity}]
│ │ │ │ --- time of performing one checker move: .0974532
│ │ │ │ --- time to make equations: .012
│ │ │ │ +-- time of performing one checker move: .129491
│ │ │ │ +-- time to make equations: .116336
│ │ │ │  Setup time: 0
│ │ │ │  Computing time:0
│ │ │ │ - -- trackHomotopy time = .0691376 sec. for [{0, 1, 3, 2}, {infinity, 1,
│ │ │ │ + -- trackHomotopy time = .00895515 sec. for [{0, 1, 3, 2}, {infinity, 1,
│ │ │ │  infinity, 2}]
│ │ │ │ --- time of performing one checker move: .109427
│ │ │ │ --- time of performing one checker move: 0
│ │ │ │ --- time of performing one checker move: .0039993
│ │ │ │ --- time to make equations: .012002
│ │ │ │ +-- time of performing one checker move: .128425
│ │ │ │ +-- time of performing one checker move: .0039221
│ │ │ │ +-- time of performing one checker move: .00404729
│ │ │ │ +-- time to make equations: .0159434
│ │ │ │  Setup time: 0
│ │ │ │  Computing time:0
│ │ │ │ - -- trackHomotopy time = .0829076 sec. for [{1, 3, 2, 0}, {infinity, 3,
│ │ │ │ + -- trackHomotopy time = .0287004 sec. for [{1, 3, 2, 0}, {infinity, 3,
│ │ │ │  infinity, 1}]
│ │ │ │ --- time of performing one checker move: .111029
│ │ │ │ +-- time of performing one checker move: .129882
│ │ │ │  -- making a recursive call to resolveNode
│ │ │ │  -- playCheckers
│ │ │ │ --- cpu time = .00799904
│ │ │ │ +-- cpu time = .00403028
│ │ │ │  -- making a recursive call to resolveNode
│ │ │ │  -- playCheckers
│ │ │ │  -- cpu time = 0
│ │ │ │  resolveNode reached node of no remaining conditions
│ │ │ │ --- time to make equations: .00399999
│ │ │ │ +-- time to make equations: .109975
│ │ │ │  Setup time: 0
│ │ │ │  Computing time:0
│ │ │ │ - -- trackHomotopy time = .00633084 sec. for [{0, 1, 2, 3}, {0, infinity, 2,
│ │ │ │ + -- trackHomotopy time = .00755955 sec. for [{0, 1, 2, 3}, {0, infinity, 2,
│ │ │ │  infinity}]
│ │ │ │ --- time of performing one checker move: .0989545
│ │ │ │ +-- time of performing one checker move: .126017
│ │ │ │  -- time of performing one checker move: 0
│ │ │ │ --- time to make equations: .00399939
│ │ │ │ +-- time to make equations: .00799392
│ │ │ │  Setup time: 0
│ │ │ │  Computing time:0
│ │ │ │ - -- trackHomotopy time = .00612709 sec. for [{0, 2, 3, 1}, {0, infinity,
│ │ │ │ + -- trackHomotopy time = .00774006 sec. for [{0, 2, 3, 1}, {0, infinity,
│ │ │ │  infinity, 2}]
│ │ │ │ --- time of performing one checker move: .0199873
│ │ │ │ --- time of performing one checker move: .0917858
│ │ │ │ --- time of performing one checker move: .00398251
│ │ │ │ +-- time of performing one checker move: .0239359
│ │ │ │ +-- time of performing one checker move: 0
│ │ │ │ +-- time of performing one checker move: 0
│ │ │ │ +-- time of performing one checker move: .00400311
│ │ │ │  -- time of performing one checker move: 0
│ │ │ │ --- time of performing one checker move: .00400001
│ │ │ │  -- time of performing one checker move: 0
│ │ │ │ --- time of performing one checker move: .00399726
│ │ │ │ +-- time of performing one checker move: .00400024
│ │ │ │  -- time of performing one checker move: 0
│ │ │ │ --- time to make equations: .0119992
│ │ │ │ +-- time to make equations: .119348
│ │ │ │  Setup time: 0
│ │ │ │  Computing time:0
│ │ │ │ - -- trackHomotopy time = .0924931 sec. for [{1, 3, 2, 0}, {1, infinity,
│ │ │ │ + -- trackHomotopy time = .00924347 sec. for [{1, 3, 2, 0}, {1, infinity,
│ │ │ │  infinity, 3}]
│ │ │ │ --- time of performing one checker move: .111872
│ │ │ │ --- time of performing one checker move: .00399915
│ │ │ │ +-- time of performing one checker move: .13532
│ │ │ │ +-- time of performing one checker move: .00401344
│ │ │ │  
│ │ │ │  o6 = {| -1.65573-.600637ii .0201935+.0437095ii   |, | -.154703+.175591ii
│ │ │ │        | -1.23037-1.66989ii -.0308057-.00120618ii |  | -.801221-.0354303ii
│ │ │ │        | 1.35971-.743988ii  -.0713133-.049047ii   |  | .325581-2.08048ii
│ │ │ │        | -.397038-1.8974ii  .0102261-.024397ii    |  | -.475895-.209388ii
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       .0376857+.0683239ii   |}
│ │ ├── ./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
│ │ │ - -- .159171s elapsed
│ │ │ + -- .116798s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .160219s elapsed
│ │ │ + -- .0968547s elapsed
│ │ │  next gb
│ │ │ - -- .000771299s elapsed
│ │ │ + -- .000851727s elapsed
│ │ │  true
│ │ │  unfolding
│ │ │ - -- .16329s elapsed
│ │ │ + -- .1129s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .137067s elapsed
│ │ │ + -- .0988441s elapsed
│ │ │  next gb
│ │ │ - -- .000599818s elapsed
│ │ │ + -- .000578211s elapsed
│ │ │  true
│ │ │ - -- 1.59938s elapsed
│ │ │ + -- 1.23829s elapsed
│ │ │  
│ │ │  o6 = {}
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : elapsedTime LL7b=select(LL7a,L->not isSmoothableSemigroup(L,0.25,0))
│ │ │ - -- 1.37691s elapsed
│ │ │ + -- 1.19822s elapsed
│ │ │  
│ │ │  o7 = {}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : LL7b=={}
│ │ │  
│ │ │ @@ -75,23 +75,23 @@
│ │ │  
│ │ │  o10 : Sequence
│ │ │  
│ │ │  i11 : elapsedTime nonWeierstrassSemigroups(m,g,Verbose=>true)
│ │ │  (13, 1)
│ │ │  {5, 8, 11, 12}
│ │ │  unfolding
│ │ │ - -- .200467s elapsed
│ │ │ + -- .192987s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .173649s elapsed
│ │ │ + -- .117053s elapsed
│ │ │  next gb
│ │ │ - -- .000926207s elapsed
│ │ │ + -- .000932633s elapsed
│ │ │  true
│ │ │ - -- .618832s elapsed
│ │ │ + -- .54854s elapsed
│ │ │  (5, 8,  all semigroups are smoothable)
│ │ │ - -- .647468s elapsed
│ │ │ + -- .591597s 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)
│ │ │ - -- .107024s elapsed
│ │ │ + -- .0765521s elapsed
│ │ │  6
│ │ │  false
│ │ │  5
│ │ │  false
│ │ │  4
│ │ │  decompose
│ │ │ - -- .304981s elapsed
│ │ │ + -- .310456s 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
│ │ │ @@ -10,15 +10,15 @@
│ │ │   -- setting random seed to 1644814534404491274313411285186041988099567563905780374824086062516559438
│ │ │  
│ │ │  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))
│ │ │ - -- 3.77995s elapsed
│ │ │ + -- 3.26708s 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)
│ │ │ - -- 1.10208s elapsed
│ │ │ + -- .822381s elapsed
│ │ │  
│ │ │  o3 = false
│ │ │  
│ │ │  i4 : elapsedTime isSmoothableSemigroup(L,0.14,0)
│ │ │ - -- 3.98405s elapsed
│ │ │ + -- 3.62086s 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)
│ │ │ - -- 4.09713s elapsed
│ │ │ + -- 3.3422s elapsed
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/example-output/_non__Weierstrass__Semigroups.out
│ │ │ @@ -1,12 +1,12 @@
│ │ │  -- -*- M2-comint -*- hash: 6860996532851631556
│ │ │  
│ │ │  i1 : elapsedTime nonWeierstrassSemigroups(6,7)
│ │ │  (6, 7,  all semigroups are smoothable)
│ │ │ - -- 1.62513s elapsed
│ │ │ + -- 1.14528s 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
│ │ │ - -- .396426s elapsed
│ │ │ + -- .379482s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .149345s elapsed
│ │ │ + -- .141966s elapsed
│ │ │  next gb
│ │ │ - -- .00186426s elapsed
│ │ │ + -- .00195322s elapsed
│ │ │  true
│ │ │ - -- .934436s elapsed
│ │ │ + -- .856537s elapsed
│ │ │  {6, 7, 9, 17}
│ │ │  unfolding
│ │ │ - -- .34282s elapsed
│ │ │ + -- .337101s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .263338s elapsed
│ │ │ + -- .177858s elapsed
│ │ │  next gb
│ │ │ - -- .00267103s elapsed
│ │ │ + -- .00276712s elapsed
│ │ │  decompose
│ │ │ - -- .122971s elapsed
│ │ │ + -- .135951s elapsed
│ │ │  number of components: 2
│ │ │  support c, codim c: {(2, 2), (5, 2)}
│ │ │  {0, -1}
│ │ │ - -- 2.82135s elapsed
│ │ │ + -- 2.57508s elapsed
│ │ │  {6, 8, 9, 10}
│ │ │  unfolding
│ │ │ - -- .173434s elapsed
│ │ │ + -- .118005s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .133545s elapsed
│ │ │ + -- .103052s elapsed
│ │ │  next gb
│ │ │ - -- .000440471s elapsed
│ │ │ + -- .000505392s elapsed
│ │ │  true
│ │ │ - -- .746963s elapsed
│ │ │ + -- .617495s elapsed
│ │ │  {6, 8, 10, 11, 13}
│ │ │  unfolding
│ │ │ - -- .532889s elapsed
│ │ │ + -- .518928s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .194284s elapsed
│ │ │ + -- .208067s elapsed
│ │ │  next gb
│ │ │ - -- .00451218s elapsed
│ │ │ + -- .00522249s elapsed
│ │ │  decompose
│ │ │ - -- 1.02026s elapsed
│ │ │ + -- .751344s elapsed
│ │ │  number of components: 1
│ │ │  support c, codim c: {(5, 1)}
│ │ │  {-1}
│ │ │ - -- 2.95467s elapsed
│ │ │ - -- 7.45755s elapsed
│ │ │ + -- 2.46593s elapsed
│ │ │ + -- 6.51519s elapsed
│ │ │  0
│ │ │  
│ │ │  {}
│ │ │ - -- .000003637s elapsed
│ │ │ - -- 7.49256s elapsed
│ │ │ + -- .000004612s elapsed
│ │ │ + -- 6.55119s elapsed
│ │ │  
│ │ │  o3 = {{6, 8, 9, 11}}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/html/___Lab__Book__Protocol.html
│ │ │ @@ -96,38 +96,38 @@
│ │ │  o5 = 2
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : elapsedTime LL7b=select(LL7a,L->not isSmoothableSemigroup(L,0.25,0,Verbose=>true))
│ │ │  unfolding
│ │ │ - -- .159171s elapsed
│ │ │ + -- .116798s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .160219s elapsed
│ │ │ + -- .0968547s elapsed
│ │ │  next gb
│ │ │ - -- .000771299s elapsed
│ │ │ + -- .000851727s elapsed
│ │ │  true
│ │ │  unfolding
│ │ │ - -- .16329s elapsed
│ │ │ + -- .1129s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .137067s elapsed
│ │ │ + -- .0988441s elapsed
│ │ │  next gb
│ │ │ - -- .000599818s elapsed
│ │ │ + -- .000578211s elapsed
│ │ │  true
│ │ │ - -- 1.59938s elapsed
│ │ │ + -- 1.23829s elapsed
│ │ │  
│ │ │  o6 = {}
│ │ │  
│ │ │  o6 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : elapsedTime LL7b=select(LL7a,L->not isSmoothableSemigroup(L,0.25,0))
│ │ │ - -- 1.37691s elapsed
│ │ │ + -- 1.19822s elapsed
│ │ │  
│ │ │  o7 = {}
│ │ │  
│ │ │  o7 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -184,23 +184,23 @@
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i11 : elapsedTime nonWeierstrassSemigroups(m,g,Verbose=>true)
│ │ │  (13, 1)
│ │ │  {5, 8, 11, 12}
│ │ │  unfolding
│ │ │ - -- .200467s elapsed
│ │ │ + -- .192987s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .173649s elapsed
│ │ │ + -- .117053s elapsed
│ │ │  next gb
│ │ │ - -- .000926207s elapsed
│ │ │ + -- .000932633s elapsed
│ │ │  true
│ │ │ - -- .618832s elapsed
│ │ │ + -- .54854s elapsed
│ │ │  (5, 8,  all semigroups are smoothable)
│ │ │ - -- .647468s elapsed
│ │ │ + -- .591597s elapsed
│ │ │  
│ │ │  o11 = {}
│ │ │  
│ │ │  o11 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │ @@ -223,22 +223,22 @@
│ │ │  
│ │ │  o13 = 8
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : isWeierstrassSemigroup(L,0.2,Verbose=>true)
│ │ │ - -- .107024s elapsed
│ │ │ + -- .0765521s elapsed
│ │ │  6
│ │ │  false
│ │ │  5
│ │ │  false
│ │ │  4
│ │ │  decompose
│ │ │ - -- .304981s elapsed
│ │ │ + -- .310456s elapsed
│ │ │  number of components: 2
│ │ │  support c, codim c: {(1, 1), (16, 3)}
│ │ │  {0, -1}
│ │ │  
│ │ │  o14 = true
    
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,34 +26,34 @@
│ │ │ │  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
│ │ │ │ - -- .159171s elapsed
│ │ │ │ + -- .116798s elapsed
│ │ │ │  flatteningRelations
│ │ │ │ - -- .160219s elapsed
│ │ │ │ + -- .0968547s elapsed
│ │ │ │  next gb
│ │ │ │ - -- .000771299s elapsed
│ │ │ │ + -- .000851727s elapsed
│ │ │ │  true
│ │ │ │  unfolding
│ │ │ │ - -- .16329s elapsed
│ │ │ │ + -- .1129s elapsed
│ │ │ │  flatteningRelations
│ │ │ │ - -- .137067s elapsed
│ │ │ │ + -- .0988441s elapsed
│ │ │ │  next gb
│ │ │ │ - -- .000599818s elapsed
│ │ │ │ + -- .000578211s elapsed
│ │ │ │  true
│ │ │ │ - -- 1.59938s elapsed
│ │ │ │ + -- 1.23829s elapsed
│ │ │ │  
│ │ │ │  o6 = {}
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : elapsedTime LL7b=select(LL7a,L->not isSmoothableSemigroup(L,0.25,0))
│ │ │ │ - -- 1.37691s elapsed
│ │ │ │ + -- 1.19822s elapsed
│ │ │ │  
│ │ │ │  o7 = {}
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  i8 : LL7b=={}
│ │ │ │  
│ │ │ │  o8 = true
│ │ │ │ @@ -92,23 +92,23 @@
│ │ │ │  o10 = (5, 8)
│ │ │ │  
│ │ │ │  o10 : Sequence
│ │ │ │  i11 : elapsedTime nonWeierstrassSemigroups(m,g,Verbose=>true)
│ │ │ │  (13, 1)
│ │ │ │  {5, 8, 11, 12}
│ │ │ │  unfolding
│ │ │ │ - -- .200467s elapsed
│ │ │ │ + -- .192987s elapsed
│ │ │ │  flatteningRelations
│ │ │ │ - -- .173649s elapsed
│ │ │ │ + -- .117053s elapsed
│ │ │ │  next gb
│ │ │ │ - -- .000926207s elapsed
│ │ │ │ + -- .000932633s elapsed
│ │ │ │  true
│ │ │ │ - -- .618832s elapsed
│ │ │ │ + -- .54854s elapsed
│ │ │ │  (5, 8,  all semigroups are smoothable)
│ │ │ │ - -- .647468s elapsed
│ │ │ │ + -- .591597s 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
│ │ │ │  multiplicity 5 and genus 8 of which only 1 is not flagged as smoothable by the
│ │ │ │ @@ -120,22 +120,22 @@
│ │ │ │  o12 = {6, 8, 9, 11}
│ │ │ │  
│ │ │ │  o12 : List
│ │ │ │  i13 : genus L
│ │ │ │  
│ │ │ │  o13 = 8
│ │ │ │  i14 : isWeierstrassSemigroup(L,0.2,Verbose=>true)
│ │ │ │ - -- .107024s elapsed
│ │ │ │ + -- .0765521s elapsed
│ │ │ │  6
│ │ │ │  false
│ │ │ │  5
│ │ │ │  false
│ │ │ │  4
│ │ │ │  decompose
│ │ │ │ - -- .304981s elapsed
│ │ │ │ + -- .310456s 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
│ │ │ @@ -94,15 +94,15 @@
│ │ │            
│ │ │              
│ │ │                
    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))
│ │ │ - -- 3.77995s elapsed
│ │ │ + -- 3.26708s elapsed
│ │ │  
│ │ │  o4 = Tally{false => 6}
│ │ │             true => 4
│ │ │  
│ │ │  o4 : Tally
    
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,15 +27,15 @@
│ │ │ │   -- setting random seed to
│ │ │ │  1644814534404491274313411285186041988099567563905780374824086062516559438
│ │ │ │  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))
│ │ │ │ - -- 3.77995s elapsed
│ │ │ │ + -- 3.26708s 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
│ │ │ @@ -95,23 +95,23 @@
│ │ │  
│ │ │  o2 = 8
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : elapsedTime isSmoothableSemigroup(L,0.30,0)
│ │ │ - -- 1.10208s elapsed
│ │ │ + -- .822381s elapsed
│ │ │  
│ │ │  o3 = false
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : elapsedTime isSmoothableSemigroup(L,0.14,0)
│ │ │ - -- 3.98405s elapsed
│ │ │ + -- 3.62086s elapsed
│ │ │  
│ │ │  o4 = true
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,19 +29,19 @@
│ │ │ │  o1 = {6, 8, 9, 11}
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  i2 : genus L
│ │ │ │  
│ │ │ │  o2 = 8
│ │ │ │  i3 : elapsedTime isSmoothableSemigroup(L,0.30,0)
│ │ │ │ - -- 1.10208s elapsed
│ │ │ │ + -- .822381s elapsed
│ │ │ │  
│ │ │ │  o3 = false
│ │ │ │  i4 : elapsedTime isSmoothableSemigroup(L,0.14,0)
│ │ │ │ - -- 3.98405s elapsed
│ │ │ │ + -- 3.62086s 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
│ │ │ @@ -94,15 +94,15 @@
│ │ │  
│ │ │  o2 = 8
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : elapsedTime isWeierstrassSemigroup(L,0.15)
│ │ │ - -- 4.09713s elapsed
│ │ │ + -- 3.3422s elapsed
│ │ │  
│ │ │  o3 = true
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │  o1 = {6, 8, 9, 11}
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  i2 : genus L
│ │ │ │  
│ │ │ │  o2 = 8
│ │ │ │  i3 : elapsedTime isWeierstrassSemigroup(L,0.15)
│ │ │ │ - -- 4.09713s elapsed
│ │ │ │ + -- 3.3422s 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
│ │ │ @@ -79,15 +79,15 @@
│ │ │            
    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.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -101,62 +101,62 @@
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : elapsedTime nonWeierstrassSemigroups(6,7)
│ │ │  (6, 7,  all semigroups are smoothable)
│ │ │ - -- 1.62513s elapsed
│ │ │ + -- 1.14528s elapsed
│ │ │  
│ │ │  o1 = {}
│ │ │  
│ │ │  o1 : List
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime nonWeierstrassSemigroups(6,8,LLdifficult,Verbose=>true)
│ │ │  (17, 5)
│ │ │  {6, 7, 8, 17}
│ │ │  unfolding
│ │ │ - -- .396426s elapsed
│ │ │ + -- .379482s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .149345s elapsed
│ │ │ + -- .141966s elapsed
│ │ │  next gb
│ │ │ - -- .00186426s elapsed
│ │ │ + -- .00195322s elapsed
│ │ │  true
│ │ │ - -- .934436s elapsed
│ │ │ + -- .856537s elapsed
│ │ │  {6, 7, 9, 17}
│ │ │  unfolding
│ │ │ - -- .34282s elapsed
│ │ │ + -- .337101s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .263338s elapsed
│ │ │ + -- .177858s elapsed
│ │ │  next gb
│ │ │ - -- .00267103s elapsed
│ │ │ + -- .00276712s elapsed
│ │ │  decompose
│ │ │ - -- .122971s elapsed
│ │ │ + -- .135951s elapsed
│ │ │  number of components: 2
│ │ │  support c, codim c: {(2, 2), (5, 2)}
│ │ │  {0, -1}
│ │ │ - -- 2.82135s elapsed
│ │ │ + -- 2.57508s elapsed
│ │ │  {6, 8, 9, 10}
│ │ │  unfolding
│ │ │ - -- .173434s elapsed
│ │ │ + -- .118005s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .133545s elapsed
│ │ │ + -- .103052s elapsed
│ │ │  next gb
│ │ │ - -- .000440471s elapsed
│ │ │ + -- .000505392s elapsed
│ │ │  true
│ │ │ - -- .746963s elapsed
│ │ │ + -- .617495s elapsed
│ │ │  {6, 8, 10, 11, 13}
│ │ │  unfolding
│ │ │ - -- .532889s elapsed
│ │ │ + -- .518928s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .194284s elapsed
│ │ │ + -- .208067s elapsed
│ │ │  next gb
│ │ │ - -- .00451218s elapsed
│ │ │ + -- .00522249s elapsed
│ │ │  decompose
│ │ │ - -- 1.02026s elapsed
│ │ │ + -- .751344s elapsed
│ │ │  number of components: 1
│ │ │  support c, codim c: {(5, 1)}
│ │ │  {-1}
│ │ │ - -- 2.95467s elapsed
│ │ │ - -- 7.45755s elapsed
│ │ │ + -- 2.46593s elapsed
│ │ │ + -- 6.51519s elapsed
│ │ │  0
│ │ │  
│ │ │  {}
│ │ │ - -- .000003637s elapsed
│ │ │ - -- 7.49256s elapsed
│ │ │ + -- .000004612s elapsed
│ │ │ + -- 6.55119s elapsed
│ │ │  
│ │ │  o3 = {{6, 8, 9, 11}}
│ │ │  
│ │ │  o3 : List
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -22,76 +22,76 @@
│ │ │ │  ********** 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)
│ │ │ │ - -- 1.62513s elapsed
│ │ │ │ + -- 1.14528s 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
│ │ │ │ - -- .396426s elapsed
│ │ │ │ + -- .379482s elapsed
│ │ │ │  flatteningRelations
│ │ │ │ - -- .149345s elapsed
│ │ │ │ + -- .141966s elapsed
│ │ │ │  next gb
│ │ │ │ - -- .00186426s elapsed
│ │ │ │ + -- .00195322s elapsed
│ │ │ │  true
│ │ │ │ - -- .934436s elapsed
│ │ │ │ + -- .856537s elapsed
│ │ │ │  {6, 7, 9, 17}
│ │ │ │  unfolding
│ │ │ │ - -- .34282s elapsed
│ │ │ │ + -- .337101s elapsed
│ │ │ │  flatteningRelations
│ │ │ │ - -- .263338s elapsed
│ │ │ │ + -- .177858s elapsed
│ │ │ │  next gb
│ │ │ │ - -- .00267103s elapsed
│ │ │ │ + -- .00276712s elapsed
│ │ │ │  decompose
│ │ │ │ - -- .122971s elapsed
│ │ │ │ + -- .135951s elapsed
│ │ │ │  number of components: 2
│ │ │ │  support c, codim c: {(2, 2), (5, 2)}
│ │ │ │  {0, -1}
│ │ │ │ - -- 2.82135s elapsed
│ │ │ │ + -- 2.57508s elapsed
│ │ │ │  {6, 8, 9, 10}
│ │ │ │  unfolding
│ │ │ │ - -- .173434s elapsed
│ │ │ │ + -- .118005s elapsed
│ │ │ │  flatteningRelations
│ │ │ │ - -- .133545s elapsed
│ │ │ │ + -- .103052s elapsed
│ │ │ │  next gb
│ │ │ │ - -- .000440471s elapsed
│ │ │ │ + -- .000505392s elapsed
│ │ │ │  true
│ │ │ │ - -- .746963s elapsed
│ │ │ │ + -- .617495s elapsed
│ │ │ │  {6, 8, 10, 11, 13}
│ │ │ │  unfolding
│ │ │ │ - -- .532889s elapsed
│ │ │ │ + -- .518928s elapsed
│ │ │ │  flatteningRelations
│ │ │ │ - -- .194284s elapsed
│ │ │ │ + -- .208067s elapsed
│ │ │ │  next gb
│ │ │ │ - -- .00451218s elapsed
│ │ │ │ + -- .00522249s elapsed
│ │ │ │  decompose
│ │ │ │ - -- 1.02026s elapsed
│ │ │ │ + -- .751344s elapsed
│ │ │ │  number of components: 1
│ │ │ │  support c, codim c: {(5, 1)}
│ │ │ │  {-1}
│ │ │ │ - -- 2.95467s elapsed
│ │ │ │ - -- 7.45755s elapsed
│ │ │ │ + -- 2.46593s elapsed
│ │ │ │ + -- 6.51519s elapsed
│ │ │ │  0
│ │ │ │  
│ │ │ │  {}
│ │ │ │ - -- .000003637s elapsed
│ │ │ │ - -- 7.49256s elapsed
│ │ │ │ + -- .000004612s elapsed
│ │ │ │ + -- 6.55119s 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
│ │ │ │  multiplicity 6 and genus 8 of which only 5 is not flagged as smoothable by the
│ │ ├── ./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 0.0808093s (cpu); 0.080808s (thread); 0s (gc)
│ │ │ + -- used 0.0989345s (cpu); 0.0989332s (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 0.177563s (cpu); 0.116485s (thread); 0s (gc)
│ │ │ + -- used 0.241946s (cpu); 0.121097s (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 0.469171s (cpu); 0.295248s (thread); 0s (gc)
│ │ │ + -- used 0.538017s (cpu); 0.32744s (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 0.217854s (cpu); 0.121538s (thread); 0s (gc)
│ │ │ + -- used 0.268385s (cpu); 0.134118s (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__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 0.0817194s (cpu); 0.0817214s (thread); 0s (gc)
│ │ │ + -- used 0.108686s (cpu); 0.108687s (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 0.307575s (cpu); 0.249063s (thread); 0s (gc)
│ │ │ + -- used 0.375192s (cpu); 0.285173s (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 0.0245101s (cpu); 0.0245105s (thread); 0s (gc)
│ │ │ + -- used 0.0306742s (cpu); 0.0306751s (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 0.0271838s (cpu); 0.0271842s (thread); 0s (gc)
│ │ │ + -- used 0.03187s (cpu); 0.0318687s (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
│ │ ├── ./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 0.22637s (cpu); 0.115683s (thread); 0s (gc)
│ │ │ + -- used 0.257029s (cpu); 0.133878s (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 0.0274721s (cpu); 0.0274727s (thread); 0s (gc)
│ │ │ + -- used 0.0314835s (cpu); 0.0314824s (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 0.470899s (cpu); 0.28403s (thread); 0s (gc)
│ │ │ + -- used 0.549502s (cpu); 0.313293s (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 0.143893s (cpu); 0.112508s (thread); 0s (gc)
│ │ │ + -- used 0.21209s (cpu); 0.140248s (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
│ │ │ @@ -74,15 +74,15 @@
│ │ │              
│ │ │                
    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 0.0808093s (cpu); 0.080808s (thread); 0s (gc)
│ │ │ + -- used 0.0989345s (cpu); 0.0989332s (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 0.0808093s (cpu); 0.080808s (thread); 0s (gc)
│ │ │ │ + -- used 0.0989345s (cpu); 0.0989332s (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
│ │ │ @@ -92,15 +92,15 @@
│ │ │              
│ │ │                
    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 0.177563s (cpu); 0.116485s (thread); 0s (gc)
    
│ │ │ + -- used 0.241946s (cpu); 0.121097s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : C_0
│ │ │  
│ │ │  o6 = Basis symbol: e0
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,15 +16,15 @@
│ │ │ │  ********** 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 0.177563s (cpu); 0.116485s (thread); 0s (gc)
│ │ │ │ + -- used 0.241946s (cpu); 0.121097s (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
│ │ │ @@ -74,15 +74,15 @@
│ │ │              
│ │ │                
    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 0.469171s (cpu); 0.295248s (thread); 0s (gc)
│ │ │ + -- used 0.538017s (cpu); 0.32744s (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 0.469171s (cpu); 0.295248s (thread); 0s (gc)
│ │ │ │ + -- used 0.538017s (cpu); 0.32744s (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
│ │ │ @@ -90,15 +90,15 @@
│ │ │              
│ │ │                
    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 0.217854s (cpu); 0.121538s (thread); 0s (gc)
    
│ │ │ + -- used 0.268385s (cpu); 0.134118s (thread); 0s (gc)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : describeFull C
│ │ │  
│ │ │  o6 = 0: Module: Basis symbol: e0
│ │ │ ├── html2text {}
│ │ │ │ @@ -14,15 +14,15 @@
│ │ │ │  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 0.217854s (cpu); 0.121538s (thread); 0s (gc)
│ │ │ │ + -- used 0.268385s (cpu); 0.134118s (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__O__I__Resolution_rp.html
│ │ │ @@ -91,15 +91,15 @@
│ │ │              
│ │ │                
    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 0.0817194s (cpu); 0.0817214s (thread); 0s (gc)
    
│ │ │ + -- used 0.108686s (cpu); 0.108687s (thread); 0s (gc)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : describe C
│ │ │  
│ │ │  o6 = 0: Module: Basis symbol: e0
│ │ │ ├── html2text {}
│ │ │ │ @@ -14,15 +14,15 @@
│ │ │ │  ********** 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 0.0817194s (cpu); 0.0817214s (thread); 0s (gc)
│ │ │ │ + -- used 0.108686s (cpu); 0.108687s (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
│ │ │ @@ -94,15 +94,15 @@
│ │ │              
│ │ │                
    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 0.307575s (cpu); 0.249063s (thread); 0s (gc)
│ │ │ + -- used 0.375192s (cpu); 0.285173s (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 {}
│ │ │ │ @@ -17,15 +17,15 @@
│ │ │ │  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 0.307575s (cpu); 0.249063s (thread); 0s (gc)
│ │ │ │ + -- used 0.375192s (cpu); 0.285173s (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
│ │ │ @@ -116,15 +116,15 @@
│ │ │  
│ │ │  o10 = false
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i11 : time B = oiGB {b1, b2}
│ │ │ - -- used 0.0245101s (cpu); 0.0245105s (thread); 0s (gc)
│ │ │ + -- used 0.0306742s (cpu); 0.0306751s (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 {}
│ │ │ │ @@ -24,15 +24,15 @@
│ │ │ │  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 0.0245101s (cpu); 0.0245105s (thread); 0s (gc)
│ │ │ │ + -- used 0.0306742s (cpu); 0.0306751s (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
│ │ │ @@ -109,15 +109,15 @@
│ │ │              
│ │ │                
    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 : time B = oiGB {b1, b2}
│ │ │ - -- used 0.0271838s (cpu); 0.0271842s (thread); 0s (gc)
│ │ │ + -- used 0.03187s (cpu); 0.0318687s (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
│ │ │ ├── html2text {}
│ │ │ │ @@ -21,15 +21,15 @@
│ │ │ │  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 0.0271838s (cpu); 0.0271842s (thread); 0s (gc)
│ │ │ │ + -- used 0.03187s (cpu); 0.0318687s (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
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_net_lp__O__I__Resolution_rp.html
│ │ │ @@ -91,15 +91,15 @@
│ │ │              
│ │ │                
    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 0.22637s (cpu); 0.115683s (thread); 0s (gc)
    
│ │ │ + -- used 0.257029s (cpu); 0.133878s (thread); 0s (gc)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : net C
│ │ │  
│ │ │  o6 = 0: (e0, {2}, {-2})
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,15 +15,15 @@
│ │ │ │  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 0.22637s (cpu); 0.115683s (thread); 0s (gc)
│ │ │ │ + -- used 0.257029s (cpu); 0.133878s (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
│ │ │ @@ -111,15 +111,15 @@
│ │ │              
│ │ │                
    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 0.0274721s (cpu); 0.0274727s (thread); 0s (gc)
│ │ │ + -- used 0.0314835s (cpu); 0.0314824s (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 {}
│ │ │ │ @@ -28,15 +28,15 @@
│ │ │ │  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 0.0274721s (cpu); 0.0274727s (thread); 0s (gc)
│ │ │ │ + -- used 0.0314835s (cpu); 0.0314824s (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
│ │ │ @@ -106,15 +106,15 @@
│ │ │              
│ │ │                
    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 0.470899s (cpu); 0.28403s (thread); 0s (gc)
│ │ │ + -- used 0.549502s (cpu); 0.313293s (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 {}
│ │ │ │ @@ -33,15 +33,15 @@
│ │ │ │  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 0.470899s (cpu); 0.28403s (thread); 0s (gc)
│ │ │ │ + -- used 0.549502s (cpu); 0.313293s (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
│ │ │ @@ -104,15 +104,15 @@
│ │ │              
│ │ │                
    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 0.143893s (cpu); 0.112508s (thread); 0s (gc)
│ │ │ + -- used 0.21209s (cpu); 0.140248s (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 {}
│ │ │ │ @@ -20,15 +20,15 @@
│ │ │ │  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 0.143893s (cpu); 0.112508s (thread); 0s (gc)
│ │ │ │ + -- used 0.21209s (cpu); 0.140248s (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/OldChainComplexes/example-output/___Fast__Nonminimal.out
│ │ │ @@ -9,25 +9,25 @@
│ │ │  i2 : S = ring I
│ │ │  
│ │ │  o2 = S
│ │ │  
│ │ │  o2 : PolynomialRing
│ │ │  
│ │ │  i3 : elapsedTime C = res(I, FastNonminimal => true)
│ │ │ - -- 2.18598s elapsed
│ │ │ + -- 2.56566s 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_*)
│ │ │ - -- 1.44714s elapsed
│ │ │ + -- 1.3738s 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/OldChainComplexes/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)
│ │ │ - -- 2.33951s elapsed
│ │ │ + -- 2.56804s 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/OldChainComplexes/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 1.04017s (cpu); 0.888714s (thread); 0s (gc)
│ │ │ - -- used 0.565405s (cpu); 0.491861s (thread); 0s (gc)
│ │ │ + -- used 1.11923s (cpu); 0.982238s (thread); 0s (gc)
│ │ │ + -- used 0.777515s (cpu); 0.695772s (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/OldChainComplexes/html/___Fast__Nonminimal.html
│ │ │ @@ -89,28 +89,28 @@
│ │ │  
│ │ │  o2 : PolynomialRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : elapsedTime C = res(I, FastNonminimal => true)
│ │ │ - -- 2.18598s elapsed
│ │ │ + -- 2.56566s 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_*)
│ │ │ - -- 1.44714s elapsed
│ │ │ + -- 1.3738s 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 {}
│ │ │ │ @@ -29,28 +29,28 @@
│ │ │ │  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)
│ │ │ │ - -- 2.18598s elapsed
│ │ │ │ + -- 2.56566s 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_*)
│ │ │ │ - -- 1.44714s elapsed
│ │ │ │ + -- 1.3738s 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/OldChainComplexes/html/_betti_lp..._cm__Minimize_eq_gt..._rp.html
│ │ │ @@ -88,15 +88,15 @@
│ │ │  
│ │ │  o2 : PolynomialRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : elapsedTime C = res(I, FastNonminimal => true)
│ │ │ - -- 2.33951s elapsed
│ │ │ + -- 2.56804s 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 {}
│ │ │ │ @@ -26,15 +26,15 @@
│ │ │ │  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)
│ │ │ │ - -- 2.33951s elapsed
│ │ │ │ + -- 2.56804s 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/OldChainComplexes/html/_computing_spresolutions.html
│ │ │ @@ -112,16 +112,16 @@
│ │ │            << res M << endl << endl;
│ │ │            break;
│ │ │            ) else (
│ │ │            << "-- computation interrupted" << endl;
│ │ │            status M.cache.resolution;
│ │ │            << "-- continuing the computation" << endl;
│ │ │            ))
│ │ │ - -- used 1.04017s (cpu); 0.888714s (thread); 0s (gc)
│ │ │ - -- used 0.565405s (cpu); 0.491861s (thread); 0s (gc)
│ │ │ + -- used 1.11923s (cpu); 0.982238s (thread); 0s (gc)
│ │ │ + -- used 0.777515s (cpu); 0.695772s (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 1.04017s (cpu); 0.888714s (thread); 0s (gc)
│ │ │ │ - -- used 0.565405s (cpu); 0.491861s (thread); 0s (gc)
│ │ │ │ + -- used 1.11923s (cpu); 0.982238s (thread); 0s (gc)
│ │ │ │ + -- used 0.777515s (cpu); 0.695772s (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/Oscillators/example-output/___Checking_spthe_spcodimension_spand_spirreducible_spdecomposition_spof_spthe_sp__I__G_spideal.out
│ │ │ @@ -182,25 +182,25 @@
│ │ │  o15 = 4
│ │ │  
│ │ │  i16 : for G in Gs list (
│ │ │            IG = oscQuadrics(G, R);
│ │ │            elapsedTime comps := decompose IG;
│ │ │            {comps/codim, comps/degree}
│ │ │            );
│ │ │ - -- .297177s elapsed
│ │ │ - -- .37676s elapsed
│ │ │ - -- .596031s elapsed
│ │ │ - -- .268268s elapsed
│ │ │ - -- .292728s elapsed
│ │ │ - -- .331552s elapsed
│ │ │ - -- .579031s elapsed
│ │ │ - -- .465787s elapsed
│ │ │ - -- .494728s elapsed
│ │ │ - -- .310933s elapsed
│ │ │ - -- .17989s elapsed
│ │ │ + -- .260087s elapsed
│ │ │ + -- .293127s elapsed
│ │ │ + -- .486587s elapsed
│ │ │ + -- .258292s elapsed
│ │ │ + -- .266921s elapsed
│ │ │ + -- .34115s elapsed
│ │ │ + -- .563796s elapsed
│ │ │ + -- .517735s elapsed
│ │ │ + -- .489887s elapsed
│ │ │ + -- .290566s elapsed
│ │ │ + -- .188196s elapsed
│ │ │  
│ │ │  i17 : netList oo
│ │ │  
│ │ │        +---------------+---------------+
│ │ │  o17 = |{3, 4, 4}      |{2, 3, 5}      |
│ │ │        +---------------+---------------+
│ │ │        |{3, 4, 4}      |{2, 3, 5}      |
│ │ │ @@ -242,75 +242,75 @@
│ │ │  o22 = 15
│ │ │  
│ │ │  i23 : allcomps = for G in Gs list (
│ │ │            IG = oscQuadrics(G, R);
│ │ │            elapsedTime comps := decompose IG;
│ │ │            {comps/codim, comps/degree}
│ │ │            );
│ │ │ - -- .476223s elapsed
│ │ │ - -- .491179s elapsed
│ │ │ - -- .899896s elapsed
│ │ │ - -- 1.24686s elapsed
│ │ │ - -- .681713s elapsed
│ │ │ - -- .854075s elapsed
│ │ │ - -- .972388s elapsed
│ │ │ - -- 1.05083s elapsed
│ │ │ - -- .827284s elapsed
│ │ │ - -- .660979s elapsed
│ │ │ - -- .342135s elapsed
│ │ │ - -- .455566s elapsed
│ │ │ - -- .514713s elapsed
│ │ │ - -- .835284s elapsed
│ │ │ - -- 1.08006s elapsed
│ │ │ - -- 1.31144s elapsed
│ │ │ - -- 1.02818s elapsed
│ │ │ - -- .894288s elapsed
│ │ │ - -- 1.29053s elapsed
│ │ │ - -- 1.03534s elapsed
│ │ │ - -- .731355s elapsed
│ │ │ - -- .912171s elapsed
│ │ │ - -- 1.40395s elapsed
│ │ │ - -- 1.25725s elapsed
│ │ │ - -- .532629s elapsed
│ │ │ - -- .656184s elapsed
│ │ │ - -- 1.26939s elapsed
│ │ │ - -- .684363s elapsed
│ │ │ - -- .643036s elapsed
│ │ │ - -- .897991s elapsed
│ │ │ - -- 1.03754s elapsed
│ │ │ - -- .918405s elapsed
│ │ │ - -- .635213s elapsed
│ │ │ - -- 1.02454s elapsed
│ │ │ - -- .803326s elapsed
│ │ │ - -- 1.1079s elapsed
│ │ │ - -- 1.0033s elapsed
│ │ │ - -- 1.07175s elapsed
│ │ │ - -- 1.36626s elapsed
│ │ │ - -- .806481s elapsed
│ │ │ - -- .632335s elapsed
│ │ │ - -- 1.04315s elapsed
│ │ │ - -- 1.52989s elapsed
│ │ │ - -- 1.87148s elapsed
│ │ │ - -- 1.37285s elapsed
│ │ │ - -- 1.02596s elapsed
│ │ │ - -- 1.43312s elapsed
│ │ │ - -- 1.20975s elapsed
│ │ │ - -- 1.03285s elapsed
│ │ │ - -- 1.0795s elapsed
│ │ │ - -- 1.13227s elapsed
│ │ │ - -- .852037s elapsed
│ │ │ - -- .700127s elapsed
│ │ │ - -- .943143s elapsed
│ │ │ - -- .960376s elapsed
│ │ │ - -- 1.59259s elapsed
│ │ │ - -- 1.31491s elapsed
│ │ │ - -- 1.24035s elapsed
│ │ │ - -- .704216s elapsed
│ │ │ - -- .519728s elapsed
│ │ │ - -- .42436s elapsed
│ │ │ + -- .449995s elapsed
│ │ │ + -- .439429s elapsed
│ │ │ + -- .867096s elapsed
│ │ │ + -- 1.10789s elapsed
│ │ │ + -- .628726s elapsed
│ │ │ + -- .848615s elapsed
│ │ │ + -- .927509s elapsed
│ │ │ + -- .985926s elapsed
│ │ │ + -- .724786s elapsed
│ │ │ + -- .668831s elapsed
│ │ │ + -- .378473s elapsed
│ │ │ + -- .433955s elapsed
│ │ │ + -- .453068s elapsed
│ │ │ + -- .609691s elapsed
│ │ │ + -- .826274s elapsed
│ │ │ + -- 1.19237s elapsed
│ │ │ + -- .930002s elapsed
│ │ │ + -- .843757s elapsed
│ │ │ + -- 1.13241s elapsed
│ │ │ + -- .99653s elapsed
│ │ │ + -- .765161s elapsed
│ │ │ + -- .894716s elapsed
│ │ │ + -- 1.31167s elapsed
│ │ │ + -- 1.13838s elapsed
│ │ │ + -- .511821s elapsed
│ │ │ + -- .650433s elapsed
│ │ │ + -- 1.2942s elapsed
│ │ │ + -- .673693s elapsed
│ │ │ + -- .576774s elapsed
│ │ │ + -- .755925s elapsed
│ │ │ + -- 1.04848s elapsed
│ │ │ + -- .759095s elapsed
│ │ │ + -- .514935s elapsed
│ │ │ + -- .972763s elapsed
│ │ │ + -- .84237s elapsed
│ │ │ + -- 1.28561s elapsed
│ │ │ + -- .953937s elapsed
│ │ │ + -- 1.09201s elapsed
│ │ │ + -- 1.20741s elapsed
│ │ │ + -- .797653s elapsed
│ │ │ + -- .634924s elapsed
│ │ │ + -- 1.19929s elapsed
│ │ │ + -- 1.48117s elapsed
│ │ │ + -- 1.7506s elapsed
│ │ │ + -- 1.10955s elapsed
│ │ │ + -- 1.08286s elapsed
│ │ │ + -- 1.36015s elapsed
│ │ │ + -- 1.20491s elapsed
│ │ │ + -- .958492s elapsed
│ │ │ + -- 1.02931s elapsed
│ │ │ + -- 1.01476s elapsed
│ │ │ + -- .77346s elapsed
│ │ │ + -- .786829s elapsed
│ │ │ + -- 1.01282s elapsed
│ │ │ + -- .608904s elapsed
│ │ │ + -- 1.08759s elapsed
│ │ │ + -- 1.3022s elapsed
│ │ │ + -- 1.24452s elapsed
│ │ │ + -- .695067s elapsed
│ │ │ + -- .440503s elapsed
│ │ │ + -- .326071s elapsed
│ │ │  
│ │ │  i24 : netList ({{"codimensions", "degrees"}} | allcomps)
│ │ │  
│ │ │        +------------------------+------------------------+
│ │ │  o24 = |codimensions            |degrees                 |
│ │ │        +------------------------+------------------------+
│ │ │        |{3, 5, 5}               |{2, 4, 6}               |
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/example-output/___Example_sp4.2_co_spa_sp__K5_spand_sppentagon_spglued_spalong_span_spedge.out
│ │ │ @@ -39,15 +39,15 @@
│ │ │       .98, .98, .101, -.98, -.298, .393, .201, .201, .201, -.995, -.201,
│ │ │       ------------------------------------------------------------------------
│ │ │       .954}}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : elapsedTime stablesolsPent = showExoticSolutions Pent
│ │ │ - -- .778s elapsed
│ │ │ + -- .992s elapsed
│ │ │  -- found extra exotic solutions for graph Graph{0 => {1, 4}} --
│ │ │                                                  1 => {0, 2}
│ │ │                                                  2 => {1, 3}
│ │ │                                                  3 => {2, 4}
│ │ │                                                  4 => {0, 3}
│ │ │  +----+-----+-----+----+-----+-----+-----+-----+
│ │ │  |.309|-.809|-.809|.309|.951 |.588 |-.588|-.951|
│ │ │ @@ -60,15 +60,15 @@
│ │ │  +---+---+---+---+
│ │ │  |72 |144|216|288|
│ │ │  +---+---+---+---+
│ │ │  |288|216|144|72 |
│ │ │  +---+---+---+---+
│ │ │  |0  |0  |0  |0  |
│ │ │  +---+---+---+---+
│ │ │ - -- .84s elapsed
│ │ │ + -- 1.04s elapsed
│ │ │  
│ │ │  o6 = {{.309, -.809, -.809, .309, .951, .588, -.588, -.951}, {.309, -.809,
│ │ │       ------------------------------------------------------------------------
│ │ │       -.809, .309, -.951, -.588, .588, .951}, {1, 1, 1, 1, 0, 0, 0, 0}}
│ │ │  
│ │ │  o6 : List
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/example-output/___S__C__T_spgraphs_spwith_spexotic_spsolutions.out
│ │ │ @@ -44,19 +44,19 @@
│ │ │  
│ │ │  i5 : printingPrecision = 3
│ │ │  
│ │ │  o5 = 3
│ │ │  
│ │ │  i6 : for G in Gs list showExoticSolutions G;
│ │ │  warning: some solutions are not regular: {37, 38, 40, 41, 42, 43, 44, 45, 46, 48, 50, 52, 53, 54, 55, 59, 60, 64, 65, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 81, 83, 85, 86, 87, 88, 89}
│ │ │ - -- .707s elapsed
│ │ │ + -- .816s elapsed
│ │ │  warning: some solutions are not regular: {43, 44, 47, 49, 50, 51, 52, 53, 55, 57, 59, 61, 62, 63, 64, 65, 66, 68, 72, 74, 76, 77, 78, 79, 80, 84, 85, 87, 88, 89, 90, 91, 97}
│ │ │ - -- .738s elapsed
│ │ │ - -- .909s elapsed
│ │ │ - -- .956s elapsed
│ │ │ + -- .732s elapsed
│ │ │ + -- .983s elapsed
│ │ │ + -- 1.3s elapsed
│ │ │  -- found extra exotic solutions for graph Graph{0 => {2, 3}} --
│ │ │                                                  1 => {3, 4}
│ │ │                                                  2 => {0, 4}
│ │ │                                                  3 => {0, 1}
│ │ │                                                  4 => {2, 1}
│ │ │  +-----+----+----+-----+-----+-----+-----+-----+
│ │ │  |-.809|.309|.309|-.809|.588 |-.951|.951 |-.588|
│ │ │ @@ -69,20 +69,20 @@
│ │ │  +---+---+---+---+
│ │ │  |144|288|72 |216|
│ │ │  +---+---+---+---+
│ │ │  |0  |0  |0  |0  |
│ │ │  +---+---+---+---+
│ │ │  |216|72 |288|144|
│ │ │  +---+---+---+---+
│ │ │ - -- 1.19s elapsed
│ │ │ - -- 1.4s elapsed
│ │ │ + -- 1.35s elapsed
│ │ │ + -- 1.49s elapsed
│ │ │  warning: some solutions are not regular: {27, 31, 32, 34, 37, 38, 44, 46, 47, 53, 54, 56, 57, 59, 60}
│ │ │ - -- 1.6s elapsed
│ │ │ + -- 1.82s elapsed
│ │ │  warning: some solutions are not regular: {16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 34}
│ │ │ - -- 1.46s elapsed
│ │ │ - -- 1.4s elapsed
│ │ │ + -- 1.51s elapsed
│ │ │ + -- 1.52s elapsed
│ │ │  warning: some solutions are not regular: {26, 29, 30, 32, 33}
│ │ │ - -- 1.56s elapsed
│ │ │ + -- 1.72s elapsed
│ │ │  warning: some solutions are not regular: {38, 40, 42, 53, 54, 55, 62, 63, 67, 72, 77, 78}
│ │ │ - -- 1.33s elapsed
│ │ │ + -- 1.52s elapsed
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/example-output/_get__Linearly__Stable__Solutions.out
│ │ │ @@ -1,15 +1,15 @@
│ │ │  -- -*- M2-comint -*- hash: 1729328129346969841
│ │ │  
│ │ │  i1 : G = graph({0,1,2,3}, {{0,1},{1,2},{2,3},{0,3}});
│ │ │  
│ │ │  i2 : getLinearlyStableSolutions(G)
│ │ │  -- warning: experimental computation over inexact field begun
│ │ │  --          results not reliable (one warning given per session)
│ │ │ - -- .377588s elapsed
│ │ │ + -- .248892s elapsed
│ │ │  warning: some solutions are not regular: {4, 5, 7, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21}
│ │ │  
│ │ │  o2 = {{1, 1, 1, 0, 0, 0}}
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/example-output/_show__Exotic__Solutions.out
│ │ │ @@ -9,15 +9,15 @@
│ │ │             4 => {0, 3}
│ │ │  
│ │ │  o1 : Graph
│ │ │  
│ │ │  i2 : showExoticSolutions G
│ │ │  -- warning: experimental computation over inexact field begun
│ │ │  --          results not reliable (one warning given per session)
│ │ │ - -- .916628s elapsed
│ │ │ + -- 1.04782s elapsed
│ │ │  -- found extra exotic solutions for graph Graph{0 => {1, 4}} --
│ │ │                                                  1 => {0, 2}
│ │ │                                                  2 => {1, 3}
│ │ │                                                  3 => {2, 4}
│ │ │                                                  4 => {0, 3}
│ │ │  +-------+--------+--------+-------+--------+--------+--------+--------+
│ │ │  |.309017|-.809017|-.809017|.309017|.951057 |.587785 |-.587785|-.951057|
│ │ │ @@ -50,14 +50,14 @@
│ │ │             2 => {1, 3, 4}
│ │ │             3 => {2, 4}
│ │ │             4 => {0, 2, 3}
│ │ │  
│ │ │  o3 : Graph
│ │ │  
│ │ │  i4 : showExoticSolutions G
│ │ │ - -- 1.20627s elapsed
│ │ │ + -- 1.32658s elapsed
│ │ │  
│ │ │  o4 = {{1, 1, 1, 1, 0, 0, 0, 0}}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/html/___Checking_spthe_spcodimension_spand_spirreducible_spdecomposition_spof_spthe_sp__I__G_spideal.html
│ │ │ @@ -295,25 +295,25 @@
│ │ │            
│ │ │              
│ │ │                
    i16 : for G in Gs list (
│ │ │            IG = oscQuadrics(G, R);
│ │ │            elapsedTime comps := decompose IG;
│ │ │            {comps/codim, comps/degree}
│ │ │            );
│ │ │ - -- .297177s elapsed
│ │ │ - -- .37676s elapsed
│ │ │ - -- .596031s elapsed
│ │ │ - -- .268268s elapsed
│ │ │ - -- .292728s elapsed
│ │ │ - -- .331552s elapsed
│ │ │ - -- .579031s elapsed
│ │ │ - -- .465787s elapsed
│ │ │ - -- .494728s elapsed
│ │ │ - -- .310933s elapsed
│ │ │ - -- .17989s elapsed
    
│ │ │ + -- .260087s elapsed
│ │ │ + -- .293127s elapsed
│ │ │ + -- .486587s elapsed
│ │ │ + -- .258292s elapsed
│ │ │ + -- .266921s elapsed
│ │ │ + -- .34115s elapsed
│ │ │ + -- .563796s elapsed
│ │ │ + -- .517735s elapsed
│ │ │ + -- .489887s elapsed
│ │ │ + -- .290566s elapsed
│ │ │ + -- .188196s elapsed
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i17 : netList oo
│ │ │  
│ │ │        +---------------+---------------+
│ │ │ @@ -380,75 +380,75 @@
│ │ │            
│ │ │              
│ │ │                
    i23 : allcomps = for G in Gs list (
│ │ │            IG = oscQuadrics(G, R);
│ │ │            elapsedTime comps := decompose IG;
│ │ │            {comps/codim, comps/degree}
│ │ │            );
│ │ │ - -- .476223s elapsed
│ │ │ - -- .491179s elapsed
│ │ │ - -- .899896s elapsed
│ │ │ - -- 1.24686s elapsed
│ │ │ - -- .681713s elapsed
│ │ │ - -- .854075s elapsed
│ │ │ - -- .972388s elapsed
│ │ │ - -- 1.05083s elapsed
│ │ │ - -- .827284s elapsed
│ │ │ - -- .660979s elapsed
│ │ │ - -- .342135s elapsed
│ │ │ - -- .455566s elapsed
│ │ │ - -- .514713s elapsed
│ │ │ - -- .835284s elapsed
│ │ │ - -- 1.08006s elapsed
│ │ │ - -- 1.31144s elapsed
│ │ │ - -- 1.02818s elapsed
│ │ │ - -- .894288s elapsed
│ │ │ - -- 1.29053s elapsed
│ │ │ - -- 1.03534s elapsed
│ │ │ - -- .731355s elapsed
│ │ │ - -- .912171s elapsed
│ │ │ - -- 1.40395s elapsed
│ │ │ - -- 1.25725s elapsed
│ │ │ - -- .532629s elapsed
│ │ │ - -- .656184s elapsed
│ │ │ - -- 1.26939s elapsed
│ │ │ - -- .684363s elapsed
│ │ │ - -- .643036s elapsed
│ │ │ - -- .897991s elapsed
│ │ │ - -- 1.03754s elapsed
│ │ │ - -- .918405s elapsed
│ │ │ - -- .635213s elapsed
│ │ │ - -- 1.02454s elapsed
│ │ │ - -- .803326s elapsed
│ │ │ - -- 1.1079s elapsed
│ │ │ - -- 1.0033s elapsed
│ │ │ - -- 1.07175s elapsed
│ │ │ - -- 1.36626s elapsed
│ │ │ - -- .806481s elapsed
│ │ │ - -- .632335s elapsed
│ │ │ - -- 1.04315s elapsed
│ │ │ - -- 1.52989s elapsed
│ │ │ - -- 1.87148s elapsed
│ │ │ - -- 1.37285s elapsed
│ │ │ - -- 1.02596s elapsed
│ │ │ - -- 1.43312s elapsed
│ │ │ - -- 1.20975s elapsed
│ │ │ - -- 1.03285s elapsed
│ │ │ - -- 1.0795s elapsed
│ │ │ - -- 1.13227s elapsed
│ │ │ - -- .852037s elapsed
│ │ │ - -- .700127s elapsed
│ │ │ - -- .943143s elapsed
│ │ │ - -- .960376s elapsed
│ │ │ - -- 1.59259s elapsed
│ │ │ - -- 1.31491s elapsed
│ │ │ - -- 1.24035s elapsed
│ │ │ - -- .704216s elapsed
│ │ │ - -- .519728s elapsed
│ │ │ - -- .42436s elapsed
    
│ │ │ + -- .449995s elapsed
│ │ │ + -- .439429s elapsed
│ │ │ + -- .867096s elapsed
│ │ │ + -- 1.10789s elapsed
│ │ │ + -- .628726s elapsed
│ │ │ + -- .848615s elapsed
│ │ │ + -- .927509s elapsed
│ │ │ + -- .985926s elapsed
│ │ │ + -- .724786s elapsed
│ │ │ + -- .668831s elapsed
│ │ │ + -- .378473s elapsed
│ │ │ + -- .433955s elapsed
│ │ │ + -- .453068s elapsed
│ │ │ + -- .609691s elapsed
│ │ │ + -- .826274s elapsed
│ │ │ + -- 1.19237s elapsed
│ │ │ + -- .930002s elapsed
│ │ │ + -- .843757s elapsed
│ │ │ + -- 1.13241s elapsed
│ │ │ + -- .99653s elapsed
│ │ │ + -- .765161s elapsed
│ │ │ + -- .894716s elapsed
│ │ │ + -- 1.31167s elapsed
│ │ │ + -- 1.13838s elapsed
│ │ │ + -- .511821s elapsed
│ │ │ + -- .650433s elapsed
│ │ │ + -- 1.2942s elapsed
│ │ │ + -- .673693s elapsed
│ │ │ + -- .576774s elapsed
│ │ │ + -- .755925s elapsed
│ │ │ + -- 1.04848s elapsed
│ │ │ + -- .759095s elapsed
│ │ │ + -- .514935s elapsed
│ │ │ + -- .972763s elapsed
│ │ │ + -- .84237s elapsed
│ │ │ + -- 1.28561s elapsed
│ │ │ + -- .953937s elapsed
│ │ │ + -- 1.09201s elapsed
│ │ │ + -- 1.20741s elapsed
│ │ │ + -- .797653s elapsed
│ │ │ + -- .634924s elapsed
│ │ │ + -- 1.19929s elapsed
│ │ │ + -- 1.48117s elapsed
│ │ │ + -- 1.7506s elapsed
│ │ │ + -- 1.10955s elapsed
│ │ │ + -- 1.08286s elapsed
│ │ │ + -- 1.36015s elapsed
│ │ │ + -- 1.20491s elapsed
│ │ │ + -- .958492s elapsed
│ │ │ + -- 1.02931s elapsed
│ │ │ + -- 1.01476s elapsed
│ │ │ + -- .77346s elapsed
│ │ │ + -- .786829s elapsed
│ │ │ + -- 1.01282s elapsed
│ │ │ + -- .608904s elapsed
│ │ │ + -- 1.08759s elapsed
│ │ │ + -- 1.3022s elapsed
│ │ │ + -- 1.24452s elapsed
│ │ │ + -- .695067s elapsed
│ │ │ + -- .440503s elapsed
│ │ │ + -- .326071s elapsed
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i24 : netList ({{"codimensions", "degrees"}} | allcomps)
│ │ │  
│ │ │        +------------------------+------------------------+
│ │ │ ├── html2text {}
│ │ │ │ @@ -180,25 +180,25 @@
│ │ │ │  
│ │ │ │  o15 = 4
│ │ │ │  i16 : for G in Gs list (
│ │ │ │            IG = oscQuadrics(G, R);
│ │ │ │            elapsedTime comps := decompose IG;
│ │ │ │            {comps/codim, comps/degree}
│ │ │ │            );
│ │ │ │ - -- .297177s elapsed
│ │ │ │ - -- .37676s elapsed
│ │ │ │ - -- .596031s elapsed
│ │ │ │ - -- .268268s elapsed
│ │ │ │ - -- .292728s elapsed
│ │ │ │ - -- .331552s elapsed
│ │ │ │ - -- .579031s elapsed
│ │ │ │ - -- .465787s elapsed
│ │ │ │ - -- .494728s elapsed
│ │ │ │ - -- .310933s elapsed
│ │ │ │ - -- .17989s elapsed
│ │ │ │ + -- .260087s elapsed
│ │ │ │ + -- .293127s elapsed
│ │ │ │ + -- .486587s elapsed
│ │ │ │ + -- .258292s elapsed
│ │ │ │ + -- .266921s elapsed
│ │ │ │ + -- .34115s elapsed
│ │ │ │ + -- .563796s elapsed
│ │ │ │ + -- .517735s elapsed
│ │ │ │ + -- .489887s elapsed
│ │ │ │ + -- .290566s elapsed
│ │ │ │ + -- .188196s elapsed
│ │ │ │  i17 : netList oo
│ │ │ │  
│ │ │ │        +---------------+---------------+
│ │ │ │  o17 = |{3, 4, 4}      |{2, 3, 5}      |
│ │ │ │        +---------------+---------------+
│ │ │ │        |{3, 4, 4}      |{2, 3, 5}      |
│ │ │ │        +---------------+---------------+
│ │ │ │ @@ -233,75 +233,75 @@
│ │ │ │  
│ │ │ │  o22 = 15
│ │ │ │  i23 : allcomps = for G in Gs list (
│ │ │ │            IG = oscQuadrics(G, R);
│ │ │ │            elapsedTime comps := decompose IG;
│ │ │ │            {comps/codim, comps/degree}
│ │ │ │            );
│ │ │ │ - -- .476223s elapsed
│ │ │ │ - -- .491179s elapsed
│ │ │ │ - -- .899896s elapsed
│ │ │ │ - -- 1.24686s elapsed
│ │ │ │ - -- .681713s elapsed
│ │ │ │ - -- .854075s elapsed
│ │ │ │ - -- .972388s elapsed
│ │ │ │ - -- 1.05083s elapsed
│ │ │ │ - -- .827284s elapsed
│ │ │ │ - -- .660979s elapsed
│ │ │ │ - -- .342135s elapsed
│ │ │ │ - -- .455566s elapsed
│ │ │ │ - -- .514713s elapsed
│ │ │ │ - -- .835284s elapsed
│ │ │ │ - -- 1.08006s elapsed
│ │ │ │ - -- 1.31144s elapsed
│ │ │ │ - -- 1.02818s elapsed
│ │ │ │ - -- .894288s elapsed
│ │ │ │ - -- 1.29053s elapsed
│ │ │ │ - -- 1.03534s elapsed
│ │ │ │ - -- .731355s elapsed
│ │ │ │ - -- .912171s elapsed
│ │ │ │ - -- 1.40395s elapsed
│ │ │ │ - -- 1.25725s elapsed
│ │ │ │ - -- .532629s elapsed
│ │ │ │ - -- .656184s elapsed
│ │ │ │ - -- 1.26939s elapsed
│ │ │ │ - -- .684363s elapsed
│ │ │ │ - -- .643036s elapsed
│ │ │ │ - -- .897991s elapsed
│ │ │ │ - -- 1.03754s elapsed
│ │ │ │ - -- .918405s elapsed
│ │ │ │ - -- .635213s elapsed
│ │ │ │ - -- 1.02454s elapsed
│ │ │ │ - -- .803326s elapsed
│ │ │ │ - -- 1.1079s elapsed
│ │ │ │ - -- 1.0033s elapsed
│ │ │ │ - -- 1.07175s elapsed
│ │ │ │ - -- 1.36626s elapsed
│ │ │ │ - -- .806481s elapsed
│ │ │ │ - -- .632335s elapsed
│ │ │ │ - -- 1.04315s elapsed
│ │ │ │ - -- 1.52989s elapsed
│ │ │ │ - -- 1.87148s elapsed
│ │ │ │ - -- 1.37285s elapsed
│ │ │ │ - -- 1.02596s elapsed
│ │ │ │ - -- 1.43312s elapsed
│ │ │ │ - -- 1.20975s elapsed
│ │ │ │ - -- 1.03285s elapsed
│ │ │ │ - -- 1.0795s elapsed
│ │ │ │ - -- 1.13227s elapsed
│ │ │ │ - -- .852037s elapsed
│ │ │ │ - -- .700127s elapsed
│ │ │ │ - -- .943143s elapsed
│ │ │ │ - -- .960376s elapsed
│ │ │ │ - -- 1.59259s elapsed
│ │ │ │ - -- 1.31491s elapsed
│ │ │ │ - -- 1.24035s elapsed
│ │ │ │ - -- .704216s elapsed
│ │ │ │ - -- .519728s elapsed
│ │ │ │ - -- .42436s elapsed
│ │ │ │ + -- .449995s elapsed
│ │ │ │ + -- .439429s elapsed
│ │ │ │ + -- .867096s elapsed
│ │ │ │ + -- 1.10789s elapsed
│ │ │ │ + -- .628726s elapsed
│ │ │ │ + -- .848615s elapsed
│ │ │ │ + -- .927509s elapsed
│ │ │ │ + -- .985926s elapsed
│ │ │ │ + -- .724786s elapsed
│ │ │ │ + -- .668831s elapsed
│ │ │ │ + -- .378473s elapsed
│ │ │ │ + -- .433955s elapsed
│ │ │ │ + -- .453068s elapsed
│ │ │ │ + -- .609691s elapsed
│ │ │ │ + -- .826274s elapsed
│ │ │ │ + -- 1.19237s elapsed
│ │ │ │ + -- .930002s elapsed
│ │ │ │ + -- .843757s elapsed
│ │ │ │ + -- 1.13241s elapsed
│ │ │ │ + -- .99653s elapsed
│ │ │ │ + -- .765161s elapsed
│ │ │ │ + -- .894716s elapsed
│ │ │ │ + -- 1.31167s elapsed
│ │ │ │ + -- 1.13838s elapsed
│ │ │ │ + -- .511821s elapsed
│ │ │ │ + -- .650433s elapsed
│ │ │ │ + -- 1.2942s elapsed
│ │ │ │ + -- .673693s elapsed
│ │ │ │ + -- .576774s elapsed
│ │ │ │ + -- .755925s elapsed
│ │ │ │ + -- 1.04848s elapsed
│ │ │ │ + -- .759095s elapsed
│ │ │ │ + -- .514935s elapsed
│ │ │ │ + -- .972763s elapsed
│ │ │ │ + -- .84237s elapsed
│ │ │ │ + -- 1.28561s elapsed
│ │ │ │ + -- .953937s elapsed
│ │ │ │ + -- 1.09201s elapsed
│ │ │ │ + -- 1.20741s elapsed
│ │ │ │ + -- .797653s elapsed
│ │ │ │ + -- .634924s elapsed
│ │ │ │ + -- 1.19929s elapsed
│ │ │ │ + -- 1.48117s elapsed
│ │ │ │ + -- 1.7506s elapsed
│ │ │ │ + -- 1.10955s elapsed
│ │ │ │ + -- 1.08286s elapsed
│ │ │ │ + -- 1.36015s elapsed
│ │ │ │ + -- 1.20491s elapsed
│ │ │ │ + -- .958492s elapsed
│ │ │ │ + -- 1.02931s elapsed
│ │ │ │ + -- 1.01476s elapsed
│ │ │ │ + -- .77346s elapsed
│ │ │ │ + -- .786829s elapsed
│ │ │ │ + -- 1.01282s elapsed
│ │ │ │ + -- .608904s elapsed
│ │ │ │ + -- 1.08759s elapsed
│ │ │ │ + -- 1.3022s elapsed
│ │ │ │ + -- 1.24452s elapsed
│ │ │ │ + -- .695067s elapsed
│ │ │ │ + -- .440503s elapsed
│ │ │ │ + -- .326071s elapsed
│ │ │ │  i24 : netList ({{"codimensions", "degrees"}} | allcomps)
│ │ │ │  
│ │ │ │        +------------------------+------------------------+
│ │ │ │  o24 = |codimensions            |degrees                 |
│ │ │ │        +------------------------+------------------------+
│ │ │ │        |{3, 5, 5}               |{2, 4, 6}               |
│ │ │ │        +------------------------+------------------------+
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/html/___Example_sp4.2_co_spa_sp__K5_spand_sppentagon_spglued_spalong_span_spedge.html
│ │ │ @@ -110,15 +110,15 @@
│ │ │  
│ │ │  o5 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : elapsedTime stablesolsPent = showExoticSolutions Pent
│ │ │ - -- .778s elapsed
│ │ │ + -- .992s elapsed
│ │ │  -- found extra exotic solutions for graph Graph{0 => {1, 4}} --
│ │ │                                                  1 => {0, 2}
│ │ │                                                  2 => {1, 3}
│ │ │                                                  3 => {2, 4}
│ │ │                                                  4 => {0, 3}
│ │ │  +----+-----+-----+----+-----+-----+-----+-----+
│ │ │  |.309|-.809|-.809|.309|.951 |.588 |-.588|-.951|
│ │ │ @@ -131,15 +131,15 @@
│ │ │  +---+---+---+---+
│ │ │  |72 |144|216|288|
│ │ │  +---+---+---+---+
│ │ │  |288|216|144|72 |
│ │ │  +---+---+---+---+
│ │ │  |0  |0  |0  |0  |
│ │ │  +---+---+---+---+
│ │ │ - -- .84s elapsed
│ │ │ + -- 1.04s elapsed
│ │ │  
│ │ │  o6 = {{.309, -.809, -.809, .309, .951, .588, -.588, -.951}, {.309, -.809,
│ │ │       ------------------------------------------------------------------------
│ │ │       -.809, .309, -.951, -.588, .588, .951}, {1, 1, 1, 1, 0, 0, 0, 0}}
│ │ │  
│ │ │  o6 : List
    
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -43,15 +43,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       .98, .98, .101, -.98, -.298, .393, .201, .201, .201, -.995, -.201,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       .954}}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : elapsedTime stablesolsPent = showExoticSolutions Pent
│ │ │ │ - -- .778s elapsed
│ │ │ │ + -- .992s elapsed
│ │ │ │  -- found extra exotic solutions for graph Graph{0 => {1, 4}} --
│ │ │ │                                                  1 => {0, 2}
│ │ │ │                                                  2 => {1, 3}
│ │ │ │                                                  3 => {2, 4}
│ │ │ │                                                  4 => {0, 3}
│ │ │ │  +----+-----+-----+----+-----+-----+-----+-----+
│ │ │ │  |.309|-.809|-.809|.309|.951 |.588 |-.588|-.951|
│ │ │ │ @@ -64,15 +64,15 @@
│ │ │ │  +---+---+---+---+
│ │ │ │  |72 |144|216|288|
│ │ │ │  +---+---+---+---+
│ │ │ │  |288|216|144|72 |
│ │ │ │  +---+---+---+---+
│ │ │ │  |0  |0  |0  |0  |
│ │ │ │  +---+---+---+---+
│ │ │ │ - -- .84s elapsed
│ │ │ │ + -- 1.04s elapsed
│ │ │ │  
│ │ │ │  o6 = {{.309, -.809, -.809, .309, .951, .588, -.588, -.951}, {.309, -.809,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       -.809, .309, -.951, -.588, .588, .951}, {1, 1, 1, 1, 0, 0, 0, 0}}
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  Computing the (linearly) stable solutions for K5C5 takes a minute or two:
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/html/___S__C__T_spgraphs_spwith_spexotic_spsolutions.html
│ │ │ @@ -115,19 +115,19 @@
│ │ │  o5 = 3
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : for G in Gs list showExoticSolutions G;
│ │ │  warning: some solutions are not regular: {37, 38, 40, 41, 42, 43, 44, 45, 46, 48, 50, 52, 53, 54, 55, 59, 60, 64, 65, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 81, 83, 85, 86, 87, 88, 89}
│ │ │ - -- .707s elapsed
│ │ │ + -- .816s elapsed
│ │ │  warning: some solutions are not regular: {43, 44, 47, 49, 50, 51, 52, 53, 55, 57, 59, 61, 62, 63, 64, 65, 66, 68, 72, 74, 76, 77, 78, 79, 80, 84, 85, 87, 88, 89, 90, 91, 97}
│ │ │ - -- .738s elapsed
│ │ │ - -- .909s elapsed
│ │ │ - -- .956s elapsed
│ │ │ + -- .732s elapsed
│ │ │ + -- .983s elapsed
│ │ │ + -- 1.3s elapsed
│ │ │  -- found extra exotic solutions for graph Graph{0 => {2, 3}} --
│ │ │                                                  1 => {3, 4}
│ │ │                                                  2 => {0, 4}
│ │ │                                                  3 => {0, 1}
│ │ │                                                  4 => {2, 1}
│ │ │  +-----+----+----+-----+-----+-----+-----+-----+
│ │ │  |-.809|.309|.309|-.809|.588 |-.951|.951 |-.588|
│ │ │ @@ -140,25 +140,25 @@
│ │ │  +---+---+---+---+
│ │ │  |144|288|72 |216|
│ │ │  +---+---+---+---+
│ │ │  |0  |0  |0  |0  |
│ │ │  +---+---+---+---+
│ │ │  |216|72 |288|144|
│ │ │  +---+---+---+---+
│ │ │ - -- 1.19s elapsed
│ │ │ - -- 1.4s elapsed
│ │ │ + -- 1.35s elapsed
│ │ │ + -- 1.49s elapsed
│ │ │  warning: some solutions are not regular: {27, 31, 32, 34, 37, 38, 44, 46, 47, 53, 54, 56, 57, 59, 60}
│ │ │ - -- 1.6s elapsed
│ │ │ + -- 1.82s elapsed
│ │ │  warning: some solutions are not regular: {16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 34}
│ │ │ - -- 1.46s elapsed
│ │ │ - -- 1.4s elapsed
│ │ │ + -- 1.51s elapsed
│ │ │ + -- 1.52s elapsed
│ │ │  warning: some solutions are not regular: {26, 29, 30, 32, 33}
│ │ │ - -- 1.56s elapsed
│ │ │ + -- 1.72s elapsed
│ │ │  warning: some solutions are not regular: {38, 40, 42, 53, 54, 55, 62, 63, 67, 72, 77, 78}
│ │ │ - -- 1.33s elapsed
    
│ │ │ + -- 1.52s elapsed
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │          
    
│ │ │          
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -48,21 +48,21 @@
│ │ │ │  i5 : printingPrecision = 3
│ │ │ │  
│ │ │ │  o5 = 3
│ │ │ │  i6 : for G in Gs list showExoticSolutions G;
│ │ │ │  warning: some solutions are not regular: {37, 38, 40, 41, 42, 43, 44, 45, 46,
│ │ │ │  48, 50, 52, 53, 54, 55, 59, 60, 64, 65, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78,
│ │ │ │  79, 81, 83, 85, 86, 87, 88, 89}
│ │ │ │ - -- .707s elapsed
│ │ │ │ + -- .816s elapsed
│ │ │ │  warning: some solutions are not regular: {43, 44, 47, 49, 50, 51, 52, 53, 55,
│ │ │ │  57, 59, 61, 62, 63, 64, 65, 66, 68, 72, 74, 76, 77, 78, 79, 80, 84, 85, 87, 88,
│ │ │ │  89, 90, 91, 97}
│ │ │ │ - -- .738s elapsed
│ │ │ │ - -- .909s elapsed
│ │ │ │ - -- .956s elapsed
│ │ │ │ + -- .732s elapsed
│ │ │ │ + -- .983s elapsed
│ │ │ │ + -- 1.3s elapsed
│ │ │ │  -- found extra exotic solutions for graph Graph{0 => {2, 3}} --
│ │ │ │                                                  1 => {3, 4}
│ │ │ │                                                  2 => {0, 4}
│ │ │ │                                                  3 => {0, 1}
│ │ │ │                                                  4 => {2, 1}
│ │ │ │  +-----+----+----+-----+-----+-----+-----+-----+
│ │ │ │  |-.809|.309|.309|-.809|.588 |-.951|.951 |-.588|
│ │ │ │ @@ -75,24 +75,24 @@
│ │ │ │  +---+---+---+---+
│ │ │ │  |144|288|72 |216|
│ │ │ │  +---+---+---+---+
│ │ │ │  |0  |0  |0  |0  |
│ │ │ │  +---+---+---+---+
│ │ │ │  |216|72 |288|144|
│ │ │ │  +---+---+---+---+
│ │ │ │ - -- 1.19s elapsed
│ │ │ │ - -- 1.4s elapsed
│ │ │ │ + -- 1.35s elapsed
│ │ │ │ + -- 1.49s elapsed
│ │ │ │  warning: some solutions are not regular: {27, 31, 32, 34, 37, 38, 44, 46, 47,
│ │ │ │  53, 54, 56, 57, 59, 60}
│ │ │ │ - -- 1.6s elapsed
│ │ │ │ + -- 1.82s elapsed
│ │ │ │  warning: some solutions are not regular: {16, 17, 18, 19, 20, 21, 22, 23, 24,
│ │ │ │  25, 26, 27, 28, 29, 31, 34}
│ │ │ │ - -- 1.46s elapsed
│ │ │ │ - -- 1.4s elapsed
│ │ │ │ + -- 1.51s elapsed
│ │ │ │ + -- 1.52s elapsed
│ │ │ │  warning: some solutions are not regular: {26, 29, 30, 32, 33}
│ │ │ │ - -- 1.56s elapsed
│ │ │ │ + -- 1.72s elapsed
│ │ │ │  warning: some solutions are not regular: {38, 40, 42, 53, 54, 55, 62, 63, 67,
│ │ │ │  72, 77, 78}
│ │ │ │ - -- 1.33s elapsed
│ │ │ │ + -- 1.52s elapsed
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/Oscillators/Documentation.m2:812:0.
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/html/_get__Linearly__Stable__Solutions.html
│ │ │ @@ -77,15 +77,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : getLinearlyStableSolutions(G)
│ │ │  -- warning: experimental computation over inexact field begun
│ │ │  --          results not reliable (one warning given per session)
│ │ │ - -- .377588s elapsed
│ │ │ + -- .248892s elapsed
│ │ │  warning: some solutions are not regular: {4, 5, 7, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21}
│ │ │  
│ │ │  o2 = {{1, 1, 1, 0, 0, 0}}
│ │ │  
│ │ │  o2 : List
    
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,15 +19,15 @@
│ │ │ │  of each oscillator is given by the Kuramoto model. The linear stability of a
│ │ │ │  solution is determined by the eigenvalues of the Jacobian matrix of the system
│ │ │ │  evaluated at the solution.
│ │ │ │  i1 : G = graph({0,1,2,3}, {{0,1},{1,2},{2,3},{0,3}});
│ │ │ │  i2 : getLinearlyStableSolutions(G)
│ │ │ │  -- warning: experimental computation over inexact field begun
│ │ │ │  --          results not reliable (one warning given per session)
│ │ │ │ - -- .377588s elapsed
│ │ │ │ + -- .248892s elapsed
│ │ │ │  warning: some solutions are not regular: {4, 5, 7, 9, 10, 12, 13, 14, 15, 16,
│ │ │ │  17, 18, 19, 20, 21}
│ │ │ │  
│ │ │ │  o2 = {{1, 1, 1, 0, 0, 0}}
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/html/_show__Exotic__Solutions.html
│ │ │ @@ -95,15 +95,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : showExoticSolutions G
│ │ │  -- warning: experimental computation over inexact field begun
│ │ │  --          results not reliable (one warning given per session)
│ │ │ - -- .916628s elapsed
│ │ │ + -- 1.04782s elapsed
│ │ │  -- found extra exotic solutions for graph Graph{0 => {1, 4}} --
│ │ │                                                  1 => {0, 2}
│ │ │                                                  2 => {1, 3}
│ │ │                                                  3 => {2, 4}
│ │ │                                                  4 => {0, 3}
│ │ │  +-------+--------+--------+-------+--------+--------+--------+--------+
│ │ │  |.309017|-.809017|-.809017|.309017|.951057 |.587785 |-.587785|-.951057|
│ │ │ @@ -147,15 +147,15 @@
│ │ │  
│ │ │  o3 : Graph
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : showExoticSolutions G
│ │ │ - -- 1.20627s elapsed
│ │ │ + -- 1.32658s elapsed
│ │ │  
│ │ │  o4 = {{1, 1, 1, 1, 0, 0, 0, 0}}
│ │ │  
│ │ │  o4 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -36,15 +36,15 @@
│ │ │ │             3 => {2, 4}
│ │ │ │             4 => {0, 3}
│ │ │ │  
│ │ │ │  o1 : Graph
│ │ │ │  i2 : showExoticSolutions G
│ │ │ │  -- warning: experimental computation over inexact field begun
│ │ │ │  --          results not reliable (one warning given per session)
│ │ │ │ - -- .916628s elapsed
│ │ │ │ + -- 1.04782s elapsed
│ │ │ │  -- found extra exotic solutions for graph Graph{0 => {1, 4}} --
│ │ │ │                                                  1 => {0, 2}
│ │ │ │                                                  2 => {1, 3}
│ │ │ │                                                  3 => {2, 4}
│ │ │ │                                                  4 => {0, 3}
│ │ │ │  +-------+--------+--------+-------+--------+--------+--------+--------+
│ │ │ │  |.309017|-.809017|-.809017|.309017|.951057 |.587785 |-.587785|-.951057|
│ │ │ │ @@ -78,15 +78,15 @@
│ │ │ │             1 => {0, 2}
│ │ │ │             2 => {1, 3, 4}
│ │ │ │             3 => {2, 4}
│ │ │ │             4 => {0, 2, 3}
│ │ │ │  
│ │ │ │  o3 : Graph
│ │ │ │  i4 : showExoticSolutions G
│ │ │ │ - -- 1.20627s elapsed
│ │ │ │ + -- 1.32658s elapsed
│ │ │ │  
│ │ │ │  o4 = {{1, 1, 1, 1, 0, 0, 0, 0}}
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _g_e_t_L_i_n_e_a_r_l_y_S_t_a_b_l_e_S_o_l_u_t_i_o_n_s -- Compute linearly stable solutions for the
│ │ │ │        Kuramoto oscillator system associated to a graph
│ │ ├── ./usr/share/doc/Macaulay2/PackageTemplate/dump/rawdocumentation.dump
│ │ │ @@ -1,8 +1,8 @@
│ │ │ -# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Fri Nov 14 16:08:07 2025
│ │ │ +# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Fri Nov 14 16:08:08 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │  #:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=14
│ │ │  c2Vjb25kRnVuY3Rpb24=
│ │ ├── ./usr/share/doc/Macaulay2/PathSignatures/example-output/___A_spfamily_spof_sppaths_spon_spa_spcone.out
│ │ │ @@ -80,20 +80,20 @@
│ │ │  i19 : needsPackage "MultigradedImplicitization";
│ │ │  
│ │ │  i20 : I = sub(ideal flatten values componentsOfKernel(2, m, Grading => matrix {toList(9:1)}), S);
│ │ │  warning: computation begun over finite field. resulting polynomials may not lie in the ideal
│ │ │  computing total degree: 1
│ │ │  number of monomials = 9
│ │ │  number of distinct multidegrees = 1
│ │ │ - -- .00921305s elapsed
│ │ │ + -- .0124117s elapsed
│ │ │  WARNING: There are linear relations. You may want to reduce the number of variables to speed up the computation.
│ │ │  computing total degree: 2
│ │ │  number of monomials = 45
│ │ │  number of distinct multidegrees = 1
│ │ │ - -- .602557s elapsed
│ │ │ + -- .588698s elapsed
│ │ │  
│ │ │  o20 : Ideal of S
│ │ │  
│ │ │  i21 : dim I
│ │ │  
│ │ │  o21 = 5
│ │ ├── ./usr/share/doc/Macaulay2/PathSignatures/html/___A_spfamily_spof_sppaths_spon_spa_spcone.html
│ │ │ @@ -208,20 +208,20 @@
│ │ │            
│ │ │              
│ │ │                
    i20 : I = sub(ideal flatten values componentsOfKernel(2, m, Grading => matrix {toList(9:1)}), S);
│ │ │  warning: computation begun over finite field. resulting polynomials may not lie in the ideal
│ │ │  computing total degree: 1
│ │ │  number of monomials = 9
│ │ │  number of distinct multidegrees = 1
│ │ │ - -- .00921305s elapsed
│ │ │ + -- .0124117s elapsed
│ │ │  WARNING: There are linear relations. You may want to reduce the number of variables to speed up the computation.
│ │ │  computing total degree: 2
│ │ │  number of monomials = 45
│ │ │  number of distinct multidegrees = 1
│ │ │ - -- .602557s elapsed
│ │ │ + -- .588698s elapsed
│ │ │  
│ │ │  o20 : Ideal of S
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i21 : dim I
│ │ │ ├── html2text {}
│ │ │ │ @@ -77,21 +77,21 @@
│ │ │ │  i20 : I = sub(ideal flatten values componentsOfKernel(2, m, Grading => matrix
│ │ │ │  {toList(9:1)}), S);
│ │ │ │  warning: computation begun over finite field. resulting polynomials may not lie
│ │ │ │  in the ideal
│ │ │ │  computing total degree: 1
│ │ │ │  number of monomials = 9
│ │ │ │  number of distinct multidegrees = 1
│ │ │ │ - -- .00921305s elapsed
│ │ │ │ + -- .0124117s elapsed
│ │ │ │  WARNING: There are linear relations. You may want to reduce the number of
│ │ │ │  variables to speed up the computation.
│ │ │ │  computing total degree: 2
│ │ │ │  number of monomials = 45
│ │ │ │  number of distinct multidegrees = 1
│ │ │ │ - -- .602557s elapsed
│ │ │ │ + -- .588698s elapsed
│ │ │ │  
│ │ │ │  o20 : Ideal of S
│ │ │ │  i21 : dim I
│ │ │ │  
│ │ │ │  o21 = 5
│ │ │ │  i22 : isPrime I
│ │ ├── ./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);
│ │ │ - -- 1.1868s elapsed
│ │ │ + -- 1.08954s 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 ();
│ │ │ - -- .454724s elapsed
│ │ │ + -- .379283s 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
│ │ │ - -- 21.1759s elapsed
│ │ │ + -- 13.4468s 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
│ │ │ @@ -46,30 +46,30 @@
│ │ │  i11 : M=cliffordModule(Mu1,Mu2,R)
│ │ │  
│ │ │  o11 = CliffordModule{...6...}
│ │ │  
│ │ │  o11 : CliffordModule
│ │ │  
│ │ │  i12 : elapsedTime Ulr = searchUlrich(M,S);
│ │ │ - -- .686347s elapsed
│ │ │ + -- .614044s elapsed
│ │ │  
│ │ │  i13 : betti freeResolution 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);
│ │ │ - -- 2.46859s elapsed
│ │ │ + -- 1.79539s elapsed
│ │ │  
│ │ │  i16 : betti freeResolution Ulr3
│ │ │  
│ │ │                0  1  2
│ │ │  o16 = total: 12 24 12
│ │ │            0: 12 24 12
│ │ ├── ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/___Lab__Book__Protocol.html
│ │ │ @@ -128,15 +128,15 @@
│ │ │              
│ │ │                
    i4 : kk= ZZ/101;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : elapsedTime (S,qq,R,u, M1,M2, Mu1, Mu2)=randomNicePencil(kk,g);
│ │ │ - -- 1.1868s elapsed
    
│ │ │ + -- 1.08954s elapsed
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : M=cliffordModule(Mu1,Mu2,R)
│ │ │  
│ │ │  o6 = CliffordModule{...6...}
│ │ │ @@ -172,15 +172,15 @@
│ │ │            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 ();
│ │ │ - -- .454724s elapsed
    
│ │ │ + -- .379283s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : betti Ul.yAction, betti Ul1.yAction
│ │ │  
│ │ │                 0  1          0  1
│ │ │ @@ -193,15 +193,15 @@
│ │ │  
│ │ │  o12 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i13 : elapsedTime Ul = tensorProduct(M,V); -- the heaviest part computing the actions of generators
│ │ │ - -- 21.1759s elapsed
    
│ │ │ + -- 13.4468s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : M1Ul=sum(#Ul.oddOperators,i->S_i*sub(Ul.oddOperators_i,S));
│ │ │  
│ │ │                32      32
│ │ │ ├── html2text {}
│ │ │ │ @@ -55,15 +55,15 @@
│ │ │ │              -- 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);
│ │ │ │ - -- 1.1868s elapsed
│ │ │ │ + -- 1.08954s elapsed
│ │ │ │  i6 : M=cliffordModule(Mu1,Mu2,R)
│ │ │ │  
│ │ │ │  o6 = CliffordModule{...6...}
│ │ │ │  
│ │ │ │  o6 : CliffordModule
│ │ │ │  i7 : Mor = vectorBundleOnE M.evenCenter;
│ │ │ │  i8 : Mor1= vectorBundleOnE M.oddCenter;
│ │ │ │ @@ -75,29 +75,29 @@
│ │ │ │            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 ();
│ │ │ │ - -- .454724s elapsed
│ │ │ │ + -- .379283s 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
│ │ │ │ - -- 21.1759s elapsed
│ │ │ │ + -- 13.4468s 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
│ │ │ @@ -161,15 +161,15 @@
│ │ │  
│ │ │  o11 : CliffordModule
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : elapsedTime Ulr = searchUlrich(M,S);
│ │ │ - -- .686347s elapsed
    
│ │ │ + -- .614044s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i13 : betti freeResolution Ulr
│ │ │  
│ │ │               0  1 2
│ │ │ @@ -185,15 +185,15 @@
│ │ │  
│ │ │  o14 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i15 : elapsedTime Ulr3 = searchUlrich(M,S,3);
│ │ │ - -- 2.46859s elapsed
    
│ │ │ + -- 1.79539s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i16 : betti freeResolution Ulr3
│ │ │  
│ │ │                0  1  2
│ │ │ ├── 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);
│ │ │ │ - -- .686347s elapsed
│ │ │ │ + -- .614044s elapsed
│ │ │ │  i13 : betti freeResolution 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);
│ │ │ │ - -- 2.46859s elapsed
│ │ │ │ + -- 1.79539s elapsed
│ │ │ │  i16 : betti freeResolution Ulr3
│ │ │ │  
│ │ │ │                0  1  2
│ │ │ │  o16 = total: 12 24 12
│ │ │ │            0: 12 24 12
│ │ │ │  
│ │ │ │  o16 : BettiTally
│ │ ├── ./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);
│ │ │ - -- 2.66266s elapsed
│ │ │ + -- 1.97316s elapsed
│ │ │  
│ │ │  i11 : elapsedTime H = affineFatPointsByIntersection(M,mults,R);
│ │ │ - -- 4.51474s elapsed
│ │ │ + -- 4.14979s elapsed
│ │ │  
│ │ │  i12 : G==H
│ │ │  
│ │ │  o12 = true
│ │ │  
│ │ │  i13 :
│ │ ├── ./usr/share/doc/Macaulay2/Points/html/_affine__Fat__Points.html
│ │ │ @@ -177,21 +177,21 @@
│ │ │  
│ │ │  o9 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : elapsedTime (Q,inG,G) = affineFatPoints(M,mults,R);
│ │ │ - -- 2.66266s elapsed
    
│ │ │ + -- 1.97316s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i11 : elapsedTime H = affineFatPointsByIntersection(M,mults,R);
│ │ │ - -- 4.51474s elapsed
    
│ │ │ + -- 4.14979s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : G==H
│ │ │  
│ │ │  o12 = true
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -81,17 +81,17 @@
│ │ │ │  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);
│ │ │ │ - -- 2.66266s elapsed
│ │ │ │ + -- 1.97316s elapsed
│ │ │ │  i11 : elapsedTime H = affineFatPointsByIntersection(M,mults,R);
│ │ │ │ - -- 4.51474s elapsed
│ │ │ │ + -- 4.14979s 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/Posets/example-output/___Precompute.out
│ │ │ @@ -31,27 +31,27 @@
│ │ │  o5 = CacheTable{name => P}
│ │ │  
│ │ │  i6 : C == P
│ │ │  
│ │ │  o6 = true
│ │ │  
│ │ │  i7 : time isDistributive C
│ │ │ - -- used 1.2633e-05s (cpu); 8.386e-06s (thread); 0s (gc)
│ │ │ + -- used 1.563e-05s (cpu); 7.389e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = true
│ │ │  
│ │ │  i8 : time isDistributive P
│ │ │ - -- used 6.21358s (cpu); 4.20688s (thread); 0s (gc)
│ │ │ + -- used 7.28779s (cpu); 4.43968s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
│ │ │  
│ │ │  i9 : C' = dual C;
│ │ │  
│ │ │  i10 : time isDistributive C'
│ │ │ - -- used 7.304e-06s (cpu); 6.462e-06s (thread); 0s (gc)
│ │ │ + -- used 8.485e-06s (cpu); 5.461e-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 0.414882s (cpu); 0.246902s (thread); 0s (gc)
│ │ │ + -- used 0.478288s (cpu); 0.249546s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = Partition{9, 2}
│ │ │  
│ │ │  o4 : Partition
│ │ │  
│ │ │  i5 : time greeneKleitmanPartition(D, Strategy => "chains")
│ │ │ - -- used 1.2734e-05s (cpu); 1.2373e-05s (thread); 0s (gc)
│ │ │ + -- used 1.2181e-05s (cpu); 1.1673e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = Partition{9, 2}
│ │ │  
│ │ │  o5 : Partition
│ │ │  
│ │ │  i6 : greeneKleitmanPartition chain 10
│ │ ├── ./usr/share/doc/Macaulay2/Posets/html/___Precompute.html
│ │ │ @@ -107,23 +107,23 @@
│ │ │  
│ │ │  o6 = true
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : time isDistributive C
│ │ │ - -- used 1.2633e-05s (cpu); 8.386e-06s (thread); 0s (gc)
│ │ │ + -- used 1.563e-05s (cpu); 7.389e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : time isDistributive P
│ │ │ - -- used 6.21358s (cpu); 4.20688s (thread); 0s (gc)
│ │ │ + -- used 7.28779s (cpu); 4.43968s (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.
    
│ │ │ @@ -133,15 +133,15 @@
│ │ │              
│ │ │                
    i9 : C' = dual C;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : time isDistributive C'
│ │ │ - -- used 7.304e-06s (cpu); 6.462e-06s (thread); 0s (gc)
│ │ │ + -- used 8.485e-06s (cpu); 5.461e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i11 : peek C'.cache
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,26 +41,26 @@
│ │ │ │  i5 : peek P.cache
│ │ │ │  
│ │ │ │  o5 = CacheTable{name => P}
│ │ │ │  i6 : C == P
│ │ │ │  
│ │ │ │  o6 = true
│ │ │ │  i7 : time isDistributive C
│ │ │ │ - -- used 1.2633e-05s (cpu); 8.386e-06s (thread); 0s (gc)
│ │ │ │ + -- used 1.563e-05s (cpu); 7.389e-06s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = true
│ │ │ │  i8 : time isDistributive P
│ │ │ │ - -- used 6.21358s (cpu); 4.20688s (thread); 0s (gc)
│ │ │ │ + -- used 7.28779s (cpu); 4.43968s (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 7.304e-06s (cpu); 6.462e-06s (thread); 0s (gc)
│ │ │ │ + -- used 8.485e-06s (cpu); 5.461e-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
│ │ │ @@ -100,25 +100,25 @@
│ │ │              
│ │ │                
    i3 : D = dominanceLattice 6;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time greeneKleitmanPartition(D, Strategy => "antichains")
│ │ │ - -- used 0.414882s (cpu); 0.246902s (thread); 0s (gc)
│ │ │ + -- used 0.478288s (cpu); 0.249546s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = Partition{9, 2}
│ │ │  
│ │ │  o4 : Partition
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time greeneKleitmanPartition(D, Strategy => "chains")
│ │ │ - -- used 1.2734e-05s (cpu); 1.2373e-05s (thread); 0s (gc)
│ │ │ + -- used 1.2181e-05s (cpu); 1.1673e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = Partition{9, 2}
│ │ │  
│ │ │  o5 : Partition
    
│ │ │              
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,21 +28,21 @@
│ │ │ │  
│ │ │ │  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 0.414882s (cpu); 0.246902s (thread); 0s (gc)
│ │ │ │ + -- used 0.478288s (cpu); 0.249546s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = Partition{9, 2}
│ │ │ │  
│ │ │ │  o4 : Partition
│ │ │ │  i5 : time greeneKleitmanPartition(D, Strategy => "chains")
│ │ │ │ - -- used 1.2734e-05s (cpu); 1.2373e-05s (thread); 0s (gc)
│ │ │ │ + -- used 1.2181e-05s (cpu); 1.1673e-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/PrimaryDecomposition/example-output/_associated__Primes.out
│ │ │ @@ -125,24 +125,24 @@
│ │ │        -----------------------------------------------------------------------
│ │ │        ideal (a, b, c, e), ideal (a, b, d, e), ideal (a, b, c, d, e)}
│ │ │  
│ │ │  o19 : List
│ │ │  
│ │ │  i20 : M1 = set apply(L1, I -> sort flatten entries gens I)
│ │ │  
│ │ │ -o20 = set {{e, c, b, a}, {e, d, c, b, a}, {d, b, a}, {e, a}, {c, b, a}, {d,
│ │ │ +o20 = set {{e, d, c, b, a}, {e, a}, {d, c, b, a}, {d, b, a}, {e, d, b, a},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      c, b, a}, {e, d, b, a}}
│ │ │ +      {c, b, a}, {e, c, b, a}}
│ │ │  
│ │ │  o20 : Set
│ │ │  
│ │ │  i21 : M2 = set apply(L2, I -> sort flatten entries gens I)
│ │ │  
│ │ │ -o21 = set {{e, c, b, a}, {e, d, c, b, a}, {d, b, a}, {e, a}, {c, b, a}, {d,
│ │ │ +o21 = set {{e, d, c, b, a}, {e, a}, {d, c, b, a}, {d, b, a}, {e, d, b, a},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      c, b, a}, {e, d, b, a}}
│ │ │ +      {c, b, a}, {e, c, b, a}}
│ │ │  
│ │ │  o21 : Set
│ │ │  
│ │ │  i22 : assert(M1 === M2)
│ │ │  
│ │ │  i23 :
│ │ ├── ./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)
│ │ │ - -- .113801s elapsed
│ │ │ + -- .100627s 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)
│ │ │ - -- .0380783s elapsed
│ │ │ + -- .0227558s 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)
│ │ │ - -- .120101s elapsed
│ │ │ + -- .0556313s 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
│ │ │ - -- .087123s elapsed
│ │ │ + -- .0833017s 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)
│ │ │ - -- .00815253s elapsed
│ │ │ + -- .0093776s 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/_associated__Primes.html
│ │ │ @@ -286,28 +286,28 @@
│ │ │  o19 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i20 : M1 = set apply(L1, I -> sort flatten entries gens I)
│ │ │  
│ │ │ -o20 = set {{e, c, b, a}, {e, d, c, b, a}, {d, b, a}, {e, a}, {c, b, a}, {d,
│ │ │ +o20 = set {{e, d, c, b, a}, {e, a}, {d, c, b, a}, {d, b, a}, {e, d, b, a},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      c, b, a}, {e, d, b, a}}
│ │ │ +      {c, b, a}, {e, c, b, a}}
│ │ │  
│ │ │  o20 : Set
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i21 : M2 = set apply(L2, I -> sort flatten entries gens I)
│ │ │  
│ │ │ -o21 = set {{e, c, b, a}, {e, d, c, b, a}, {d, b, a}, {e, a}, {c, b, a}, {d,
│ │ │ +o21 = set {{e, d, c, b, a}, {e, a}, {d, c, b, a}, {d, b, a}, {e, d, b, a},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      c, b, a}, {e, d, b, a}}
│ │ │ +      {c, b, a}, {e, c, b, a}}
│ │ │  
│ │ │  o21 : Set
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i22 : assert(M1 === M2)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -155,24 +155,24 @@
│ │ │ │  o19 = {ideal (a, e), ideal (a, b, c), ideal (a, b, d), ideal (a, b, c, d),
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        ideal (a, b, c, e), ideal (a, b, d, e), ideal (a, b, c, d, e)}
│ │ │ │  
│ │ │ │  o19 : List
│ │ │ │  i20 : M1 = set apply(L1, I -> sort flatten entries gens I)
│ │ │ │  
│ │ │ │ -o20 = set {{e, c, b, a}, {e, d, c, b, a}, {d, b, a}, {e, a}, {c, b, a}, {d,
│ │ │ │ +o20 = set {{e, d, c, b, a}, {e, a}, {d, c, b, a}, {d, b, a}, {e, d, b, a},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      c, b, a}, {e, d, b, a}}
│ │ │ │ +      {c, b, a}, {e, c, b, a}}
│ │ │ │  
│ │ │ │  o20 : Set
│ │ │ │  i21 : M2 = set apply(L2, I -> sort flatten entries gens I)
│ │ │ │  
│ │ │ │ -o21 = set {{e, c, b, a}, {e, d, c, b, a}, {d, b, a}, {e, a}, {c, b, a}, {d,
│ │ │ │ +o21 = set {{e, d, c, b, a}, {e, a}, {d, c, b, a}, {d, b, a}, {e, d, b, a},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      c, b, a}, {e, d, b, a}}
│ │ │ │ +      {c, b, a}, {e, c, b, a}}
│ │ │ │  
│ │ │ │  o21 : Set
│ │ │ │  i22 : assert(M1 === M2)
│ │ │ │  The method using Ext modules comes from Eisenbud-Huneke-Vasconcelos, Invent.
│ │ │ │  Math 110 (1992) 207-235.
│ │ │ │  Original author (for ideals): _C_._ _Y_a_c_k_e_l. Updated for modules by J. Chen.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_kernel__Of__Localization.html
│ │ │ @@ -107,41 +107,41 @@
│ │ │                              3
│ │ │  o3 : R-module, quotient of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : elapsedTime kernelOfLocalization(M, I1)
│ │ │ - -- .113801s elapsed
│ │ │ + -- .100627s 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)
│ │ │ - -- .0380783s elapsed
│ │ │ + -- .0227558s 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)
│ │ │ - -- .120101s elapsed
│ │ │ + -- .0556313s 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 {}
│ │ │ │ @@ -41,39 +41,39 @@
│ │ │ │  |
│ │ │ │                | 0            0             0            0              x_1^5-
│ │ │ │  x_0x_2^4 |
│ │ │ │  
│ │ │ │                              3
│ │ │ │  o3 : R-module, quotient of R
│ │ │ │  i4 : elapsedTime kernelOfLocalization(M, I1)
│ │ │ │ - -- .113801s elapsed
│ │ │ │ + -- .100627s 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)
│ │ │ │ - -- .0380783s elapsed
│ │ │ │ + -- .0227558s 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)
│ │ │ │ - -- .120101s elapsed
│ │ │ │ + -- .0556313s 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 @@
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : elapsedTime regSeqInIdeal I
│ │ │ - -- .087123s elapsed
│ │ │ + -- .0833017s 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
    
│ │ │              
│ │ │            
│ │ │ @@ -148,15 +148,15 @@
│ │ │  
│ │ │  o6 = true
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : elapsedTime regSeqInIdeal(I, 3, 3, 1)
│ │ │ - -- .00815253s elapsed
│ │ │ + -- .0093776s 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 {}
│ │ │ │ @@ -40,15 +40,15 @@
│ │ │ │               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
│ │ │ │ - -- .087123s elapsed
│ │ │ │ + -- .0833017s 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
│ │ │ │ @@ -70,15 +70,15 @@
│ │ │ │       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)
│ │ │ │ - -- .00815253s elapsed
│ │ │ │ + -- .0093776s 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/Python/example-output/_iterator_lp__Python__Object_rp.out
│ │ │ @@ -10,12 +10,12 @@
│ │ │  
│ │ │  o2 = range(0, 3)
│ │ │  
│ │ │  o2 : PythonObject of class range
│ │ │  
│ │ │  i3 : i = iterator x
│ │ │  
│ │ │ -o3 = 
│ │ │ +o3 = 
│ │ │  
│ │ │  o3 : PythonObject of class range_iterator
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/Python/example-output/_next_lp__Python__Object_rp.out
│ │ │ @@ -10,15 +10,15 @@
│ │ │  
│ │ │  o2 = range(0, 3)
│ │ │  
│ │ │  o2 : PythonObject of class range
│ │ │  
│ │ │  i3 : i = iterator x
│ │ │  
│ │ │ -o3 = 
│ │ │ +o3 = 
│ │ │  
│ │ │  o3 : PythonObject of class range_iterator
│ │ │  
│ │ │  i4 : next i
│ │ │  
│ │ │  o4 = 0
│ │ ├── ./usr/share/doc/Macaulay2/Python/example-output/_python__Run__Script.out
│ │ │ @@ -1,22 +1,22 @@
│ │ │  -- -*- M2-comint -*- hash: 447449196062331972
│ │ │  
│ │ │  i1 : pyfile = temporaryFileName() | ".py"
│ │ │  
│ │ │ -o1 = /tmp/M2-47339-0/0.py
│ │ │ +o1 = /tmp/M2-73709-0/0.py
│ │ │  
│ │ │  i2 : pyfile << "import math" << endl
│ │ │  
│ │ │ -o2 = /tmp/M2-47339-0/0.py
│ │ │ +o2 = /tmp/M2-73709-0/0.py
│ │ │  
│ │ │  o2 : File
│ │ │  
│ │ │  i3 : pyfile << "x = math.sin(3.4)" << endl << close
│ │ │  
│ │ │ -o3 = /tmp/M2-47339-0/0.py
│ │ │ +o3 = /tmp/M2-73709-0/0.py
│ │ │  
│ │ │  o3 : File
│ │ │  
│ │ │  i4 : get pyfile
│ │ │  
│ │ │  o4 = import math
│ │ │       x = math.sin(3.4)
│ │ ├── ./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/example-output/_use_lp__Python__Context_rp.out
│ │ │ @@ -30,15 +30,15 @@
│ │ │  
│ │ │  o7 : Symbol
│ │ │  
│ │ │  i8 : use ctx
│ │ │  
│ │ │  i9 : f
│ │ │  
│ │ │ -o9 =  at 0x7f92fa0eccc0>
│ │ │ +o9 =  at 0x7f9e37c2ccc0>
│ │ │  
│ │ │  o9 : PythonObject of class function
│ │ │  
│ │ │  i10 : x
│ │ │  
│ │ │  o10 = 5
│ │ ├── ./usr/share/doc/Macaulay2/Python/html/_iterator_lp__Python__Object_rp.html
│ │ │ @@ -90,15 +90,15 @@
│ │ │  o2 : PythonObject of class range
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : i = iterator x
│ │ │  
│ │ │ -o3 = <range_iterator object at 0x7f92fa135650>
│ │ │ +o3 = <range_iterator object at 0x7f9e37c75740>
│ │ │  
│ │ │  o3 : PythonObject of class range_iterator
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -22,15 +22,15 @@
│ │ │ │  i2 : x = builtins@@range 3
│ │ │ │  
│ │ │ │  o2 = range(0, 3)
│ │ │ │  
│ │ │ │  o2 : PythonObject of class range
│ │ │ │  i3 : i = iterator x
│ │ │ │  
│ │ │ │ -o3 = 
│ │ │ │ +o3 = 
│ │ │ │  
│ │ │ │  o3 : 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
│ │ │ @@ -86,15 +86,15 @@
│ │ │  o2 : PythonObject of class range
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : i = iterator x
│ │ │  
│ │ │ -o3 = <range_iterator object at 0x7f92fa129da0>
│ │ │ +o3 = <range_iterator object at 0x7f9e37c69e90>
│ │ │  
│ │ │  o3 : PythonObject of class range_iterator
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : next i
│ │ │ ├── html2text {}
│ │ │ │ @@ -21,15 +21,15 @@
│ │ │ │  i2 : x = builtins@@range 3
│ │ │ │  
│ │ │ │  o2 = range(0, 3)
│ │ │ │  
│ │ │ │  o2 : PythonObject of class range
│ │ │ │  i3 : i = iterator x
│ │ │ │  
│ │ │ │ -o3 = 
│ │ │ │ +o3 = 
│ │ │ │  
│ │ │ │  o3 : PythonObject of class range_iterator
│ │ │ │  i4 : next i
│ │ │ │  
│ │ │ │  o4 = 0
│ │ │ │  
│ │ │ │  o4 : PythonObject of class int
│ │ ├── ./usr/share/doc/Macaulay2/Python/html/_python__Run__Script.html
│ │ │ @@ -76,31 +76,31 @@
│ │ │            
    The return value is a Python dictionary containing all the variables defined in the global scope.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : pyfile = temporaryFileName() | ".py"
│ │ │  
│ │ │ -o1 = /tmp/M2-47339-0/0.py
    
│ │ │ +o1 = /tmp/M2-73709-0/0.py
│ │ │              
    
│ │ │                
    i2 : pyfile << "import math" << endl
│ │ │  
│ │ │ -o2 = /tmp/M2-47339-0/0.py
│ │ │ +o2 = /tmp/M2-73709-0/0.py
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │                
    i3 : pyfile << "x = math.sin(3.4)" << endl << close
│ │ │  
│ │ │ -o3 = /tmp/M2-47339-0/0.py
│ │ │ +o3 = /tmp/M2-73709-0/0.py
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : get pyfile
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,23 +16,23 @@
│ │ │ │  Execute a sequence of statements as if they were read from a Python file. This
│ │ │ │  is for multi-line code that might contain definitions, control structures,
│ │ │ │  imports, etc. It is great for running Python code from a file.
│ │ │ │  The return value is a Python dictionary containing all the variables defined in
│ │ │ │  the global scope.
│ │ │ │  i1 : pyfile = temporaryFileName() | ".py"
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-47339-0/0.py
│ │ │ │ +o1 = /tmp/M2-73709-0/0.py
│ │ │ │  i2 : pyfile << "import math" << endl
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-47339-0/0.py
│ │ │ │ +o2 = /tmp/M2-73709-0/0.py
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : pyfile << "x = math.sin(3.4)" << endl << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-47339-0/0.py
│ │ │ │ +o3 = /tmp/M2-73709-0/0.py
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : get pyfile
│ │ │ │  
│ │ │ │  o4 = import math
│ │ │ │       x = math.sin(3.4)
│ │ │ │  i5 : pythonRunScript oo
│ │ ├── ./usr/share/doc/Macaulay2/Python/html/_to__Python.html
│ │ │ @@ -181,15 +181,15 @@
│ │ │  o12 : FunctionClosure
    
│ │ │              
    
│ │ │                
    i13 : pysqrt = toPython m2sqrt
│ │ │  
│ │ │ -o13 = <built-in method m2sqrt of PyCapsule object at 0x7f92fa11a9d0>
│ │ │ +o13 = <built-in method m2sqrt of PyCapsule object at 0x7f9e37c5aa20>
│ │ │  
│ │ │  o13 : PythonObject of class builtin_function_or_method
    
│ │ │              
    
│ │ │                
    i14 : pysqrt 2
│ │ │ ├── html2text {}
│ │ │ │ @@ -72,15 +72,15 @@
│ │ │ │            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/Python/html/_use_lp__Python__Context_rp.html
│ │ │ @@ -124,15 +124,15 @@
│ │ │                
    i8 : use ctx
    
│ │ │              
    
│ │ │                
    i9 : f
│ │ │  
│ │ │ -o9 = <function <lambda> at 0x7f92fa0eccc0>
│ │ │ +o9 = <function <lambda> at 0x7f9e37c2ccc0>
│ │ │  
│ │ │  o9 : PythonObject of class function
    
│ │ │              
    
│ │ │                
    i10 : x
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,15 +34,15 @@
│ │ │ │  
│ │ │ │  o7 = y
│ │ │ │  
│ │ │ │  o7 : Symbol
│ │ │ │  i8 : use ctx
│ │ │ │  i9 : f
│ │ │ │  
│ │ │ │ -o9 =  at 0x7f92fa0eccc0>
│ │ │ │ +o9 =  at 0x7f9e37c2ccc0>
│ │ │ │  
│ │ │ │  o9 : PythonObject of class function
│ │ │ │  i10 : x
│ │ │ │  
│ │ │ │  o10 = 5
│ │ │ │  
│ │ │ │  o10 : PythonObject of class int
│ │ ├── ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.out
│ │ │ @@ -180,15 +180,15 @@
│ │ │  i21 : L = trim groebnerStratum F;
│ │ │  
│ │ │  o21 : Ideal of T
│ │ │  
│ │ │  i22 : assert(dim L == 18)
│ │ │  
│ │ │  i23 : elapsedTime isPrime L
│ │ │ - -- 2.47531s elapsed
│ │ │ + -- 2.30383s 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 -
│ │ │ @@ -302,15 +302,15 @@
│ │ │  o38 = true
│ │ │  
│ │ │  i39 : L441 = trim(L + ideal M1);
│ │ │  
│ │ │  o39 : Ideal of T
│ │ │  
│ │ │  i40 : elapsedTime compsL441 = decompose L441;
│ │ │ - -- 1.52419s elapsed
│ │ │ + -- 1.25749s elapsed
│ │ │  
│ │ │  i41 : #compsL441
│ │ │  
│ │ │  o41 = 2
│ │ │  
│ │ │  i42 : compsL441/dim -- two components, of dimensions 14 and 16.
│ │ │  
│ │ │ @@ -320,37 +320,37 @@
│ │ │  
│ │ │  i43 : compsL441/dim == {16, 14}
│ │ │  
│ │ │  o43 = true
│ │ │  
│ │ │  i44 : pta = randomPointOnRationalVariety compsL441_0
│ │ │  
│ │ │ -o44 = | 22 -10 -8 -1 34 44 -21 -1 25 -41 6 -11 -50 -50 43 -28 -6 45 -28 22 42
│ │ │ +o44 = | 32 -41 22 15 22 -46 43 42 -27 -27 -13 10 -24 19 -25 48 31 10 41 49 39
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -29 -32 -28 5 -10 34 15 19 37 26 49 19 5 10 18 |
│ │ │ +      -28 -29 -10 -48 5 15 18 45 19 49 37 -32 34 26 -50 |
│ │ │  
│ │ │                 1       36
│ │ │  o44 : Matrix kk  <-- kk
│ │ │  
│ │ │  i45 : Fa = sub(F, (vars S) | pta)
│ │ │  
│ │ │ -              2              2                             2              
│ │ │ -o45 = ideal (a  + 43b*c - 41c  - 11a*d + 44b*d - 8c*d + 22d , a*b + 5b*c -
│ │ │ +              2              2                              2               
│ │ │ +o45 = ideal (a  - 25b*c - 27c  + 10a*d - 46b*d + 22c*d + 32d , a*b - 48b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │           2                              2   2              2                
│ │ │ -      28c  + 42a*d - 50b*d + 25c*d - 10d , b  + 10b*c + 15c  + 19a*d - 29b*d
│ │ │ +      41c  + 39a*d - 24b*d - 27c*d - 41d , b  + 26b*c + 18c  - 32a*d - 28b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2                   2                              2     2
│ │ │ -      - 28c*d - 21d , a*c + 49b*c - 10c  + 19a*d + 22b*d - 50c*d + 34d , b*c 
│ │ │ +                   2                  2                              2     2
│ │ │ +      + 48c*d + 43d , a*c + 37b*c + 5c  + 45a*d + 49b*d + 19c*d + 22d , b*c 
│ │ │        -----------------------------------------------------------------------
│ │ │ -                     2         2       2       2    3   3                2   
│ │ │ -      + 37b*c*d - 32c d + 34a*d  - 6b*d  + 6c*d  - d , c  + 18b*c*d + 26c d +
│ │ │ +                     2         2        2        2      3   3            
│ │ │ +      + 19b*c*d - 29c d + 15a*d  + 31b*d  - 13c*d  + 15d , c  - 50b*c*d +
│ │ │        -----------------------------------------------------------------------
│ │ │ -          2        2        2    3
│ │ │ -      5a*d  - 28b*d  + 45c*d  - d )
│ │ │ +         2         2        2        2      3
│ │ │ +      49c d + 34a*d  - 10b*d  + 10c*d  + 42d )
│ │ │  
│ │ │  o45 : Ideal of S
│ │ │  
│ │ │  i46 : betti res Fa
│ │ │  
│ │ │               0 1 2 3
│ │ │  o46 = total: 1 6 8 3
│ │ │ @@ -358,81 +358,83 @@
│ │ │            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 + 19d, b - 37d, a)                                                                                                                                        |
│ │ │ -      +-------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                             2              2                      2   3                2         2       2     3     2                2         2        2      3 |
│ │ │ -      |ideal (a + 49b - 10c + 39d, b  + 10b*c + 15c  + 50b*d - 40c*d + 46d , c  + 18b*c*d + 26c d + 30b*d  - 6c*d  + 6d , b*c  + 37b*c*d - 32c d + 45b*d  + 43c*d  - 14d )|
│ │ │ -      +-------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      +----------------------------------------------------------------------------------------------+
│ │ │ +o47 = |ideal (c + 45d, b - 34d, a - 35d)                                                             |
│ │ │ +      +----------------------------------------------------------------------------------------------+
│ │ │ +      |ideal (c + 18d, b - 12d, a + 47d)                                                             |
│ │ │ +      +----------------------------------------------------------------------------------------------+
│ │ │ +      |                                   2                      2   2      2                      2 |
│ │ │ +      |ideal (a + 37b + 5c - 4d, b*c - 13c  + 45b*d - 28c*d - 29d , b  - 48c  - 14b*d + 27c*d - 38d )|
│ │ │ +      +----------------------------------------------------------------------------------------------+
│ │ │  
│ │ │  i48 : CFa = minimalPrimes Fa
│ │ │  
│ │ │ -                                                                 2          
│ │ │ -o48 = {ideal (c + 19d, b - 37d, a), ideal (a + 49b - 10c + 39d, b  + 10b*c +
│ │ │ +                                                                            
│ │ │ +o48 = {ideal (c + 45d, b - 34d, a - 35d), ideal (c + 18d, b - 12d, a + 47d),
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                      2   3                2         2       2  
│ │ │ -      15c  + 50b*d - 40c*d + 46d , c  + 18b*c*d + 26c d + 30b*d  - 6c*d  +
│ │ │ +                                         2                      2   2      2
│ │ │ +      ideal (a + 37b + 5c - 4d, b*c - 13c  + 45b*d - 28c*d - 29d , b  - 48c 
│ │ │        -----------------------------------------------------------------------
│ │ │ -        3     2                2         2        2      3
│ │ │ -      6d , b*c  + 37b*c*d - 32c d + 45b*d  + 43c*d  - 14d )}
│ │ │ +                           2
│ │ │ +      - 14b*d + 27c*d - 38d )}
│ │ │  
│ │ │  o48 : List
│ │ │  
│ │ │  i49 : lin = CFa_1_0 -- a linear form, defining a plane.
│ │ │  
│ │ │ -o49 = a + 49b - 10c + 39d
│ │ │ +o49 = c + 18d
│ │ │  
│ │ │  o49 : S
│ │ │  
│ │ │  i50 : CFa/degree
│ │ │  
│ │ │ -o50 = {1, 5}
│ │ │ +o50 = {1, 1, 4}
│ │ │  
│ │ │  o50 : List
│ │ │  
│ │ │  i51 : CFa/(I -> lin % I == 0) -- so 5 points on the plane.
│ │ │  
│ │ │ -o51 = {false, true}
│ │ │ +o51 = {false, true, false}
│ │ │  
│ │ │  o51 : List
│ │ │  
│ │ │  i52 : degree(Fa : (Fa : lin))  -- somewhat simpler(?) way to see the ideal of the 5 points
│ │ │  
│ │ │ -o52 = 5
│ │ │ +o52 = 1
│ │ │  
│ │ │  i53 : ptb = randomPointOnRationalVariety compsL441_1
│ │ │  
│ │ │ -o53 = | 27 12 -34 9 -19 -43 -32 27 40 45 -13 29 -41 -13 22 -49 -4 -4 9 -23 43
│ │ │ +o53 = | -31 -3 -29 -17 -21 5 -32 33 -24 2 26 -26 -45 -4 16 -22 2 -37 16 -23
│ │ │        -----------------------------------------------------------------------
│ │ │ -      18 -9 -47 43 21 38 17 -20 21 -29 47 0 2 -37 9 |
│ │ │ +      -42 19 -29 21 7 2 17 9 -15 -9 -47 -13 0 38 47 21 |
│ │ │  
│ │ │                 1       36
│ │ │  o53 : Matrix kk  <-- kk
│ │ │  
│ │ │  i54 : Fb = sub(F, (vars S) | ptb)
│ │ │  
│ │ │ -              2              2                              2               
│ │ │ -o54 = ideal (a  + 22b*c + 45c  + 29a*d - 43b*d - 34c*d + 27d , a*b + 43b*c +
│ │ │ +              2             2                             2              
│ │ │ +o54 = ideal (a  + 16b*c + 2c  - 26a*d + 5b*d - 29c*d - 31d , a*b + 7b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -        2                              2   2              2                  
│ │ │ -      9c  + 43a*d - 41b*d + 40c*d + 12d , b  - 37b*c + 17c  + 18b*d - 49c*d -
│ │ │ +         2                             2   2             2                  
│ │ │ +      16c  - 42a*d - 45b*d - 24c*d - 3d , b  + 47b*c + 9c  + 19b*d - 22c*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           
│ │ │ +      32d , a*c - 13b*c + 2c  - 15a*d - 23b*d - 4c*d - 21d , b*c  - 9b*c*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       2        2      3   3                2         2  
│ │ │ +      29c d + 17a*d  + 2b*d  + 26c*d  - 17d , c  + 21b*c*d - 47c d + 38a*d  +
│ │ │        -----------------------------------------------------------------------
│ │ │ -           2       2      3
│ │ │ -      47b*d  - 4c*d  + 27d )
│ │ │ +           2        2      3
│ │ │ +      21b*d  - 37c*d  + 33d )
│ │ │  
│ │ │  o54 : Ideal of S
│ │ │  
│ │ │  i55 : betti res Fb
│ │ │  
│ │ │               0 1 2 3
│ │ │  o55 = total: 1 6 8 3
│ │ │ @@ -440,80 +442,84 @@
│ │ │            1: . 4 4 1
│ │ │            2: . 2 4 2
│ │ │  
│ │ │  o55 : BettiTally
│ │ │  
│ │ │  i56 : netList decompose Fb --
│ │ │  
│ │ │ -      +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                                      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 )|
│ │ │ -      +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      +---------------------------------------------------------------+
│ │ │ +o56 = |ideal (c - 17d, b - 26d, a - 49d)                              |
│ │ │ +      +---------------------------------------------------------------+
│ │ │ +      |                                     3      2         2      3 |
│ │ │ +      |ideal (b + 39c - 21d, a + 4c - 27d, c  + 43c d + 39c*d  - 15d )|
│ │ │ +      +---------------------------------------------------------------+
│ │ │ +      |                                    2             2            |
│ │ │ +      |ideal (b + 8c + 40d, a + 5c - 33d, c  + 4c*d + 45d )           |
│ │ │ +      +---------------------------------------------------------------+
│ │ │  
│ │ │  i57 : netList for x in subsets(decompose Fb, 3) list intersect(x#0, x#1, x#2)
│ │ │  
│ │ │ -o57 = ++
│ │ │ -      ++
│ │ │ +      +-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                       2                             2   2             2                      2                  2                             2   2             2                             2   3                2         2        2        2      3     2               2         2       2        2      3 |
│ │ │ +o57 = |ideal (a*c - 13b*c + 2c  - 15a*d - 23b*d - 4c*d - 21d , b  + 47b*c + 9c  + 19b*d - 22c*d - 32d , a*b + 7b*c + 16c  - 42a*d - 45b*d - 24c*d - 3d , a  + 16b*c + 2c  - 26a*d + 5b*d - 29c*d - 31d , c  + 21b*c*d - 47c d + 38a*d  + 21b*d  - 37c*d  + 33d , b*c  - 9b*c*d - 29c d + 17a*d  + 2b*d  + 26c*d  - 17d )|
│ │ │ +      +-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  
│ │ │  i58 : pt0 = randomPointOnRationalVariety(compsL441_0)
│ │ │  
│ │ │ -o58 = | 32 -46 33 -7 -2 -29 -20 10 -23 -26 5 -16 1 -18 -3 46 13 -21 5 -22 17
│ │ │ +o58 = | 49 0 -30 -36 -1 0 -9 17 37 29 34 13 19 8 -10 -47 21 -24 -44 42 9 46
│ │ │        -----------------------------------------------------------------------
│ │ │ -      15 -33 46 -2 -29 -23 18 -42 -2 -13 39 8 -40 -24 -22 |
│ │ │ +      15 -29 35 -40 18 -22 -21 -42 39 -2 -33 -23 -13 -18 |
│ │ │  
│ │ │                 1       36
│ │ │  o58 : Matrix kk  <-- kk
│ │ │  
│ │ │  i59 : pt1 = randomPointOnRationalVariety(compsL441_1)
│ │ │  
│ │ │ -o59 = | -8 41 28 -44 50 33 -38 33 -23 1 -2 -47 32 46 30 -22 -2 -14 27 37 15
│ │ │ +o59 = | -5 -40 -6 3 -28 -8 -25 15 15 29 26 -37 11 -14 31 14 1 -50 43 37 5 50
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -25 -15 33 -23 3 21 -18 -9 3 10 -49 0 -35 -50 32 |
│ │ │ +      10 3 -3 -35 -18 32 -7 -15 33 46 0 21 -49 3 |
│ │ │  
│ │ │                 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  - 3b*c - 26c  - 16a*d - 29b*d + 33c*d + 32d , a*b - 2b*c +
│ │ │ +              2              2                      2                   2  
│ │ │ +o60 = ideal (a  - 10b*c + 29c  + 13a*d - 30c*d + 49d , a*b + 35b*c - 44c  +
│ │ │        -----------------------------------------------------------------------
│ │ │ -        2                            2   2              2                 
│ │ │ -      5c  + 17a*d + b*d - 23c*d - 46d , b  - 24b*c + 18c  + 8a*d + 15b*d +
│ │ │ +                             2              2                             2 
│ │ │ +      9a*d + 19b*d + 37c*d, b  - 13b*c - 22c  - 33a*d + 46b*d - 47c*d - 9d ,
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 2                   2                             2     2  
│ │ │ -      46c*d - 20d , a*c + 39b*c - 29c  - 42a*d - 22b*d - 18c*d - 2d , b*c  -
│ │ │ +                      2                           2     2                2   
│ │ │ +      a*c - 2b*c - 40c  - 21a*d + 42b*d + 8c*d - d , b*c  - 42b*c*d + 15c d +
│ │ │        -----------------------------------------------------------------------
│ │ │ -                  2         2        2       2     3   3                2   
│ │ │ -      2b*c*d - 33c d - 23a*d  + 13b*d  + 5c*d  - 7d , c  - 22b*c*d - 13c d -
│ │ │ +           2        2        2      3   3                2         2        2
│ │ │ +      18a*d  + 21b*d  + 34c*d  - 36d , c  - 18b*c*d + 39c d - 23a*d  - 29b*d 
│ │ │        -----------------------------------------------------------------------
│ │ │ -           2        2        2      3
│ │ │ -      40a*d  + 46b*d  - 21c*d  + 10d )
│ │ │ +             2      3
│ │ │ +      - 24c*d  + 17d )
│ │ │  
│ │ │  o60 : Ideal of S
│ │ │  
│ │ │  i61 : I1 = sub(sub(F, (vars ring F) | sub(pt1, ring F)), S)
│ │ │  
│ │ │ -              2            2                             2               
│ │ │ -o61 = ideal (a  + 30b*c + c  - 47a*d + 33b*d + 28c*d - 8d , a*b - 23b*c +
│ │ │ +              2              2                           2                  2
│ │ │ +o61 = ideal (a  + 31b*c + 29c  - 37a*d - 8b*d - 6c*d - 5d , a*b - 3b*c + 43c 
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                              2   2              2                
│ │ │ -      27c  + 15a*d + 32b*d - 23c*d + 41d , b  - 50b*c - 18c  - 25b*d - 22c*d
│ │ │ +                                  2   2              2                  
│ │ │ +      + 5a*d + 11b*d + 15c*d - 40d , b  - 49b*c + 32c  + 50b*d + 14c*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          
│ │ │ +      25d , a*c + 46b*c - 35c  - 7a*d + 37b*d - 14c*d - 28d , b*c  - 15b*c*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      2        2     3   3               2         2  
│ │ │ +      + 10c d - 18a*d  + b*d  + 26c*d  + 3d , c  + 3b*c*d + 33c d + 21a*d  +
│ │ │        -----------------------------------------------------------------------
│ │ │ -             2        2      3
│ │ │ -      + 33b*d  - 14c*d  + 33d )
│ │ │ +          2        2      3
│ │ │ +      3b*d  - 50c*d  + 15d )
│ │ │  
│ │ │  o61 : Ideal of S
│ │ │  
│ │ │  i62 : betti res I0
│ │ │  
│ │ │               0 1 2 3
│ │ │  o62 = total: 1 6 8 3
│ │ │ @@ -531,36 +537,33 @@
│ │ │            1: . 4 4 1
│ │ │            2: . 2 4 2
│ │ │  
│ │ │  o63 : BettiTally
│ │ │  
│ │ │  i64 : netList decompose I0
│ │ │  
│ │ │ -      +------------------------------------------------------------------------------------------------+
│ │ │ -o64 = |ideal (c - 42d, b + 26d, a - 30d)                                                               |
│ │ │ -      +------------------------------------------------------------------------------------------------+
│ │ │ -      |ideal (c - 47d, b + 7d, a - 44d)                                                                |
│ │ │ -      +------------------------------------------------------------------------------------------------+
│ │ │ -      |                                     2                      2   2      2                      2 |
│ │ │ -      |ideal (a + 39b - 29c - 24d, b*c - 15c  - 29b*d - 38c*d + 16d , b  - 39c  + 17b*d - 28c*d - 50d )|
│ │ │ -      +------------------------------------------------------------------------------------------------+
│ │ │ +      +---------------------------------------------------+
│ │ │ +o64 = |ideal (c - 21d, b - 26d, a - 50d)                  |
│ │ │ +      +---------------------------------------------------+
│ │ │ +      |ideal (c - 49d, b - 33d, a - 30d)                  |
│ │ │ +      +---------------------------------------------------+
│ │ │ +      |ideal (c + 41d, b + 33d, a - 35d)                  |
│ │ │ +      +---------------------------------------------------+
│ │ │ +      |ideal (c + d, b + 40d, a - 5d)                     |
│ │ │ +      +---------------------------------------------------+
│ │ │ +      |                                    2            2 |
│ │ │ +      |ideal (b + c - 47d, a - 38c - 17d, c  + 4c*d - 8d )|
│ │ │ +      +---------------------------------------------------+
│ │ │  
│ │ │  i65 : netList decompose I1
│ │ │  
│ │ │ -      +---------------------------------+
│ │ │ -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)|
│ │ │ -      +---------------------------------+
│ │ │ +      +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                        2                             2   2              2                      2                  2                             2   2              2                           2   3               2         2       2        2      3     2                2         2      2        2     3 |
│ │ │ +o65 = |ideal (a*c + 46b*c - 35c  - 7a*d + 37b*d - 14c*d - 28d , b  - 49b*c + 32c  + 50b*d + 14c*d - 25d , a*b - 3b*c + 43c  + 5a*d + 11b*d + 15c*d - 40d , a  + 31b*c + 29c  - 37a*d - 8b*d - 6c*d - 5d , c  + 3b*c*d + 33c d + 21a*d  + 3b*d  - 50c*d  + 15d , b*c  - 15b*c*d + 10c d - 18a*d  + b*d  + 26c*d  + 3d )|
│ │ │ +      +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  
│ │ │  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
│ │ │ @@ -344,15 +344,15 @@
│ │ │              
    		
│ │ │                
    i22 : assert(dim L == 18)
    
│ │ │              
    
│ │ │                
    i23 : elapsedTime isPrime L
│ │ │ - -- 2.47531s elapsed
│ │ │ + -- 2.30383s elapsed
│ │ │  
│ │ │  o23 = true
    
│ │ │              
    
│ │ │          
    
│ │ │            
    The Schreyer resolution and minimal Betti numbers
    
│ │ │ @@ -556,15 +556,15 @@
│ │ │  
│ │ │  o39 : Ideal of T    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i40 : elapsedTime compsL441 = decompose L441;
│ │ │ - -- 1.52419s elapsed
    
│ │ │ + -- 1.25749s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i41 : #compsL441
│ │ │  
│ │ │  o41 = 2
    
│ │ │ @@ -591,40 +591,40 @@
│ │ │            
    Both components are rational, and here are random points, one on each component:
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i44 : pta = randomPointOnRationalVariety compsL441_0
│ │ │  
│ │ │ -o44 = | 22 -10 -8 -1 34 44 -21 -1 25 -41 6 -11 -50 -50 43 -28 -6 45 -28 22 42
│ │ │ +o44 = | 32 -41 22 15 22 -46 43 42 -27 -27 -13 10 -24 19 -25 48 31 10 41 49 39
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -29 -32 -28 5 -10 34 15 19 37 26 49 19 5 10 18 |
│ │ │ +      -28 -29 -10 -48 5 15 18 45 19 49 37 -32 34 26 -50 |
│ │ │  
│ │ │                 1       36
│ │ │  o44 : Matrix kk  <-- kk
    
│ │ │              
    
│ │ │                
    i45 : Fa = sub(F, (vars S) | pta)
│ │ │  
│ │ │ -              2              2                             2              
│ │ │ -o45 = ideal (a  + 43b*c - 41c  - 11a*d + 44b*d - 8c*d + 22d , a*b + 5b*c -
│ │ │ +              2              2                              2               
│ │ │ +o45 = ideal (a  - 25b*c - 27c  + 10a*d - 46b*d + 22c*d + 32d , a*b - 48b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │           2                              2   2              2                
│ │ │ -      28c  + 42a*d - 50b*d + 25c*d - 10d , b  + 10b*c + 15c  + 19a*d - 29b*d
│ │ │ +      41c  + 39a*d - 24b*d - 27c*d - 41d , b  + 26b*c + 18c  - 32a*d - 28b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2                   2                              2     2
│ │ │ -      - 28c*d - 21d , a*c + 49b*c - 10c  + 19a*d + 22b*d - 50c*d + 34d , b*c 
│ │ │ +                   2                  2                              2     2
│ │ │ +      + 48c*d + 43d , a*c + 37b*c + 5c  + 45a*d + 49b*d + 19c*d + 22d , b*c 
│ │ │        -----------------------------------------------------------------------
│ │ │ -                     2         2       2       2    3   3                2   
│ │ │ -      + 37b*c*d - 32c d + 34a*d  - 6b*d  + 6c*d  - d , c  + 18b*c*d + 26c d +
│ │ │ +                     2         2        2        2      3   3            
│ │ │ +      + 19b*c*d - 29c d + 15a*d  + 31b*d  - 13c*d  + 15d , c  - 50b*c*d +
│ │ │        -----------------------------------------------------------------------
│ │ │ -          2        2        2    3
│ │ │ -      5a*d  - 28b*d  + 45c*d  - d )
│ │ │ +         2         2        2        2      3
│ │ │ +      49c d + 34a*d  - 10b*d  + 10c*d  + 42d )
│ │ │  
│ │ │  o45 : Ideal of S
    
│ │ │              
    
│ │ │                
    i46 : betti res Fa
│ │ │ @@ -638,104 +638,106 @@
│ │ │  o46 : BettiTally
    
│ │ │              
    
│ │ │                
    i47 : netList decompose Fa -- this one is 5 points on a plane, and another point
│ │ │  
│ │ │ -      +-------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -o47 = |ideal (c + 19d, b - 37d, a)                                                                                                                                        |
│ │ │ -      +-------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                             2              2                      2   3                2         2       2     3     2                2         2        2      3 |
│ │ │ -      |ideal (a + 49b - 10c + 39d, b  + 10b*c + 15c  + 50b*d - 40c*d + 46d , c  + 18b*c*d + 26c d + 30b*d  - 6c*d  + 6d , b*c  + 37b*c*d - 32c d + 45b*d  + 43c*d  - 14d )|
│ │ │ -      +-------------------------------------------------------------------------------------------------------------------------------------------------------------------+
    
│ │ │ +      +----------------------------------------------------------------------------------------------+
│ │ │ +o47 = |ideal (c + 45d, b - 34d, a - 35d)                                                             |
│ │ │ +      +----------------------------------------------------------------------------------------------+
│ │ │ +      |ideal (c + 18d, b - 12d, a + 47d)                                                             |
│ │ │ +      +----------------------------------------------------------------------------------------------+
│ │ │ +      |                                   2                      2   2      2                      2 |
│ │ │ +      |ideal (a + 37b + 5c - 4d, b*c - 13c  + 45b*d - 28c*d - 29d , b  - 48c  - 14b*d + 27c*d - 38d )|
│ │ │ +      +----------------------------------------------------------------------------------------------+
│ │ │              
    
│ │ │                
    i48 : CFa = minimalPrimes Fa
│ │ │  
│ │ │ -                                                                 2          
│ │ │ -o48 = {ideal (c + 19d, b - 37d, a), ideal (a + 49b - 10c + 39d, b  + 10b*c +
│ │ │ +                                                                            
│ │ │ +o48 = {ideal (c + 45d, b - 34d, a - 35d), ideal (c + 18d, b - 12d, a + 47d),
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                      2   3                2         2       2  
│ │ │ -      15c  + 50b*d - 40c*d + 46d , c  + 18b*c*d + 26c d + 30b*d  - 6c*d  +
│ │ │ +                                         2                      2   2      2
│ │ │ +      ideal (a + 37b + 5c - 4d, b*c - 13c  + 45b*d - 28c*d - 29d , b  - 48c 
│ │ │        -----------------------------------------------------------------------
│ │ │ -        3     2                2         2        2      3
│ │ │ -      6d , b*c  + 37b*c*d - 32c d + 45b*d  + 43c*d  - 14d )}
│ │ │ +                           2
│ │ │ +      - 14b*d + 27c*d - 38d )}
│ │ │  
│ │ │  o48 : List
    
│ │ │              
    
│ │ │                
    i49 : lin = CFa_1_0 -- a linear form, defining a plane.
│ │ │  
│ │ │ -o49 = a + 49b - 10c + 39d
│ │ │ +o49 = c + 18d
│ │ │  
│ │ │  o49 : S
    
│ │ │              
    
│ │ │                
    i50 : CFa/degree
│ │ │  
│ │ │ -o50 = {1, 5}
│ │ │ +o50 = {1, 1, 4}
│ │ │  
│ │ │  o50 : List
    
│ │ │              
    
│ │ │                
    i51 : CFa/(I -> lin % I == 0) -- so 5 points on the plane.
│ │ │  
│ │ │ -o51 = {false, true}
│ │ │ +o51 = {false, true, false}
│ │ │  
│ │ │  o51 : List
    
│ │ │              
    
│ │ │                
    i52 : degree(Fa : (Fa : lin))  -- somewhat simpler(?) way to see the ideal of the 5 points
│ │ │  
│ │ │ -o52 = 5
    
│ │ │ +o52 = 1
│ │ │              
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i53 : ptb = randomPointOnRationalVariety compsL441_1
│ │ │  
│ │ │ -o53 = | 27 12 -34 9 -19 -43 -32 27 40 45 -13 29 -41 -13 22 -49 -4 -4 9 -23 43
│ │ │ +o53 = | -31 -3 -29 -17 -21 5 -32 33 -24 2 26 -26 -45 -4 16 -22 2 -37 16 -23
│ │ │        -----------------------------------------------------------------------
│ │ │ -      18 -9 -47 43 21 38 17 -20 21 -29 47 0 2 -37 9 |
│ │ │ +      -42 19 -29 21 7 2 17 9 -15 -9 -47 -13 0 38 47 21 |
│ │ │  
│ │ │                 1       36
│ │ │  o53 : Matrix kk  <-- kk
    
│ │ │              
    
│ │ │                
    i54 : Fb = sub(F, (vars S) | ptb)
│ │ │  
│ │ │ -              2              2                              2               
│ │ │ -o54 = ideal (a  + 22b*c + 45c  + 29a*d - 43b*d - 34c*d + 27d , a*b + 43b*c +
│ │ │ +              2             2                             2              
│ │ │ +o54 = ideal (a  + 16b*c + 2c  - 26a*d + 5b*d - 29c*d - 31d , a*b + 7b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -        2                              2   2              2                  
│ │ │ -      9c  + 43a*d - 41b*d + 40c*d + 12d , b  - 37b*c + 17c  + 18b*d - 49c*d -
│ │ │ +         2                             2   2             2                  
│ │ │ +      16c  - 42a*d - 45b*d - 24c*d - 3d , b  + 47b*c + 9c  + 19b*d - 22c*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           
│ │ │ +      32d , a*c - 13b*c + 2c  - 15a*d - 23b*d - 4c*d - 21d , b*c  - 9b*c*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       2        2      3   3                2         2  
│ │ │ +      29c d + 17a*d  + 2b*d  + 26c*d  - 17d , c  + 21b*c*d - 47c d + 38a*d  +
│ │ │        -----------------------------------------------------------------------
│ │ │ -           2       2      3
│ │ │ -      47b*d  - 4c*d  + 27d )
│ │ │ +           2        2      3
│ │ │ +      21b*d  - 37c*d  + 33d )
│ │ │  
│ │ │  o54 : Ideal of S
    
│ │ │              
    
│ │ │                
    i55 : betti res Fb
│ │ │ @@ -749,102 +751,106 @@
│ │ │  o55 : BettiTally
    
│ │ │              
    
│ │ │                
    i56 : netList decompose Fb --
│ │ │  
│ │ │ -      +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                                      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 )|
│ │ │ -      +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
    
│ │ │ +      +---------------------------------------------------------------+
│ │ │ +o56 = |ideal (c - 17d, b - 26d, a - 49d)                              |
│ │ │ +      +---------------------------------------------------------------+
│ │ │ +      |                                     3      2         2      3 |
│ │ │ +      |ideal (b + 39c - 21d, a + 4c - 27d, c  + 43c d + 39c*d  - 15d )|
│ │ │ +      +---------------------------------------------------------------+
│ │ │ +      |                                    2             2            |
│ │ │ +      |ideal (b + 8c + 40d, a + 5c - 33d, c  + 4c*d + 45d )           |
│ │ │ +      +---------------------------------------------------------------+
│ │ │              
    
│ │ │                
    i57 : netList for x in subsets(decompose Fb, 3) list intersect(x#0, x#1, x#2)
│ │ │  
│ │ │ -o57 = ++
│ │ │ -      ++
    
│ │ │ +      +-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                       2                             2   2             2                      2                  2                             2   2             2                             2   3                2         2        2        2      3     2               2         2       2        2      3 |
│ │ │ +o57 = |ideal (a*c - 13b*c + 2c  - 15a*d - 23b*d - 4c*d - 21d , b  + 47b*c + 9c  + 19b*d - 22c*d - 32d , a*b + 7b*c + 16c  - 42a*d - 45b*d - 24c*d - 3d , a  + 16b*c + 2c  - 26a*d + 5b*d - 29c*d - 31d , c  + 21b*c*d - 47c d + 38a*d  + 21b*d  - 37c*d  + 33d , b*c  - 9b*c*d - 29c d + 17a*d  + 2b*d  + 26c*d  - 17d )|
│ │ │ +      +-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │              
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i58 : pt0 = randomPointOnRationalVariety(compsL441_0)
│ │ │  
│ │ │ -o58 = | 32 -46 33 -7 -2 -29 -20 10 -23 -26 5 -16 1 -18 -3 46 13 -21 5 -22 17
│ │ │ +o58 = | 49 0 -30 -36 -1 0 -9 17 37 29 34 13 19 8 -10 -47 21 -24 -44 42 9 46
│ │ │        -----------------------------------------------------------------------
│ │ │ -      15 -33 46 -2 -29 -23 18 -42 -2 -13 39 8 -40 -24 -22 |
│ │ │ +      15 -29 35 -40 18 -22 -21 -42 39 -2 -33 -23 -13 -18 |
│ │ │  
│ │ │                 1       36
│ │ │  o58 : Matrix kk  <-- kk
    
│ │ │              
    
│ │ │                
    i59 : pt1 = randomPointOnRationalVariety(compsL441_1)
│ │ │  
│ │ │ -o59 = | -8 41 28 -44 50 33 -38 33 -23 1 -2 -47 32 46 30 -22 -2 -14 27 37 15
│ │ │ +o59 = | -5 -40 -6 3 -28 -8 -25 15 15 29 26 -37 11 -14 31 14 1 -50 43 37 5 50
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -25 -15 33 -23 3 21 -18 -9 3 10 -49 0 -35 -50 32 |
│ │ │ +      10 3 -3 -35 -18 32 -7 -15 33 46 0 21 -49 3 |
│ │ │  
│ │ │                 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  - 3b*c - 26c  - 16a*d - 29b*d + 33c*d + 32d , a*b - 2b*c +
│ │ │ +              2              2                      2                   2  
│ │ │ +o60 = ideal (a  - 10b*c + 29c  + 13a*d - 30c*d + 49d , a*b + 35b*c - 44c  +
│ │ │        -----------------------------------------------------------------------
│ │ │ -        2                            2   2              2                 
│ │ │ -      5c  + 17a*d + b*d - 23c*d - 46d , b  - 24b*c + 18c  + 8a*d + 15b*d +
│ │ │ +                             2              2                             2 
│ │ │ +      9a*d + 19b*d + 37c*d, b  - 13b*c - 22c  - 33a*d + 46b*d - 47c*d - 9d ,
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 2                   2                             2     2  
│ │ │ -      46c*d - 20d , a*c + 39b*c - 29c  - 42a*d - 22b*d - 18c*d - 2d , b*c  -
│ │ │ +                      2                           2     2                2   
│ │ │ +      a*c - 2b*c - 40c  - 21a*d + 42b*d + 8c*d - d , b*c  - 42b*c*d + 15c d +
│ │ │        -----------------------------------------------------------------------
│ │ │ -                  2         2        2       2     3   3                2   
│ │ │ -      2b*c*d - 33c d - 23a*d  + 13b*d  + 5c*d  - 7d , c  - 22b*c*d - 13c d -
│ │ │ +           2        2        2      3   3                2         2        2
│ │ │ +      18a*d  + 21b*d  + 34c*d  - 36d , c  - 18b*c*d + 39c d - 23a*d  - 29b*d 
│ │ │        -----------------------------------------------------------------------
│ │ │ -           2        2        2      3
│ │ │ -      40a*d  + 46b*d  - 21c*d  + 10d )
│ │ │ +             2      3
│ │ │ +      - 24c*d  + 17d )
│ │ │  
│ │ │  o60 : Ideal of S
    
│ │ │              
    
│ │ │                
    i61 : I1 = sub(sub(F, (vars ring F) | sub(pt1, ring F)), S)
│ │ │  
│ │ │ -              2            2                             2               
│ │ │ -o61 = ideal (a  + 30b*c + c  - 47a*d + 33b*d + 28c*d - 8d , a*b - 23b*c +
│ │ │ +              2              2                           2                  2
│ │ │ +o61 = ideal (a  + 31b*c + 29c  - 37a*d - 8b*d - 6c*d - 5d , a*b - 3b*c + 43c 
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                              2   2              2                
│ │ │ -      27c  + 15a*d + 32b*d - 23c*d + 41d , b  - 50b*c - 18c  - 25b*d - 22c*d
│ │ │ +                                  2   2              2                  
│ │ │ +      + 5a*d + 11b*d + 15c*d - 40d , b  - 49b*c + 32c  + 50b*d + 14c*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          
│ │ │ +      25d , a*c + 46b*c - 35c  - 7a*d + 37b*d - 14c*d - 28d , b*c  - 15b*c*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      2        2     3   3               2         2  
│ │ │ +      + 10c d - 18a*d  + b*d  + 26c*d  + 3d , c  + 3b*c*d + 33c d + 21a*d  +
│ │ │        -----------------------------------------------------------------------
│ │ │ -             2        2      3
│ │ │ -      + 33b*d  - 14c*d  + 33d )
│ │ │ +          2        2      3
│ │ │ +      3b*d  - 50c*d  + 15d )
│ │ │  
│ │ │  o61 : Ideal of S
    
│ │ │              
    
│ │ │                
    i62 : betti res I0
│ │ │ @@ -873,39 +879,36 @@
│ │ │            
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i64 : netList decompose I0
│ │ │  
│ │ │ -      +------------------------------------------------------------------------------------------------+
│ │ │ -o64 = |ideal (c - 42d, b + 26d, a - 30d)                                                               |
│ │ │ -      +------------------------------------------------------------------------------------------------+
│ │ │ -      |ideal (c - 47d, b + 7d, a - 44d)                                                                |
│ │ │ -      +------------------------------------------------------------------------------------------------+
│ │ │ -      |                                     2                      2   2      2                      2 |
│ │ │ -      |ideal (a + 39b - 29c - 24d, b*c - 15c  - 29b*d - 38c*d + 16d , b  - 39c  + 17b*d - 28c*d - 50d )|
│ │ │ -      +------------------------------------------------------------------------------------------------+
    
│ │ │ +      +---------------------------------------------------+
│ │ │ +o64 = |ideal (c - 21d, b - 26d, a - 50d)                  |
│ │ │ +      +---------------------------------------------------+
│ │ │ +      |ideal (c - 49d, b - 33d, a - 30d)                  |
│ │ │ +      +---------------------------------------------------+
│ │ │ +      |ideal (c + 41d, b + 33d, a - 35d)                  |
│ │ │ +      +---------------------------------------------------+
│ │ │ +      |ideal (c + d, b + 40d, a - 5d)                     |
│ │ │ +      +---------------------------------------------------+
│ │ │ +      |                                    2            2 |
│ │ │ +      |ideal (b + c - 47d, a - 38c - 17d, c  + 4c*d - 8d )|
│ │ │ +      +---------------------------------------------------+
│ │ │              
    
│ │ │                
    i65 : netList decompose I1
│ │ │  
│ │ │ -      +---------------------------------+
│ │ │ -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)|
│ │ │ -      +---------------------------------+
    
│ │ │ +      +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                        2                             2   2              2                      2                  2                             2   2              2                           2   3               2         2       2        2      3     2                2         2      2        2     3 |
│ │ │ +o65 = |ideal (a*c + 46b*c - 35c  - 7a*d + 37b*d - 14c*d - 28d , b  - 49b*c + 32c  + 50b*d + 14c*d - 25d , a*b - 3b*c + 43c  + 5a*d + 11b*d + 15c*d - 40d , a  + 31b*c + 29c  - 37a*d - 8b*d - 6c*d - 5d , c  + 3b*c*d + 33c d + 21a*d  + 3b*d  - 50c*d  + 15d , b*c  - 15b*c*d + 10c d - 18a*d  + b*d  + 26c*d  + 3d )|
│ │ │ +      +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │              
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i66 : L430 = (trim minors(2, M1)) + groebnerStratum F;
│ │ │ ├── html2text {}
│ │ │ │ @@ -251,15 +251,15 @@
│ │ │ │        |      31         33        32       34        35        36    |
│ │ │ │        +--------------------------------------------------------------+
│ │ │ │  i21 : L = trim groebnerStratum F;
│ │ │ │  
│ │ │ │  o21 : Ideal of T
│ │ │ │  i22 : assert(dim L == 18)
│ │ │ │  i23 : elapsedTime isPrime L
│ │ │ │ - -- 2.47531s elapsed
│ │ │ │ + -- 2.30383s 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
│ │ │ │ @@ -415,15 +415,15 @@
│ │ │ │  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;
│ │ │ │ - -- 1.52419s elapsed
│ │ │ │ + -- 1.25749s elapsed
│ │ │ │  i41 : #compsL441
│ │ │ │  
│ │ │ │  o41 = 2
│ │ │ │  i42 : compsL441/dim -- two components, of dimensions 14 and 16.
│ │ │ │  
│ │ │ │  o42 = {16, 14}
│ │ │ │  
│ │ │ │ @@ -431,36 +431,36 @@
│ │ │ │  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 = | 22 -10 -8 -1 34 44 -21 -1 25 -41 6 -11 -50 -50 43 -28 -6 45 -28 22 42
│ │ │ │ +o44 = | 32 -41 22 15 22 -46 43 42 -27 -27 -13 10 -24 19 -25 48 31 10 41 49 39
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      -29 -32 -28 5 -10 34 15 19 37 26 49 19 5 10 18 |
│ │ │ │ +      -28 -29 -10 -48 5 15 18 45 19 49 37 -32 34 26 -50 |
│ │ │ │  
│ │ │ │                 1       36
│ │ │ │  o44 : Matrix kk  <-- kk
│ │ │ │  i45 : Fa = sub(F, (vars S) | pta)
│ │ │ │  
│ │ │ │ -              2              2                             2
│ │ │ │ -o45 = ideal (a  + 43b*c - 41c  - 11a*d + 44b*d - 8c*d + 22d , a*b + 5b*c -
│ │ │ │ +              2              2                              2
│ │ │ │ +o45 = ideal (a  - 25b*c - 27c  + 10a*d - 46b*d + 22c*d + 32d , a*b - 48b*c +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │           2                              2   2              2
│ │ │ │ -      28c  + 42a*d - 50b*d + 25c*d - 10d , b  + 10b*c + 15c  + 19a*d - 29b*d
│ │ │ │ +      41c  + 39a*d - 24b*d - 27c*d - 41d , b  + 26b*c + 18c  - 32a*d - 28b*d
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                   2                   2                              2     2
│ │ │ │ -      - 28c*d - 21d , a*c + 49b*c - 10c  + 19a*d + 22b*d - 50c*d + 34d , b*c
│ │ │ │ +                   2                  2                              2     2
│ │ │ │ +      + 48c*d + 43d , a*c + 37b*c + 5c  + 45a*d + 49b*d + 19c*d + 22d , b*c
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                     2         2       2       2    3   3                2
│ │ │ │ -      + 37b*c*d - 32c d + 34a*d  - 6b*d  + 6c*d  - d , c  + 18b*c*d + 26c d +
│ │ │ │ +                     2         2        2        2      3   3
│ │ │ │ +      + 19b*c*d - 29c d + 15a*d  + 31b*d  - 13c*d  + 15d , c  - 50b*c*d +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -          2        2        2    3
│ │ │ │ -      5a*d  - 28b*d  + 45c*d  - d )
│ │ │ │ +         2         2        2        2      3
│ │ │ │ +      49c d + 34a*d  - 10b*d  + 10c*d  + 42d )
│ │ │ │  
│ │ │ │  o45 : Ideal of S
│ │ │ │  i46 : betti res Fa
│ │ │ │  
│ │ │ │               0 1 2 3
│ │ │ │  o46 = total: 1 6 8 3
│ │ │ │            0: 1 . . .
│ │ │ │ @@ -468,172 +468,177 @@
│ │ │ │            2: . 2 4 2
│ │ │ │  
│ │ │ │  o46 : BettiTally
│ │ │ │  i47 : netList decompose Fa -- this one is 5 points on a plane, and another
│ │ │ │  point
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------------------------
│ │ │ │ -------------+
│ │ │ │ -o47 = |ideal (c + 19d, b - 37d, a)
│ │ │ │ +----------------------+
│ │ │ │ +o47 = |ideal (c + 45d, b - 34d, a - 35d)
│ │ │ │  |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------------------------
│ │ │ │ -------------+
│ │ │ │ -      |                             2              2                      2   3
│ │ │ │ -2         2       2     3     2                2         2        2      3 |
│ │ │ │ -      |ideal (a + 49b - 10c + 39d, b  + 10b*c + 15c  + 50b*d - 40c*d + 46d , c
│ │ │ │ -+ 18b*c*d + 26c d + 30b*d  - 6c*d  + 6d , b*c  + 37b*c*d - 32c d + 45b*d  +
│ │ │ │ -43c*d  - 14d )|
│ │ │ │ +----------------------+
│ │ │ │ +      |ideal (c + 18d, b - 12d, a + 47d)
│ │ │ │ +|
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------------------------
│ │ │ │ -------------+
│ │ │ │ +----------------------+
│ │ │ │ +      |                                   2                      2   2      2
│ │ │ │ +2 |
│ │ │ │ +      |ideal (a + 37b + 5c - 4d, b*c - 13c  + 45b*d - 28c*d - 29d , b  - 48c  -
│ │ │ │ +14b*d + 27c*d - 38d )|
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +----------------------+
│ │ │ │  i48 : CFa = minimalPrimes Fa
│ │ │ │  
│ │ │ │ -                                                                 2
│ │ │ │ -o48 = {ideal (c + 19d, b - 37d, a), ideal (a + 49b - 10c + 39d, b  + 10b*c +
│ │ │ │ +
│ │ │ │ +o48 = {ideal (c + 45d, b - 34d, a - 35d), ideal (c + 18d, b - 12d, a + 47d),
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -         2                      2   3                2         2       2
│ │ │ │ -      15c  + 50b*d - 40c*d + 46d , c  + 18b*c*d + 26c d + 30b*d  - 6c*d  +
│ │ │ │ +                                         2                      2   2      2
│ │ │ │ +      ideal (a + 37b + 5c - 4d, b*c - 13c  + 45b*d - 28c*d - 29d , b  - 48c
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -        3     2                2         2        2      3
│ │ │ │ -      6d , b*c  + 37b*c*d - 32c d + 45b*d  + 43c*d  - 14d )}
│ │ │ │ +                           2
│ │ │ │ +      - 14b*d + 27c*d - 38d )}
│ │ │ │  
│ │ │ │  o48 : List
│ │ │ │  i49 : lin = CFa_1_0 -- a linear form, defining a plane.
│ │ │ │  
│ │ │ │ -o49 = a + 49b - 10c + 39d
│ │ │ │ +o49 = c + 18d
│ │ │ │  
│ │ │ │  o49 : S
│ │ │ │  i50 : CFa/degree
│ │ │ │  
│ │ │ │ -o50 = {1, 5}
│ │ │ │ +o50 = {1, 1, 4}
│ │ │ │  
│ │ │ │  o50 : List
│ │ │ │  i51 : CFa/(I -> lin % I == 0) -- so 5 points on the plane.
│ │ │ │  
│ │ │ │ -o51 = {false, true}
│ │ │ │ +o51 = {false, true, false}
│ │ │ │  
│ │ │ │  o51 : List
│ │ │ │  i52 : degree(Fa : (Fa : lin))  -- somewhat simpler(?) way to see the ideal of
│ │ │ │  the 5 points
│ │ │ │  
│ │ │ │ -o52 = 5
│ │ │ │ +o52 = 1
│ │ │ │  i53 : ptb = randomPointOnRationalVariety compsL441_1
│ │ │ │  
│ │ │ │ -o53 = | 27 12 -34 9 -19 -43 -32 27 40 45 -13 29 -41 -13 22 -49 -4 -4 9 -23 43
│ │ │ │ +o53 = | -31 -3 -29 -17 -21 5 -32 33 -24 2 26 -26 -45 -4 16 -22 2 -37 16 -23
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      18 -9 -47 43 21 38 17 -20 21 -29 47 0 2 -37 9 |
│ │ │ │ +      -42 19 -29 21 7 2 17 9 -15 -9 -47 -13 0 38 47 21 |
│ │ │ │  
│ │ │ │                 1       36
│ │ │ │  o53 : Matrix kk  <-- kk
│ │ │ │  i54 : Fb = sub(F, (vars S) | ptb)
│ │ │ │  
│ │ │ │ -              2              2                              2
│ │ │ │ -o54 = ideal (a  + 22b*c + 45c  + 29a*d - 43b*d - 34c*d + 27d , a*b + 43b*c +
│ │ │ │ +              2             2                             2
│ │ │ │ +o54 = ideal (a  + 16b*c + 2c  - 26a*d + 5b*d - 29c*d - 31d , a*b + 7b*c +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -        2                              2   2              2
│ │ │ │ -      9c  + 43a*d - 41b*d + 40c*d + 12d , b  - 37b*c + 17c  + 18b*d - 49c*d -
│ │ │ │ +         2                             2   2             2
│ │ │ │ +      16c  - 42a*d - 45b*d - 24c*d - 3d , b  + 47b*c + 9c  + 19b*d - 22c*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
│ │ │ │ +      32d , a*c - 13b*c + 2c  - 15a*d - 23b*d - 4c*d - 21d , b*c  - 9b*c*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       2        2      3   3                2         2
│ │ │ │ +      29c d + 17a*d  + 2b*d  + 26c*d  - 17d , c  + 21b*c*d - 47c d + 38a*d  +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -           2       2      3
│ │ │ │ -      47b*d  - 4c*d  + 27d )
│ │ │ │ +           2        2      3
│ │ │ │ +      21b*d  - 37c*d  + 33d )
│ │ │ │  
│ │ │ │  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 - 17d, b - 26d, a - 49d)                              |
│ │ │ │ +      +---------------------------------------------------------------+
│ │ │ │ +      |                                     3      2         2      3 |
│ │ │ │ +      |ideal (b + 39c - 21d, a + 4c - 27d, c  + 43c d + 39c*d  - 15d )|
│ │ │ │ +      +---------------------------------------------------------------+
│ │ │ │ +      |                                    2             2            |
│ │ │ │ +      |ideal (b + 8c + 40d, a + 5c - 33d, c  + 4c*d + 45d )           |
│ │ │ │ +      +---------------------------------------------------------------+
│ │ │ │ +i57 : netList for x in subsets(decompose Fb, 3) list intersect(x#0, x#1, x#2)
│ │ │ │ +
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │ ---------------------------------------------------------------+
│ │ │ │ -      |                                      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 )|
│ │ │ │ +---------------------------------------------------------------------------+
│ │ │ │ +      |                       2                             2   2             2
│ │ │ │ +2                  2                             2   2             2
│ │ │ │ +2   3                2         2        2        2      3     2               2
│ │ │ │ +2       2        2      3 |
│ │ │ │ +o57 = |ideal (a*c - 13b*c + 2c  - 15a*d - 23b*d - 4c*d - 21d , b  + 47b*c + 9c
│ │ │ │ ++ 19b*d - 22c*d - 32d , a*b + 7b*c + 16c  - 42a*d - 45b*d - 24c*d - 3d , a  +
│ │ │ │ +16b*c + 2c  - 26a*d + 5b*d - 29c*d - 31d , c  + 21b*c*d - 47c d + 38a*d  +
│ │ │ │ +21b*d  - 37c*d  + 33d , b*c  - 9b*c*d - 29c d + 17a*d  + 2b*d  + 26c*d  - 17d
│ │ │ │ +)|
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │ ---------------------------------------------------------------+
│ │ │ │ -i57 : netList for x in subsets(decompose Fb, 3) list intersect(x#0, x#1, x#2)
│ │ │ │ -
│ │ │ │ -o57 = ++
│ │ │ │ -      ++
│ │ │ │ +-------------------------------------------------------------------------------
│ │ │ │ +---------------------------------------------------------------------------+
│ │ │ │  i58 : pt0 = randomPointOnRationalVariety(compsL441_0)
│ │ │ │  
│ │ │ │ -o58 = | 32 -46 33 -7 -2 -29 -20 10 -23 -26 5 -16 1 -18 -3 46 13 -21 5 -22 17
│ │ │ │ +o58 = | 49 0 -30 -36 -1 0 -9 17 37 29 34 13 19 8 -10 -47 21 -24 -44 42 9 46
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      15 -33 46 -2 -29 -23 18 -42 -2 -13 39 8 -40 -24 -22 |
│ │ │ │ +      15 -29 35 -40 18 -22 -21 -42 39 -2 -33 -23 -13 -18 |
│ │ │ │  
│ │ │ │                 1       36
│ │ │ │  o58 : Matrix kk  <-- kk
│ │ │ │  i59 : pt1 = randomPointOnRationalVariety(compsL441_1)
│ │ │ │  
│ │ │ │ -o59 = | -8 41 28 -44 50 33 -38 33 -23 1 -2 -47 32 46 30 -22 -2 -14 27 37 15
│ │ │ │ +o59 = | -5 -40 -6 3 -28 -8 -25 15 15 29 26 -37 11 -14 31 14 1 -50 43 37 5 50
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      -25 -15 33 -23 3 21 -18 -9 3 10 -49 0 -35 -50 32 |
│ │ │ │ +      10 3 -3 -35 -18 32 -7 -15 33 46 0 21 -49 3 |
│ │ │ │  
│ │ │ │                 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  - 3b*c - 26c  - 16a*d - 29b*d + 33c*d + 32d , a*b - 2b*c +
│ │ │ │ +              2              2                      2                   2
│ │ │ │ +o60 = ideal (a  - 10b*c + 29c  + 13a*d - 30c*d + 49d , a*b + 35b*c - 44c  +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -        2                            2   2              2
│ │ │ │ -      5c  + 17a*d + b*d - 23c*d - 46d , b  - 24b*c + 18c  + 8a*d + 15b*d +
│ │ │ │ +                             2              2                             2
│ │ │ │ +      9a*d + 19b*d + 37c*d, b  - 13b*c - 22c  - 33a*d + 46b*d - 47c*d - 9d ,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                 2                   2                             2     2
│ │ │ │ -      46c*d - 20d , a*c + 39b*c - 29c  - 42a*d - 22b*d - 18c*d - 2d , b*c  -
│ │ │ │ +                      2                           2     2                2
│ │ │ │ +      a*c - 2b*c - 40c  - 21a*d + 42b*d + 8c*d - d , b*c  - 42b*c*d + 15c d +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                  2         2        2       2     3   3                2
│ │ │ │ -      2b*c*d - 33c d - 23a*d  + 13b*d  + 5c*d  - 7d , c  - 22b*c*d - 13c d -
│ │ │ │ +           2        2        2      3   3                2         2        2
│ │ │ │ +      18a*d  + 21b*d  + 34c*d  - 36d , c  - 18b*c*d + 39c d - 23a*d  - 29b*d
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -           2        2        2      3
│ │ │ │ -      40a*d  + 46b*d  - 21c*d  + 10d )
│ │ │ │ +             2      3
│ │ │ │ +      - 24c*d  + 17d )
│ │ │ │  
│ │ │ │  o60 : Ideal of S
│ │ │ │  i61 : I1 = sub(sub(F, (vars ring F) | sub(pt1, ring F)), S)
│ │ │ │  
│ │ │ │ -              2            2                             2
│ │ │ │ -o61 = ideal (a  + 30b*c + c  - 47a*d + 33b*d + 28c*d - 8d , a*b - 23b*c +
│ │ │ │ +              2              2                           2                  2
│ │ │ │ +o61 = ideal (a  + 31b*c + 29c  - 37a*d - 8b*d - 6c*d - 5d , a*b - 3b*c + 43c
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -         2                              2   2              2
│ │ │ │ -      27c  + 15a*d + 32b*d - 23c*d + 41d , b  - 50b*c - 18c  - 25b*d - 22c*d
│ │ │ │ +                                  2   2              2
│ │ │ │ +      + 5a*d + 11b*d + 15c*d - 40d , b  - 49b*c + 32c  + 50b*d + 14c*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
│ │ │ │ +      25d , a*c + 46b*c - 35c  - 7a*d + 37b*d - 14c*d - 28d , b*c  - 15b*c*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      2        2     3   3               2         2
│ │ │ │ +      + 10c d - 18a*d  + b*d  + 26c*d  + 3d , c  + 3b*c*d + 33c d + 21a*d  +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -             2        2      3
│ │ │ │ -      + 33b*d  - 14c*d  + 33d )
│ │ │ │ +          2        2      3
│ │ │ │ +      3b*d  - 50c*d  + 15d )
│ │ │ │  
│ │ │ │  o61 : Ideal of S
│ │ │ │  i62 : betti res I0
│ │ │ │  
│ │ │ │               0 1 2 3
│ │ │ │  o62 = total: 1 6 8 3
│ │ │ │            0: 1 . . .
│ │ │ │ @@ -648,43 +653,44 @@
│ │ │ │            0: 1 . . .
│ │ │ │            1: . 4 4 1
│ │ │ │            2: . 2 4 2
│ │ │ │  
│ │ │ │  o63 : BettiTally
│ │ │ │  i64 : netList decompose I0
│ │ │ │  
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ -------------------------+
│ │ │ │ -o64 = |ideal (c - 42d, b + 26d, a - 30d)
│ │ │ │ -|
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ -------------------------+
│ │ │ │ -      |ideal (c - 47d, b + 7d, a - 44d)
│ │ │ │ -|
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ -------------------------+
│ │ │ │ -      |                                     2                      2   2      2
│ │ │ │ -2 |
│ │ │ │ -      |ideal (a + 39b - 29c - 24d, b*c - 15c  - 29b*d - 38c*d + 16d , b  - 39c
│ │ │ │ -+ 17b*d - 28c*d - 50d )|
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ -------------------------+
│ │ │ │ +      +---------------------------------------------------+
│ │ │ │ +o64 = |ideal (c - 21d, b - 26d, a - 50d)                  |
│ │ │ │ +      +---------------------------------------------------+
│ │ │ │ +      |ideal (c - 49d, b - 33d, a - 30d)                  |
│ │ │ │ +      +---------------------------------------------------+
│ │ │ │ +      |ideal (c + 41d, b + 33d, a - 35d)                  |
│ │ │ │ +      +---------------------------------------------------+
│ │ │ │ +      |ideal (c + d, b + 40d, a - 5d)                     |
│ │ │ │ +      +---------------------------------------------------+
│ │ │ │ +      |                                    2            2 |
│ │ │ │ +      |ideal (b + c - 47d, a - 38c - 17d, c  + 4c*d - 8d )|
│ │ │ │ +      +---------------------------------------------------+
│ │ │ │  i65 : netList decompose I1
│ │ │ │  
│ │ │ │ -      +---------------------------------+
│ │ │ │ -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)|
│ │ │ │ -      +---------------------------------+
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------------------+
│ │ │ │ +      |                        2                             2   2
│ │ │ │ +2                      2                  2                             2   2
│ │ │ │ +2                           2   3               2         2       2        2
│ │ │ │ +3     2                2         2      2        2     3 |
│ │ │ │ +o65 = |ideal (a*c + 46b*c - 35c  - 7a*d + 37b*d - 14c*d - 28d , b  - 49b*c +
│ │ │ │ +32c  + 50b*d + 14c*d - 25d , a*b - 3b*c + 43c  + 5a*d + 11b*d + 15c*d - 40d , a
│ │ │ │ ++ 31b*c + 29c  - 37a*d - 8b*d - 6c*d - 5d , c  + 3b*c*d + 33c d + 21a*d  + 3b*d
│ │ │ │ +- 50c*d  + 15d , b*c  - 15b*c*d + 10c d - 18a*d  + b*d  + 26c*d  + 3d )|
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------------------+
│ │ │ │  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/RandomCanonicalCurves/example-output/_canonical__Curve.out
│ │ │ @@ -6,15 +6,15 @@
│ │ │  i2 : g=14;
│ │ │  
│ │ │  i3 : FF=ZZ/10007;
│ │ │  
│ │ │  i4 : R=FF[x_0..x_(g-1)];
│ │ │  
│ │ │  i5 : time betti(I=(random canonicalCurve)(g,R))
│ │ │ - -- used 8.8703s (cpu); 6.22891s (thread); 0s (gc)
│ │ │ + -- used 7.58381s (cpu); 5.859s (thread); 0s (gc)
│ │ │  
│ │ │              0  1
│ │ │  o5 = total: 1 66
│ │ │           0: 1  .
│ │ │           1: . 66
│ │ │  
│ │ │  o5 : BettiTally
│ │ ├── ./usr/share/doc/Macaulay2/RandomCanonicalCurves/html/_canonical__Curve.html
│ │ │ @@ -92,15 +92,15 @@
│ │ │              
    		
│ │ │                
    i4 : R=FF[x_0..x_(g-1)];
    
│ │ │              
    
│ │ │                
    i5 : time betti(I=(random canonicalCurve)(g,R))
│ │ │ - -- used 8.8703s (cpu); 6.22891s (thread); 0s (gc)
│ │ │ + -- used 7.58381s (cpu); 5.859s (thread); 0s (gc)
│ │ │  
│ │ │              0  1
│ │ │  o5 = total: 1 66
│ │ │           0: 1  .
│ │ │           1: . 66
│ │ │  
│ │ │  o5 : BettiTally
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,15 +17,15 @@
│ │ │ │  unirationality of $M_g$ by Severi, Sernesi, Chang-Ran and Verra.
│ │ │ │  i1 : setRandomSeed "alpha";
│ │ │ │   -- setting random seed to 10206284518
│ │ │ │  i2 : g=14;
│ │ │ │  i3 : FF=ZZ/10007;
│ │ │ │  i4 : R=FF[x_0..x_(g-1)];
│ │ │ │  i5 : time betti(I=(random canonicalCurve)(g,R))
│ │ │ │ - -- used 8.8703s (cpu); 6.22891s (thread); 0s (gc)
│ │ │ │ + -- used 7.58381s (cpu); 5.859s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              0  1
│ │ │ │  o5 = total: 1 66
│ │ │ │           0: 1  .
│ │ │ │           1: . 66
│ │ │ │  
│ │ │ │  o5 : BettiTally
│ │ ├── ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_test__Time__For__L__L__Lon__Syzygies.out
│ │ │ @@ -7,42 +7,42 @@
│ │ │  
│ │ │  o2 = (10, 20)
│ │ │  
│ │ │  o2 : Sequence
│ │ │  
│ │ │  i3 : (m,t1,t2)=testTimeForLLLonSyzygies(r,n,Height=>11)
│ │ │  
│ │ │ -o3 = ({5, 2.91596e52, 9}, .00212284, .000873057)
│ │ │ +o3 = ({5, 2.91596e52, 9}, .00203576, .000940587)
│ │ │  
│ │ │  o3 : Sequence
│ │ │  
│ │ │  i4 : (m,t1,t2)=testTimeForLLLonSyzygies(15,30,Height=>100)
│ │ │  
│ │ │ -o4 = ({50, 2.30853e454, 98}, .00576907, .0352356)
│ │ │ +o4 = ({50, 2.30853e454, 98}, .00542753, .0406366)
│ │ │  
│ │ │  o4 : Sequence
│ │ │  
│ │ │  i5 : L=apply(10,c->(testTimeForLLLonSyzygies(15,30))_{1,2})
│ │ │  
│ │ │ -o5 = {{.00542734, .0121802}, {.00514022, .00415768}, {.00512656, .00663411},
│ │ │ +o5 = {{.00537418, .0137539}, {.00574346, .0047277}, {.00615455, .00775135},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {.0049702, .00989477}, {.00503716, .0131842}, {.0883312, .0123666},
│ │ │ +     {.00584512, .0114041}, {.0065868, .0151811}, {.0993418, .0142239},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {.00508916, .00815041}, {.00494047, .00757079}, {.00439367, .00535668},
│ │ │ +     {.00446696, .00903898}, {.00484054, .00864619}, {.00396867, .00620799},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {.0060391, .00810552}}
│ │ │ +     {.00548647, .00920139}}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : 1/10*sum(L,t->t_0)
│ │ │  
│ │ │ -o6 = .01344950969999994
│ │ │ +o6 = .01478085150000004
│ │ │  
│ │ │  o6 : RR (of precision 53)
│ │ │  
│ │ │  i7 : 1/10*sum(L,t->t_1)
│ │ │  
│ │ │ -o7 = .008760095100000065
│ │ │ +o7 = .01001365240000034
│ │ │  
│ │ │  o7 : RR (of precision 53)
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/RandomComplexes/html/_test__Time__For__L__L__Lon__Syzygies.html
│ │ │ @@ -93,57 +93,57 @@
│ │ │  o2 : Sequence
│ │ │              
    
│ │ │                
    i3 : (m,t1,t2)=testTimeForLLLonSyzygies(r,n,Height=>11)
│ │ │  
│ │ │ -o3 = ({5, 2.91596e52, 9}, .00212284, .000873057)
│ │ │ +o3 = ({5, 2.91596e52, 9}, .00203576, .000940587)
│ │ │  
│ │ │  o3 : Sequence
    
│ │ │              
    
│ │ │                
    i4 : (m,t1,t2)=testTimeForLLLonSyzygies(15,30,Height=>100)
│ │ │  
│ │ │ -o4 = ({50, 2.30853e454, 98}, .00576907, .0352356)
│ │ │ +o4 = ({50, 2.30853e454, 98}, .00542753, .0406366)
│ │ │  
│ │ │  o4 : Sequence
    
│ │ │              
    
│ │ │                
    i5 : L=apply(10,c->(testTimeForLLLonSyzygies(15,30))_{1,2})
│ │ │  
│ │ │ -o5 = {{.00542734, .0121802}, {.00514022, .00415768}, {.00512656, .00663411},
│ │ │ +o5 = {{.00537418, .0137539}, {.00574346, .0047277}, {.00615455, .00775135},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {.0049702, .00989477}, {.00503716, .0131842}, {.0883312, .0123666},
│ │ │ +     {.00584512, .0114041}, {.0065868, .0151811}, {.0993418, .0142239},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {.00508916, .00815041}, {.00494047, .00757079}, {.00439367, .00535668},
│ │ │ +     {.00446696, .00903898}, {.00484054, .00864619}, {.00396867, .00620799},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {.0060391, .00810552}}
│ │ │ +     {.00548647, .00920139}}
│ │ │  
│ │ │  o5 : List
    
│ │ │              
    
│ │ │                
    i6 : 1/10*sum(L,t->t_0)
│ │ │  
│ │ │ -o6 = .01344950969999994
│ │ │ +o6 = .01478085150000004
│ │ │  
│ │ │  o6 : RR (of precision 53)
    
│ │ │              
    
│ │ │                
    i7 : 1/10*sum(L,t->t_1)
│ │ │  
│ │ │ -o7 = .008760095100000065
│ │ │ +o7 = .01001365240000034
│ │ │  
│ │ │  o7 : RR (of precision 53)
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,41 +25,41 @@
│ │ │ │  i2 : r=10,n=20
│ │ │ │  
│ │ │ │  o2 = (10, 20)
│ │ │ │  
│ │ │ │  o2 : Sequence
│ │ │ │  i3 : (m,t1,t2)=testTimeForLLLonSyzygies(r,n,Height=>11)
│ │ │ │  
│ │ │ │ -o3 = ({5, 2.91596e52, 9}, .00212284, .000873057)
│ │ │ │ +o3 = ({5, 2.91596e52, 9}, .00203576, .000940587)
│ │ │ │  
│ │ │ │  o3 : Sequence
│ │ │ │  i4 : (m,t1,t2)=testTimeForLLLonSyzygies(15,30,Height=>100)
│ │ │ │  
│ │ │ │ -o4 = ({50, 2.30853e454, 98}, .00576907, .0352356)
│ │ │ │ +o4 = ({50, 2.30853e454, 98}, .00542753, .0406366)
│ │ │ │  
│ │ │ │  o4 : Sequence
│ │ │ │  i5 : L=apply(10,c->(testTimeForLLLonSyzygies(15,30))_{1,2})
│ │ │ │  
│ │ │ │ -o5 = {{.00542734, .0121802}, {.00514022, .00415768}, {.00512656, .00663411},
│ │ │ │ +o5 = {{.00537418, .0137539}, {.00574346, .0047277}, {.00615455, .00775135},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {.0049702, .00989477}, {.00503716, .0131842}, {.0883312, .0123666},
│ │ │ │ +     {.00584512, .0114041}, {.0065868, .0151811}, {.0993418, .0142239},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {.00508916, .00815041}, {.00494047, .00757079}, {.00439367, .00535668},
│ │ │ │ +     {.00446696, .00903898}, {.00484054, .00864619}, {.00396867, .00620799},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {.0060391, .00810552}}
│ │ │ │ +     {.00548647, .00920139}}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : 1/10*sum(L,t->t_0)
│ │ │ │  
│ │ │ │ -o6 = .01344950969999994
│ │ │ │ +o6 = .01478085150000004
│ │ │ │  
│ │ │ │  o6 : RR (of precision 53)
│ │ │ │  i7 : 1/10*sum(L,t->t_1)
│ │ │ │  
│ │ │ │ -o7 = .008760095100000065
│ │ │ │ +o7 = .01001365240000034
│ │ │ │  
│ │ │ │  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/RandomCurvesOverVerySmallFiniteFields/example-output/_smooth__Canonical__Curve.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 11549527689790345152
│ │ │  
│ │ │  i1 : time ICan = smoothCanonicalCurve(11,5);
│ │ │ - -- used 1.37715s (cpu); 1.10031s (thread); 0s (gc)
│ │ │ + -- used 1.65124s (cpu); 1.26297s (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
│ │ │ @@ -82,15 +82,15 @@
│ │ │            
    If the option Printing is set to true then printings about the current step in the construction are displayed.
    
│ │ │            
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │  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 1.37715s (cpu); 1.10031s (thread); 0s (gc)
│ │ │ │ + -- used 1.65124s (cpu); 1.26297s (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/example-output/_random__Curve__Genus14__Degree18in__P6.out
│ │ │ @@ -4,15 +4,15 @@
│ │ │   -- setting random seed to 10206284518
│ │ │  
│ │ │  i2 : FF=ZZ/10007;
│ │ │  
│ │ │  i3 : S=FF[x_0..x_6];
│ │ │  
│ │ │  i4 : time I=randomCurveGenus14Degree18inP6(S);
│ │ │ - -- used 1.70428s (cpu); 1.37292s (thread); 0s (gc)
│ │ │ + -- used 1.81318s (cpu); 1.51322s (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
│ │ │ @@ -93,15 +93,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : time ICan = smoothCanonicalCurve(11,5);
│ │ │ - -- used 1.37715s (cpu); 1.10031s (thread); 0s (gc)
│ │ │ + -- used 1.65124s (cpu); 1.26297s (thread); 0s (gc)
│ │ │  
│ │ │                ZZ
│ │ │  o1 : Ideal of --[t ..t  ]
│ │ │                 5  0   10
    
│ │ │              
    
│ │ │                
    i3 : S=FF[x_0..x_6];
    
│ │ │              
    
│ │ │                
    i4 : time I=randomCurveGenus14Degree18inP6(S);
│ │ │ - -- used 1.70428s (cpu); 1.37292s (thread); 0s (gc)
│ │ │ + -- used 1.81318s (cpu); 1.51322s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : Ideal of S
    
│ │ │              
    
│ │ │                
    i5 : betti res I
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,15 +28,15 @@
│ │ │ │  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");
│ │ │ │   -- setting random seed to 10206284518
│ │ │ │  i2 : FF=ZZ/10007;
│ │ │ │  i3 : S=FF[x_0..x_6];
│ │ │ │  i4 : time I=randomCurveGenus14Degree18inP6(S);
│ │ │ │ - -- used 1.70428s (cpu); 1.37292s (thread); 0s (gc)
│ │ │ │ + -- used 1.81318s (cpu); 1.51322s (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/example-output/___Random__Ideals.out
│ │ │ @@ -1,25 +1,24 @@
│ │ │  -- -*- M2-comint -*- hash: 9542801742429495161
│ │ │  
│ │ │  i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1763142113
│ │ │ + -- setting random seed to 1766074331
│ │ │  
│ │ │ -o1 = 1763142113
│ │ │ +o1 = 1766074331
│ │ │  
│ │ │  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 3.09133s (cpu); 1.64886s (thread); 0s (gc)
│ │ │ + -- used 3.4514s (cpu); 1.67086s (thread); 0s (gc)
│ │ │  
│ │ │ -o4 = Tally{4 => 45 }
│ │ │ -           5 => 189
│ │ │ -           6 => 186
│ │ │ -           7 => 71
│ │ │ -           8 => 7
│ │ │ +o4 = Tally{4 => 41 }
│ │ │ +           5 => 216
│ │ │ +           6 => 166
│ │ │ +           7 => 65
│ │ │ +           8 => 11
│ │ │             9 => 1
│ │ │ -           10 => 1
│ │ │  
│ │ │  o4 : Tally
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Monomial.out
│ │ │ @@ -1,13 +1,13 @@
│ │ │  -- -*- M2-comint -*- hash: 5959465567197821046
│ │ │  
│ │ │  i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1763142118
│ │ │ + -- setting random seed to 1766074334
│ │ │  
│ │ │ -o1 = 1763142118
│ │ │ +o1 = 1766074334
│ │ │  
│ │ │  i2 : kk=ZZ/101
│ │ │  
│ │ │  o2 = kk
│ │ │  
│ │ │  o2 : QuotientRing
│ │ │  
│ │ │ @@ -15,13 +15,13 @@
│ │ │  
│ │ │  o3 = S
│ │ │  
│ │ │  o3 : PolynomialRing
│ │ │  
│ │ │  i4 : randomMonomial(3,S)
│ │ │  
│ │ │ -      3
│ │ │ -o4 = a
│ │ │ +        2
│ │ │ +o4 = b*c
│ │ │  
│ │ │  o4 : S
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Square__Free__Monomial__Ideal.out
│ │ │ @@ -1,13 +1,13 @@
│ │ │  -- -*- M2-comint -*- hash: 8876340562021865447
│ │ │  
│ │ │  i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1763142127
│ │ │ + -- setting random seed to 1766074341
│ │ │  
│ │ │ -o1 = 1763142127
│ │ │ +o1 = 1766074341
│ │ │  
│ │ │  i2 : kk=ZZ/101
│ │ │  
│ │ │  o2 = kk
│ │ │  
│ │ │  o2 : QuotientRing
│ │ │  
│ │ │ @@ -28,12 +28,12 @@
│ │ │  
│ │ │  o5 = ideal(b*d*e)
│ │ │  
│ │ │  o5 : Ideal of S
│ │ │  
│ │ │  i6 : randomSquareFreeMonomialIdeal(5:2, S)
│ │ │  
│ │ │ -o6 = ideal (b*c, a*c, a*e, c*e, c*d)
│ │ │ +o6 = ideal (a*d, b*c, b*e, a*b, a*c)
│ │ │  
│ │ │  o6 : Ideal of S
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Square__Free__Step.out
│ │ │ @@ -1,13 +1,13 @@
│ │ │  -- -*- M2-comint -*- hash: 10504911213508281315
│ │ │  
│ │ │  i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1763142123
│ │ │ + -- setting random seed to 1766074337
│ │ │  
│ │ │ -o1 = 1763142123
│ │ │ +o1 = 1766074337
│ │ │  
│ │ │  i2 : S=ZZ/2[vars(0..3)]
│ │ │  
│ │ │  o2 = S
│ │ │  
│ │ │  o2 : PolynomialRing
│ │ │  
│ │ │ @@ -15,17 +15,15 @@
│ │ │  
│ │ │  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*d), {a*b, a*c, a*d, b*d}, {c*d, b*c,
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     a}}
│ │ │ +o4 = {monomialIdeal (a*b, a*d), {a*b, a*d}, {b*c*d, a*c}}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : setRandomSeed(1)
│ │ │   -- setting random seed to 1
│ │ │  
│ │ │  o5 = 1
│ │ │ @@ -39,15 +37,15 @@
│ │ │  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 4.19238s (cpu); 2.83348s (thread); 0s (gc)
│ │ │ + -- used 4.95884s (cpu); 3.26982s (thread); 0s (gc)
│ │ │  
│ │ │  i9 : #T
│ │ │  
│ │ │  o9 = 168
│ │ │  
│ │ │  i10 : T
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Monomial.html
│ │ │ @@ -71,17 +71,17 @@
│ │ │          
    
│ │ │            
    Chooses a random monomial.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1763142118
│ │ │ + -- setting random seed to 1766074334
│ │ │  
│ │ │ -o1 = 1763142118
    
│ │ │ +o1 = 1766074334
│ │ │              
    
│ │ │                
    i2 : kk=ZZ/101
│ │ │  
│ │ │  o2 = kk
│ │ │ @@ -98,16 +98,16 @@
│ │ │  o3 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i4 : randomMonomial(3,S)
│ │ │  
│ │ │ -      3
│ │ │ -o4 = a
│ │ │ +        2
│ │ │ +o4 = b*c
│ │ │  
│ │ │  o4 : S
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,31 +11,31 @@
│ │ │ │            o d, an _i_n_t_e_g_e_r, non-negative
│ │ │ │            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())
│ │ │ │ - -- setting random seed to 1763142118
│ │ │ │ + -- setting random seed to 1766074334
│ │ │ │  
│ │ │ │ -o1 = 1763142118
│ │ │ │ +o1 = 1766074334
│ │ │ │  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 = b*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
│ │ │ @@ -71,17 +71,17 @@
│ │ │          
    
│ │ │            
    Choose a random square-free monomial ideal whose generators have the degrees specified by the list or sequence L.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1763142127
│ │ │ + -- setting random seed to 1766074341
│ │ │  
│ │ │ -o1 = 1763142127
    
│ │ │ +o1 = 1766074341
│ │ │              
    
│ │ │                
    i2 : kk=ZZ/101
│ │ │  
│ │ │  o2 = kk
│ │ │ @@ -117,15 +117,15 @@
│ │ │  o5 : Ideal of S
    
│ │ │              
    
│ │ │                
    i6 : randomSquareFreeMonomialIdeal(5:2, S)
│ │ │  
│ │ │ -o6 = ideal (b*c, a*c, a*e, c*e, c*d)
│ │ │ +o6 = ideal (a*d, b*c, b*e, a*b, a*c)
│ │ │  
│ │ │  o6 : Ideal of S
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,17 +13,17 @@
│ │ │ │      * Outputs:
│ │ │ │            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())
│ │ │ │ - -- setting random seed to 1763142127
│ │ │ │ + -- setting random seed to 1766074341
│ │ │ │  
│ │ │ │ -o1 = 1763142127
│ │ │ │ +o1 = 1766074341
│ │ │ │  i2 : kk=ZZ/101
│ │ │ │  
│ │ │ │  o2 = kk
│ │ │ │  
│ │ │ │  o2 : QuotientRing
│ │ │ │  i3 : S=kk[a..e]
│ │ │ │  
│ │ │ │ @@ -39,15 +39,15 @@
│ │ │ │  low degree gens generated everything
│ │ │ │  
│ │ │ │  o5 = ideal(b*d*e)
│ │ │ │  
│ │ │ │  o5 : Ideal of S
│ │ │ │  i6 : randomSquareFreeMonomialIdeal(5:2, S)
│ │ │ │  
│ │ │ │ -o6 = ideal (b*c, a*c, a*e, c*e, c*d)
│ │ │ │ +o6 = ideal (a*d, b*c, b*e, a*b, a*c)
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -79,17 +79,17 @@
│ │ │            
    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())
│ │ │ - -- setting random seed to 1763142123
│ │ │ + -- setting random seed to 1766074337
│ │ │  
│ │ │ -o1 = 1763142123
    
│ │ │ +o1 = 1766074337
│ │ │              
    
│ │ │                
    i2 : S=ZZ/2[vars(0..3)]
│ │ │  
│ │ │  o2 = S
│ │ │ @@ -106,17 +106,15 @@
│ │ │  o3 : MonomialIdeal of S
    
│ │ │              
    
│ │ │                
    i4 : randomSquareFreeStep J
│ │ │  
│ │ │ -o4 = {monomialIdeal (a*b, a*c, a*d, b*d), {a*b, a*c, a*d, b*d}, {c*d, b*c,
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     a}}
│ │ │ +o4 = {monomialIdeal (a*b, a*d), {a*b, a*d}, {b*c*d, a*c}}
│ │ │  
│ │ │  o4 : List
    
│ │ │              
    
│ │ │          
    
│ │ │            
    With 4 variables and 168 possible monomial ideals, a run of 5000 takes less than 6 seconds on a reasonably fast machine. With 10 variables a run of 1000 takes about 2 seconds.
    
│ │ │ @@ -147,15 +145,15 @@
│ │ │  
│ │ │  o7 : MonomialIdeal of S
    
│ │ │              
    
│ │ │                
    i8 : time T=tally for t from 1 to 5000 list first (J=rsfs(J,AlexanderProbability => .01));
│ │ │ - -- used 4.19238s (cpu); 2.83348s (thread); 0s (gc)
    
│ │ │ + -- used 4.95884s (cpu); 3.26982s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i9 : #T
│ │ │  
│ │ │  o9 = 168
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,32 +35,30 @@
│ │ │ │  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())
│ │ │ │ - -- setting random seed to 1763142123
│ │ │ │ + -- setting random seed to 1766074337
│ │ │ │  
│ │ │ │ -o1 = 1763142123
│ │ │ │ +o1 = 1766074337
│ │ │ │  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*d), {a*b, a*c, a*d, b*d}, {c*d, b*c,
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     a}}
│ │ │ │ +o4 = {monomialIdeal (a*b, a*d), {a*b, a*d}, {b*c*d, a*c}}
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  With 4 variables and 168 possible monomial ideals, a run of 5000 takes less
│ │ │ │  than 6 seconds on a reasonably fast machine. With 10 variables a run of 1000
│ │ │ │  takes about 2 seconds.
│ │ │ │  i5 : setRandomSeed(1)
│ │ │ │   -- setting random seed to 1
│ │ │ │ @@ -74,15 +72,15 @@
│ │ │ │  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 4.19238s (cpu); 2.83348s (thread); 0s (gc)
│ │ │ │ + -- used 4.95884s (cpu); 3.26982s (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
│ │ │ @@ -54,17 +54,17 @@
│ │ │          
    
│ │ │            
    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.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -72,23 +72,22 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1763142113
│ │ │ + -- setting random seed to 1766074331
│ │ │  
│ │ │ -o1 = 1763142113
    
│ │ │ +o1 = 1766074331
│ │ │              
    
│ │ │                
    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 3.09133s (cpu); 1.64886s (thread); 0s (gc)
│ │ │ + -- used 3.4514s (cpu); 1.67086s (thread); 0s (gc)
│ │ │  
│ │ │ -o4 = Tally{4 => 45 }
│ │ │ -           5 => 189
│ │ │ -           6 => 186
│ │ │ -           7 => 71
│ │ │ -           8 => 7
│ │ │ +o4 = Tally{4 => 41 }
│ │ │ +           5 => 216
│ │ │ +           6 => 166
│ │ │ +           7 => 65
│ │ │ +           8 => 11
│ │ │             9 => 1
│ │ │ -           10 => 1
│ │ │  
│ │ │  o4 : Tally
    
│ │ │              
    
│ │ │          
    
│ │ │            
    How does this compare with the case of binomial ideals? or pure binomial ideals? We invite the reader to experiment, replacing "randomMonomialIdeal" above with "randomBinomialIdeal" or "randomPureBinomialIdeal", or taking larger numbers of examples. Click the link "Finding Extreme Examples" below to see some other, more elaborate ways to search.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -9,29 +9,28 @@
│ │ │ │  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 : setRandomSeed(currentTime())
│ │ │ │ - -- setting random seed to 1763142113
│ │ │ │ + -- setting random seed to 1766074331
│ │ │ │  
│ │ │ │ -o1 = 1763142113
│ │ │ │ +o1 = 1766074331
│ │ │ │  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 3.09133s (cpu); 1.64886s (thread); 0s (gc)
│ │ │ │ + -- used 3.4514s (cpu); 1.67086s (thread); 0s (gc)
│ │ │ │  
│ │ │ │ -o4 = Tally{4 => 45 }
│ │ │ │ -           5 => 189
│ │ │ │ -           6 => 186
│ │ │ │ -           7 => 71
│ │ │ │ -           8 => 7
│ │ │ │ +o4 = Tally{4 => 41 }
│ │ │ │ +           5 => 216
│ │ │ │ +           6 => 166
│ │ │ │ +           7 => 65
│ │ │ │ +           8 => 11
│ │ │ │             9 => 1
│ │ │ │ -           10 => 1
│ │ │ │  
│ │ │ │  o4 : Tally
│ │ │ │  How does this compare with the case of binomial ideals? or pure binomial
│ │ │ │  ideals? We invite the reader to experiment, replacing "randomMonomialIdeal"
│ │ │ │  above with "randomBinomialIdeal" or "randomPureBinomialIdeal", or taking larger
│ │ │ │  numbers of examples. Click the link "Finding Extreme Examples" below to see
│ │ │ │  some other, more elaborate ways to search.
│ │ ├── ./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)
│ │ │ - -- 1.48313s elapsed
│ │ │ + -- 1.3149s elapsed
│ │ │  
│ │ │  o4 = 4
│ │ │  
│ │ │  i5 : elapsedTime dim I
│ │ │ - -- 3.03377s elapsed
│ │ │ + -- 3.17624s 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)) );
│ │ │ - -- 2.18511s elapsed
│ │ │ + -- 1.63541s 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)
│ │ │ - -- 3.48603s elapsed
│ │ │ + -- 2.77832s 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)
│ │ │ - -- 2.33464s elapsed
│ │ │ + -- 1.98206s 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
│ │ │ @@ -95,23 +95,23 @@
│ │ │  
│ │ │  o3 : Ideal of S
│ │ │              
    
│ │ │                
    i4 : elapsedTime dimViaBezout(I)
│ │ │ - -- 1.48313s elapsed
│ │ │ + -- 1.3149s elapsed
│ │ │  
│ │ │  o4 = 4
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime dim I
│ │ │ - -- 3.03377s elapsed
│ │ │ + -- 3.17624s 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 {}
│ │ │ │ @@ -32,19 +32,19 @@
│ │ │ │  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)
│ │ │ │ - -- 1.48313s elapsed
│ │ │ │ + -- 1.3149s elapsed
│ │ │ │  
│ │ │ │  o4 = 4
│ │ │ │  i5 : elapsedTime dim I
│ │ │ │ - -- 3.03377s elapsed
│ │ │ │ + -- 3.17624s 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
│ │ │ @@ -155,15 +155,15 @@
│ │ │  
│ │ │  o9 : Ideal of T
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : elapsedTime(while (i < 10) and dim J > 1 do (i = i+1; J = extendIdealByNonZeroMinor(4, M, J)) );
│ │ │ - -- 2.18511s elapsed
    
│ │ │ + -- 1.63541s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i11 : dim J
│ │ │  
│ │ │  o11 = 1
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -79,15 +79,15 @@
│ │ │ │  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)) );
│ │ │ │ - -- 2.18511s elapsed
│ │ │ │ + -- 1.63541s 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
│ │ │ @@ -144,27 +144,27 @@
│ │ │  
│ │ │  o7 : Ideal of S
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : elapsedTime randomPoints(I, Strategy=>LinearIntersection, DecompositionStrategy=>MultiplicationTable)
│ │ │ - -- 3.48603s elapsed
│ │ │ + -- 2.77832s 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)
│ │ │ - -- 2.33464s elapsed
│ │ │ + -- 1.98206s 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 {}
│ │ │ │ @@ -66,24 +66,24 @@
│ │ │ │  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)
│ │ │ │ - -- 3.48603s elapsed
│ │ │ │ + -- 2.77832s 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)
│ │ │ │ - -- 2.33464s elapsed
│ │ │ │ + -- 1.98206s 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/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 0.616843s (cpu); 0.437628s (thread); 0s (gc)
│ │ │ + -- used 0.916303s (cpu); 0.465656s (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
│ │ │ @@ -189,15 +189,15 @@
│ │ │  
│ │ │  o13 : RingMap Q <-- Q
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : time inverseOfMap(phi,CheckBirational=>false, Verbosity=>0)
│ │ │ - -- used 0.616843s (cpu); 0.437628s (thread); 0s (gc)
│ │ │ + -- used 0.916303s (cpu); 0.465656s (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 {}
│ │ │ │ @@ -94,15 +94,15 @@
│ │ │ │  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 0.616843s (cpu); 0.437628s (thread); 0s (gc)
│ │ │ │ + -- used 0.916303s (cpu); 0.465656s (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/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 0.00326124s (cpu); 0.00325846s (thread); 0s (gc)
│ │ │ + -- used 0.00361549s (cpu); 0.00361302s (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]
│ │ │  
│ │ │ @@ -142,23 +142,23 @@
│ │ │  
│ │ │  i31 : nodes = I + ideal jacobian I;
│ │ │  
│ │ │  o31 : Ideal of R
│ │ │  
│ │ │  i32 : time rationalPoints(variety nodes, Split=>true, Verbose=>true);
│ │ │  -- base change to the field QQ[a]/(a^8-40*a^6+230*a^4-200*a^2+25)
│ │ │ - -- used 1.00756s (cpu); 0.82979s (thread); 0s (gc)
│ │ │ + -- used 1.16142s (cpu); 0.898393s (thread); 0s (gc)
│ │ │  
│ │ │  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 0.288685s (cpu); 0.211506s (thread); 0s (gc)
│ │ │ + -- used 0.362927s (cpu); 0.270577s (thread); 0s (gc)
│ │ │  
│ │ │  o35 = 31
│ │ │  
│ │ │  i36 :
│ │ ├── ./usr/share/doc/Macaulay2/RationalPoints2/html/_rational__Points.html
│ │ │ @@ -178,15 +178,15 @@
│ │ │  o13 : Ideal of ---[u ..u  ]
│ │ │                 101  0   10
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : time rationalPoints(I, Amount => true)
│ │ │ - -- used 0.00326124s (cpu); 0.00325846s (thread); 0s (gc)
│ │ │ + -- used 0.00361549s (cpu); 0.00361302s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = 110462212541120451001
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    Over number fields
    
│ │ │ @@ -348,15 +348,15 @@
│ │ │  o31 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i32 : time rationalPoints(variety nodes, Split=>true, Verbose=>true);
│ │ │  -- base change to the field QQ[a]/(a^8-40*a^6+230*a^4-200*a^2+25)
│ │ │ - -- used 1.00756s (cpu); 0.82979s (thread); 0s (gc)
    
│ │ │ + -- used 1.16142s (cpu); 0.898393s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i33 : #oo
│ │ │  
│ │ │  o33 = 31
    
│ │ │ @@ -373,15 +373,15 @@
│ │ │  
│ │ │  o34 : Ideal of GF 1048969271299456081[x..z, w]
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i35 : time #rationalPoints(variety nodes', Split=>true, Verbose=>true)
│ │ │ - -- used 0.288685s (cpu); 0.211506s (thread); 0s (gc)
│ │ │ + -- used 0.362927s (cpu); 0.270577s (thread); 0s (gc)
│ │ │  
│ │ │  o35 = 31
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -89,15 +89,15 @@
│ │ │ │  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 0.00326124s (cpu); 0.00325846s (thread); 0s (gc)
│ │ │ │ + -- used 0.00361549s (cpu); 0.00361302s (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);
│ │ │ │ @@ -197,24 +197,24 @@
│ │ │ │  
│ │ │ │  o30 : Ideal of R
│ │ │ │  i31 : nodes = I + ideal jacobian I;
│ │ │ │  
│ │ │ │  o31 : Ideal of R
│ │ │ │  i32 : time rationalPoints(variety nodes, Split=>true, Verbose=>true);
│ │ │ │  -- base change to the field QQ[a]/(a^8-40*a^6+230*a^4-200*a^2+25)
│ │ │ │ - -- used 1.00756s (cpu); 0.82979s (thread); 0s (gc)
│ │ │ │ + -- used 1.16142s (cpu); 0.898393s (thread); 0s (gc)
│ │ │ │  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 0.288685s (cpu); 0.211506s (thread); 0s (gc)
│ │ │ │ + -- used 0.362927s (cpu); 0.270577s (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.
│ │ │ │  ******** MMeennuu ********
│ │ ├── ./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 0.84218s (cpu); 0.713177s (thread); 0s (gc)
│ │ │ + -- used 1.01257s (cpu); 0.811113s (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
│ │ │ @@ -58,15 +58,15 @@
│ │ │  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 1.03324s (cpu); 0.7369s (thread); 0s (gc)
│ │ │ + -- used 1.07044s (cpu); 0.825906s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of S[w ..w ]
│ │ │                   0   4
│ │ │  
│ │ │  i8 : kk = ZZ/101;
│ │ │  
│ │ │  i9 : S = kk[x,y,z];
│ │ │ @@ -77,19 +77,19 @@
│ │ │  o10 : Matrix S  <-- S
│ │ │  
│ │ │  i11 : I = minors(3,m);
│ │ │  
│ │ │  o11 : Ideal of S
│ │ │  
│ │ │  i12 : time reesIdeal (I, I_0);
│ │ │ - -- used 1.49044s (cpu); 1.21436s (thread); 0s (gc)
│ │ │ + -- used 1.813s (cpu); 1.44855s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of S[w ..w ]
│ │ │                    0   3
│ │ │  
│ │ │  i13 : time reesIdeal (I, I_0, Jacobian =>false);
│ │ │ - -- used 1.63351s (cpu); 1.25312s (thread); 0s (gc)
│ │ │ + -- used 1.68593s (cpu); 1.44874s (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 0.0281685s (cpu); 0.0259639s (thread); 0s (gc)
│ │ │ + -- used 0.486058s (cpu); 0.0833724s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : Ideal of S[w ..w ]
│ │ │                   0   6
│ │ │  
│ │ │  i5 : time V2 = reesIdeal(i,i_0);
│ │ │ - -- used 0.116569s (cpu); 0.115819s (thread); 0s (gc)
│ │ │ + -- used 0.205529s (cpu); 0.14747s (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 0.0193057s (cpu); 0.0182587s (thread); 0s (gc)
│ │ │ + -- used 0.188879s (cpu); 0.0391355s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : Ideal of S[w ..w ]
│ │ │                   0   2
│ │ │  
│ │ │  i10 : time I2 = reesIdeal(i,i_0);
│ │ │ - -- used 0.00768211s (cpu); 0.00741424s (thread); 0s (gc)
│ │ │ + -- used 0.0218421s (cpu); 0.0100216s (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
│ │ │ @@ -587,15 +587,15 @@
│ │ │          
    
│ │ │            
    We compute the singular locus once again:
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── 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 0.84218s (cpu); 0.713177s (thread); 0s (gc)
│ │ │ │ + -- used 1.01257s (cpu); 0.811113s (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
│ │ │ @@ -151,15 +151,15 @@
│ │ │  
│ │ │  o6 : Ideal of S
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i48 : time sing2 = ideal singularLocus strictTransform2;
│ │ │ - -- used 0.84218s (cpu); 0.713177s (thread); 0s (gc)
│ │ │ + -- used 1.01257s (cpu); 0.811113s (thread); 0s (gc)
│ │ │  
│ │ │                   ZZ
│ │ │  o48 : Ideal of -----[p ..p , w ..w , x..y]
│ │ │                 32003  0   2   0   1
    
│ │ │              
    
│ │ │                
    i7 : time rI = expectedReesIdeal I; -- n= 5 case takes < 1 sec.
│ │ │ - -- used 1.03324s (cpu); 0.7369s (thread); 0s (gc)
│ │ │ + -- used 1.07044s (cpu); 0.825906s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of S[w ..w ]
│ │ │                   0   4
    
│ │ │              
    
│ │ │ @@ -185,24 +185,24 @@
│ │ │  
│ │ │  o11 : Ideal of S
│ │ │              
    
│ │ │                
    i12 : time reesIdeal (I, I_0);
│ │ │ - -- used 1.49044s (cpu); 1.21436s (thread); 0s (gc)
│ │ │ + -- used 1.813s (cpu); 1.44855s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of S[w ..w ]
│ │ │                    0   3
    
│ │ │              
    
│ │ │                
    i13 : time reesIdeal (I, I_0, Jacobian =>false);
│ │ │ - -- used 1.63351s (cpu); 1.25312s (thread); 0s (gc)
│ │ │ + -- used 1.68593s (cpu); 1.44874s (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 1.03324s (cpu); 0.7369s (thread); 0s (gc)
│ │ │ │ + -- used 1.07044s (cpu); 0.825906s (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 1.49044s (cpu); 1.21436s (thread); 0s (gc)
│ │ │ │ + -- used 1.813s (cpu); 1.44855s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o12 : Ideal of S[w ..w ]
│ │ │ │                    0   3
│ │ │ │  i13 : time reesIdeal (I, I_0, Jacobian =>false);
│ │ │ │ - -- used 1.63351s (cpu); 1.25312s (thread); 0s (gc)
│ │ │ │ + -- used 1.68593s (cpu); 1.44874s (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
│ │ │ @@ -110,24 +110,24 @@
│ │ │  
│ │ │  o3 : Ideal of S
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time V1 = reesIdeal i;
│ │ │ - -- used 0.0281685s (cpu); 0.0259639s (thread); 0s (gc)
│ │ │ + -- used 0.486058s (cpu); 0.0833724s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : Ideal of S[w ..w ]
│ │ │                   0   6
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time V2 = reesIdeal(i,i_0);
│ │ │ - -- used 0.116569s (cpu); 0.115819s (thread); 0s (gc)
│ │ │ + -- used 0.205529s (cpu); 0.14747s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : Ideal of S[w ..w ]
│ │ │                   0   6
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │ @@ -164,24 +164,24 @@
│ │ │  
│ │ │  o8 : Ideal of S
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : time I1 = reesIdeal i;
│ │ │ - -- used 0.0193057s (cpu); 0.0182587s (thread); 0s (gc)
│ │ │ + -- used 0.188879s (cpu); 0.0391355s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : Ideal of S[w ..w ]
│ │ │                   0   2
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : time I2 = reesIdeal(i,i_0);
│ │ │ - -- used 0.00768211s (cpu); 0.00741424s (thread); 0s (gc)
│ │ │ + -- used 0.0218421s (cpu); 0.0100216s (thread); 0s (gc)
│ │ │  
│ │ │  o10 : Ideal of S[w ..w ]
│ │ │                    0   2
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -51,20 +51,20 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                  3    2
│ │ │ │       - x x x , x  - x x )
│ │ │ │          0 2 4   1    0 4
│ │ │ │  
│ │ │ │  o3 : Ideal of S
│ │ │ │  i4 : time V1 = reesIdeal i;
│ │ │ │ - -- used 0.0281685s (cpu); 0.0259639s (thread); 0s (gc)
│ │ │ │ + -- used 0.486058s (cpu); 0.0833724s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : Ideal of S[w ..w ]
│ │ │ │                   0   6
│ │ │ │  i5 : time V2 = reesIdeal(i,i_0);
│ │ │ │ - -- used 0.116569s (cpu); 0.115819s (thread); 0s (gc)
│ │ │ │ + -- used 0.205529s (cpu); 0.14747s (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
│ │ │ │ @@ -81,20 +81,20 @@
│ │ │ │  i8 : i=minors(2,m)
│ │ │ │  
│ │ │ │               2        2
│ │ │ │  o8 = ideal (a , a*b, b )
│ │ │ │  
│ │ │ │  o8 : Ideal of S
│ │ │ │  i9 : time I1 = reesIdeal i;
│ │ │ │ - -- used 0.0193057s (cpu); 0.0182587s (thread); 0s (gc)
│ │ │ │ + -- used 0.188879s (cpu); 0.0391355s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 : Ideal of S[w ..w ]
│ │ │ │                   0   2
│ │ │ │  i10 : time I2 = reesIdeal(i,i_0);
│ │ │ │ - -- used 0.00768211s (cpu); 0.00741424s (thread); 0s (gc)
│ │ │ │ + -- used 0.0218421s (cpu); 0.0100216s (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/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 = .2611710043999999
│ │ │ +o8 = .2713961083636363
│ │ │  
│ │ │  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 = .07145842478873242
│ │ │ +o11 = .08430326596551725
│ │ │  
│ │ │  o11 : RR (of precision 53)
│ │ │  
│ │ │  i12 : time regularity I2  
│ │ │ - -- used 0.00248761s (cpu); 0.00248695s (thread); 0s (gc)
│ │ │ + -- used 0.00258513s (cpu); 0.00258901s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = 4
│ │ │  
│ │ │  i13 :
│ │ ├── ./usr/share/doc/Macaulay2/Regularity/html/_m__Regularity.html
│ │ │ @@ -176,15 +176,15 @@
│ │ │  o7 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : benchmark "mRegularity I1"
│ │ │  
│ │ │ -o8 = .2611710043999999
│ │ │ +o8 = .2713961083636363
│ │ │  
│ │ │  o8 : RR (of precision 53)
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    This is an example where regularity is faster than mRegularity.
    
│ │ │          
│ │ │ @@ -204,23 +204,23 @@
│ │ │  o10 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i11 : benchmark " mRegularity I2"
│ │ │  
│ │ │ -o11 = .07145842478873242
│ │ │ +o11 = .08430326596551725
│ │ │  
│ │ │  o11 : RR (of precision 53)
    
│ │ │              
    
│ │ │                
    i12 : time regularity I2  
│ │ │ - -- used 0.00248761s (cpu); 0.00248695s (thread); 0s (gc)
│ │ │ + -- used 0.00258513s (cpu); 0.00258901s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = 4
    
│ │ │              
    
│ │ │          
    This symbol is provided by the package Regularity.
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -94,34 +94,34 @@
│ │ │ │                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 = .2611710043999999
│ │ │ │ +o8 = .2713961083636363
│ │ │ │  
│ │ │ │  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 = .07145842478873242
│ │ │ │ +o11 = .08430326596551725
│ │ │ │  
│ │ │ │  o11 : RR (of precision 53)
│ │ │ │  i12 : time regularity I2
│ │ │ │ - -- used 0.00248761s (cpu); 0.00248695s (thread); 0s (gc)
│ │ │ │ + -- used 0.00258513s (cpu); 0.00258901s (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/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 0.105384s (cpu); 0.0634894s (thread); 0s (gc)
│ │ │ + -- used 0.141133s (cpu); 0.0772861s (thread); 0s (gc)
│ │ │  
│ │ │  i4 : time (P1xP1xP1,P1xP1xP1') = cayleyTrick(P1xP1,1)
│ │ │ - -- used 0.120535s (cpu); 0.0680862s (thread); 0s (gc)
│ │ │ + -- used 0.155746s (cpu); 0.0860477s (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 0.128164s (cpu); 0.0856389s (thread); 0s (gc)
│ │ │ + -- used 0.184604s (cpu); 0.123887s (thread); 0s (gc)
│ │ │  
│ │ │  i6 : assert(oo == (P1xP1xP1,P1xP1xP1'))
│ │ │  
│ │ │  i7 : time cayleyTrick(P1xP1,2,Duality=>true);
│ │ │ - -- used 0.140744s (cpu); 0.0923886s (thread); 0s (gc)
│ │ │ + -- used 0.173958s (cpu); 0.114587s (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 0.122683s (cpu); 0.065425s (thread); 0s (gc)
│ │ │ + -- used 0.151088s (cpu); 0.0727082s (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 0.110296s (cpu); 0.0572802s (thread); 0s (gc)
│ │ │ + -- used 0.143305s (cpu); 0.0655288s (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 0.222628s (cpu); 0.114447s (thread); 0s (gc)
│ │ │ + -- used 0.280202s (cpu); 0.133291s (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 0.148849s (cpu); 0.0872414s (thread); 0s (gc)
│ │ │ + -- used 0.178663s (cpu); 0.0975334s (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 5.02713s (cpu); 4.55844s (thread); 0s (gc)
│ │ │ + -- used 5.32126s (cpu); 4.96784s (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 1.12217s (cpu); 1.02004s (thread); 0s (gc)
│ │ │ + -- used 1.19952s (cpu); 1.13884s (thread); 0s (gc)
│ │ │  
│ │ │  i5 : -- X-resultant of V
│ │ │       time Xres = fromPluckerToStiefel dualize ChowV;
│ │ │ - -- used 0.305813s (cpu); 0.217168s (thread); 0s (gc)
│ │ │ + -- used 0.282821s (cpu); 0.223351s (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 0.206977s (cpu); 0.161989s (thread); 0s (gc)
│ │ │ + -- used 0.308846s (cpu); 0.242229s (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 0.00923004s (cpu); 0.00922235s (thread); 0s (gc)
│ │ │ + -- used 0.0110453s (cpu); 0.0110454s (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 0.0095562s (cpu); 0.00955663s (thread); 0s (gc)
│ │ │ + -- used 0.0115445s (cpu); 0.0115459s (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 0.488929s (cpu); 0.431808s (thread); 0s (gc)
│ │ │ + -- used 0.627095s (cpu); 0.568205s (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 0.177816s (cpu); 0.127633s (thread); 0s (gc)
│ │ │ + -- used 0.25687s (cpu); 0.187415s (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 0.183756s (cpu); 0.142182s (thread); 0s (gc)
│ │ │ + -- used 0.235299s (cpu); 0.170649s (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 0.0560351s (cpu); 0.0560385s (thread); 0s (gc)
│ │ │ + -- used 0.0652985s (cpu); 0.0652985s (thread); 0s (gc)
│ │ │  
│ │ │                 ZZ
│ │ │  o5 : Ideal of ----[a ..a ]
│ │ │                3331  0   9
│ │ │  
│ │ │  i6 : time Z = dualVariety(veronese(2,3,ZZ/3331),AssumeOrdinary=>true);
│ │ │ - -- used 0.590731s (cpu); 0.548602s (thread); 0s (gc)
│ │ │ + -- used 0.797107s (cpu); 0.713324s (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 0.109761s (cpu); 0.0701715s (thread); 0s (gc)
│ │ │ + -- used 0.14531s (cpu); 0.0812359s (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 0.04042s (cpu); 0.0404231s (thread); 0s (gc)
│ │ │ + -- used 0.0513505s (cpu); 0.0513533s (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 0.0192665s (cpu); 0.0192642s (thread); 0s (gc)
│ │ │ + -- used 0.0257401s (cpu); 0.0257415s (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 0.0171298s (cpu); 0.0171312s (thread); 0s (gc)
│ │ │ + -- used 0.021545s (cpu); 0.0215465s (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  + 7p p  + 6p p  + -p p  + --p )
│ │ │         4 3   9 0 4     1 4     2 4   9 3 4   10 4
│ │ │  
│ │ │  o1 : Ideal of QQ[p ..p ]
│ │ │                    0   4
│ │ │  
│ │ │  i2 : time hurwitzForm Q
│ │ │ - -- used 0.0374164s (cpu); 0.0374172s (thread); 0s (gc)
│ │ │ + -- used 0.0437654s (cpu); 0.0437713s (thread); 0s (gc)
│ │ │  
│ │ │                2                                 2                      
│ │ │  o2 = 11966535p    + 14645610p   p    + 11354175p    + 1666980p   p    +
│ │ │                0,1            0,1 0,2            0,2           0,1 1,2  
│ │ │       ------------------------------------------------------------------------
│ │ │                                 2                                          
│ │ │       4456620p   p    + 1127196p    + 54176850p   p    + 20326950p   p    +
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_is__Coisotropic.out
│ │ │ @@ -26,15 +26,15 @@
│ │ │       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 0.00831348s (cpu); 0.00831097s (thread); 0s (gc)
│ │ │ + -- used 0.0100823s (cpu); 0.0100822s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = true
│ │ │  
│ │ │  i3 : -- random quadric in G(1,3)
│ │ │       w' = random(2,Grass(1,3))
│ │ │  
│ │ │         2     5           10 2     2                          2      3        
│ │ │ @@ -56,12 +56,12 @@
│ │ │       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 0.00672256s (cpu); 0.00672294s (thread); 0s (gc)
│ │ │ + -- used 0.00886348s (cpu); 0.00886589s (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 0.0197561s (cpu); 0.019757s (thread); 0s (gc)
│ │ │ + -- used 0.0234582s (cpu); 0.023472s (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 0.00363368s (cpu); 0.00363028s (thread); 0s (gc)
│ │ │ + -- used 0.00391973s (cpu); 0.00391665s (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 @@
│ │ │       10   2   7   2   5 3
│ │ │       --p p  + -p p  + -p }
│ │ │        9 0 2   8 1 2   6 2
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : time (D,D') = macaulayFormula F
│ │ │ - -- used 0.00232045s (cpu); 0.00232055s (thread); 0s (gc)
│ │ │ + -- used 0.00279512s (cpu); 0.00279304s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = (| 9/2 9/4 3/4 7/4  7/9  7/10 0    0    0    0    0   0   0    0    0   
│ │ │        | 0   9/2 0   9/4  3/4  0    7/4  7/9  7/10 0    0   0   0    0    0   
│ │ │        | 0   0   9/2 0    9/4  3/4  0    7/4  7/9  7/10 0   0   0    0    0   
│ │ │        | 0   0   0   9/2  0    0    9/4  3/4  0    0    7/4 7/9 7/10 0    0   
│ │ │        | 0   0   0   0    9/2  0    0    9/4  3/4  0    0   7/4 7/9  7/10 0   
│ │ │        | 0   0   0   0    0    9/2  0    0    9/4  3/4  0   0   7/4  7/9  7/10
│ │ ├── ./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 0.00446275s (cpu); 0.00446052s (thread); 0s (gc)
│ │ │ + -- used 0.0063302s (cpu); 0.00633055s (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 0.101231s (cpu); 0.0491833s (thread); 0s (gc)
│ │ │ + -- used 0.131861s (cpu); 0.0664408s (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 0.0311593s (cpu); 0.0311572s (thread); 0s (gc)
│ │ │ + -- used 0.0421751s (cpu); 0.0421748s (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 0.144036s (cpu); 0.144016s (thread); 0s (gc)
│ │ │ + -- used 0.190179s (cpu); 0.189993s (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 0.286935s (cpu); 0.170618s (thread); 0s (gc)
│ │ │ + -- used 0.344301s (cpu); 0.217012s (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 0.145456s (cpu); 0.0909893s (thread); 0s (gc)
│ │ │ + -- used 0.168853s (cpu); 0.104261s (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 0.376249s (cpu); 0.321284s (thread); 0s (gc)
│ │ │ + -- used 0.455063s (cpu); 0.394962s (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 0.62408s (cpu); 0.567743s (thread); 0s (gc)
│ │ │ + -- used 0.672402s (cpu); 0.615665s (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 0.024024s (cpu); 0.0240227s (thread); 0s (gc)
│ │ │ + -- used 0.0323992s (cpu); 0.0323992s (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 2.64631s (cpu); 2.04937s (thread); 0s (gc)
│ │ │ + -- used 2.68354s (cpu); 2.12845s (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 0.588475s (cpu); 0.418407s (thread); 0s (gc)
│ │ │ + -- used 0.458825s (cpu); 0.390027s (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 0.0288231s (cpu); 0.0288233s (thread); 0s (gc)
│ │ │ + -- used 0.0576008s (cpu); 0.0576002s (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 0.1176s (cpu); 0.0698098s (thread); 0s (gc)
│ │ │ + -- used 0.168496s (cpu); 0.102076s (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 0.0316428s (cpu); 0.0316457s (thread); 0s (gc)
│ │ │ + -- used 0.0387531s (cpu); 0.0387574s (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 0.103735s (cpu); 0.0542823s (thread); 0s (gc)
│ │ │ + -- used 0.123886s (cpu); 0.0674976s (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 0.149537s (cpu); 0.110227s (thread); 0s (gc)
│ │ │ + -- used 0.164202s (cpu); 0.108623s (thread); 0s (gc)
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_cayley__Trick.html
│ │ │ @@ -86,24 +86,24 @@
│ │ │  o2 : Ideal of QQ[x ..x ]
│ │ │                    0   3
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time (P1xP1xP2,P1xP1xP2') = cayleyTrick(P1xP1,2);
│ │ │ - -- used 0.105384s (cpu); 0.0634894s (thread); 0s (gc)
    
│ │ │ + -- used 0.141133s (cpu); 0.0772861s (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 0.120535s (cpu); 0.0680862s (thread); 0s (gc)
│ │ │ + -- used 0.155746s (cpu); 0.0860477s (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   
│ │ │ @@ -130,26 +130,26 @@
│ │ │            
    
│ │ │          
    If the option Duality is set to true, then the method applies the so-called "dual Cayley trick".
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -38,20 +38,20 @@
│ │ │ │  
│ │ │ │  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 0.105384s (cpu); 0.0634894s (thread); 0s (gc)
│ │ │ │ + -- used 0.141133s (cpu); 0.0772861s (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 0.120535s (cpu); 0.0680862s (thread); 0s (gc)
│ │ │ │ + -- used 0.155746s (cpu); 0.0860477s (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
│ │ │ │ @@ -73,18 +73,18 @@
│ │ │ │       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 0.128164s (cpu); 0.0856389s (thread); 0s (gc)
│ │ │ │ + -- used 0.184604s (cpu); 0.123887s (thread); 0s (gc)
│ │ │ │  i6 : assert(oo == (P1xP1xP1,P1xP1xP1'))
│ │ │ │  i7 : time cayleyTrick(P1xP1,2,Duality=>true);
│ │ │ │ - -- used 0.140744s (cpu); 0.0923886s (thread); 0s (gc)
│ │ │ │ + -- used 0.173958s (cpu); 0.114587s (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
│ │ │ @@ -90,15 +90,15 @@
│ │ │  o2 : Ideal of P3
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i5 : time cayleyTrick(P1xP1,1,Duality=>true);
│ │ │ - -- used 0.128164s (cpu); 0.0856389s (thread); 0s (gc)
    
│ │ │ + -- used 0.184604s (cpu); 0.123887s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i6 : assert(oo == (P1xP1xP1,P1xP1xP1'))
    
│ │ │              
    
│ │ │                
    i7 : time cayleyTrick(P1xP1,2,Duality=>true);
│ │ │ - -- used 0.140744s (cpu); 0.0923886s (thread); 0s (gc)
    
│ │ │ + -- used 0.173958s (cpu); 0.114587s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i8 : assert(oo == (P1xP1xP2,P1xP1xP2'))
    
│ │ │              
    
│ │ │                
    i3 : -- Chow equations of C
│ │ │       time eqsC = chowEquations chowForm C
│ │ │ - -- used 0.122683s (cpu); 0.065425s (thread); 0s (gc)
│ │ │ + -- used 0.151088s (cpu); 0.0727082s (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 
│ │ │ @@ -162,15 +162,15 @@
│ │ │  o5 : Ideal of P3
    
│ │ │              
    
│ │ │                
    i6 : -- Chow equations of D
│ │ │       time eqsD = chowEquations chowForm D
│ │ │ - -- used 0.110296s (cpu); 0.0572802s (thread); 0s (gc)
│ │ │ + -- used 0.143305s (cpu); 0.0655288s (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  -
│ │ │ @@ -222,30 +222,30 @@
│ │ │  o9 : Ideal of P3
    
│ │ │              
    
│ │ │                
    i10 : -- tangential Chow forms of Q
│ │ │        time (W0,W1,W2) = (tangentialChowForm(Q,0),tangentialChowForm(Q,1),tangentialChowForm(Q,2))
│ │ │ - -- used 0.222628s (cpu); 0.114447s (thread); 0s (gc)
│ │ │ + -- used 0.280202s (cpu); 0.133291s (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 0.148849s (cpu); 0.0872414s (thread); 0s (gc)
│ │ │ + -- used 0.178663s (cpu); 0.0975334s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = true
    
│ │ │              
    
│ │ │          
    Note that chowEquations(W,0) is not the same as chowEquations W.
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,15 +28,15 @@
│ │ │ │               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 0.122683s (cpu); 0.065425s (thread); 0s (gc)
│ │ │ │ + -- used 0.151088s (cpu); 0.0727082s (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
│ │ │ │ @@ -88,15 +88,15 @@
│ │ │ │               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 0.110296s (cpu); 0.0572802s (thread); 0s (gc)
│ │ │ │ + -- used 0.143305s (cpu); 0.0655288s (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  -
│ │ │ │ @@ -135,27 +135,27 @@
│ │ │ │  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 0.222628s (cpu); 0.114447s (thread); 0s (gc)
│ │ │ │ + -- used 0.280202s (cpu); 0.133291s (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 0.148849s (cpu); 0.0872414s (thread); 0s (gc)
│ │ │ │ + -- used 0.178663s (cpu); 0.0975334s (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
│ │ │ @@ -97,15 +97,15 @@
│ │ │                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 5.02713s (cpu); 4.55844s (thread); 0s (gc)
│ │ │ + -- used 5.32126s (cpu); 4.96784s (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      +
│ │ │ @@ -227,22 +227,22 @@
│ │ │         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 1.12217s (cpu); 1.02004s (thread); 0s (gc)
    
│ │ │ + -- used 1.19952s (cpu); 1.13884s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : -- X-resultant of V
│ │ │       time Xres = fromPluckerToStiefel dualize ChowV;
│ │ │ - -- used 0.305813s (cpu); 0.217168s (thread); 0s (gc)
    
│ │ │ + -- used 0.282821s (cpu); 0.223351s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : -- three generic ternary quadrics
│ │ │       F = genericPolynomials({2,2,2},ZZ/3331)
│ │ │  
│ │ │ @@ -257,15 +257,15 @@
│ │ │  o6 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : -- resultant of the three forms
│ │ │       time resF = resultant F;
│ │ │ - -- used 0.206977s (cpu); 0.161989s (thread); 0s (gc)
    
│ │ │ + -- used 0.308846s (cpu); 0.242229s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : assert(resF === sub(Xres,vars ring resF) and Xres === sub(resF,vars ring Xres))
    
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,15 +41,15 @@
│ │ │ │  
│ │ │ │                 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 5.02713s (cpu); 4.55844s (thread); 0s (gc)
│ │ │ │ + -- used 5.32126s (cpu); 4.96784s (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      +
│ │ │ │ @@ -234,33 +234,33 @@
│ │ │ │  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 1.12217s (cpu); 1.02004s (thread); 0s (gc)
│ │ │ │ + -- used 1.19952s (cpu); 1.13884s (thread); 0s (gc)
│ │ │ │  i5 : -- X-resultant of V
│ │ │ │       time Xres = fromPluckerToStiefel dualize ChowV;
│ │ │ │ - -- used 0.305813s (cpu); 0.217168s (thread); 0s (gc)
│ │ │ │ + -- used 0.282821s (cpu); 0.223351s (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 0.206977s (cpu); 0.161989s (thread); 0s (gc)
│ │ │ │ + -- used 0.308846s (cpu); 0.242229s (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 @@
│ │ │  
│ │ │  o2 : ZZ[a..c][x..y]
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time discriminant F
│ │ │ - -- used 0.00923004s (cpu); 0.00922235s (thread); 0s (gc)
│ │ │ + -- used 0.0110453s (cpu); 0.0110454s (thread); 0s (gc)
│ │ │  
│ │ │          2
│ │ │  o3 = - b  + 4a*c
│ │ │  
│ │ │  o3 : ZZ[a..c]
    
│ │ │              
│ │ │            
│ │ │ @@ -104,15 +104,15 @@
│ │ │  
│ │ │  o5 : ZZ[a..d][x..y]
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time discriminant F
│ │ │ - -- used 0.0095562s (cpu); 0.00955663s (thread); 0s (gc)
│ │ │ + -- used 0.0115445s (cpu); 0.0115459s (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]
    
│ │ │              
│ │ │            
│ │ │ @@ -165,15 +165,15 @@
│ │ │  
│ │ │  o12 : R'
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i13 : time D=discriminant pencil
│ │ │ - -- used 0.488929s (cpu); 0.431808s (thread); 0s (gc)
│ │ │ + -- used 0.627095s (cpu); 0.568205s (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 {}
│ │ │ │ @@ -23,28 +23,28 @@
│ │ │ │  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 0.00923004s (cpu); 0.00922235s (thread); 0s (gc)
│ │ │ │ + -- used 0.0110453s (cpu); 0.0110454s (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 0.0095562s (cpu); 0.00955663s (thread); 0s (gc)
│ │ │ │ + -- used 0.0115445s (cpu); 0.0115459s (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
│ │ │ │ @@ -74,15 +74,15 @@
│ │ │ │  
│ │ │ │                  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 0.488929s (cpu); 0.431808s (thread); 0s (gc)
│ │ │ │ + -- used 0.627095s (cpu); 0.568205s (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
│ │ │ @@ -90,28 +90,28 @@
│ │ │  o1 : Ideal of QQ[x ..x ]
│ │ │                    0   5
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time V' = dualVariety V
│ │ │ - -- used 0.177816s (cpu); 0.127633s (thread); 0s (gc)
│ │ │ + -- used 0.25687s (cpu); 0.187415s (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 0.183756s (cpu); 0.142182s (thread); 0s (gc)
│ │ │ + -- used 0.235299s (cpu); 0.170649s (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 $\mathbb{P}^9$
    
│ │ │          
│ │ │ @@ -131,25 +131,25 @@
│ │ │  o4 : ----[a ..a ][x ..x ]
│ │ │       3331  0   9   0   2
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -31,24 +31,24 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       x x )
│ │ │ │        0 3
│ │ │ │  
│ │ │ │  o1 : Ideal of QQ[x ..x ]
│ │ │ │                    0   5
│ │ │ │  i2 : time V' = dualVariety V
│ │ │ │ - -- used 0.177816s (cpu); 0.127633s (thread); 0s (gc)
│ │ │ │ + -- used 0.25687s (cpu); 0.187415s (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 0.183756s (cpu); 0.142182s (thread); 0s (gc)
│ │ │ │ + -- used 0.235299s (cpu); 0.170649s (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)
│ │ │ │  
│ │ │ │ @@ -60,21 +60,21 @@
│ │ │ │       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 0.0560351s (cpu); 0.0560385s (thread); 0s (gc)
│ │ │ │ + -- used 0.0652985s (cpu); 0.0652985s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                 ZZ
│ │ │ │  o5 : Ideal of ----[a ..a ]
│ │ │ │                3331  0   9
│ │ │ │  i6 : time Z = dualVariety(veronese(2,3,ZZ/3331),AssumeOrdinary=>true);
│ │ │ │ - -- used 0.590731s (cpu); 0.548602s (thread); 0s (gc)
│ │ │ │ + -- used 0.797107s (cpu); 0.713324s (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 @@
│ │ │  o1 : Ideal of QQ[x ..x ]
│ │ │                    0   3
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i5 : time discF = ideal discriminant F;
│ │ │ - -- used 0.0560351s (cpu); 0.0560385s (thread); 0s (gc)
│ │ │ + -- used 0.0652985s (cpu); 0.0652985s (thread); 0s (gc)
│ │ │  
│ │ │                 ZZ
│ │ │  o5 : Ideal of ----[a ..a ]
│ │ │                3331  0   9
    
│ │ │              
    
│ │ │                
    i6 : time Z = dualVariety(veronese(2,3,ZZ/3331),AssumeOrdinary=>true);
│ │ │ - -- used 0.590731s (cpu); 0.548602s (thread); 0s (gc)
│ │ │ + -- used 0.797107s (cpu); 0.713324s (thread); 0s (gc)
│ │ │  
│ │ │                 ZZ
│ │ │  o6 : Ideal of ----[x ..x ]
│ │ │                3331  0   9
    
│ │ │              
    
│ │ │                
    i2 : time fromPluckerToStiefel dualize chowForm C
│ │ │ - -- used 0.109761s (cpu); 0.0701715s (thread); 0s (gc)
│ │ │ + -- used 0.14531s (cpu); 0.0812359s (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    -
│ │ │ @@ -138,15 +138,15 @@
│ │ │  o2 : QQ[x   ..x   ]
│ │ │           0,0   1,3
    
│ │ │              
    
│ │ │                
    i3 : time fromPluckerToStiefel(dualize chowForm C,AffineChartGrass=>{0,1})
│ │ │ - -- used 0.04042s (cpu); 0.0404231s (thread); 0s (gc)
│ │ │ + -- used 0.0513505s (cpu); 0.0513533s (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
│ │ │ @@ -179,15 +179,15 @@
│ │ │              
    		
│ │ │                
    i5 : w = chowForm C;
    
│ │ │              
    
│ │ │                
    i6 : time U = apply(subsets(4,2),s->ideal fromPluckerToStiefel(w,AffineChartGrass=>s))
│ │ │ - -- used 0.0192665s (cpu); 0.0192642s (thread); 0s (gc)
│ │ │ + -- used 0.0257401s (cpu); 0.0257415s (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    +
│ │ │ @@ -227,15 +227,15 @@
│ │ │  
│ │ │  o6 : List
    
│ │ │              
    
│ │ │                
    i7 : time apply(U,u->dim singularLocus u)
│ │ │ - -- used 0.0171298s (cpu); 0.0171312s (thread); 0s (gc)
│ │ │ + -- used 0.021545s (cpu); 0.0215465s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {2, 2, 2, 2, 2, 2}
│ │ │  
│ │ │  o7 : List
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,15 +26,15 @@
│ │ │ │               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 0.109761s (cpu); 0.0701715s (thread); 0s (gc)
│ │ │ │ + -- used 0.14531s (cpu); 0.0812359s (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    -
│ │ │ │ @@ -75,15 +75,15 @@
│ │ │ │            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 0.04042s (cpu); 0.0404231s (thread); 0s (gc)
│ │ │ │ + -- used 0.0513505s (cpu); 0.0513533s (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
│ │ │ │ @@ -105,15 +105,15 @@
│ │ │ │  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 0.0192665s (cpu); 0.0192642s (thread); 0s (gc)
│ │ │ │ + -- used 0.0257401s (cpu); 0.0257415s (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    +
│ │ │ │ @@ -149,15 +149,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │             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 0.0171298s (cpu); 0.0171312s (thread); 0s (gc)
│ │ │ │ + -- used 0.021545s (cpu); 0.0215465s (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
│ │ │ @@ -92,15 +92,15 @@
│ │ │  o1 : Ideal of QQ[p ..p ]
│ │ │                    0   4
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time hurwitzForm Q
│ │ │ - -- used 0.0374164s (cpu); 0.0374172s (thread); 0s (gc)
│ │ │ + -- used 0.0437654s (cpu); 0.0437713s (thread); 0s (gc)
│ │ │  
│ │ │                2                                 2                      
│ │ │  o2 = 11966535p    + 14645610p   p    + 11354175p    + 1666980p   p    +
│ │ │                0,1            0,1 0,2            0,2           0,1 1,2  
│ │ │       ------------------------------------------------------------------------
│ │ │                                 2                                          
│ │ │       4456620p   p    + 1127196p    + 54176850p   p    + 20326950p   p    +
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,15 +34,15 @@
│ │ │ │         5 2   7                       2        3 2
│ │ │ │       + -p  + -p p  + 7p p  + 6p p  + -p p  + --p )
│ │ │ │         4 3   9 0 4     1 4     2 4   9 3 4   10 4
│ │ │ │  
│ │ │ │  o1 : Ideal of QQ[p ..p ]
│ │ │ │                    0   4
│ │ │ │  i2 : time hurwitzForm Q
│ │ │ │ - -- used 0.0374164s (cpu); 0.0374172s (thread); 0s (gc)
│ │ │ │ + -- used 0.0437654s (cpu); 0.0437713s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                2                                 2
│ │ │ │  o2 = 11966535p    + 14645610p   p    + 11354175p    + 1666980p   p    +
│ │ │ │                0,1            0,1 0,2            0,2           0,1 1,2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                                 2
│ │ │ │       4456620p   p    + 1127196p    + 54176850p   p    + 20326950p   p    +
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_is__Coisotropic.html
│ │ │ @@ -104,15 +104,15 @@
│ │ │           p   p    - p   p    + p   p
│ │ │            1,2 0,3    0,2 1,3    0,1 2,3
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time isCoisotropic w
│ │ │ - -- used 0.00831348s (cpu); 0.00831097s (thread); 0s (gc)
│ │ │ + -- used 0.0100823s (cpu); 0.0100822s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : -- random quadric in G(1,3)
│ │ │ @@ -140,15 +140,15 @@
│ │ │           p   p    - p   p    + p   p
│ │ │            1,2 0,3    0,2 1,3    0,1 2,3
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time isCoisotropic w'
│ │ │ - -- used 0.00672256s (cpu); 0.00672294s (thread); 0s (gc)
│ │ │ + -- used 0.00886348s (cpu); 0.00886589s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = false
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -44,15 +44,15 @@
│ │ │ │  
│ │ │ │       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 0.00831348s (cpu); 0.00831097s (thread); 0s (gc)
│ │ │ │ + -- used 0.0100823s (cpu); 0.0100822s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = true
│ │ │ │  i3 : -- random quadric in G(1,3)
│ │ │ │       w' = random(2,Grass(1,3))
│ │ │ │  
│ │ │ │         2     5           10 2     2                          2      3
│ │ │ │  o3 = 6p    + -p   p    + --p    + -p   p    + 10p   p    + 5p    + --p   p
│ │ │ │ @@ -72,15 +72,15 @@
│ │ │ │  
│ │ │ │       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 0.00672256s (cpu); 0.00672294s (thread); 0s (gc)
│ │ │ │ + -- used 0.00886348s (cpu); 0.00886589s (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
│ │ │ @@ -117,15 +117,15 @@
│ │ │  o3 : Ideal of -----[x ..x ]
│ │ │                33331  0   5
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time isInCoisotropic(L,I) -- whether L belongs to Z_1(V(I))
│ │ │ - -- used 0.0197561s (cpu); 0.019757s (thread); 0s (gc)
│ │ │ + -- used 0.0234582s (cpu); 0.023472s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = true
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -54,15 +54,15 @@
│ │ │ │       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 0.0197561s (cpu); 0.019757s (thread); 0s (gc)
│ │ │ │ + -- used 0.0234582s (cpu); 0.023472s (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
│ │ │ @@ -87,15 +87,15 @@
│ │ │  
│ │ │  o1 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time (D,D') = macaulayFormula F
│ │ │ - -- used 0.00363368s (cpu); 0.00363028s (thread); 0s (gc)
│ │ │ + -- used 0.00391973s (cpu); 0.00391665s (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  
│ │ │ @@ -158,15 +158,15 @@
│ │ │  
│ │ │  o3 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time (D,D') = macaulayFormula F
│ │ │ - -- used 0.00232045s (cpu); 0.00232055s (thread); 0s (gc)
│ │ │ + -- used 0.00279512s (cpu); 0.00279304s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = (| 9/2 9/4 3/4 7/4  7/9  7/10 0    0    0    0    0   0   0    0    0   
│ │ │        | 0   9/2 0   9/4  3/4  0    7/4  7/9  7/10 0    0   0   0    0    0   
│ │ │        | 0   0   9/2 0    9/4  3/4  0    7/4  7/9  7/10 0   0   0    0    0   
│ │ │        | 0   0   0   9/2  0    0    9/4  3/4  0    0    7/4 7/9 7/10 0    0   
│ │ │        | 0   0   0   0    9/2  0    0    9/4  3/4  0    0   7/4 7/9  7/10 0   
│ │ │        | 0   0   0   0    0    9/2  0    0    9/4  3/4  0   0   7/4  7/9  7/10
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,15 +28,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                     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 0.00363368s (cpu); 0.00363028s (thread); 0s (gc)
│ │ │ │ + -- used 0.00391973s (cpu); 0.00391665s (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
│ │ │ │ @@ -91,15 +91,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       10   2   7   2   5 3
│ │ │ │       --p p  + -p p  + -p }
│ │ │ │        9 0 2   8 1 2   6 2
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : time (D,D') = macaulayFormula F
│ │ │ │ - -- used 0.00232045s (cpu); 0.00232055s (thread); 0s (gc)
│ │ │ │ + -- used 0.00279512s (cpu); 0.00279304s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = (| 9/2 9/4 3/4 7/4  7/9  7/10 0    0    0    0    0   0   0    0    0
│ │ │ │        | 0   9/2 0   9/4  3/4  0    7/4  7/9  7/10 0    0   0   0    0    0
│ │ │ │        | 0   0   9/2 0    9/4  3/4  0    7/4  7/9  7/10 0   0   0    0    0
│ │ │ │        | 0   0   0   9/2  0    0    9/4  3/4  0    0    7/4 7/9 7/10 0    0
│ │ │ │        | 0   0   0   0    9/2  0    0    9/4  3/4  0    0   7/4 7/9  7/10 0
│ │ │ │        | 0   0   0   0    0    9/2  0    0    9/4  3/4  0   0   7/4  7/9  7/10
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_plucker.html
│ │ │ @@ -92,15 +92,15 @@
│ │ │  
│ │ │  o3 : Ideal of P4
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time p = plucker L
│ │ │ - -- used 0.00446275s (cpu); 0.00446052s (thread); 0s (gc)
│ │ │ + -- used 0.0063302s (cpu); 0.00633055s (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  
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -109,15 +109,15 @@
│ │ │  
│ │ │  o4 : Ideal of G'1'4
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time L' = plucker p
│ │ │ - -- used 0.101231s (cpu); 0.0491833s (thread); 0s (gc)
│ │ │ + -- used 0.131861s (cpu); 0.0664408s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = ideal (x  + 8480x  - 11656x , x  - 6727x  + 14853x , x  + 15777x  -
│ │ │               2        3         4   1        3         4   0         3  
│ │ │       ------------------------------------------------------------------------
│ │ │       664x )
│ │ │           4
│ │ │  
│ │ │ @@ -138,15 +138,15 @@
│ │ │  
│ │ │  o7 : Ideal of G'1'4
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : time W = plucker Y; -- surface swept out by the lines of Y
│ │ │ - -- used 0.0311593s (cpu); 0.0311572s (thread); 0s (gc)
│ │ │ + -- used 0.0421751s (cpu); 0.0421748s (thread); 0s (gc)
│ │ │  
│ │ │  o8 : Ideal of P4
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : (codim W,degree W)
│ │ │ @@ -158,15 +158,15 @@
│ │ │            
│ │ │          
│ │ │          
    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 0.144036s (cpu); 0.144016s (thread); 0s (gc)
│ │ │ + -- used 0.190179s (cpu); 0.189993s (thread); 0s (gc)
│ │ │  
│ │ │  o10 : Ideal of G'1'4
    
│ │ │              
    
│ │ │                
    i11 : assert(Y' == Y)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,28 +28,28 @@
│ │ │ │               2        3         4   1        3         4   0         3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       664x )
│ │ │ │           4
│ │ │ │  
│ │ │ │  o3 : Ideal of P4
│ │ │ │  i4 : time p = plucker L
│ │ │ │ - -- used 0.00446275s (cpu); 0.00446052s (thread); 0s (gc)
│ │ │ │ + -- used 0.0063302s (cpu); 0.00633055s (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 0.101231s (cpu); 0.0491833s (thread); 0s (gc)
│ │ │ │ + -- used 0.131861s (cpu); 0.0664408s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = ideal (x  + 8480x  - 11656x , x  - 6727x  + 14853x , x  + 15777x  -
│ │ │ │               2        3         4   1        3         4   0         3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       664x )
│ │ │ │           4
│ │ │ │  
│ │ │ │ @@ -60,26 +60,26 @@
│ │ │ │  $W\subset\mathbb{P}^n$ 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 0.0311593s (cpu); 0.0311572s (thread); 0s (gc)
│ │ │ │ + -- used 0.0421751s (cpu); 0.0421748s (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 0.144036s (cpu); 0.144016s (thread); 0s (gc)
│ │ │ │ + -- used 0.190179s (cpu); 0.189993s (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
│ │ │ @@ -108,15 +108,15 @@
│ │ │  
│ │ │  o2 : List
│ │ │              
    
│ │ │                
    i3 : time resultant(F,Algorithm=>"Poisson2")
│ │ │ - -- used 0.286935s (cpu); 0.170618s (thread); 0s (gc)
│ │ │ + -- used 0.344301s (cpu); 0.217012s (thread); 0s (gc)
│ │ │  
│ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4   
│ │ │  o3 = - -----------------------------a  - -------------------------------a b -
│ │ │             2222549728809984000000            12700284164628480000000         
│ │ │       ------------------------------------------------------------------------
│ │ │       348237304382147063838108483692249 3 2  
│ │ │       ---------------------------------a b  -
│ │ │ @@ -132,15 +132,15 @@
│ │ │  
│ │ │  o3 : QQ[a..b]
    
│ │ │              
    
│ │ │                
    i4 : time resultant(F,Algorithm=>"Macaulay2")
│ │ │ - -- used 0.145456s (cpu); 0.0909893s (thread); 0s (gc)
│ │ │ + -- used 0.168853s (cpu); 0.104261s (thread); 0s (gc)
│ │ │  
│ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4   
│ │ │  o4 = - -----------------------------a  - -------------------------------a b -
│ │ │             2222549728809984000000            12700284164628480000000         
│ │ │       ------------------------------------------------------------------------
│ │ │       348237304382147063838108483692249 3 2  
│ │ │       ---------------------------------a b  -
│ │ │ @@ -156,15 +156,15 @@
│ │ │  
│ │ │  o4 : QQ[a..b]
    
│ │ │              
    
│ │ │                
    i5 : time resultant(F,Algorithm=>"Poisson")
│ │ │ - -- used 0.376249s (cpu); 0.321284s (thread); 0s (gc)
│ │ │ + -- used 0.455063s (cpu); 0.394962s (thread); 0s (gc)
│ │ │  
│ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4   
│ │ │  o5 = - -----------------------------a  - -------------------------------a b -
│ │ │             2222549728809984000000            12700284164628480000000         
│ │ │       ------------------------------------------------------------------------
│ │ │       348237304382147063838108483692249 3 2  
│ │ │       ---------------------------------a b  -
│ │ │ @@ -180,15 +180,15 @@
│ │ │  
│ │ │  o5 : QQ[a..b]
    
│ │ │              
    
│ │ │                
    i6 : time resultant(F,Algorithm=>"Macaulay")
│ │ │ - -- used 0.62408s (cpu); 0.567743s (thread); 0s (gc)
│ │ │ + -- used 0.672402s (cpu); 0.615665s (thread); 0s (gc)
│ │ │  
│ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4   
│ │ │  o6 = - -----------------------------a  - -------------------------------a b -
│ │ │             2222549728809984000000            12700284164628480000000         
│ │ │       ------------------------------------------------------------------------
│ │ │       348237304382147063838108483692249 3 2  
│ │ │       ---------------------------------a b  -
│ │ │ ├── html2text {}
│ │ │ │ @@ -58,15 +58,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       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 0.286935s (cpu); 0.170618s (thread); 0s (gc)
│ │ │ │ + -- used 0.344301s (cpu); 0.217012s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4
│ │ │ │  o3 = - -----------------------------a  - -------------------------------a b -
│ │ │ │             2222549728809984000000            12700284164628480000000
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       348237304382147063838108483692249 3 2
│ │ │ │       ---------------------------------a b  -
│ │ │ │ @@ -78,15 +78,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       1146977327343523453866040839029   4   194441910898734675845094443 5
│ │ │ │       -------------------------------a*b  - ---------------------------b
│ │ │ │            1119954511872000000000                895963609497600000
│ │ │ │  
│ │ │ │  o3 : QQ[a..b]
│ │ │ │  i4 : time resultant(F,Algorithm=>"Macaulay2")
│ │ │ │ - -- used 0.145456s (cpu); 0.0909893s (thread); 0s (gc)
│ │ │ │ + -- used 0.168853s (cpu); 0.104261s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4
│ │ │ │  o4 = - -----------------------------a  - -------------------------------a b -
│ │ │ │             2222549728809984000000            12700284164628480000000
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       348237304382147063838108483692249 3 2
│ │ │ │       ---------------------------------a b  -
│ │ │ │ @@ -98,15 +98,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       1146977327343523453866040839029   4   194441910898734675845094443 5
│ │ │ │       -------------------------------a*b  - ---------------------------b
│ │ │ │            1119954511872000000000                895963609497600000
│ │ │ │  
│ │ │ │  o4 : QQ[a..b]
│ │ │ │  i5 : time resultant(F,Algorithm=>"Poisson")
│ │ │ │ - -- used 0.376249s (cpu); 0.321284s (thread); 0s (gc)
│ │ │ │ + -- used 0.455063s (cpu); 0.394962s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4
│ │ │ │  o5 = - -----------------------------a  - -------------------------------a b -
│ │ │ │             2222549728809984000000            12700284164628480000000
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       348237304382147063838108483692249 3 2
│ │ │ │       ---------------------------------a b  -
│ │ │ │ @@ -118,15 +118,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       1146977327343523453866040839029   4   194441910898734675845094443 5
│ │ │ │       -------------------------------a*b  - ---------------------------b
│ │ │ │            1119954511872000000000                895963609497600000
│ │ │ │  
│ │ │ │  o5 : QQ[a..b]
│ │ │ │  i6 : time resultant(F,Algorithm=>"Macaulay")
│ │ │ │ - -- used 0.62408s (cpu); 0.567743s (thread); 0s (gc)
│ │ │ │ + -- used 0.672402s (cpu); 0.615665s (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
│ │ │ @@ -92,15 +92,15 @@
│ │ │  
│ │ │  o2 : List
    
│ │ │              
    
│ │ │                
    i3 : time resultant F
│ │ │ - -- used 0.024024s (cpu); 0.0240227s (thread); 0s (gc)
│ │ │ + -- used 0.0323992s (cpu); 0.0323992s (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  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -155,15 +155,15 @@
│ │ │  
│ │ │  o4 : List
    
│ │ │              
    
│ │ │                
    i5 : time resultant F
│ │ │ - -- used 2.64631s (cpu); 2.04937s (thread); 0s (gc)
│ │ │ + -- used 2.68354s (cpu); 2.12845s (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  -
│ │ │ @@ -1790,15 +1790,15 @@
│ │ │  
│ │ │  o6 : List
    
│ │ │              
    
│ │ │                
    i7 : time # terms resultant F
│ │ │ - -- used 0.588475s (cpu); 0.418407s (thread); 0s (gc)
│ │ │ + -- used 0.458825s (cpu); 0.390027s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = 21894
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -32,15 +32,15 @@
│ │ │ │  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 0.024024s (cpu); 0.0240227s (thread); 0s (gc)
│ │ │ │ + -- used 0.0323992s (cpu); 0.0323992s (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  +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -86,15 +86,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │            3
│ │ │ │       + c x }
│ │ │ │          9 2
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : time resultant F
│ │ │ │ - -- used 2.64631s (cpu); 2.04937s (thread); 0s (gc)
│ │ │ │ + -- used 2.68354s (cpu); 2.12845s (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  -
│ │ │ │ @@ -1712,15 +1712,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                            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 0.588475s (cpu); 0.418407s (thread); 0s (gc)
│ │ │ │ + -- used 0.458825s (cpu); 0.390027s (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
│ │ │ @@ -97,15 +97,15 @@
│ │ │                    0   4
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : -- 0-th associated hypersurface of S in G(1,4) (Chow form)
│ │ │       time tangentialChowForm(S,0)
│ │ │ - -- used 0.0288231s (cpu); 0.0288233s (thread); 0s (gc)
│ │ │ + -- used 0.0576008s (cpu); 0.0576002s (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
│ │ │ @@ -118,15 +118,15 @@
│ │ │         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 0.1176s (cpu); 0.0698098s (thread); 0s (gc)
│ │ │ + -- used 0.168496s (cpu); 0.102076s (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      +
│ │ │ @@ -163,43 +163,43 @@
│ │ │         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 0.0316428s (cpu); 0.0316457s (thread); 0s (gc)
│ │ │ + -- used 0.0387531s (cpu); 0.0387574s (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 0.103735s (cpu); 0.0542823s (thread); 0s (gc)
│ │ │ + -- used 0.123886s (cpu); 0.0674976s (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 0.149537s (cpu); 0.110227s (thread); 0s (gc)
    
│ │ │ + -- used 0.164202s (cpu); 0.108623s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -63,15 +63,15 @@
│ │ │ │  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 0.0288231s (cpu); 0.0288233s (thread); 0s (gc)
│ │ │ │ + -- used 0.0576008s (cpu); 0.0576002s (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
│ │ │ │ @@ -88,15 +88,15 @@
│ │ │ │  - 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 0.1176s (cpu); 0.0698098s (thread); 0s (gc)
│ │ │ │ + -- used 0.168496s (cpu); 0.102076s (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      +
│ │ │ │ @@ -138,35 +138,35 @@
│ │ │ │  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 0.0316428s (cpu); 0.0316457s (thread); 0s (gc)
│ │ │ │ + -- used 0.0387531s (cpu); 0.0387574s (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 0.103735s (cpu); 0.0542823s (thread); 0s (gc)
│ │ │ │ + -- used 0.123886s (cpu); 0.0674976s (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 0.149537s (cpu); 0.110227s (thread); 0s (gc)
│ │ │ │ + -- used 0.164202s (cpu); 0.108623s (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/example-output/_resource_splimits.out
│ │ │ @@ -4,15 +4,15 @@
│ │ │  time(seconds)        700
│ │ │  file(blocks)         unlimited
│ │ │  data(kbytes)         unlimited
│ │ │  stack(kbytes)        8192
│ │ │  coredump(blocks)     unlimited
│ │ │  memory(kbytes)       850000
│ │ │  locked memory(kbytes) 8192
│ │ │ -process              63817
│ │ │ +process              63521
│ │ │  nofiles              512
│ │ │  vmemory(kbytes)      unlimited
│ │ │  locks                unlimited
│ │ │  rtprio               0
│ │ │  
│ │ │  o1 = 0
│ │ ├── ./usr/share/doc/Macaulay2/RunExternalM2/example-output/_run__External__M2.out
│ │ │ @@ -1,137 +1,137 @@
│ │ │  -- -*- M2-comint -*- hash: 2927978066455787395
│ │ │  
│ │ │  i1 : fn=temporaryFileName()|".m2"
│ │ │  
│ │ │ -o1 = /tmp/M2-29954-0/0.m2
│ │ │ +o1 = /tmp/M2-43124-0/0.m2
│ │ │  
│ │ │  i2 : fn< (stderr<<"Running"< ( exit(27); ); ///< (stderr<<"Spinning!!"<"/tmp/M2-29954-0/1.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/1.m2" >"/tmp/M2-43124-0/1.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  
│ │ │  i7 : h
│ │ │  
│ │ │  o7 = HashTable{"answer file" => null}
│ │ │                 "exit code" => 0
│ │ │                 "output file" => null
│ │ │                 "return code" => 0
│ │ │                 "statistics" => null
│ │ │ -               "time used" => 2
│ │ │ +               "time used" => 3
│ │ │                 value => 16
│ │ │  
│ │ │  o7 : HashTable
│ │ │  
│ │ │  i8 : h#value===4^2
│ │ │  
│ │ │  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/M2-29954-0/2.m2" >"/tmp/M2-29954-0/2.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/2.m2" >"/tmp/M2-43124-0/2.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
│ │ │  
│ │ │  i11 : h
│ │ │  
│ │ │ -o11 = HashTable{"answer file" => /tmp/M2-29954-0/2.ans}
│ │ │ +o11 = HashTable{"answer file" => /tmp/M2-43124-0/2.ans}
│ │ │                  "exit code" => 27
│ │ │ -                "output file" => /tmp/M2-29954-0/2.out
│ │ │ +                "output file" => /tmp/M2-43124-0/2.out
│ │ │                  "return code" => 6912
│ │ │                  "statistics" => null
│ │ │ -                "time used" => 2
│ │ │ +                "time used" => 1
│ │ │                  value => null
│ │ │  
│ │ │  o11 : HashTable
│ │ │  
│ │ │  i12 : fileExists(h#"output file")
│ │ │  
│ │ │  o12 = true
│ │ │  
│ │ │  i13 : fileExists(h#"answer file")
│ │ │  
│ │ │  o13 = false
│ │ │  
│ │ │  i14 : h=runExternalM2(fn,"spin",10,PreRunScript=>"ulimit -t 2");
│ │ │ -Running (ulimit -t 2 && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-29954-0/3.m2" >"/tmp/M2-29954-0/3.out" 2>&1 ))
│ │ │ +Running (ulimit -t 2 && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/3.m2" >"/tmp/M2-43124-0/3.out" 2>&1 ))
│ │ │  Killed
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
│ │ │  
│ │ │  i15 : h
│ │ │  
│ │ │ -o15 = HashTable{"answer file" => /tmp/M2-29954-0/3.ans}
│ │ │ +o15 = HashTable{"answer file" => /tmp/M2-43124-0/3.ans}
│ │ │                  "exit code" => 0
│ │ │ -                "output file" => /tmp/M2-29954-0/3.out
│ │ │ +                "output file" => /tmp/M2-43124-0/3.out
│ │ │                  "return code" => 9
│ │ │                  "statistics" => null
│ │ │                  "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/M2-29954-0/3.m2 automatically generated by RunExternalM2
│ │ │ +      i1 : -- Script /tmp/M2-43124-0/3.m2 automatically generated by RunExternalM2
│ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │  
│ │ │ -      i2 : load "/tmp/M2-29954-0/0.m2";
│ │ │ +      i2 : load "/tmp/M2-43124-0/0.m2";
│ │ │  
│ │ │ -      i3 : runExternalM2ReturnAnswer("/tmp/M2-29954-0/3.ans",spin (10));
│ │ │ +      i3 : runExternalM2ReturnAnswer("/tmp/M2-43124-0/3.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/M2-29954-0/4.m2" >"/tmp/M2-29954-0/4.out" 2>&1') >"/tmp/M2-29954-0/4.stat" 2>&1 ))
│ │ │ +Running (true && ( (/usr/bin/time --verbose sh -c '/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/4.m2" >"/tmp/M2-43124-0/4.out" 2>&1') >"/tmp/M2-43124-0/4.stat" 2>&1 ))
│ │ │  Finished running.
│ │ │  
│ │ │  i19 : h#"statistics"
│ │ │  
│ │ │ -o19 =         Command being timed: "sh -c /usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-29954-0/4.m2" >"/tmp/M2-29954-0/4.out" 2>&1"
│ │ │ -              User time (seconds): 5.28
│ │ │ -              System time (seconds): 0.08
│ │ │ -              Percent of CPU this job got: 95%
│ │ │ -              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:05.62
│ │ │ +o19 =         Command being timed: "sh -c /usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/4.m2" >"/tmp/M2-43124-0/4.out" 2>&1"
│ │ │ +              User time (seconds): 4.74
│ │ │ +              System time (seconds): 0.27
│ │ │ +              Percent of CPU this job got: 121%
│ │ │ +              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:04.14
│ │ │                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): 252580
│ │ │ +              Maximum resident set size (kbytes): 339096
│ │ │                Average resident set size (kbytes): 0
│ │ │                Major (requiring I/O) page faults: 0
│ │ │ -              Minor (reclaiming a frame) page faults: 9521
│ │ │ -              Voluntary context switches: 2087
│ │ │ -              Involuntary context switches: 1715
│ │ │ +              Minor (reclaiming a frame) page faults: 10623
│ │ │ +              Voluntary context switches: 6159
│ │ │ +              Involuntary context switches: 1241
│ │ │                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/M2-29954-0/6.m2" >"/tmp/M2-29954-0/6.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/6.m2" >"/tmp/M2-43124-0/6.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+9/4y 7/9x+7/10y 7x+3/7y |
│ │ │                 | 3/4x+7/4y 7/10x+7/3y 6/7x+6y |
│ │ │  
│ │ │                               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/M2-29954-0/7.m2" >"/tmp/M2-29954-0/7.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/7.m2" >"/tmp/M2-43124-0/7.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
│ │ │  
│ │ │ -o24 = HashTable{"answer file" => /tmp/M2-29954-0/7.ans}
│ │ │ +o24 = HashTable{"answer file" => /tmp/M2-43124-0/7.ans}
│ │ │                  "exit code" => 1
│ │ │ -                "output file" => /tmp/M2-29954-0/7.out
│ │ │ +                "output file" => /tmp/M2-43124-0/7.out
│ │ │                  "return code" => 256
│ │ │                  "statistics" => null
│ │ │ -                "time used" => 2
│ │ │ +                "time used" => 1
│ │ │                  value => null
│ │ │  
│ │ │  o24 : HashTable
│ │ │  
│ │ │  i25 : get(h#"output file")
│ │ │  
│ │ │  o25 = 
│ │ │ -      i1 : -- Script /tmp/M2-29954-0/7.m2 automatically generated by RunExternalM2
│ │ │ +      i1 : -- Script /tmp/M2-43124-0/7.m2 automatically generated by RunExternalM2
│ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │  
│ │ │ -      i2 : load "/tmp/M2-29954-0/0.m2";
│ │ │ +      i2 : load "/tmp/M2-43124-0/0.m2";
│ │ │  
│ │ │ -      i3 : runExternalM2ReturnAnswer("/tmp/M2-29954-0/7.ans",identity (cokernel(map(R^2,R^{3:{-1}},{{(9/2)*x+(9/4)*y, (7/9)*x+(7/10)*y, 7*x+(3/7)*y}, {(3/4)*x+(7/4)*y, (7/10)*x+(7/3)*y, (6/7)*x+6*y}}))));
│ │ │ +      i3 : runExternalM2ReturnAnswer("/tmp/M2-43124-0/7.ans",identity (cokernel(map(R^2,R^{3:{-1}},{{(9/2)*x+(9/4)*y, (7/9)*x+(7/10)*y, 7*x+(3/7)*y}, {(3/4)*x+(7/4)*y, (7/10)*x+(7/3)*y, (6/7)*x+6*y}}))));
│ │ │        stdio:4:74:(3):[1]: error: no method for binary operator ^ applied to objects:
│ │ │                    R (of class Symbol)
│ │ │              ^     2 (of class ZZ)
│ │ │  
│ │ │  
│ │ │  i26 : fn<"/tmp/M2-29954-0/8.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/8.m2" >"/tmp/M2-43124-0/8.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/M2-29954-0/9.m2" >"/tmp/M2-29954-0/9.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/9.m2" >"/tmp/M2-43124-0/9.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  
│ │ │  i30 : h#value
│ │ │  
│ │ │  o30 = QQ[x..y]
│ │ │  
│ │ │  o30 : PolynomialRing
│ │ ├── ./usr/share/doc/Macaulay2/RunExternalM2/html/_resource_splimits.html
│ │ │ @@ -75,15 +75,15 @@
│ │ │  time(seconds)        700
│ │ │  file(blocks)         unlimited
│ │ │  data(kbytes)         unlimited
│ │ │  stack(kbytes)        8192
│ │ │  coredump(blocks)     unlimited
│ │ │  memory(kbytes)       850000
│ │ │  locked memory(kbytes) 8192
│ │ │ -process              63817
│ │ │ +process              63521
│ │ │  nofiles              512
│ │ │  vmemory(kbytes)      unlimited
│ │ │  locks                unlimited
│ │ │  rtprio               0
│ │ │  
│ │ │  o1 = 0
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,15 +35,15 @@
│ │ │ │  time(seconds)        700
│ │ │ │  file(blocks)         unlimited
│ │ │ │  data(kbytes)         unlimited
│ │ │ │  stack(kbytes)        8192
│ │ │ │  coredump(blocks)     unlimited
│ │ │ │  memory(kbytes)       850000
│ │ │ │  locked memory(kbytes) 8192
│ │ │ │ -process              63817
│ │ │ │ +process              63521
│ │ │ │  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
│ │ │ @@ -84,15 +84,15 @@
│ │ │            
      For example, we can write a few functions to a temporary file:
      
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -115,28 +115,28 @@
│ │ │          
    
│ │ │            
    and then call them:
    
│ │ │          
    
│ │ │          
    
│ │ │                
    i1 : fn=temporaryFileName()|".m2"
│ │ │  
│ │ │ -o1 = /tmp/M2-29954-0/0.m2
    
│ │ │ +o1 = /tmp/M2-43124-0/0.m2
│ │ │              
    
│ │ │                
    i2 : fn<</// square = (x) -> (stderr<<"Running"<<endl; sleep(1); x^2); ///<<endl;
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i6 : h=runExternalM2(fn,"square",(4));
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-29954-0/1.m2" >"/tmp/M2-29954-0/1.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/1.m2" >"/tmp/M2-43124-0/1.out" 2>&1 ))
│ │ │  Finished running.
    
│ │ │              
    
│ │ │                
    i7 : h
│ │ │  
│ │ │  o7 = HashTable{"answer file" => null}
│ │ │                 "exit code" => 0
│ │ │                 "output file" => null
│ │ │                 "return code" => 0
│ │ │                 "statistics" => null
│ │ │ -               "time used" => 2
│ │ │ +               "time used" => 3
│ │ │                 value => 16
│ │ │  
│ │ │  o7 : HashTable
    
│ │ │              
    
│ │ │ @@ -157,29 +157,29 @@
│ │ │            
    
│ │ │            
    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/M2-29954-0/2.m2" >"/tmp/M2-29954-0/2.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/2.m2" >"/tmp/M2-43124-0/2.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
    
│ │ │              
    
│ │ │                
    i11 : h
│ │ │  
│ │ │ -o11 = HashTable{"answer file" => /tmp/M2-29954-0/2.ans}
│ │ │ +o11 = HashTable{"answer file" => /tmp/M2-43124-0/2.ans}
│ │ │                  "exit code" => 27
│ │ │ -                "output file" => /tmp/M2-29954-0/2.out
│ │ │ +                "output file" => /tmp/M2-43124-0/2.out
│ │ │                  "return code" => 6912
│ │ │                  "statistics" => null
│ │ │ -                "time used" => 2
│ │ │ +                "time used" => 1
│ │ │                  value => null
│ │ │  
│ │ │  o11 : HashTable
    
│ │ │              
    
│ │ │ @@ -199,46 +199,46 @@
│ │ │          
    
│ │ │            
    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:
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ @@ -248,40 +248,40 @@
│ │ │            
    
│ │ │            
    We can get quite a lot of detail on the resources used with the KeepStatistics command:
    
│ │ │          
│ │ │          
    
│ │ │                
    i14 : h=runExternalM2(fn,"spin",10,PreRunScript=>"ulimit -t 2");
│ │ │ -Running (ulimit -t 2 && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-29954-0/3.m2" >"/tmp/M2-29954-0/3.out" 2>&1 ))
│ │ │ +Running (ulimit -t 2 && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/3.m2" >"/tmp/M2-43124-0/3.out" 2>&1 ))
│ │ │  Killed
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
    
│ │ │              
    
│ │ │                
    i15 : h
│ │ │  
│ │ │ -o15 = HashTable{"answer file" => /tmp/M2-29954-0/3.ans}
│ │ │ +o15 = HashTable{"answer file" => /tmp/M2-43124-0/3.ans}
│ │ │                  "exit code" => 0
│ │ │ -                "output file" => /tmp/M2-29954-0/3.out
│ │ │ +                "output file" => /tmp/M2-43124-0/3.out
│ │ │                  "return code" => 9
│ │ │                  "statistics" => null
│ │ │                  "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/M2-29954-0/3.m2 automatically generated by RunExternalM2
│ │ │ +      i1 : -- Script /tmp/M2-43124-0/3.m2 automatically generated by RunExternalM2
│ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │  
│ │ │ -      i2 : load "/tmp/M2-29954-0/0.m2";
│ │ │ +      i2 : load "/tmp/M2-43124-0/0.m2";
│ │ │  
│ │ │ -      i3 : runExternalM2ReturnAnswer("/tmp/M2-29954-0/3.ans",spin (10));
│ │ │ +      i3 : runExternalM2ReturnAnswer("/tmp/M2-43124-0/3.ans",spin (10));
│ │ │        Spinning!!
    
│ │ │              
    
│ │ │                
    i17 : if h#"answer file" =!= null and fileExists(h#"answer file") then get(h#"answer file")
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -295,15 +295,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    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/M2-29954-0/4.m2" >"/tmp/M2-29954-0/4.out" 2>&1') >"/tmp/M2-29954-0/4.stat" 2>&1 ))
│ │ │ +Running (true && ( (/usr/bin/time --verbose sh -c '/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/4.m2" >"/tmp/M2-43124-0/4.out" 2>&1') >"/tmp/M2-43124-0/4.stat" 2>&1 ))
│ │ │  Finished running.
    
│ │ │              
    
│ │ │                
    i19 : h#"statistics"
│ │ │  
│ │ │ -o19 =         Command being timed: "sh -c /usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-29954-0/4.m2" >"/tmp/M2-29954-0/4.out" 2>&1"
│ │ │ -              User time (seconds): 5.28
│ │ │ -              System time (seconds): 0.08
│ │ │ -              Percent of CPU this job got: 95%
│ │ │ -              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:05.62
│ │ │ +o19 =         Command being timed: "sh -c /usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/4.m2" >"/tmp/M2-43124-0/4.out" 2>&1"
│ │ │ +              User time (seconds): 4.74
│ │ │ +              System time (seconds): 0.27
│ │ │ +              Percent of CPU this job got: 121%
│ │ │ +              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:04.14
│ │ │                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): 252580
│ │ │ +              Maximum resident set size (kbytes): 339096
│ │ │                Average resident set size (kbytes): 0
│ │ │                Major (requiring I/O) page faults: 0
│ │ │ -              Minor (reclaiming a frame) page faults: 9521
│ │ │ -              Voluntary context switches: 2087
│ │ │ -              Involuntary context switches: 1715
│ │ │ +              Minor (reclaiming a frame) page faults: 10623
│ │ │ +              Voluntary context switches: 6159
│ │ │ +              Involuntary context switches: 1241
│ │ │                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/M2-29954-0/6.m2" >"/tmp/M2-29954-0/6.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/6.m2" >"/tmp/M2-43124-0/6.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  
│ │ │  o21 = true
    
│ │ │              
    
│ │ │          
    
│ │ │ @@ -325,24 +325,24 @@
│ │ │                               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/M2-29954-0/7.m2" >"/tmp/M2-29954-0/7.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/7.m2" >"/tmp/M2-43124-0/7.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
│ │ │  
│ │ │ -o24 = HashTable{"answer file" => /tmp/M2-29954-0/7.ans}
│ │ │ +o24 = HashTable{"answer file" => /tmp/M2-43124-0/7.ans}
│ │ │                  "exit code" => 1
│ │ │ -                "output file" => /tmp/M2-29954-0/7.out
│ │ │ +                "output file" => /tmp/M2-43124-0/7.out
│ │ │                  "return code" => 256
│ │ │                  "statistics" => null
│ │ │ -                "time used" => 2
│ │ │ +                "time used" => 1
│ │ │                  value => null
│ │ │  
│ │ │  o24 : HashTable
    
│ │ │              
    
│ │ │          
    
│ │ │ @@ -350,20 +350,20 @@
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i25 : get(h#"output file")
│ │ │  
│ │ │  o25 = 
│ │ │ -      i1 : -- Script /tmp/M2-29954-0/7.m2 automatically generated by RunExternalM2
│ │ │ +      i1 : -- Script /tmp/M2-43124-0/7.m2 automatically generated by RunExternalM2
│ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │  
│ │ │ -      i2 : load "/tmp/M2-29954-0/0.m2";
│ │ │ +      i2 : load "/tmp/M2-43124-0/0.m2";
│ │ │  
│ │ │ -      i3 : runExternalM2ReturnAnswer("/tmp/M2-29954-0/7.ans",identity (cokernel(map(R^2,R^{3:{-1}},{{(9/2)*x+(9/4)*y, (7/9)*x+(7/10)*y, 7*x+(3/7)*y}, {(3/4)*x+(7/4)*y, (7/10)*x+(7/3)*y, (6/7)*x+6*y}}))));
│ │ │ +      i3 : runExternalM2ReturnAnswer("/tmp/M2-43124-0/7.ans",identity (cokernel(map(R^2,R^{3:{-1}},{{(9/2)*x+(9/4)*y, (7/9)*x+(7/10)*y, 7*x+(3/7)*y}, {(3/4)*x+(7/4)*y, (7/10)*x+(7/3)*y, (6/7)*x+6*y}}))));
│ │ │        stdio:4:74:(3):[1]: error: no method for binary operator ^ applied to objects:
│ │ │                    R (of class Symbol)
│ │ │              ^     2 (of class ZZ)
    
│ │ │              
    
│ │ │          
    
│ │ │ @@ -374,15 +374,15 @@
│ │ │              
    		
│ │ │                
    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/M2-29954-0/8.m2" >"/tmp/M2-29954-0/8.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/8.m2" >"/tmp/M2-43124-0/8.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  
│ │ │  o27 = true
    
│ │ │              
    
│ │ │          
    
│ │ │ @@ -394,15 +394,15 @@
│ │ │              
│ │ │                
    i28 : v=R;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i29 : h=runExternalM2(fn,identity,v);
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-29954-0/9.m2" >"/tmp/M2-29954-0/9.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/9.m2" >"/tmp/M2-43124-0/9.out" 2>&1 ))
│ │ │  Finished running.
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i30 : h#value
│ │ │ ├── html2text {}
│ │ │ │ @@ -45,201 +45,201 @@
│ │ │ │  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/M2-29954-0/0.m2
│ │ │ │ +o1 = /tmp/M2-43124-0/0.m2
│ │ │ │  i2 : fn< (stderr<<"Running"< ( exit(27); ); ///< (stderr<<"Spinning!!"<"/tmp/M2-29954-0/1.out" 2>&1 ))
│ │ │ │ +M2-43124-0/1.m2" >"/tmp/M2-43124-0/1.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  i7 : h
│ │ │ │  
│ │ │ │  o7 = HashTable{"answer file" => null}
│ │ │ │                 "exit code" => 0
│ │ │ │                 "output file" => null
│ │ │ │                 "return code" => 0
│ │ │ │                 "statistics" => null
│ │ │ │ -               "time used" => 2
│ │ │ │ +               "time used" => 3
│ │ │ │                 value => 16
│ │ │ │  
│ │ │ │  o7 : HashTable
│ │ │ │  i8 : h#value===4^2
│ │ │ │  
│ │ │ │  o8 = true
│ │ │ │  i9 : h#"exit code"===0
│ │ │ │  
│ │ │ │  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/
│ │ │ │ -M2-29954-0/2.m2" >"/tmp/M2-29954-0/2.out" 2>&1 ))
│ │ │ │ +M2-43124-0/2.m2" >"/tmp/M2-43124-0/2.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  RunExternalM2: expected answer file does not exist
│ │ │ │  i11 : h
│ │ │ │  
│ │ │ │ -o11 = HashTable{"answer file" => /tmp/M2-29954-0/2.ans}
│ │ │ │ +o11 = HashTable{"answer file" => /tmp/M2-43124-0/2.ans}
│ │ │ │                  "exit code" => 27
│ │ │ │ -                "output file" => /tmp/M2-29954-0/2.out
│ │ │ │ +                "output file" => /tmp/M2-43124-0/2.out
│ │ │ │                  "return code" => 6912
│ │ │ │                  "statistics" => null
│ │ │ │ -                "time used" => 2
│ │ │ │ +                "time used" => 1
│ │ │ │                  value => null
│ │ │ │  
│ │ │ │  o11 : HashTable
│ │ │ │  i12 : fileExists(h#"output file")
│ │ │ │  
│ │ │ │  o12 = true
│ │ │ │  i13 : fileExists(h#"answer file")
│ │ │ │  
│ │ │ │  o13 = false
│ │ │ │  Here, we use _r_e_s_o_u_r_c_e_ _l_i_m_i_t_s to limit the routine to 2 seconds of computational
│ │ │ │  time, while the system is asked to use 10 seconds of computational time:
│ │ │ │  i14 : h=runExternalM2(fn,"spin",10,PreRunScript=>"ulimit -t 2");
│ │ │ │  Running (ulimit -t 2 && (/usr/bin/M2-binary  --stop --no-debug --silent  -
│ │ │ │ -q  <"/tmp/M2-29954-0/3.m2" >"/tmp/M2-29954-0/3.out" 2>&1 ))
│ │ │ │ +q  <"/tmp/M2-43124-0/3.m2" >"/tmp/M2-43124-0/3.out" 2>&1 ))
│ │ │ │  Killed
│ │ │ │  Finished running.
│ │ │ │  RunExternalM2: expected answer file does not exist
│ │ │ │  i15 : h
│ │ │ │  
│ │ │ │ -o15 = HashTable{"answer file" => /tmp/M2-29954-0/3.ans}
│ │ │ │ +o15 = HashTable{"answer file" => /tmp/M2-43124-0/3.ans}
│ │ │ │                  "exit code" => 0
│ │ │ │ -                "output file" => /tmp/M2-29954-0/3.out
│ │ │ │ +                "output file" => /tmp/M2-43124-0/3.out
│ │ │ │                  "return code" => 9
│ │ │ │                  "statistics" => null
│ │ │ │                  "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/M2-29954-0/3.m2 automatically generated by
│ │ │ │ +      i1 : -- Script /tmp/M2-43124-0/3.m2 automatically generated by
│ │ │ │  RunExternalM2
│ │ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │ │  
│ │ │ │ -      i2 : load "/tmp/M2-29954-0/0.m2";
│ │ │ │ +      i2 : load "/tmp/M2-43124-0/0.m2";
│ │ │ │  
│ │ │ │ -      i3 : runExternalM2ReturnAnswer("/tmp/M2-29954-0/3.ans",spin (10));
│ │ │ │ +      i3 : runExternalM2ReturnAnswer("/tmp/M2-43124-0/3.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/M2-29954-0/4.m2" >"/tmp/M2-29954-0/4.out" 2>&1')
│ │ │ │ ->"/tmp/M2-29954-0/4.stat" 2>&1 ))
│ │ │ │ +-no-debug --silent  -q  <"/tmp/M2-43124-0/4.m2" >"/tmp/M2-43124-0/4.out" 2>&1')
│ │ │ │ +>"/tmp/M2-43124-0/4.stat" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  i19 : h#"statistics"
│ │ │ │  
│ │ │ │  o19 =         Command being timed: "sh -c /usr/bin/M2-binary  --stop --no-debug
│ │ │ │ ---silent  -q  <"/tmp/M2-29954-0/4.m2" >"/tmp/M2-29954-0/4.out" 2>&1"
│ │ │ │ -              User time (seconds): 5.28
│ │ │ │ -              System time (seconds): 0.08
│ │ │ │ -              Percent of CPU this job got: 95%
│ │ │ │ -              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:05.62
│ │ │ │ +--silent  -q  <"/tmp/M2-43124-0/4.m2" >"/tmp/M2-43124-0/4.out" 2>&1"
│ │ │ │ +              User time (seconds): 4.74
│ │ │ │ +              System time (seconds): 0.27
│ │ │ │ +              Percent of CPU this job got: 121%
│ │ │ │ +              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:04.14
│ │ │ │                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): 252580
│ │ │ │ +              Maximum resident set size (kbytes): 339096
│ │ │ │                Average resident set size (kbytes): 0
│ │ │ │                Major (requiring I/O) page faults: 0
│ │ │ │ -              Minor (reclaiming a frame) page faults: 9521
│ │ │ │ -              Voluntary context switches: 2087
│ │ │ │ -              Involuntary context switches: 1715
│ │ │ │ +              Minor (reclaiming a frame) page faults: 10623
│ │ │ │ +              Voluntary context switches: 6159
│ │ │ │ +              Involuntary context switches: 1241
│ │ │ │                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/
│ │ │ │ -M2-29954-0/6.m2" >"/tmp/M2-29954-0/6.out" 2>&1 ))
│ │ │ │ +M2-43124-0/6.m2" >"/tmp/M2-43124-0/6.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+9/4y 7/9x+7/10y 7x+3/7y |
│ │ │ │                 | 3/4x+7/4y 7/10x+7/3y 6/7x+6y |
│ │ │ │  
│ │ │ │                               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/
│ │ │ │ -M2-29954-0/7.m2" >"/tmp/M2-29954-0/7.out" 2>&1 ))
│ │ │ │ +M2-43124-0/7.m2" >"/tmp/M2-43124-0/7.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  RunExternalM2: expected answer file does not exist
│ │ │ │  
│ │ │ │ -o24 = HashTable{"answer file" => /tmp/M2-29954-0/7.ans}
│ │ │ │ +o24 = HashTable{"answer file" => /tmp/M2-43124-0/7.ans}
│ │ │ │                  "exit code" => 1
│ │ │ │ -                "output file" => /tmp/M2-29954-0/7.out
│ │ │ │ +                "output file" => /tmp/M2-43124-0/7.out
│ │ │ │                  "return code" => 256
│ │ │ │                  "statistics" => null
│ │ │ │ -                "time used" => 2
│ │ │ │ +                "time used" => 1
│ │ │ │                  value => null
│ │ │ │  
│ │ │ │  o24 : HashTable
│ │ │ │  To view the error message:
│ │ │ │  i25 : get(h#"output file")
│ │ │ │  
│ │ │ │  o25 =
│ │ │ │ -      i1 : -- Script /tmp/M2-29954-0/7.m2 automatically generated by
│ │ │ │ +      i1 : -- Script /tmp/M2-43124-0/7.m2 automatically generated by
│ │ │ │  RunExternalM2
│ │ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │ │  
│ │ │ │ -      i2 : load "/tmp/M2-29954-0/0.m2";
│ │ │ │ +      i2 : load "/tmp/M2-43124-0/0.m2";
│ │ │ │  
│ │ │ │ -      i3 : runExternalM2ReturnAnswer("/tmp/M2-29954-0/7.ans",identity (cokernel
│ │ │ │ +      i3 : runExternalM2ReturnAnswer("/tmp/M2-43124-0/7.ans",identity (cokernel
│ │ │ │  (map(R^2,R^{3:{-1}},{{(9/2)*x+(9/4)*y, (7/9)*x+(7/10)*y, 7*x+(3/7)*y}, {(3/
│ │ │ │  4)*x+(7/4)*y, (7/10)*x+(7/3)*y, (6/7)*x+6*y}}))));
│ │ │ │        stdio:4:74:(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/M2-29954-0/8.out" 2>&1 ))
│ │ │ │ +M2-43124-0/8.m2" >"/tmp/M2-43124-0/8.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/
│ │ │ │ -M2-29954-0/9.m2" >"/tmp/M2-29954-0/9.out" 2>&1 ))
│ │ │ │ +M2-43124-0/9.m2" >"/tmp/M2-43124-0/9.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  i30 : h#value
│ │ │ │  
│ │ │ │  o30 = QQ[x..y]
│ │ │ │  
│ │ │ │  o30 : PolynomialRing
│ │ │ │  i31 : v===h#value
│ │ ├── ./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 0.00188335s (cpu); 0.000216766s (thread); 0s (gc)
│ │ │ + -- used 0.000489259s (cpu); 0.000253062s (thread); 0s (gc)
│ │ │  
│ │ │                1       1
│ │ │  o6 : Matrix ZZ  <-- ZZ
│ │ │  
│ │ │  i7 : ZZ[y];
│ │ │  
│ │ │  i8 : time B = sub((y+1)^(2^n),{y=>1})
│ │ │ - -- used 4.80745s (cpu); 3.48244s (thread); 0s (gc)
│ │ │ + -- used 4.15827s (cpu); 3.11568s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = 104438888141315250669175271071662438257996424904738378038423348328395390
│ │ │       797155745684882681193499755834089010671443926283798757343818579360726323
│ │ │       608785136527794595697654370999834036159013438371831442807001185594622637
│ │ │       631883939771274567233468434458661749680790870580370407128404874011860911
│ │ │       446797778359802900668693897688178778594690563019026094059957945343282346
│ │ │       930302669644305902501597239986771421554169383555988529148631823791443449
│ │ ├── ./usr/share/doc/Macaulay2/SLPexpressions/html/index.html
│ │ │ @@ -104,29 +104,29 @@
│ │ │  
│ │ │  o5 : InterpretedSLProgram
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time A = evaluate(slp,matrix{{1}});
│ │ │ - -- used 0.00188335s (cpu); 0.000216766s (thread); 0s (gc)
│ │ │ + -- used 0.000489259s (cpu); 0.000253062s (thread); 0s (gc)
│ │ │  
│ │ │                1       1
│ │ │  o6 : Matrix ZZ  <-- ZZ
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : ZZ[y];
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : time B = sub((y+1)^(2^n),{y=>1})
│ │ │ - -- used 4.80745s (cpu); 3.48244s (thread); 0s (gc)
│ │ │ + -- used 4.15827s (cpu); 3.11568s (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 0.00188335s (cpu); 0.000216766s (thread); 0s (gc)
│ │ │ │ + -- used 0.000489259s (cpu); 0.000253062s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                1       1
│ │ │ │  o6 : Matrix ZZ  <-- ZZ
│ │ │ │  i7 : ZZ[y];
│ │ │ │  i8 : time B = sub((y+1)^(2^n),{y=>1})
│ │ │ │ - -- used 4.80745s (cpu); 3.48244s (thread); 0s (gc)
│ │ │ │ + -- used 4.15827s (cpu); 3.11568s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = 104438888141315250669175271071662438257996424904738378038423348328395390
│ │ │ │       797155745684882681193499755834089010671443926283798757343818579360726323
│ │ │ │       608785136527794595697654370999834036159013438371831442807001185594622637
│ │ │ │       631883939771274567233468434458661749680790870580370407128404874011860911
│ │ │ │       446797778359802900668693897688178778594690563019026094059957945343282346
│ │ │ │       930302669644305902501597239986771421554169383555988529148631823791443449
│ │ ├── ./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)
│ │ │ - -- .0071433s elapsed
│ │ │ + -- .00715486s 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;
│ │ │ - -- .00262291s elapsed
│ │ │ + -- .00300309s 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);
│ │ │ - -- .00583468s elapsed
│ │ │ + -- .00615408s 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)
│ │ │ - -- .00298752s elapsed
│ │ │ + -- .00389946s 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
│ │ │ - -- .000624815s elapsed
│ │ │ + -- .000716252s 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)
│ │ │ - -- .00154927s elapsed
│ │ │ + -- .0015537s 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
│ │ │ @@ -18,15 +18,15 @@
│ │ │  i4 : r={4,3,5}
│ │ │  
│ │ │  o4 = {4, 3, 5}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : elapsedTime C=randomChainComplex(h,r,Height=>100,ZeroMean=>true)
│ │ │ - -- .00379414s elapsed
│ │ │ + -- .00391958s 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
│ │ │ @@ -18,15 +18,15 @@
│ │ │  i4 : r={4,3,3}
│ │ │  
│ │ │  o4 = {4, 3, 3}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ - -- .00281601s elapsed
│ │ │ + -- .00298434s 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
│ │ │ @@ -18,15 +18,15 @@
│ │ │  i4 : r={4,3,3}
│ │ │  
│ │ │  o4 = {4, 3, 3}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ - -- .00287549s elapsed
│ │ │ + -- .00298918s 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
│ │ │ @@ -105,15 +105,15 @@
│ │ │  
│ │ │  o3 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : elapsedTime C=randomChainComplex(h,r,Height=>4)
│ │ │ - -- .0071433s elapsed
│ │ │ + -- .00715486s elapsed
│ │ │  
│ │ │         6       19       19       7       3
│ │ │  o4 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ  <-- ZZ
│ │ │                                          
│ │ │       0       1        2        3       4
│ │ │  
│ │ │  o4 : ChainComplex
    
│ │ │ @@ -150,15 +150,15 @@
│ │ │  
│ │ │  o6 : ChainComplex
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : elapsedTime (h,U)=SVDComplex CR;
│ │ │ - -- .00262291s elapsed
    
│ │ │ + -- .00300309s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : h
│ │ │  
│ │ │  o8 = HashTable{-1 => 1}
│ │ │ @@ -212,15 +212,15 @@
│ │ │  
│ │ │  o12 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i13 : elapsedTime (hL,U)=SVDComplex(CR,Strategy=>Laplacian);
│ │ │ - -- .00583468s elapsed
    
│ │ │ + -- .00615408s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : hL === h
│ │ │  
│ │ │  o14 = true
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -37,15 +37,15 @@
│ │ │ │  o2 : List
│ │ │ │  i3 : r={5,11,3,2}
│ │ │ │  
│ │ │ │  o3 = {5, 11, 3, 2}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : elapsedTime C=randomChainComplex(h,r,Height=>4)
│ │ │ │ - -- .0071433s elapsed
│ │ │ │ + -- .00715486s elapsed
│ │ │ │  
│ │ │ │         6       19       19       7       3
│ │ │ │  o4 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ  <-- ZZ
│ │ │ │  
│ │ │ │       0       1        2        3       4
│ │ │ │  
│ │ │ │  o4 : ChainComplex
│ │ │ │ @@ -70,15 +70,15 @@
│ │ │ │  o6 = RR    <-- RR     <-- RR     <-- RR    <-- RR
│ │ │ │         53        53         53         53        53
│ │ │ │  
│ │ │ │       -1        0          1          2         3
│ │ │ │  
│ │ │ │  o6 : ChainComplex
│ │ │ │  i7 : elapsedTime (h,U)=SVDComplex CR;
│ │ │ │ - -- .00262291s elapsed
│ │ │ │ + -- .00300309s elapsed
│ │ │ │  i8 : h
│ │ │ │  
│ │ │ │  o8 = HashTable{-1 => 1}
│ │ │ │                 0 => 3
│ │ │ │                 1 => 5
│ │ │ │                 2 => 2
│ │ │ │                 3 => 1
│ │ │ │ @@ -109,15 +109,15 @@
│ │ │ │  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);
│ │ │ │ - -- .00583468s elapsed
│ │ │ │ + -- .00615408s 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
│ │ │ @@ -107,15 +107,15 @@
│ │ │  
│ │ │  o3 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ - -- .00298752s elapsed
│ │ │ + -- .00389946s elapsed
│ │ │  
│ │ │         5       10       11       5
│ │ │  o4 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o4 : ChainComplex
    
│ │ │ @@ -148,28 +148,28 @@
│ │ │  
│ │ │  o6 : ChainComplex
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : elapsedTime (h,h1)=SVDHomology CR
│ │ │ - -- .000624815s elapsed
│ │ │ + -- .000716252s 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)
│ │ │ - -- .00154927s elapsed
│ │ │ + -- .0015537s 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 {}
│ │ │ │ @@ -40,15 +40,15 @@
│ │ │ │  o2 : List
│ │ │ │  i3 : r={4,3,3}
│ │ │ │  
│ │ │ │  o3 = {4, 3, 3}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ │ - -- .00298752s elapsed
│ │ │ │ + -- .00389946s elapsed
│ │ │ │  
│ │ │ │         5       10       11       5
│ │ │ │  o4 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │ │  
│ │ │ │       0       1        2        3
│ │ │ │  
│ │ │ │  o4 : ChainComplex
│ │ │ │ @@ -69,24 +69,24 @@
│ │ │ │  o6 = RR    <-- RR     <-- RR     <-- RR
│ │ │ │         53        53         53         53
│ │ │ │  
│ │ │ │       0         1          2          3
│ │ │ │  
│ │ │ │  o6 : ChainComplex
│ │ │ │  i7 : elapsedTime (h,h1)=SVDHomology CR
│ │ │ │ - -- .000624815s elapsed
│ │ │ │ + -- .000716252s 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)
│ │ │ │ - -- .00154927s elapsed
│ │ │ │ + -- .0015537s 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
│ │ │ @@ -110,15 +110,15 @@
│ │ │  
│ │ │  o4 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : elapsedTime C=randomChainComplex(h,r,Height=>100,ZeroMean=>true)
│ │ │ - -- .00379414s elapsed
│ │ │ + -- .00391958s 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)
│ │ │ │ - -- .00379414s elapsed
│ │ │ │ + -- .00391958s 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
│ │ │ @@ -104,15 +104,15 @@
│ │ │  
│ │ │  o4 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ - -- .00281601s elapsed
│ │ │ + -- .00298434s 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)
│ │ │ │ - -- .00281601s elapsed
│ │ │ │ + -- .00298434s 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
│ │ │ @@ -104,15 +104,15 @@
│ │ │  
│ │ │  o4 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ - -- .00287549s elapsed
│ │ │ + -- .00298918s 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)
│ │ │ │ - -- .00287549s elapsed
│ │ │ │ + -- .00298918s elapsed
│ │ │ │  
│ │ │ │         6       10       11       5
│ │ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │ │  
│ │ │ │       0       1        2        3
│ │ │ │  
│ │ │ │  o5 : ChainComplex
│ │ ├── ./usr/share/doc/Macaulay2/Saturation/example-output/_quotient_lp..._cm__Strategy_eq_gt..._rp.out
│ │ │ @@ -37,33 +37,33 @@
│ │ │  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 0.385785s (cpu); 0.315262s (thread); 0s (gc)
│ │ │ + -- used 0.383591s (cpu); 0.383591s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of S
│ │ │  
│ │ │  i8 : time quotient(I^3, J^2, Strategy => Quotient);
│ │ │ - -- used 0.459981s (cpu); 0.459987s (thread); 0s (gc)
│ │ │ + -- used 0.711663s (cpu); 0.641923s (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 0.0242176s (cpu); 0.0242182s (thread); 0s (gc)
│ │ │ + -- used 0.0268394s (cpu); 0.026842s (thread); 0s (gc)
│ │ │  
│ │ │  o11 : Ideal of S
│ │ │  
│ │ │  i12 : time quotient(I^5, I^3, Strategy => Quotient);
│ │ │ - -- used 0.00726035s (cpu); 0.00726111s (thread); 0s (gc)
│ │ │ + -- used 0.00901864s (cpu); 0.00902542s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of S
│ │ │  
│ │ │  i13 :
│ │ ├── ./usr/share/doc/Macaulay2/Saturation/html/_quotient_lp..._cm__Strategy_eq_gt..._rp.html
│ │ │ @@ -125,23 +125,23 @@
│ │ │  
│ │ │  o6 : Ideal of S
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : time quotient(I^3, J^2, Strategy => Iterate);
│ │ │ - -- used 0.385785s (cpu); 0.315262s (thread); 0s (gc)
│ │ │ + -- used 0.383591s (cpu); 0.383591s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of S
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : time quotient(I^3, J^2, Strategy => Quotient);
│ │ │ - -- used 0.459981s (cpu); 0.459987s (thread); 0s (gc)
│ │ │ + -- used 0.711663s (cpu); 0.641923s (thread); 0s (gc)
│ │ │  
│ │ │  o8 : Ideal of S
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    Strategy => Quotient is faster in other cases:
    
│ │ │ @@ -158,23 +158,23 @@
│ │ │  
│ │ │  o10 : Ideal of S
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i11 : time quotient(I^5, I^3, Strategy => Iterate);
│ │ │ - -- used 0.0242176s (cpu); 0.0242182s (thread); 0s (gc)
│ │ │ + -- used 0.0268394s (cpu); 0.026842s (thread); 0s (gc)
│ │ │  
│ │ │  o11 : Ideal of S
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : time quotient(I^5, I^3, Strategy => Quotient);
│ │ │ - -- used 0.00726035s (cpu); 0.00726111s (thread); 0s (gc)
│ │ │ + -- used 0.00901864s (cpu); 0.00902542s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of S
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -56,32 +56,32 @@
│ │ │ │  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 0.385785s (cpu); 0.315262s (thread); 0s (gc)
│ │ │ │ + -- used 0.383591s (cpu); 0.383591s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 : Ideal of S
│ │ │ │  i8 : time quotient(I^3, J^2, Strategy => Quotient);
│ │ │ │ - -- used 0.459981s (cpu); 0.459987s (thread); 0s (gc)
│ │ │ │ + -- used 0.711663s (cpu); 0.641923s (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 0.0242176s (cpu); 0.0242182s (thread); 0s (gc)
│ │ │ │ + -- used 0.0268394s (cpu); 0.026842s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 : Ideal of S
│ │ │ │  i12 : time quotient(I^5, I^3, Strategy => Quotient);
│ │ │ │ - -- used 0.00726035s (cpu); 0.00726111s (thread); 0s (gc)
│ │ │ │ + -- used 0.00901864s (cpu); 0.00902542s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o12 : Ideal of S
│ │ │ │  ********** RReeffeerreenncceess **********
│ │ │ │  For further information see for example Exercise 15.41 in Eisenbud's
│ │ │ │  Commutative Algebra with a View Towards Algebraic Geometry.
│ │ │ │  ********** FFuunnccttiioonnss wwiitthh ooppttiioonnaall aarrgguummeenntt nnaammeedd SSttrraatteeggyy:: **********
│ │ │ │      * addHook(...,Strategy=>...) -- see _a_d_d_H_o_o_k -- add a hook function to an
│ │ ├── ./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 0.00492304s (cpu); 0.00491944s (thread); 0s (gc)
│ │ │ - -- used 0.00638266s (cpu); 0.00638735s (thread); 0s (gc)
│ │ │ - -- used 0.00960665s (cpu); 0.0096079s (thread); 0s (gc)
│ │ │ - -- used 0.0166536s (cpu); 0.0166633s (thread); 0s (gc)
│ │ │ - -- used 0.0323217s (cpu); 0.0323255s (thread); 0s (gc)
│ │ │ - -- used 0.0554866s (cpu); 0.0554921s (thread); 0s (gc)
│ │ │ - -- used 0.0940579s (cpu); 0.0940639s (thread); 0s (gc)
│ │ │ - -- used 0.274754s (cpu); 0.18535s (thread); 0s (gc)
│ │ │ - -- used 0.22605s (cpu); 0.226055s (thread); 0s (gc)
│ │ │ + -- used 0.00955092s (cpu); 0.00954892s (thread); 0s (gc)
│ │ │ + -- used 0.0111424s (cpu); 0.0111508s (thread); 0s (gc)
│ │ │ + -- used 0.0166338s (cpu); 0.0166442s (thread); 0s (gc)
│ │ │ + -- used 0.0298695s (cpu); 0.0298806s (thread); 0s (gc)
│ │ │ + -- used 0.0546927s (cpu); 0.054701s (thread); 0s (gc)
│ │ │ + -- used 0.0676052s (cpu); 0.0676147s (thread); 0s (gc)
│ │ │ + -- used 0.135011s (cpu); 0.134817s (thread); 0s (gc)
│ │ │ + -- used 0.187201s (cpu); 0.187211s (thread); 0s (gc)
│ │ │ + -- used 0.600968s (cpu); 0.485218s (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
│ │ │ @@ -126,23 +126,23 @@
│ │ │  
│ │ │  o6 : FunctionClosure
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : for n from 2 to 10 list time f n
│ │ │ - -- used 0.00492304s (cpu); 0.00491944s (thread); 0s (gc)
│ │ │ - -- used 0.00638266s (cpu); 0.00638735s (thread); 0s (gc)
│ │ │ - -- used 0.00960665s (cpu); 0.0096079s (thread); 0s (gc)
│ │ │ - -- used 0.0166536s (cpu); 0.0166633s (thread); 0s (gc)
│ │ │ - -- used 0.0323217s (cpu); 0.0323255s (thread); 0s (gc)
│ │ │ - -- used 0.0554866s (cpu); 0.0554921s (thread); 0s (gc)
│ │ │ - -- used 0.0940579s (cpu); 0.0940639s (thread); 0s (gc)
│ │ │ - -- used 0.274754s (cpu); 0.18535s (thread); 0s (gc)
│ │ │ - -- used 0.22605s (cpu); 0.226055s (thread); 0s (gc)
│ │ │ + -- used 0.00955092s (cpu); 0.00954892s (thread); 0s (gc)
│ │ │ + -- used 0.0111424s (cpu); 0.0111508s (thread); 0s (gc)
│ │ │ + -- used 0.0166338s (cpu); 0.0166442s (thread); 0s (gc)
│ │ │ + -- used 0.0298695s (cpu); 0.0298806s (thread); 0s (gc)
│ │ │ + -- used 0.0546927s (cpu); 0.054701s (thread); 0s (gc)
│ │ │ + -- used 0.0676052s (cpu); 0.0676147s (thread); 0s (gc)
│ │ │ + -- used 0.135011s (cpu); 0.134817s (thread); 0s (gc)
│ │ │ + -- used 0.187201s (cpu); 0.187211s (thread); 0s (gc)
│ │ │ + -- used 0.600968s (cpu); 0.485218s (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 0.00492304s (cpu); 0.00491944s (thread); 0s (gc)
│ │ │ │ - -- used 0.00638266s (cpu); 0.00638735s (thread); 0s (gc)
│ │ │ │ - -- used 0.00960665s (cpu); 0.0096079s (thread); 0s (gc)
│ │ │ │ - -- used 0.0166536s (cpu); 0.0166633s (thread); 0s (gc)
│ │ │ │ - -- used 0.0323217s (cpu); 0.0323255s (thread); 0s (gc)
│ │ │ │ - -- used 0.0554866s (cpu); 0.0554921s (thread); 0s (gc)
│ │ │ │ - -- used 0.0940579s (cpu); 0.0940639s (thread); 0s (gc)
│ │ │ │ - -- used 0.274754s (cpu); 0.18535s (thread); 0s (gc)
│ │ │ │ - -- used 0.22605s (cpu); 0.226055s (thread); 0s (gc)
│ │ │ │ + -- used 0.00955092s (cpu); 0.00954892s (thread); 0s (gc)
│ │ │ │ + -- used 0.0111424s (cpu); 0.0111508s (thread); 0s (gc)
│ │ │ │ + -- used 0.0166338s (cpu); 0.0166442s (thread); 0s (gc)
│ │ │ │ + -- used 0.0298695s (cpu); 0.0298806s (thread); 0s (gc)
│ │ │ │ + -- used 0.0546927s (cpu); 0.054701s (thread); 0s (gc)
│ │ │ │ + -- used 0.0676052s (cpu); 0.0676147s (thread); 0s (gc)
│ │ │ │ + -- used 0.135011s (cpu); 0.134817s (thread); 0s (gc)
│ │ │ │ + -- used 0.187201s (cpu); 0.187211s (thread); 0s (gc)
│ │ │ │ + -- used 0.600968s (cpu); 0.485218s (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/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 4.62089s (cpu); 3.61525s (thread); 0s (gc)
│ │ │ + -- used 8.09691s (cpu); 3.90703s (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 49.7272s (cpu); 46.0603s (thread); 0s (gc)
│ │ │ + -- used 62.0348s (cpu); 57.052s (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 0.274879s (cpu); 0.168297s (thread); 0s (gc)
│ │ │ + -- used 0.691163s (cpu); 0.218455s (thread); 0s (gc)
│ │ │  
│ │ │         2             2
│ │ │  o6 = 2H  + 4H H  + 2H
│ │ │         1     1 2     2
│ │ │  
│ │ │  o6 : A
│ │ │  
│ │ │  i7 : time segre(X,Y,A)
│ │ │ - -- used 0.193686s (cpu); 0.116572s (thread); 0s (gc)
│ │ │ + -- used 0.323354s (cpu); 0.131946s (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
│ │ │ @@ -162,15 +162,15 @@
│ │ │  
│ │ │  o10 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i11 : time isComponentContained(X,Y)
│ │ │ - -- used 4.62089s (cpu); 3.61525s (thread); 0s (gc)
│ │ │ + -- used 8.09691s (cpu); 3.90703s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : print "we could confirm this with the computation:"
│ │ │ @@ -189,15 +189,15 @@
│ │ │  
│ │ │  o13 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : time isSubset(saturate(Y,B),saturate(X,B))
│ │ │ - -- used 49.7272s (cpu); 46.0603s (thread); 0s (gc)
│ │ │ + -- used 62.0348s (cpu); 57.052s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = true
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -68,30 +68,30 @@
│ │ │ │  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 4.62089s (cpu); 3.61525s (thread); 0s (gc)
│ │ │ │ + -- used 8.09691s (cpu); 3.90703s (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 49.7272s (cpu); 46.0603s (thread); 0s (gc)
│ │ │ │ + -- used 62.0348s (cpu); 57.052s (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
│ │ │ @@ -118,27 +118,27 @@
│ │ │  
│ │ │  o5 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time s = segreDimX(X,Y,A)
│ │ │ - -- used 0.274879s (cpu); 0.168297s (thread); 0s (gc)
│ │ │ + -- used 0.691163s (cpu); 0.218455s (thread); 0s (gc)
│ │ │  
│ │ │         2             2
│ │ │  o6 = 2H  + 4H H  + 2H
│ │ │         1     1 2     2
│ │ │  
│ │ │  o6 : A
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : time segre(X,Y,A)
│ │ │ - -- used 0.193686s (cpu); 0.116572s (thread); 0s (gc)
│ │ │ + -- used 0.323354s (cpu); 0.131946s (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 {}
│ │ │ │ @@ -48,23 +48,23 @@
│ │ │ │  o4 : Ideal of R
│ │ │ │  i5 : A = makeChowRing(R)
│ │ │ │  
│ │ │ │  o5 = A
│ │ │ │  
│ │ │ │  o5 : QuotientRing
│ │ │ │  i6 : time s = segreDimX(X,Y,A)
│ │ │ │ - -- used 0.274879s (cpu); 0.168297s (thread); 0s (gc)
│ │ │ │ + -- used 0.691163s (cpu); 0.218455s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         2             2
│ │ │ │  o6 = 2H  + 4H H  + 2H
│ │ │ │         1     1 2     2
│ │ │ │  
│ │ │ │  o6 : A
│ │ │ │  i7 : time segre(X,Y,A)
│ │ │ │ - -- used 0.193686s (cpu); 0.116572s (thread); 0s (gc)
│ │ │ │ + -- used 0.323354s (cpu); 0.131946s (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/SimpleDoc/example-output/_test__Example.out
│ │ │ @@ -1,6 +1,6 @@
│ │ │  -- -*- M2-comint -*- hash: 1331702921222
│ │ │  
│ │ │  i1 : check SimpleDoc
│ │ │ - -- capturing check(0, "SimpleDoc")           -- .389602s elapsed
│ │ │ + -- capturing check(0, "SimpleDoc")           -- .160799s elapsed
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/SimpleDoc/html/_test__Example.html
│ │ │ @@ -74,15 +74,15 @@
│ │ │          
    
│ │ │            
    The check method executes all package tests defined this way.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : check SimpleDoc
│ │ │ - -- capturing check(0, "SimpleDoc")           -- .389602s elapsed
    
│ │ │ + -- capturing check(0, "SimpleDoc")           -- .160799s elapsed
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,15 +10,15 @@
│ │ │ │  The variable testExample is a _S_t_r_i_n_g containing an example of the use of _T_E_S_T
│ │ │ │  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")           -- .389602s elapsed
│ │ │ │ + -- capturing check(0, "SimpleDoc")           -- .160799s 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/SparseResultants/example-output/_degree__Determinant.out
│ │ │ @@ -3,15 +3,15 @@
│ │ │  i1 : n = {2,3,2}
│ │ │  
│ │ │  o1 = {2, 3, 2}
│ │ │  
│ │ │  o1 : List
│ │ │  
│ │ │  i2 : time degreeDeterminant n
│ │ │ - -- used 8.2585e-05s (cpu); 7.8617e-05s (thread); 0s (gc)
│ │ │ + -- used 0.000110033s (cpu); 0.000102784s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 6
│ │ │  
│ │ │  i3 : M = genericMultidimensionalMatrix n;
│ │ │  warning: clearing value of symbol x2 to allow access to subscripted variables based on it
│ │ │         : debug with expression   debug 1368   or with command line option   --debug 1368
│ │ │  warning: clearing value of symbol x1 to allow access to subscripted variables based on it
│ │ │ @@ -19,14 +19,14 @@
│ │ │  warning: clearing value of symbol x0 to allow access to subscripted variables based on it
│ │ │         : debug with expression   debug 6010   or with command line option   --debug 6010
│ │ │  
│ │ │  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 0.0314058s (cpu); 0.0305595s (thread); 0s (gc)
│ │ │ + -- used 0.088229s (cpu); 0.0397949s (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 0.429761s (cpu); 0.235137s (thread); 0s (gc)
│ │ │ + -- used 0.501638s (cpu); 0.272668s (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 0.163415s (cpu); 0.105804s (thread); 0s (gc)
│ │ │ + -- used 0.171576s (cpu); 0.106533s (thread); 0s (gc)
│ │ │  
│ │ │  i3 : time denseResultant(f0,f1,f2,Algorithm=>"Macaulay"); -- using Macaulay formula
│ │ │ - -- used 0.292346s (cpu); 0.241712s (thread); 0s (gc)
│ │ │ + -- used 0.345345s (cpu); 0.289327s (thread); 0s (gc)
│ │ │  
│ │ │  i4 : time (denseResultant(1,2,2)) (f0,f1,f2); -- using sparseResultant
│ │ │ - -- used 0.361747s (cpu); 0.305057s (thread); 0s (gc)
│ │ │ + -- used 0.318952s (cpu); 0.306688s (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 0.0870891s (cpu); 0.0852933s (thread); 0s (gc)
│ │ │ + -- used 0.153027s (cpu); 0.103044s (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 0.539221s (cpu); 0.47049s (thread); 0s (gc)
│ │ │ + -- used 0.448063s (cpu); 0.448062s (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 2.53309s (cpu); 2.15434s (thread); 0s (gc)
│ │ │ + -- used 2.82605s (cpu); 2.49519s (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 0.496444s (cpu); 0.422067s (thread); 0s (gc)
│ │ │ + -- used 0.502446s (cpu); 0.44334s (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 0.00962779s (cpu); 0.00962822s (thread); 0s (gc)
│ │ │ + -- used 0.0106742s (cpu); 0.0106785s (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 0.0317773s (cpu); 0.0309755s (thread); 0s (gc)
│ │ │ + -- used 0.151724s (cpu); 0.0598404s (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 0.00333243s (cpu); 0.00331511s (thread); 0s (gc)
│ │ │ + -- used 0.00447146s (cpu); 0.00447218s (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 0.276884s (cpu); 0.197253s (thread); 0s (gc)
│ │ │ + -- used 0.344921s (cpu); 0.221547s (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
│ │ │ @@ -76,15 +76,15 @@
│ │ │  
│ │ │  o1 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i2 : time degreeDeterminant n
│ │ │ - -- used 8.2585e-05s (cpu); 7.8617e-05s (thread); 0s (gc)
│ │ │ + -- used 0.000110033s (cpu); 0.000102784s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 6
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i3 : M = genericMultidimensionalMatrix n;
│ │ │ @@ -98,15 +98,15 @@
│ │ │  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 0.0314058s (cpu); 0.0305595s (thread); 0s (gc)
│ │ │ + -- used 0.088229s (cpu); 0.0397949s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = {6}
│ │ │  
│ │ │  o4 : List
      
│ │ │              
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,15 +15,15 @@
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : n = {2,3,2}
│ │ │ │  
│ │ │ │  o1 = {2, 3, 2}
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  i2 : time degreeDeterminant n
│ │ │ │ - -- used 8.2585e-05s (cpu); 7.8617e-05s (thread); 0s (gc)
│ │ │ │ + -- used 0.000110033s (cpu); 0.000102784s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = 6
│ │ │ │  i3 : M = genericMultidimensionalMatrix n;
│ │ │ │  warning: clearing value of symbol x2 to allow access to subscripted variables
│ │ │ │  based on it
│ │ │ │         : debug with expression   debug 1368   or with command line option   --
│ │ │ │  debug 1368
│ │ │ │ @@ -35,15 +35,15 @@
│ │ │ │  based on it
│ │ │ │         : debug with expression   debug 6010   or with command line option   --
│ │ │ │  debug 6010
│ │ │ │  
│ │ │ │  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 0.0314058s (cpu); 0.0305595s (thread); 0s (gc)
│ │ │ │ + -- used 0.088229s (cpu); 0.0397949s (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
│ │ │ @@ -80,15 +80,15 @@
│ │ │              
│ │ │                
      i1 : (d,n) := (2,3);
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i2 : time Disc = denseDiscriminant(d,n)
│ │ │ - -- used 0.429761s (cpu); 0.235137s (thread); 0s (gc)
│ │ │ + -- used 0.501638s (cpu); 0.272668s (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 |
      
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,15 +18,15 @@
│ │ │ │      * 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 0.429761s (cpu); 0.235137s (thread); 0s (gc)
│ │ │ │ + -- used 0.501638s (cpu); 0.272668s (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
│ │ │ @@ -90,27 +90,27 @@
│ │ │  
│ │ │  o1 : Sequence
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i2 : time denseResultant(f0,f1,f2); -- using Poisson formula
│ │ │ - -- used 0.163415s (cpu); 0.105804s (thread); 0s (gc)
      
│ │ │ + -- used 0.171576s (cpu); 0.106533s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i3 : time denseResultant(f0,f1,f2,Algorithm=>"Macaulay"); -- using Macaulay formula
│ │ │ - -- used 0.292346s (cpu); 0.241712s (thread); 0s (gc)
      
│ │ │ + -- used 0.345345s (cpu); 0.289327s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i4 : time (denseResultant(1,2,2)) (f0,f1,f2); -- using sparseResultant
│ │ │ - -- used 0.361747s (cpu); 0.305057s (thread); 0s (gc)
      
│ │ │ + -- used 0.318952s (cpu); 0.306688s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i5 : assert(o2 == o3 and o3 == o4)
      
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,20 +28,20 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                   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 0.163415s (cpu); 0.105804s (thread); 0s (gc)
│ │ │ │ + -- used 0.171576s (cpu); 0.106533s (thread); 0s (gc)
│ │ │ │  i3 : time denseResultant(f0,f1,f2,Algorithm=>"Macaulay"); -- using Macaulay
│ │ │ │  formula
│ │ │ │ - -- used 0.292346s (cpu); 0.241712s (thread); 0s (gc)
│ │ │ │ + -- used 0.345345s (cpu); 0.289327s (thread); 0s (gc)
│ │ │ │  i4 : time (denseResultant(1,2,2)) (f0,f1,f2); -- using sparseResultant
│ │ │ │ - -- used 0.361747s (cpu); 0.305057s (thread); 0s (gc)
│ │ │ │ + -- used 0.318952s (cpu); 0.306688s (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 @@
│ │ │  
│ │ │  o1 : 4-dimensional matrix of shape 2 x 2 x 2 x 2 over ZZ
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i2 : time det M
│ │ │ - -- used 0.0870891s (cpu); 0.0852933s (thread); 0s (gc)
│ │ │ + -- used 0.153027s (cpu); 0.103044s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 9698337990421512192
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i3 : M = randomMultidimensionalMatrix(2,2,2,2,5)
│ │ │ @@ -109,15 +109,15 @@
│ │ │  
│ │ │  o3 : 5-dimensional matrix of shape 2 x 2 x 2 x 2 x 5 over ZZ
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i4 : time det M
│ │ │ - -- used 0.539221s (cpu); 0.47049s (thread); 0s (gc)
│ │ │ + -- used 0.448063s (cpu); 0.448062s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 912984499996938980479447727885644530753184525786986940737407301278806287
│ │ │       9257139493926586400187927813888
      
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,15 +25,15 @@
│ │ │ │  
│ │ │ │  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 0.0870891s (cpu); 0.0852933s (thread); 0s (gc)
│ │ │ │ + -- used 0.153027s (cpu); 0.103044s (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,
│ │ │ │ @@ -42,15 +42,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       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 0.539221s (cpu); 0.47049s (thread); 0s (gc)
│ │ │ │ + -- used 0.448063s (cpu); 0.448062s (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
│ │ │ @@ -90,15 +90,15 @@
│ │ │  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 2.53309s (cpu); 2.15434s (thread); 0s (gc)
│ │ │ + -- used 2.82605s (cpu); 2.49519s (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 {}
│ │ │ │ @@ -37,15 +37,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       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 2.53309s (cpu); 2.15434s (thread); 0s (gc)
│ │ │ │ + -- used 2.82605s (cpu); 2.49519s (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
│ │ │ @@ -74,15 +74,15 @@
│ │ │          
    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 $f = c_{(1,1)}+c_{(1,2)} x y+c_{(1,3)} x^2 y+c_{(1,4)} x, g = c_{(2,1)} y+c_{(2,2)} x^2 y^2+c_{(2,3)} x^2 y+c_{(2,4)} x, h = c_{(3,1)}+c_{(3,2)} y+c_{(3,3)} x y+c_{(3,4)} x$.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -104,15 +104,15 @@
│ │ │  
│ │ │  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 0.496444s (cpu); 0.422067s (thread); 0s (gc)
│ │ │ + -- used 0.502446s (cpu); 0.44334s (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 0.00962779s (cpu); 0.00962822s (thread); 0s (gc)
│ │ │ + -- used 0.0106742s (cpu); 0.0106785s (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    +
│ │ │ @@ -825,15 +825,15 @@
│ │ │            
    
│ │ │          
    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 $\mathbb{Z}/3331$.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -68,15 +68,15 @@
│ │ │ │  
│ │ │ │  o4 = | 0 1 |
│ │ │ │       | 2 3 |
│ │ │ │       | 4 |
│ │ │ │  
│ │ │ │  o4 : YoungTableau
│ │ │ │  i5 : time higherSpechtPolynomial(S,T,R)
│ │ │ │ - -- used 0.00149283s (cpu); 0.00148952s (thread); 0s (gc)
│ │ │ │ + -- used 0.00167486s (cpu); 0.00167197s (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  -
│ │ │ │ @@ -88,15 +88,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │          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 0.00128378s (cpu); 0.00128396s (thread); 0s (gc)
│ │ │ │ + -- used 0.00165147s (cpu); 0.00165278s (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  -
│ │ │ │ @@ -108,15 +108,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │          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 0.00197708s (cpu); 0.00197739s (thread); 0s (gc)
│ │ │ │ + -- used 0.00229176s (cpu); 0.00229227s (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
│ │ │ @@ -126,15 +126,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -111,15 +111,15 @@
│ │ │  
│ │ │  o4 : ProjectiveVariety, a point in PP^5
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -121,15 +121,15 @@
│ │ │  
│ │ │  o5 : ProjectiveVariety, a point in PP^8
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i6 : time Res = sparseResultant(matrix{{0,0,1,1},{0,1,0,1}},CoefficientRing=>ZZ/3331);
│ │ │ - -- used 0.0317773s (cpu); 0.0309755s (thread); 0s (gc)
│ │ │ + -- used 0.151724s (cpu); 0.0598404s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : SparseResultant (sparse unmixed resultant associated to | 0 0 1 1 | over ZZ/3331)
│ │ │                                                               | 0 1 0 1 |
    
│ │ │              
    
│ │ │ @@ -849,15 +849,15 @@
│ │ │  
│ │ │  o8 : Sequence
│ │ │              
    
│ │ │                
    i9 : time Res(f,g,h)
│ │ │ - -- used 0.00333243s (cpu); 0.00331511s (thread); 0s (gc)
│ │ │ + -- used 0.00447146s (cpu); 0.00447218s (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  +
│ │ │ @@ -938,15 +938,15 @@
│ │ │  
│ │ │  o11 : Sequence
    
│ │ │              
    
│ │ │                
    i12 : time (MixedRes,UnmixedRes) = (sparseResultant(f,g,h),sparseResultant(f,g,h,Unmixed=>true));
│ │ │ - -- used 0.276884s (cpu); 0.197253s (thread); 0s (gc)
    
│ │ │ + -- used 0.344921s (cpu); 0.221547s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i13 : quotientRemainder(UnmixedRes,MixedRes)
│ │ │  
│ │ │          2 2                   2    2                               2 2
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,15 +34,15 @@
│ │ │ │  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 0.496444s (cpu); 0.422067s (thread); 0s (gc)
│ │ │ │ + -- used 0.502446s (cpu); 0.44334s (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 |
│ │ │ │ @@ -55,15 +55,15 @@
│ │ │ │         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 0.00962779s (cpu); 0.00962822s (thread); 0s (gc)
│ │ │ │ + -- used 0.0106742s (cpu); 0.0106785s (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    +
│ │ │ │ @@ -771,29 +771,29 @@
│ │ │ │  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 0.0317773s (cpu); 0.0309755s (thread); 0s (gc)
│ │ │ │ + -- used 0.151724s (cpu); 0.0598404s (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 0.00333243s (cpu); 0.00331511s (thread); 0s (gc)
│ │ │ │ + -- used 0.00447146s (cpu); 0.00447218s (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  +
│ │ │ │ @@ -863,15 +863,15 @@
│ │ │ │                    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 0.276884s (cpu); 0.197253s (thread); 0s (gc)
│ │ │ │ + -- used 0.344921s (cpu); 0.221547s (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/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 0.00149283s (cpu); 0.00148952s (thread); 0s (gc)
│ │ │ + -- used 0.00167486s (cpu); 0.00167197s (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 0.00128378s (cpu); 0.00128396s (thread); 0s (gc)
│ │ │ + -- used 0.00165147s (cpu); 0.00165278s (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 0.00197708s (cpu); 0.00197739s (thread); 0s (gc)
│ │ │ + -- used 0.00229176s (cpu); 0.00229227s (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 0.339216s (cpu); 0.297324s (thread); 0s (gc)
│ │ │ + -- used 0.385296s (cpu); 0.323421s (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
│ │ │ @@ -20,15 +20,15 @@
│ │ │  (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)
│ │ │  (Partition{2, 2, 2}, Ambient_Dimension, 5, Rank, 1)
│ │ │  (Partition{2, 2, 1, 1}, Ambient_Dimension, 9, Rank, 1)
│ │ │  (Partition{2, 1, 1, 1, 1}, Ambient_Dimension, 5, Rank, 0)
│ │ │  (Partition{1, 1, 1, 1, 1, 1}, Ambient_Dimension, 1, Rank, 1)
│ │ │ - -- used 0.860244s (cpu); 0.695269s (thread); 0s (gc)
│ │ │ + -- used 0.765041s (cpu); 0.541557s (thread); 0s (gc)
│ │ │  
│ │ │  i4 : seco#(new Partition from {2,2,2})
│ │ │  
│ │ │                                                        2 2 2       4 2   2     2   2 2     2 2     2   4   2   2   2     2 2   1 2 2       2 2   2     1   2 2     1 2 2       2 2 2       1 2 2       1 2   2     2   2 2     1 2   2     2 2   2     1   2 2     1   2 2     1 2     2   2   2   2   1     2 2   2 2     2   1   2   2   1 2     2   1 2     2   1   2   2   2   2   2   1     2 2   2     2 2   1     2 2   1 2 2       2 2   2     1   2 2     1 2 2       2 2 2       1 2 2       1 2   2     2   2 2     1 2   2     2 2   2     1   2 2     1   2 2     2 2 2       4 2 2       2 2 2       2 2 2       4 2 2       2 2 2       1 2   2     1   2 2     2 2   2     1 2   2     2   2 2     1   2 2     1 2   2     2 2   2     1 2   2     1   2 2     2   2 2     1   2 2     1 2     2   2   2   2   1     2 2   2 2     2   1   2   2   1 2     2   1 2     2   1   2   2   2   2   2   1     2 2   2     2 2   1     2 2   1 2     2   1   2   2   2 2     2   1 2     2   2   2   2   1   2   2   1 2     2   2 2     2   1 2     2   1   2   2   2   2   2   1   2   2   2     2 2   4     2 2   2     2 2   2     2 2   4     2 2   2     2 2
│ │ │  o4 = HashTable{{0, 1, 2, 3, 4, 5} => HashTable{0 => - -x x x x  + -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x }                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        }
│ │ │                                                        3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 5   3 1 2 3 5   3 1 2 3 5   3 1 2 4 5   3 1 3 4 5   3 2 3 4 5   3 1 2 4 5   3 1 2 4 5   3 1 3 4 5   3 2 3 4 5   3 1 3 4 5   3 2 3 4 5   3 1 2 3 5   3 1 2 3 5   3 1 2 3 5   3 1 2 4 5   3 1 2 4 5   3 1 3 4 5   3 2 3 4 5   3 1 3 4 5   3 2 3 4 5   3 1 2 4 5   3 1 3 4 5   3 2 3 4 5   3 1 2 3 6   3 1 2 3 6   3 1 2 3 6   3 1 2 4 6   3 1 3 4 6   3 2 3 4 6   3 1 2 4 6   3 1 2 4 6   3 1 3 4 6   3 2 3 4 6   3 1 3 4 6   3 2 3 4 6   3 1 2 5 6   3 1 3 5 6   3 2 3 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 2 5 6   3 1 2 5 6   3 1 3 5 6   3 2 3 5 6   3 1 3 5 6   3 2 3 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 2 3 6   3 1 2 3 6   3 1 2 3 6   3 1 2 4 6   3 1 2 4 6   3 1 3 4 6   3 2 3 4 6   3 1 3 4 6   3 2 3 4 6   3 1 2 4 6   3 1 3 4 6   3 2 3 4 6   3 1 2 5 6   3 1 2 5 6   3 1 3 5 6   3 2 3 5 6   3 1 3 5 6   3 2 3 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 2 5 6   3 1 3 5 6   3 2 3 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6
│ │ │                                                      2 3 2 2     4 2 3 2     2 2 2 3     4 3 2   2   2 2 3   2   2 3   2 2   2   3 2 2   2 2   3 2   4   2 3 2   2 2 2   3   4 2   2 3   2   2 2 3   1 3 2 2     2 2 3 2     1 2 2 3     2 3 2 2     1 2 3 2     1 3 2 2     1 3 2 2     1 2 3 2     2 2 3 2     1 2 2 3     2 2 2 3     1 2 2 3     2 3 2   2   1 2 3   2   1 3   2 2   1   3 2 2   1 2   3 2   2   2 3 2   1 3 2   2   2 2 3   2   1 3 2   2   2 3 2   2   1 2 3   2   1 2 3   2   1 3   2 2   1   3 2 2   2 3   2 2   1 3   2 2   2   3 2 2   1   3 2 2   2 2   3 2   1   2 3 2   1 2   3 2   1 2   3 2   1   2 3 2   2   2 3 2   1 2 2   3   2 2   2 3   1   2 2 3   1 2 2   3   2 2 2   3   1 2 2   3   1 2   2 3   2   2 2 3   1 2   2 3   2 2   2 3   1   2 2 3   1   2 2 3   1 3 2 2     2 2 3 2     1 2 2 3     2 3 2 2     1 2 3 2     1 3 2 2     1 3 2 2     1 2 3 2     2 2 3 2     1 2 2 3     2 2 2 3     1 2 2 3     1 3 2 2     1 2 3 2     2 3 2 2     1 3 2 2     2 2 3 2     1 2 3 2     1 3 2 2     2 3 2 2     1 3 2 2     1 2 3 2     2 2 3 2     1 2 3 2     2 2 2 3     4 2 2 3     2 2 2 3     2 2 2 3     4 2 2 3     2 2 2 3     2 3 2   2   1 2 3   2   1 3   2 2   1   3 2 2   1 2   3 2   2   2 3 2   1 3 2   2   2 2 3   2   1 3 2   2   2 3 2   2   1 2 3   2   1 2 3   2   1 3   2 2   1   3 2 2   2 3   2 2   1 3   2 2   2   3 2 2   1   3 2 2   2 2   3 2   1   2 3 2   1 2   3 2   1 2   3 2   1   2 3 2   2   2 3 2   1 3 2   2   1 2 3   2   2 3 2   2   1 3 2   2   2 2 3   2   1 2 3   2   1 3 2   2   2 3 2   2   1 3 2   2   1 2 3   2   2 2 3   2   1 2 3   2   2 3   2 2   2   3 2 2   4 3   2 2   2 3   2 2   4   3 2 2   2   3 2 2   2 3   2 2   4 3   2 2   2 3   2 2   2   3 2 2   4   3 2 2   2   3 2 2   1 2   3 2   1   2 3 2   2 2   3 2   1 2   3 2   2   2 3 2   1   2 3 2   1 2   3 2   2 2   3 2   1 2   3 2   1   2 3 2   2   2 3 2   1   2 3 2   1 2 2   3   2 2   2 3   1   2 2 3   1 2 2   3   2 2 2   3   1 2 2   3   1 2   2 3   2   2 2 3   1 2   2 3   2 2   2 3   1   2 2 3   1   2 2 3   2 2 2   3   4 2 2   3   2 2 2   3   2 2 2   3   4 2 2   3   2 2 2   3   1 2   2 3   1   2 2 3   2 2   2 3   1 2   2 3   2   2 2 3   1   2 2 3   1 2   2 3   2 2   2 3   1 2   2 3   1   2 2 3   2   2 2 3   1   2 2 3
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/html/_higher__Specht__Polynomial_lp__Young__Tableau_cm__Young__Tableau_cm__Polynomial__Ring_rp.html
│ │ │ @@ -125,15 +125,15 @@
│ │ │  
│ │ │  o4 : YoungTableau
    
│ │ │              
    
│ │ │                
    i5 : time higherSpechtPolynomial(S,T,R)
│ │ │ - -- used 0.00149283s (cpu); 0.00148952s (thread); 0s (gc)
│ │ │ + -- used 0.00167486s (cpu); 0.00167197s (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  -
│ │ │ @@ -149,15 +149,15 @@
│ │ │  
│ │ │  o5 : R
    
│ │ │              
    
│ │ │                
    i6 : time higherSpechtPolynomial(S,T,R, Robust => false)
│ │ │ - -- used 0.00128378s (cpu); 0.00128396s (thread); 0s (gc)
│ │ │ + -- used 0.00165147s (cpu); 0.00165278s (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  -
│ │ │ @@ -173,15 +173,15 @@
│ │ │  
│ │ │  o6 : R
    
│ │ │              
    
│ │ │                
    i7 : time higherSpechtPolynomial(S,T,R, Robust => false, AsExpression => true)
│ │ │ - -- used 0.00197708s (cpu); 0.00197739s (thread); 0s (gc)
│ │ │ + -- used 0.00229176s (cpu); 0.00229227s (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
    
│ │ │              
    
│ │ │                
    i4 : partis = partitions 6;
    
│ │ │              
    
│ │ │                
    i5 : time multi = hashTable apply (partis, p-> p=> representationMultiplicity(tal,p))
│ │ │ - -- used 0.339216s (cpu); 0.297324s (thread); 0s (gc)
│ │ │ + -- used 0.385296s (cpu); 0.323421s (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 {}
│ │ │ │ @@ -63,15 +63,15 @@
│ │ │ │  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 0.339216s (cpu); 0.297324s (thread); 0s (gc)
│ │ │ │ + -- used 0.385296s (cpu); 0.323421s (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
│ │ │ @@ -109,15 +109,15 @@
│ │ │  (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)
│ │ │  (Partition{2, 2, 2}, Ambient_Dimension, 5, Rank, 1)
│ │ │  (Partition{2, 2, 1, 1}, Ambient_Dimension, 9, Rank, 1)
│ │ │  (Partition{2, 1, 1, 1, 1}, Ambient_Dimension, 5, Rank, 0)
│ │ │  (Partition{1, 1, 1, 1, 1, 1}, Ambient_Dimension, 1, Rank, 1)
│ │ │ - -- used 0.860244s (cpu); 0.695269s (thread); 0s (gc)
    
│ │ │ + -- used 0.765041s (cpu); 0.541557s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i4 : seco#(new Partition from {2,2,2})
│ │ │  
│ │ │                                                        2 2 2       4 2   2     2   2 2     2 2     2   4   2   2   2     2 2   1 2 2       2 2   2     1   2 2     1 2 2       2 2 2       1 2 2       1 2   2     2   2 2     1 2   2     2 2   2     1   2 2     1   2 2     1 2     2   2   2   2   1     2 2   2 2     2   1   2   2   1 2     2   1 2     2   1   2   2   2   2   2   1     2 2   2     2 2   1     2 2   1 2 2       2 2   2     1   2 2     1 2 2       2 2 2       1 2 2       1 2   2     2   2 2     1 2   2     2 2   2     1   2 2     1   2 2     2 2 2       4 2 2       2 2 2       2 2 2       4 2 2       2 2 2       1 2   2     1   2 2     2 2   2     1 2   2     2   2 2     1   2 2     1 2   2     2 2   2     1 2   2     1   2 2     2   2 2     1   2 2     1 2     2   2   2   2   1     2 2   2 2     2   1   2   2   1 2     2   1 2     2   1   2   2   2   2   2   1     2 2   2     2 2   1     2 2   1 2     2   1   2   2   2 2     2   1 2     2   2   2   2   1   2   2   1 2     2   2 2     2   1 2     2   1   2   2   2   2   2   1   2   2   2     2 2   4     2 2   2     2 2   2     2 2   4     2 2   2     2 2
│ │ │ ├── html2text {}
│ │ │ │ @@ -56,15 +56,15 @@
│ │ │ │  (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)
│ │ │ │  (Partition{2, 2, 2}, Ambient_Dimension, 5, Rank, 1)
│ │ │ │  (Partition{2, 2, 1, 1}, Ambient_Dimension, 9, Rank, 1)
│ │ │ │  (Partition{2, 1, 1, 1, 1}, Ambient_Dimension, 5, Rank, 0)
│ │ │ │  (Partition{1, 1, 1, 1, 1, 1}, Ambient_Dimension, 1, Rank, 1)
│ │ │ │ - -- used 0.860244s (cpu); 0.695269s (thread); 0s (gc)
│ │ │ │ + -- used 0.765041s (cpu); 0.541557s (thread); 0s (gc)
│ │ │ │  i4 : seco#(new Partition from {2,2,2})
│ │ │ │  
│ │ │ │                                                        2 2 2       4 2   2     2
│ │ │ │  2 2     2 2     2   4   2   2   2     2 2   1 2 2       2 2   2     1   2 2
│ │ │ │  1 2 2       2 2 2       1 2 2       1 2   2     2   2 2     1 2   2     2 2   2
│ │ │ │  1   2 2     1   2 2     1 2     2   2   2   2   1     2 2   2 2     2   1   2
│ │ │ │  2   1 2     2   1 2     2   1   2   2   2   2   2   1     2 2   2     2 2   1
│ │ ├── ./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 2.03131s (cpu); 1.06704s (thread); 0s (gc)
│ │ │ + -- used 2.73142s (cpu); 1.10853s (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 1.94686s (cpu); 1.08366s (thread); 0s (gc)
│ │ │ + -- used 2.71338s (cpu); 1.06295s (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 6.18185s (cpu); 3.89639s (thread); 0s (gc)
│ │ │ + -- used 8.19386s (cpu); 4.56168s (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 3.88786s (cpu); 2.13569s (thread); 0s (gc)
│ │ │ + -- used 3.28863s (cpu); 1.96592s (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 0.372515s (cpu); 0.293278s (thread); 0s (gc)
│ │ │ + -- used 0.383174s (cpu); 0.345902s (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 12.44s (cpu); 6.54122s (thread); 0s (gc)
│ │ │ + -- used 21.0838s (cpu); 7.96065s (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 0.442939s (cpu); 0.267302s (thread); 0s (gc)
│ │ │ + -- used 0.653979s (cpu); 0.304604s (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 0.455756s (cpu); 0.177026s (thread); 0s (gc)
│ │ │ + -- used 0.372043s (cpu); 0.120614s (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 0.961803s (cpu); 0.441922s (thread); 0s (gc)
│ │ │ + -- used 1.06774s (cpu); 0.491515s (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 0.295634s (cpu); 0.202129s (thread); 0s (gc)
│ │ │ + -- used 0.473492s (cpu); 0.26936s (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 0.655776s (cpu); 0.413174s (thread); 0s (gc)
│ │ │ + -- used 0.748694s (cpu); 0.441106s (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 3.58482s (cpu); 2.29779s (thread); 0s (gc)
│ │ │ + -- used 4.08846s (cpu); 2.88086s (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 1.60219s (cpu); 0.786489s (thread); 0s (gc)
│ │ │ + -- used 1.59845s (cpu); 0.773506s (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 0.00810325s (cpu); 0.00921378s (thread); 0s (gc)
│ │ │ + -- used 0.0121259s (cpu); 0.0129517s (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 0.45505s (cpu); 0.218762s (thread); 0s (gc)
│ │ │ + -- used 0.831207s (cpu); 0.237716s (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 2.32867s (cpu); 1.58006s (thread); 0s (gc)
│ │ │ + -- used 1.94654s (cpu); 1.60837s (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 5.28433s (cpu); 3.03701s (thread); 0s (gc)
│ │ │ + -- used 6.44517s (cpu); 3.34763s (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 4.33255s (cpu); 2.58524s (thread); 0s (gc)
│ │ │ + -- used 5.67342s (cpu); 3.07795s (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 4.71829s (cpu); 2.90582s (thread); 0s (gc)
│ │ │ + -- used 5.66204s (cpu); 3.126s (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 1.21143s (cpu); 0.601591s (thread); 0s (gc)
│ │ │ + -- used 1.50286s (cpu); 0.741151s (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
│ │ │ @@ -106,15 +106,15 @@
│ │ │       cut out by 5 hypersurfaces of degree 1
│ │ │       (This is a classical example of rational fourfold)
    
│ │ │              
    
│ │ │                
    i3 : time U' = associatedCastelnuovoSurface X;
│ │ │ - -- used 2.03131s (cpu); 1.06704s (thread); 0s (gc)
│ │ │ + -- used 2.73142s (cpu); 1.10853s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : ProjectiveVariety, Castelnuovo surface associated to X
    
│ │ │              
    
│ │ │                
    i4 : (mu,U,C,f) = building U';
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,15 +41,15 @@
│ │ │ │  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 2.03131s (cpu); 1.06704s (thread); 0s (gc)
│ │ │ │ + -- used 2.73142s (cpu); 1.10853s (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
│ │ │ @@ -104,15 +104,15 @@
│ │ │       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 1.94686s (cpu); 1.08366s (thread); 0s (gc)
│ │ │ + -- used 2.71338s (cpu); 1.06295s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : ProjectiveVariety, K3 surface associated to X
    
│ │ │              
    
│ │ │                
    i4 : (mu,U,C,f) = building U';
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,15 +41,15 @@
│ │ │ │  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 1.94686s (cpu); 1.08366s (thread); 0s (gc)
│ │ │ │ + -- used 2.71338s (cpu); 1.06295s (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
│ │ │ @@ -107,15 +107,15 @@
│ │ │       Type: ordinary
│ │ │       (case 1 of Table 1 in arXiv:2002.07026)
│ │ │              
    
│ │ │                
    i3 : time U' = associatedK3surface X;
│ │ │ - -- used 6.18185s (cpu); 3.89639s (thread); 0s (gc)
│ │ │ + -- used 8.19386s (cpu); 4.56168s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : ProjectiveVariety, K3 surface associated to X
    
│ │ │              
    
│ │ │                
    i4 : (mu,U,C,f) = building U';
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -43,15 +43,15 @@
│ │ │ │  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 6.18185s (cpu); 3.89639s (thread); 0s (gc)
│ │ │ │ + -- used 8.19386s (cpu); 4.56168s (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
│ │ │ @@ -91,15 +91,15 @@
│ │ │       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 3.88786s (cpu); 2.13569s (thread); 0s (gc)
│ │ │ + -- used 3.28863s (cpu); 1.96592s (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 0.372515s (cpu); 0.293278s (thread); 0s (gc)
│ │ │ + -- used 0.383174s (cpu); 0.345902s (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 {}
│ │ │ │ @@ -30,28 +30,28 @@
│ │ │ │  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 3.88786s (cpu); 2.13569s (thread); 0s (gc)
│ │ │ │ + -- used 3.28863s (cpu); 1.96592s (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 0.372515s (cpu); 0.293278s (thread); 0s (gc)
│ │ │ │ + -- used 0.383174s (cpu); 0.345902s (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
│ │ │ @@ -94,15 +94,15 @@
│ │ │       Type: ordinary
│ │ │       (case 17 of Table 1 in arXiv:2002.07026)
│ │ │              
    
│ │ │                
    i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ - -- used 12.44s (cpu); 6.54122s (thread); 0s (gc)
│ │ │ + -- used 21.0838s (cpu); 7.96065s (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
    
│ │ │              
    
│ │ │                
    i6 : time C = f p; -- 3-secant conic to the surface
│ │ │ - -- used 0.442939s (cpu); 0.267302s (thread); 0s (gc)
│ │ │ + -- used 0.653979s (cpu); 0.304604s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : ProjectiveVariety, curve in PP^8 (subvariety of codimension 4 in Y)
    
│ │ │              
    
│ │ │                
    i7 : S = surface X;
│ │ │ ├── html2text {}
│ │ │ │ @@ -36,15 +36,15 @@
│ │ │ │  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 12.44s (cpu); 6.54122s (thread); 0s (gc)
│ │ │ │ + -- used 21.0838s (cpu); 7.96065s (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
│ │ │ │ @@ -53,15 +53,15 @@
│ │ │ │  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 0.442939s (cpu); 0.267302s (thread); 0s (gc)
│ │ │ │ + -- used 0.653979s (cpu); 0.304604s (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 @@
│ │ │  
│ │ │  o1 : ProjectiveVariety, cubic fourfold containing a surface of degree 5 and sectional genus 1
    
│ │ │              
    
│ │ │                
    i2 : time discriminant X
│ │ │ - -- used 0.455756s (cpu); 0.177026s (thread); 0s (gc)
│ │ │ + -- used 0.372043s (cpu); 0.120614s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 14
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,15 +20,15 @@
│ │ │ │  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 0.455756s (cpu); 0.177026s (thread); 0s (gc)
│ │ │ │ + -- used 0.372043s (cpu); 0.120614s (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 @@
│ │ │  
│ │ │  o1 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and sectional genus 0
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time discriminant X
│ │ │ - -- used 0.961803s (cpu); 0.441922s (thread); 0s (gc)
│ │ │ + -- used 1.06774s (cpu); 0.491515s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 10
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,15 +20,15 @@
│ │ │ │  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 0.961803s (cpu); 0.441922s (thread); 0s (gc)
│ │ │ │ + -- used 1.06774s (cpu); 0.491515s (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
│ │ │ @@ -88,15 +88,15 @@
│ │ │  
│ │ │  o3 : ProjectiveVariety, surface in PP^5
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time parameterCount(S,X,Verbose=>true)
│ │ │ - -- used 0.295634s (cpu); 0.202129s (thread); 0s (gc)
│ │ │ + -- used 0.473492s (cpu); 0.26936s (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 {}
│ │ │ │ @@ -23,15 +23,15 @@
│ │ │ │  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 0.295634s (cpu); 0.202129s (thread); 0s (gc)
│ │ │ │ + -- used 0.473492s (cpu); 0.26936s (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
│ │ │ @@ -89,15 +89,15 @@
│ │ │  
│ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and sectional genus 0
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time parameterCount(X,Verbose=>true)
│ │ │ - -- used 0.655776s (cpu); 0.413174s (thread); 0s (gc)
│ │ │ + -- used 0.748694s (cpu); 0.441106s (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 {}
│ │ │ │ @@ -33,15 +33,15 @@
│ │ │ │  
│ │ │ │  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 0.655776s (cpu); 0.413174s (thread); 0s (gc)
│ │ │ │ + -- used 0.748694s (cpu); 0.441106s (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
│ │ │ @@ -98,15 +98,15 @@
│ │ │  
│ │ │  o3 : ProjectiveVariety, GM fourfold containing a surface of degree 3 and sectional genus 0
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time parameterCount(X,Verbose=>true)
│ │ │ - -- used 3.58482s (cpu); 2.29779s (thread); 0s (gc)
│ │ │ + -- used 4.08846s (cpu); 2.88086s (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 {}
│ │ │ │ @@ -35,15 +35,15 @@
│ │ │ │  
│ │ │ │  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 3.58482s (cpu); 2.29779s (thread); 0s (gc)
│ │ │ │ + -- used 4.08846s (cpu); 2.88086s (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
│ │ │ @@ -88,15 +88,15 @@
│ │ │  o3 = 4-dimensional subvariety of PP^9 cut out by 7 hypersurfaces of degrees
│ │ │       1^2 2^5
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time parametrizeFanoFourfold X
│ │ │ - -- used 1.60219s (cpu); 0.786489s (thread); 0s (gc)
│ │ │ + -- used 1.59845s (cpu); 0.773506s (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 {}
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │  
│ │ │ │  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 1.60219s (cpu); 0.786489s (thread); 0s (gc)
│ │ │ │ + -- used 1.59845s (cpu); 0.773506s (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
│ │ │ @@ -95,25 +95,25 @@
│ │ │  
│ │ │  o4 : ProjectiveVariety, hypersurface in PP^5
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time F = specialCubicFourfold(S,X,NumNodes=>3);
│ │ │ - -- used 0.00810325s (cpu); 0.00921378s (thread); 0s (gc)
│ │ │ + -- used 0.0121259s (cpu); 0.0129517s (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 0.45505s (cpu); 0.218762s (thread); 0s (gc)
│ │ │ + -- used 0.831207s (cpu); 0.237716s (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
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -115,24 +115,24 @@
│ │ │ │  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 0.00810325s (cpu); 0.00921378s (thread); 0s (gc)
│ │ │ │ + -- used 0.0121259s (cpu); 0.0129517s (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 0.45505s (cpu); 0.218762s (thread); 0s (gc)
│ │ │ │ + -- used 0.831207s (cpu); 0.237716s (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
│ │ │ @@ -93,25 +93,25 @@
│ │ │  
│ │ │  o4 : ProjectiveVariety, 4-dimensional subvariety of PP^8
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time F = specialGushelMukaiFourfold(S,X);
│ │ │ - -- used 2.32867s (cpu); 1.58006s (thread); 0s (gc)
│ │ │ + -- used 1.94654s (cpu); 1.60837s (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 5.28433s (cpu); 3.03701s (thread); 0s (gc)
│ │ │ + -- used 6.44517s (cpu); 3.34763s (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 {}
│ │ │ │ @@ -33,24 +33,24 @@
│ │ │ │  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 2.32867s (cpu); 1.58006s (thread); 0s (gc)
│ │ │ │ + -- used 1.94654s (cpu); 1.60837s (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 5.28433s (cpu); 3.03701s (thread); 0s (gc)
│ │ │ │ + -- used 6.44517s (cpu); 3.34763s (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
│ │ │ @@ -81,15 +81,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    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 4.33255s (cpu); 2.58524s (thread); 0s (gc)
│ │ │ + -- used 5.67342s (cpu); 3.07795s (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 {}
│ │ │ │ @@ -26,15 +26,15 @@
│ │ │ │  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 4.33255s (cpu); 2.58524s (thread); 0s (gc)
│ │ │ │ + -- used 5.67342s (cpu); 3.07795s (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
│ │ │ @@ -82,15 +82,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    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 4.71829s (cpu); 2.90582s (thread); 0s (gc)
│ │ │ + -- used 5.66204s (cpu); 3.126s (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 {}
│ │ │ │ @@ -25,15 +25,15 @@
│ │ │ │  
│ │ │ │  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 4.71829s (cpu); 2.90582s (thread); 0s (gc)
│ │ │ │ + -- used 5.66204s (cpu); 3.126s (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
│ │ │ @@ -82,15 +82,15 @@
│ │ │  
│ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and sectional genus 0
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time f = unirationalParametrization X;
│ │ │ - -- used 1.21143s (cpu); 0.601591s (thread); 0s (gc)
│ │ │ + -- used 1.50286s (cpu); 0.741151s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : MultirationalMap (rational map from PP^4 to X)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : degreeSequence f
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,15 +18,15 @@
│ │ │ │  
│ │ │ │  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 1.21143s (cpu); 0.601591s (thread); 0s (gc)
│ │ │ │ + -- used 1.50286s (cpu); 0.741151s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : MultirationalMap (rational map from PP^4 to X)
│ │ │ │  i5 : degreeSequence f
│ │ │ │  
│ │ │ │  o5 = {[10]}
│ │ │ │  
│ │ │ │  o5 : List
│ │ ├── ./usr/share/doc/Macaulay2/StatGraphs/example-output/_graph_lp__Mixed__Graph_rp.out
│ │ │ @@ -30,15 +30,15 @@
│ │ │                                b => {a, c}
│ │ │                                c => {b}
│ │ │  
│ │ │  o2 : HashTable
│ │ │  
│ │ │  i3 : keys (graph G)
│ │ │  
│ │ │ -o3 = {Graph, Bigraph, Digraph}
│ │ │ +o3 = {Digraph, Graph, Bigraph}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : (graph G)#Bigraph === bigraph G
│ │ │  
│ │ │  o4 = true
│ │ ├── ./usr/share/doc/Macaulay2/StatGraphs/example-output/_to__String_lp__Mixed__Graph_rp.out
│ │ │ @@ -11,12 +11,12 @@
│ │ │                  Graph => Graph{1 => {3}}
│ │ │                                 3 => {1}
│ │ │  
│ │ │  o1 : MixedGraph
│ │ │  
│ │ │  i2 : toString G
│ │ │  
│ │ │ -o2 = new HashTable from {Digraph => digraph ({1, 2, 3}, {{1, 2}, {2, 3}}),
│ │ │ -     Graph => graph ({3, 1}, {{1, 3}}), Bigraph => bigraph ({3, 4, 2}, {{4,
│ │ │ -     3}, {4, 2}})}
│ │ │ +o2 = new HashTable from {Graph => graph ({3, 1}, {{1, 3}}), Bigraph =>
│ │ │ +     bigraph ({3, 4, 2}, {{4, 3}, {4, 2}}), Digraph => digraph ({1, 2, 3},
│ │ │ +     {{1, 2}, {2, 3}})}
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/StatGraphs/html/_graph_lp__Mixed__Graph_rp.html
│ │ │ @@ -116,15 +116,15 @@
│ │ │  o2 : HashTable
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : keys (graph G)
│ │ │  
│ │ │ -o3 = {Graph, Bigraph, Digraph}
│ │ │ +o3 = {Digraph, Graph, Bigraph}
│ │ │  
│ │ │  o3 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : (graph G)#Bigraph === bigraph G
│ │ │ ├── html2text {}
│ │ │ │ @@ -46,15 +46,15 @@
│ │ │ │                 Graph => Graph{a => {b}   }
│ │ │ │                                b => {a, c}
│ │ │ │                                c => {b}
│ │ │ │  
│ │ │ │  o2 : HashTable
│ │ │ │  i3 : keys (graph G)
│ │ │ │  
│ │ │ │ -o3 = {Graph, Bigraph, Digraph}
│ │ │ │ +o3 = {Digraph, Graph, Bigraph}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : (graph G)#Bigraph === bigraph G
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _M_i_x_e_d_G_r_a_p_h -- a graph that has undirected, directed and bidirected edges
│ │ ├── ./usr/share/doc/Macaulay2/StatGraphs/html/_to__String_lp__Mixed__Graph_rp.html
│ │ │ @@ -88,17 +88,17 @@
│ │ │  o1 : MixedGraph
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : toString G
│ │ │  
│ │ │ -o2 = new HashTable from {Digraph => digraph ({1, 2, 3}, {{1, 2}, {2, 3}}),
│ │ │ -     Graph => graph ({3, 1}, {{1, 3}}), Bigraph => bigraph ({3, 4, 2}, {{4,
│ │ │ -     3}, {4, 2}})}
    
│ │ │ +o2 = new HashTable from {Graph => graph ({3, 1}, {{1, 3}}), Bigraph =>
│ │ │ +     bigraph ({3, 4, 2}, {{4, 3}, {4, 2}}), Digraph => digraph ({1, 2, 3},
│ │ │ +     {{1, 2}, {2, 3}})}
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,17 +23,17 @@
│ │ │ │                                     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}}),
│ │ │ │ -     Graph => graph ({3, 1}, {{1, 3}}), Bigraph => bigraph ({3, 4, 2}, {{4,
│ │ │ │ -     3}, {4, 2}})}
│ │ │ │ +o2 = new HashTable from {Graph => graph ({3, 1}, {{1, 3}}), Bigraph =>
│ │ │ │ +     bigraph ({3, 4, 2}, {{4, 3}, {4, 2}}), 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/Style/example-output/_generate__Grammar.out
│ │ │ @@ -1,16 +1,16 @@
│ │ │  -- -*- M2-comint -*- hash: 3455701143666534588
│ │ │  
│ │ │  i1 : outfile = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10069-0/0
│ │ │ +o1 = /tmp/M2-10109-0/0
│ │ │  
│ │ │  i2 : template = outfile | ".in"
│ │ │  
│ │ │ -o2 = /tmp/M2-10069-0/0.in
│ │ │ +o2 = /tmp/M2-10109-0/0.in
│ │ │  
│ │ │  i3 : template << "@M2BANNER@" << endl << endl;
│ │ │  
│ │ │  i4 : template << "This is an example file for the generateGrammar method!";
│ │ │  
│ │ │  i5 : template << endl;
│ │ │  
│ │ │ @@ -30,15 +30,15 @@
│ │ │        String regex: @M2STRINGS@
│ │ │        List of keywords: {
│ │ │            @M2KEYWORDS@
│ │ │        }
│ │ │  
│ │ │  
│ │ │  i11 : generateGrammar(template, outfile, x -> demark(",\n    ", x))
│ │ │ - -- generating /tmp/M2-10069-0/0
│ │ │ + -- generating /tmp/M2-10109-0/0
│ │ │  
│ │ │  i12 : get outfile
│ │ │  
│ │ │  o12 = Auto-generated for Macaulay2-1.25.11. Do not modify this file manually.
│ │ │  
│ │ │        This is an example file for the generateGrammar method!
│ │ │        String regex: "///\\(/?/?[^/]\\|\\(//\\)*////[^/]\\)*\\(//\\)*///"
│ │ ├── ./usr/share/doc/Macaulay2/Style/html/_generate__Grammar.html
│ │ │ @@ -82,22 +82,22 @@
│ │ │            
      The function demarkf indicates how the elements of each of the lists will be demarked in the resulting file.   The file outfile will then be generated, replacing each of these strings as indicated above.
      
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -143,15 +143,15 @@
│ │ │            @M2KEYWORDS@
│ │ │        }
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : outfile = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10069-0/0
    
│ │ │ +o1 = /tmp/M2-10109-0/0
│ │ │              
    
│ │ │                
    i2 : template = outfile | ".in"
│ │ │  
│ │ │ -o2 = /tmp/M2-10069-0/0.in
    
│ │ │ +o2 = /tmp/M2-10109-0/0.in
│ │ │              
    
│ │ │                
    i3 : template << "@M2BANNER@" << endl << endl;
    
│ │ │              
    
│ │ │                
    i11 : generateGrammar(template, outfile, x -> demark(",\n    ", x))
│ │ │ - -- generating /tmp/M2-10069-0/0
    
│ │ │ + -- generating /tmp/M2-10109-0/0
│ │ │              
    
│ │ │                
    i12 : get outfile
│ │ │  
│ │ │  o12 = Auto-generated for Macaulay2-1.25.11. Do not modify this file manually.
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,18 +26,18 @@
│ │ │ │      * @M2CONSTANTS@, for a list of Macaulay2 symbols and packages.
│ │ │ │      * @M2STRINGS@, for a regular expression that matches Macaulay2 strings.
│ │ │ │  The function demarkf indicates how the elements of each of the lists will be
│ │ │ │  demarked in the resulting file. The file outfile will then be generated,
│ │ │ │  replacing each of these strings as indicated above.
│ │ │ │  i1 : outfile = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-10069-0/0
│ │ │ │ +o1 = /tmp/M2-10109-0/0
│ │ │ │  i2 : template = outfile | ".in"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10069-0/0.in
│ │ │ │ +o2 = /tmp/M2-10109-0/0.in
│ │ │ │  i3 : template << "@M2BANNER@" << endl << endl;
│ │ │ │  i4 : template << "This is an example file for the generateGrammar method!";
│ │ │ │  i5 : template << endl;
│ │ │ │  i6 : template << "String regex: @M2STRINGS@" << endl;
│ │ │ │  i7 : template << "List of keywords: {" << endl;
│ │ │ │  i8 : template << "    @M2KEYWORDS@" << endl;
│ │ │ │  i9 : template << "}" << endl << close;
│ │ │ │ @@ -47,15 +47,15 @@
│ │ │ │  
│ │ │ │        This is an example file for the generateGrammar method!
│ │ │ │        String regex: @M2STRINGS@
│ │ │ │        List of keywords: {
│ │ │ │            @M2KEYWORDS@
│ │ │ │        }
│ │ │ │  i11 : generateGrammar(template, outfile, x -> demark(",\n    ", x))
│ │ │ │ - -- generating /tmp/M2-10069-0/0
│ │ │ │ + -- generating /tmp/M2-10109-0/0
│ │ │ │  i12 : get outfile
│ │ │ │  
│ │ │ │  o12 = Auto-generated for Macaulay2-1.25.11. Do not modify this file manually.
│ │ │ │  
│ │ │ │        This is an example file for the generateGrammar method!
│ │ │ │        String regex: "///\\(/?/?[^/]\\|\\(//\\)*////[^/]\\)*\\(//\\)*///"
│ │ │ │        List of keywords: {
│ │ ├── ./usr/share/doc/Macaulay2/SubalgebraBases/dump/rawdocumentation.dump
│ │ │ @@ -1,8 +1,8 @@
│ │ │ -# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Fri Nov 14 16:08:08 2025
│ │ │ +# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Fri Nov 14 16:08:07 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │  #:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=24
│ │ │  c2FnYmkoLi4uLFN0cmF0ZWd5PT4uLi4p
│ │ ├── ./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 0.257575s (cpu); 0.158536s (thread); 0s (gc)
│ │ │ + -- used 0.326193s (cpu); 0.185784s (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 0.0335686s (cpu); 0.0335728s (thread); 0s (gc)
│ │ │ + -- used 0.0365242s (cpu); 0.0365245s (thread); 0s (gc)
│ │ │  
│ │ │  o10 : Ideal of QQ[x..z]
│ │ │  
│ │ │  i11 :
│ │ ├── ./usr/share/doc/Macaulay2/SymbolicPowers/html/_symbolic__Power.html
│ │ │ @@ -141,15 +141,15 @@
│ │ │  
│ │ │  o6 = false
    
│ │ │              
    
│ │ │                
    i7 : time symbolicPower(P,4);
│ │ │ - -- used 0.257575s (cpu); 0.158536s (thread); 0s (gc)
│ │ │ + -- used 0.326193s (cpu); 0.185784s (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})
│ │ │ @@ -166,15 +166,15 @@
│ │ │  
│ │ │  o9 = true
    
│ │ │              
    
│ │ │                
    i10 : time symbolicPower(Q,4);
│ │ │ - -- used 0.0335686s (cpu); 0.0335728s (thread); 0s (gc)
│ │ │ + -- used 0.0365242s (cpu); 0.0365245s (thread); 0s (gc)
│ │ │  
│ │ │  o10 : Ideal of QQ[x..z]
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -59,28 +59,28 @@
│ │ │ │  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 0.257575s (cpu); 0.158536s (thread); 0s (gc)
│ │ │ │ + -- used 0.326193s (cpu); 0.185784s (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 0.0335686s (cpu); 0.0335728s (thread); 0s (gc)
│ │ │ │ + -- used 0.0365242s (cpu); 0.0365245s (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/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 0.118011s (cpu); 0.118013s (thread); 0s (gc)
│ │ │ + -- used 0.130802s (cpu); 0.130804s (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
│ │ │ @@ -92,15 +92,15 @@
│ │ │  
│ │ │  o3 : ChainComplex
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time T=tateExtension W;
│ │ │ - -- used 0.118011s (cpu); 0.118013s (thread); 0s (gc)
    
│ │ │ + -- used 0.130802s (cpu); 0.130804s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : cohomologyMatrix(T,-{3,3},{3,3})
│ │ │  
│ │ │  o5 = | 8h  4h  0 4  8  12 16 |
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,15 +23,15 @@
│ │ │ │               1
│ │ │ │  o3 = 0  <-- E  <-- 0
│ │ │ │  
│ │ │ │       -1     0      1
│ │ │ │  
│ │ │ │  o3 : ChainComplex
│ │ │ │  i4 : time T=tateExtension W;
│ │ │ │ - -- used 0.118011s (cpu); 0.118013s (thread); 0s (gc)
│ │ │ │ + -- used 0.130802s (cpu); 0.130804s (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/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 0.852713s (cpu); 0.64916s (thread); 0s (gc)
│ │ │ + -- used 1.05884s (cpu); 0.791334s (thread); 0s (gc)
│ │ │  
│ │ │  o17 : Ideal of R
│ │ │  
│ │ │  i18 : time J2 = frobeniusRoot(1, I1^8*I2^10*I3^12);
│ │ │ - -- used 2.97145s (cpu); 2.57894s (thread); 0s (gc)
│ │ │ + -- used 2.85632s (cpu); 2.33024s (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 0.00319701s (cpu); 0.00319364s (thread); 0s (gc)
│ │ │ + -- used 0.00259043s (cpu); 0.00258722s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = true
│ │ │  
│ │ │  i6 : time isCohenMacaulay(R, AtOrigin => true)
│ │ │ - -- used 0.00756402s (cpu); 0.00756525s (thread); 0s (gc)
│ │ │ + -- used 0.0057236s (cpu); 0.00572996s (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 0.026s (cpu); 0.0259991s (thread); 0s (gc)
│ │ │ + -- used 0.0318857s (cpu); 0.0317219s (thread); 0s (gc)
│ │ │  
│ │ │  o20 = true
│ │ │  
│ │ │  i21 : time isFInjective(R, CanonicalStrategy => null)
│ │ │ - -- used 2.29738s (cpu); 1.38959s (thread); 0s (gc)
│ │ │ + -- used 2.7042s (cpu); 1.51101s (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 0.146035s (cpu); 0.0901309s (thread); 0s (gc)
│ │ │ + -- used 0.170929s (cpu); 0.1023s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = false
│ │ │  
│ │ │  i24 : time isFInjective(R, AtOrigin => true)
│ │ │ - -- used 0.15607s (cpu); 0.0971744s (thread); 0s (gc)
│ │ │ + -- used 0.186976s (cpu); 0.115021s (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 0.875806s (cpu); 0.712389s (thread); 0s (gc)
│ │ │ + -- used 1.08087s (cpu); 0.862123s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = false
│ │ │  
│ │ │  i30 : time isFInjective(R, AssumeCM => true)
│ │ │ - -- used 0.362851s (cpu); 0.249689s (thread); 0s (gc)
│ │ │ + -- used 0.445819s (cpu); 0.301281s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = true
│ │ │  
│ │ │  i31 :
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Regular.out
│ │ │ @@ -80,19 +80,19 @@
│ │ │  
│ │ │  o25 : Ideal of S
│ │ │  
│ │ │  i26 : debugLevel = 1;
│ │ │  
│ │ │  i27 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 1)
│ │ │  isFRegular: This ring does not appear to be F-regular.  Increasing DepthOfSearch will let the function search more deeply.
│ │ │ - -- used 0.124736s (cpu); 0.0763298s (thread); 0s (gc)
│ │ │ + -- used 0.144598s (cpu); 0.0737087s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = false
│ │ │  
│ │ │  i28 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 2)
│ │ │ - -- used 0.229358s (cpu); 0.169449s (thread); 0s (gc)
│ │ │ + -- used 0.270865s (cpu); 0.202105s (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 0.331885s (cpu); 0.176341s (thread); 0s (gc)
│ │ │ + -- used 0.413895s (cpu); 0.209s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = ideal (y, x)
│ │ │  
│ │ │  o23 : Ideal of R
│ │ │  
│ │ │  i24 : time testIdeal(1/60, x^45*y^40*(x + y)^36)
│ │ │ - -- used 0.371358s (cpu); 0.249305s (thread); 0s (gc)
│ │ │ + -- used 0.496294s (cpu); 0.286593s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = ideal (y, x)
│ │ │  
│ │ │  o24 : Ideal of R
│ │ │  
│ │ │  i25 :
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Root.html
│ │ │ @@ -226,23 +226,23 @@
│ │ │  
│ │ │  o16 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i17 : time J1 = frobeniusRoot(1, {8, 10, 12}, {I1, I2, I3});
│ │ │ - -- used 0.852713s (cpu); 0.64916s (thread); 0s (gc)
│ │ │ + -- used 1.05884s (cpu); 0.791334s (thread); 0s (gc)
│ │ │  
│ │ │  o17 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i18 : time J2 = frobeniusRoot(1, I1^8*I2^10*I3^12);
│ │ │ - -- used 2.97145s (cpu); 2.57894s (thread); 0s (gc)
│ │ │ + -- used 2.85632s (cpu); 2.33024s (thread); 0s (gc)
│ │ │  
│ │ │  o18 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i19 : J1 == J2
│ │ │ ├── html2text {}
│ │ │ │ @@ -106,19 +106,19 @@
│ │ │ │  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 0.852713s (cpu); 0.64916s (thread); 0s (gc)
│ │ │ │ + -- used 1.05884s (cpu); 0.791334s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o17 : Ideal of R
│ │ │ │  i18 : time J2 = frobeniusRoot(1, I1^8*I2^10*I3^12);
│ │ │ │ - -- used 2.97145s (cpu); 2.57894s (thread); 0s (gc)
│ │ │ │ + -- used 2.85632s (cpu); 2.33024s (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
│ │ │ @@ -96,23 +96,23 @@
│ │ │              
│ │ │                
    i4 : R = S/(ker g);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time isCohenMacaulay(R)
│ │ │ - -- used 0.00319701s (cpu); 0.00319364s (thread); 0s (gc)
│ │ │ + -- used 0.00259043s (cpu); 0.00258722s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time isCohenMacaulay(R, AtOrigin => true)
│ │ │ - -- used 0.00756402s (cpu); 0.00756525s (thread); 0s (gc)
│ │ │ + -- used 0.0057236s (cpu); 0.00572996s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = true
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,19 +23,19 @@
│ │ │ │  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 0.00319701s (cpu); 0.00319364s (thread); 0s (gc)
│ │ │ │ + -- used 0.00259043s (cpu); 0.00258722s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = true
│ │ │ │  i6 : time isCohenMacaulay(R, AtOrigin => true)
│ │ │ │ - -- used 0.00756402s (cpu); 0.00756525s (thread); 0s (gc)
│ │ │ │ + -- used 0.0057236s (cpu); 0.00572996s (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
│ │ │ @@ -214,23 +214,23 @@
│ │ │  
│ │ │  o19 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i20 : time isFInjective(R)
│ │ │ - -- used 0.026s (cpu); 0.0259991s (thread); 0s (gc)
│ │ │ + -- used 0.0318857s (cpu); 0.0317219s (thread); 0s (gc)
│ │ │  
│ │ │  o20 = true
    
│ │ │              
    
│ │ │                
    i21 : time isFInjective(R, CanonicalStrategy => null)
│ │ │ - -- used 2.29738s (cpu); 1.38959s (thread); 0s (gc)
│ │ │ + -- used 2.7042s (cpu); 1.51101s (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.
    
│ │ │ @@ -240,23 +240,23 @@
│ │ │              
│ │ │                
    i22 : R = ZZ/7[x,y,z]/((x-1)^5 + (y+1)^5 + z^5);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i23 : time isFInjective(R)
│ │ │ - -- used 0.146035s (cpu); 0.0901309s (thread); 0s (gc)
│ │ │ + -- used 0.170929s (cpu); 0.1023s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = false
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i24 : time isFInjective(R, AtOrigin => true)
│ │ │ - -- used 0.15607s (cpu); 0.0971744s (thread); 0s (gc)
│ │ │ + -- used 0.186976s (cpu); 0.115021s (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 $\mathbb{P}^1$.
    
│ │ │ @@ -283,23 +283,23 @@
│ │ │              
│ │ │                
    i28 : R = S/(ker f);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i29 : time isFInjective(R)
│ │ │ - -- used 0.875806s (cpu); 0.712389s (thread); 0s (gc)
│ │ │ + -- used 1.08087s (cpu); 0.862123s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = false
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i30 : time isFInjective(R, AssumeCM => true)
│ │ │ - -- used 0.362851s (cpu); 0.249689s (thread); 0s (gc)
│ │ │ + -- used 0.445819s (cpu); 0.301281s (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 {}
│ │ │ │ @@ -81,52 +81,52 @@
│ │ │ │  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 0.026s (cpu); 0.0259991s (thread); 0s (gc)
│ │ │ │ + -- used 0.0318857s (cpu); 0.0317219s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o20 = true
│ │ │ │  i21 : time isFInjective(R, CanonicalStrategy => null)
│ │ │ │ - -- used 2.29738s (cpu); 1.38959s (thread); 0s (gc)
│ │ │ │ + -- used 2.7042s (cpu); 1.51101s (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 0.146035s (cpu); 0.0901309s (thread); 0s (gc)
│ │ │ │ + -- used 0.170929s (cpu); 0.1023s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = false
│ │ │ │  i24 : time isFInjective(R, AtOrigin => true)
│ │ │ │ - -- used 0.15607s (cpu); 0.0971744s (thread); 0s (gc)
│ │ │ │ + -- used 0.186976s (cpu); 0.115021s (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 0.875806s (cpu); 0.712389s (thread); 0s (gc)
│ │ │ │ + -- used 1.08087s (cpu); 0.862123s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o29 = false
│ │ │ │  i30 : time isFInjective(R, AssumeCM => true)
│ │ │ │ - -- used 0.362851s (cpu); 0.249689s (thread); 0s (gc)
│ │ │ │ + -- used 0.445819s (cpu); 0.301281s (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
│ │ │ @@ -273,23 +273,23 @@
│ │ │                
    i26 : debugLevel = 1;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i27 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 1)
│ │ │  isFRegular: This ring does not appear to be F-regular.  Increasing DepthOfSearch will let the function search more deeply.
│ │ │ - -- used 0.124736s (cpu); 0.0763298s (thread); 0s (gc)
│ │ │ + -- used 0.144598s (cpu); 0.0737087s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = false
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i28 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 2)
│ │ │ - -- used 0.229358s (cpu); 0.169449s (thread); 0s (gc)
│ │ │ + -- used 0.270865s (cpu); 0.202105s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i29 : debugLevel = 0;
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -114,19 +114,19 @@
│ │ │ │  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)
│ │ │ │  isFRegular: This ring does not appear to be F-regular.  Increasing
│ │ │ │  DepthOfSearch will let the function search more deeply.
│ │ │ │ - -- used 0.124736s (cpu); 0.0763298s (thread); 0s (gc)
│ │ │ │ + -- used 0.144598s (cpu); 0.0737087s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o27 = false
│ │ │ │  i28 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 2)
│ │ │ │ - -- used 0.229358s (cpu); 0.169449s (thread); 0s (gc)
│ │ │ │ + -- used 0.270865s (cpu); 0.202105s (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
│ │ │ @@ -255,25 +255,25 @@
│ │ │          
    
│ │ │            
    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 0.331885s (cpu); 0.176341s (thread); 0s (gc)
│ │ │ + -- used 0.413895s (cpu); 0.209s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = ideal (y, x)
│ │ │  
│ │ │  o23 : Ideal of R
    
│ │ │              
    
│ │ │                
    i24 : time testIdeal(1/60, x^45*y^40*(x + y)^36)
│ │ │ - -- used 0.371358s (cpu); 0.249305s (thread); 0s (gc)
│ │ │ + -- used 0.496294s (cpu); 0.286593s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = ideal (y, x)
│ │ │  
│ │ │  o24 : Ideal of R
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -100,21 +100,21 @@
│ │ │ │  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 0.331885s (cpu); 0.176341s (thread); 0s (gc)
│ │ │ │ + -- used 0.413895s (cpu); 0.209s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = ideal (y, x)
│ │ │ │  
│ │ │ │  o23 : Ideal of R
│ │ │ │  i24 : time testIdeal(1/60, x^45*y^40*(x + y)^36)
│ │ │ │ - -- used 0.371358s (cpu); 0.249305s (thread); 0s (gc)
│ │ │ │ + -- used 0.496294s (cpu); 0.286593s (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/ThreadedGB/example-output/___Minimal.out
│ │ │ @@ -2,17 +2,16 @@
│ │ │  
│ │ │  i1 : S = ZZ/101[a,b,c];
│ │ │  
│ │ │  i2 : allowableThreads= 2;
│ │ │  
│ │ │  i3 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2", Minimal=>true)
│ │ │  
│ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ -                  ((0, 1), 0) => null
│ │ │ -                  ((0, 1), 1) => null
│ │ │ +o3 = LineageTable{((0, 2), 0) => null}
│ │ │ +                  ((0, 2), 1) => null
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_matrix_lp__Lineage__Table_rp.out
│ │ │ @@ -2,16 +2,16 @@
│ │ │  
│ │ │  i1 : R = ZZ/101[a,b,c];
│ │ │  
│ │ │  i2 : allowableThreads= 2;
│ │ │  
│ │ │  i3 : T = reduce tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │  
│ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ -                  ((0, 1), 0) => null
│ │ │ +o3 = LineageTable{((0, 2), 0) => null}
│ │ │ +                  ((0, 2), 1) => null
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_minimize_lp__Lineage__Table_rp.out
│ │ │ @@ -2,18 +2,16 @@
│ │ │  
│ │ │  i1 : R = ZZ/101[a,b,c];
│ │ │  
│ │ │  i2 : allowableThreads= 2;
│ │ │  
│ │ │  i3 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │  
│ │ │ -                                        3
│ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => -c }
│ │ │ -                                     2
│ │ │ -                  ((0, 1), 0) => -a*c
│ │ │ +                                   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
│ │ │ @@ -24,16 +22,15 @@
│ │ │                          2
│ │ │                    2 => b
│ │ │  
│ │ │  o3 : LineageTable
│ │ │  
│ │ │  i4 : minimize T
│ │ │  
│ │ │ -o4 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ -                  ((0, 1), 0) => null
│ │ │ +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
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_reduce.out
│ │ │ @@ -2,18 +2,18 @@
│ │ │  
│ │ │  i1 : R = ZZ/101[a,b,c];
│ │ │  
│ │ │  i2 : allowableThreads= 2;
│ │ │  
│ │ │  i3 : T = tgb ideal "abc+c2,ab2-b3c+ac,b2"
│ │ │  
│ │ │ -                                        3
│ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => -c }
│ │ │                                       2
│ │ │ -                  ((0, 1), 0) => -a*c
│ │ │ +o3 = LineageTable{((0, 1), 0) => -a*c    }
│ │ │ +                                   3
│ │ │ +                  ((0, 2), 0) => -c
│ │ │                                     2
│ │ │                    ((1, 2), 0) => -c
│ │ │                               2
│ │ │                    (0, 1) => a c
│ │ │                                 2
│ │ │                    (0, 2) => b*c
│ │ │                    (1, 2) => -a*c
│ │ │ @@ -24,16 +24,16 @@
│ │ │                          2
│ │ │                    2 => b
│ │ │  
│ │ │  o3 : LineageTable
│ │ │  
│ │ │  i4 : reduce T
│ │ │  
│ │ │ -o4 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ -                  ((0, 1), 0) => null
│ │ │ +o4 = LineageTable{((0, 1), 0) => null}
│ │ │ +                  ((0, 2), 0) => null
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_tgb.out
│ │ │ @@ -6,84 +6,42 @@
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : allowableThreads  = 4;
│ │ │  
│ │ │  i4 : H = tgb I
│ │ │  
│ │ │ -                                                       2 13
│ │ │ -o4 = LineageTable{((((((0, 1), 2), 1), 2), 2), 2) => 9y z                  }
│ │ │ -                                                       2 12
│ │ │ -                  ((((((0, 1), 2), 1), 3), 2), 2) => 9y z
│ │ │ -                                                                 2 8
│ │ │ -                  ((((((0, 1), 2), 2), ((0, 1), 2)), 2), 2) => 9y z
│ │ │ -                                                       2 11
│ │ │ -                  ((((((0, 1), 2), 2), 1), 2), 2) => 9y z
│ │ │ -                                                       2 10
│ │ │ -                  ((((((0, 1), 2), 2), 2), 2), 2) => 9y z
│ │ │ -                                                  3 11
│ │ │ -                  (((((0, 1), 2), 1), 2), 2) => 9y z
│ │ │ -                                                  3 10
│ │ │ -                  (((((0, 1), 2), 1), 3), 2) => 9y z
│ │ │ -                                                            3 6
│ │ │ -                  (((((0, 1), 2), 2), ((0, 1), 2)), 2) => 9y z
│ │ │ -                                                  3 9
│ │ │ -                  (((((0, 1), 2), 2), 1), 2) => 9y z
│ │ │ -                                                  3 8
│ │ │ -                  (((((0, 1), 2), 2), 2), 2) => 9y z
│ │ │ -                                                  2 25
│ │ │ -                  (((((0, 1), 3), 1), 2), 2) => 9y z
│ │ │ -                                                  2 15
│ │ │ -                  (((((0, 1), 3), 2), 2), 2) => 9y z
│ │ │ -                                                          4 7      4 6
│ │ │ -                  ((((0, 1), 2), 1), ((0, 1), 3)) => - 16y z  + 40y z
│ │ │ -                                             4 9      4 7
│ │ │ -                  ((((0, 1), 2), 1), 2) => 6y z  + 30y z
│ │ │ -                                              4 8      4 7
│ │ │ -                  ((((0, 1), 2), 1), 3) => 27y z  + 30y z
│ │ │ -                                                        4 4
│ │ │ -                  ((((0, 1), 2), 2), ((0, 1), 2)) => 13y z
│ │ │ -                                               4 7
│ │ │ -                  ((((0, 1), 2), 2), 1) => -45y z
│ │ │ -                                             4 6
│ │ │ -                  ((((0, 1), 2), 2), 2) => 9y z
│ │ │ -                                                       3 12
│ │ │ -                  ((((0, 1), 3), ((0, 1), 2)), 2) => 9y z
│ │ │ -                                             3 23
│ │ │ -                  ((((0, 1), 3), 1), 2) => 9y z
│ │ │ -                                             3 13
│ │ │ -                  ((((0, 1), 3), 2), 2) => 9y z
│ │ │ -                                         6 4      4 6
│ │ │ -                  (((0, 1), 2), 1) => 19y z  - 30y z
│ │ │ -                                         5 4     4 7
│ │ │ -                  (((0, 1), 2), 2) => 37y z  + 9y z
│ │ │ -                                         4 14      4 11
│ │ │ -                  (((0, 1), 2), 3) => 27y z   - 16y z
│ │ │ -                                                   4 10      4 7
│ │ │ -                  (((0, 1), 3), ((0, 1), 2)) => 17y z   - 40y z
│ │ │ -                                           4 21     4 18
│ │ │ -                  (((0, 1), 3), 1) => - 37y z   - 8y z
│ │ │ -                                         4 11     4 10
│ │ │ -                  (((0, 1), 3), 2) => 44y z   + 4y z
│ │ │ -                                      5 5     4 4
│ │ │ -                  ((0, 1), 2) => - 24y z  + 9y z
│ │ │ -                                    7 2     4 4
│ │ │ -                  ((0, 1), 3) => 28y z  - 9y z
│ │ │ -                                                                        2 5
│ │ │ -                  ((0, 2), (((((0, 1), 2), 2), ((0, 1), 2)), 2)) => -13y z
│ │ │ -                                                                  2 4
│ │ │ -                  ((0, 2), ((((0, 1), 2), 2), ((0, 1), 2))) => 41y z
│ │ │ -                                   2 7
│ │ │ -                  ((0, 2), 2) => 9y z
│ │ │ +                                    2 6      2 4
│ │ │ +o4 = LineageTable{((0, 1), 2) => 23y z  + 19y z   }
│ │ │ +                                   2 5      2 4
│ │ │ +                  ((0, 1), 3) => 7y z  + 19y z
│ │ │ +                                     2 5     2 4
│ │ │ +                  ((0, 2), 1) => - 2y z  + 9y z
│ │ │ +                                    3 10      3 9
│ │ │ +                  ((0, 3), 1) => 43y z   - 42y z
│ │ │ +                                    2 6      2 4
│ │ │ +                  ((0, 3), 2) => 23y z  + 19y z
│ │ │ +                                         2 4
│ │ │ +                  ((1, 2), (0, 3)) => 18y z
│ │ │ +                                      2 5      2 4
│ │ │ +                  ((1, 2), 1) => - 25y z  + 12y z
│ │ │ +                                    2 4
│ │ │ +                  ((2, 3), 1) => -7y z
│ │ │ +                                     2 4
│ │ │ +                  ((2, 3), 2) => -20y z
│ │ │                                   5 2      3 4
│ │ │                    (0, 1) => - 25y z  - 19y z
│ │ │ -                                 3 5     2 4
│ │ │ -                  (0, 2) => - 24y z  + 9y z
│ │ │ -                               5       3 4
│ │ │ -                  (0, 3) => 28y z - 24y z
│ │ │ +                              5 3     2 4
│ │ │ +                  (0, 2) => 5y z  + 9y z
│ │ │ +                              5 2      5
│ │ │ +                  (0, 3) => 5y z  + 28y z
│ │ │ +                               5 6      4 5
│ │ │ +                  (1, 2) => 19y z  - 45y z
│ │ │ +                               3 7      3 6
│ │ │ +                  (1, 3) => 30y z  - 34y z
│ │ │                                3 4     2 4
│ │ │                    (2, 3) => 7y z  - 9y z
│ │ │                                 2
│ │ │                    0 => 2x + 10y z
│ │ │                           2           3
│ │ │                    1 => 8x y + 10x*y*z
│ │ │                             3 2       3
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/html/___Minimal.html
│ │ │ @@ -73,17 +73,16 @@
│ │ │                
    i2 : allowableThreads= 2;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2", Minimal=>true)
│ │ │  
│ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ -                  ((0, 1), 0) => null
│ │ │ -                  ((0, 1), 1) => null
│ │ │ +o3 = LineageTable{((0, 2), 0) => null}
│ │ │ +                  ((0, 2), 1) => null
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,17 +12,16 @@
│ │ │ │  Gr\"obner basis is minimized. Lineages of non-minimal Gr\"obner basis elements
│ │ │ │  that were added to the basis during the distributed computation are saved, with
│ │ │ │  the corresponding entry in the table being null.
│ │ │ │  i1 : S = ZZ/101[a,b,c];
│ │ │ │  i2 : allowableThreads= 2;
│ │ │ │  i3 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2", Minimal=>true)
│ │ │ │  
│ │ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ │ -                  ((0, 1), 0) => null
│ │ │ │ -                  ((0, 1), 1) => null
│ │ │ │ +o3 = LineageTable{((0, 2), 0) => null}
│ │ │ │ +                  ((0, 2), 1) => null
│ │ │ │                                    2
│ │ │ │                    ((1, 2), 0) => c
│ │ │ │                    (0, 1) => null
│ │ │ │                    (0, 2) => null
│ │ │ │                    (1, 2) => a*c
│ │ │ │                    0 => null
│ │ │ │                    1 => null
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/html/_matrix_lp__Lineage__Table_rp.html
│ │ │ @@ -86,16 +86,16 @@
│ │ │                
    i2 : allowableThreads= 2;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : T = reduce tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │  
│ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ -                  ((0, 1), 0) => null
│ │ │ +o3 = LineageTable{((0, 2), 0) => null}
│ │ │ +                  ((0, 2), 1) => null
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,16 +19,16 @@
│ │ │ │  This simple function just returns the Gr\"obner basis computed with threaded
│ │ │ │  Gr\"obner basis function _t_g_b in the expected Macaulay2 format, so that further
│ │ │ │  computation are one step easier to set up.
│ │ │ │  i1 : R = ZZ/101[a,b,c];
│ │ │ │  i2 : allowableThreads= 2;
│ │ │ │  i3 : T = reduce tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │ │  
│ │ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ │ -                  ((0, 1), 0) => null
│ │ │ │ +o3 = LineageTable{((0, 2), 0) => null}
│ │ │ │ +                  ((0, 2), 1) => null
│ │ │ │                                    2
│ │ │ │                    ((1, 2), 0) => c
│ │ │ │                    (0, 1) => null
│ │ │ │                    (0, 2) => null
│ │ │ │                    (1, 2) => a*c
│ │ │ │                    0 => null
│ │ │ │                    1 => null
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/html/_minimize_lp__Lineage__Table_rp.html
│ │ │ @@ -82,18 +82,16 @@
│ │ │                
    i2 : allowableThreads= 2;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │  
│ │ │ -                                        3
│ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => -c }
│ │ │ -                                     2
│ │ │ -                  ((0, 1), 0) => -a*c
│ │ │ +                                   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
│ │ │ @@ -107,16 +105,15 @@
│ │ │  o3 : LineageTable
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : minimize T
│ │ │  
│ │ │ -o4 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ -                  ((0, 1), 0) => null
│ │ │ +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
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,18 +19,16 @@
│ │ │ │  minimal generators of the ideal generated by the leading terms of the values of
│ │ │ │  H. If the values of H constitute a Gr\"obner basis of the ideal they generate,
│ │ │ │  this method returns a minimal Gr\"obner basis.
│ │ │ │  i1 : R = ZZ/101[a,b,c];
│ │ │ │  i2 : allowableThreads= 2;
│ │ │ │  i3 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │ │  
│ │ │ │ -                                        3
│ │ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => -c }
│ │ │ │ -                                     2
│ │ │ │ -                  ((0, 1), 0) => -a*c
│ │ │ │ +                                   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
│ │ │ │ @@ -40,16 +38,15 @@
│ │ │ │                    1 => - b c + a*b  + a*c
│ │ │ │                          2
│ │ │ │                    2 => b
│ │ │ │  
│ │ │ │  o3 : LineageTable
│ │ │ │  i4 : minimize T
│ │ │ │  
│ │ │ │ -o4 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ │ -                  ((0, 1), 0) => null
│ │ │ │ +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
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/html/_reduce.html
│ │ │ @@ -82,18 +82,18 @@
│ │ │                
    i2 : allowableThreads= 2;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : T = tgb ideal "abc+c2,ab2-b3c+ac,b2"
│ │ │  
│ │ │ -                                        3
│ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => -c }
│ │ │                                       2
│ │ │ -                  ((0, 1), 0) => -a*c
│ │ │ +o3 = LineageTable{((0, 1), 0) => -a*c    }
│ │ │ +                                   3
│ │ │ +                  ((0, 2), 0) => -c
│ │ │                                     2
│ │ │                    ((1, 2), 0) => -c
│ │ │                               2
│ │ │                    (0, 1) => a c
│ │ │                                 2
│ │ │                    (0, 2) => b*c
│ │ │                    (1, 2) => -a*c
│ │ │ @@ -107,16 +107,16 @@
│ │ │  o3 : LineageTable
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : reduce T
│ │ │  
│ │ │ -o4 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ -                  ((0, 1), 0) => null
│ │ │ +o4 = LineageTable{((0, 1), 0) => null}
│ │ │ +                  ((0, 2), 0) => null
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,18 +20,18 @@
│ │ │ │  remainder on the division by the remaining values H.
│ │ │ │  If values H constitute a Gr\"obner basis of the ideal they generate, this
│ │ │ │  method returns a reduced Gr\"obner basis.
│ │ │ │  i1 : R = ZZ/101[a,b,c];
│ │ │ │  i2 : allowableThreads= 2;
│ │ │ │  i3 : T = tgb ideal "abc+c2,ab2-b3c+ac,b2"
│ │ │ │  
│ │ │ │ -                                        3
│ │ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => -c }
│ │ │ │                                       2
│ │ │ │ -                  ((0, 1), 0) => -a*c
│ │ │ │ +o3 = LineageTable{((0, 1), 0) => -a*c    }
│ │ │ │ +                                   3
│ │ │ │ +                  ((0, 2), 0) => -c
│ │ │ │                                     2
│ │ │ │                    ((1, 2), 0) => -c
│ │ │ │                               2
│ │ │ │                    (0, 1) => a c
│ │ │ │                                 2
│ │ │ │                    (0, 2) => b*c
│ │ │ │                    (1, 2) => -a*c
│ │ │ │ @@ -41,16 +41,16 @@
│ │ │ │                    1 => - b c + a*b  + a*c
│ │ │ │                          2
│ │ │ │                    2 => b
│ │ │ │  
│ │ │ │  o3 : LineageTable
│ │ │ │  i4 : reduce T
│ │ │ │  
│ │ │ │ -o4 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ │ -                  ((0, 1), 0) => null
│ │ │ │ +o4 = LineageTable{((0, 1), 0) => null}
│ │ │ │ +                  ((0, 2), 0) => null
│ │ │ │                                    2
│ │ │ │                    ((1, 2), 0) => c
│ │ │ │                    (0, 1) => null
│ │ │ │                    (0, 2) => null
│ │ │ │                    (1, 2) => a*c
│ │ │ │                    0 => null
│ │ │ │                    1 => null
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/html/_tgb.html
│ │ │ @@ -95,84 +95,42 @@
│ │ │                
    i3 : allowableThreads  = 4;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : H = tgb I
│ │ │  
│ │ │ -                                                       2 13
│ │ │ -o4 = LineageTable{((((((0, 1), 2), 1), 2), 2), 2) => 9y z                  }
│ │ │ -                                                       2 12
│ │ │ -                  ((((((0, 1), 2), 1), 3), 2), 2) => 9y z
│ │ │ -                                                                 2 8
│ │ │ -                  ((((((0, 1), 2), 2), ((0, 1), 2)), 2), 2) => 9y z
│ │ │ -                                                       2 11
│ │ │ -                  ((((((0, 1), 2), 2), 1), 2), 2) => 9y z
│ │ │ -                                                       2 10
│ │ │ -                  ((((((0, 1), 2), 2), 2), 2), 2) => 9y z
│ │ │ -                                                  3 11
│ │ │ -                  (((((0, 1), 2), 1), 2), 2) => 9y z
│ │ │ -                                                  3 10
│ │ │ -                  (((((0, 1), 2), 1), 3), 2) => 9y z
│ │ │ -                                                            3 6
│ │ │ -                  (((((0, 1), 2), 2), ((0, 1), 2)), 2) => 9y z
│ │ │ -                                                  3 9
│ │ │ -                  (((((0, 1), 2), 2), 1), 2) => 9y z
│ │ │ -                                                  3 8
│ │ │ -                  (((((0, 1), 2), 2), 2), 2) => 9y z
│ │ │ -                                                  2 25
│ │ │ -                  (((((0, 1), 3), 1), 2), 2) => 9y z
│ │ │ -                                                  2 15
│ │ │ -                  (((((0, 1), 3), 2), 2), 2) => 9y z
│ │ │ -                                                          4 7      4 6
│ │ │ -                  ((((0, 1), 2), 1), ((0, 1), 3)) => - 16y z  + 40y z
│ │ │ -                                             4 9      4 7
│ │ │ -                  ((((0, 1), 2), 1), 2) => 6y z  + 30y z
│ │ │ -                                              4 8      4 7
│ │ │ -                  ((((0, 1), 2), 1), 3) => 27y z  + 30y z
│ │ │ -                                                        4 4
│ │ │ -                  ((((0, 1), 2), 2), ((0, 1), 2)) => 13y z
│ │ │ -                                               4 7
│ │ │ -                  ((((0, 1), 2), 2), 1) => -45y z
│ │ │ -                                             4 6
│ │ │ -                  ((((0, 1), 2), 2), 2) => 9y z
│ │ │ -                                                       3 12
│ │ │ -                  ((((0, 1), 3), ((0, 1), 2)), 2) => 9y z
│ │ │ -                                             3 23
│ │ │ -                  ((((0, 1), 3), 1), 2) => 9y z
│ │ │ -                                             3 13
│ │ │ -                  ((((0, 1), 3), 2), 2) => 9y z
│ │ │ -                                         6 4      4 6
│ │ │ -                  (((0, 1), 2), 1) => 19y z  - 30y z
│ │ │ -                                         5 4     4 7
│ │ │ -                  (((0, 1), 2), 2) => 37y z  + 9y z
│ │ │ -                                         4 14      4 11
│ │ │ -                  (((0, 1), 2), 3) => 27y z   - 16y z
│ │ │ -                                                   4 10      4 7
│ │ │ -                  (((0, 1), 3), ((0, 1), 2)) => 17y z   - 40y z
│ │ │ -                                           4 21     4 18
│ │ │ -                  (((0, 1), 3), 1) => - 37y z   - 8y z
│ │ │ -                                         4 11     4 10
│ │ │ -                  (((0, 1), 3), 2) => 44y z   + 4y z
│ │ │ -                                      5 5     4 4
│ │ │ -                  ((0, 1), 2) => - 24y z  + 9y z
│ │ │ -                                    7 2     4 4
│ │ │ -                  ((0, 1), 3) => 28y z  - 9y z
│ │ │ -                                                                        2 5
│ │ │ -                  ((0, 2), (((((0, 1), 2), 2), ((0, 1), 2)), 2)) => -13y z
│ │ │ -                                                                  2 4
│ │ │ -                  ((0, 2), ((((0, 1), 2), 2), ((0, 1), 2))) => 41y z
│ │ │ -                                   2 7
│ │ │ -                  ((0, 2), 2) => 9y z
│ │ │ +                                    2 6      2 4
│ │ │ +o4 = LineageTable{((0, 1), 2) => 23y z  + 19y z   }
│ │ │ +                                   2 5      2 4
│ │ │ +                  ((0, 1), 3) => 7y z  + 19y z
│ │ │ +                                     2 5     2 4
│ │ │ +                  ((0, 2), 1) => - 2y z  + 9y z
│ │ │ +                                    3 10      3 9
│ │ │ +                  ((0, 3), 1) => 43y z   - 42y z
│ │ │ +                                    2 6      2 4
│ │ │ +                  ((0, 3), 2) => 23y z  + 19y z
│ │ │ +                                         2 4
│ │ │ +                  ((1, 2), (0, 3)) => 18y z
│ │ │ +                                      2 5      2 4
│ │ │ +                  ((1, 2), 1) => - 25y z  + 12y z
│ │ │ +                                    2 4
│ │ │ +                  ((2, 3), 1) => -7y z
│ │ │ +                                     2 4
│ │ │ +                  ((2, 3), 2) => -20y z
│ │ │                                   5 2      3 4
│ │ │                    (0, 1) => - 25y z  - 19y z
│ │ │ -                                 3 5     2 4
│ │ │ -                  (0, 2) => - 24y z  + 9y z
│ │ │ -                               5       3 4
│ │ │ -                  (0, 3) => 28y z - 24y z
│ │ │ +                              5 3     2 4
│ │ │ +                  (0, 2) => 5y z  + 9y z
│ │ │ +                              5 2      5
│ │ │ +                  (0, 3) => 5y z  + 28y z
│ │ │ +                               5 6      4 5
│ │ │ +                  (1, 2) => 19y z  - 45y z
│ │ │ +                               3 7      3 6
│ │ │ +                  (1, 3) => 30y z  - 34y z
│ │ │                                3 4     2 4
│ │ │                    (2, 3) => 7y z  - 9y z
│ │ │                                 2
│ │ │                    0 => 2x + 10y z
│ │ │                           2           3
│ │ │                    1 => 8x y + 10x*y*z
│ │ │                             3 2       3
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,84 +26,42 @@
│ │ │ │  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 13
│ │ │ │ -o4 = LineageTable{((((((0, 1), 2), 1), 2), 2), 2) => 9y z                  }
│ │ │ │ -                                                       2 12
│ │ │ │ -                  ((((((0, 1), 2), 1), 3), 2), 2) => 9y z
│ │ │ │ -                                                                 2 8
│ │ │ │ -                  ((((((0, 1), 2), 2), ((0, 1), 2)), 2), 2) => 9y z
│ │ │ │ -                                                       2 11
│ │ │ │ -                  ((((((0, 1), 2), 2), 1), 2), 2) => 9y z
│ │ │ │ -                                                       2 10
│ │ │ │ -                  ((((((0, 1), 2), 2), 2), 2), 2) => 9y z
│ │ │ │ -                                                  3 11
│ │ │ │ -                  (((((0, 1), 2), 1), 2), 2) => 9y z
│ │ │ │ -                                                  3 10
│ │ │ │ -                  (((((0, 1), 2), 1), 3), 2) => 9y z
│ │ │ │ -                                                            3 6
│ │ │ │ -                  (((((0, 1), 2), 2), ((0, 1), 2)), 2) => 9y z
│ │ │ │ -                                                  3 9
│ │ │ │ -                  (((((0, 1), 2), 2), 1), 2) => 9y z
│ │ │ │ -                                                  3 8
│ │ │ │ -                  (((((0, 1), 2), 2), 2), 2) => 9y z
│ │ │ │ -                                                  2 25
│ │ │ │ -                  (((((0, 1), 3), 1), 2), 2) => 9y z
│ │ │ │ -                                                  2 15
│ │ │ │ -                  (((((0, 1), 3), 2), 2), 2) => 9y z
│ │ │ │ -                                                          4 7      4 6
│ │ │ │ -                  ((((0, 1), 2), 1), ((0, 1), 3)) => - 16y z  + 40y z
│ │ │ │ -                                             4 9      4 7
│ │ │ │ -                  ((((0, 1), 2), 1), 2) => 6y z  + 30y z
│ │ │ │ -                                              4 8      4 7
│ │ │ │ -                  ((((0, 1), 2), 1), 3) => 27y z  + 30y z
│ │ │ │ -                                                        4 4
│ │ │ │ -                  ((((0, 1), 2), 2), ((0, 1), 2)) => 13y z
│ │ │ │ -                                               4 7
│ │ │ │ -                  ((((0, 1), 2), 2), 1) => -45y z
│ │ │ │ -                                             4 6
│ │ │ │ -                  ((((0, 1), 2), 2), 2) => 9y z
│ │ │ │ -                                                       3 12
│ │ │ │ -                  ((((0, 1), 3), ((0, 1), 2)), 2) => 9y z
│ │ │ │ -                                             3 23
│ │ │ │ -                  ((((0, 1), 3), 1), 2) => 9y z
│ │ │ │ -                                             3 13
│ │ │ │ -                  ((((0, 1), 3), 2), 2) => 9y z
│ │ │ │ -                                         6 4      4 6
│ │ │ │ -                  (((0, 1), 2), 1) => 19y z  - 30y z
│ │ │ │ -                                         5 4     4 7
│ │ │ │ -                  (((0, 1), 2), 2) => 37y z  + 9y z
│ │ │ │ -                                         4 14      4 11
│ │ │ │ -                  (((0, 1), 2), 3) => 27y z   - 16y z
│ │ │ │ -                                                   4 10      4 7
│ │ │ │ -                  (((0, 1), 3), ((0, 1), 2)) => 17y z   - 40y z
│ │ │ │ -                                           4 21     4 18
│ │ │ │ -                  (((0, 1), 3), 1) => - 37y z   - 8y z
│ │ │ │ -                                         4 11     4 10
│ │ │ │ -                  (((0, 1), 3), 2) => 44y z   + 4y z
│ │ │ │ -                                      5 5     4 4
│ │ │ │ -                  ((0, 1), 2) => - 24y z  + 9y z
│ │ │ │ -                                    7 2     4 4
│ │ │ │ -                  ((0, 1), 3) => 28y z  - 9y z
│ │ │ │ -                                                                        2 5
│ │ │ │ -                  ((0, 2), (((((0, 1), 2), 2), ((0, 1), 2)), 2)) => -13y z
│ │ │ │ -                                                                  2 4
│ │ │ │ -                  ((0, 2), ((((0, 1), 2), 2), ((0, 1), 2))) => 41y z
│ │ │ │ -                                   2 7
│ │ │ │ -                  ((0, 2), 2) => 9y z
│ │ │ │ +                                    2 6      2 4
│ │ │ │ +o4 = LineageTable{((0, 1), 2) => 23y z  + 19y z   }
│ │ │ │ +                                   2 5      2 4
│ │ │ │ +                  ((0, 1), 3) => 7y z  + 19y z
│ │ │ │ +                                     2 5     2 4
│ │ │ │ +                  ((0, 2), 1) => - 2y z  + 9y z
│ │ │ │ +                                    3 10      3 9
│ │ │ │ +                  ((0, 3), 1) => 43y z   - 42y z
│ │ │ │ +                                    2 6      2 4
│ │ │ │ +                  ((0, 3), 2) => 23y z  + 19y z
│ │ │ │ +                                         2 4
│ │ │ │ +                  ((1, 2), (0, 3)) => 18y z
│ │ │ │ +                                      2 5      2 4
│ │ │ │ +                  ((1, 2), 1) => - 25y z  + 12y z
│ │ │ │ +                                    2 4
│ │ │ │ +                  ((2, 3), 1) => -7y z
│ │ │ │ +                                     2 4
│ │ │ │ +                  ((2, 3), 2) => -20y z
│ │ │ │                                   5 2      3 4
│ │ │ │                    (0, 1) => - 25y z  - 19y z
│ │ │ │ -                                 3 5     2 4
│ │ │ │ -                  (0, 2) => - 24y z  + 9y z
│ │ │ │ -                               5       3 4
│ │ │ │ -                  (0, 3) => 28y z - 24y z
│ │ │ │ +                              5 3     2 4
│ │ │ │ +                  (0, 2) => 5y z  + 9y z
│ │ │ │ +                              5 2      5
│ │ │ │ +                  (0, 3) => 5y z  + 28y z
│ │ │ │ +                               5 6      4 5
│ │ │ │ +                  (1, 2) => 19y z  - 45y z
│ │ │ │ +                               3 7      3 6
│ │ │ │ +                  (1, 3) => 30y z  - 34y z
│ │ │ │                                3 4     2 4
│ │ │ │                    (2, 3) => 7y z  - 9y z
│ │ │ │                                 2
│ │ │ │                    0 => 2x + 10y z
│ │ │ │                           2           3
│ │ │ │                    1 => 8x y + 10x*y*z
│ │ │ │                             3 2       3
│ │ ├── ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_ed__Deg.out
│ │ │ @@ -40,15 +40,15 @@
│ │ │  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
│ │ │ - -- used 1.18281s (cpu); 0.815142s (thread); 0s (gc)
│ │ │ + -- used 1.38413s (cpu); 0.887032s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 252
│ │ │  
│ │ │  o4 : QQ
│ │ │  
│ │ │  i5 : time edDeg(A,ForceAmat=>true)
│ │ │  
│ │ │ @@ -56,14 +56,14 @@
│ │ │  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
│ │ │ - -- used 4.85555s (cpu); 3.16599s (thread); 0s (gc)
│ │ │ + -- used 5.55924s (cpu); 3.33596s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 252
│ │ │  
│ │ │  o5 : QQ
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/ToricInvariants/html/_ed__Deg.html
│ │ │ @@ -131,15 +131,15 @@
│ │ │  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
│ │ │ - -- used 1.18281s (cpu); 0.815142s (thread); 0s (gc)
│ │ │ + -- used 1.38413s (cpu); 0.887032s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 252
│ │ │  
│ │ │  o4 : QQ
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -150,15 +150,15 @@
│ │ │  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
│ │ │ - -- used 4.85555s (cpu); 3.16599s (thread); 0s (gc)
│ │ │ + -- used 5.55924s (cpu); 3.33596s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 252
│ │ │  
│ │ │  o5 : QQ
    
│ │ │              
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -66,30 +66,30 @@
│ │ │ │  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
│ │ │ │ - -- used 1.18281s (cpu); 0.815142s (thread); 0s (gc)
│ │ │ │ + -- used 1.38413s (cpu); 0.887032s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 252
│ │ │ │  
│ │ │ │  o4 : QQ
│ │ │ │  i5 : time edDeg(A,ForceAmat=>true)
│ │ │ │  
│ │ │ │  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
│ │ │ │ - -- used 4.85555s (cpu); 3.16599s (thread); 0s (gc)
│ │ │ │ + -- used 5.55924s (cpu); 3.33596s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 252
│ │ │ │  
│ │ │ │  o5 : QQ
│ │ │ │  ********** WWaayyss ttoo uussee eeddDDeegg:: **********
│ │ │ │      * edDeg(Matrix)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./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);
│ │ │ - -- .14823s elapsed
│ │ │ + -- .132955s 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;
│ │ │ - -- 1.22082s elapsed
│ │ │ + -- .954958s elapsed
│ │ │  
│ │ │  i9 : #Ts2 == #Ts
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 :
│ │ ├── ./usr/share/doc/Macaulay2/Triangulations/example-output/_generate__Triangulations.out
│ │ │ @@ -21,87 +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, 4, 7}, {0, 4, 5, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, 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}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {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}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ +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},
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -227,275 +155,275 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 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, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7},
│ │ │ +     {2, 5, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │ +     {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 6, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 3, 5, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │ +     triangulation {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 5},
│ │ │ +     {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 5, 7},
│ │ │ +     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7},
│ │ │ +     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6},
│ │ │ +     {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 5, 6},
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},
│ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {2, 3, 5, 7},
│ │ │ +     {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5},
│ │ │ +     {0, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │ +     {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6}, {2, 4, 5, 6},
│ │ │ +     {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6},
│ │ │ +     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │ +     {3, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7}, {1, 4, 5, 7},
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7},
│ │ │ +     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 4, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7},
│ │ │ +     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     {3, 4, 5, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 7},
│ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}, {1, 2, 3, 7}},
│ │ │ +     triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6},
│ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │ +     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 4},
│ │ │ +     {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},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     {1, 2, 3, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6}, {0, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 6},
│ │ │ +     {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7},
│ │ │ +     {1, 2, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6},
│ │ │ +     {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 6},
│ │ │ +     {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ +     {1, 2, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 5},
│ │ │ +     {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6},
│ │ │ +     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 2, 7},
│ │ │ +     {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}},
│ │ │ +     {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ +     {2, 3, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 5, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7},
│ │ │ +     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6},
│ │ │ +     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6},
│ │ │ +     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 3, 5, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     {1, 2, 3, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7},
│ │ │ +     {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7},
│ │ │ +     {0, 4, 6, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6},
│ │ │ +     {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 2, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │ +     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, {{0, 1, 3, 5},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     {2, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 5, 6}, {2, 4, 5, 6},
│ │ │ +     {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │ +     {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6},
│ │ │ +     {0, 2, 3, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6},
│ │ │ +     {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}},
│ │ │ +     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {2, 3, 5, 6},
│ │ │ +     {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7},
│ │ │ +     {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7},
│ │ │ +     {1, 2, 3, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ +     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}},
│ │ │ +     {1, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 6, 7},
│ │ │ +     {2, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7},
│ │ │ +     {1, 2, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ +     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 5}, {2, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4},
│ │ │ +     {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, {{0, 1, 3, 5},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │ +     {0, 5, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 6}, {0, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 6},
│ │ │ +     {0, 4, 6, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 6},
│ │ │ +     {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5},
│ │ │ +     {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │ +     {3, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6},
│ │ │ +     {3, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7},
│ │ │ +     {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {2, 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, 4, 7}, {0, 4, 5, 7},
│ │ │ +     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +     {0, 1, 4, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │ +     {2, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7},
│ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5},
│ │ │ +     {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7},
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     {0, 4, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │ +     {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │ +     {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ +     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ +     {1, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation
│ │ │ +     {1, 3, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │ +     {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ +     {0, 2, 5, 6}, {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6},
│ │ │ +     {2, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │ +     {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ +     {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ +     {0, 2, 4, 5}, {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5},
│ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ +     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7},
│ │ │ +     {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4},
│ │ │ +     {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     {0, 3, 4, 6}, {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6},
│ │ │ +     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {1, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6}, {1, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     {0, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ +     {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},
│ │ │       ------------------------------------------------------------------------
│ │ │       {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},
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -621,93 +549,93 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}}
│ │ │ -
│ │ │ -o6 : List
│ │ │ -
│ │ │ -i7 : Ts4 = generateTriangulations tri -- list of Triangulations
│ │ │ -
│ │ │ -o7 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ +     {2, 5, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +     {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │ +     triangulation {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7},
│ │ │ +     {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5},
│ │ │ +     {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7},
│ │ │ +     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +     {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │ +     triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ +     {0, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ +     {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation
│ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │ +     triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ +     {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6},
│ │ │ +     {3, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │ +     {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5},
│ │ │ +     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7},
│ │ │ +     {3, 4, 5, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4},
│ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ +     triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6},
│ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ +     {{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}},
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ +     {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},
│ │ │       ------------------------------------------------------------------------
│ │ │       {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},
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -833,15 +761,87 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}}
│ │ │ +     {2, 5, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {3, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     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}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {3, 4, 5, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {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}}}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : all(Ts4, isFine)
│ │ │  
│ │ │  o8 = true
│ │ │  
│ │ │ @@ -858,191 +858,193 @@
│ │ │  o11 = Tally{false => 66}
│ │ │              true => 8
│ │ │  
│ │ │  o11 : Tally
│ │ │  
│ │ │  i12 : Ts4/gkzVector
│ │ │  
│ │ │ -        16     16  4     8  4       20     8  8     4  8          4  8     8 
│ │ │ -o12 = {{--, 4, --, -, 4, -, -, 8}, {--, 4, -, -, 4, -, -, 8}, {8, -, -, 4, -,
│ │ │ -         3      3  3     3  3        3     3  3     3  3          3  3     3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         8  20       16  16  4  16  4  4       8        4     16  4  16      
│ │ │ -      4, -, --}, {4, --, --, -, --, -, -, 8}, {-, 4, 8, -, 4, --, -, --}, {4,
│ │ │ -         3   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, 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    20        4  8  8  8          8  8  8  4        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     4  8          8  4     8  8     20       20     4  4 
│ │ │ -      {--, --, 4, -, 4, -, -, 8}, {8, -, -, 4, -, -, 4, --}, {4, --, 4, -, -,
│ │ │ -        3   3     3     3  3          3  3     3  3      3        3     3  3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         20       20  4              4  20       4  20        20  4       8 
│ │ │ -      4, --, 4}, {--, -, 4, 4, 4, 4, -, --}, {4, -, --, 4, 4, --, -, 4}, {-,
│ │ │ -          3        3  3              3   3       3   3         3  3       3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         8  20     4  8          16  4  16  8        4    4     16  16     8 
│ │ │ -      4, -, --, 8, -, -, 4}, {4, --, -, --, -, 4, 8, -}, {-, 4, --, --, 8, -,
│ │ │ -         3   3     3  3           3  3   3  3        3    3      3   3     3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4          4  8     16  16     4    4     20        20     4    8  8 
│ │ │ -      -, 4}, {4, -, -, 8, --, --, 4, -}, {-, 4, --, 4, 4, --, 4, -}, {-, -,
│ │ │ -      3          3  3      3   3     3    3      3         3     3    3  3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      8        8  8  8    8  20  8     8        4       16  8     4  16    
│ │ │ -      -, 8, 8, -, -, -}, {-, --, -, 4, -, 4, 8, -}, {4, --, -, 4, -, --, 8,
│ │ │ -      3        3  3  3    3   3  3     3        3        3  3     3   3    
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4    4        8  16  4  16       16  4     16  4     16  4    4     16 
│ │ │ -      -}, {-, 8, 4, -, --, -, --, 4}, {--, -, 4, --, -, 8, --, -}, {-, 8, --,
│ │ │ -      3    3        3   3  3   3        3  3      3  3      3  3    3      3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4  16  4     16       8  8  8  8  8  8       20  8     8     8  4     
│ │ │ -      -, --, -, 4, --}, {8, -, -, -, -, -, -, 8}, {--, -, 4, -, 4, -, -, 8},
│ │ │ -      3   3  3      3       3  3  3  3  3  3        3  3     3     3  3     
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -          4  8     4  16     16    20     4        4     20       4  4  16 
│ │ │ -      {8, -, -, 4, -, --, 4, --}, {--, 4, -, 4, 4, -, 4, --}, {8, -, -, --,
│ │ │ -          3  3     3   3      3     3     3        3      3       3  3   3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4  16  16          8  16     4     16  4    8  8     20     8  4     
│ │ │ -      -, --, --, 4}, {4, -, --, 4, -, 8, --, -}, {-, -, 4, --, 8, -, -, 4},
│ │ │ -      3   3   3          3   3     3      3  3    3  3      3     3  3     
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -          4  8     20     8  8       8  4     16     16  4    8  8  20     8 
│ │ │ -      {4, -, -, 8, --, 4, -, -}, {4, -, -, 8, --, 4, --, -}, {-, -, --, 4, -,
│ │ │ -          3  3      3     3  3       3  3      3      3  3    3  3   3     3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -            4    4        8     20  8  8    8     8  8  8  8     8    4  16 
│ │ │ -      8, 4, -}, {-, 4, 8, -, 4, --, -, -}, {-, 8, -, -, -, -, 8, -}, {-, --,
│ │ │ -            3    3        3      3  3  3    3     3  3  3  3     3    3   3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      16        4  4  16       20  4     8  8     8    8        4  8  20  8
│ │ │ -      --, 4, 8, -, -, --}, {4, --, -, 4, -, -, 8, -}, {-, 4, 8, -, -, --, -,
│ │ │ -       3        3  3   3        3  3     3  3     3    3        3  3   3  3
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4}}
│ │ │ +        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
│ │ │  
│ │ │  o12 : List
│ │ │  
│ │ │  i13 : volume convexHull A -- 8
│ │ │  
│ │ │  o13 = 8
│ │ │  
│ │ │  o13 : QQ
│ │ │  
│ │ │  i14 : stars1 = select(Ts4, t -> (gkzVector t)#-1 == 8)
│ │ │  
│ │ │ -o14 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ +o14 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +      {0, 4, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ +      {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7},
│ │ │ +      {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7},
│ │ │ +      {0, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7},
│ │ │ +      {0, 2, 4, 7}, {0, 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}, {1, 2, 3, 7}},
│ │ │ +      {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7},
│ │ │ +      triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ +      {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation
│ │ │ +      {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ +      {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │        {2, 4, 6, 7}}}
│ │ │  
│ │ │  o14 : List
│ │ │  
│ │ │  i15 : stars2 = select(Ts4, isStar)
│ │ │  
│ │ │ -o15 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ +o15 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +      {0, 4, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ +      {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7},
│ │ │ +      {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7},
│ │ │ +      {0, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7},
│ │ │ +      {0, 2, 4, 7}, {0, 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}, {1, 2, 3, 7}},
│ │ │ +      {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7},
│ │ │ +      triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ +      {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation
│ │ │ +      {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ +      {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │        {2, 4, 6, 7}}}
│ │ │  
│ │ │  o15 : List
│ │ │  
│ │ │  i16 : stars1 == stars2
│ │ ├── ./usr/share/doc/Macaulay2/Triangulations/html/_generate__Triangulations.html
│ │ │ @@ -117,87 +117,15 @@
│ │ │  o3 : Triangulation
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : Ts1 = generateTriangulations A -- list of Triangulation's.
│ │ │  
│ │ │ -o4 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, 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}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {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}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ +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},
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -323,281 +251,281 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 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, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7},
│ │ │ +     {2, 5, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │ +     {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 6, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 3, 5, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │ +     triangulation {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 5},
│ │ │ +     {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 5, 7},
│ │ │ +     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7},
│ │ │ +     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6},
│ │ │ +     {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 5, 6},
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},
│ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {2, 3, 5, 7},
│ │ │ +     {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5},
│ │ │ +     {0, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │ +     {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6}, {2, 4, 5, 6},
│ │ │ +     {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6},
│ │ │ +     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │ +     {3, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7}, {1, 4, 5, 7},
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7},
│ │ │ +     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 4, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7},
│ │ │ +     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     {3, 4, 5, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 7},
│ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}, {1, 2, 3, 7}},
│ │ │ +     triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6},
│ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │ +     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 4},
│ │ │ +     {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},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     {1, 2, 3, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6}, {0, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 6},
│ │ │ +     {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7},
│ │ │ +     {1, 2, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6},
│ │ │ +     {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 6},
│ │ │ +     {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ +     {1, 2, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 5},
│ │ │ +     {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6},
│ │ │ +     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 2, 7},
│ │ │ +     {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}},
│ │ │ +     {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ +     {2, 3, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 5, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7},
│ │ │ +     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6},
│ │ │ +     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6},
│ │ │ +     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 3, 5, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     {1, 2, 3, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7},
│ │ │ +     {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7},
│ │ │ +     {0, 4, 6, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6},
│ │ │ +     {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 2, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │ +     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, {{0, 1, 3, 5},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     {2, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 5, 6}, {2, 4, 5, 6},
│ │ │ +     {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │ +     {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6},
│ │ │ +     {0, 2, 3, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6},
│ │ │ +     {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}},
│ │ │ +     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {2, 3, 5, 6},
│ │ │ +     {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7},
│ │ │ +     {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7},
│ │ │ +     {1, 2, 3, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ +     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}},
│ │ │ +     {1, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 6, 7},
│ │ │ +     {2, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7},
│ │ │ +     {1, 2, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ +     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 5}, {2, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4},
│ │ │ +     {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, {{0, 1, 3, 5},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │ +     {0, 5, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 6}, {0, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 6},
│ │ │ +     {0, 4, 6, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 6},
│ │ │ +     {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5},
│ │ │ +     {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │ +     {3, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6},
│ │ │ +     {3, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7},
│ │ │ +     {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {2, 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, 4, 7}, {0, 4, 5, 7},
│ │ │ +     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +     {0, 1, 4, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │ +     {2, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7},
│ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5},
│ │ │ +     {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7},
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     {0, 4, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │ +     {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │ +     {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ +     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ +     {1, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation
│ │ │ +     {1, 3, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │ +     {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ +     {0, 2, 5, 6}, {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6},
│ │ │ +     {2, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │ +     {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ +     {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ +     {0, 2, 4, 5}, {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5},
│ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ +     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7},
│ │ │ +     {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4},
│ │ │ +     {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     {0, 3, 4, 6}, {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6},
│ │ │ +     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {1, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6}, {1, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     {0, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ +     {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},
│ │ │       ------------------------------------------------------------------------
│ │ │       {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},
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -723,96 +651,96 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}}
│ │ │ -
│ │ │ -o6 : List
    
│ │ │ -            
│ │ │ -          
│ │ │ -          
│ │ │ -            
│ │ │ -              
    i7 : Ts4 = generateTriangulations tri -- list of Triangulations
│ │ │ -
│ │ │ -o7 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ +     {2, 5, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +     {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │ +     triangulation {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7},
│ │ │ +     {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5},
│ │ │ +     {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7},
│ │ │ +     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +     {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │ +     triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ +     {0, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ +     {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation
│ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │ +     triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ +     {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6},
│ │ │ +     {3, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │ +     {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5},
│ │ │ +     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7},
│ │ │ +     {3, 4, 5, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4},
│ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ +     triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6},
│ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ +     {{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}},
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ +     {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},
│ │ │       ------------------------------------------------------------------------
│ │ │       {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},
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -938,15 +866,87 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}}
│ │ │ +     {2, 5, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {3, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     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}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {3, 4, 5, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {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}}}
│ │ │  
│ │ │  o7 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : all(Ts4, isFine)
│ │ │ @@ -978,131 +978,133 @@
│ │ │  o11 : Tally
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : Ts4/gkzVector
│ │ │  
│ │ │ -        16     16  4     8  4       20     8  8     4  8          4  8     8 
│ │ │ -o12 = {{--, 4, --, -, 4, -, -, 8}, {--, 4, -, -, 4, -, -, 8}, {8, -, -, 4, -,
│ │ │ -         3      3  3     3  3        3     3  3     3  3          3  3     3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         8  20       16  16  4  16  4  4       8        4     16  4  16      
│ │ │ -      4, -, --}, {4, --, --, -, --, -, -, 8}, {-, 4, 8, -, 4, --, -, --}, {4,
│ │ │ -         3   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, 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    20        4  8  8  8          8  8  8  4        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     4  8          8  4     8  8     20       20     4  4 
│ │ │ -      {--, --, 4, -, 4, -, -, 8}, {8, -, -, 4, -, -, 4, --}, {4, --, 4, -, -,
│ │ │ -        3   3     3     3  3          3  3     3  3      3        3     3  3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         20       20  4              4  20       4  20        20  4       8 
│ │ │ -      4, --, 4}, {--, -, 4, 4, 4, 4, -, --}, {4, -, --, 4, 4, --, -, 4}, {-,
│ │ │ -          3        3  3              3   3       3   3         3  3       3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         8  20     4  8          16  4  16  8        4    4     16  16     8 
│ │ │ -      4, -, --, 8, -, -, 4}, {4, --, -, --, -, 4, 8, -}, {-, 4, --, --, 8, -,
│ │ │ -         3   3     3  3           3  3   3  3        3    3      3   3     3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4          4  8     16  16     4    4     20        20     4    8  8 
│ │ │ -      -, 4}, {4, -, -, 8, --, --, 4, -}, {-, 4, --, 4, 4, --, 4, -}, {-, -,
│ │ │ -      3          3  3      3   3     3    3      3         3     3    3  3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      8        8  8  8    8  20  8     8        4       16  8     4  16    
│ │ │ -      -, 8, 8, -, -, -}, {-, --, -, 4, -, 4, 8, -}, {4, --, -, 4, -, --, 8,
│ │ │ -      3        3  3  3    3   3  3     3        3        3  3     3   3    
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4    4        8  16  4  16       16  4     16  4     16  4    4     16 
│ │ │ -      -}, {-, 8, 4, -, --, -, --, 4}, {--, -, 4, --, -, 8, --, -}, {-, 8, --,
│ │ │ -      3    3        3   3  3   3        3  3      3  3      3  3    3      3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4  16  4     16       8  8  8  8  8  8       20  8     8     8  4     
│ │ │ -      -, --, -, 4, --}, {8, -, -, -, -, -, -, 8}, {--, -, 4, -, 4, -, -, 8},
│ │ │ -      3   3  3      3       3  3  3  3  3  3        3  3     3     3  3     
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -          4  8     4  16     16    20     4        4     20       4  4  16 
│ │ │ -      {8, -, -, 4, -, --, 4, --}, {--, 4, -, 4, 4, -, 4, --}, {8, -, -, --,
│ │ │ -          3  3     3   3      3     3     3        3      3       3  3   3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4  16  16          8  16     4     16  4    8  8     20     8  4     
│ │ │ -      -, --, --, 4}, {4, -, --, 4, -, 8, --, -}, {-, -, 4, --, 8, -, -, 4},
│ │ │ -      3   3   3          3   3     3      3  3    3  3      3     3  3     
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -          4  8     20     8  8       8  4     16     16  4    8  8  20     8 
│ │ │ -      {4, -, -, 8, --, 4, -, -}, {4, -, -, 8, --, 4, --, -}, {-, -, --, 4, -,
│ │ │ -          3  3      3     3  3       3  3      3      3  3    3  3   3     3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -            4    4        8     20  8  8    8     8  8  8  8     8    4  16 
│ │ │ -      8, 4, -}, {-, 4, 8, -, 4, --, -, -}, {-, 8, -, -, -, -, 8, -}, {-, --,
│ │ │ -            3    3        3      3  3  3    3     3  3  3  3     3    3   3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      16        4  4  16       20  4     8  8     8    8        4  8  20  8
│ │ │ -      --, 4, 8, -, -, --}, {4, --, -, 4, -, -, 8, -}, {-, 4, 8, -, -, --, -,
│ │ │ -       3        3  3   3        3  3     3  3     3    3        3  3   3  3
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4}}
│ │ │ +        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
│ │ │  
│ │ │  o12 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i13 : volume convexHull A -- 8
│ │ │ @@ -1112,66 +1114,66 @@
│ │ │  o13 : QQ
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : stars1 = select(Ts4, t -> (gkzVector t)#-1 == 8)
│ │ │  
│ │ │ -o14 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ +o14 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +      {0, 4, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ +      {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7},
│ │ │ +      {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7},
│ │ │ +      {0, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7},
│ │ │ +      {0, 2, 4, 7}, {0, 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}, {1, 2, 3, 7}},
│ │ │ +      {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7},
│ │ │ +      triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ +      {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation
│ │ │ +      {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ +      {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │        {2, 4, 6, 7}}}
│ │ │  
│ │ │  o14 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i15 : stars2 = select(Ts4, isStar)
│ │ │  
│ │ │ -o15 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ +o15 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +      {0, 4, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ +      {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7},
│ │ │ +      {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7},
│ │ │ +      {0, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7},
│ │ │ +      {0, 2, 4, 7}, {0, 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}, {1, 2, 3, 7}},
│ │ │ +      {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7},
│ │ │ +      triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ +      {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation
│ │ │ +      {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ +      {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │        {2, 4, 6, 7}}}
│ │ │  
│ │ │  o15 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -57,87 +57,15 @@
│ │ │ │  
│ │ │ │  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, 4, 7}, {0, 4, 5, 7},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, 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}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {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}},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ │ +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},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -263,273 +191,273 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 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, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7},
│ │ │ │ +     {2, 5, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │ │ +     {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 4, 6, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 3, 5, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │ │ +     triangulation {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 5},
│ │ │ │ +     {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ │ +     {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 5, 7},
│ │ │ │ +     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 5, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7},
│ │ │ │ +     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6},
│ │ │ │ +     {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 5, 6},
│ │ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 3, 5},
│ │ │ │ +     triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},
│ │ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {2, 3, 5, 7},
│ │ │ │ +     {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7},
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5},
│ │ │ │ +     {0, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │ │ +     {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 5},
│ │ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ │ +     triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6}, {2, 4, 5, 6},
│ │ │ │ +     {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6},
│ │ │ │ +     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5},
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │ │ +     {3, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 5},
│ │ │ │ +     {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7}, {1, 4, 5, 7},
│ │ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {3, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7},
│ │ │ │ +     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 3, 4, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7},
│ │ │ │ +     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 5},
│ │ │ │ +     {3, 4, 5, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 7},
│ │ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}, {1, 2, 3, 7}},
│ │ │ │ +     triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6},
│ │ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │ │ +     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7},
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 4},
│ │ │ │ +     {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},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ │ +     {1, 2, 3, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6}, {0, 3, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 6},
│ │ │ │ +     {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7},
│ │ │ │ +     {1, 2, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 2, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6},
│ │ │ │ +     {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 6},
│ │ │ │ +     {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}},
│ │ │ │ +     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ │ +     {1, 2, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 5},
│ │ │ │ +     {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 3, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6},
│ │ │ │ +     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 2, 7},
│ │ │ │ +     {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}},
│ │ │ │ +     {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ │ +     {2, 3, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 5, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7},
│ │ │ │ +     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6},
│ │ │ │ +     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6},
│ │ │ │ +     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 3, 5},
│ │ │ │ +     {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 3, 5, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ │ +     {1, 2, 3, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7},
│ │ │ │ +     {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7},
│ │ │ │ +     {0, 4, 6, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6},
│ │ │ │ +     {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 2, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │ │ +     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, {{0, 1, 3, 5},
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ │ +     {2, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 5, 6}, {2, 4, 5, 6},
│ │ │ │ +     {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {3, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │ │ +     {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6},
│ │ │ │ +     {0, 2, 3, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6},
│ │ │ │ +     {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}},
│ │ │ │ +     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {2, 3, 5, 6},
│ │ │ │ +     {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7},
│ │ │ │ +     {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7},
│ │ │ │ +     {1, 2, 3, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ │ +     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 5},
│ │ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}},
│ │ │ │ +     {1, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 6, 7},
│ │ │ │ +     {2, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 5, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7},
│ │ │ │ +     {1, 2, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ │ +     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 5}, {2, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4},
│ │ │ │ +     {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, {{0, 1, 3, 5},
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │ │ +     {0, 5, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 6}, {0, 3, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 6},
│ │ │ │ +     {0, 4, 6, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 6},
│ │ │ │ +     {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5},
│ │ │ │ +     {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │ │ +     {3, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6},
│ │ │ │ +     {3, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7},
│ │ │ │ +     {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 4, 5, 6}, {2, 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, 4, 7}, {0, 4, 5, 7},
│ │ │ │ +     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ │ +     {0, 1, 4, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │ │ +     {2, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7},
│ │ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5},
│ │ │ │ +     {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ │ +     {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7},
│ │ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │ +     {0, 4, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, {1, 2, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │ │ +     {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │ │ +     {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ │ +     {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},
│ │ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ │ +     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ │ +     {1, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation
│ │ │ │ +     {1, 3, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │ │ +     {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ │ +     {0, 2, 5, 6}, {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ │ +     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6},
│ │ │ │ +     {2, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │ │ +     {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ │ +     {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ │ +     {0, 2, 4, 5}, {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5},
│ │ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ │ +     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7},
│ │ │ │ +     {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4},
│ │ │ │ +     {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │ +     {0, 3, 4, 6}, {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6},
│ │ │ │ +     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ │ +     {1, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6}, {1, 3, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ │ +     {0, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ │ +     {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},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {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},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -655,92 +583,92 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 5, 6, 7}}}
│ │ │ │ -
│ │ │ │ -o6 : List
│ │ │ │ -i7 : Ts4 = generateTriangulations tri -- list of Triangulations
│ │ │ │ -
│ │ │ │ -o7 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ │ +     {2, 5, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ │ +     {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │ │ +     triangulation {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7},
│ │ │ │ +     {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5},
│ │ │ │ +     {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ │ +     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7},
│ │ │ │ +     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ │ +     {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │ │ +     triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ │ +     {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ │ +     {0, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ │ +     {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation
│ │ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │ │ +     triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ │ +     {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ │ +     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6},
│ │ │ │ +     {3, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │ │ +     {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5},
│ │ │ │ +     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ │ +     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7},
│ │ │ │ +     {3, 4, 5, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4},
│ │ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ │ +     triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6},
│ │ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ │ +     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ │ +     {{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}},
│ │ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ │ +     {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},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {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},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -866,15 +794,87 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 5, 6, 7}}}
│ │ │ │ +     {2, 5, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     triangulation {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {0, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {3, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     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}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {3, 4, 5, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {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}}}
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  i8 : all(Ts4, isFine)
│ │ │ │  
│ │ │ │  o8 = true
│ │ │ │  i9 : all(Ts4, isStar)
│ │ │ │  
│ │ │ │ @@ -886,188 +886,190 @@
│ │ │ │  
│ │ │ │  o11 = Tally{false => 66}
│ │ │ │              true => 8
│ │ │ │  
│ │ │ │  o11 : Tally
│ │ │ │  i12 : Ts4/gkzVector
│ │ │ │  
│ │ │ │ -        16     16  4     8  4       20     8  8     4  8          4  8     8
│ │ │ │ -o12 = {{--, 4, --, -, 4, -, -, 8}, {--, 4, -, -, 4, -, -, 8}, {8, -, -, 4, -,
│ │ │ │ -         3      3  3     3  3        3     3  3     3  3          3  3     3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -         8  20       16  16  4  16  4  4       8        4     16  4  16
│ │ │ │ -      4, -, --}, {4, --, --, -, --, -, -, 8}, {-, 4, 8, -, 4, --, -, --}, {4,
│ │ │ │ -         3   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, 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    20        4  8  8  8          8  8  8  4        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     4  8          8  4     8  8     20       20     4  4
│ │ │ │ -      {--, --, 4, -, 4, -, -, 8}, {8, -, -, 4, -, -, 4, --}, {4, --, 4, -, -,
│ │ │ │ -        3   3     3     3  3          3  3     3  3      3        3     3  3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -         20       20  4              4  20       4  20        20  4       8
│ │ │ │ -      4, --, 4}, {--, -, 4, 4, 4, 4, -, --}, {4, -, --, 4, 4, --, -, 4}, {-,
│ │ │ │ -          3        3  3              3   3       3   3         3  3       3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -         8  20     4  8          16  4  16  8        4    4     16  16     8
│ │ │ │ -      4, -, --, 8, -, -, 4}, {4, --, -, --, -, 4, 8, -}, {-, 4, --, --, 8, -,
│ │ │ │ -         3   3     3  3           3  3   3  3        3    3      3   3     3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      4          4  8     16  16     4    4     20        20     4    8  8
│ │ │ │ -      -, 4}, {4, -, -, 8, --, --, 4, -}, {-, 4, --, 4, 4, --, 4, -}, {-, -,
│ │ │ │ -      3          3  3      3   3     3    3      3         3     3    3  3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      8        8  8  8    8  20  8     8        4       16  8     4  16
│ │ │ │ -      -, 8, 8, -, -, -}, {-, --, -, 4, -, 4, 8, -}, {4, --, -, 4, -, --, 8,
│ │ │ │ -      3        3  3  3    3   3  3     3        3        3  3     3   3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      4    4        8  16  4  16       16  4     16  4     16  4    4     16
│ │ │ │ -      -}, {-, 8, 4, -, --, -, --, 4}, {--, -, 4, --, -, 8, --, -}, {-, 8, --,
│ │ │ │ -      3    3        3   3  3   3        3  3      3  3      3  3    3      3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      4  16  4     16       8  8  8  8  8  8       20  8     8     8  4
│ │ │ │ -      -, --, -, 4, --}, {8, -, -, -, -, -, -, 8}, {--, -, 4, -, 4, -, -, 8},
│ │ │ │ -      3   3  3      3       3  3  3  3  3  3        3  3     3     3  3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -          4  8     4  16     16    20     4        4     20       4  4  16
│ │ │ │ -      {8, -, -, 4, -, --, 4, --}, {--, 4, -, 4, 4, -, 4, --}, {8, -, -, --,
│ │ │ │ -          3  3     3   3      3     3     3        3      3       3  3   3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      4  16  16          8  16     4     16  4    8  8     20     8  4
│ │ │ │ -      -, --, --, 4}, {4, -, --, 4, -, 8, --, -}, {-, -, 4, --, 8, -, -, 4},
│ │ │ │ -      3   3   3          3   3     3      3  3    3  3      3     3  3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -          4  8     20     8  8       8  4     16     16  4    8  8  20     8
│ │ │ │ -      {4, -, -, 8, --, 4, -, -}, {4, -, -, 8, --, 4, --, -}, {-, -, --, 4, -,
│ │ │ │ -          3  3      3     3  3       3  3      3      3  3    3  3   3     3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -            4    4        8     20  8  8    8     8  8  8  8     8    4  16
│ │ │ │ -      8, 4, -}, {-, 4, 8, -, 4, --, -, -}, {-, 8, -, -, -, -, 8, -}, {-, --,
│ │ │ │ -            3    3        3      3  3  3    3     3  3  3  3     3    3   3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      16        4  4  16       20  4     8  8     8    8        4  8  20  8
│ │ │ │ -      --, 4, 8, -, -, --}, {4, --, -, 4, -, -, 8, -}, {-, 4, 8, -, -, --, -,
│ │ │ │ -       3        3  3   3        3  3     3  3     3    3        3  3   3  3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      4}}
│ │ │ │ +        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
│ │ │ │  
│ │ │ │  o12 : List
│ │ │ │  i13 : volume convexHull A -- 8
│ │ │ │  
│ │ │ │  o13 = 8
│ │ │ │  
│ │ │ │  o13 : QQ
│ │ │ │  i14 : stars1 = select(Ts4, t -> (gkzVector t)#-1 == 8)
│ │ │ │  
│ │ │ │ -o14 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ │ +o14 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ │ +      {0, 4, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ │ +      {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7},
│ │ │ │ +      {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7},
│ │ │ │ +      {0, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7},
│ │ │ │ +      {0, 2, 4, 7}, {0, 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}, {1, 2, 3, 7}},
│ │ │ │ +      {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7},
│ │ │ │ +      triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ │ +      {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation
│ │ │ │ +      {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ │ +      {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        {2, 4, 6, 7}}}
│ │ │ │  
│ │ │ │  o14 : List
│ │ │ │  i15 : stars2 = select(Ts4, isStar)
│ │ │ │  
│ │ │ │ -o15 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ │ +o15 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ │ +      {0, 4, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ │ +      {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7},
│ │ │ │ +      {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7},
│ │ │ │ +      {0, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7},
│ │ │ │ +      {0, 2, 4, 7}, {0, 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}, {1, 2, 3, 7}},
│ │ │ │ +      {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7},
│ │ │ │ +      triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ │ +      {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation
│ │ │ │ +      {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ │ +      {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        {2, 4, 6, 7}}}
│ │ │ │  
│ │ │ │  o15 : List
│ │ │ │  i16 : stars1 == stars2
│ │ │ │  
│ │ │ │  o16 = true
│ │ ├── ./usr/share/doc/Macaulay2/Triangulations/html/index.html
│ │ │ @@ -150,15 +150,15 @@
│ │ │                4       10
│ │ │  o2 : Matrix ZZ  <-- ZZ
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : elapsedTime Ts = allTriangulations(A, Fine => true);
│ │ │ - -- .14823s elapsed
    
│ │ │ + -- .132955s 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,
│ │ │ @@ -198,15 +198,15 @@
│ │ │  
│ │ │  o7 : Triangulation
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : elapsedTime Ts2 = generateTriangulations T;
│ │ │ - -- 1.22082s elapsed
    
│ │ │ + -- .954958s 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);
│ │ │ │ - -- .14823s elapsed
│ │ │ │ + -- .132955s 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;
│ │ │ │ - -- 1.22082s elapsed
│ │ │ │ + -- .954958s 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/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 0.928671s (cpu); 0.578647s (thread); 0s (gc)
│ │ │ + -- used 1.0789s (cpu); 0.677271s (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 0.732527s (cpu); 0.472537s (thread); 0s (gc)
│ │ │ + -- used 0.831289s (cpu); 0.485528s (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
│ │ │ @@ -71,15 +71,15 @@
│ │ │            
│ │ │          
│ │ │          
    With the default setting SmartLift=>true we get very nice equations for the base space:
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -98,15 +98,15 @@
│ │ │            
│ │ │          
    
│ │ │                
    i3 : time (F,R,G,C)=localHilbertScheme(F0);
│ │ │ - -- used 0.928671s (cpu); 0.578647s (thread); 0s (gc)
    
│ │ │ + -- used 1.0789s (cpu); 0.677271s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i4 : T=ring first G;
    
│ │ │              
    
│ │ │          
    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 0.732527s (cpu); 0.472537s (thread); 0s (gc)
    
│ │ │ + -- used 0.831289s (cpu); 0.485528s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i7 : sum G
│ │ │  
│ │ │  o7 = | t_1t_16
│ │ │ ├── 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 0.928671s (cpu); 0.578647s (thread); 0s (gc)
│ │ │ │ + -- used 1.0789s (cpu); 0.677271s (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 0.732527s (cpu); 0.472537s (thread); 0s (gc)
│ │ │ │ + -- used 0.831289s (cpu); 0.485528s (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/WeilDivisors/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 0.12276s (cpu); 0.0663951s (thread); 0s (gc)
│ │ │ + -- used 0.17623s (cpu); 0.0843247s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of R
│ │ │  
│ │ │  i13 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.408131s (cpu); 0.408137s (thread); 0s (gc)
│ │ │ + -- used 0.548685s (cpu); 0.548521s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of R
│ │ │  
│ │ │  i14 : time dualize(M, Strategy=>IdealStrategy);
│ │ │ - -- used 0.54272s (cpu); 0.47665s (thread); 0s (gc)
│ │ │ + -- used 0.685048s (cpu); 0.600493s (thread); 0s (gc)
│ │ │  
│ │ │  i15 : time dualize(M, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.002816s (cpu); 0.00281656s (thread); 0s (gc)
│ │ │ + -- used 0.00318349s (cpu); 0.00319061s (thread); 0s (gc)
│ │ │  
│ │ │  i16 : time embedAsIdeal dualize(M, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00217764s (cpu); 0.00217846s (thread); 0s (gc)
│ │ │ + -- used 0.00336767s (cpu); 0.00337819s (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 0.232709s (cpu); 0.117318s (thread); 0s (gc)
│ │ │ + -- used 0.336527s (cpu); 0.15727s (thread); 0s (gc)
│ │ │  
│ │ │  o20 : Ideal of R
│ │ │  
│ │ │  i21 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00588895s (cpu); 0.00588973s (thread); 0s (gc)
│ │ │ + -- used 0.00722865s (cpu); 0.00723496s (thread); 0s (gc)
│ │ │  
│ │ │  o21 : Ideal of R
│ │ │  
│ │ │  i22 : R = QQ[x,y]/ideal(x*y);
│ │ │  
│ │ │  i23 : J = ideal(x,y);
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/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 0.265292s (cpu); 0.211481s (thread); 0s (gc)
│ │ │ + -- used 0.292936s (cpu); 0.218099s (thread); 0s (gc)
│ │ │  
│ │ │  o23 : Ideal of R
│ │ │  
│ │ │  i24 : time reflexify(J*R^1);
│ │ │ - -- used 0.508363s (cpu); 0.374589s (thread); 0s (gc)
│ │ │ + -- used 0.44693s (cpu); 0.37241s (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 0.270151s (cpu); 0.134196s (thread); 0s (gc)
│ │ │ + -- used 0.317624s (cpu); 0.153567s (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 6.77561s (cpu); 4.70658s (thread); 0s (gc)
│ │ │ + -- used 5.96442s (cpu); 4.68701s (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 5.83438s (cpu); 4.55312s (thread); 0s (gc)
│ │ │ + -- used 6.07438s (cpu); 4.85471s (thread); 0s (gc)
│ │ │  
│ │ │  i33 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ - -- used 0.553639s (cpu); 0.383614s (thread); 0s (gc)
│ │ │ + -- used 0.604544s (cpu); 0.431667s (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 1.11106s (cpu); 0.416741s (thread); 0s (gc)
│ │ │ + -- used 1.18326s (cpu); 0.423418s (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 0.0142796s (cpu); 0.0142766s (thread); 0s (gc)
│ │ │ + -- used 0.247183s (cpu); 0.0630558s (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 0.270201s (cpu); 0.0924585s (thread); 0s (gc)
│ │ │ + -- used 0.288817s (cpu); 0.0969406s (thread); 0s (gc)
│ │ │  
│ │ │  i41 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ - -- used 0.00697604s (cpu); 0.00697697s (thread); 0s (gc)
│ │ │ + -- used 0.00759587s (cpu); 0.0076018s (thread); 0s (gc)
│ │ │  
│ │ │  i42 : R = QQ[x,y]/ideal(x*y);
│ │ │  
│ │ │  i43 : I = ideal(x,y);
│ │ │  
│ │ │  o43 : Ideal of R
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/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 0.0328143s (cpu); 0.0327921s (thread); 0s (gc)
│ │ │ + -- used 0.0397678s (cpu); 0.0396471s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of R
│ │ │  
│ │ │  i8 : I20 = I^20;
│ │ │  
│ │ │  o8 : Ideal of R
│ │ │  
│ │ │  i9 : time J20b = reflexify(I20);
│ │ │ - -- used 0.208458s (cpu); 0.147573s (thread); 0s (gc)
│ │ │ + -- used 0.241436s (cpu); 0.163184s (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 0.0296356s (cpu); 0.029641s (thread); 0s (gc)
│ │ │ + -- used 0.0394464s (cpu); 0.0394504s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of R
│ │ │  
│ │ │  i14 : time J2 = reflexivePower(20, I, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.152316s (cpu); 0.0807728s (thread); 0s (gc)
│ │ │ + -- used 0.16848s (cpu); 0.0972662s (thread); 0s (gc)
│ │ │  
│ │ │  o14 : Ideal of R
│ │ │  
│ │ │  i15 : J1 == J2
│ │ │  
│ │ │  o15 = true
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_dualize.html
│ │ │ @@ -163,43 +163,43 @@
│ │ │              
    		
│ │ │                
    i11 : M = J*R^1;
    
│ │ │              
    
│ │ │                
    i12 : time dualize(J, Strategy=>IdealStrategy);
│ │ │ - -- used 0.12276s (cpu); 0.0663951s (thread); 0s (gc)
│ │ │ + -- used 0.17623s (cpu); 0.0843247s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of R
    
│ │ │              
    
│ │ │                
    i13 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.408131s (cpu); 0.408137s (thread); 0s (gc)
│ │ │ + -- used 0.548685s (cpu); 0.548521s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of R
    
│ │ │              
    
│ │ │                
    i14 : time dualize(M, Strategy=>IdealStrategy);
│ │ │ - -- used 0.54272s (cpu); 0.47665s (thread); 0s (gc)
    
│ │ │ + -- used 0.685048s (cpu); 0.600493s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i15 : time dualize(M, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.002816s (cpu); 0.00281656s (thread); 0s (gc)
    
│ │ │ + -- used 0.00318349s (cpu); 0.00319061s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i16 : time embedAsIdeal dualize(M, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00217764s (cpu); 0.00217846s (thread); 0s (gc)
│ │ │ + -- used 0.00336767s (cpu); 0.00337819s (thread); 0s (gc)
│ │ │  
│ │ │  o16 : Ideal of R
    
│ │ │              
    
│ │ │          
    
│ │ │            
    For monomial ideals in toric rings, frequently ModuleStrategy appears faster.
    
│ │ │ @@ -223,23 +223,23 @@
│ │ │  
│ │ │  o19 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i20 : time dualize(J, Strategy=>IdealStrategy);
│ │ │ - -- used 0.232709s (cpu); 0.117318s (thread); 0s (gc)
│ │ │ + -- used 0.336527s (cpu); 0.15727s (thread); 0s (gc)
│ │ │  
│ │ │  o20 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i21 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00588895s (cpu); 0.00588973s (thread); 0s (gc)
│ │ │ + -- used 0.00722865s (cpu); 0.00723496s (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 {}
│ │ │ │ @@ -60,43 +60,43 @@
│ │ │ │  
│ │ │ │  o9 : Ideal of R
│ │ │ │  i10 : J = m^9;
│ │ │ │  
│ │ │ │  o10 : Ideal of R
│ │ │ │  i11 : M = J*R^1;
│ │ │ │  i12 : time dualize(J, Strategy=>IdealStrategy);
│ │ │ │ - -- used 0.12276s (cpu); 0.0663951s (thread); 0s (gc)
│ │ │ │ + -- used 0.17623s (cpu); 0.0843247s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o12 : Ideal of R
│ │ │ │  i13 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ │ - -- used 0.408131s (cpu); 0.408137s (thread); 0s (gc)
│ │ │ │ + -- used 0.548685s (cpu); 0.548521s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o13 : Ideal of R
│ │ │ │  i14 : time dualize(M, Strategy=>IdealStrategy);
│ │ │ │ - -- used 0.54272s (cpu); 0.47665s (thread); 0s (gc)
│ │ │ │ + -- used 0.685048s (cpu); 0.600493s (thread); 0s (gc)
│ │ │ │  i15 : time dualize(M, Strategy=>ModuleStrategy);
│ │ │ │ - -- used 0.002816s (cpu); 0.00281656s (thread); 0s (gc)
│ │ │ │ + -- used 0.00318349s (cpu); 0.00319061s (thread); 0s (gc)
│ │ │ │  i16 : time embedAsIdeal dualize(M, Strategy=>ModuleStrategy);
│ │ │ │ - -- used 0.00217764s (cpu); 0.00217846s (thread); 0s (gc)
│ │ │ │ + -- used 0.00336767s (cpu); 0.00337819s (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 0.232709s (cpu); 0.117318s (thread); 0s (gc)
│ │ │ │ + -- used 0.336527s (cpu); 0.15727s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o20 : Ideal of R
│ │ │ │  i21 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ │ - -- used 0.00588895s (cpu); 0.00588973s (thread); 0s (gc)
│ │ │ │ + -- used 0.00722865s (cpu); 0.00723496s (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/WeilDivisors/html/_reflexify.html
│ │ │ @@ -267,23 +267,23 @@
│ │ │  
│ │ │  o22 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i23 : time reflexify(J);
│ │ │ - -- used 0.265292s (cpu); 0.211481s (thread); 0s (gc)
│ │ │ + -- used 0.292936s (cpu); 0.218099s (thread); 0s (gc)
│ │ │  
│ │ │  o23 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i24 : time reflexify(J*R^1);
│ │ │ - -- used 0.508363s (cpu); 0.374589s (thread); 0s (gc)
    
│ │ │ + -- used 0.44693s (cpu); 0.37241s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    Because of this, there are two strategies for computing a reflexification (at least if the module embeds as an ideal).
    
│ │ │          
    
│ │ │          
    
│ │ │ @@ -319,26 +319,26 @@
│ │ │              
│ │ │                
    i28 : M = J*R^1;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i29 : J1 = time reflexify( J, Strategy=>IdealStrategy )
│ │ │ - -- used 0.270151s (cpu); 0.134196s (thread); 0s (gc)
│ │ │ + -- used 0.317624s (cpu); 0.153567s (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 6.77561s (cpu); 4.70658s (thread); 0s (gc)
│ │ │ + -- used 5.96442s (cpu); 4.68701s (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
    
│ │ │              
│ │ │            
│ │ │ @@ -348,21 +348,21 @@
│ │ │  
│ │ │  o31 = true
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i32 : time reflexify( M, Strategy=>IdealStrategy );
│ │ │ - -- used 5.83438s (cpu); 4.55312s (thread); 0s (gc)
    
│ │ │ + -- used 6.07438s (cpu); 4.85471s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i33 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ - -- used 0.553639s (cpu); 0.383614s (thread); 0s (gc)
    
│ │ │ + -- used 0.604544s (cpu); 0.431667s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    However, sometimes ModuleStrategy is faster, especially for Monomial ideals.
    
│ │ │          
    
│ │ │          
│ │ │ @@ -389,15 +389,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i37 : M = I^20*R^1;
    
│ │ │              
    
│ │ │                
    i38 : time reflexify( J, Strategy=>IdealStrategy )
│ │ │ - -- used 1.11106s (cpu); 0.416741s (thread); 0s (gc)
│ │ │ + -- used 1.18326s (cpu); 0.423418s (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 ,
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -406,15 +406,15 @@
│ │ │  
│ │ │  o38 : Ideal of R
    
│ │ │              
    
│ │ │                
    i39 : time reflexify( J, Strategy=>ModuleStrategy )
│ │ │ - -- used 0.0142796s (cpu); 0.0142766s (thread); 0s (gc)
│ │ │ + -- used 0.247183s (cpu); 0.0630558s (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 ,
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -423,21 +423,21 @@
│ │ │  
│ │ │  o39 : Ideal of R
    
│ │ │              
    
│ │ │                
    i40 : time reflexify( M, Strategy=>IdealStrategy );
│ │ │ - -- used 0.270201s (cpu); 0.0924585s (thread); 0s (gc)
    
│ │ │ + -- used 0.288817s (cpu); 0.0969406s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i41 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ - -- used 0.00697604s (cpu); 0.00697697s (thread); 0s (gc)
    
│ │ │ + -- used 0.00759587s (cpu); 0.0076018s (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.
    
│ │ │          
    
│ │ │          
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -114,19 +114,19 @@
│ │ │ │  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 0.265292s (cpu); 0.211481s (thread); 0s (gc)
│ │ │ │ + -- used 0.292936s (cpu); 0.218099s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 : Ideal of R
│ │ │ │  i24 : time reflexify(J*R^1);
│ │ │ │ - -- used 0.508363s (cpu); 0.374589s (thread); 0s (gc)
│ │ │ │ + -- used 0.44693s (cpu); 0.37241s (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
│ │ │ │ @@ -139,73 +139,73 @@
│ │ │ │  
│ │ │ │  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 0.270151s (cpu); 0.134196s (thread); 0s (gc)
│ │ │ │ + -- used 0.317624s (cpu); 0.153567s (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 6.77561s (cpu); 4.70658s (thread); 0s (gc)
│ │ │ │ + -- used 5.96442s (cpu); 4.68701s (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 5.83438s (cpu); 4.55312s (thread); 0s (gc)
│ │ │ │ + -- used 6.07438s (cpu); 4.85471s (thread); 0s (gc)
│ │ │ │  i33 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ │ - -- used 0.553639s (cpu); 0.383614s (thread); 0s (gc)
│ │ │ │ + -- used 0.604544s (cpu); 0.431667s (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 1.11106s (cpu); 0.416741s (thread); 0s (gc)
│ │ │ │ + -- used 1.18326s (cpu); 0.423418s (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 0.0142796s (cpu); 0.0142766s (thread); 0s (gc)
│ │ │ │ + -- used 0.247183s (cpu); 0.0630558s (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 0.270201s (cpu); 0.0924585s (thread); 0s (gc)
│ │ │ │ + -- used 0.288817s (cpu); 0.0969406s (thread); 0s (gc)
│ │ │ │  i41 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ │ - -- used 0.00697604s (cpu); 0.00697697s (thread); 0s (gc)
│ │ │ │ + -- used 0.00759587s (cpu); 0.0076018s (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/WeilDivisors/html/_reflexive__Power.html
│ │ │ @@ -124,30 +124,30 @@
│ │ │  
│ │ │  o6 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : time J20a = reflexivePower(20, I);
│ │ │ - -- used 0.0328143s (cpu); 0.0327921s (thread); 0s (gc)
│ │ │ + -- used 0.0397678s (cpu); 0.0396471s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : I20 = I^20;
│ │ │  
│ │ │  o8 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : time J20b = reflexify(I20);
│ │ │ - -- used 0.208458s (cpu); 0.147573s (thread); 0s (gc)
│ │ │ + -- used 0.241436s (cpu); 0.163184s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : J20a == J20b
│ │ │ @@ -171,23 +171,23 @@
│ │ │  
│ │ │  o12 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i13 : time J1 = reflexivePower(20, I, Strategy=>IdealStrategy);
│ │ │ - -- used 0.0296356s (cpu); 0.029641s (thread); 0s (gc)
│ │ │ + -- used 0.0394464s (cpu); 0.0394504s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : time J2 = reflexivePower(20, I, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.152316s (cpu); 0.0807728s (thread); 0s (gc)
│ │ │ + -- used 0.16848s (cpu); 0.0972662s (thread); 0s (gc)
│ │ │  
│ │ │  o14 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i15 : J1 == J2
│ │ │ ├── html2text {}
│ │ │ │ @@ -40,39 +40,39 @@
│ │ │ │  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 0.0328143s (cpu); 0.0327921s (thread); 0s (gc)
│ │ │ │ + -- used 0.0397678s (cpu); 0.0396471s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 : Ideal of R
│ │ │ │  i8 : I20 = I^20;
│ │ │ │  
│ │ │ │  o8 : Ideal of R
│ │ │ │  i9 : time J20b = reflexify(I20);
│ │ │ │ - -- used 0.208458s (cpu); 0.147573s (thread); 0s (gc)
│ │ │ │ + -- used 0.241436s (cpu); 0.163184s (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 0.0296356s (cpu); 0.029641s (thread); 0s (gc)
│ │ │ │ + -- used 0.0394464s (cpu); 0.0394504s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o13 : Ideal of R
│ │ │ │  i14 : time J2 = reflexivePower(20, I, Strategy=>ModuleStrategy);
│ │ │ │ - -- used 0.152316s (cpu); 0.0807728s (thread); 0s (gc)
│ │ │ │ + -- used 0.16848s (cpu); 0.0972662s (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/WeylAlgebras/example-output/_factor__Weyl__Algebra.out
│ │ │ @@ -4,19 +4,19 @@
│ │ │  
│ │ │  o1 = R
│ │ │  
│ │ │  o1 : PolynomialRing, 1 differential variable(s)
│ │ │  
│ │ │  i2 : factorWA(x^5*dx^2+7*x^4*dx+8*x^3-x*dx^2+dx)
│ │ │  
│ │ │ -                                    2         3       2                  2  
│ │ │ -o2 = {(x*dx - 1)(dx)(x - 1)(x + 1)(x  + 1), (x dx + 3x  - x*dx + 1)(dx)(x  +
│ │ │ +                                    2         3       2                    
│ │ │ +o2 = {(x*dx - 1)(dx)(x - 1)(x + 1)(x  + 1), (x dx + 3x  + x*dx - 1)(dx)(x -
│ │ │       ------------------------------------------------------------------------
│ │ │ -               3                2         3       2                         
│ │ │ -     1), (dx)(x dx - x*dx + 2)(x  + 1), (x dx + 3x  + x*dx - 1)(dx)(x - 1)(x
│ │ │ +                      3                                3       2         
│ │ │ +     1)(x + 1), (dx)(x dx + x*dx - 2)(x - 1)(x + 1), (x dx + 3x  - x*dx +
│ │ │       ------------------------------------------------------------------------
│ │ │ -                 3
│ │ │ -     + 1), (dx)(x dx + x*dx - 2)(x - 1)(x + 1)}
│ │ │ +             2             3                2
│ │ │ +     1)(dx)(x  + 1), (dx)(x dx - x*dx + 2)(x  + 1)}
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylAlgebras/html/_factor__Weyl__Algebra.html
│ │ │ @@ -84,22 +84,22 @@
│ │ │  o1 : PolynomialRing, 1 differential variable(s)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : factorWA(x^5*dx^2+7*x^4*dx+8*x^3-x*dx^2+dx)
│ │ │  
│ │ │ -                                    2         3       2                  2  
│ │ │ -o2 = {(x*dx - 1)(dx)(x - 1)(x + 1)(x  + 1), (x dx + 3x  - x*dx + 1)(dx)(x  +
│ │ │ +                                    2         3       2                    
│ │ │ +o2 = {(x*dx - 1)(dx)(x - 1)(x + 1)(x  + 1), (x dx + 3x  + x*dx - 1)(dx)(x -
│ │ │       ------------------------------------------------------------------------
│ │ │ -               3                2         3       2                         
│ │ │ -     1), (dx)(x dx - x*dx + 2)(x  + 1), (x dx + 3x  + x*dx - 1)(dx)(x - 1)(x
│ │ │ +                      3                                3       2         
│ │ │ +     1)(x + 1), (dx)(x dx + x*dx - 2)(x - 1)(x + 1), (x dx + 3x  - x*dx +
│ │ │       ------------------------------------------------------------------------
│ │ │ -                 3
│ │ │ -     + 1), (dx)(x dx + x*dx - 2)(x - 1)(x + 1)}
│ │ │ +             2             3                2
│ │ │ +     1)(dx)(x  + 1), (dx)(x dx - x*dx + 2)(x  + 1)}
│ │ │  
│ │ │  o2 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    To reduce their number, two factorisations are considered equivalent if they can be related by (1)	switching commuting irreducible factors or (2) switching monomials and degree 0 factors; a normal order is chosen where commuting factors are sorted, and monomials are pushed to the right/left if they're differential/not.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,22 +20,22 @@
│ │ │ │  i1 : R = makeWA(QQ[x])
│ │ │ │  
│ │ │ │  o1 = R
│ │ │ │  
│ │ │ │  o1 : PolynomialRing, 1 differential variable(s)
│ │ │ │  i2 : factorWA(x^5*dx^2+7*x^4*dx+8*x^3-x*dx^2+dx)
│ │ │ │  
│ │ │ │ -                                    2         3       2                  2
│ │ │ │ -o2 = {(x*dx - 1)(dx)(x - 1)(x + 1)(x  + 1), (x dx + 3x  - x*dx + 1)(dx)(x  +
│ │ │ │ +                                    2         3       2
│ │ │ │ +o2 = {(x*dx - 1)(dx)(x - 1)(x + 1)(x  + 1), (x dx + 3x  + x*dx - 1)(dx)(x -
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -               3                2         3       2
│ │ │ │ -     1), (dx)(x dx - x*dx + 2)(x  + 1), (x dx + 3x  + x*dx - 1)(dx)(x - 1)(x
│ │ │ │ +                      3                                3       2
│ │ │ │ +     1)(x + 1), (dx)(x dx + x*dx - 2)(x - 1)(x + 1), (x dx + 3x  - x*dx +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -                 3
│ │ │ │ -     + 1), (dx)(x dx + x*dx - 2)(x - 1)(x + 1)}
│ │ │ │ +             2             3                2
│ │ │ │ +     1)(dx)(x  + 1), (dx)(x dx - x*dx + 2)(x  + 1)}
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  To reduce their number, two factorisations are considered equivalent if they
│ │ │ │  can be related by (1) switching commuting irreducible factors or (2) switching
│ │ │ │  monomials and degree 0 factors; a normal order is chosen where commuting
│ │ │ │  factors are sorted, and monomials are pushed to the right/left if they're
│ │ │ │  differential/not.
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/example-output/_above__Bruhat_lp__Basic__List_rp.out
│ │ │ @@ -36,34 +36,34 @@
│ │ │                                             |  2 |
│ │ │                                             | -1 |
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : aboveBruhat(L1)
│ │ │  
│ │ │ -o4 = {{WeylGroupElement{RootSystem{...8...}, |  3 |}, {{1, |  0 |}, {2, | -1
│ │ │ -                                             | -2 |        | -1 |       |  1
│ │ │ -                                             |  1 |        |  2 |       |  1
│ │ │ +o4 = {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1, | -1 |}, {2, | -1
│ │ │ +                                             |  1 |        |  1 |       |  2
│ │ │ +                                             | -2 |        |  1 |       | -1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {1,
│ │ │ +     |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{1, |  0 |}, {2,
│ │ │ +     |                                            | -2 |        | -1 |      
│ │ │       |                                            |  1 |        |  2 |      
│ │ │ -     |                                            |  2 |        | -1 |      
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0 |},
│ │ │ -     |  1 |                                            |  3 |        | -1 |  
│ │ │ -     | -1 |                                            | -1 |        |  2 |  
│ │ │ +     | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |},
│ │ │ +     |  1 |                                            |  1 |        |  2 |  
│ │ │ +     |  1 |                                            |  2 |        | -1 |  
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  1
│ │ │ -         | -1 |                                            | -2 |        |  1
│ │ │ -         |  0 |                                            |  3 |        | -1
│ │ │ +     {1, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0
│ │ │ +         |  1 |                                            |  3 |        | -1
│ │ │ +         | -1 |                                            | -1 |        |  2
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}, {1, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1,
│ │ │ -     |       | -1 |                                            |  1 |       
│ │ │ -     |       |  0 |                                            | -2 |       
│ │ │ +     |}, {2, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0,
│ │ │ +     |       | -1 |                                            | -2 |       
│ │ │ +     |       |  0 |                                            |  3 |       
│ │ │       ------------------------------------------------------------------------
│ │ │ -     | -1 |}, {2, | -1 |}}}}
│ │ │ -     |  1 |       |  2 |
│ │ │ +     |  1 |}, {1, |  2 |}}}}
│ │ │       |  1 |       | -1 |
│ │ │ +     | -1 |       |  0 |
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/example-output/_eval_lp__Root__System_cm__Weight_cm__Root_rp.out
│ │ │ @@ -4,33 +4,33 @@
│ │ │  
│ │ │  o1 = RootSystem{...8...}
│ │ │  
│ │ │  o1 : RootSystem
│ │ │  
│ │ │  i2 : L=toList(positiveRoots(R))
│ │ │  
│ │ │ -o2 = {| 1 |, |  2 |, | -1 |}
│ │ │ -      | 1 |  | -1 |  |  2 |
│ │ │ +o2 = {| -1 |, | 1 |, |  2 |}
│ │ │ +      |  2 |  | 1 |  | -1 |
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 : v=weight(R,{1,2})
│ │ │  
│ │ │  o3 = | 1 |
│ │ │       | 2 |
│ │ │  
│ │ │         2
│ │ │  o3 : ZZ
│ │ │  
│ │ │  i4 : eval(R,v,L#0)
│ │ │  
│ │ │ -o4 = 3
│ │ │ +o4 = 2
│ │ │  
│ │ │  i5 : eval(R,v,L#1)
│ │ │  
│ │ │ -o5 = 1
│ │ │ +o5 = 3
│ │ │  
│ │ │  i6 : eval(R,v,L#2)
│ │ │  
│ │ │ -o6 = 2
│ │ │ +o6 = 1
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/example-output/_eval_lp__Root__System_cm__Z__Z_cm__Root_rp.out
│ │ │ @@ -4,25 +4,25 @@
│ │ │  
│ │ │  o1 = RootSystem{...8...}
│ │ │  
│ │ │  o1 : RootSystem
│ │ │  
│ │ │  i2 : L=toList(positiveRoots(R))
│ │ │  
│ │ │ -o2 = {| 1 |, |  2 |, | -1 |}
│ │ │ -      | 1 |  | -1 |  |  2 |
│ │ │ +o2 = {| -1 |, | 1 |, |  2 |}
│ │ │ +      |  2 |  | 1 |  | -1 |
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 : eval(R,1,L#0)
│ │ │  
│ │ │ -o3 = 1
│ │ │ +o3 = 0
│ │ │  
│ │ │  i4 : eval(R,1,L#1)
│ │ │  
│ │ │  o4 = 1
│ │ │  
│ │ │  i5 : eval(R,1,L#2)
│ │ │  
│ │ │ -o5 = 0
│ │ │ +o5 = 1
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/example-output/_hasse__Diagram__To__Graph_lp__Hasse__Diagram_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 |}, {2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | -1 |}, {1, |  1 |}, {3, | 1 |}, {4, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{1, | 1 |}, {2, | -1 |}, {3, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {2, |  2 |}, {4, |  0 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, |  0 |}, {2, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{1, |  0 |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{2, | 1 |}, {3, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {3, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {3, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{0, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ -                                                          | -2 |        | -1 |       |  1 |       |  2 |                                              | -3 |        |  2 |       |  1 |       | 0 |       |  1 |                                            |  2 |        | 0 |       |  2 |       | -1 |                                            | -1 |        |  1 |       | -1 |       | -1 |                                              | -3 |        | -1 |       | -1 |                                            | -1 |        | -1 |       |  2 |                                            |  1 |        | 0 |       | -1 |                                            |  3 |        |  1 |       |  1 |                                            | -1 |        |  2 |       | -1 |                                              | -2 |        | -1 |                                            |  1 |        |  1 |                                            | -2 |        | -1 |                                            |  1 |        |  1 |                                              | -1 |
│ │ │ -                                                          |  1 |        |  2 |       | -1 |       | -1 |                                              |  1 |        | -1 |       | -1 |       | 1 |       |  1 |                                            | -1 |        | 1 |       | -1 |       |  2 |                                            |  2 |        |  1 |       |  0 |       |  2 |                                              |  2 |        |  2 |       |  0 |                                            | -1 |        |  2 |       | -1 |                                            |  1 |        | 1 |       |  2 |                                            | -2 |        | -1 |       |  1 |                                            |  3 |        | -1 |       |  0 |                                              |  3 |        |  0 |                                            | -2 |        |  1 |                                            |  1 |        |  2 |                                            |  2 |        | -1 |                                              |  2 |
│ │ │ +o4 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0 |}, {1, |  1 |}, {2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | -1 |}, {1, | -1 |}, {2, |  1 |}, {4, | 1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{2, | 1 |}, {3, | -1 |}, {4, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  0 |}, {1, | -1 |}, {3, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {3, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, |  0 |}, {2, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{1, |  0 |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{2, | 1 |}, {3, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {3, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{0, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ +                                                          | -2 |        | -1 |       |  1 |       |  2 |                                              | -3 |        |  1 |       |  2 |       |  1 |       | 0 |                                            |  2 |        | 0 |       |  2 |       | -1 |                                            | -1 |        | -1 |       |  1 |       | -1 |                                              | -1 |        |  2 |       | -1 |                                            | -3 |        | -1 |       | -1 |                                            | -1 |        | -1 |       |  2 |                                            |  1 |        | 0 |       | -1 |                                            |  3 |        |  1 |       |  1 |                                              | -2 |        | -1 |                                            |  1 |        |  1 |                                            | -2 |        | -1 |                                            |  1 |        |  1 |                                              | -1 |
│ │ │ +                                                          |  1 |        |  2 |       | -1 |       | -1 |                                              |  1 |        |  1 |       | -1 |       | -1 |       | 1 |                                            | -1 |        | 1 |       | -1 |       |  2 |                                            |  2 |        |  2 |       |  1 |       |  0 |                                              |  3 |        | -1 |       |  0 |                                            |  2 |        |  2 |       |  0 |                                            | -1 |        |  2 |       | -1 |                                            |  1 |        | 1 |       |  2 |                                            | -2 |        | -1 |       |  1 |                                              |  3 |        |  0 |                                            | -2 |        |  1 |                                            |  1 |        |  2 |                                            |  2 |        | -1 |                                              |  2 |
│ │ │  
│ │ │  o4 : HasseDiagram
│ │ │  
│ │ │  i5 : hasseDiagramToGraph(myInterval)
│ │ │  
│ │ │ -o5 = HasseGraph{{{, {{, 0}, {, 1}, {, 2}}}}, {{, {{, 0}, {, 1}, {, 3}, {, 4}}}, {, {{, 1}, {, 2}, {, 3}}}, {, {{, 0}, {, 2}, {, 4}}}}, {{, {{, 0}, {, 2}}}, {, {{, 1}, {, 2}}}, {, {{, 2}, {, 3}}}, {, {{, 1}, {, 3}}}, {, {{, 0}, {, 3}}}}, {{, {{, 0}}}, {, {{, 0}}}, {, {{, 0}}}, {, {{, 0}}}}, {{, {}}}}
│ │ │ +o5 = HasseGraph{{{, {{, 0}, {, 1}, {, 2}}}}, {{, {{, 0}, {, 1}, {, 2}, {, 4}}}, {, {{, 2}, {, 3}, {, 4}}}, {, {{, 0}, {, 1}, {, 3}}}}, {{, {{, 0}, {, 3}}}, {, {{, 0}, {, 2}}}, {, {{, 1}, {, 2}}}, {, {{, 2}, {, 3}}}, {, {{, 1}, {, 3}}}}, {{, {{, 0}}}, {, {{, 0}}}, {, {{, 0}}}, {, {{, 0}}}}, {{, {}}}}
│ │ │  
│ │ │  o5 : HasseGraph
│ │ │  
│ │ │  i6 : hasseDiagramToGraph(myInterval,"labels"=>"reduced decomposition")
│ │ │  
│ │ │ -o6 = HasseGraph{{{12132, {{3, 0}, {121, 1}, {2, 2}}}}, {{2132, {{2, 0}, {121, 1}, {12321, 3}, {232, 4}}}, {1232, {{12321, 1}, {2, 2}, {3, 3}}}, {1213, {{232, 0}, {1, 2}, {3, 4}}}}, {{213, {{3, 0}, {1, 2}}}, {232, {{3, 1}, {2, 2}}}, {123, {{12321, 2}, {3, 3}}}, {132, {{121, 1}, {232, 3}}}, {121, {{2, 0}, {1, 3}}}}, {{21, {{1, 0}}}, {32, {{232, 0}}}, {23, {{3, 0}}}, {12, {{121, 0}}}}, {{2, {}}}}
│ │ │ +o6 = HasseGraph{{{12132, {{3, 0}, {121, 1}, {2, 2}}}}, {{2132, {{232, 0}, {2, 1}, {121, 2}, {12321, 4}}}, {1232, {{12321, 2}, {2, 3}, {3, 4}}}, {1213, {{3, 0}, {232, 1}, {1, 3}}}}, {{121, {{2, 0}, {1, 3}}}, {213, {{3, 0}, {1, 2}}}, {232, {{3, 1}, {2, 2}}}, {123, {{12321, 2}, {3, 3}}}, {132, {{121, 1}, {232, 3}}}}, {{21, {{1, 0}}}, {32, {{232, 0}}}, {23, {{3, 0}}}, {12, {{121, 0}}}}, {{2, {}}}}
│ │ │  
│ │ │  o6 : HasseGraph
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/example-output/_interval__Bruhat_lp__Weyl__Group__Left__Coset_cm__Weyl__Group__Left__Coset_rp.out
│ │ │ @@ -26,30 +26,30 @@
│ │ │                                             | -2 |
│ │ │                                             |  1 |
│ │ │  
│ │ │  o4 : WeylGroupElement
│ │ │  
│ │ │  i5 : myInterval=intervalBruhat(w1 % P,w2 % P)
│ │ │  
│ │ │ -o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | 1 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {2, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ -                                                          | -3 |        | 0 |       |  1 |                                              |  3 |        |  1 |       |  1 |                                            | -1 |        |  2 |       | -1 |                                              | -2 |        | -1 |                                            |  1 |        |  1 |                                            |  1 |        |  1 |                                              | -1 |
│ │ │ -                                                          |  1 |        | 1 |       |  1 |                                              | -2 |        | -1 |       |  1 |                                            |  3 |        | -1 |       |  0 |                                              |  3 |        |  0 |                                            | -2 |        |  1 |                                            |  2 |        | -1 |                                              |  2 |
│ │ │ +o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | -1 |}, {1, | 1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {2, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ +                                                          | -3 |        |  1 |       | 0 |                                              | -1 |        |  2 |       | -1 |                                            |  3 |        |  1 |       |  1 |                                              | -2 |        | -1 |                                            |  1 |        |  1 |                                            |  1 |        |  1 |                                              | -1 |
│ │ │ +                                                          |  1 |        |  1 |       | 1 |                                              |  3 |        | -1 |       |  0 |                                            | -2 |        | -1 |       |  1 |                                              |  3 |        |  0 |                                            | -2 |        |  1 |                                            |  2 |        | -1 |                                              |  2 |
│ │ │  
│ │ │  o5 : HasseDiagram
│ │ │  
│ │ │  i6 : myInterval#1
│ │ │  
│ │ │ -o6 = {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {2, | -1
│ │ │ -                                             |  3 |        |  1 |       |  1
│ │ │ -                                             | -2 |        | -1 |       |  1
│ │ │ +o6 = {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {2, |  2
│ │ │ +                                             | -1 |        |  2 |       | -1
│ │ │ +                                             |  3 |        | -1 |       |  0
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {2,
│ │ │ -     |                                            | -1 |        |  2 |      
│ │ │ -     |                                            |  3 |        | -1 |      
│ │ │ +     |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {2,
│ │ │ +     |                                            |  3 |        |  1 |      
│ │ │ +     |                                            | -2 |        | -1 |      
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |  2 |}}}}
│ │ │ -     | -1 |
│ │ │ -     |  0 |
│ │ │ +     | -1 |}}}}
│ │ │ +     |  1 |
│ │ │ +     |  1 |
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/example-output/_interval__Bruhat_lp__Weyl__Group__Right__Coset_cm__Weyl__Group__Right__Coset_rp.out
│ │ │ @@ -26,17 +26,17 @@
│ │ │                                             | -2 |
│ │ │                                             |  1 |
│ │ │  
│ │ │  o4 : WeylGroupElement
│ │ │  
│ │ │  i5 : myInterval=intervalBruhat(P % w1,P % w2)
│ │ │  
│ │ │ -o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{1, | -1 |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  2 |}, {1, |  0 |}, {2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, |  0 |}, {2, | 1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  2 |}, {1, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1, |  0 |}, {2, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{0, |  0 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ -                                                          | -2 |        | -1 |       |  2 |                                              | -3 |        |  1 |       |  2 |                                            | -1 |        | -1 |       | -1 |       |  1 |                                              |  1 |        | -1 |       | 0 |                                            | -1 |        | -1 |       |  2 |                                            | -3 |        | -1 |       | -1 |                                              |  1 |        |  1 |                                            | -2 |        | -1 |                                            | -2 |        | -1 |                                              | -1 |
│ │ │ -                                                          |  1 |        |  2 |       | -1 |                                              |  1 |        |  1 |       | -1 |                                            |  2 |        |  0 |       |  2 |       |  1 |                                              |  1 |        |  2 |       | 1 |                                            |  3 |        |  0 |       | -1 |                                            |  2 |        |  2 |       |  0 |                                              |  2 |        | -1 |                                            |  3 |        |  0 |                                            |  1 |        |  2 |                                              |  2 |
│ │ │ +o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{1, | -1 |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  2 |}, {1, |  0 |}, {2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, | 1 |}, {1, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1, |  2 |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, |  2 |}, {2, |  0 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  3 |}, {{0, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ +                                                          | -2 |        | -1 |       |  2 |                                              | -3 |        |  1 |       |  2 |                                            | -1 |        | -1 |       | -1 |       |  1 |                                              |  1 |        | 0 |       | -1 |                                            | -1 |        | -1 |       |  2 |                                            | -3 |        | -1 |       | -1 |                                              | -2 |        | -1 |                                            |  1 |        |  1 |                                            | -2 |        | -1 |                                              | -1 |
│ │ │ +                                                          |  1 |        |  2 |       | -1 |                                              |  1 |        |  1 |       | -1 |                                            |  2 |        |  0 |       |  2 |       |  1 |                                              |  1 |        | 1 |       |  2 |                                            |  3 |        |  0 |       | -1 |                                            |  2 |        |  0 |       |  2 |                                              |  1 |        |  2 |                                            |  2 |        | -1 |                                            |  3 |        |  0 |                                              |  2 |
│ │ │  
│ │ │  o5 : HasseDiagram
│ │ │  
│ │ │  i6 : myInterval#1
│ │ │  
│ │ │  o6 = {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{1, | -1 |}, {2, | -1
│ │ │                                               | -3 |        |  1 |       |  2
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/example-output/_poincare__Series_lp__Hasse__Diagram_cm__Ring__Element_rp.out
│ │ │ @@ -4,16 +4,16 @@
│ │ │  
│ │ │  o1 = RootSystem{...8...}
│ │ │  
│ │ │  o1 : RootSystem
│ │ │  
│ │ │  i2 : H=intervalBruhat(neutralWeylGroupElement R, longestWeylGroupElement R)
│ │ │  
│ │ │ -o2 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {1, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | 1 |}, {1, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {1, | 1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | 1 |}, {}}}}
│ │ │ -                                                          | -1 |        |  2 |       | -1 |                                              | -2 |        | 1 |       | -1 |                                            |  1 |        |  2 |       | 1 |                                              |  2 |        | -1 |                                            | -1 |        |  2 |                                              | 1 |
│ │ │ +o2 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  2 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | 1 |}, {1, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}, {1, | 1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | 1 |}, {}}}}
│ │ │ +                                                          | -1 |        | -1 |       |  2 |                                              |  1 |        | 1 |       |  2 |                                            | -2 |        | -1 |       | 1 |                                              | -1 |        |  2 |                                            |  2 |        | -1 |                                              | 1 |
│ │ │  
│ │ │  o2 : HasseDiagram
│ │ │  
│ │ │  i3 : ZZ[x]
│ │ │  
│ │ │  o3 = ZZ[x]
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/example-output/_positive__Roots_lp__Root__System_cm__Parabolic_rp.out
│ │ │ @@ -10,15 +10,15 @@
│ │ │  
│ │ │  o2 = set {1, 2}
│ │ │  
│ │ │  o2 : Parabolic
│ │ │  
│ │ │  i3 : positiveRoots(R,P)
│ │ │  
│ │ │ -o3 = set {|  2 |, | -1 |, |  1 |}
│ │ │ -          | -1 |  |  2 |  |  1 |
│ │ │ -          |  0 |  | -1 |  | -1 |
│ │ │ +o3 = set {|  1 |, |  2 |, | -1 |}
│ │ │ +          |  1 |  | -1 |  |  2 |
│ │ │ +          | -1 |  |  0 |  | -1 |
│ │ │            |  0 |  |  0 |  |  0 |
│ │ │  
│ │ │  o3 : Set
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/example-output/_positive__Roots_lp__Root__System_rp.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 1330744940387
│ │ │  
│ │ │  i1 : positiveRoots(rootSystemA(3))
│ │ │  
│ │ │ -o1 = set {|  0 |, | -1 |, | -1 |, | 1 |, |  1 |, |  2 |}
│ │ │ -          | -1 |  |  1 |  |  2 |  | 0 |  |  1 |  | -1 |
│ │ │ -          |  2 |  |  1 |  | -1 |  | 1 |  | -1 |  |  0 |
│ │ │ +o1 = set {|  2 |, |  0 |, | -1 |, | -1 |, | 1 |, |  1 |}
│ │ │ +          | -1 |  | -1 |  |  1 |  |  2 |  | 0 |  |  1 |
│ │ │ +          |  0 |  |  2 |  |  1 |  | -1 |  | 1 |  | -1 |
│ │ │  
│ │ │  o1 : Set
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/example-output/_under__Bruhat_lp__Basic__List_rp.out
│ │ │ @@ -36,34 +36,34 @@
│ │ │                                             | -2 |
│ │ │                                             | -1 |
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : underBruhat(L1)
│ │ │  
│ │ │ -o4 = {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  0 |}, {2, |  2
│ │ │ -                                             | -3 |        | -1 |       | -1
│ │ │ -                                             |  1 |        |  2 |       |  0
│ │ │ +o4 = {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1, |  0 |}, {2, | -1
│ │ │ +                                             |  2 |        | -1 |       |  1
│ │ │ +                                             | -3 |        |  2 |       |  1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1, | -1 |}, {2,
│ │ │ -     |                                            | -1 |        |  1 |      
│ │ │ -     |                                            | -2 |        |  1 |      
│ │ │ +     |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  0 |}, {2,
│ │ │ +     |                                            | -3 |        | -1 |      
│ │ │ +     |                                            |  1 |        |  2 |      
│ │ │       ------------------------------------------------------------------------
│ │ │ -     | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, |  1 |},
│ │ │ -     |  2 |                                            |  2 |        |  1 |  
│ │ │ -     | -1 |                                            | -1 |        | -1 |  
│ │ │ +     |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1, | -1 |},
│ │ │ +     | -1 |                                            | -1 |        |  1 |  
│ │ │ +     |  0 |                                            | -2 |        |  1 |  
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1
│ │ │ -         | -1 |                                            | -1 |        |  2
│ │ │ -         |  0 |                                            |  2 |        | -1
│ │ │ +     {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, |  1
│ │ │ +         |  2 |                                            |  2 |        |  1
│ │ │ +         | -1 |                                            | -1 |        | -1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}, {1, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1,
│ │ │ -     |       |  1 |                                            |  2 |       
│ │ │ -     |       | -1 |                                            | -3 |       
│ │ │ +     |}, {1, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0,
│ │ │ +     |       | -1 |                                            | -1 |       
│ │ │ +     |       |  0 |                                            |  2 |       
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |  0 |}, {2, | -1 |}}}}
│ │ │ -     | -1 |       |  1 |
│ │ │ +     | -1 |}, {1, |  1 |}}}}
│ │ │       |  2 |       |  1 |
│ │ │ +     | -1 |       | -1 |
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/html/_above__Bruhat_lp__Basic__List_rp.html
│ │ │ @@ -116,37 +116,37 @@
│ │ │  o3 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : aboveBruhat(L1)
│ │ │  
│ │ │ -o4 = {{WeylGroupElement{RootSystem{...8...}, |  3 |}, {{1, |  0 |}, {2, | -1
│ │ │ -                                             | -2 |        | -1 |       |  1
│ │ │ -                                             |  1 |        |  2 |       |  1
│ │ │ +o4 = {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1, | -1 |}, {2, | -1
│ │ │ +                                             |  1 |        |  1 |       |  2
│ │ │ +                                             | -2 |        |  1 |       | -1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {1,
│ │ │ +     |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{1, |  0 |}, {2,
│ │ │ +     |                                            | -2 |        | -1 |      
│ │ │       |                                            |  1 |        |  2 |      
│ │ │ -     |                                            |  2 |        | -1 |      
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0 |},
│ │ │ -     |  1 |                                            |  3 |        | -1 |  
│ │ │ -     | -1 |                                            | -1 |        |  2 |  
│ │ │ +     | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |},
│ │ │ +     |  1 |                                            |  1 |        |  2 |  
│ │ │ +     |  1 |                                            |  2 |        | -1 |  
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  1
│ │ │ -         | -1 |                                            | -2 |        |  1
│ │ │ -         |  0 |                                            |  3 |        | -1
│ │ │ +     {1, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0
│ │ │ +         |  1 |                                            |  3 |        | -1
│ │ │ +         | -1 |                                            | -1 |        |  2
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}, {1, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1,
│ │ │ -     |       | -1 |                                            |  1 |       
│ │ │ -     |       |  0 |                                            | -2 |       
│ │ │ +     |}, {2, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0,
│ │ │ +     |       | -1 |                                            | -2 |       
│ │ │ +     |       |  0 |                                            |  3 |       
│ │ │       ------------------------------------------------------------------------
│ │ │ -     | -1 |}, {2, | -1 |}}}}
│ │ │ -     |  1 |       |  2 |
│ │ │ +     |  1 |}, {1, |  2 |}}}}
│ │ │       |  1 |       | -1 |
│ │ │ +     | -1 |       |  0 |
│ │ │  
│ │ │  o4 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -48,37 +48,37 @@
│ │ │ │       WeylGroupElement{RootSystem{...8...}, |  1 |}}
│ │ │ │                                             |  2 |
│ │ │ │                                             | -1 |
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : aboveBruhat(L1)
│ │ │ │  
│ │ │ │ -o4 = {{WeylGroupElement{RootSystem{...8...}, |  3 |}, {{1, |  0 |}, {2, | -1
│ │ │ │ -                                             | -2 |        | -1 |       |  1
│ │ │ │ -                                             |  1 |        |  2 |       |  1
│ │ │ │ +o4 = {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1, | -1 |}, {2, | -1
│ │ │ │ +                                             |  1 |        |  1 |       |  2
│ │ │ │ +                                             | -2 |        |  1 |       | -1
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {1,
│ │ │ │ +     |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{1, |  0 |}, {2,
│ │ │ │ +     |                                            | -2 |        | -1 |
│ │ │ │       |                                            |  1 |        |  2 |
│ │ │ │ -     |                                            |  2 |        | -1 |
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0 |},
│ │ │ │ -     |  1 |                                            |  3 |        | -1 |
│ │ │ │ -     | -1 |                                            | -1 |        |  2 |
│ │ │ │ +     | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |},
│ │ │ │ +     |  1 |                                            |  1 |        |  2 |
│ │ │ │ +     |  1 |                                            |  2 |        | -1 |
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  1
│ │ │ │ -         | -1 |                                            | -2 |        |  1
│ │ │ │ -         |  0 |                                            |  3 |        | -1
│ │ │ │ +     {1, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0
│ │ │ │ +         |  1 |                                            |  3 |        | -1
│ │ │ │ +         | -1 |                                            | -1 |        |  2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     |}, {1, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1,
│ │ │ │ -     |       | -1 |                                            |  1 |
│ │ │ │ -     |       |  0 |                                            | -2 |
│ │ │ │ +     |}, {2, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0,
│ │ │ │ +     |       | -1 |                                            | -2 |
│ │ │ │ +     |       |  0 |                                            |  3 |
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     | -1 |}, {2, | -1 |}}}}
│ │ │ │ -     |  1 |       |  2 |
│ │ │ │ +     |  1 |}, {1, |  2 |}}}}
│ │ │ │       |  1 |       | -1 |
│ │ │ │ +     | -1 |       |  0 |
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _a_b_o_v_e_B_r_u_h_a_t_(_B_a_s_i_c_L_i_s_t_) -- The Weyl group elements just under the ones in
│ │ │ │        the list for the Bruhat order
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/html/_eval_lp__Root__System_cm__Weight_cm__Root_rp.html
│ │ │ @@ -80,16 +80,16 @@
│ │ │  o1 : RootSystem
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : L=toList(positiveRoots(R))
│ │ │  
│ │ │ -o2 = {| 1 |, |  2 |, | -1 |}
│ │ │ -      | 1 |  | -1 |  |  2 |
│ │ │ +o2 = {| -1 |, | 1 |, |  2 |}
│ │ │ +      |  2 |  | 1 |  | -1 |
│ │ │  
│ │ │  o2 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : v=weight(R,{1,2})
│ │ │ @@ -101,29 +101,29 @@
│ │ │  o3 : ZZ
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : eval(R,v,L#0)
│ │ │  
│ │ │ -o4 = 3
    
│ │ │ +o4 = 2
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : eval(R,v,L#1)
│ │ │  
│ │ │ -o5 = 1
    
│ │ │ +o5 = 3
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : eval(R,v,L#2)
│ │ │  
│ │ │ -o6 = 2
    
│ │ │ +o6 = 1
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │          
    
│ │ │            
    Ways to use this method:
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,32 +18,32 @@
│ │ │ │  i1 : R=rootSystemA(2)
│ │ │ │  
│ │ │ │  o1 = RootSystem{...8...}
│ │ │ │  
│ │ │ │  o1 : RootSystem
│ │ │ │  i2 : L=toList(positiveRoots(R))
│ │ │ │  
│ │ │ │ -o2 = {| 1 |, |  2 |, | -1 |}
│ │ │ │ -      | 1 |  | -1 |  |  2 |
│ │ │ │ +o2 = {| -1 |, | 1 |, |  2 |}
│ │ │ │ +      |  2 |  | 1 |  | -1 |
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  i3 : v=weight(R,{1,2})
│ │ │ │  
│ │ │ │  o3 = | 1 |
│ │ │ │       | 2 |
│ │ │ │  
│ │ │ │         2
│ │ │ │  o3 : ZZ
│ │ │ │  i4 : eval(R,v,L#0)
│ │ │ │  
│ │ │ │ -o4 = 3
│ │ │ │ +o4 = 2
│ │ │ │  i5 : eval(R,v,L#1)
│ │ │ │  
│ │ │ │ -o5 = 1
│ │ │ │ +o5 = 3
│ │ │ │  i6 : eval(R,v,L#2)
│ │ │ │  
│ │ │ │ -o6 = 2
│ │ │ │ +o6 = 1
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _e_v_a_l_(_R_o_o_t_S_y_s_t_e_m_,_W_e_i_g_h_t_,_R_o_o_t_) -- evaluate the dual of a root at a Weight
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/WeylGroups.m2:2488:0.
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/html/_eval_lp__Root__System_cm__Z__Z_cm__Root_rp.html
│ │ │ @@ -80,39 +80,39 @@
│ │ │  o1 : RootSystem
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : L=toList(positiveRoots(R))
│ │ │  
│ │ │ -o2 = {| 1 |, |  2 |, | -1 |}
│ │ │ -      | 1 |  | -1 |  |  2 |
│ │ │ +o2 = {| -1 |, | 1 |, |  2 |}
│ │ │ +      |  2 |  | 1 |  | -1 |
│ │ │  
│ │ │  o2 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : eval(R,1,L#0)
│ │ │  
│ │ │ -o3 = 1
    
│ │ │ +o3 = 0
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : eval(R,1,L#1)
│ │ │  
│ │ │  o4 = 1
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : eval(R,1,L#2)
│ │ │  
│ │ │ -o5 = 0
    
│ │ │ +o5 = 1
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │          
    
│ │ │            
    Ways to use this method:
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,26 +19,26 @@
│ │ │ │  i1 : R=rootSystemA(2)
│ │ │ │  
│ │ │ │  o1 = RootSystem{...8...}
│ │ │ │  
│ │ │ │  o1 : RootSystem
│ │ │ │  i2 : L=toList(positiveRoots(R))
│ │ │ │  
│ │ │ │ -o2 = {| 1 |, |  2 |, | -1 |}
│ │ │ │ -      | 1 |  | -1 |  |  2 |
│ │ │ │ +o2 = {| -1 |, | 1 |, |  2 |}
│ │ │ │ +      |  2 |  | 1 |  | -1 |
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  i3 : eval(R,1,L#0)
│ │ │ │  
│ │ │ │ -o3 = 1
│ │ │ │ +o3 = 0
│ │ │ │  i4 : eval(R,1,L#1)
│ │ │ │  
│ │ │ │  o4 = 1
│ │ │ │  i5 : eval(R,1,L#2)
│ │ │ │  
│ │ │ │ -o5 = 0
│ │ │ │ +o5 = 1
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _e_v_a_l_(_R_o_o_t_S_y_s_t_e_m_,_Z_Z_,_R_o_o_t_) -- evaluate the dual of a root at a fundamental
│ │ │ │        weight
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/WeylGroups.m2:2512:0.
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/html/_hasse__Diagram__To__Graph_lp__Hasse__Diagram_rp.html
│ │ │ @@ -107,40 +107,40 @@
│ │ │  o3 : WeylGroupElement
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : myInterval=intervalBruhat(w1,w2)
│ │ │  
│ │ │ -o4 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0 |}, {1, |  1 |}, {2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | -1 |}, {1, |  1 |}, {3, | 1 |}, {4, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{1, | 1 |}, {2, | -1 |}, {3, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {2, |  2 |}, {4, |  0 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, |  0 |}, {2, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{1, |  0 |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{2, | 1 |}, {3, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {3, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {3, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{0, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ -                                                          | -2 |        | -1 |       |  1 |       |  2 |                                              | -3 |        |  2 |       |  1 |       | 0 |       |  1 |                                            |  2 |        | 0 |       |  2 |       | -1 |                                            | -1 |        |  1 |       | -1 |       | -1 |                                              | -3 |        | -1 |       | -1 |                                            | -1 |        | -1 |       |  2 |                                            |  1 |        | 0 |       | -1 |                                            |  3 |        |  1 |       |  1 |                                            | -1 |        |  2 |       | -1 |                                              | -2 |        | -1 |                                            |  1 |        |  1 |                                            | -2 |        | -1 |                                            |  1 |        |  1 |                                              | -1 |
│ │ │ -                                                          |  1 |        |  2 |       | -1 |       | -1 |                                              |  1 |        | -1 |       | -1 |       | 1 |       |  1 |                                            | -1 |        | 1 |       | -1 |       |  2 |                                            |  2 |        |  1 |       |  0 |       |  2 |                                              |  2 |        |  2 |       |  0 |                                            | -1 |        |  2 |       | -1 |                                            |  1 |        | 1 |       |  2 |                                            | -2 |        | -1 |       |  1 |                                            |  3 |        | -1 |       |  0 |                                              |  3 |        |  0 |                                            | -2 |        |  1 |                                            |  1 |        |  2 |                                            |  2 |        | -1 |                                              |  2 |
│ │ │ +o4 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0 |}, {1, |  1 |}, {2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | -1 |}, {1, | -1 |}, {2, |  1 |}, {4, | 1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{2, | 1 |}, {3, | -1 |}, {4, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  0 |}, {1, | -1 |}, {3, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {3, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, |  0 |}, {2, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{1, |  0 |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{2, | 1 |}, {3, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {3, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{0, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ +                                                          | -2 |        | -1 |       |  1 |       |  2 |                                              | -3 |        |  1 |       |  2 |       |  1 |       | 0 |                                            |  2 |        | 0 |       |  2 |       | -1 |                                            | -1 |        | -1 |       |  1 |       | -1 |                                              | -1 |        |  2 |       | -1 |                                            | -3 |        | -1 |       | -1 |                                            | -1 |        | -1 |       |  2 |                                            |  1 |        | 0 |       | -1 |                                            |  3 |        |  1 |       |  1 |                                              | -2 |        | -1 |                                            |  1 |        |  1 |                                            | -2 |        | -1 |                                            |  1 |        |  1 |                                              | -1 |
│ │ │ +                                                          |  1 |        |  2 |       | -1 |       | -1 |                                              |  1 |        |  1 |       | -1 |       | -1 |       | 1 |                                            | -1 |        | 1 |       | -1 |       |  2 |                                            |  2 |        |  2 |       |  1 |       |  0 |                                              |  3 |        | -1 |       |  0 |                                            |  2 |        |  2 |       |  0 |                                            | -1 |        |  2 |       | -1 |                                            |  1 |        | 1 |       |  2 |                                            | -2 |        | -1 |       |  1 |                                              |  3 |        |  0 |                                            | -2 |        |  1 |                                            |  1 |        |  2 |                                            |  2 |        | -1 |                                              |  2 |
│ │ │  
│ │ │  o4 : HasseDiagram
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : hasseDiagramToGraph(myInterval)
│ │ │  
│ │ │ -o5 = HasseGraph{{{, {{, 0}, {, 1}, {, 2}}}}, {{, {{, 0}, {, 1}, {, 3}, {, 4}}}, {, {{, 1}, {, 2}, {, 3}}}, {, {{, 0}, {, 2}, {, 4}}}}, {{, {{, 0}, {, 2}}}, {, {{, 1}, {, 2}}}, {, {{, 2}, {, 3}}}, {, {{, 1}, {, 3}}}, {, {{, 0}, {, 3}}}}, {{, {{, 0}}}, {, {{, 0}}}, {, {{, 0}}}, {, {{, 0}}}}, {{, {}}}}
│ │ │ +o5 = HasseGraph{{{, {{, 0}, {, 1}, {, 2}}}}, {{, {{, 0}, {, 1}, {, 2}, {, 4}}}, {, {{, 2}, {, 3}, {, 4}}}, {, {{, 0}, {, 1}, {, 3}}}}, {{, {{, 0}, {, 3}}}, {, {{, 0}, {, 2}}}, {, {{, 1}, {, 2}}}, {, {{, 2}, {, 3}}}, {, {{, 1}, {, 3}}}}, {{, {{, 0}}}, {, {{, 0}}}, {, {{, 0}}}, {, {{, 0}}}}, {{, {}}}}
│ │ │  
│ │ │  o5 : HasseGraph
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    It is also possible to ask for reduced decompositions as labels by changing the option "labels" as below.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i6 : hasseDiagramToGraph(myInterval,"labels"=>"reduced decomposition")
│ │ │  
│ │ │ -o6 = HasseGraph{{{12132, {{3, 0}, {121, 1}, {2, 2}}}}, {{2132, {{2, 0}, {121, 1}, {12321, 3}, {232, 4}}}, {1232, {{12321, 1}, {2, 2}, {3, 3}}}, {1213, {{232, 0}, {1, 2}, {3, 4}}}}, {{213, {{3, 0}, {1, 2}}}, {232, {{3, 1}, {2, 2}}}, {123, {{12321, 2}, {3, 3}}}, {132, {{121, 1}, {232, 3}}}, {121, {{2, 0}, {1, 3}}}}, {{21, {{1, 0}}}, {32, {{232, 0}}}, {23, {{3, 0}}}, {12, {{121, 0}}}}, {{2, {}}}}
│ │ │ +o6 = HasseGraph{{{12132, {{3, 0}, {121, 1}, {2, 2}}}}, {{2132, {{232, 0}, {2, 1}, {121, 2}, {12321, 4}}}, {1232, {{12321, 2}, {2, 3}, {3, 4}}}, {1213, {{3, 0}, {232, 1}, {1, 3}}}}, {{121, {{2, 0}, {1, 3}}}, {213, {{3, 0}, {1, 2}}}, {232, {{3, 1}, {2, 2}}}, {123, {{12321, 2}, {3, 3}}}, {132, {{121, 1}, {232, 3}}}}, {{21, {{1, 0}}}, {32, {{232, 0}}}, {23, {{3, 0}}}, {12, {{121, 0}}}}, {{2, {}}}}
│ │ │  
│ │ │  o6 : HasseGraph
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,72 +35,73 @@
│ │ │ │                                             |  1 |
│ │ │ │  
│ │ │ │  o3 : WeylGroupElement
│ │ │ │  i4 : myInterval=intervalBruhat(w1,w2)
│ │ │ │  
│ │ │ │  o4 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0
│ │ │ │  |}, {1, |  1 |}, {2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1
│ │ │ │ -|}, {{0, | -1 |}, {1, |  1 |}, {3, | 1 |}, {4, | -1 |}}}, {WeylGroupElement
│ │ │ │ -{RootSystem{...8...}, | -3 |}, {{1, | 1 |}, {2, | -1 |}, {3, |  0 |}}},
│ │ │ │ -{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {2, |  2 |}, {4,
│ │ │ │ -|  0 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, |  0 |}, {2,
│ │ │ │ -|  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{1, |  0 |}, {2, |
│ │ │ │ --1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{2, | 1 |}, {3, |  0
│ │ │ │ +|}, {{0, | -1 |}, {1, | -1 |}, {2, |  1 |}, {4, | 1 |}}}, {WeylGroupElement
│ │ │ │ +{RootSystem{...8...}, | -3 |}, {{2, | 1 |}, {3, | -1 |}, {4, |  0 |}}},
│ │ │ │ +{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  0 |}, {1, | -1 |}, {3,
│ │ │ │ +|  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {3,
│ │ │ │ +|  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, |  0 |}, {2, |
│ │ │ │ +2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{1, |  0 |}, {2, | -
│ │ │ │ +1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{2, | 1 |}, {3, |  0
│ │ │ │  |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {3, | -
│ │ │ │ -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {3, |  2
│ │ │ │ -|}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}},
│ │ │ │ +1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}},
│ │ │ │  {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}},
│ │ │ │  {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{0, |  0 |}}},
│ │ │ │  {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}}, {
│ │ │ │  {WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ │                                                            | -2 |        | -1 |
│ │ │ │  |  1 |       |  2 |                                              | -3 |
│ │ │ │ -|  2 |       |  1 |       | 0 |       |  1 |
│ │ │ │ +|  1 |       |  2 |       |  1 |       | 0 |
│ │ │ │  |  2 |        | 0 |       |  2 |       | -1 |
│ │ │ │ -| -1 |        |  1 |       | -1 |       | -1 |
│ │ │ │ -| -3 |        | -1 |       | -1 |                                            |
│ │ │ │ --1 |        | -1 |       |  2 |                                            |  1
│ │ │ │ +| -1 |        | -1 |       |  1 |       | -1 |
│ │ │ │ +| -1 |        |  2 |       | -1 |                                            |
│ │ │ │ +-3 |        | -1 |       | -1 |                                            | -
│ │ │ │ +1 |        | -1 |       |  2 |                                            |  1
│ │ │ │  |        | 0 |       | -1 |                                            |  3 |
│ │ │ │ -|  1 |       |  1 |                                            | -1 |        |
│ │ │ │ -2 |       | -1 |                                              | -2 |        | -
│ │ │ │ -1 |                                            |  1 |        |  1 |
│ │ │ │ +|  1 |       |  1 |                                              | -2 |
│ │ │ │ +| -1 |                                            |  1 |        |  1 |
│ │ │ │  | -2 |        | -1 |                                            |  1 |        |
│ │ │ │  1 |                                              | -1 |
│ │ │ │                                                            |  1 |        |  2 |
│ │ │ │  | -1 |       | -1 |                                              |  1 |
│ │ │ │ -| -1 |       | -1 |       | 1 |       |  1 |
│ │ │ │ +|  1 |       | -1 |       | -1 |       | 1 |
│ │ │ │  | -1 |        | 1 |       | -1 |       |  2 |
│ │ │ │ -|  2 |        |  1 |       |  0 |       |  2 |
│ │ │ │ -|  2 |        |  2 |       |  0 |                                            |
│ │ │ │ --1 |        |  2 |       | -1 |                                            |  1
│ │ │ │ +|  2 |        |  2 |       |  1 |       |  0 |
│ │ │ │ +|  3 |        | -1 |       |  0 |                                            |
│ │ │ │ +2 |        |  2 |       |  0 |                                            | -
│ │ │ │ +1 |        |  2 |       | -1 |                                            |  1
│ │ │ │  |        | 1 |       |  2 |                                            | -2 |
│ │ │ │ -| -1 |       |  1 |                                            |  3 |        |
│ │ │ │ --1 |       |  0 |                                              |  3 |        |
│ │ │ │ -0 |                                            | -2 |        |  1 |
│ │ │ │ +| -1 |       |  1 |                                              |  3 |
│ │ │ │ +|  0 |                                            | -2 |        |  1 |
│ │ │ │  |  1 |        |  2 |                                            |  2 |        |
│ │ │ │  -1 |                                              |  2 |
│ │ │ │  
│ │ │ │  o4 : HasseDiagram
│ │ │ │  i5 : hasseDiagramToGraph(myInterval)
│ │ │ │  
│ │ │ │ -o5 = HasseGraph{{{, {{, 0}, {, 1}, {, 2}}}}, {{, {{, 0}, {, 1}, {, 3}, {, 4}}},
│ │ │ │ -{, {{, 1}, {, 2}, {, 3}}}, {, {{, 0}, {, 2}, {, 4}}}}, {{, {{, 0}, {, 2}}}, {,
│ │ │ │ -{{, 1}, {, 2}}}, {, {{, 2}, {, 3}}}, {, {{, 1}, {, 3}}}, {, {{, 0}, {, 3}}}}, {
│ │ │ │ +o5 = HasseGraph{{{, {{, 0}, {, 1}, {, 2}}}}, {{, {{, 0}, {, 1}, {, 2}, {, 4}}},
│ │ │ │ +{, {{, 2}, {, 3}, {, 4}}}, {, {{, 0}, {, 1}, {, 3}}}}, {{, {{, 0}, {, 3}}}, {,
│ │ │ │ +{{, 0}, {, 2}}}, {, {{, 1}, {, 2}}}, {, {{, 2}, {, 3}}}, {, {{, 1}, {, 3}}}}, {
│ │ │ │  {, {{, 0}}}, {, {{, 0}}}, {, {{, 0}}}, {, {{, 0}}}}, {{, {}}}}
│ │ │ │  
│ │ │ │  o5 : HasseGraph
│ │ │ │  It is also possible to ask for reduced decompositions as labels by changing the
│ │ │ │  option "labels" as below.
│ │ │ │  i6 : hasseDiagramToGraph(myInterval,"labels"=>"reduced decomposition")
│ │ │ │  
│ │ │ │ -o6 = HasseGraph{{{12132, {{3, 0}, {121, 1}, {2, 2}}}}, {{2132, {{2, 0}, {121,
│ │ │ │ -1}, {12321, 3}, {232, 4}}}, {1232, {{12321, 1}, {2, 2}, {3, 3}}}, {1213, {{232,
│ │ │ │ -0}, {1, 2}, {3, 4}}}}, {{213, {{3, 0}, {1, 2}}}, {232, {{3, 1}, {2, 2}}}, {123,
│ │ │ │ -{{12321, 2}, {3, 3}}}, {132, {{121, 1}, {232, 3}}}, {121, {{2, 0}, {1, 3}}}}, {
│ │ │ │ -{21, {{1, 0}}}, {32, {{232, 0}}}, {23, {{3, 0}}}, {12, {{121, 0}}}}, {{2, {}}}}
│ │ │ │ +o6 = HasseGraph{{{12132, {{3, 0}, {121, 1}, {2, 2}}}}, {{2132, {{232, 0}, {2,
│ │ │ │ +1}, {121, 2}, {12321, 4}}}, {1232, {{12321, 2}, {2, 3}, {3, 4}}}, {1213, {{3,
│ │ │ │ +0}, {232, 1}, {1, 3}}}}, {{121, {{2, 0}, {1, 3}}}, {213, {{3, 0}, {1, 2}}},
│ │ │ │ +{232, {{3, 1}, {2, 2}}}, {123, {{12321, 2}, {3, 3}}}, {132, {{121, 1}, {232,
│ │ │ │ +3}}}}, {{21, {{1, 0}}}, {32, {{232, 0}}}, {23, {{3, 0}}}, {12, {{121, 0}}}}, {
│ │ │ │ +{2, {}}}}
│ │ │ │  
│ │ │ │  o6 : HasseGraph
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _h_a_s_s_e_D_i_a_g_r_a_m_T_o_G_r_a_p_h_(_H_a_s_s_e_D_i_a_g_r_a_m_) -- turning a hasse diagram into a graph
│ │ │ │        (intended for graphic representation)
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/html/_interval__Bruhat_lp__Weyl__Group__Left__Coset_cm__Weyl__Group__Left__Coset_rp.html
│ │ │ @@ -110,41 +110,41 @@
│ │ │  o4 : WeylGroupElement
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : myInterval=intervalBruhat(w1 % P,w2 % P)
│ │ │  
│ │ │ -o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | 1 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {2, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ -                                                          | -3 |        | 0 |       |  1 |                                              |  3 |        |  1 |       |  1 |                                            | -1 |        |  2 |       | -1 |                                              | -2 |        | -1 |                                            |  1 |        |  1 |                                            |  1 |        |  1 |                                              | -1 |
│ │ │ -                                                          |  1 |        | 1 |       |  1 |                                              | -2 |        | -1 |       |  1 |                                            |  3 |        | -1 |       |  0 |                                              |  3 |        |  0 |                                            | -2 |        |  1 |                                            |  2 |        | -1 |                                              |  2 |
│ │ │ +o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | -1 |}, {1, | 1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {2, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ +                                                          | -3 |        |  1 |       | 0 |                                              | -1 |        |  2 |       | -1 |                                            |  3 |        |  1 |       |  1 |                                              | -2 |        | -1 |                                            |  1 |        |  1 |                                            |  1 |        |  1 |                                              | -1 |
│ │ │ +                                                          |  1 |        |  1 |       | 1 |                                              |  3 |        | -1 |       |  0 |                                            | -2 |        | -1 |       |  1 |                                              |  3 |        |  0 |                                            | -2 |        |  1 |                                            |  2 |        | -1 |                                              |  2 |
│ │ │  
│ │ │  o5 : HasseDiagram
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    Each row of the Hasse diagram contains the elements of a certain length together with their links to the next row.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i6 : myInterval#1
│ │ │  
│ │ │ -o6 = {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {2, | -1
│ │ │ -                                             |  3 |        |  1 |       |  1
│ │ │ -                                             | -2 |        | -1 |       |  1
│ │ │ +o6 = {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {2, |  2
│ │ │ +                                             | -1 |        |  2 |       | -1
│ │ │ +                                             |  3 |        | -1 |       |  0
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {2,
│ │ │ -     |                                            | -1 |        |  2 |      
│ │ │ -     |                                            |  3 |        | -1 |      
│ │ │ +     |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {2,
│ │ │ +     |                                            |  3 |        |  1 |      
│ │ │ +     |                                            | -2 |        | -1 |      
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |  2 |}}}}
│ │ │ -     | -1 |
│ │ │ -     |  0 |
│ │ │ +     | -1 |}}}}
│ │ │ +     |  1 |
│ │ │ +     |  1 |
│ │ │  
│ │ │  o6 : List
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -40,50 +40,50 @@
│ │ │ │  o4 = WeylGroupElement{RootSystem{...8...}, | -1 |}
│ │ │ │                                             | -2 |
│ │ │ │                                             |  1 |
│ │ │ │  
│ │ │ │  o4 : WeylGroupElement
│ │ │ │  i5 : myInterval=intervalBruhat(w1 % P,w2 % P)
│ │ │ │  
│ │ │ │ -o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | 1 |},
│ │ │ │ -{1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |},
│ │ │ │ -{2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |},
│ │ │ │ -{2, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2
│ │ │ │ +o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | -
│ │ │ │ +1 |}, {1, | 1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -
│ │ │ │ +1 |}, {2, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1
│ │ │ │ +|}, {2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2
│ │ │ │  |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}},
│ │ │ │  {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}}, {
│ │ │ │  {WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ │ -                                                          | -3 |        | 0 |
│ │ │ │ -|  1 |                                              |  3 |        |  1 |
│ │ │ │ -|  1 |                                            | -1 |        |  2 |       |
│ │ │ │ --1 |                                              | -2 |        | -1 |
│ │ │ │ +                                                          | -3 |        |  1 |
│ │ │ │ +| 0 |                                              | -1 |        |  2 |       |
│ │ │ │ +-1 |                                            |  3 |        |  1 |       |  1
│ │ │ │ +|                                              | -2 |        | -1 |
│ │ │ │  |  1 |        |  1 |                                            |  1 |        |
│ │ │ │  1 |                                              | -1 |
│ │ │ │ -                                                          |  1 |        | 1 |
│ │ │ │ -|  1 |                                              | -2 |        | -1 |
│ │ │ │ -|  1 |                                            |  3 |        | -1 |       |
│ │ │ │ -0 |                                              |  3 |        |  0 |
│ │ │ │ +                                                          |  1 |        |  1 |
│ │ │ │ +| 1 |                                              |  3 |        | -1 |       |
│ │ │ │ +0 |                                            | -2 |        | -1 |       |  1
│ │ │ │ +|                                              |  3 |        |  0 |
│ │ │ │  | -2 |        |  1 |                                            |  2 |        |
│ │ │ │  -1 |                                              |  2 |
│ │ │ │  
│ │ │ │  o5 : HasseDiagram
│ │ │ │  Each row of the Hasse diagram contains the elements of a certain length
│ │ │ │  together with their links to the next row.
│ │ │ │  i6 : myInterval#1
│ │ │ │  
│ │ │ │ -o6 = {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {2, | -1
│ │ │ │ -                                             |  3 |        |  1 |       |  1
│ │ │ │ -                                             | -2 |        | -1 |       |  1
│ │ │ │ +o6 = {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {2, |  2
│ │ │ │ +                                             | -1 |        |  2 |       | -1
│ │ │ │ +                                             |  3 |        | -1 |       |  0
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {2,
│ │ │ │ -     |                                            | -1 |        |  2 |
│ │ │ │ -     |                                            |  3 |        | -1 |
│ │ │ │ +     |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {2,
│ │ │ │ +     |                                            |  3 |        |  1 |
│ │ │ │ +     |                                            | -2 |        | -1 |
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     |  2 |}}}}
│ │ │ │ -     | -1 |
│ │ │ │ -     |  0 |
│ │ │ │ +     | -1 |}}}}
│ │ │ │ +     |  1 |
│ │ │ │ +     |  1 |
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _i_n_t_e_r_v_a_l_B_r_u_h_a_t_(_W_e_y_l_G_r_o_u_p_L_e_f_t_C_o_s_e_t_,_W_e_y_l_G_r_o_u_p_L_e_f_t_C_o_s_e_t_) -- elements between
│ │ │ │        two given ones for the Bruhat order on a quotient of a Weyl group
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/html/_interval__Bruhat_lp__Weyl__Group__Right__Coset_cm__Weyl__Group__Right__Coset_rp.html
│ │ │ @@ -110,17 +110,17 @@
│ │ │  o4 : WeylGroupElement
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : myInterval=intervalBruhat(P % w1,P % w2)
│ │ │  
│ │ │ -o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{1, | -1 |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  2 |}, {1, |  0 |}, {2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, |  0 |}, {2, | 1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  2 |}, {1, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1, |  0 |}, {2, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{0, |  0 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ -                                                          | -2 |        | -1 |       |  2 |                                              | -3 |        |  1 |       |  2 |                                            | -1 |        | -1 |       | -1 |       |  1 |                                              |  1 |        | -1 |       | 0 |                                            | -1 |        | -1 |       |  2 |                                            | -3 |        | -1 |       | -1 |                                              |  1 |        |  1 |                                            | -2 |        | -1 |                                            | -2 |        | -1 |                                              | -1 |
│ │ │ -                                                          |  1 |        |  2 |       | -1 |                                              |  1 |        |  1 |       | -1 |                                            |  2 |        |  0 |       |  2 |       |  1 |                                              |  1 |        |  2 |       | 1 |                                            |  3 |        |  0 |       | -1 |                                            |  2 |        |  2 |       |  0 |                                              |  2 |        | -1 |                                            |  3 |        |  0 |                                            |  1 |        |  2 |                                              |  2 |
│ │ │ +o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{1, | -1 |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  2 |}, {1, |  0 |}, {2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, | 1 |}, {1, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1, |  2 |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, |  2 |}, {2, |  0 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  3 |}, {{0, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ +                                                          | -2 |        | -1 |       |  2 |                                              | -3 |        |  1 |       |  2 |                                            | -1 |        | -1 |       | -1 |       |  1 |                                              |  1 |        | 0 |       | -1 |                                            | -1 |        | -1 |       |  2 |                                            | -3 |        | -1 |       | -1 |                                              | -2 |        | -1 |                                            |  1 |        |  1 |                                            | -2 |        | -1 |                                              | -1 |
│ │ │ +                                                          |  1 |        |  2 |       | -1 |                                              |  1 |        |  1 |       | -1 |                                            |  2 |        |  0 |       |  2 |       |  1 |                                              |  1 |        | 1 |       |  2 |                                            |  3 |        |  0 |       | -1 |                                            |  2 |        |  0 |       |  2 |                                              |  1 |        |  2 |                                            |  2 |        | -1 |                                            |  3 |        |  0 |                                              |  2 |
│ │ │  
│ │ │  o5 : HasseDiagram
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    Each row of the Hasse diagram contains the elements of a certain length together with their links to the next row.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -45,38 +45,38 @@
│ │ │ │  o4 : WeylGroupElement
│ │ │ │  i5 : myInterval=intervalBruhat(P % w1,P % w2)
│ │ │ │  
│ │ │ │  o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0
│ │ │ │  |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{1, | -
│ │ │ │  1 |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  2
│ │ │ │  |}, {1, |  0 |}, {2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -
│ │ │ │ -3 |}, {{0, |  0 |}, {2, | 1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -
│ │ │ │ -1 |}, {{0, |  2 |}, {1, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2
│ │ │ │ -|}, {{1, |  0 |}, {2, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -
│ │ │ │ -2 |}, {{0, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2
│ │ │ │ -|}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{0, |  0 |}}}}, {
│ │ │ │ +3 |}, {{0, | 1 |}, {1, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -
│ │ │ │ +1 |}, {{1, |  2 |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2
│ │ │ │ +|}, {{0, |  2 |}, {2, |  0 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  3
│ │ │ │ +|}, {{0, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1
│ │ │ │ +|}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}}, {
│ │ │ │  {WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ │                                                            | -2 |        | -1 |
│ │ │ │  |  2 |                                              | -3 |        |  1 |
│ │ │ │  |  2 |                                            | -1 |        | -1 |       |
│ │ │ │  -1 |       |  1 |                                              |  1 |        |
│ │ │ │ --1 |       | 0 |                                            | -1 |        | -
│ │ │ │ +0 |       | -1 |                                            | -1 |        | -
│ │ │ │  1 |       |  2 |                                            | -3 |        | -
│ │ │ │ -1 |       | -1 |                                              |  1 |        |
│ │ │ │ -1 |                                            | -2 |        | -1 |
│ │ │ │ +1 |       | -1 |                                              | -2 |        | -
│ │ │ │ +1 |                                            |  1 |        |  1 |
│ │ │ │  | -2 |        | -1 |                                              | -1 |
│ │ │ │                                                            |  1 |        |  2 |
│ │ │ │  | -1 |                                              |  1 |        |  1 |
│ │ │ │  | -1 |                                            |  2 |        |  0 |       |
│ │ │ │ -2 |       |  1 |                                              |  1 |        |
│ │ │ │ -2 |       | 1 |                                            |  3 |        |  0 |
│ │ │ │ -| -1 |                                            |  2 |        |  2 |       |
│ │ │ │ -0 |                                              |  2 |        | -1 |
│ │ │ │ -|  3 |        |  0 |                                            |  1 |        |
│ │ │ │ -2 |                                              |  2 |
│ │ │ │ +2 |       |  1 |                                              |  1 |        | 1
│ │ │ │ +|       |  2 |                                            |  3 |        |  0 |
│ │ │ │ +| -1 |                                            |  2 |        |  0 |       |
│ │ │ │ +2 |                                              |  1 |        |  2 |
│ │ │ │ +|  2 |        | -1 |                                            |  3 |        |
│ │ │ │ +0 |                                              |  2 |
│ │ │ │  
│ │ │ │  o5 : HasseDiagram
│ │ │ │  Each row of the Hasse diagram contains the elements of a certain length
│ │ │ │  together with their links to the next row.
│ │ │ │  i6 : myInterval#1
│ │ │ │  
│ │ │ │  o6 = {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{1, | -1 |}, {2, | -1
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/html/_poincare__Series_lp__Hasse__Diagram_cm__Ring__Element_rp.html
│ │ │ @@ -79,16 +79,16 @@
│ │ │  o1 : RootSystem
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : H=intervalBruhat(neutralWeylGroupElement R, longestWeylGroupElement R)
│ │ │  
│ │ │ -o2 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {1, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | 1 |}, {1, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {1, | 1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | 1 |}, {}}}}
│ │ │ -                                                          | -1 |        |  2 |       | -1 |                                              | -2 |        | 1 |       | -1 |                                            |  1 |        |  2 |       | 1 |                                              |  2 |        | -1 |                                            | -1 |        |  2 |                                              | 1 |
│ │ │ +o2 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  2 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | 1 |}, {1, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}, {1, | 1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | 1 |}, {}}}}
│ │ │ +                                                          | -1 |        | -1 |       |  2 |                                              |  1 |        | 1 |       |  2 |                                            | -2 |        | -1 |       | 1 |                                              | -1 |        |  2 |                                            |  2 |        | -1 |                                              | 1 |
│ │ │  
│ │ │  o2 : HasseDiagram
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : ZZ[x]
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,25 +19,25 @@
│ │ │ │  i1 : R=rootSystemA(2)
│ │ │ │  
│ │ │ │  o1 = RootSystem{...8...}
│ │ │ │  
│ │ │ │  o1 : RootSystem
│ │ │ │  i2 : H=intervalBruhat(neutralWeylGroupElement R, longestWeylGroupElement R)
│ │ │ │  
│ │ │ │ -o2 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -
│ │ │ │ -1 |}, {1, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | 1
│ │ │ │ -|}, {1, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -
│ │ │ │ -1 |}, {1, | 1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  2
│ │ │ │ -|}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}}, {
│ │ │ │ +o2 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  2
│ │ │ │ +|}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | 1
│ │ │ │ +|}, {1, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2
│ │ │ │ +|}, {1, | 1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -
│ │ │ │ +1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  2 |}}}}, {
│ │ │ │  {WeylGroupElement{RootSystem{...8...}, | 1 |}, {}}}}
│ │ │ │ -                                                          | -1 |        |  2 |
│ │ │ │ -| -1 |                                              | -2 |        | 1 |       |
│ │ │ │ --1 |                                            |  1 |        |  2 |       | 1
│ │ │ │ -|                                              |  2 |        | -1 |
│ │ │ │ -| -1 |        |  2 |                                              | 1 |
│ │ │ │ +                                                          | -1 |        | -1 |
│ │ │ │ +|  2 |                                              |  1 |        | 1 |       |
│ │ │ │ +2 |                                            | -2 |        | -1 |       | 1 |
│ │ │ │ +| -1 |        |  2 |                                            |  2 |        |
│ │ │ │ +-1 |                                              | 1 |
│ │ │ │  
│ │ │ │  o2 : HasseDiagram
│ │ │ │  i3 : ZZ[x]
│ │ │ │  
│ │ │ │  o3 = ZZ[x]
│ │ │ │  
│ │ │ │  o3 : PolynomialRing
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/html/_positive__Roots_lp__Root__System_cm__Parabolic_rp.html
│ │ │ @@ -88,17 +88,17 @@
│ │ │  o2 : Parabolic
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : positiveRoots(R,P)
│ │ │  
│ │ │ -o3 = set {|  2 |, | -1 |, |  1 |}
│ │ │ -          | -1 |  |  2 |  |  1 |
│ │ │ -          |  0 |  | -1 |  | -1 |
│ │ │ +o3 = set {|  1 |, |  2 |, | -1 |}
│ │ │ +          |  1 |  | -1 |  |  2 |
│ │ │ +          | -1 |  |  0 |  | -1 |
│ │ │            |  0 |  |  0 |  |  0 |
│ │ │  
│ │ │  o3 : Set
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,17 +23,17 @@
│ │ │ │  i2 : P=parabolic(R,set{1,2})
│ │ │ │  
│ │ │ │  o2 = set {1, 2}
│ │ │ │  
│ │ │ │  o2 : Parabolic
│ │ │ │  i3 : positiveRoots(R,P)
│ │ │ │  
│ │ │ │ -o3 = set {|  2 |, | -1 |, |  1 |}
│ │ │ │ -          | -1 |  |  2 |  |  1 |
│ │ │ │ -          |  0 |  | -1 |  | -1 |
│ │ │ │ +o3 = set {|  1 |, |  2 |, | -1 |}
│ │ │ │ +          |  1 |  | -1 |  |  2 |
│ │ │ │ +          | -1 |  |  0 |  | -1 |
│ │ │ │            |  0 |  |  0 |  |  0 |
│ │ │ │  
│ │ │ │  o3 : Set
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _p_o_s_i_t_i_v_e_R_o_o_t_s_(_R_o_o_t_S_y_s_t_e_m_,_P_a_r_a_b_o_l_i_c_) -- the set of all positive roots in a
│ │ │ │        parabolic sugroups
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/html/_positive__Roots_lp__Root__System_rp.html
│ │ │ @@ -69,17 +69,17 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : positiveRoots(rootSystemA(3))
│ │ │  
│ │ │ -o1 = set {|  0 |, | -1 |, | -1 |, | 1 |, |  1 |, |  2 |}
│ │ │ -          | -1 |  |  1 |  |  2 |  | 0 |  |  1 |  | -1 |
│ │ │ -          |  2 |  |  1 |  | -1 |  | 1 |  | -1 |  |  0 |
│ │ │ +o1 = set {|  2 |, |  0 |, | -1 |, | -1 |, | 1 |, |  1 |}
│ │ │ +          | -1 |  | -1 |  |  1 |  |  2 |  | 0 |  |  1 |
│ │ │ +          |  0 |  |  2 |  |  1 |  | -1 |  | 1 |  | -1 |
│ │ │  
│ │ │  o1 : Set
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,17 +10,17 @@
│ │ │ │      * Inputs:
│ │ │ │            o R, an instance of the type _R_o_o_t_S_y_s_t_e_m,
│ │ │ │      * Outputs:
│ │ │ │            o a _s_e_t, of all positive roots of R
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : positiveRoots(rootSystemA(3))
│ │ │ │  
│ │ │ │ -o1 = set {|  0 |, | -1 |, | -1 |, | 1 |, |  1 |, |  2 |}
│ │ │ │ -          | -1 |  |  1 |  |  2 |  | 0 |  |  1 |  | -1 |
│ │ │ │ -          |  2 |  |  1 |  | -1 |  | 1 |  | -1 |  |  0 |
│ │ │ │ +o1 = set {|  2 |, |  0 |, | -1 |, | -1 |, | 1 |, |  1 |}
│ │ │ │ +          | -1 |  | -1 |  |  1 |  |  2 |  | 0 |  |  1 |
│ │ │ │ +          |  0 |  |  2 |  |  1 |  | -1 |  | 1 |  | -1 |
│ │ │ │  
│ │ │ │  o1 : Set
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _p_o_s_i_t_i_v_e_R_o_o_t_s_(_R_o_o_t_S_y_s_t_e_m_) -- the set of all positive roots
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/WeylGroups.m2:2604:0.
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/html/_under__Bruhat_lp__Basic__List_rp.html
│ │ │ @@ -116,37 +116,37 @@
│ │ │  o3 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : underBruhat(L1)
│ │ │  
│ │ │ -o4 = {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  0 |}, {2, |  2
│ │ │ -                                             | -3 |        | -1 |       | -1
│ │ │ -                                             |  1 |        |  2 |       |  0
│ │ │ +o4 = {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1, |  0 |}, {2, | -1
│ │ │ +                                             |  2 |        | -1 |       |  1
│ │ │ +                                             | -3 |        |  2 |       |  1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1, | -1 |}, {2,
│ │ │ -     |                                            | -1 |        |  1 |      
│ │ │ -     |                                            | -2 |        |  1 |      
│ │ │ +     |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  0 |}, {2,
│ │ │ +     |                                            | -3 |        | -1 |      
│ │ │ +     |                                            |  1 |        |  2 |      
│ │ │       ------------------------------------------------------------------------
│ │ │ -     | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, |  1 |},
│ │ │ -     |  2 |                                            |  2 |        |  1 |  
│ │ │ -     | -1 |                                            | -1 |        | -1 |  
│ │ │ +     |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1, | -1 |},
│ │ │ +     | -1 |                                            | -1 |        |  1 |  
│ │ │ +     |  0 |                                            | -2 |        |  1 |  
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1
│ │ │ -         | -1 |                                            | -1 |        |  2
│ │ │ -         |  0 |                                            |  2 |        | -1
│ │ │ +     {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, |  1
│ │ │ +         |  2 |                                            |  2 |        |  1
│ │ │ +         | -1 |                                            | -1 |        | -1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}, {1, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1,
│ │ │ -     |       |  1 |                                            |  2 |       
│ │ │ -     |       | -1 |                                            | -3 |       
│ │ │ +     |}, {1, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0,
│ │ │ +     |       | -1 |                                            | -1 |       
│ │ │ +     |       |  0 |                                            |  2 |       
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |  0 |}, {2, | -1 |}}}}
│ │ │ -     | -1 |       |  1 |
│ │ │ +     | -1 |}, {1, |  1 |}}}}
│ │ │       |  2 |       |  1 |
│ │ │ +     | -1 |       | -1 |
│ │ │  
│ │ │  o4 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -48,37 +48,37 @@
│ │ │ │       WeylGroupElement{RootSystem{...8...}, |  1 |}}
│ │ │ │                                             | -2 |
│ │ │ │                                             | -1 |
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : underBruhat(L1)
│ │ │ │  
│ │ │ │ -o4 = {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  0 |}, {2, |  2
│ │ │ │ -                                             | -3 |        | -1 |       | -1
│ │ │ │ -                                             |  1 |        |  2 |       |  0
│ │ │ │ +o4 = {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1, |  0 |}, {2, | -1
│ │ │ │ +                                             |  2 |        | -1 |       |  1
│ │ │ │ +                                             | -3 |        |  2 |       |  1
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1, | -1 |}, {2,
│ │ │ │ -     |                                            | -1 |        |  1 |
│ │ │ │ -     |                                            | -2 |        |  1 |
│ │ │ │ +     |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  0 |}, {2,
│ │ │ │ +     |                                            | -3 |        | -1 |
│ │ │ │ +     |                                            |  1 |        |  2 |
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, |  1 |},
│ │ │ │ -     |  2 |                                            |  2 |        |  1 |
│ │ │ │ -     | -1 |                                            | -1 |        | -1 |
│ │ │ │ +     |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1, | -1 |},
│ │ │ │ +     | -1 |                                            | -1 |        |  1 |
│ │ │ │ +     |  0 |                                            | -2 |        |  1 |
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1
│ │ │ │ -         | -1 |                                            | -1 |        |  2
│ │ │ │ -         |  0 |                                            |  2 |        | -1
│ │ │ │ +     {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, |  1
│ │ │ │ +         |  2 |                                            |  2 |        |  1
│ │ │ │ +         | -1 |                                            | -1 |        | -1
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     |}, {1, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1,
│ │ │ │ -     |       |  1 |                                            |  2 |
│ │ │ │ -     |       | -1 |                                            | -3 |
│ │ │ │ +     |}, {1, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0,
│ │ │ │ +     |       | -1 |                                            | -1 |
│ │ │ │ +     |       |  0 |                                            |  2 |
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     |  0 |}, {2, | -1 |}}}}
│ │ │ │ -     | -1 |       |  1 |
│ │ │ │ +     | -1 |}, {1, |  1 |}}}}
│ │ │ │       |  2 |       |  1 |
│ │ │ │ +     | -1 |       | -1 |
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _u_n_d_e_r_B_r_u_h_a_t_(_B_a_s_i_c_L_i_s_t_) -- Weyl group elements just under the ones in the
│ │ │ │        list for the Bruhat order
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/WhitneyStratifications/example-output/_map__Stratify.out
│ │ │ @@ -122,15 +122,15 @@
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  loop over components of JY=ideal 1
│ │ │ - -- used 1.96289s (cpu); 1.23405s (thread); 0s (gc)
│ │ │ + -- used 2.42997s (cpu); 1.31756s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │  
│ │ │  o23 : List
│ │ │  
│ │ │  i24 : peek last ms
│ │ │  
│ │ │ @@ -142,15 +142,15 @@
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  loop over components of JY=ideal 1
│ │ │ - -- used 4.52445s (cpu); 2.71961s (thread); 0s (gc)
│ │ │ + -- used 7.10048s (cpu); 3.06806s (thread); 0s (gc)
│ │ │  
│ │ │  o25 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │  
│ │ │  o25 : List
│ │ │  
│ │ │  i26 : peek last ms
│ │ │  
│ │ │ @@ -162,15 +162,15 @@
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  loop over components of JY=ideal 1
│ │ │ - -- used 5.27688s (cpu); 3.13311s (thread); 0s (gc)
│ │ │ + -- used 7.88749s (cpu); 3.33197s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │  
│ │ │  o27 : List
│ │ │  
│ │ │  i28 : peek last ms
│ │ ├── ./usr/share/doc/Macaulay2/WhitneyStratifications/html/_map__Stratify.html
│ │ │ @@ -292,15 +292,15 @@
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  loop over components of JY=ideal 1
│ │ │ - -- used 1.96289s (cpu); 1.23405s (thread); 0s (gc)
│ │ │ + -- used 2.42997s (cpu); 1.31756s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │  
│ │ │  o23 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -318,15 +318,15 @@
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  loop over components of JY=ideal 1
│ │ │ - -- used 4.52445s (cpu); 2.71961s (thread); 0s (gc)
│ │ │ + -- used 7.10048s (cpu); 3.06806s (thread); 0s (gc)
│ │ │  
│ │ │  o25 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │  
│ │ │  o25 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -344,15 +344,15 @@
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  loop over components of JY=ideal 1
│ │ │ - -- used 5.27688s (cpu); 3.13311s (thread); 0s (gc)
│ │ │ + -- used 7.88749s (cpu); 3.33197s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │  
│ │ │  o27 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -184,15 +184,15 @@
│ │ │ │  for j2....
│ │ │ │  loop over components of JY=ideal 1
│ │ │ │  for j2....
│ │ │ │  loop over components of JY=ideal 1
│ │ │ │  for j2....
│ │ │ │  loop over components of JY=ideal 1
│ │ │ │  loop over components of JY=ideal 1
│ │ │ │ - -- used 1.96289s (cpu); 1.23405s (thread); 0s (gc)
│ │ │ │ + -- used 2.42997s (cpu); 1.31756s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │ │  
│ │ │ │  o23 : List
│ │ │ │  i24 : peek last ms
│ │ │ │  
│ │ │ │  o24 = MutableHashTable{0 => {ideal (P, M1)}                    }
│ │ │ │ @@ -202,15 +202,15 @@
│ │ │ │  for j2....
│ │ │ │  loop over components of JY=ideal 1
│ │ │ │  for j2....
│ │ │ │  loop over components of JY=ideal 1
│ │ │ │  for j2....
│ │ │ │  loop over components of JY=ideal 1
│ │ │ │  loop over components of JY=ideal 1
│ │ │ │ - -- used 4.52445s (cpu); 2.71961s (thread); 0s (gc)
│ │ │ │ + -- used 7.10048s (cpu); 3.06806s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o25 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │ │  
│ │ │ │  o25 : List
│ │ │ │  i26 : peek last ms
│ │ │ │  
│ │ │ │  o26 = MutableHashTable{0 => {ideal (P, M1)}                    }
│ │ │ │ @@ -220,15 +220,15 @@
│ │ │ │  for j2....
│ │ │ │  loop over components of JY=ideal 1
│ │ │ │  for j2....
│ │ │ │  loop over components of JY=ideal 1
│ │ │ │  for j2....
│ │ │ │  loop over components of JY=ideal 1
│ │ │ │  loop over components of JY=ideal 1
│ │ │ │ - -- used 5.27688s (cpu); 3.13311s (thread); 0s (gc)
│ │ │ │ + -- used 7.88749s (cpu); 3.33197s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o27 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │ │  
│ │ │ │  o27 : List
│ │ │ │  i28 : peek last ms
│ │ │ │  
│ │ │ │  o28 = MutableHashTable{0 => {ideal (P, M1)}                    }
│ │ ├── ./usr/share/doc/Macaulay2/gfanInterface/example-output/___Installation_spand_sp__Configuration_spof_spgfan__Interface.out
│ │ │ @@ -19,15 +19,15 @@
│ │ │  i4 : prefixDirectory | currentLayout#"programs"
│ │ │  
│ │ │  o4 = /usr/x86_64-Linux-
│ │ │       Debian-forky/libexec/Macaulay2/bin/
│ │ │  
│ │ │  i5 : loadPackage("gfanInterface", Configuration => { "keepfiles" => true, "verbose" => true}, Reload => true);
│ │ │   -- warning: reloading gfanInterface; recreate instances of types from this package
│ │ │ - -- running: /usr/bin/gfan gfan --help < /tmp/M2-16975-0/172
│ │ │ + -- running: /usr/bin/gfan gfan --help < /tmp/M2-22611-0/172
│ │ │  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.
│ │ │ @@ -38,16 +38,16 @@
│ │ │  --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/M2-16975-0/172
│ │ │ - -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-16975-0/174
│ │ │ +using temporary file /tmp/M2-22611-0/172
│ │ │ + -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-22611-0/174
│ │ │  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.
│ │ │ @@ -56,69 +56,69 @@
│ │ │   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/M2-16975-0/174
│ │ │ - -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-16975-0/176
│ │ │ +using temporary file /tmp/M2-22611-0/174
│ │ │ + -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-22611-0/176
│ │ │  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/M2-16975-0/176
│ │ │ - -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-16975-0/178
│ │ │ +using temporary file /tmp/M2-22611-0/176
│ │ │ + -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-22611-0/178
│ │ │  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/M2-16975-0/178
│ │ │ - -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-16975-0/180
│ │ │ +using temporary file /tmp/M2-22611-0/178
│ │ │ + -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-22611-0/180
│ │ │  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/M2-16975-0/180
│ │ │ - -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-16975-0/182
│ │ │ +using temporary file /tmp/M2-22611-0/180
│ │ │ + -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-22611-0/182
│ │ │  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/M2-16975-0/182
│ │ │ - -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-16975-0/184
│ │ │ +using temporary file /tmp/M2-22611-0/182
│ │ │ + -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-22611-0/184
│ │ │  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/M2-16975-0/184
│ │ │ - -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-16975-0/186
│ │ │ +using temporary file /tmp/M2-22611-0/184
│ │ │ + -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-22611-0/186
│ │ │  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/M2-16975-0/186
│ │ │ - -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-16975-0/188
│ │ │ +using temporary file /tmp/M2-22611-0/186
│ │ │ + -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-22611-0/188
│ │ │  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}
│ │ │ @@ -126,30 +126,30 @@
│ │ │  -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/M2-16975-0/188
│ │ │ - -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-16975-0/190
│ │ │ +using temporary file /tmp/M2-22611-0/188
│ │ │ + -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-22611-0/190
│ │ │  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/M2-16975-0/190
│ │ │ - -- running: /usr/bin/gfan _interactive --help < /tmp/M2-16975-0/192
│ │ │ +using temporary file /tmp/M2-22611-0/190
│ │ │ + -- running: /usr/bin/gfan _interactive --help < /tmp/M2-22611-0/192
│ │ │  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.
│ │ │ @@ -164,57 +164,57 @@
│ │ │   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/M2-16975-0/192
│ │ │ - -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-16975-0/194
│ │ │ +using temporary file /tmp/M2-22611-0/192
│ │ │ + -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-22611-0/194
│ │ │  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/M2-16975-0/194
│ │ │ - -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-16975-0/196
│ │ │ +using temporary file /tmp/M2-22611-0/194
│ │ │ + -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-22611-0/196
│ │ │  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/M2-16975-0/196
│ │ │ - -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-16975-0/198
│ │ │ +using temporary file /tmp/M2-22611-0/196
│ │ │ + -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-22611-0/198
│ │ │  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/M2-16975-0/198
│ │ │ - -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-16975-0/200
│ │ │ +using temporary file /tmp/M2-22611-0/198
│ │ │ + -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-22611-0/200
│ │ │  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/M2-16975-0/200
│ │ │ - -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-16975-0/202
│ │ │ +using temporary file /tmp/M2-22611-0/200
│ │ │ + -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-22611-0/202
│ │ │  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/M2-16975-0/202
│ │ │ - -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-16975-0/204
│ │ │ +using temporary file /tmp/M2-22611-0/202
│ │ │ + -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-22611-0/204
│ │ │  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/M2-16975-0/204
│ │ │ - -- running: /usr/bin/gfan _minors --help < /tmp/M2-16975-0/206
│ │ │ +using temporary file /tmp/M2-22611-0/204
│ │ │ + -- running: /usr/bin/gfan _minors --help < /tmp/M2-22611-0/206
│ │ │  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:
│ │ │ @@ -229,16 +229,16 @@
│ │ │   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/M2-16975-0/206
│ │ │ - -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-16975-0/208
│ │ │ +using temporary file /tmp/M2-22611-0/206
│ │ │ + -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-22611-0/208
│ │ │  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:
│ │ │ @@ -249,44 +249,44 @@
│ │ │   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/M2-16975-0/208
│ │ │ - -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-16975-0/210
│ │ │ +using temporary file /tmp/M2-22611-0/208
│ │ │ + -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-22611-0/210
│ │ │  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/M2-16975-0/210
│ │ │ - -- running: /usr/bin/gfan _render --help < /tmp/M2-16975-0/212
│ │ │ +using temporary file /tmp/M2-22611-0/210
│ │ │ + -- running: /usr/bin/gfan _render --help < /tmp/M2-22611-0/212
│ │ │  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/M2-16975-0/212
│ │ │ - -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-16975-0/214
│ │ │ +using temporary file /tmp/M2-22611-0/212
│ │ │ + -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-22611-0/214
│ │ │  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/M2-16975-0/214
│ │ │ - -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-16975-0/216
│ │ │ +using temporary file /tmp/M2-22611-0/214
│ │ │ + -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-22611-0/216
│ │ │  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.
│ │ │ @@ -299,25 +299,25 @@
│ │ │  
│ │ │  --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/M2-16975-0/216
│ │ │ - -- running: /usr/bin/gfan _saturation --help < /tmp/M2-16975-0/218
│ │ │ +using temporary file /tmp/M2-22611-0/216
│ │ │ + -- running: /usr/bin/gfan _saturation --help < /tmp/M2-22611-0/218
│ │ │  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/M2-16975-0/218
│ │ │ - -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-16975-0/220
│ │ │ +using temporary file /tmp/M2-22611-0/218
│ │ │ + -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-22611-0/220
│ │ │  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:
│ │ │ @@ -326,70 +326,70 @@
│ │ │  --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/M2-16975-0/220
│ │ │ - -- running: /usr/bin/gfan _stats --help < /tmp/M2-16975-0/222
│ │ │ +using temporary file /tmp/M2-22611-0/220
│ │ │ + -- running: /usr/bin/gfan _stats --help < /tmp/M2-22611-0/222
│ │ │  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/M2-16975-0/222
│ │ │ - -- running: /usr/bin/gfan _substitute --help < /tmp/M2-16975-0/224
│ │ │ +using temporary file /tmp/M2-22611-0/222
│ │ │ + -- running: /usr/bin/gfan _substitute --help < /tmp/M2-22611-0/224
│ │ │  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/M2-16975-0/224
│ │ │ - -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-16975-0/226
│ │ │ +using temporary file /tmp/M2-22611-0/224
│ │ │ + -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-22611-0/226
│ │ │  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/M2-16975-0/226
│ │ │ - -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-16975-0/228
│ │ │ +using temporary file /tmp/M2-22611-0/226
│ │ │ + -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-22611-0/228
│ │ │  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/M2-16975-0/228
│ │ │ - -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-16975-0/230
│ │ │ +using temporary file /tmp/M2-22611-0/228
│ │ │ + -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-22611-0/230
│ │ │  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/M2-16975-0/230
│ │ │ - -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-16975-0/232
│ │ │ +using temporary file /tmp/M2-22611-0/230
│ │ │ + -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-22611-0/232
│ │ │  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/M2-16975-0/232
│ │ │ - -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-16975-0/234
│ │ │ +using temporary file /tmp/M2-22611-0/232
│ │ │ + -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-22611-0/234
│ │ │  This program evaluates a tropical polynomial function in a given set of points.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16975-0/234
│ │ │ - -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-16975-0/236
│ │ │ +using temporary file /tmp/M2-22611-0/234
│ │ │ + -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-22611-0/236
│ │ │  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/M2-16975-0/236
│ │ │ - -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-16975-0/238
│ │ │ +using temporary file /tmp/M2-22611-0/236
│ │ │ + -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-22611-0/238
│ │ │  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/M2-16975-0/238
│ │ │ - -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-16975-0/240
│ │ │ +using temporary file /tmp/M2-22611-0/238
│ │ │ + -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-22611-0/240
│ │ │  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.
│ │ │ @@ -401,16 +401,16 @@
│ │ │   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/M2-16975-0/240
│ │ │ - -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-16975-0/242
│ │ │ +using temporary file /tmp/M2-22611-0/240
│ │ │ + -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-22611-0/242
│ │ │  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
│ │ │  
│ │ │ @@ -435,48 +435,48 @@
│ │ │  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/M2-16975-0/242
│ │ │ - -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-16975-0/244
│ │ │ +using temporary file /tmp/M2-22611-0/242
│ │ │ + -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-22611-0/244
│ │ │  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/M2-16975-0/244
│ │ │ - -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-16975-0/246
│ │ │ +using temporary file /tmp/M2-22611-0/244
│ │ │ + -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-22611-0/246
│ │ │  This program computes the multiplicity of a tropical cone given a marked reduced Groebner basis for its initial ideal.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16975-0/246
│ │ │ - -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-16975-0/248
│ │ │ +using temporary file /tmp/M2-22611-0/246
│ │ │ + -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-22611-0/248
│ │ │  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/M2-16975-0/248
│ │ │ - -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-16975-0/250
│ │ │ +using temporary file /tmp/M2-22611-0/248
│ │ │ + -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-22611-0/250
│ │ │  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/M2-16975-0/250
│ │ │ - -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-16975-0/252
│ │ │ +using temporary file /tmp/M2-22611-0/250
│ │ │ + -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-22611-0/252
│ │ │  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:
│ │ │ @@ -484,24 +484,24 @@
│ │ │  --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/M2-16975-0/252
│ │ │ - -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-16975-0/254
│ │ │ +using temporary file /tmp/M2-22611-0/252
│ │ │ + -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-22611-0/254
│ │ │  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/M2-16975-0/254
│ │ │ - -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-16975-0/256
│ │ │ +using temporary file /tmp/M2-22611-0/254
│ │ │ + -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-22611-0/256
│ │ │  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.
│ │ │ @@ -521,21 +521,21 @@
│ │ │  --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/M2-16975-0/256
│ │ │ +using temporary file /tmp/M2-22611-0/256
│ │ │  
│ │ │  i6 : QQ[x,y];
│ │ │  
│ │ │  i7 : gfan {x,y};
│ │ │ - -- running: /usr/bin/gfan _bases < /tmp/M2-16975-0/258
│ │ │ + -- running: /usr/bin/gfan _bases < /tmp/M2-22611-0/258
│ │ │  Q[x1,x2]
│ │ │  {{
│ │ │  x2,
│ │ │  x1}
│ │ │  }
│ │ │ -using temporary file /tmp/M2-16975-0/258
│ │ │ +using temporary file /tmp/M2-22611-0/258
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/gfanInterface/html/___Installation_spand_sp__Configuration_spof_spgfan__Interface.html
│ │ │ @@ -109,15 +109,15 @@
│ │ │            
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i5 : loadPackage("gfanInterface", Configuration => { "keepfiles" => true, "verbose" => true}, Reload => true);
│ │ │   -- warning: reloading gfanInterface; recreate instances of types from this package
│ │ │ - -- running: /usr/bin/gfan gfan --help < /tmp/M2-16975-0/172
│ │ │ + -- running: /usr/bin/gfan gfan --help < /tmp/M2-22611-0/172
│ │ │  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.
│ │ │ @@ -128,16 +128,16 @@
│ │ │  --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/M2-16975-0/172
│ │ │ - -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-16975-0/174
│ │ │ +using temporary file /tmp/M2-22611-0/172
│ │ │ + -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-22611-0/174
│ │ │  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.
│ │ │ @@ -146,69 +146,69 @@
│ │ │   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/M2-16975-0/174
│ │ │ - -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-16975-0/176
│ │ │ +using temporary file /tmp/M2-22611-0/174
│ │ │ + -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-22611-0/176
│ │ │  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/M2-16975-0/176
│ │ │ - -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-16975-0/178
│ │ │ +using temporary file /tmp/M2-22611-0/176
│ │ │ + -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-22611-0/178
│ │ │  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/M2-16975-0/178
│ │ │ - -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-16975-0/180
│ │ │ +using temporary file /tmp/M2-22611-0/178
│ │ │ + -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-22611-0/180
│ │ │  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/M2-16975-0/180
│ │ │ - -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-16975-0/182
│ │ │ +using temporary file /tmp/M2-22611-0/180
│ │ │ + -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-22611-0/182
│ │ │  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/M2-16975-0/182
│ │ │ - -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-16975-0/184
│ │ │ +using temporary file /tmp/M2-22611-0/182
│ │ │ + -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-22611-0/184
│ │ │  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/M2-16975-0/184
│ │ │ - -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-16975-0/186
│ │ │ +using temporary file /tmp/M2-22611-0/184
│ │ │ + -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-22611-0/186
│ │ │  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/M2-16975-0/186
│ │ │ - -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-16975-0/188
│ │ │ +using temporary file /tmp/M2-22611-0/186
│ │ │ + -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-22611-0/188
│ │ │  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}
│ │ │ @@ -216,30 +216,30 @@
│ │ │  -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/M2-16975-0/188
│ │ │ - -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-16975-0/190
│ │ │ +using temporary file /tmp/M2-22611-0/188
│ │ │ + -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-22611-0/190
│ │ │  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/M2-16975-0/190
│ │ │ - -- running: /usr/bin/gfan _interactive --help < /tmp/M2-16975-0/192
│ │ │ +using temporary file /tmp/M2-22611-0/190
│ │ │ + -- running: /usr/bin/gfan _interactive --help < /tmp/M2-22611-0/192
│ │ │  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.
│ │ │ @@ -254,57 +254,57 @@
│ │ │   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/M2-16975-0/192
│ │ │ - -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-16975-0/194
│ │ │ +using temporary file /tmp/M2-22611-0/192
│ │ │ + -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-22611-0/194
│ │ │  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/M2-16975-0/194
│ │ │ - -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-16975-0/196
│ │ │ +using temporary file /tmp/M2-22611-0/194
│ │ │ + -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-22611-0/196
│ │ │  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/M2-16975-0/196
│ │ │ - -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-16975-0/198
│ │ │ +using temporary file /tmp/M2-22611-0/196
│ │ │ + -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-22611-0/198
│ │ │  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/M2-16975-0/198
│ │ │ - -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-16975-0/200
│ │ │ +using temporary file /tmp/M2-22611-0/198
│ │ │ + -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-22611-0/200
│ │ │  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/M2-16975-0/200
│ │ │ - -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-16975-0/202
│ │ │ +using temporary file /tmp/M2-22611-0/200
│ │ │ + -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-22611-0/202
│ │ │  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/M2-16975-0/202
│ │ │ - -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-16975-0/204
│ │ │ +using temporary file /tmp/M2-22611-0/202
│ │ │ + -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-22611-0/204
│ │ │  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/M2-16975-0/204
│ │ │ - -- running: /usr/bin/gfan _minors --help < /tmp/M2-16975-0/206
│ │ │ +using temporary file /tmp/M2-22611-0/204
│ │ │ + -- running: /usr/bin/gfan _minors --help < /tmp/M2-22611-0/206
│ │ │  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:
│ │ │ @@ -319,16 +319,16 @@
│ │ │   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/M2-16975-0/206
│ │ │ - -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-16975-0/208
│ │ │ +using temporary file /tmp/M2-22611-0/206
│ │ │ + -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-22611-0/208
│ │ │  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:
│ │ │ @@ -339,44 +339,44 @@
│ │ │   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/M2-16975-0/208
│ │ │ - -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-16975-0/210
│ │ │ +using temporary file /tmp/M2-22611-0/208
│ │ │ + -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-22611-0/210
│ │ │  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/M2-16975-0/210
│ │ │ - -- running: /usr/bin/gfan _render --help < /tmp/M2-16975-0/212
│ │ │ +using temporary file /tmp/M2-22611-0/210
│ │ │ + -- running: /usr/bin/gfan _render --help < /tmp/M2-22611-0/212
│ │ │  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/M2-16975-0/212
│ │ │ - -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-16975-0/214
│ │ │ +using temporary file /tmp/M2-22611-0/212
│ │ │ + -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-22611-0/214
│ │ │  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/M2-16975-0/214
│ │ │ - -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-16975-0/216
│ │ │ +using temporary file /tmp/M2-22611-0/214
│ │ │ + -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-22611-0/216
│ │ │  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.
│ │ │ @@ -389,25 +389,25 @@
│ │ │  
│ │ │  --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/M2-16975-0/216
│ │ │ - -- running: /usr/bin/gfan _saturation --help < /tmp/M2-16975-0/218
│ │ │ +using temporary file /tmp/M2-22611-0/216
│ │ │ + -- running: /usr/bin/gfan _saturation --help < /tmp/M2-22611-0/218
│ │ │  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/M2-16975-0/218
│ │ │ - -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-16975-0/220
│ │ │ +using temporary file /tmp/M2-22611-0/218
│ │ │ + -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-22611-0/220
│ │ │  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:
│ │ │ @@ -416,70 +416,70 @@
│ │ │  --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/M2-16975-0/220
│ │ │ - -- running: /usr/bin/gfan _stats --help < /tmp/M2-16975-0/222
│ │ │ +using temporary file /tmp/M2-22611-0/220
│ │ │ + -- running: /usr/bin/gfan _stats --help < /tmp/M2-22611-0/222
│ │ │  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/M2-16975-0/222
│ │ │ - -- running: /usr/bin/gfan _substitute --help < /tmp/M2-16975-0/224
│ │ │ +using temporary file /tmp/M2-22611-0/222
│ │ │ + -- running: /usr/bin/gfan _substitute --help < /tmp/M2-22611-0/224
│ │ │  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/M2-16975-0/224
│ │ │ - -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-16975-0/226
│ │ │ +using temporary file /tmp/M2-22611-0/224
│ │ │ + -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-22611-0/226
│ │ │  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/M2-16975-0/226
│ │ │ - -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-16975-0/228
│ │ │ +using temporary file /tmp/M2-22611-0/226
│ │ │ + -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-22611-0/228
│ │ │  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/M2-16975-0/228
│ │ │ - -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-16975-0/230
│ │ │ +using temporary file /tmp/M2-22611-0/228
│ │ │ + -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-22611-0/230
│ │ │  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/M2-16975-0/230
│ │ │ - -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-16975-0/232
│ │ │ +using temporary file /tmp/M2-22611-0/230
│ │ │ + -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-22611-0/232
│ │ │  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/M2-16975-0/232
│ │ │ - -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-16975-0/234
│ │ │ +using temporary file /tmp/M2-22611-0/232
│ │ │ + -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-22611-0/234
│ │ │  This program evaluates a tropical polynomial function in a given set of points.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16975-0/234
│ │ │ - -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-16975-0/236
│ │ │ +using temporary file /tmp/M2-22611-0/234
│ │ │ + -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-22611-0/236
│ │ │  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/M2-16975-0/236
│ │ │ - -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-16975-0/238
│ │ │ +using temporary file /tmp/M2-22611-0/236
│ │ │ + -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-22611-0/238
│ │ │  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/M2-16975-0/238
│ │ │ - -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-16975-0/240
│ │ │ +using temporary file /tmp/M2-22611-0/238
│ │ │ + -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-22611-0/240
│ │ │  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.
│ │ │ @@ -491,16 +491,16 @@
│ │ │   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/M2-16975-0/240
│ │ │ - -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-16975-0/242
│ │ │ +using temporary file /tmp/M2-22611-0/240
│ │ │ + -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-22611-0/242
│ │ │  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
│ │ │  
│ │ │ @@ -525,48 +525,48 @@
│ │ │  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/M2-16975-0/242
│ │ │ - -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-16975-0/244
│ │ │ +using temporary file /tmp/M2-22611-0/242
│ │ │ + -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-22611-0/244
│ │ │  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/M2-16975-0/244
│ │ │ - -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-16975-0/246
│ │ │ +using temporary file /tmp/M2-22611-0/244
│ │ │ + -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-22611-0/246
│ │ │  This program computes the multiplicity of a tropical cone given a marked reduced Groebner basis for its initial ideal.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16975-0/246
│ │ │ - -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-16975-0/248
│ │ │ +using temporary file /tmp/M2-22611-0/246
│ │ │ + -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-22611-0/248
│ │ │  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/M2-16975-0/248
│ │ │ - -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-16975-0/250
│ │ │ +using temporary file /tmp/M2-22611-0/248
│ │ │ + -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-22611-0/250
│ │ │  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/M2-16975-0/250
│ │ │ - -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-16975-0/252
│ │ │ +using temporary file /tmp/M2-22611-0/250
│ │ │ + -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-22611-0/252
│ │ │  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:
│ │ │ @@ -574,24 +574,24 @@
│ │ │  --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/M2-16975-0/252
│ │ │ - -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-16975-0/254
│ │ │ +using temporary file /tmp/M2-22611-0/252
│ │ │ + -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-22611-0/254
│ │ │  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/M2-16975-0/254
│ │ │ - -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-16975-0/256
│ │ │ +using temporary file /tmp/M2-22611-0/254
│ │ │ + -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-22611-0/256
│ │ │  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.
│ │ │ @@ -611,32 +611,32 @@
│ │ │  --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/M2-16975-0/256
    
│ │ │ +using temporary file /tmp/M2-22611-0/256
│ │ │              
    
│ │ │                
    i6 : QQ[x,y];
    
│ │ │              
    
│ │ │                
    i7 : gfan {x,y};
│ │ │ - -- running: /usr/bin/gfan _bases < /tmp/M2-16975-0/258
│ │ │ + -- running: /usr/bin/gfan _bases < /tmp/M2-22611-0/258
│ │ │  Q[x1,x2]
│ │ │  {{
│ │ │  x2,
│ │ │  x1}
│ │ │  }
│ │ │ -using temporary file /tmp/M2-16975-0/258
    
│ │ │ +using temporary file /tmp/M2-22611-0/258
│ │ │              
    
│ │ │          
    
│ │ │            
    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 {}
│ │ │ │ @@ -43,15 +43,15 @@
│ │ │ │  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);
│ │ │ │   -- warning: reloading gfanInterface; recreate instances of types from this
│ │ │ │  package
│ │ │ │ - -- running: /usr/bin/gfan gfan --help < /tmp/M2-16975-0/172
│ │ │ │ + -- running: /usr/bin/gfan gfan --help < /tmp/M2-22611-0/172
│ │ │ │  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.
│ │ │ │ @@ -81,16 +81,16 @@
│ │ │ │   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/M2-16975-0/172
│ │ │ │ - -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-16975-0/174
│ │ │ │ +using temporary file /tmp/M2-22611-0/172
│ │ │ │ + -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-22611-0/174
│ │ │ │  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
│ │ │ │ @@ -111,63 +111,63 @@
│ │ │ │  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/M2-16975-0/174
│ │ │ │ - -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-16975-0/176
│ │ │ │ +using temporary file /tmp/M2-22611-0/174
│ │ │ │ + -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-22611-0/176
│ │ │ │  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/M2-16975-0/176
│ │ │ │ - -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-16975-0/178
│ │ │ │ +using temporary file /tmp/M2-22611-0/176
│ │ │ │ + -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-22611-0/178
│ │ │ │  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/M2-16975-0/178
│ │ │ │ - -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-16975-0/180
│ │ │ │ +using temporary file /tmp/M2-22611-0/178
│ │ │ │ + -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-22611-0/180
│ │ │ │  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/M2-16975-0/180
│ │ │ │ - -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-16975-0/182
│ │ │ │ +using temporary file /tmp/M2-22611-0/180
│ │ │ │ + -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-22611-0/182
│ │ │ │  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/M2-16975-0/182
│ │ │ │ - -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-16975-0/184
│ │ │ │ +using temporary file /tmp/M2-22611-0/182
│ │ │ │ + -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-22611-0/184
│ │ │ │  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.
│ │ │ │ @@ -184,24 +184,24 @@
│ │ │ │  --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/M2-16975-0/184
│ │ │ │ - -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-16975-0/186
│ │ │ │ +using temporary file /tmp/M2-22611-0/184
│ │ │ │ + -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-22611-0/186
│ │ │ │  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/M2-16975-0/186
│ │ │ │ - -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-16975-0/188
│ │ │ │ +using temporary file /tmp/M2-22611-0/186
│ │ │ │ + -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-22611-0/188
│ │ │ │  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}
│ │ │ │ @@ -217,16 +217,16 @@
│ │ │ │   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/M2-16975-0/188
│ │ │ │ - -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-16975-0/190
│ │ │ │ +using temporary file /tmp/M2-22611-0/188
│ │ │ │ + -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-22611-0/190
│ │ │ │  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
│ │ │ │ @@ -242,16 +242,16 @@
│ │ │ │  --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/M2-16975-0/190
│ │ │ │ - -- running: /usr/bin/gfan _interactive --help < /tmp/M2-16975-0/192
│ │ │ │ +using temporary file /tmp/M2-22611-0/190
│ │ │ │ + -- running: /usr/bin/gfan _interactive --help < /tmp/M2-22611-0/192
│ │ │ │  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
│ │ │ │ @@ -281,54 +281,54 @@
│ │ │ │  -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/M2-16975-0/192
│ │ │ │ - -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-16975-0/194
│ │ │ │ +using temporary file /tmp/M2-22611-0/192
│ │ │ │ + -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-22611-0/194
│ │ │ │  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/M2-16975-0/194
│ │ │ │ - -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-16975-0/196
│ │ │ │ +using temporary file /tmp/M2-22611-0/194
│ │ │ │ + -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-22611-0/196
│ │ │ │  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/M2-16975-0/196
│ │ │ │ - -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-16975-0/198
│ │ │ │ +using temporary file /tmp/M2-22611-0/196
│ │ │ │ + -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-22611-0/198
│ │ │ │  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/M2-16975-0/198
│ │ │ │ - -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-16975-0/200
│ │ │ │ +using temporary file /tmp/M2-22611-0/198
│ │ │ │ + -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-22611-0/200
│ │ │ │  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/M2-16975-0/200
│ │ │ │ - -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-16975-0/202
│ │ │ │ +using temporary file /tmp/M2-22611-0/200
│ │ │ │ + -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-22611-0/202
│ │ │ │  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/M2-16975-0/202
│ │ │ │ - -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-16975-0/204
│ │ │ │ +using temporary file /tmp/M2-22611-0/202
│ │ │ │ + -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-22611-0/204
│ │ │ │  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
│ │ │ │ @@ -338,16 +338,16 @@
│ │ │ │  --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/M2-16975-0/204
│ │ │ │ - -- running: /usr/bin/gfan _minors --help < /tmp/M2-16975-0/206
│ │ │ │ +using temporary file /tmp/M2-22611-0/204
│ │ │ │ + -- running: /usr/bin/gfan _minors --help < /tmp/M2-22611-0/206
│ │ │ │  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:
│ │ │ │ @@ -365,16 +365,16 @@
│ │ │ │   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/M2-16975-0/206
│ │ │ │ - -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-16975-0/208
│ │ │ │ +using temporary file /tmp/M2-22611-0/206
│ │ │ │ + -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-22611-0/208
│ │ │ │  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.
│ │ │ │ @@ -388,25 +388,25 @@
│ │ │ │   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/M2-16975-0/208
│ │ │ │ - -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-16975-0/210
│ │ │ │ +using temporary file /tmp/M2-22611-0/208
│ │ │ │ + -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-22611-0/210
│ │ │ │  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/M2-16975-0/210
│ │ │ │ - -- running: /usr/bin/gfan _render --help < /tmp/M2-16975-0/212
│ │ │ │ +using temporary file /tmp/M2-22611-0/210
│ │ │ │ + -- running: /usr/bin/gfan _render --help < /tmp/M2-22611-0/212
│ │ │ │  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.
│ │ │ │ @@ -414,16 +414,16 @@
│ │ │ │  -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/M2-16975-0/212
│ │ │ │ - -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-16975-0/214
│ │ │ │ +using temporary file /tmp/M2-22611-0/212
│ │ │ │ + -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-22611-0/214
│ │ │ │  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:
│ │ │ │ @@ -436,16 +436,16 @@
│ │ │ │  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/M2-16975-0/214
│ │ │ │ - -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-16975-0/216
│ │ │ │ +using temporary file /tmp/M2-22611-0/214
│ │ │ │ + -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-22611-0/216
│ │ │ │  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.
│ │ │ │ @@ -473,28 +473,28 @@
│ │ │ │  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/M2-16975-0/216
│ │ │ │ - -- running: /usr/bin/gfan _saturation --help < /tmp/M2-16975-0/218
│ │ │ │ +using temporary file /tmp/M2-22611-0/216
│ │ │ │ + -- running: /usr/bin/gfan _saturation --help < /tmp/M2-22611-0/218
│ │ │ │  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/M2-16975-0/218
│ │ │ │ - -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-16975-0/220
│ │ │ │ +using temporary file /tmp/M2-22611-0/218
│ │ │ │ + -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-22611-0/220
│ │ │ │  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
│ │ │ │ @@ -523,103 +523,103 @@
│ │ │ │   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/M2-16975-0/220
│ │ │ │ - -- running: /usr/bin/gfan _stats --help < /tmp/M2-16975-0/222
│ │ │ │ +using temporary file /tmp/M2-22611-0/220
│ │ │ │ + -- running: /usr/bin/gfan _stats --help < /tmp/M2-22611-0/222
│ │ │ │  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/M2-16975-0/222
│ │ │ │ - -- running: /usr/bin/gfan _substitute --help < /tmp/M2-16975-0/224
│ │ │ │ +using temporary file /tmp/M2-22611-0/222
│ │ │ │ + -- running: /usr/bin/gfan _substitute --help < /tmp/M2-22611-0/224
│ │ │ │  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/M2-16975-0/224
│ │ │ │ - -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-16975-0/226
│ │ │ │ +using temporary file /tmp/M2-22611-0/224
│ │ │ │ + -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-22611-0/226
│ │ │ │  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/M2-16975-0/226
│ │ │ │ - -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-16975-0/228
│ │ │ │ +using temporary file /tmp/M2-22611-0/226
│ │ │ │ + -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-22611-0/228
│ │ │ │  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/M2-16975-0/228
│ │ │ │ - -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-16975-0/230
│ │ │ │ +using temporary file /tmp/M2-22611-0/228
│ │ │ │ + -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-22611-0/230
│ │ │ │  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/M2-16975-0/230
│ │ │ │ - -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-16975-0/232
│ │ │ │ +using temporary file /tmp/M2-22611-0/230
│ │ │ │ + -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-22611-0/232
│ │ │ │  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/M2-16975-0/232
│ │ │ │ - -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-16975-0/234
│ │ │ │ +using temporary file /tmp/M2-22611-0/232
│ │ │ │ + -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-22611-0/234
│ │ │ │  This program evaluates a tropical polynomial function in a given set of points.
│ │ │ │  Options:
│ │ │ │ -using temporary file /tmp/M2-16975-0/234
│ │ │ │ - -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-16975-0/236
│ │ │ │ +using temporary file /tmp/M2-22611-0/234
│ │ │ │ + -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-22611-0/236
│ │ │ │  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/M2-16975-0/236
│ │ │ │ - -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-16975-0/238
│ │ │ │ +using temporary file /tmp/M2-22611-0/236
│ │ │ │ + -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-22611-0/238
│ │ │ │  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/M2-16975-0/238
│ │ │ │ - -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-16975-0/240
│ │ │ │ +using temporary file /tmp/M2-22611-0/238
│ │ │ │ + -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-22611-0/240
│ │ │ │  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:
│ │ │ │ @@ -656,16 +656,16 @@
│ │ │ │  --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/M2-16975-0/240
│ │ │ │ - -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-16975-0/242
│ │ │ │ +using temporary file /tmp/M2-22611-0/240
│ │ │ │ + -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-22611-0/242
│ │ │ │  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
│ │ │ │  
│ │ │ │ @@ -693,54 +693,54 @@
│ │ │ │  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/M2-16975-0/242
│ │ │ │ - -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-16975-0/244
│ │ │ │ +using temporary file /tmp/M2-22611-0/242
│ │ │ │ + -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-22611-0/244
│ │ │ │  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/M2-16975-0/244
│ │ │ │ - -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-16975-0/246
│ │ │ │ +using temporary file /tmp/M2-22611-0/244
│ │ │ │ + -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-22611-0/246
│ │ │ │  This program computes the multiplicity of a tropical cone given a marked
│ │ │ │  reduced Groebner basis for its initial ideal.
│ │ │ │  Options:
│ │ │ │ -using temporary file /tmp/M2-16975-0/246
│ │ │ │ - -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-16975-0/248
│ │ │ │ +using temporary file /tmp/M2-22611-0/246
│ │ │ │ + -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-22611-0/248
│ │ │ │  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/M2-16975-0/248
│ │ │ │ - -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-16975-0/250
│ │ │ │ +using temporary file /tmp/M2-22611-0/248
│ │ │ │ + -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-22611-0/250
│ │ │ │  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/M2-16975-0/250
│ │ │ │ - -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-16975-0/252
│ │ │ │ +using temporary file /tmp/M2-22611-0/250
│ │ │ │ + -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-22611-0/252
│ │ │ │  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
│ │ │ │ @@ -770,27 +770,27 @@
│ │ │ │  --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/M2-16975-0/252
│ │ │ │ - -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-16975-0/254
│ │ │ │ +using temporary file /tmp/M2-22611-0/252
│ │ │ │ + -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-22611-0/254
│ │ │ │  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/M2-16975-0/254
│ │ │ │ - -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-16975-0/256
│ │ │ │ +using temporary file /tmp/M2-22611-0/254
│ │ │ │ + -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-22611-0/256
│ │ │ │  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,
│ │ │ │ @@ -825,23 +825,23 @@
│ │ │ │   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/M2-16975-0/256
│ │ │ │ +using temporary file /tmp/M2-22611-0/256
│ │ │ │  i6 : QQ[x,y];
│ │ │ │  i7 : gfan {x,y};
│ │ │ │ - -- running: /usr/bin/gfan _bases < /tmp/M2-16975-0/258
│ │ │ │ + -- running: /usr/bin/gfan _bases < /tmp/M2-22611-0/258
│ │ │ │  Q[x1,x2]
│ │ │ │  {{
│ │ │ │  x2,
│ │ │ │  x1}
│ │ │ │  }
│ │ │ │ -using temporary file /tmp/M2-16975-0/258
│ │ │ │ +using temporary file /tmp/M2-22611-0/258
│ │ │ │  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.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/gfanInterface.m2:2630:0.
│ │ ├── ./usr/share/info/AInfinity.info.gz
│ │ │ ├── AInfinity.info
│ │ │ │ @@ -6133,16 +6133,16 @@
│ │ │ │  00017f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017f70: 2b0a 7c69 3320 3a20 656c 6170 7365 6454  +.|i3 : elapsedT
│ │ │ │  00017f80: 696d 6520 6275 726b 6552 6573 6f6c 7574  ime burkeResolut
│ │ │ │  00017f90: 696f 6e28 4d2c 2037 2c20 4368 6563 6b20  ion(M, 7, Check 
│ │ │ │  00017fa0: 3d3e 2066 616c 7365 2920 2020 2020 2020  => false)       
│ │ │ │ -00017fb0: 2020 2020 7c0a 7c20 2d2d 2032 2e30 3138      |.| -- 2.018
│ │ │ │ -00017fc0: 3035 7320 656c 6170 7365 6420 2020 2020  05s elapsed     
│ │ │ │ +00017fb0: 2020 2020 7c0a 7c20 2d2d 2031 2e34 3035      |.| -- 1.405
│ │ │ │ +00017fc0: 3634 7320 656c 6170 7365 6420 2020 2020  64s elapsed     
│ │ │ │  00017fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017ff0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  00018000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018030: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ @@ -6176,15 +6176,15 @@
│ │ │ │  000181f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018210: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20  --------+.|i4 : 
│ │ │ │  00018220: 656c 6170 7365 6454 696d 6520 6275 726b  elapsedTime burk
│ │ │ │  00018230: 6552 6573 6f6c 7574 696f 6e28 4d2c 2037  eResolution(M, 7
│ │ │ │  00018240: 2c20 4368 6563 6b20 3d3e 2074 7275 6529  , Check => true)
│ │ │ │  00018250: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00018260: 2d2d 2032 2e33 3631 3335 7320 656c 6170  -- 2.36135s elap
│ │ │ │ +00018260: 2d2d 2031 2e38 3133 3132 7320 656c 6170  -- 1.81312s elap
│ │ │ │  00018270: 7365 6420 2020 2020 2020 2020 2020 2020  sed             
│ │ │ │  00018280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000182a0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000182b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000182c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000182d0: 2020 2020 2020 2020 2020 2020 2020 2020
│ │ ├── ./usr/share/info/AdjunctionForSurfaces.info.gz
│ │ │ ├── AdjunctionForSurfaces.info
│ │ │ │ @@ -741,16 +741,16 @@
│ │ │ │  00002e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  00002e70: 7c69 3130 203a 2065 6c61 7073 6564 5469  |i10 : elapsedTi
│ │ │ │  00002e80: 6d65 2066 493d 7265 7320 4920 2020 2020  me fI=res I     
│ │ │ │  00002e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00002eb0: 2020 207c 0a7c 202d 2d20 2e30 3231 3636     |.| -- .02166
│ │ │ │ -00002ec0: 3731 7320 656c 6170 7365 6420 2020 2020  71s elapsed     
│ │ │ │ +00002eb0: 2020 207c 0a7c 202d 2d20 2e30 3237 3435     |.| -- .02745
│ │ │ │ +00002ec0: 3936 7320 656c 6170 7365 6420 2020 2020  96s elapsed     
│ │ │ │  00002ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002ef0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  00002f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002f30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ @@ -1596,15 +1596,15 @@
│ │ │ │  000063b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000063c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000063d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  000063e0: 7c69 3135 203a 2065 6c61 7073 6564 5469  |i15 : elapsedTi
│ │ │ │  000063f0: 6d65 2062 6574 7469 2849 273d 7472 696d  me betti(I'=trim
│ │ │ │  00006400: 206b 6572 2070 6869 2920 2020 2020 2020   ker phi)       
│ │ │ │  00006410: 2020 2020 2020 2020 207c 0a7c 202d 2d20           |.| -- 
│ │ │ │ -00006420: 2e35 3835 3435 3573 2065 6c61 7073 6564  .585455s elapsed
│ │ │ │ +00006420: 2e35 3630 3838 3973 2065 6c61 7073 6564  .560889s elapsed
│ │ │ │  00006430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006450: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00006460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006480: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00006490: 0a7c 2020 2020 2020 2020 2020 2020 2030  .|             0
│ │ │ │ @@ -1651,15 +1651,15 @@
│ │ │ │  00006720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00006730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00006740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00006750: 2d2d 2d2b 0a7c 6931 3720 3a20 656c 6170  ---+.|i17 : elap
│ │ │ │  00006760: 7365 6454 696d 6520 6261 7365 5074 733d  sedTime basePts=
│ │ │ │  00006770: 7072 696d 6172 7944 6563 6f6d 706f 7369  primaryDecomposi
│ │ │ │  00006780: 7469 6f6e 2069 6465 616c 2048 3b20 7c0a  tion ideal H; |.
│ │ │ │ -00006790: 7c20 2d2d 2035 2e37 3936 3235 7320 656c  | -- 5.79625s el
│ │ │ │ +00006790: 7c20 2d2d 2035 2e30 3232 3534 7320 656c  | -- 5.02254s el
│ │ │ │  000067a0: 6170 7365 6420 2020 2020 2020 2020 2020  apsed           
│ │ │ │  000067b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000067c0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │  000067d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000067e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000067f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00006800: 2d2d 2d2d 2b0a 7c69 3138 203a 2074 616c  ----+.|i18 : tal
│ │ │ │ @@ -2608,15 +2608,15 @@
│ │ │ │  0000a2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │  0000a320: 3134 203a 2065 6c61 7073 6564 5469 6d65  14 : elapsedTime
│ │ │ │  0000a330: 2073 7562 2849 2c48 2920 2020 2020 2020   sub(I,H)       
│ │ │ │  0000a340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a350: 2020 2020 2020 207c 0a7c 202d 2d20 2e30         |.| -- .0
│ │ │ │ -0000a360: 3133 3537 3633 7320 656c 6170 7365 6420  135763s elapsed 
│ │ │ │ +0000a360: 3139 3630 3937 7320 656c 6170 7365 6420  196097s elapsed 
│ │ │ │  0000a370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a390: 2020 7c0a 7c20 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 207c 0a7c               |.|
│ │ │ │  0000a3d0: 6f31 3420 3d20 6964 6561 6c20 2830 2c20  o14 = ideal (0, 
│ │ │ │ @@ -2648,15 +2648,15 @@
│ │ │ │  0000a570: 2d2d 2d2d 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 2d2b 0a7c 6931 3620 3a20 656c  -----+.|i16 : el
│ │ │ │  0000a5b0: 6170 7365 6454 696d 6520 6265 7474 6928  apsedTime betti(
│ │ │ │  0000a5c0: 4927 3d74 7269 6d20 6b65 7220 7068 6929  I'=trim ker phi)
│ │ │ │  0000a5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a5e0: 7c0a 7c20 2d2d 202e 3035 3439 3531 3373  |.| -- .0549513s
│ │ │ │ +0000a5e0: 7c0a 7c20 2d2d 202e 3037 3230 3630 3373  |.| -- .0720603s
│ │ │ │  0000a5f0: 2065 6c61 7073 6564 2020 2020 2020 2020   elapsed        
│ │ │ │  0000a600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a610: 2020 2020 2020 2020 2020 207c 0a7c 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 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ @@ -2700,15 +2700,15 @@
│ │ │ │  0000a8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  0000a8e0: 0a7c 6931 3820 3a20 656c 6170 7365 6454  .|i18 : elapsedT
│ │ │ │  0000a8f0: 696d 6520 6261 7365 5074 733d 7072 696d  ime basePts=prim
│ │ │ │  0000a900: 6172 7944 6563 6f6d 706f 7369 7469 6f6e  aryDecomposition
│ │ │ │  0000a910: 2069 6465 616c 2048 3b20 7c0a 7c20 2d2d   ideal H; |.| --
│ │ │ │ -0000a920: 2032 2e31 3232 3573 2065 6c61 7073 6564   2.1225s elapsed
│ │ │ │ +0000a920: 2031 2e35 3830 3973 2065 6c61 7073 6564   1.5809s elapsed
│ │ │ │  0000a930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a950: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  0000a960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a990: 2b0a 7c69 3139 203a 2074 616c 6c79 2061  +.|i19 : tally a
│ │ ├── ./usr/share/info/BGG.info.gz
│ │ │ ├── BGG.info
│ │ │ │ @@ -4338,16 +4338,16 @@
│ │ │ │  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 2b0a 7c69  ------------+.|i
│ │ │ │  00010f40: 3134 203a 2074 696d 6520 6265 7474 6920  14 : time betti 
│ │ │ │  00010f50: 2846 203d 2070 7572 6552 6573 6f6c 7574  (F = pureResolut
│ │ │ │  00010f60: 696f 6e28 4d2c 7b30 2c32 2c34 7d29 2920  ion(M,{0,2,4})) 
│ │ │ │  00010f70: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ -00010f80: 302e 3434 3737 3339 7320 2863 7075 293b  0.447739s (cpu);
│ │ │ │ -00010f90: 2030 2e33 3637 3033 3673 2028 7468 7265   0.367036s (thre
│ │ │ │ +00010f80: 302e 3534 3536 3534 7320 2863 7075 293b  0.545654s (cpu);
│ │ │ │ +00010f90: 2030 2e34 3437 3231 3673 2028 7468 7265   0.447216s (thre
│ │ │ │  00010fa0: 6164 293b 2030 7320 2867 6329 7c0a 7c20  ad); 0s (gc)|.| 
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00010ff0: 2020 2020 3020 3120 3220 2020 2020 2020      0 1 2       
│ │ │ │  00011000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -4403,16 +4403,16 @@
│ │ │ │  00011320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  00011350: 0a7c 6931 3520 3a20 7469 6d65 2062 6574  .|i15 : time bet
│ │ │ │  00011360: 7469 2028 4620 3d20 7075 7265 5265 736f  ti (F = pureReso
│ │ │ │  00011370: 6c75 7469 6f6e 2831 312c 342c 7b30 2c32  lution(11,4,{0,2
│ │ │ │  00011380: 2c34 7d29 2920 207c 0a7c 202d 2d20 7573  ,4}))  |.| -- us
│ │ │ │ -00011390: 6564 2030 2e34 3830 3138 3573 2028 6370  ed 0.480185s (cp
│ │ │ │ -000113a0: 7529 3b20 302e 3430 3335 3037 7320 2874  u); 0.403507s (t
│ │ │ │ +00011390: 6564 2030 2e35 3735 3235 3673 2028 6370  ed 0.575256s (cp
│ │ │ │ +000113a0: 7529 3b20 302e 3438 3338 3934 7320 2874  u); 0.483894s (t
│ │ │ │  000113b0: 6872 6561 6429 3b20 3073 2028 6763 297c  hread); 0s (gc)|
│ │ │ │  000113c0: 0a7c 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 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00011400: 2020 2020 2020 2030 2031 2032 2020 2020         0 1 2    
│ │ │ │  00011410: 2020 2020 2020 2020 2020 2020 2020 2020
│ │ ├── ./usr/share/info/Benchmark.info.gz
│ │ │ ├── Benchmark.info
│ │ │ │ @@ -200,71 +200,76 @@
│ │ │ │  00000c70: 2d2d 2d2d 2b0a 7c69 3120 3a20 7275 6e42  ----+.|i1 : runB
│ │ │ │  00000c80: 656e 6368 6d61 726b 7320 2272 6573 3339  enchmarks "res39
│ │ │ │  00000c90: 2220 2020 2020 2020 2020 2020 2020 2020  "               
│ │ │ │  00000ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00000cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00000cc0: 2020 2020 7c0a 7c2d 2d20 6265 6769 6e6e      |.|-- beginn
│ │ │ │  00000cd0: 696e 6720 636f 6d70 7574 6174 696f 6e20  ing computation 
│ │ │ │ -00000ce0: 4672 6920 4e6f 7620 3134 2031 373a 3331  Fri Nov 14 17:31
│ │ │ │ -00000cf0: 3a31 3120 5554 4320 3230 3235 2020 2020  :11 UTC 2025    
│ │ │ │ +00000ce0: 5468 7520 4465 6320 3138 2031 363a 3033  Thu Dec 18 16:03
│ │ │ │ +00000cf0: 3a32 3920 5554 4320 3230 3235 2020 2020  :29 UTC 2025    
│ │ │ │  00000d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00000d10: 2020 2020 7c0a 7c2d 2d20 4c69 6e75 7820      |.|-- Linux 
│ │ │ │ -00000d20: 7362 7569 6c64 2036 2e31 322e 3438 2b64  sbuild 6.12.48+d
│ │ │ │ -00000d30: 6562 3133 2d61 6d64 3634 2023 3120 534d  eb13-amd64 #1 SM
│ │ │ │ -00000d40: 5020 5052 4545 4d50 545f 4459 4e41 4d49  P PREEMPT_DYNAMI
│ │ │ │ -00000d50: 4320 4465 6269 616e 2036 2e31 322e 3438  C Debian 6.12.48
│ │ │ │ -00000d60: 2d31 2020 7c0a 7c2d 2d20 414d 4420 4550  -1  |.|-- AMD EP
│ │ │ │ -00000d70: 5943 2037 3730 3250 2036 342d 436f 7265  YC 7702P 64-Core
│ │ │ │ -00000d80: 2050 726f 6365 7373 6f72 2020 4175 7468   Processor  Auth
│ │ │ │ -00000d90: 656e 7469 6341 4d44 2020 6370 7520 4d48  enticAMD  cpu MH
│ │ │ │ -00000da0: 7a20 3139 3936 2e32 3530 2020 2020 2020  z 1996.250      
│ │ │ │ +00000d20: 7362 7569 6c64 2036 2e31 322e 3537 2b64  sbuild 6.12.57+d
│ │ │ │ +00000d30: 6562 3133 2d63 6c6f 7564 2d61 6d64 3634  eb13-cloud-amd64
│ │ │ │ +00000d40: 2023 3120 534d 5020 5052 4545 4d50 545f   #1 SMP PREEMPT_
│ │ │ │ +00000d50: 4459 4e41 4d49 4320 4465 6269 616e 2020  DYNAMIC Debian  
│ │ │ │ +00000d60: 2020 2020 7c0a 7c2d 2d20 496e 7465 6c20      |.|-- Intel 
│ │ │ │ +00000d70: 5865 6f6e 2050 726f 6365 7373 6f72 2028  Xeon Processor (
│ │ │ │ +00000d80: 536b 796c 616b 652c 2049 4252 5329 2020  Skylake, IBRS)  
│ │ │ │ +00000d90: 4765 6e75 696e 6549 6e74 656c 2020 6370  GenuineIntel  cp
│ │ │ │ +00000da0: 7520 4d48 7a20 3230 3939 2e39 3938 2020  u MHz 2099.998  
│ │ │ │  00000db0: 2020 2020 7c0a 7c2d 2d20 4d61 6361 756c      |.|-- Macaul
│ │ │ │  00000dc0: 6179 3220 312e 3235 2e31 312c 2063 6f6d  ay2 1.25.11, com
│ │ │ │  00000dd0: 7069 6c65 6420 7769 7468 2067 6363 2031  piled with gcc 1
│ │ │ │  00000de0: 352e 322e 3020 2020 2020 2020 2020 2020  5.2.0           
│ │ │ │  00000df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00000e00: 2020 2020 7c0a 7c2d 2d20 7265 7333 393a      |.|-- res39:
│ │ │ │  00000e10: 2072 6573 206f 6620 6120 6765 6e65 7269   res of a generi
│ │ │ │  00000e20: 6320 3320 6279 2039 206d 6174 7269 7820  c 3 by 9 matrix 
│ │ │ │ -00000e30: 6f76 6572 205a 5a2f 3130 313a 202e 3131  over ZZ/101: .11
│ │ │ │ -00000e40: 3535 3235 2073 6563 6f6e 6473 2020 2020  5525 seconds    
│ │ │ │ +00000e30: 6f76 6572 205a 5a2f 3130 313a 202e 3136  over ZZ/101: .16
│ │ │ │ +00000e40: 3432 3034 2073 6563 6f6e 6473 2020 2020  4204 seconds    
│ │ │ │  00000e50: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
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│ │ │ │  00000e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00000e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00000ea0: 2d2d 2d2d 7c0a 7c28 3230 3235 2d30 392d  ----|.|(2025-09-
│ │ │ │ -00000eb0: 3230 2920 7838 365f 3634 2047 4e55 2f4c  20) x86_64 GNU/L
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│ │ │ │ +00000ea0: 2d2d 2d2d 7c0a 7c36 2e31 322e 3537 2d31  ----|.|6.12.57-1
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│ │ │ │  00000ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00000ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00000ef0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
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│ │ │ │ -00000f40: 2d2d 2d2d 2b0a 0a46 6f72 2074 6865 2070  ----+..For the p
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│ │ │ │ -00000f70: 6520 6f62 6a65 6374 202a 6e6f 7465 2072  e object *note r
│ │ │ │ -00000f80: 756e 4265 6e63 686d 6172 6b73 3a20 7275  unBenchmarks: ru
│ │ │ │ -00000f90: 6e42 656e 6368 6d61 726b 732c 2069 7320  nBenchmarks, is 
│ │ │ │ -00000fa0: 6120 2a6e 6f74 6520 636f 6d6d 616e 643a  a *note command:
│ │ │ │ -00000fb0: 0a28 4d61 6361 756c 6179 3244 6f63 2943  .(Macaulay2Doc)C
│ │ │ │ -00000fc0: 6f6d 6d61 6e64 2c2e 0a0a 2d2d 2d2d 2d2d  ommand,...------
│ │ │ │ -00000fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00000fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00000ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00001000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00001010: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073  ---------..The s
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│ │ │ │ -00001030: 6375 6d65 6e74 2069 7320 696e 0a2f 6275  cument is in./bu
│ │ │ │ -00001040: 696c 642f 7265 7072 6f64 7563 6962 6c65  ild/reproducible
│ │ │ │ -00001050: 2d70 6174 682f 6d61 6361 756c 6179 322d  -path/macaulay2-
│ │ │ │ -00001060: 312e 3235 2e31 312b 6473 2f4d 322f 4d61  1.25.11+ds/M2/Ma
│ │ │ │ -00001070: 6361 756c 6179 322f 7061 636b 6167 6573  caulay2/packages
│ │ │ │ -00001080: 2f42 656e 6368 6d61 726b 2e0a 6d32 3a32  /Benchmark..m2:2
│ │ │ │ -00001090: 3937 3a30 2e0a 1f0a 5461 6720 5461 626c  97:0....Tag Tabl
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│ │ │ │ +00000f40: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
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│ │ │ │ +00000f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00000f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00000f90: 2d2d 2d2d 2b0a 0a46 6f72 2074 6865 2070  ----+..For the p
│ │ │ │ +00000fa0: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d  rogrammer.======
│ │ │ │ +00000fb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  ============..Th
│ │ │ │ +00000fc0: 6520 6f62 6a65 6374 202a 6e6f 7465 2072  e object *note r
│ │ │ │ +00000fd0: 756e 4265 6e63 686d 6172 6b73 3a20 7275  unBenchmarks: ru
│ │ │ │ +00000fe0: 6e42 656e 6368 6d61 726b 732c 2069 7320  nBenchmarks, is 
│ │ │ │ +00000ff0: 6120 2a6e 6f74 6520 636f 6d6d 616e 643a  a *note command:
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│ │ │ │ +00001040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00001080: 6375 6d65 6e74 2069 7320 696e 0a2f 6275  cument is in./bu
│ │ │ │ +00001090: 696c 642f 7265 7072 6f64 7563 6962 6c65  ild/reproducible
│ │ │ │ +000010a0: 2d70 6174 682f 6d61 6361 756c 6179 322d  -path/macaulay2-
│ │ │ │ +000010b0: 312e 3235 2e31 312b 6473 2f4d 322f 4d61  1.25.11+ds/M2/Ma
│ │ │ │ +000010c0: 6361 756c 6179 322f 7061 636b 6167 6573  caulay2/packages
│ │ │ │ +000010d0: 2f42 656e 6368 6d61 726b 2e0a 6d32 3a32  /Benchmark..m2:2
│ │ │ │ +000010e0: 3937 3a30 2e0a 1f0a 5461 6720 5461 626c  97:0....Tag Tabl
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│ │ │ │ +00001120: 5461 6720 5461 626c 650a                 Tag Table.
│ │ ├── ./usr/share/info/Bertini.info.gz
│ │ │ ├── Bertini.info
│ │ │ │ @@ -2253,16 +2253,16 @@
│ │ │ │  00008cc0: 616c 206e 756d 6265 720a 2020 2020 2020  al number.      
│ │ │ │  00008cd0: 2020 6f72 2072 616e 646f 6d20 636f 6d70    or random comp
│ │ │ │  00008ce0: 6c65 7820 6e75 6d62 6572 0a20 2020 2020  lex number.     
│ │ │ │  00008cf0: 202a 202a 6e6f 7465 2054 6f70 4469 7265   * *note TopDire
│ │ │ │  00008d00: 6374 6f72 793a 2054 6f70 4469 7265 6374  ctory: TopDirect
│ │ │ │  00008d10: 6f72 792c 203d 3e20 2e2e 2e2c 2064 6566  ory, => ..., def
│ │ │ │  00008d20: 6175 6c74 2076 616c 7565 0a20 2020 2020  ault value.     
│ │ │ │ -00008d30: 2020 2022 2f74 6d70 2f4d 322d 3238 3638     "/tmp/M2-2868
│ │ │ │ -00008d40: 372d 302f 3022 2c20 4f70 7469 6f6e 2074  7-0/0", Option t
│ │ │ │ +00008d30: 2020 2022 2f74 6d70 2f4d 322d 3430 3939     "/tmp/M2-4099
│ │ │ │ +00008d40: 382d 302f 3022 2c20 4f70 7469 6f6e 2074  8-0/0", Option t
│ │ │ │  00008d50: 6f20 6368 616e 6765 2064 6972 6563 746f  o change directo
│ │ │ │  00008d60: 7279 2066 6f72 2066 696c 6520 7374 6f72  ry for file stor
│ │ │ │  00008d70: 6167 652e 0a20 2020 2020 202a 202a 6e6f  age..      * *no
│ │ │ │  00008d80: 7465 2056 6572 626f 7365 3a20 6265 7274  te Verbose: bert
│ │ │ │  00008d90: 696e 6954 7261 636b 486f 6d6f 746f 7079  iniTrackHomotopy
│ │ │ │  00008da0: 5f6c 705f 7064 5f70 645f 7064 5f63 6d56  _lp_pd_pd_pd_cmV
│ │ │ │  00008db0: 6572 626f 7365 3d3e 5f70 645f 7064 5f70  erbose=>_pd_pd_p
│ │ │ │ @@ -4971,15 +4971,15 @@
│ │ │ │  000136a0: 6e74 6174 696f 6e29 203d 3e20 2e2e 2e2c  ntation) => ...,
│ │ │ │  000136b0: 2064 6566 6175 6c74 2076 616c 7565 207b   default value {
│ │ │ │  000136c0: 7d2c 200a 2020 2020 2020 2a20 2a6e 6f74  }, .      * *not
│ │ │ │  000136d0: 6520 546f 7044 6972 6563 746f 7279 3a20  e TopDirectory: 
│ │ │ │  000136e0: 546f 7044 6972 6563 746f 7279 2c20 3d3e  TopDirectory, =>
│ │ │ │  000136f0: 202e 2e2e 2c20 6465 6661 756c 7420 7661   ..., default va
│ │ │ │  00013700: 6c75 650a 2020 2020 2020 2020 222f 746d  lue.        "/tm
│ │ │ │ -00013710: 702f 4d32 2d32 3836 3837 2d30 2f30 222c  p/M2-28687-0/0",
│ │ │ │ +00013710: 702f 4d32 2d34 3039 3938 2d30 2f30 222c  p/M2-40998-0/0",
│ │ │ │  00013720: 204f 7074 696f 6e20 746f 2063 6861 6e67   Option to chang
│ │ │ │  00013730: 6520 6469 7265 6374 6f72 7920 666f 7220  e directory for 
│ │ │ │  00013740: 6669 6c65 2073 746f 7261 6765 2e0a 2020  file storage..  
│ │ │ │  00013750: 2020 2020 2a20 2a6e 6f74 6520 5665 7262      * *note Verb
│ │ │ │  00013760: 6f73 653a 2062 6572 7469 6e69 5472 6163  ose: bertiniTrac
│ │ │ │  00013770: 6b48 6f6d 6f74 6f70 795f 6c70 5f70 645f  kHomotopy_lp_pd_
│ │ │ │  00013780: 7064 5f70 645f 636d 5665 7262 6f73 653d  pd_pd_cmVerbose=
│ │ │ │ @@ -5472,16 +5472,16 @@
│ │ │ │  000155f0: 6561 6c20 6e75 6d62 6572 0a20 2020 2020  eal number.     
│ │ │ │  00015600: 2020 206f 7220 7261 6e64 6f6d 2063 6f6d     or random com
│ │ │ │  00015610: 706c 6578 206e 756d 6265 720a 2020 2020  plex number.    
│ │ │ │  00015620: 2020 2a20 2a6e 6f74 6520 546f 7044 6972    * *note TopDir
│ │ │ │  00015630: 6563 746f 7279 3a20 546f 7044 6972 6563  ectory: TopDirec
│ │ │ │  00015640: 746f 7279 2c20 3d3e 202e 2e2e 2c20 6465  tory, => ..., de
│ │ │ │  00015650: 6661 756c 7420 7661 6c75 650a 2020 2020  fault value.    
│ │ │ │ -00015660: 2020 2020 222f 746d 702f 4d32 2d32 3836      "/tmp/M2-286
│ │ │ │ -00015670: 3837 2d30 2f30 222c 204f 7074 696f 6e20  87-0/0", Option 
│ │ │ │ +00015660: 2020 2020 222f 746d 702f 4d32 2d34 3039      "/tmp/M2-409
│ │ │ │ +00015670: 3938 2d30 2f30 222c 204f 7074 696f 6e20  98-0/0", Option 
│ │ │ │  00015680: 746f 2063 6861 6e67 6520 6469 7265 6374  to change direct
│ │ │ │  00015690: 6f72 7920 666f 7220 6669 6c65 2073 746f  ory for file sto
│ │ │ │  000156a0: 7261 6765 2e0a 2020 2020 2020 2a20 5573  rage..      * Us
│ │ │ │  000156b0: 6552 6567 656e 6572 6174 696f 6e20 286d  eRegeneration (m
│ │ │ │  000156c0: 6973 7369 6e67 2064 6f63 756d 656e 7461  issing documenta
│ │ │ │  000156d0: 7469 6f6e 2920 3d3e 202e 2e2e 2c20 6465  tion) => ..., de
│ │ │ │  000156e0: 6661 756c 7420 7661 6c75 6520 2d31 2c20  fault value -1,
│ │ ├── ./usr/share/info/BettiCharacters.info.gz
│ │ │ ├── BettiCharacters.info
│ │ │ │ @@ -12972,15 +12972,15 @@
│ │ │ │  00032ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  00032ad0: 0a7c 6939 203a 2065 6c61 7073 6564 5469  .|i9 : elapsedTi
│ │ │ │  00032ae0: 6d65 2063 203d 2063 6861 7261 6374 6572  me c = character
│ │ │ │  00032af0: 2041 2020 2020 2020 2020 2020 2020 2020   A              
│ │ │ │  00032b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032b10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00032b20: 0a7c 202d 2d20 2e34 3730 3738 3573 2065  .| -- .470785s e
│ │ │ │ +00032b20: 0a7c 202d 2d20 2e34 3230 3131 3973 2065  .| -- .420119s e
│ │ │ │  00032b30: 6c61 7073 6564 2020 2020 2020 2020 2020  lapsed          
│ │ │ │  00032b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032b60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00032b70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00032b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -14183,15 +14183,15 @@
│ │ │ │  00037660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037680: 2b0a 7c69 3720 3a20 656c 6170 7365 6454  +.|i7 : elapsedT
│ │ │ │  00037690: 696d 6520 633d 6368 6172 6163 7465 7220  ime c=character 
│ │ │ │  000376a0: 4120 2020 2020 2020 2020 2020 2020 2020  A               
│ │ │ │  000376b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000376c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000376d0: 7c0a 7c20 2d2d 202e 3339 3139 3939 7320  |.| -- .391999s 
│ │ │ │ +000376d0: 7c0a 7c20 2d2d 202e 3436 3934 3533 7320  |.| -- .469453s 
│ │ │ │  000376e0: 656c 6170 7365 6420 2020 2020 2020 2020  elapsed         
│ │ │ │  000376f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037720: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00037730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -15614,16 +15614,16 @@
│ │ │ │  0003cfd0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │  0003cfe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003cff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003d000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003d010: 2d2b 0a7c 6932 3020 3a20 656c 6170 7365  -+.|i20 : elapse
│ │ │ │  0003d020: 6454 696d 6520 6131 203d 2063 6861 7261  dTime a1 = chara
│ │ │ │  0003d030: 6374 6572 2041 3120 2020 2020 2020 2020  cter A1         
│ │ │ │ -0003d040: 2020 2020 2020 7c0a 7c20 2d2d 202e 3735        |.| -- .75
│ │ │ │ -0003d050: 3034 3539 7320 656c 6170 7365 6420 2020  0459s elapsed   
│ │ │ │ +0003d040: 2020 2020 2020 7c0a 7c20 2d2d 202e 3832        |.| -- .82
│ │ │ │ +0003d050: 3539 3738 7320 656c 6170 7365 6420 2020  5978s elapsed   
│ │ │ │  0003d060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d070: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0003d080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d0b0: 7c0a 7c6f 3230 203d 2043 6861 7261 6374  |.|o20 = Charact
│ │ │ │  0003d0c0: 6572 206f 7665 7220 5220 2020 2020 2020  er over R       
│ │ │ │ @@ -15654,15 +15654,15 @@
│ │ │ │  0003d250: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │  0003d260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003d270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003d280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │  0003d290: 6932 3120 3a20 656c 6170 7365 6454 696d  i21 : elapsedTim
│ │ │ │  0003d2a0: 6520 6132 203d 2063 6861 7261 6374 6572  e a2 = character
│ │ │ │  0003d2b0: 2041 3220 2020 2020 2020 2020 2020 2020   A2             
│ │ │ │ -0003d2c0: 2020 7c0a 7c20 2d2d 2033 342e 3031 3035    |.| -- 34.0105
│ │ │ │ +0003d2c0: 2020 7c0a 7c20 2d2d 2032 392e 3338 3638    |.| -- 29.3868
│ │ │ │  0003d2d0: 7320 656c 6170 7365 6420 2020 2020 2020  s elapsed       
│ │ │ │  0003d2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d2f0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0003d300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d320: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │  0003d330: 3231 203d 2043 6861 7261 6374 6572 206f  21 = Character o
│ │ │ │ @@ -16112,16 +16112,16 @@
│ │ │ │  0003eef0: 6f33 3120 3a20 4163 7469 6f6e 4f6e 4772  o31 : ActionOnGr
│ │ │ │  0003ef00: 6164 6564 4d6f 6475 6c65 2020 2020 2020  adedModule      
│ │ │ │  0003ef10: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  0003ef20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ef30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │  0003ef40: 6933 3220 3a20 656c 6170 7365 6454 696d  i32 : elapsedTim
│ │ │ │  0003ef50: 6520 6220 3d20 6368 6172 6163 7465 7228  e b = character(
│ │ │ │ -0003ef60: 422c 3231 297c 0a7c 202d 2d20 3134 2e30  B,21)|.| -- 14.0
│ │ │ │ -0003ef70: 3833 3473 2065 6c61 7073 6564 2020 2020  834s elapsed    
│ │ │ │ +0003ef60: 422c 3231 297c 0a7c 202d 2d20 3132 2e34  B,21)|.| -- 12.4
│ │ │ │ +0003ef70: 3739 3273 2065 6c61 7073 6564 2020 2020  792s elapsed    
│ │ │ │  0003ef80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0003ef90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003efa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003efb0: 2020 2020 207c 0a7c 6f33 3220 3d20 4368       |.|o32 = Ch
│ │ │ │  0003efc0: 6172 6163 7465 7220 6f76 6572 2052 2020  aracter over R  
│ │ │ │  0003efd0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0003efe0: 2020 2020 2020 2020 2020 2020 2020 2020
│ │ ├── ./usr/share/info/Bruns.info.gz
│ │ │ ├── Bruns.info
│ │ │ │ @@ -1095,18 +1095,18 @@
│ │ │ │  00004460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00004470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │  00004480: 6932 3320 3a20 7469 6d65 206a 3d62 7275  i23 : time j=bru
│ │ │ │  00004490: 6e73 2046 2e64 645f 333b 2020 2020 2020  ns F.dd_3;      
│ │ │ │  000044a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000044b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000044c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000044d0: 202d 2d20 7573 6564 2030 2e32 3332 3833   -- used 0.23283
│ │ │ │ -000044e0: 3773 2028 6370 7529 3b20 302e 3137 3938  7s (cpu); 0.1798
│ │ │ │ -000044f0: 3036 7320 2874 6872 6561 6429 3b20 3073  06s (thread); 0s
│ │ │ │ -00004500: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │ +000044d0: 202d 2d20 7573 6564 2030 2e32 3939 3433   -- used 0.29943
│ │ │ │ +000044e0: 7320 2863 7075 293b 2030 2e32 3234 3739  s (cpu); 0.22479
│ │ │ │ +000044f0: 3873 2028 7468 7265 6164 293b 2030 7320  8s (thread); 0s 
│ │ │ │ +00004500: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │  00004510: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00004520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004560: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00004570: 6f32 3320 3a20 4964 6561 6c20 6f66 2053  o23 : Ideal of S
│ │ ├── ./usr/share/info/CellularResolutions.info.gz
│ │ │ ├── CellularResolutions.info
│ │ │ │ @@ -2973,27 +2973,27 @@
│ │ │ │  0000b9c0: 2020 2020 2020 2020 207c 0a7c 6f38 203d           |.|o8 =
│ │ │ │  0000b9d0: 2048 6173 6854 6162 6c65 7b30 203d 3e20   HashTable{0 => 
│ │ │ │  0000b9e0: 7b78 202c 2078 2079 2c20 7820 7920 2c20  {x , x y, x y , 
│ │ │ │  0000b9f0: 7820 7920 2c20 782a 7920 2c20 7820 7d20  x y , x*y , x } 
│ │ │ │  0000ba00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ba10: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0000ba20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000ba30: 2020 3520 3320 2020 3520 3420 2020 3420    5 3   5 4   4 
│ │ │ │ -0000ba40: 3220 2020 3420 3420 2020 3520 2020 2033  2   4 4   5    3
│ │ │ │ -0000ba50: 2033 2020 2035 2032 2020 2032 2034 2020   3   5 2   2 4  
│ │ │ │ -0000ba60: 2035 2033 2020 2020 207c 0a7c 2020 2020   5 3     |.|    
│ │ │ │ +0000ba30: 2020 3220 3420 2020 3520 3320 2020 3520    2 4   5 3   5 
│ │ │ │ +0000ba40: 3420 2020 3520 2020 2035 2032 2020 2035  4   5    5 2   5
│ │ │ │ +0000ba50: 2033 2020 2035 2034 2020 2034 2032 2020   3   5 4   4 2  
│ │ │ │ +0000ba60: 2034 2034 2020 2020 207c 0a7c 2020 2020   4 4     |.|    
│ │ │ │  0000ba70: 2020 2020 2020 2020 2020 2031 203d 3e20             1 => 
│ │ │ │  0000ba80: 7b78 2079 202c 2078 2079 202c 2078 2079  {x y , x y , x y
│ │ │ │ -0000ba90: 202c 2078 2079 202c 2078 2079 2c20 7820   , x y , x y, x 
│ │ │ │ +0000ba90: 202c 2078 2079 2c20 7820 7920 2c20 7820   , x y, x y , x 
│ │ │ │  0000baa0: 7920 2c20 7820 7920 2c20 7820 7920 2c20  y , x y , x y , 
│ │ │ │  0000bab0: 7820 7920 2c20 2020 207c 0a7c 2020 2020  x y ,    |.|    
│ │ │ │  0000bac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000bad0: 2020 3520 3420 2020 3520 3320 2020 3520    5 4   5 3   5 
│ │ │ │ -0000bae0: 3420 2020 3520 3220 2020 3520 3420 2020  4   5 2   5 4   
│ │ │ │ -0000baf0: 3520 3320 2020 3520 3420 2020 3520 3220  5 3   5 4   5 2 
│ │ │ │ +0000bad0: 2020 3520 3220 2020 3520 3420 2020 3520    5 2   5 4   5 
│ │ │ │ +0000bae0: 3320 2020 3520 3420 2020 3520 3220 2020  3   5 4   5 2   
│ │ │ │ +0000baf0: 3520 3420 2020 3520 3320 2020 3520 3420  5 4   5 3   5 4 
│ │ │ │  0000bb00: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0000bb10: 2020 2020 2020 2020 2020 2032 203d 3e20             2 => 
│ │ │ │  0000bb20: 7b78 2079 202c 2078 2079 202c 2078 2079  {x y , x y , x y
│ │ │ │  0000bb30: 202c 2078 2079 202c 2078 2079 202c 2078   , x y , x y , x
│ │ │ │  0000bb40: 2079 202c 2078 2079 202c 2078 2079 207d   y , x y , x y }
│ │ │ │  0000bb50: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0000bb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -3011,21 +3011,21 @@
│ │ │ │  0000bc20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000bc30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000bc40: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020  ---------|.|    
│ │ │ │  0000bc50: 2020 2020 2020 2020 2020 2020 7d20 2020              }   
│ │ │ │  0000bc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000bc90: 2020 2020 2020 2020 207c 0a7c 2035 2034           |.| 5 4
│ │ │ │ -0000bca0: 2020 2035 2020 2020 3520 3220 2020 2020     5    5 2     
│ │ │ │ +0000bc90: 2020 2020 2020 2020 207c 0a7c 2035 2020           |.| 5  
│ │ │ │ +0000bca0: 2020 3320 3320 2020 3520 3220 2020 2020    3 3   5 2     
│ │ │ │  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 207c 0a7c 7820 7920           |.|x y 
│ │ │ │ -0000bcf0: 2c20 7820 792c 2078 2079 207d 2020 2020  , x y, x y }    
│ │ │ │ +0000bce0: 2020 2020 2020 2020 207c 0a7c 7820 792c           |.|x y,
│ │ │ │ +0000bcf0: 2078 2079 202c 2078 2079 207d 2020 2020   x y , x y }    
│ │ │ │  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 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │  0000bd40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000bd50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000bd60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ @@ -6321,22 +6321,22 @@
│ │ │ │  00018b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018b10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00018b20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00018b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018b60: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00018b70: 2020 2020 3420 3520 2020 3220 2020 2020      4 5   2     
│ │ │ │ -00018b80: 2020 2032 2020 2020 3420 3420 2020 2035     2    4 4    5
│ │ │ │ -00018b90: 2033 2020 2020 3520 3420 2020 3320 3520   3    5 4   3 5 
│ │ │ │ +00018b70: 2020 2020 3520 3320 2020 2035 2034 2020      5 3    5 4  
│ │ │ │ +00018b80: 2033 2035 2020 2020 3420 3520 2020 3220   3 5    4 5   2 
│ │ │ │ +00018b90: 2020 2020 2020 2032 2020 2020 3420 3420         2    4 4 
│ │ │ │  00018ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018bb0: 2020 2020 207c 0a7c 6f35 203d 207b 7820       |.|o5 = {x 
│ │ │ │ -00018bc0: 7920 2c20 7820 792a 7a2c 2078 2a79 207a  y , x y*z, x*y z
│ │ │ │ -00018bd0: 2c20 7820 7920 7a2c 2078 2079 207a 2c20  , x y z, x y z, 
│ │ │ │ -00018be0: 7820 7920 2c20 7820 7920 7a7d 2020 2020  x y , x y z}    
│ │ │ │ +00018bc0: 7920 7a2c 2078 2079 202c 2078 2079 207a  y z, x y , x y z
│ │ │ │ +00018bd0: 2c20 7820 7920 2c20 7820 792a 7a2c 2078  , x y , x y*z, x
│ │ │ │ +00018be0: 2a79 207a 2c20 7820 7920 7a7d 2020 2020  *y z, x y z}    
│ │ │ │  00018bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018c00: 7c0a 7c20 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 2020                  
│ │ │ │  00018c40: 2020 2020 2020 2020 2020 207c 0a7c 6f35             |.|o5
│ │ │ │  00018c50: 203a 204c 6973 7420 2020 2020 2020 2020   : List
│ │ ├── ./usr/share/info/ChainComplexExtras.info.gz
│ │ │ ├── ChainComplexExtras.info
│ │ │ │ @@ -4819,16 +4819,16 @@
│ │ │ │  00012d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00012d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00012d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │  00012d50: 3133 203a 2074 696d 6520 6d20 3d20 6d69  13 : time m = mi
│ │ │ │  00012d60: 6e69 6d69 7a65 2028 455b 315d 293b 2020  nimize (E[1]);  
│ │ │ │  00012d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012d80: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ -00012d90: 302e 3237 3337 3032 7320 2863 7075 293b  0.273702s (cpu);
│ │ │ │ -00012da0: 2030 2e32 3133 3536 3973 2028 7468 7265   0.213569s (thre
│ │ │ │ +00012d90: 302e 3335 3434 3732 7320 2863 7075 293b  0.354472s (cpu);
│ │ │ │ +00012da0: 2030 2e32 3730 3037 3173 2028 7468 7265   0.270071s (thre
│ │ │ │  00012db0: 6164 293b 2030 7320 2867 6329 7c0a 2b2d  ad); 0s (gc)|.+-
│ │ │ │  00012dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00012dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00012de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00012df0: 2d2d 2d2d 2b0a 7c69 3134 203a 2069 7351  ----+.|i14 : isQ
│ │ │ │  00012e00: 7561 7369 4973 6f6d 6f72 7068 6973 6d20  uasiIsomorphism 
│ │ │ │  00012e10: 6d20 2020 2020 2020 2020 2020 2020 2020  m               
│ │ │ │ @@ -6579,33 +6579,33 @@
│ │ │ │  00019b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │  00019b40: 3820 3a20 7469 6d65 206d 203d 2072 6573  8 : time m = res
│ │ │ │  00019b50: 6f6c 7574 696f 6e4f 6643 6861 696e 436f  olutionOfChainCo
│ │ │ │  00019b60: 6d70 6c65 7820 433b 2020 2020 2020 2020  mplex C;        
│ │ │ │  00019b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019b80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00019b90: 2d2d 2075 7365 6420 302e 3039 3437 3737  -- used 0.094777
│ │ │ │ -00019ba0: 3473 2028 6370 7529 3b20 302e 3039 3437  4s (cpu); 0.0947
│ │ │ │ -00019bb0: 3737 3173 2028 7468 7265 6164 293b 2030  771s (thread); 0
│ │ │ │ -00019bc0: 7320 2867 6329 2020 2020 2020 2020 2020  s (gc)          
│ │ │ │ +00019b90: 2d2d 2075 7365 6420 302e 3130 3436 3731  -- used 0.104671
│ │ │ │ +00019ba0: 7320 2863 7075 293b 2030 2e31 3034 3637  s (cpu); 0.10467
│ │ │ │ +00019bb0: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ +00019bc0: 6763 2920 2020 2020 2020 2020 2020 2020  gc)             
│ │ │ │  00019bd0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │  00019be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │  00019c30: 3920 3a20 7469 6d65 206e 203d 2063 6172  9 : time n = car
│ │ │ │  00019c40: 7461 6e45 696c 656e 6265 7267 5265 736f  tanEilenbergReso
│ │ │ │  00019c50: 6c75 7469 6f6e 2043 3b20 2020 2020 2020  lution C;       
│ │ │ │  00019c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019c70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00019c80: 2d2d 2075 7365 6420 302e 3231 3532 3432  -- used 0.215242
│ │ │ │ -00019c90: 7320 2863 7075 293b 2030 2e31 3734 3238  s (cpu); 0.17428
│ │ │ │ -00019ca0: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ -00019cb0: 6763 2920 2020 2020 2020 2020 2020 2020  gc)             
│ │ │ │ +00019c80: 2d2d 2075 7365 6420 302e 3235 3238 3135  -- used 0.252815
│ │ │ │ +00019c90: 7320 2863 7075 293b 2030 2e31 3637 3737  s (cpu); 0.16777
│ │ │ │ +00019ca0: 3673 2028 7468 7265 6164 293b 2030 7320  6s (thread); 0s 
│ │ │ │ +00019cb0: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │  00019cc0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │  00019cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │  00019d20: 3130 203a 2062 6574 7469 2073 6f75 7263  10 : betti sourc
│ │ ├── ./usr/share/info/CharacteristicClasses.info.gz
│ │ │ ├── CharacteristicClasses.info
│ │ │ │ @@ -1215,17 +1215,17 @@
│ │ │ │  00004be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00004bf0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a  ---------+.|i3 :
│ │ │ │  00004c00: 2074 696d 6520 4353 4d20 5520 2020 2020   time CSM U     
│ │ │ │  00004c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004c40: 2020 2020 2020 2020 207c 0a7c 202d 2d20           |.| -- 
│ │ │ │ -00004c50: 7573 6564 2030 2e32 3332 3737 7320 2863  used 0.23277s (c
│ │ │ │ -00004c60: 7075 293b 2030 2e31 3534 3631 7320 2874  pu); 0.15461s (t
│ │ │ │ -00004c70: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │ +00004c50: 7573 6564 2030 2e32 3635 3831 7320 2863  used 0.26581s (c
│ │ │ │ +00004c60: 7075 293b 2030 2e31 3634 3230 3373 2028  pu); 0.164203s (
│ │ │ │ +00004c70: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 0s (gc)
│ │ │ │  00004c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004c90: 2020 2020 2020 2020 207c 0a7c 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 2020                  
│ │ │ │  00004cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004ce0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ @@ -1300,16 +1300,16 @@
│ │ │ │  00005130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00005140: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a  ---------+.|i4 :
│ │ │ │  00005150: 2074 696d 6520 4353 4d28 552c 4368 6563   time CSM(U,Chec
│ │ │ │  00005160: 6b53 6d6f 6f74 683d 3e66 616c 7365 2920  kSmooth=>false) 
│ │ │ │  00005170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005190: 2020 2020 2020 2020 207c 0a7c 202d 2d20           |.| -- 
│ │ │ │ -000051a0: 7573 6564 2030 2e33 3539 3132 3473 2028  used 0.359124s (
│ │ │ │ -000051b0: 6370 7529 3b20 302e 3239 3135 3238 7320  cpu); 0.291528s 
│ │ │ │ +000051a0: 7573 6564 2030 2e34 3134 3934 3473 2028  used 0.414944s (
│ │ │ │ +000051b0: 6370 7529 3b20 302e 3331 3136 3432 7320  cpu); 0.311642s 
│ │ │ │  000051c0: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │  000051d0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  000051e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000051f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -4341,18 +4341,18 @@
│ │ │ │  00010f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010f60: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a  ---------+.|i5 :
│ │ │ │  00010f70: 2074 696d 6520 4353 4d28 492c 436f 6d70   time CSM(I,Comp
│ │ │ │  00010f80: 4d65 7468 6f64 3d3e 5072 6f6a 6563 7469  Method=>Projecti
│ │ │ │  00010f90: 7665 4465 6772 6565 2920 2020 2020 2020  veDegree)       
│ │ │ │  00010fa0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00010fb0: 2d2d 2075 7365 6420 302e 3632 3839 3531  -- used 0.628951
│ │ │ │ -00010fc0: 7320 2863 7075 293b 2030 2e33 3036 3237  s (cpu); 0.30627
│ │ │ │ -00010fd0: 3973 2028 7468 7265 6164 293b 2030 7320  9s (thread); 0s 
│ │ │ │ -00010fe0: 2867 6329 2020 2020 2020 2020 2020 207c  (gc)           |
│ │ │ │ +00010fb0: 2d2d 2075 7365 6420 312e 3236 3635 3573  -- used 1.26655s
│ │ │ │ +00010fc0: 2028 6370 7529 3b20 302e 3432 3730 3034   (cpu); 0.427004
│ │ │ │ +00010fd0: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ +00010fe0: 6763 2920 2020 2020 2020 2020 2020 207c  gc)            |
│ │ │ │  00010ff0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00011000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011030: 2020 7c0a 7c20 2020 2020 2020 3520 2020    |.|       5   
│ │ │ │  00011040: 2020 2034 2020 2020 2020 3320 2020 2020     4      3     
│ │ │ │  00011050: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ @@ -4400,16 +4400,16 @@
│ │ │ │  000112f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011310: 2d2d 2d2b 0a7c 6936 203a 2074 696d 6520  ---+.|i6 : time 
│ │ │ │  00011320: 4353 4d28 492c 436f 6d70 4d65 7468 6f64  CSM(I,CompMethod
│ │ │ │  00011330: 3d3e 506e 5265 7369 6475 616c 2920 2020  =>PnResidual)   
│ │ │ │  00011340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011350: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365        |.| -- use
│ │ │ │ -00011360: 6420 322e 3237 3532 3973 2028 6370 7529  d 2.27529s (cpu)
│ │ │ │ -00011370: 3b20 312e 3837 3834 3373 2028 7468 7265  ; 1.87843s (thre
│ │ │ │ +00011360: 6420 322e 3530 3337 3673 2028 6370 7529  d 2.50376s (cpu)
│ │ │ │ +00011370: 3b20 322e 3138 3235 3973 2028 7468 7265  ; 2.18259s (thre
│ │ │ │  00011380: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │  00011390: 2020 2020 2020 2020 207c 0a7c 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 2020                  
│ │ │ │  000113d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000113e0: 2020 2020 2020 3520 2020 2020 2034 2020        5      4  
│ │ │ │ @@ -4488,17 +4488,17 @@
│ │ │ │  00011870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011890: 2d2d 2b0a 7c69 3130 203a 2074 696d 6520  --+.|i10 : time 
│ │ │ │  000118a0: 4353 4d28 4b2c 436f 6d70 4d65 7468 6f64  CSM(K,CompMethod
│ │ │ │  000118b0: 3d3e 5072 6f6a 6563 7469 7665 4465 6772  =>ProjectiveDegr
│ │ │ │  000118c0: 6565 2920 2020 2020 2020 2020 2020 2020  ee)             
│ │ │ │  000118d0: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ -000118e0: 2030 2e32 3737 3838 3573 2028 6370 7529   0.277885s (cpu)
│ │ │ │ -000118f0: 3b20 302e 3230 3430 3373 2028 7468 7265  ; 0.20403s (thre
│ │ │ │ -00011900: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │ +000118e0: 2030 2e33 3637 3032 3373 2028 6370 7529   0.367023s (cpu)
│ │ │ │ +000118f0: 3b20 302e 3234 3634 3239 7320 2874 6872  ; 0.246429s (thr
│ │ │ │ +00011900: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │  00011910: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  00011920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011950: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00011960: 2020 2020 2020 3320 2020 2020 3220 2020        3     2   
│ │ │ │  00011970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -4546,18 +4546,18 @@
│ │ │ │  00011c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │  00011c40: 3131 203a 2074 696d 6520 4353 4d28 4b2c  11 : time CSM(K,
│ │ │ │  00011c50: 436f 6d70 4d65 7468 6f64 3d3e 506e 5265  CompMethod=>PnRe
│ │ │ │  00011c60: 7369 6475 616c 2920 2020 2020 2020 2020  sidual)         
│ │ │ │  00011c70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00011c80: 0a7c 202d 2d20 7573 6564 2030 2e30 3831  .| -- used 0.081
│ │ │ │ -00011c90: 3931 3534 7320 2863 7075 293b 2030 2e30  9154s (cpu); 0.0
│ │ │ │ -00011ca0: 3831 3932 3332 7320 2874 6872 6561 6429  819232s (thread)
│ │ │ │ -00011cb0: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │ +00011c80: 0a7c 202d 2d20 7573 6564 2030 2e31 3030  .| -- used 0.100
│ │ │ │ +00011c90: 3339 3973 2028 6370 7529 3b20 302e 3130  399s (cpu); 0.10
│ │ │ │ +00011ca0: 3034 3039 7320 2874 6872 6561 6429 3b20  0409s (thread); 
│ │ │ │ +00011cb0: 3073 2028 6763 2920 2020 2020 2020 2020  0s (gc)         
│ │ │ │  00011cc0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00011cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011d00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00011d10: 3320 2020 2020 3220 2020 2020 2020 2020  3     2         
│ │ │ │  00011d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -5543,17 +5543,17 @@
│ │ │ │  00015a60: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00015a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015a90: 2d2d 2d2d 2d2d 2b0a 7c69 3135 203a 2074  ------+.|i15 : t
│ │ │ │  00015aa0: 696d 6520 6373 6d4b 3d43 534d 2841 2c4b  ime csmK=CSM(A,K
│ │ │ │  00015ab0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  00015ac0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00015ad0: 202d 2d20 7573 6564 2030 2e36 3630 3035   -- used 0.66005
│ │ │ │ -00015ae0: 7320 2863 7075 293b 2030 2e33 3836 3930  s (cpu); 0.38690
│ │ │ │ -00015af0: 3573 2028 7468 7265 6164 293b 2030 7320  5s (thread); 0s 
│ │ │ │ +00015ad0: 202d 2d20 7573 6564 2031 2e32 3633 3936   -- used 1.26396
│ │ │ │ +00015ae0: 7320 2863 7075 293b 2030 2e34 3438 3230  s (cpu); 0.44820
│ │ │ │ +00015af0: 3873 2028 7468 7265 6164 293b 2030 7320  8s (thread); 0s 
│ │ │ │  00015b00: 2867 6329 7c0a 7c20 2020 2020 2020 2020  (gc)|.|         
│ │ │ │  00015b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015b30: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00015b40: 2020 2020 2020 3220 3220 2020 2020 3220        2 2     2 
│ │ │ │  00015b50: 2020 2020 2020 2020 3220 2020 2032 2020          2    2  
│ │ │ │  00015b60: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ @@ -5721,17 +5721,17 @@
│ │ │ │  00016580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000165a0: 2d2d 2d2d 2d2d 2b0a 7c69 3232 203a 2074  ------+.|i22 : t
│ │ │ │  000165b0: 696d 6520 4353 4d28 412c 4b2c 6d29 2020  ime CSM(A,K,m)  
│ │ │ │  000165c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000165d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000165e0: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ -000165f0: 302e 3039 3135 3831 3973 2028 6370 7529  0.0915819s (cpu)
│ │ │ │ -00016600: 3b20 302e 3036 3030 3634 3273 2028 7468  ; 0.0600642s (th
│ │ │ │ -00016610: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │ +000165f0: 302e 3130 3533 3436 7320 2863 7075 293b  0.105346s (cpu);
│ │ │ │ +00016600: 2030 2e30 3733 3430 3239 7320 2874 6872   0.0734029s (thr
│ │ │ │ +00016610: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │  00016620: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00016630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016660: 7c0a 7c20 2020 2020 2020 2032 2032 2020  |.|        2 2  
│ │ │ │  00016670: 2020 2032 2020 2020 2020 2020 2032 2020     2         2  
│ │ │ │  00016680: 2020 3220 2020 2020 2020 2020 2020 2032    2            2
│ │ │ │ @@ -6691,17 +6691,17 @@
│ │ │ │  0001a220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a230: 2d2d 2b0a 7c69 3420 3a20 7469 6d65 2045  --+.|i4 : time E
│ │ │ │  0001a240: 756c 6572 2849 2c49 6e70 7574 4973 536d  uler(I,InputIsSm
│ │ │ │  0001a250: 6f6f 7468 3d3e 7472 7565 2920 2020 2020  ooth=>true)     
│ │ │ │  0001a260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a280: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e    |.| -- used 0.
│ │ │ │ -0001a290: 3037 3836 3232 3373 2028 6370 7529 3b20  0786223s (cpu); 
│ │ │ │ -0001a2a0: 302e 3033 3539 3036 3973 2028 7468 7265  0.0359069s (thre
│ │ │ │ -0001a2b0: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │ +0001a290: 3039 3235 3131 7320 2863 7075 293b 2030  092511s (cpu); 0
│ │ │ │ +0001a2a0: 2e30 3631 3036 3632 7320 2874 6872 6561  .0610662s (threa
│ │ │ │ +0001a2b0: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │  0001a2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a2d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0001a2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a320: 2020 7c0a 7c6f 3420 3d20 3420 2020 2020    |.|o4 = 4     
│ │ │ │ @@ -6716,16 +6716,16 @@
│ │ │ │  0001a3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a3c0: 2d2d 2b0a 7c69 3520 3a20 7469 6d65 2045  --+.|i5 : time E
│ │ │ │  0001a3d0: 756c 6572 2049 2020 2020 2020 2020 2020  uler I          
│ │ │ │  0001a3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a410: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e    |.| -- used 0.
│ │ │ │ -0001a420: 3235 3639 3636 7320 2863 7075 293b 2030  256966s (cpu); 0
│ │ │ │ -0001a430: 2e31 3532 3633 3673 2028 7468 7265 6164  .152636s (thread
│ │ │ │ +0001a420: 3331 3330 3238 7320 2863 7075 293b 2030  313028s (cpu); 0
│ │ │ │ +0001a430: 2e31 3736 3639 3473 2028 7468 7265 6164  .176694s (thread
│ │ │ │  0001a440: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │  0001a450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a460: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0001a470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -6919,16 +6919,16 @@
│ │ │ │  0001b060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b080: 2d2d 2d2d 2d2d 2b0a 7c69 3130 203a 2074  ------+.|i10 : t
│ │ │ │  0001b090: 696d 6520 4575 6c65 7228 4a2c 4d65 7468  ime Euler(J,Meth
│ │ │ │  0001b0a0: 6f64 3d3e 4469 7265 6374 436f 6d70 6c65  od=>DirectComple
│ │ │ │  0001b0b0: 7465 496e 7429 2020 2020 2020 2020 2020  teInt)          
│ │ │ │  0001b0c0: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d            |.| --
│ │ │ │ -0001b0d0: 2075 7365 6420 302e 3133 3336 3731 7320   used 0.133671s 
│ │ │ │ -0001b0e0: 2863 7075 293b 2030 2e30 3734 3832 3937  (cpu); 0.0748297
│ │ │ │ +0001b0d0: 2075 7365 6420 302e 3233 3032 3237 7320   used 0.230227s 
│ │ │ │ +0001b0e0: 2863 7075 293b 2030 2e30 3933 3135 3736  (cpu); 0.0931576
│ │ │ │  0001b0f0: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │  0001b100: 6763 2920 2020 2020 2020 2020 2020 7c0a  gc)           |.
│ │ │ │  0001b110: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0001b120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b150: 2020 7c0a 7c6f 3130 203d 2032 2020 2020    |.|o10 = 2    
│ │ │ │ @@ -6940,18 +6940,18 @@
│ │ │ │  0001b1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131  ----------+.|i11
│ │ │ │  0001b1e0: 203a 2074 696d 6520 4575 6c65 7228 4a2c   : time Euler(J,
│ │ │ │  0001b1f0: 4d65 7468 6f64 3d3e 4469 7265 6374 436f  Method=>DirectCo
│ │ │ │  0001b200: 6d70 6c65 7465 496e 742c 496e 6473 4f66  mpleteInt,IndsOf
│ │ │ │  0001b210: 536d 6f6f 7468 3d3e 7b30 2c31 7d29 7c0a  Smooth=>{0,1})|.
│ │ │ │ -0001b220: 7c20 2d2d 2075 7365 6420 302e 3232 3439  | -- used 0.2249
│ │ │ │ -0001b230: 3533 7320 2863 7075 293b 2030 2e31 3032  53s (cpu); 0.102
│ │ │ │ -0001b240: 3936 3773 2028 7468 7265 6164 293b 2030  967s (thread); 0
│ │ │ │ -0001b250: 7320 2867 6329 2020 2020 2020 2020 2020  s (gc)          
│ │ │ │ +0001b220: 7c20 2d2d 2075 7365 6420 302e 3238 3230  | -- used 0.2820
│ │ │ │ +0001b230: 3838 7320 2863 7075 293b 2030 2e31 3132  88s (cpu); 0.112
│ │ │ │ +0001b240: 3233 7320 2874 6872 6561 6429 3b20 3073  23s (thread); 0s
│ │ │ │ +0001b250: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │  0001b260: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0001b270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b2a0: 2020 2020 2020 7c0a 7c6f 3131 203d 2032        |.|o11 = 2
│ │ │ │  0001b2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -7574,16 +7574,16 @@
│ │ │ │  0001d950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  0001d980: 0a7c 6934 203a 2074 696d 6520 4575 6c65  .|i4 : time Eule
│ │ │ │  0001d990: 7241 6666 696e 6520 4920 2020 2020 2020  rAffine I       
│ │ │ │  0001d9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d9b0: 2020 2020 2020 2020 207c 0a7c 202d 2d20           |.| -- 
│ │ │ │ -0001d9c0: 7573 6564 2030 2e30 3630 3336 3833 7320  used 0.0603683s 
│ │ │ │ -0001d9d0: 2863 7075 293b 2030 2e30 3535 3038 3331  (cpu); 0.0550831
│ │ │ │ +0001d9c0: 7573 6564 2030 2e30 3930 3430 3631 7320  used 0.0904061s 
│ │ │ │ +0001d9d0: 2863 7075 293b 2030 2e30 3637 3235 3039  (cpu); 0.0672509
│ │ │ │  0001d9e0: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │  0001d9f0: 6763 297c 0a7c 2020 2020 2020 2020 2020  gc)|.|          
│ │ │ │  0001da00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001da10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001da20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0001da30: 6f34 203d 2032 2020 2020 2020 2020 2020  o4 = 2          
│ │ │ │  0001da40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -7717,17 +7717,17 @@
│ │ │ │  0001e240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e260: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │  0001e270: 7469 6d65 2043 534d 2849 2c4d 6574 686f  time CSM(I,Metho
│ │ │ │  0001e280: 643d 3e44 6972 6563 7443 6f6d 706c 6574  d=>DirectComplet
│ │ │ │  0001e290: 496e 7429 2020 2020 2020 2020 2020 2020  Int)            
│ │ │ │  0001e2a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001e2b0: 0a7c 202d 2d20 7573 6564 2032 2e32 3638  .| -- used 2.268
│ │ │ │ -0001e2c0: 3537 7320 2863 7075 293b 2031 2e31 3631  57s (cpu); 1.161
│ │ │ │ -0001e2d0: 3936 7320 2874 6872 6561 6429 3b20 3073  96s (thread); 0s
│ │ │ │ +0001e2b0: 0a7c 202d 2d20 7573 6564 2036 2e37 3931  .| -- used 6.791
│ │ │ │ +0001e2c0: 3031 7320 2863 7075 293b 2031 2e34 3638  01s (cpu); 1.468
│ │ │ │ +0001e2d0: 3836 7320 2874 6872 6561 6429 3b20 3073  86s (thread); 0s
│ │ │ │  0001e2e0: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │  0001e2f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001e300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e330: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0001e340: 2020 2020 2020 2032 2032 2020 2020 2032         2 2     2
│ │ │ │ @@ -7779,18 +7779,18 @@
│ │ │ │  0001e620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e640: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420  ----------+.|i4 
│ │ │ │  0001e650: 3a20 7469 6d65 2043 534d 2849 2c4d 6574  : time CSM(I,Met
│ │ │ │  0001e660: 686f 643d 3e44 6972 6563 7443 6f6d 706c  hod=>DirectCompl
│ │ │ │  0001e670: 6574 496e 742c 496e 6473 4f66 536d 6f6f  etInt,IndsOfSmoo
│ │ │ │  0001e680: 7468 3d3e 7b31 2c32 7d29 2020 2020 2020  th=>{1,2})      
│ │ │ │ -0001e690: 207c 0a7c 202d 2d20 7573 6564 2032 2e38   |.| -- used 2.8
│ │ │ │ -0001e6a0: 3539 3834 7320 2863 7075 293b 2031 2e32  5984s (cpu); 1.2
│ │ │ │ -0001e6b0: 3337 3736 7320 2874 6872 6561 6429 3b20  3776s (thread); 
│ │ │ │ -0001e6c0: 3073 2028 6763 2920 2020 2020 2020 2020  0s (gc)         
│ │ │ │ +0001e690: 207c 0a7c 202d 2d20 7573 6564 2036 2e36   |.| -- used 6.6
│ │ │ │ +0001e6a0: 3031 3631 7320 2863 7075 293b 2031 2e35  0161s (cpu); 1.5
│ │ │ │ +0001e6b0: 3237 3873 2028 7468 7265 6164 293b 2030  278s (thread); 0
│ │ │ │ +0001e6c0: 7320 2867 6329 2020 2020 2020 2020 2020  s (gc)          
│ │ │ │  0001e6d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  0001e6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e710: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0001e720: 0a7c 2020 2020 2020 2032 2032 2020 2020  .|       2 2    
│ │ │ │  0001e730: 2032 2020 2020 2020 2020 2032 2020 2020   2         2    
│ │ │ │ @@ -7930,17 +7930,17 @@
│ │ │ │  0001ef90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001efa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001efb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933  -----------+.|i3
│ │ │ │  0001efc0: 203a 2074 696d 6520 4353 4d20 4920 2020   : time CSM I   
│ │ │ │  0001efd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001efe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001eff0: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ -0001f000: 2030 2e36 3538 3336 3873 2028 6370 7529   0.658368s (cpu)
│ │ │ │ -0001f010: 3b20 302e 3339 3639 3132 7320 2874 6872  ; 0.396912s (thr
│ │ │ │ -0001f020: 6561 6429 3b20 3073 2028 6763 2920 207c  ead); 0s (gc)  |
│ │ │ │ +0001f000: 2031 2e31 3830 3538 7320 2863 7075 293b   1.18058s (cpu);
│ │ │ │ +0001f010: 2030 2e35 3133 3935 3273 2028 7468 7265   0.513952s (thre
│ │ │ │ +0001f020: 6164 293b 2030 7320 2867 6329 2020 207c  ad); 0s (gc)   |
│ │ │ │  0001f030: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0001f040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f060: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0001f070: 2020 2033 2020 2020 2020 2020 2020 2020     3            
│ │ │ │  0001f080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -7981,16 +7981,16 @@
│ │ │ │  0001f2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f2e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2074  -------+.|i4 : t
│ │ │ │  0001f2f0: 696d 6520 4353 4d28 492c 496e 7075 7449  ime CSM(I,InputI
│ │ │ │  0001f300: 7353 6d6f 6f74 683d 3e74 7275 6529 2020  sSmooth=>true)  
│ │ │ │  0001f310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f320: 207c 0a7c 202d 2d20 7573 6564 2030 2e30   |.| -- used 0.0
│ │ │ │ -0001f330: 3436 3432 3131 7320 2863 7075 293b 2030  464211s (cpu); 0
│ │ │ │ -0001f340: 2e30 3332 3733 3436 7320 2874 6872 6561  .0327346s (threa
│ │ │ │ +0001f330: 3939 3236 3936 7320 2863 7075 293b 2030  992696s (cpu); 0
│ │ │ │ +0001f340: 2e30 3430 3934 3832 7320 2874 6872 6561  .0409482s (threa
│ │ │ │  0001f350: 6429 3b20 3073 2028 6763 297c 0a7c 2020  d); 0s (gc)|.|  
│ │ │ │  0001f360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f390: 2020 2020 207c 0a7c 2020 2020 2020 2033       |.|       3
│ │ │ │  0001f3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -8036,4200 +8036,4199 @@
│ │ │ │  0001f630: 616c 656e 746c 792c 2075 7365 2074 6865  alently, use the
│ │ │ │  0001f640: 2063 6f6d 6d61 6e64 202a 6e6f 7465 2043   command *note C
│ │ │ │  0001f650: 6865 726e 3a20 4368 6572 6e2c 2069 6e73  hern: Chern, ins
│ │ │ │  0001f660: 7465 6164 0a69 6e20 7468 6973 2063 6173  tead.in this cas
│ │ │ │  0001f670: 652e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  e...+-----------
│ │ │ │  0001f680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0001f6b0: 3520 3a20 7469 6d65 2043 6865 726e 2049  5 : time Chern I
│ │ │ │ +0001f6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935  -----------+.|i5
│ │ │ │ +0001f6b0: 203a 2074 696d 6520 4368 6572 6e20 4920   : time Chern I 
│ │ │ │  0001f6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f6e0: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365        |.| -- use
│ │ │ │ -0001f6f0: 6420 302e 3033 3830 3232 3873 2028 6370  d 0.0380228s (cp
│ │ │ │ -0001f700: 7529 3b20 302e 3033 3431 3539 3973 2028  u); 0.0341599s (
│ │ │ │ -0001f710: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 0s (gc)
│ │ │ │ -0001f720: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0001f6e0: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ +0001f6f0: 302e 3037 3231 3931 3773 2028 6370 7529  0.0721917s (cpu)
│ │ │ │ +0001f700: 3b20 302e 3034 3331 3738 7320 2874 6872  ; 0.043178s (thr
│ │ │ │ +0001f710: 6561 6429 3b20 3073 2028 6763 297c 0a7c  ead); 0s (gc)|.|
│ │ │ │ +0001f720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f750: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0001f760: 2020 2020 3320 2020 2020 2020 2020 2020      3           
│ │ │ │ +0001f750: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001f760: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │  0001f770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f790: 2020 2020 7c0a 7c6f 3520 3d20 3468 2020      |.|o5 = 4h  
│ │ │ │ +0001f780: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001f790: 0a7c 6f35 203d 2034 6820 2020 2020 2020  .|o5 = 4h       
│ │ │ │  0001f7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f7c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001f7d0: 7c20 2020 2020 2020 3120 2020 2020 2020  |       1       
│ │ │ │ +0001f7c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001f7d0: 2020 3120 2020 2020 2020 2020 2020 2020    1             
│ │ │ │  0001f7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f800: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001f800: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001f810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f840: 2020 7c0a 7c20 2020 2020 5a5a 5b68 205d    |.|     ZZ[h ]
│ │ │ │ +0001f830: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001f840: 2020 5a5a 5b68 205d 2020 2020 2020 2020    ZZ[h ]        
│ │ │ │  0001f850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f870: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001f880: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ +0001f870: 2020 207c 0a7c 2020 2020 2020 2020 2031     |.|         1
│ │ │ │ +0001f880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f8b0: 2020 2020 2020 7c0a 7c6f 3520 3a20 2d2d        |.|o5 : --
│ │ │ │ -0001f8c0: 2d2d 2d2d 2020 2020 2020 2020 2020 2020  ----            
│ │ │ │ +0001f8a0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0001f8b0: 3520 3a20 2d2d 2d2d 2d2d 2020 2020 2020  5 : ------      
│ │ │ │ +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: 7c0a 7c20 2020 2020 2020 2035 2020 2020  |.|        5    
│ │ │ │ +0001f8e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001f8f0: 3520 2020 2020 2020 2020 2020 2020 2020  5               
│ │ │ │  0001f900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f920: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0001f930: 2020 2020 6820 2020 2020 2020 2020 2020      h           
│ │ │ │ +0001f910: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001f920: 7c20 2020 2020 2020 6820 2020 2020 2020  |       h       
│ │ │ │ +0001f930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f960: 2020 2020 7c0a 7c20 2020 2020 2020 2031      |.|        1
│ │ │ │ +0001f950: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001f960: 2020 3120 2020 2020 2020 2020 2020 2020    1             
│ │ │ │  0001f970: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0001ffd0: 6473 2074 6f20 636f 6d70 7574 650a 6368  ds to compute.ch
│ │ │ │ +0001ffe0: 6172 6163 7465 7269 7374 6963 2063 6c61  aracteristic cla
│ │ │ │ +0001fff0: 7373 6573 2e0a 0a54 6869 7320 6d65 7468  sses...This meth
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│ │ │ │ +00020010: 616c 7320 696e 2074 6865 2067 7261 6465  als in the grade
│ │ │ │ +00020020: 6420 636f 6f72 6469 6e61 7465 2072 696e  d coordinate rin
│ │ │ │ +00020030: 6773 206f 6620 746f 7269 6320 7661 7269  gs of toric vari
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│ │ │ │ +00020050: 2066 6f72 2070 726f 6475 6374 7320 6f66   for products of
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│ │ │ │ +00020070: 6573 2e20 5468 6573 6520 6361 6e20 6265  es. These can be
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│ │ │ │ +00020090: 792c 206f 720a 7573 696e 6720 6d65 7468  y, or.using meth
│ │ │ │ +000200a0: 6f64 7320 7468 6520 2a6e 6f74 6520 4d75  ods the *note Mu
│ │ │ │ +000200b0: 6c74 6950 726f 6a43 6f6f 7264 5269 6e67  ltiProjCoordRing
│ │ │ │ +000200c0: 3a20 4d75 6c74 6950 726f 6a43 6f6f 7264  : MultiProjCoord
│ │ │ │ +000200d0: 5269 6e67 2c20 6d65 7468 6f64 206f 6620  Ring, method of 
│ │ │ │ +000200e0: 7468 6973 0a70 6163 6b61 6765 2c20 6f72  this.package, or
│ │ │ │ +000200f0: 2077 6974 6820 6d65 7468 6f64 7320 6672   with methods fr
│ │ │ │ +00020100: 6f6d 2074 6865 204e 6f72 6d61 6c54 6f72  om the NormalTor
│ │ │ │ +00020110: 6963 5661 7269 6574 6965 7320 5061 636b  icVarieties Pack
│ │ │ │ +00020120: 6167 652e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d  age...+---------
│ │ │ │ +00020130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020170: 2d2d 2d2d 2d2b 0a7c 6931 203a 2052 3d4d  -----+.|i1 : R=M
│ │ │ │ -00020180: 756c 7469 5072 6f6a 436f 6f72 6452 696e  ultiProjCoordRin
│ │ │ │ -00020190: 6728 7b31 2c32 2c31 7d29 2020 2020 2020  g({1,2,1})      
│ │ │ │ -000201a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000201b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00020160: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 523d  ------+.|i1 : R=
│ │ │ │ +00020170: 4d75 6c74 6950 726f 6a43 6f6f 7264 5269  MultiProjCoordRi
│ │ │ │ +00020180: 6e67 287b 312c 322c 317d 2920 2020 2020  ng({1,2,1})     
│ │ │ │ +00020190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000201a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000201b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000201c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000201d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000201e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000201f0: 2020 2020 2020 2020 207c 0a7c 6f31 203d           |.|o1 =
│ │ │ │ -00020200: 2052 2020 2020 2020 2020 2020 2020 2020   R              
│ │ │ │ +000201e0: 2020 2020 2020 2020 2020 7c0a 7c6f 3120            |.|o1 
│ │ │ │ +000201f0: 3d20 5220 2020 2020 2020 2020 2020 2020  = R             
│ │ │ │ +00020200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020230: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00020220: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00020230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020270: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00020280: 6f31 203a 2050 6f6c 796e 6f6d 6961 6c52  o1 : PolynomialR
│ │ │ │ -00020290: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │ +00020260: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00020270: 7c6f 3120 3a20 506f 6c79 6e6f 6d69 616c  |o1 : Polynomial
│ │ │ │ +00020280: 5269 6e67 2020 2020 2020 2020 2020 2020  Ring            
│ │ │ │ +00020290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000202a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000202b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000202c0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000202b0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +000202c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000202d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000202e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000202f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020300: 2d2b 0a7c 6932 203a 2078 3d67 656e 7328  -+.|i2 : x=gens(
│ │ │ │ -00020310: 5229 2020 2020 2020 2020 2020 2020 2020  R)              
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│ │ │ │ +00020300: 2852 2920 2020 2020 2020 2020 2020 2020  (R)             
│ │ │ │ +00020310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020340: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00020330: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00020340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020380: 2020 2020 207c 0a7c 6f32 203d 207b 7820       |.|o2 = {x 
│ │ │ │ -00020390: 2c20 7820 2c20 7820 2c20 7820 2c20 7820  , x , x , x , x 
│ │ │ │ -000203a0: 2c20 7820 2c20 7820 7d20 2020 2020 2020  , x , x }       
│ │ │ │ -000203b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000203c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000203d0: 2030 2020 2031 2020 2032 2020 2033 2020   0   1   2   3  
│ │ │ │ -000203e0: 2034 2020 2035 2020 2036 2020 2020 2020   4   5   6      
│ │ │ │ -000203f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020400: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00020370: 2020 2020 2020 7c0a 7c6f 3220 3d20 7b78        |.|o2 = {x
│ │ │ │ +00020380: 202c 2078 202c 2078 202c 2078 202c 2078   , x , x , x , x
│ │ │ │ +00020390: 202c 2078 202c 2078 207d 2020 2020 2020   , x , x }      
│ │ │ │ +000203a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000203b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000203c0: 2020 3020 2020 3120 2020 3220 2020 3320    0   1   2   3 
│ │ │ │ +000203d0: 2020 3420 2020 3520 2020 3620 2020 2020    4   5   6     
│ │ │ │ +000203e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000203f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00020400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020440: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ -00020450: 203a 204c 6973 7420 2020 2020 2020 2020   : List         
│ │ │ │ +00020430: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00020440: 3220 3a20 4c69 7374 2020 2020 2020 2020  2 : List        
│ │ │ │ +00020450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020480: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00020470: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00020480: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00020490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000204a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000204b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000204c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000204d0: 0a7c 6933 203a 2049 3d69 6465 616c 2878  .|i3 : I=ideal(x
│ │ │ │ -000204e0: 5f30 5e32 2a78 5f33 2d78 5f31 2a78 5f30  _0^2*x_3-x_1*x_0
│ │ │ │ -000204f0: 2a78 5f34 2c78 5f36 5e33 2920 2020 2020  *x_4,x_6^3)     
│ │ │ │ -00020500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020510: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000204c0: 2b0a 7c69 3320 3a20 493d 6964 6561 6c28  +.|i3 : I=ideal(
│ │ │ │ +000204d0: 785f 305e 322a 785f 332d 785f 312a 785f  x_0^2*x_3-x_1*x_
│ │ │ │ +000204e0: 302a 785f 342c 785f 365e 3329 2020 2020  0*x_4,x_6^3)    
│ │ │ │ +000204f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020500: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00020510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020550: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00020560: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -00020570: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │ -00020580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020590: 2020 2020 207c 0a7c 6f33 203d 2069 6465       |.|o3 = ide
│ │ │ │ -000205a0: 616c 2028 7820 7820 202d 2078 2078 2078  al (x x  - x x x
│ │ │ │ -000205b0: 202c 2078 2029 2020 2020 2020 2020 2020   , x )          
│ │ │ │ -000205c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000205d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000205e0: 2020 2020 2020 2030 2033 2020 2020 3020         0 3    0 
│ │ │ │ -000205f0: 3120 3420 2020 3620 2020 2020 2020 2020  1 4   6         
│ │ │ │ -00020600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020610: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00020540: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00020550: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +00020560: 2020 2033 2020 2020 2020 2020 2020 2020     3            
│ │ │ │ +00020570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020580: 2020 2020 2020 7c0a 7c6f 3320 3d20 6964        |.|o3 = id
│ │ │ │ +00020590: 6561 6c20 2878 2078 2020 2d20 7820 7820  eal (x x  - x x 
│ │ │ │ +000205a0: 7820 2c20 7820 2920 2020 2020 2020 2020  x , x )         
│ │ │ │ +000205b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000205c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000205d0: 2020 2020 2020 2020 3020 3320 2020 2030          0 3    0
│ │ │ │ +000205e0: 2031 2034 2020 2036 2020 2020 2020 2020   1 4   6        
│ │ │ │ +000205f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020600: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00020610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020650: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ -00020660: 203a 2049 6465 616c 206f 6620 5220 2020   : Ideal of R   
│ │ │ │ +00020640: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00020650: 3320 3a20 4964 6561 6c20 6f66 2052 2020  3 : Ideal of R  
│ │ │ │ +00020660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020690: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00020680: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00020690: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  000206a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000206b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000206c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000206d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000206e0: 0a7c 6934 203a 2069 734d 756c 7469 486f  .|i4 : isMultiHo
│ │ │ │ -000206f0: 6d6f 6765 6e65 6f75 7320 4920 2020 2020  mogeneous I     
│ │ │ │ +000206d0: 2b0a 7c69 3420 3a20 6973 4d75 6c74 6948  +.|i4 : isMultiH
│ │ │ │ +000206e0: 6f6d 6f67 656e 656f 7573 2049 2020 2020  omogeneous I    
│ │ │ │ +000206f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020720: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00020710: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00020720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020760: 2020 207c 0a7c 6f34 203d 2074 7275 6520     |.|o4 = true 
│ │ │ │ +00020750: 2020 2020 7c0a 7c6f 3420 3d20 7472 7565      |.|o4 = true
│ │ │ │ +00020760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000207a0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00020790: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +000207a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000207b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000207c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000207d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000207e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2069  -------+.|i5 : i
│ │ │ │ -000207f0: 734d 756c 7469 486f 6d6f 6765 6e65 6f75  sMultiHomogeneou
│ │ │ │ -00020800: 7320 6964 6561 6c28 785f 302a 785f 332d  s ideal(x_0*x_3-
│ │ │ │ -00020810: 785f 312a 785f 302a 785f 342c 785f 365e  x_1*x_0*x_4,x_6^
│ │ │ │ -00020820: 3329 2020 2020 2020 207c 0a7c 496e 7075  3)       |.|Inpu
│ │ │ │ -00020830: 7420 7465 726d 2062 656c 6f77 2069 7320  t term below is 
│ │ │ │ -00020840: 6e6f 7420 686f 6d6f 6765 6e65 6f75 7320  not homogeneous 
│ │ │ │ -00020850: 7769 7468 2072 6573 7065 6374 2074 6f20  with respect to 
│ │ │ │ -00020860: 7468 6520 6772 6164 696e 677c 0a7c 2d20  the grading|.|- 
│ │ │ │ -00020870: 7820 7820 7820 202b 2078 2078 2020 2020  x x x  + x x    
│ │ │ │ +000207d0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20  --------+.|i5 : 
│ │ │ │ +000207e0: 6973 4d75 6c74 6948 6f6d 6f67 656e 656f  isMultiHomogeneo
│ │ │ │ +000207f0: 7573 2069 6465 616c 2878 5f30 2a78 5f33  us ideal(x_0*x_3
│ │ │ │ +00020800: 2d78 5f31 2a78 5f30 2a78 5f34 2c78 5f36  -x_1*x_0*x_4,x_6
│ │ │ │ +00020810: 5e33 2920 2020 2020 2020 7c0a 7c49 6e70  ^3)       |.|Inp
│ │ │ │ +00020820: 7574 2074 6572 6d20 6265 6c6f 7720 6973  ut term below is
│ │ │ │ +00020830: 206e 6f74 2068 6f6d 6f67 656e 656f 7573   not homogeneous
│ │ │ │ +00020840: 2077 6974 6820 7265 7370 6563 7420 746f   with respect to
│ │ │ │ +00020850: 2074 6865 2067 7261 6469 6e67 7c0a 7c2d   the grading|.|-
│ │ │ │ +00020860: 2078 2078 2078 2020 2b20 7820 7820 2020   x x x  + x x   
│ │ │ │ +00020870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000208a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000208b0: 2020 2030 2031 2034 2020 2020 3020 3320     0 1 4    0 3 
│ │ │ │ +00020890: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000208a0: 7c20 2020 3020 3120 3420 2020 2030 2033  |   0 1 4    0 3
│ │ │ │ +000208b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000208c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000208d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000208e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000208f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000208e0: 7c0a 7c20 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 2020 2020 2020 2020 2020                  
│ │ │ │ -00020920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020930: 207c 0a7c 6f35 203d 2066 616c 7365 2020   |.|o5 = false  
│ │ │ │ +00020920: 2020 7c0a 7c6f 3520 3d20 6661 6c73 6520    |.|o5 = false 
│ │ │ │ +00020930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020970: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00020960: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00020970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -000209a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00020c80: 2069 734d 756c 7469 486f 6d6f 6765 6e65   isMultiHomogene
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│ │ │ │ -00020ca0: 7468 6f64 0a2a 2a2a 2a2a 2a0a 0a44 6573  thod.******..Des
│ │ │ │ -00020cb0: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ -00020cc0: 3d3d 3d3d 0a0a 5468 6520 6f70 7469 6f6e  ====..The option
│ │ │ │ -00020cd0: 204d 6574 686f 6420 6973 206f 6e6c 7920   Method is only 
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│ │ │ │ -00020d00: 4353 4d2c 2061 6e64 202a 6e6f 7465 2045  CSM, and *note E
│ │ │ │ -00020d10: 756c 6572 3a0a 4575 6c65 722c 2061 6e64  uler:.Euler, and
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│ │ │ │ -00020d80: 7863 6c75 7369 6f6e 2077 696c 6c20 616c  xclusion will al
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│ │ │ │ -00020ec0: 636f 6d62 696e 6174 696f 6e20 7769 7468  combination with
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│ │ │ │ +00020d70: 4578 636c 7573 696f 6e20 7769 6c6c 2061  Exclusion will a
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│ │ │ │ +00020ea0: 6c65 722c 2061 6e64 206f 6e6c 7920 696e  ler, and only in
│ │ │ │ +00020eb0: 2063 6f6d 6269 6e61 7469 6f6e 2077 6974   combination wit
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│ │ │ │ +00020f40: 4d65 7468 6f64 2c20 506e 5265 7369 6475  Method, PnResidu
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│ │ │ │ -00021000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021010: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00021020: 7c6f 3120 3d20 5220 2020 2020 2020 2020  |o1 = R         
│ │ │ │ +00021000: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00021010: 0a7c 6f31 203d 2052 2020 2020 2020 2020  .|o1 = R        
│ │ │ │ +00021020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021050: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00021040: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00021050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021090: 2020 2020 7c0a 7c6f 3120 3a20 506f 6c79      |.|o1 : Poly
│ │ │ │ -000210a0: 6e6f 6d69 616c 5269 6e67 2020 2020 2020  nomialRing      
│ │ │ │ +00021080: 2020 2020 207c 0a7c 6f31 203a 2050 6f6c       |.|o1 : Pol
│ │ │ │ +00021090: 796e 6f6d 6961 6c52 696e 6720 2020 2020  ynomialRing     
│ │ │ │ +000210a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000210b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000210c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000210d0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000210c0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +000210d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000210e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000210f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021100: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220  ----------+.|i2 
│ │ │ │ -00021110: 3a20 493d 6964 6561 6c28 7261 6e64 6f6d  : I=ideal(random
│ │ │ │ -00021120: 2832 2c52 292c 7261 6e64 6f6d 2831 2c52  (2,R),random(1,R
│ │ │ │ -00021130: 292c 525f 302a 525f 312a 525f 362d 525f  ),R_0*R_1*R_6-R_
│ │ │ │ -00021140: 305e 3329 3b7c 0a7c 2020 2020 2020 2020  0^3);|.|        
│ │ │ │ +000210f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +00021100: 203a 2049 3d69 6465 616c 2872 616e 646f   : I=ideal(rando
│ │ │ │ +00021110: 6d28 322c 5229 2c72 616e 646f 6d28 312c  m(2,R),random(1,
│ │ │ │ +00021120: 5229 2c52 5f30 2a52 5f31 2a52 5f36 2d52  R),R_0*R_1*R_6-R
│ │ │ │ +00021130: 5f30 5e33 293b 7c0a 7c20 2020 2020 2020  _0^3);|.|       
│ │ │ │ +00021140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021180: 7c0a 7c6f 3220 3a20 4964 6561 6c20 6f66  |.|o2 : Ideal of
│ │ │ │ -00021190: 2052 2020 2020 2020 2020 2020 2020 2020   R              
│ │ │ │ -000211a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000211b0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00021170: 207c 0a7c 6f32 203a 2049 6465 616c 206f   |.|o2 : Ideal o
│ │ │ │ +00021180: 6620 5220 2020 2020 2020 2020 2020 2020  f R             
│ │ │ │ +00021190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000211a0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +000211b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000211c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000211d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000211e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000211f0: 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 7469  ------+.|i3 : ti
│ │ │ │ -00021200: 6d65 2043 534d 2049 2020 2020 2020 2020  me CSM I        
│ │ │ │ +000211e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2074  -------+.|i3 : t
│ │ │ │ +000211f0: 696d 6520 4353 4d20 4920 2020 2020 2020  ime CSM I       
│ │ │ │ +00021200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021230: 207c 0a7c 202d 2d20 7573 6564 2031 2e36   |.| -- used 1.6
│ │ │ │ -00021240: 3133 3931 7320 2863 7075 293b 2030 2e39  1391s (cpu); 0.9
│ │ │ │ -00021250: 3230 3235 3273 2028 7468 7265 6164 293b  20252s (thread);
│ │ │ │ -00021260: 2030 7320 2867 6329 2020 2020 7c0a 7c20   0s (gc)    |.| 
│ │ │ │ +00021220: 2020 7c0a 7c20 2d2d 2075 7365 6420 342e    |.| -- used 4.
│ │ │ │ +00021230: 3231 3633 7320 2863 7075 293b 2031 2e34  2163s (cpu); 1.4
│ │ │ │ +00021240: 3833 3038 7320 2874 6872 6561 6429 3b20  8308s (thread); 
│ │ │ │ +00021250: 3073 2028 6763 2920 2020 2020 207c 0a7c  0s (gc)      |.|
│ │ │ │ +00021260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000212a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000212b0: 2020 3520 2020 2020 2034 2020 2020 2033    5      4     3
│ │ │ │ +00021290: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000212a0: 2020 2035 2020 2020 2020 3420 2020 2020     5      4     
│ │ │ │ +000212b0: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │  000212c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000212d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000212e0: 2020 7c0a 7c6f 3320 3d20 3132 6820 202b    |.|o3 = 12h  +
│ │ │ │ -000212f0: 2031 3068 2020 2b20 3668 2020 2020 2020   10h  + 6h      
│ │ │ │ -00021300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021310: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00021320: 2020 2020 2020 2020 3120 2020 2020 2031          1      1
│ │ │ │ -00021330: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ -00021340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021350: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000212d0: 2020 207c 0a7c 6f33 203d 2031 3268 2020     |.|o3 = 12h  
│ │ │ │ +000212e0: 2b20 3130 6820 202b 2036 6820 2020 2020  + 10h  + 6h     
│ │ │ │ +000212f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021300: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00021310: 7c20 2020 2020 2020 2031 2020 2020 2020  |        1      
│ │ │ │ +00021320: 3120 2020 2020 3120 2020 2020 2020 2020  1     1         
│ │ │ │ +00021330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021340: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +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 207c 0a7c 2020 2020 205a 5a5b 6820     |.|     ZZ[h 
│ │ │ │ -000213a0: 5d20 2020 2020 2020 2020 2020 2020 2020  ]               
│ │ │ │ -000213b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000213c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000213d0: 7c20 2020 2020 2020 2020 3120 2020 2020  |         1     
│ │ │ │ +00021380: 2020 2020 7c0a 7c20 2020 2020 5a5a 5b68      |.|     ZZ[h
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│ │ │ │ +000213a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000213b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000213c0: 0a7c 2020 2020 2020 2020 2031 2020 2020  .|         1    
│ │ │ │ +000213d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000213e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000213f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021400: 2020 2020 2020 2020 207c 0a7c 6f33 203a           |.|o3 :
│ │ │ │ -00021410: 202d 2d2d 2d2d 2d20 2020 2020 2020 2020   ------         
│ │ │ │ +000213f0: 2020 2020 2020 2020 2020 7c0a 7c6f 3320            |.|o3 
│ │ │ │ +00021400: 3a20 2d2d 2d2d 2d2d 2020 2020 2020 2020  : ------        
│ │ │ │ +00021410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021440: 2020 2020 7c0a 7c20 2020 2020 2020 2037      |.|        7
│ │ │ │ +00021430: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00021440: 3720 2020 2020 2020 2020 2020 2020 2020  7               
│ │ │ │  00021450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021470: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00021480: 0a7c 2020 2020 2020 2068 2020 2020 2020  .|       h      
│ │ │ │ +00021470: 7c0a 7c20 2020 2020 2020 6820 2020 2020  |.|       h     
│ │ │ │ +00021480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000214a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000214b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -000214c0: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ +000214a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000214b0: 2020 2020 2020 3120 2020 2020 2020 2020        1         
│ │ │ │ +000214c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000214d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000214e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000214f0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +000214e0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +000214f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021530: 2b0a 7c69 3420 3a20 7469 6d65 2043 534d  +.|i4 : time CSM
│ │ │ │ -00021540: 2849 2c4d 6574 686f 643d 3e44 6972 6563  (I,Method=>Direc
│ │ │ │ -00021550: 7443 6f6d 706c 6574 6549 6e74 2920 2020  tCompleteInt)   
│ │ │ │ -00021560: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ -00021570: 2d20 7573 6564 2030 2e34 3833 3131 3273  - used 0.483112s
│ │ │ │ -00021580: 2028 6370 7529 3b20 302e 3231 3438 3734   (cpu); 0.214874
│ │ │ │ -00021590: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ -000215a0: 6763 2920 2020 7c0a 7c20 2020 2020 2020  gc)   |.|       
│ │ │ │ +00021520: 2d2b 0a7c 6934 203a 2074 696d 6520 4353  -+.|i4 : time CS
│ │ │ │ +00021530: 4d28 492c 4d65 7468 6f64 3d3e 4469 7265  M(I,Method=>Dire
│ │ │ │ +00021540: 6374 436f 6d70 6c65 7465 496e 7429 2020  ctCompleteInt)  
│ │ │ │ +00021550: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00021560: 2d2d 2075 7365 6420 302e 3834 3634 3131  -- used 0.846411
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│ │ │ │ +000215a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000215b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000215c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000215d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000215e0: 207c 0a7c 2020 2020 2020 2020 3520 2020   |.|        5   
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│ │ │ │ -00021600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021610: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00021620: 3420 3d20 3132 6820 202b 2031 3068 2020  4 = 12h  + 10h  
│ │ │ │ -00021630: 2b20 3668 2020 2020 2020 2020 2020 2020  + 6h            
│ │ │ │ -00021640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021650: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00021660: 2020 3120 2020 2020 2031 2020 2020 2031    1      1     1
│ │ │ │ +000215d0: 2020 7c0a 7c20 2020 2020 2020 2035 2020    |.|        5  
│ │ │ │ +000215e0: 2020 2020 3420 2020 2020 3320 2020 2020      4     3     
│ │ │ │ +000215f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021600: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00021610: 6f34 203d 2031 3268 2020 2b20 3130 6820  o4 = 12h  + 10h 
│ │ │ │ +00021620: 202b 2036 6820 2020 2020 2020 2020 2020   + 6h           
│ │ │ │ +00021630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021640: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00021650: 2020 2031 2020 2020 2020 3120 2020 2020     1      1     
│ │ │ │ +00021660: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │  00021670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021690: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00021680: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00021690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000216a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000216b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000216c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000216d0: 2020 2020 205a 5a5b 6820 5d20 2020 2020       ZZ[h ]     
│ │ │ │ +000216b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000216c0: 7c20 2020 2020 5a5a 5b68 205d 2020 2020  |     ZZ[h ]    
│ │ │ │ +000216d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000216e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000216f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021700: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00021710: 2020 2020 3120 2020 2020 2020 2020 2020      1           
│ │ │ │ +000216f0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00021700: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ +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 207c 0a7c 6f34 203a 202d 2d2d 2d2d     |.|o4 : -----
│ │ │ │ -00021750: 2d20 2020 2020 2020 2020 2020 2020 2020  -               
│ │ │ │ -00021760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021770: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00021780: 7c20 2020 2020 2020 2037 2020 2020 2020  |        7      
│ │ │ │ +00021730: 2020 2020 7c0a 7c6f 3420 3a20 2d2d 2d2d      |.|o4 : ----
│ │ │ │ +00021740: 2d2d 2020 2020 2020 2020 2020 2020 2020  --              
│ │ │ │ +00021750: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00021770: 0a7c 2020 2020 2020 2020 3720 2020 2020  .|        7     
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│ │ │ │  00021790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000217a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000217d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000217e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000217f0: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │  00021800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021820: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00021830: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00021820: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00021830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021860: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a57 6865  ----------+..Whe
│ │ │ │ -00021870: 6e20 7573 696e 6720 7468 6520 4469 7265  n using the Dire
│ │ │ │ -00021880: 6374 436f 6d70 6c65 7465 496e 7420 6d65  ctCompleteInt me
│ │ │ │ -00021890: 7468 6f64 206f 6e65 206d 6179 2070 6f74  thod one may pot
│ │ │ │ -000218a0: 656e 7469 616c 6c79 2066 7572 7468 6572  entially further
│ │ │ │ -000218b0: 2073 7065 6564 2075 700a 636f 6d70 7574   speed up.comput
│ │ │ │ -000218c0: 6174 696f 6e20 7469 6d65 2062 7920 7370  ation time by sp
│ │ │ │ -000218d0: 6563 6966 7969 6e67 2077 6861 7420 7375  ecifying what su
│ │ │ │ -000218e0: 6273 6574 206f 6620 7468 6520 6765 6e65  bset of the gene
│ │ │ │ -000218f0: 7261 746f 7273 206f 6620 7468 6520 696e  rators of the in
│ │ │ │ -00021900: 7075 7420 6964 6561 6c0a 6465 6669 6e65  put ideal.define
│ │ │ │ -00021910: 2061 2073 6d6f 6f74 6820 7375 6273 6368   a smooth subsch
│ │ │ │ -00021920: 656d 6520 2869 6620 7468 6973 2069 7320  eme (if this is 
│ │ │ │ -00021930: 6b6e 6f77 6e29 2c20 7365 6520 2a6e 6f74  known), see *not
│ │ │ │ -00021940: 6520 496e 6473 4f66 536d 6f6f 7468 3a0a  e IndsOfSmooth:.
│ │ │ │ -00021950: 496e 6473 4f66 536d 6f6f 7468 2c2e 0a0a  IndsOfSmooth,...
│ │ │ │ -00021960: 4675 6e63 7469 6f6e 7320 7769 7468 206f  Functions with o
│ │ │ │ -00021970: 7074 696f 6e61 6c20 6172 6775 6d65 6e74  ptional argument
│ │ │ │ -00021980: 206e 616d 6564 204d 6574 686f 643a 0a3d   named Method:.=
│ │ │ │ +00021850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5768  -----------+..Wh
│ │ │ │ +00021860: 656e 2075 7369 6e67 2074 6865 2044 6972  en using the Dir
│ │ │ │ +00021870: 6563 7443 6f6d 706c 6574 6549 6e74 206d  ectCompleteInt m
│ │ │ │ +00021880: 6574 686f 6420 6f6e 6520 6d61 7920 706f  ethod one may po
│ │ │ │ +00021890: 7465 6e74 6961 6c6c 7920 6675 7274 6865  tentially furthe
│ │ │ │ +000218a0: 7220 7370 6565 6420 7570 0a63 6f6d 7075  r speed up.compu
│ │ │ │ +000218b0: 7461 7469 6f6e 2074 696d 6520 6279 2073  tation time by s
│ │ │ │ +000218c0: 7065 6369 6679 696e 6720 7768 6174 2073  pecifying what s
│ │ │ │ +000218d0: 7562 7365 7420 6f66 2074 6865 2067 656e  ubset of the gen
│ │ │ │ +000218e0: 6572 6174 6f72 7320 6f66 2074 6865 2069  erators of the i
│ │ │ │ +000218f0: 6e70 7574 2069 6465 616c 0a64 6566 696e  nput ideal.defin
│ │ │ │ +00021900: 6520 6120 736d 6f6f 7468 2073 7562 7363  e a smooth subsc
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│ │ │ │  00021c60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00021c70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00021c80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00021c90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00021ca0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00021cb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00021cc0: 0a0a 2020 2a20 5573 6167 653a 200a 2020  ..  * Usage: .  
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│ │ │ │ +00021e90: 6e67 2074 6f20 6265 2062 7569 6c74 2062  ng to be built b
│ │ │ │ +00021ea0: 7920 7468 6520 6d65 7468 6f64 2c20 6279  y the method, by
│ │ │ │ +00021eb0: 2064 6566 6175 6c74 2074 6869 730a 2020   default this.  
│ │ │ │ +00021ec0: 2020 2020 2020 6973 205c 5a5a 2f33 3237        is \ZZ/327
│ │ │ │ +00021ed0: 3439 0a20 2020 2020 202a 2076 6172 2c20  49.      * var, 
│ │ │ │ +00021ee0: 6120 2a6e 6f74 6520 7379 6d62 6f6c 3a20  a *note symbol: 
│ │ │ │ +00021ef0: 284d 6163 6175 6c61 7932 446f 6329 5379  (Macaulay2Doc)Sy
│ │ │ │ +00021f00: 6d62 6f6c 2c2c 2074 6f20 6265 2075 7365  mbol,, to be use
│ │ │ │ +00021f10: 6420 666f 7220 7468 650a 2020 2020 2020  d for the.      
│ │ │ │ +00021f20: 2020 696e 7465 726d 6564 6961 7465 7320    intermediates 
│ │ │ │ +00021f30: 6f66 2074 6865 2067 7261 6465 6420 706f  of the graded po
│ │ │ │ +00021f40: 6c79 6e6f 6d69 616c 2072 696e 6720 746f  lynomial ring to
│ │ │ │ +00021f50: 2062 6520 6275 696c 7420 6279 2074 6865   be built by the
│ │ │ │ +00021f60: 206d 6574 686f 640a 2020 2a20 4f75 7470   method.  * Outp
│ │ │ │ +00021f70: 7574 733a 0a20 2020 2020 202a 2061 202a  uts:.      * a *
│ │ │ │ +00021f80: 6e6f 7465 2072 696e 673a 2028 4d61 6361  note ring: (Maca
│ │ │ │ +00021f90: 756c 6179 3244 6f63 2952 696e 672c 2c20  ulay2Doc)Ring,, 
│ │ │ │ +00021fa0: 7468 6520 6772 6164 6564 2063 6f6f 7264  the graded coord
│ │ │ │ +00021fb0: 696e 6174 6520 7269 6e67 206f 6620 7468  inate ring of th
│ │ │ │ +00021fc0: 650a 2020 2020 2020 2020 5c50 505e 7b6e  e.        \PP^{n
│ │ │ │ +00021fd0: 5f31 7d20 782e 2e2e 2e20 7820 5c50 505e  _1} x.... x \PP^
│ │ │ │ +00021fe0: 7b6e 5f6d 7d20 7768 6572 6520 7b6e 5f31  {n_m} where {n_1
│ │ │ │ +00021ff0: 2c2e 2e2e 2c6e 5f6d 7d20 6973 2074 6865  ,...,n_m} is the
│ │ │ │ +00022000: 2069 6e70 7574 206c 6973 7420 6f66 0a20   input list of. 
│ │ │ │ +00022010: 2020 2020 2020 2064 696d 656e 7369 6f6e         dimension
│ │ │ │ +00022020: 730a 0a44 6573 6372 6970 7469 6f6e 0a3d  s..Description.=
│ │ │ │ +00022030: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 436f 6d70  ==========..Comp
│ │ │ │ +00022040: 7574 6573 2074 6865 2067 7261 6465 6420  utes the graded 
│ │ │ │ +00022050: 636f 6f72 6469 6e61 7465 2072 696e 6720  coordinate ring 
│ │ │ │ +00022060: 6f66 2074 6865 205c 5050 5e7b 6e5f 317d  of the \PP^{n_1}
│ │ │ │ +00022070: 2078 2e2e 2e2e 2078 205c 5050 5e7b 6e5f   x.... x \PP^{n_
│ │ │ │ +00022080: 6d7d 2077 6865 7265 0a7b 6e5f 312c 2e2e  m} where.{n_1,..
│ │ │ │ +00022090: 2e2c 6e5f 6d7d 2069 7320 7468 6520 696e  .,n_m} is the in
│ │ │ │ +000220a0: 7075 7420 6c69 7374 206f 6620 6469 6d65  put list of dime
│ │ │ │ +000220b0: 6e73 696f 6e73 2e20 5468 6973 206d 6574  nsions. This met
│ │ │ │ +000220c0: 686f 6420 6973 2075 7365 6420 746f 2071  hod is used to q
│ │ │ │ +000220d0: 7569 636b 6c79 0a62 7569 6c64 2074 6865  uickly.build the
│ │ │ │ +000220e0: 2063 6f6f 7264 696e 6174 6520 7269 6e67   coordinate ring
│ │ │ │ +000220f0: 206f 6620 6120 7072 6f64 7563 7420 6f66   of a product of
│ │ │ │ +00022100: 2070 726f 6a65 6374 6976 6520 7370 6163   projective spac
│ │ │ │ +00022110: 6573 2066 6f72 2075 7365 2069 6e0a 636f  es for use in.co
│ │ │ │ +00022120: 6d70 7574 6174 696f 6e73 2e0a 0a2b 2d2d  mputations...+--
│ │ │ │ +00022130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022180: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120  ----------+.|i1 
│ │ │ │ -00022190: 3a20 533d 4d75 6c74 6950 726f 6a43 6f6f  : S=MultiProjCoo
│ │ │ │ -000221a0: 7264 5269 6e67 2851 512c 7379 6d62 6f6c  rdRing(QQ,symbol
│ │ │ │ -000221b0: 207a 2c7b 312c 332c 337d 2920 2020 2020   z,{1,3,3})     
│ │ │ │ -000221c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000221d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00022170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +00022180: 203a 2053 3d4d 756c 7469 5072 6f6a 436f   : S=MultiProjCo
│ │ │ │ +00022190: 6f72 6452 696e 6728 5151 2c73 796d 626f  ordRing(QQ,symbo
│ │ │ │ +000221a0: 6c20 7a2c 7b31 2c33 2c33 7d29 2020 2020  l z,{1,3,3})    
│ │ │ │ +000221b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000221c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000221d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000221e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000221f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022220: 2020 2020 2020 2020 2020 7c0a 7c6f 3120            |.|o1 
│ │ │ │ -00022230: 3d20 5320 2020 2020 2020 2020 2020 2020  = S             
│ │ │ │ +00022210: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +00022220: 203d 2053 2020 2020 2020 2020 2020 2020   = S            
│ │ │ │ +00022230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022270: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00022260: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00022270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000222a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000222b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000222c0: 2020 2020 2020 2020 2020 7c0a 7c6f 3120            |.|o1 
│ │ │ │ -000222d0: 3a20 506f 6c79 6e6f 6d69 616c 5269 6e67  : PolynomialRing
│ │ │ │ +000222b0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +000222c0: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e   : PolynomialRin
│ │ │ │ +000222d0: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │  000222e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000222f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022310: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00022300: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00022310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  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 2d2d 2b0a 7c69 3220  ----------+.|i2 
│ │ │ │ -00022370: 3a20 6465 6772 6565 7320 5320 2020 2020  : degrees S     
│ │ │ │ +00022350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +00022360: 203a 2064 6567 7265 6573 2053 2020 2020   : degrees S    
│ │ │ │ +00022370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000223a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000223b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000223a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000223b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000223c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000223d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000223e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000223f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022400: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ -00022410: 3d20 7b7b 312c 2030 2c20 307d 2c20 7b31  = {{1, 0, 0}, {1
│ │ │ │ -00022420: 2c20 302c 2030 7d2c 207b 302c 2031 2c20  , 0, 0}, {0, 1, 
│ │ │ │ -00022430: 307d 2c20 7b30 2c20 312c 2030 7d2c 207b  0}, {0, 1, 0}, {
│ │ │ │ -00022440: 302c 2031 2c20 307d 2c20 7b30 2c20 312c  0, 1, 0}, {0, 1,
│ │ │ │ -00022450: 2030 7d2c 207b 302c 2020 7c0a 7c20 2020   0}, {0,  |.|   
│ │ │ │ -00022460: 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d    --------------
│ │ │ │ +000223f0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +00022400: 203d 207b 7b31 2c20 302c 2030 7d2c 207b   = {{1, 0, 0}, {
│ │ │ │ +00022410: 312c 2030 2c20 307d 2c20 7b30 2c20 312c  1, 0, 0}, {0, 1,
│ │ │ │ +00022420: 2030 7d2c 207b 302c 2031 2c20 307d 2c20   0}, {0, 1, 0}, 
│ │ │ │ +00022430: 7b30 2c20 312c 2030 7d2c 207b 302c 2031  {0, 1, 0}, {0, 1
│ │ │ │ +00022440: 2c20 307d 2c20 7b30 2c20 207c 0a7c 2020  , 0}, {0,  |.|  
│ │ │ │ +00022450: 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d     -------------
│ │ │ │ +00022460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000224a0: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020  ----------|.|   
│ │ │ │ -000224b0: 2020 302c 2031 7d2c 207b 302c 2030 2c20    0, 1}, {0, 0, 
│ │ │ │ -000224c0: 317d 2c20 7b30 2c20 302c 2031 7d2c 207b  1}, {0, 0, 1}, {
│ │ │ │ -000224d0: 302c 2030 2c20 317d 7d20 2020 2020 2020  0, 0, 1}}       
│ │ │ │ -000224e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000224f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00022490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ +000224a0: 2020 2030 2c20 317d 2c20 7b30 2c20 302c     0, 1}, {0, 0,
│ │ │ │ +000224b0: 2031 7d2c 207b 302c 2030 2c20 317d 2c20   1}, {0, 0, 1}, 
│ │ │ │ +000224c0: 7b30 2c20 302c 2031 7d7d 2020 2020 2020  {0, 0, 1}}      
│ │ │ │ +000224d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000224e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000224f0: 2020 2020 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 2020                  
│ │ │ │ -00022530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022540: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ -00022550: 3a20 4c69 7374 2020 2020 2020 2020 2020  : List          
│ │ │ │ +00022530: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +00022540: 203a 204c 6973 7420 2020 2020 2020 2020   : List         
│ │ │ │ +00022550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022590: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00022580: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00022590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000225a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000225b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000225c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000225d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000225e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320  ----------+.|i3 
│ │ │ │ -000225f0: 3a20 523d 4d75 6c74 6950 726f 6a43 6f6f  : R=MultiProjCoo
│ │ │ │ -00022600: 7264 5269 6e67 207b 322c 337d 2020 2020  rdRing {2,3}    
│ │ │ │ +000225d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933  -----------+.|i3
│ │ │ │ +000225e0: 203a 2052 3d4d 756c 7469 5072 6f6a 436f   : R=MultiProjCo
│ │ │ │ +000225f0: 6f72 6452 696e 6720 7b32 2c33 7d20 2020  ordRing {2,3}   
│ │ │ │ +00022600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022630: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00022620: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00022630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022680: 2020 2020 2020 2020 2020 7c0a 7c6f 3320            |.|o3 
│ │ │ │ -00022690: 3d20 5220 2020 2020 2020 2020 2020 2020  = R             
│ │ │ │ +00022670: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ +00022680: 203d 2052 2020 2020 2020 2020 2020 2020   = R            
│ │ │ │ +00022690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000226a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000226b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000226c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000226d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000226c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000226d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000226e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000226f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022720: 2020 2020 2020 2020 2020 7c0a 7c6f 3320            |.|o3 
│ │ │ │ -00022730: 3a20 506f 6c79 6e6f 6d69 616c 5269 6e67  : PolynomialRing
│ │ │ │ +00022710: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ +00022720: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e   : PolynomialRin
│ │ │ │ +00022730: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │  00022740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022770: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00022760: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00022770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000227a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000227b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000227c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420  ----------+.|i4 
│ │ │ │ -000227d0: 3a20 636f 6566 6669 6369 656e 7452 696e  : coefficientRin
│ │ │ │ -000227e0: 6720 5220 2020 2020 2020 2020 2020 2020  g R             
│ │ │ │ +000227b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934  -----------+.|i4
│ │ │ │ +000227c0: 203a 2063 6f65 6666 6963 6965 6e74 5269   : coefficientRi
│ │ │ │ +000227d0: 6e67 2052 2020 2020 2020 2020 2020 2020  ng R            
│ │ │ │ +000227e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000227f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022810: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00022800: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00022810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022860: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00022870: 2020 2020 5a5a 2020 2020 2020 2020 2020      ZZ          
│ │ │ │ +00022850: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00022860: 2020 2020 205a 5a20 2020 2020 2020 2020       ZZ         
│ │ │ │ +00022870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000228a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000228b0: 2020 2020 2020 2020 2020 7c0a 7c6f 3420            |.|o4 
│ │ │ │ -000228c0: 3d20 2d2d 2d2d 2d20 2020 2020 2020 2020  = -----         
│ │ │ │ +000228a0: 2020 2020 2020 2020 2020 207c 0a7c 6f34             |.|o4
│ │ │ │ +000228b0: 203d 202d 2d2d 2d2d 2020 2020 2020 2020   = -----        
│ │ │ │ +000228c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000228d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000228e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000228f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022900: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00022910: 2020 3332 3734 3920 2020 2020 2020 2020    32749         
│ │ │ │ +000228f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00022900: 2020 2033 3237 3439 2020 2020 2020 2020     32749        
│ │ │ │ +00022910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022950: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00022940: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00022950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000229a0: 2020 2020 2020 2020 2020 7c0a 7c6f 3420            |.|o4 
│ │ │ │ -000229b0: 3a20 5175 6f74 6965 6e74 5269 6e67 2020  : QuotientRing  
│ │ │ │ +00022990: 2020 2020 2020 2020 2020 207c 0a7c 6f34             |.|o4
│ │ │ │ +000229a0: 203a 2051 756f 7469 656e 7452 696e 6720   : QuotientRing 
│ │ │ │ +000229b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000229c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000229d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000229e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000229f0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +000229e0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +000229f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520  ----------+.|i5 
│ │ │ │ -00022a50: 3a20 6465 7363 7269 6265 2052 2020 2020  : describe R    
│ │ │ │ +00022a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935  -----------+.|i5
│ │ │ │ +00022a40: 203a 2064 6573 6372 6962 6520 5220 2020   : describe R   
│ │ │ │ +00022a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022a90: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00022a80: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00022a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022ae0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00022af0: 2020 2020 5a5a 2020 2020 2020 2020 2020      ZZ          
│ │ │ │ +00022ad0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00022ae0: 2020 2020 205a 5a20 2020 2020 2020 2020       ZZ         
│ │ │ │ +00022af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022b30: 2020 2020 2020 2020 2020 7c0a 7c6f 3520            |.|o5 
│ │ │ │ -00022b40: 3d20 2d2d 2d2d 2d5b 7820 2e2e 7820 2c20  = -----[x ..x , 
│ │ │ │ -00022b50: 4465 6772 6565 7320 3d3e 207b 333a 7b31  Degrees => {3:{1
│ │ │ │ -00022b60: 7d2c 2034 3a7b 307d 7d2c 2048 6566 7420  }, 4:{0}}, Heft 
│ │ │ │ -00022b70: 3d3e 207b 323a 317d 5d20 2020 2020 2020  => {2:1}]       
│ │ │ │ -00022b80: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00022b90: 2020 3332 3734 3920 2030 2020 2036 2020    32749  0   6  
│ │ │ │ -00022ba0: 2020 2020 2020 2020 2020 2020 2020 7b30                {0
│ │ │ │ -00022bb0: 7d20 2020 207b 317d 2020 2020 2020 2020  }    {1}        
│ │ │ │ -00022bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022bd0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00022b20: 2020 2020 2020 2020 2020 207c 0a7c 6f35             |.|o5
│ │ │ │ +00022b30: 203d 202d 2d2d 2d2d 5b78 202e 2e78 202c   = -----[x ..x ,
│ │ │ │ +00022b40: 2044 6567 7265 6573 203d 3e20 7b33 3a7b   Degrees => {3:{
│ │ │ │ +00022b50: 317d 2c20 343a 7b30 7d7d 2c20 4865 6674  1}, 4:{0}}, Heft
│ │ │ │ +00022b60: 203d 3e20 7b32 3a31 7d5d 2020 2020 2020   => {2:1}]      
│ │ │ │ +00022b70: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00022b80: 2020 2033 3237 3439 2020 3020 2020 3620     32749  0   6 
│ │ │ │ +00022b90: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +00022ba0: 307d 2020 2020 7b31 7d20 2020 2020 2020  0}    {1}       
│ │ │ │ +00022bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022bc0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00022bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620  ----------+.|i6 
│ │ │ │ -00022c30: 3a20 413d 4368 6f77 5269 6e67 2052 2020  : A=ChowRing R  
│ │ │ │ +00022c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936  -----------+.|i6
│ │ │ │ +00022c20: 203a 2041 3d43 686f 7752 696e 6720 5220   : A=ChowRing R 
│ │ │ │ +00022c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022c70: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00022c60: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00022c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022cc0: 2020 2020 2020 2020 2020 7c0a 7c6f 3620            |.|o6 
│ │ │ │ -00022cd0: 3d20 4120 2020 2020 2020 2020 2020 2020  = A             
│ │ │ │ +00022cb0: 2020 2020 2020 2020 2020 207c 0a7c 6f36             |.|o6
│ │ │ │ +00022cc0: 203d 2041 2020 2020 2020 2020 2020 2020   = A            
│ │ │ │ +00022cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022d10: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00022d00: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00022d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022d60: 2020 2020 2020 2020 2020 7c0a 7c6f 3620            |.|o6 
│ │ │ │ -00022d70: 3a20 5175 6f74 6965 6e74 5269 6e67 2020  : QuotientRing  
│ │ │ │ +00022d50: 2020 2020 2020 2020 2020 207c 0a7c 6f36             |.|o6
│ │ │ │ +00022d60: 203a 2051 756f 7469 656e 7452 696e 6720   : QuotientRing 
│ │ │ │ +00022d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022d80: 2020 2020 2020 2020 2020 2020 2020 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 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00022da0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00022db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720  ----------+.|i7 
│ │ │ │ -00022e10: 3a20 6465 7363 7269 6265 2041 2020 2020  : describe A    
│ │ │ │ +00022df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937  -----------+.|i7
│ │ │ │ +00022e00: 203a 2064 6573 6372 6962 6520 4120 2020   : describe A   
│ │ │ │ +00022e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022e20: 2020 2020 2020 2020 2020 2020 2020 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 7c0a 7c20 2020            |.|   
│ │ │ │ +00022e40: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00022e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022e70: 2020 2020 2020 2020 2020 2020 2020 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 7c0a 7c20 2020            |.|   
│ │ │ │ -00022eb0: 2020 5a5a 5b68 202e 2e68 205d 2020 2020    ZZ[h ..h ]    
│ │ │ │ +00022e90: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00022ea0: 2020 205a 5a5b 6820 2e2e 6820 5d20 2020     ZZ[h ..h ]   
│ │ │ │ +00022eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022ec0: 2020 2020 2020 2020 2020 2020 2020 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 7c0a 7c20 2020            |.|   
│ │ │ │ -00022f00: 2020 2020 2020 3120 2020 3220 2020 2020        1   2     
│ │ │ │ +00022ee0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00022ef0: 2020 2020 2020 2031 2020 2032 2020 2020         1   2    
│ │ │ │ +00022f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022f40: 2020 2020 2020 2020 2020 7c0a 7c6f 3720            |.|o7 
│ │ │ │ -00022f50: 3d20 2d2d 2d2d 2d2d 2d2d 2d2d 2020 2020  = ----------    
│ │ │ │ +00022f30: 2020 2020 2020 2020 2020 207c 0a7c 6f37             |.|o7
│ │ │ │ +00022f40: 203d 202d 2d2d 2d2d 2d2d 2d2d 2d20 2020   = ----------   
│ │ │ │ +00022f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022f90: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00022fa0: 2020 2020 2033 2020 2034 2020 2020 2020       3   4      
│ │ │ │ +00022f80: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00022f90: 2020 2020 2020 3320 2020 3420 2020 2020        3   4     
│ │ │ │ +00022fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022fe0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00022ff0: 2020 2028 6820 2c20 6820 2920 2020 2020     (h , h )     
│ │ │ │ +00022fd0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00022fe0: 2020 2020 2868 202c 2068 2029 2020 2020      (h , h )    
│ │ │ │ +00022ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023000: 2020 2020 2020 2020 2020 2020 2020 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 7c0a 7c20 2020            |.|   
│ │ │ │ -00023040: 2020 2020 2031 2020 2032 2020 2020 2020       1   2      
│ │ │ │ +00023020: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00023030: 2020 2020 2020 3120 2020 3220 2020 2020        1   2     
│ │ │ │ +00023040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023080: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00023070: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00023080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000230a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000230b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000230c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000230d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820  ----------+.|i8 
│ │ │ │ -000230e0: 3a20 5365 6772 6528 412c 6964 6561 6c20  : Segre(A,ideal 
│ │ │ │ -000230f0: 7261 6e64 6f6d 287b 312c 317d 2c52 2929  random({1,1},R))
│ │ │ │ +000230c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938  -----------+.|i8
│ │ │ │ +000230d0: 203a 2053 6567 7265 2841 2c69 6465 616c   : Segre(A,ideal
│ │ │ │ +000230e0: 2072 616e 646f 6d28 7b31 2c31 7d2c 5229   random({1,1},R)
│ │ │ │ +000230f0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  00023100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023120: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00023110: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00023120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023170: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00023180: 2020 2020 2032 2033 2020 2020 2032 2032       2 3     2 2
│ │ │ │ -00023190: 2020 2020 2020 2033 2020 2020 2032 2020         3     2  
│ │ │ │ -000231a0: 2020 2020 2020 2032 2020 2020 3320 2020         2    3   
│ │ │ │ -000231b0: 2032 2020 2020 2020 2020 2020 2020 3220   2            2 
│ │ │ │ -000231c0: 2020 2020 2020 2020 2020 7c0a 7c6f 3820            |.|o8 
│ │ │ │ -000231d0: 3d20 3130 6820 6820 202d 2036 6820 6820  = 10h h  - 6h h 
│ │ │ │ -000231e0: 202d 2034 6820 6820 202b 2033 6820 6820   - 4h h  + 3h h 
│ │ │ │ -000231f0: 202b 2033 6820 6820 202b 2068 2020 2d20   + 3h h  + h  - 
│ │ │ │ -00023200: 6820 202d 2032 6820 6820 202d 2068 2020  h  - 2h h  - h  
│ │ │ │ -00023210: 2b20 6820 202b 2068 2020 7c0a 7c20 2020  + h  + h  |.|   
│ │ │ │ -00023220: 2020 2020 2031 2032 2020 2020 2031 2032       1 2     1 2
│ │ │ │ -00023230: 2020 2020 2031 2032 2020 2020 2031 2032       1 2     1 2
│ │ │ │ -00023240: 2020 2020 2031 2032 2020 2020 3220 2020       1 2    2   
│ │ │ │ -00023250: 2031 2020 2020 2031 2032 2020 2020 3220   1     1 2    2 
│ │ │ │ -00023260: 2020 2031 2020 2020 3220 7c0a 7c20 2020     1    2 |.|   
│ │ │ │ +00023160: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00023170: 2020 2020 2020 3220 3320 2020 2020 3220        2 3     2 
│ │ │ │ +00023180: 3220 2020 2020 2020 3320 2020 2020 3220  2       3     2 
│ │ │ │ +00023190: 2020 2020 2020 2020 3220 2020 2033 2020          2    3  
│ │ │ │ +000231a0: 2020 3220 2020 2020 2020 2020 2020 2032    2            2
│ │ │ │ +000231b0: 2020 2020 2020 2020 2020 207c 0a7c 6f38             |.|o8
│ │ │ │ +000231c0: 203d 2031 3068 2068 2020 2d20 3668 2068   = 10h h  - 6h h
│ │ │ │ +000231d0: 2020 2d20 3468 2068 2020 2b20 3368 2068    - 4h h  + 3h h
│ │ │ │ +000231e0: 2020 2b20 3368 2068 2020 2b20 6820 202d    + 3h h  + h  -
│ │ │ │ +000231f0: 2068 2020 2d20 3268 2068 2020 2d20 6820   h  - 2h h  - h 
│ │ │ │ +00023200: 202b 2068 2020 2b20 6820 207c 0a7c 2020   + h  + h  |.|  
│ │ │ │ +00023210: 2020 2020 2020 3120 3220 2020 2020 3120        1 2     1 
│ │ │ │ +00023220: 3220 2020 2020 3120 3220 2020 2020 3120  2     1 2     1 
│ │ │ │ +00023230: 3220 2020 2020 3120 3220 2020 2032 2020  2     1 2    2  
│ │ │ │ +00023240: 2020 3120 2020 2020 3120 3220 2020 2032    1     1 2    2
│ │ │ │ +00023250: 2020 2020 3120 2020 2032 207c 0a7c 2020      1    2 |.|  
│ │ │ │ +00023260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000232a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000232b0: 2020 2020 2020 2020 2020 7c0a 7c6f 3820            |.|o8 
│ │ │ │ -000232c0: 3a20 4120 2020 2020 2020 2020 2020 2020  : A             
│ │ │ │ +000232a0: 2020 2020 2020 2020 2020 207c 0a7c 6f38             |.|o8
│ │ │ │ +000232b0: 203a 2041 2020 2020 2020 2020 2020 2020   : A            
│ │ │ │ +000232c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000232d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000232e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000232f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023300: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +000232f0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00023300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023350: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a57 6179  ----------+..Way
│ │ │ │ -00023360: 7320 746f 2075 7365 204d 756c 7469 5072  s to use MultiPr
│ │ │ │ -00023370: 6f6a 436f 6f72 6452 696e 673a 0a3d 3d3d  ojCoordRing:.===
│ │ │ │ -00023380: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00023390: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020  ============..  
│ │ │ │ -000233a0: 2a20 224d 756c 7469 5072 6f6a 436f 6f72  * "MultiProjCoor
│ │ │ │ -000233b0: 6452 696e 6728 4c69 7374 2922 0a20 202a  dRing(List)".  *
│ │ │ │ -000233c0: 2022 4d75 6c74 6950 726f 6a43 6f6f 7264   "MultiProjCoord
│ │ │ │ -000233d0: 5269 6e67 2852 696e 672c 4c69 7374 2922  Ring(Ring,List)"
│ │ │ │ -000233e0: 0a20 202a 2022 4d75 6c74 6950 726f 6a43  .  * "MultiProjC
│ │ │ │ -000233f0: 6f6f 7264 5269 6e67 2852 696e 672c 5379  oordRing(Ring,Sy
│ │ │ │ -00023400: 6d62 6f6c 2c4c 6973 7429 220a 2020 2a20  mbol,List)".  * 
│ │ │ │ -00023410: 224d 756c 7469 5072 6f6a 436f 6f72 6452  "MultiProjCoordR
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│ │ │ │ -00023430: 220a 0a46 6f72 2074 6865 2070 726f 6772  "..For the progr
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│ │ │ │ -00023470: 5072 6f6a 436f 6f72 6452 696e 673a 204d  ProjCoordRing: M
│ │ │ │ -00023480: 756c 7469 5072 6f6a 436f 6f72 6452 696e  ultiProjCoordRin
│ │ │ │ -00023490: 672c 2069 7320 6120 2a6e 6f74 6520 6d65  g, is a *note me
│ │ │ │ -000234a0: 7468 6f64 0a66 756e 6374 696f 6e3a 2028  thod.function: (
│ │ │ │ -000234b0: 4d61 6361 756c 6179 3244 6f63 294d 6574  Macaulay2Doc)Met
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│ │ │ │ +00023350: 7973 2074 6f20 7573 6520 4d75 6c74 6950  ys to use MultiP
│ │ │ │ +00023360: 726f 6a43 6f6f 7264 5269 6e67 3a0a 3d3d  rojCoordRing:.==
│ │ │ │ +00023370: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00023380: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20  =============.. 
│ │ │ │ +00023390: 202a 2022 4d75 6c74 6950 726f 6a43 6f6f   * "MultiProjCoo
│ │ │ │ +000233a0: 7264 5269 6e67 284c 6973 7429 220a 2020  rdRing(List)".  
│ │ │ │ +000233b0: 2a20 224d 756c 7469 5072 6f6a 436f 6f72  * "MultiProjCoor
│ │ │ │ +000233c0: 6452 696e 6728 5269 6e67 2c4c 6973 7429  dRing(Ring,List)
│ │ │ │ +000233d0: 220a 2020 2a20 224d 756c 7469 5072 6f6a  ".  * "MultiProj
│ │ │ │ +000233e0: 436f 6f72 6452 696e 6728 5269 6e67 2c53  CoordRing(Ring,S
│ │ │ │ +000233f0: 796d 626f 6c2c 4c69 7374 2922 0a20 202a  ymbol,List)".  *
│ │ │ │ +00023400: 2022 4d75 6c74 6950 726f 6a43 6f6f 7264   "MultiProjCoord
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│ │ │ │ +00023440: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f  =========..The o
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│ │ │ │ +00023470: 4d75 6c74 6950 726f 6a43 6f6f 7264 5269  MultiProjCoordRi
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│ │ │ │ +000234a0: 284d 6163 6175 6c61 7932 446f 6329 4d65  (Macaulay2Doc)Me
│ │ │ │ +000234b0: 7468 6f64 4675 6e63 7469 6f6e 2c2e 0a0a  thodFunction,...
│ │ │ │ +000234c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000234d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000234e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000234f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a  --------------..
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│ │ │ │ -00023550: 6369 626c 652d 7061 7468 2f6d 6163 6175  cible-path/macau
│ │ │ │ -00023560: 6c61 7932 2d31 2e32 352e 3131 2b64 732f  lay2-1.25.11+ds/
│ │ │ │ -00023570: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163  M2/Macaulay2/pac
│ │ │ │ -00023580: 6b61 6765 732f 0a43 6861 7261 6374 6572  kages/.Character
│ │ │ │ -00023590: 6973 7469 6343 6c61 7373 6573 2e6d 323a  isticClasses.m2:
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│ │ │ │ -000235b0: 4368 6172 6163 7465 7269 7374 6963 436c  CharacteristicCl
│ │ │ │ -000235c0: 6173 7365 732e 696e 666f 2c20 4e6f 6465  asses.info, Node
│ │ │ │ -000235d0: 3a20 4f75 7470 7574 2c20 4e65 7874 3a20  : Output, Next: 
│ │ │ │ -000235e0: 7072 6f62 6162 696c 6973 7469 6320 616c  probabilistic al
│ │ │ │ -000235f0: 676f 7269 7468 6d2c 2050 7265 763a 204d  gorithm, Prev: M
│ │ │ │ -00023600: 756c 7469 5072 6f6a 436f 6f72 6452 696e  ultiProjCoordRin
│ │ │ │ -00023610: 672c 2055 703a 2054 6f70 0a0a 4f75 7470  g, Up: Top..Outp
│ │ │ │ -00023620: 7574 0a2a 2a2a 2a2a 2a0a 0a44 6573 6372  ut.******..Descr
│ │ │ │ -00023630: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d  iption.=========
│ │ │ │ -00023640: 3d3d 0a0a 5468 6520 6f70 7469 6f6e 204f  ==..The option O
│ │ │ │ -00023650: 7574 7075 7420 6973 206f 6e6c 7920 7573  utput is only us
│ │ │ │ -00023660: 6564 2062 7920 7468 6520 636f 6d6d 616e  ed by the comman
│ │ │ │ -00023670: 6473 202a 6e6f 7465 2043 534d 3a20 4353  ds *note CSM: CS
│ │ │ │ -00023680: 4d2c 2c20 2a6e 6f74 6520 5365 6772 653a  M,, *note Segre:
│ │ │ │ -00023690: 0a53 6567 7265 2c2c 202a 6e6f 7465 2043  .Segre,, *note C
│ │ │ │ -000236a0: 6865 726e 3a20 4368 6572 6e2c 2061 6e64  hern: Chern, and
│ │ │ │ -000236b0: 202a 6e6f 7465 2045 756c 6572 3a20 4575   *note Euler: Eu
│ │ │ │ -000236c0: 6c65 722c 2074 6f20 7370 6563 6966 7920  ler, to specify 
│ │ │ │ -000236d0: 7468 6520 7479 7065 206f 660a 6f75 7470  the type of.outp
│ │ │ │ -000236e0: 7574 2074 6f20 6265 2072 6574 7572 6e65  ut to be returne
│ │ │ │ -000236f0: 6420 746f 2074 6865 2075 7365 642e 2054  d to the used. T
│ │ │ │ -00023700: 6869 7320 6f70 7469 6f6e 2077 696c 6c20  his option will 
│ │ │ │ -00023710: 6265 2069 676e 6f72 6564 2077 6865 6e20  be ignored when 
│ │ │ │ -00023720: 7573 6564 2077 6974 680a 2a6e 6f74 6520  used with.*note 
│ │ │ │ -00023730: 436f 6d70 4d65 7468 6f64 3a20 436f 6d70  CompMethod: Comp
│ │ │ │ -00023740: 4d65 7468 6f64 2c20 506e 5265 7369 6475  Method, PnResidu
│ │ │ │ -00023750: 616c 206f 7220 6265 7274 696e 692e 2054  al or bertini. T
│ │ │ │ -00023760: 6865 206f 7074 696f 6e20 7769 6c6c 2061  he option will a
│ │ │ │ -00023770: 6c73 6f20 6265 0a69 676e 6f72 6520 7768  lso be.ignore wh
│ │ │ │ -00023780: 656e 202a 6e6f 7465 204d 6574 686f 643a  en *note Method:
│ │ │ │ -00023790: 204d 6574 686f 642c 3d3e 4469 7265 6374   Method,=>Direct
│ │ │ │ -000237a0: 436f 6d70 6c65 7465 496e 7420 6973 2075  CompleteInt is u
│ │ │ │ -000237b0: 7365 642e 2054 6865 2064 6566 6175 6c74  sed. The default
│ │ │ │ -000237c0: 0a6f 7574 7075 7420 666f 7220 616c 6c20  .output for all 
│ │ │ │ -000237d0: 7468 6573 6520 6d65 7468 6f64 7320 6973  these methods is
│ │ │ │ -000237e0: 2043 686f 7752 696e 6745 6c65 6c6d 656e   ChowRingElelmen
│ │ │ │ -000237f0: 7420 7768 6963 6820 7769 6c6c 2072 6574  t which will ret
│ │ │ │ -00023800: 7572 6e20 616e 2065 6c65 6d65 6e74 0a6f  urn an element.o
│ │ │ │ -00023810: 6620 7468 6520 6170 7072 6f70 7269 6174  f the appropriat
│ │ │ │ -00023820: 6520 4368 6f77 2072 696e 672e 2041 6c6c  e Chow ring. All
│ │ │ │ -00023830: 206d 6574 686f 6473 2061 6c73 6f20 6861   methods also ha
│ │ │ │ -00023840: 7665 2061 6e20 6f70 7469 6f6e 2048 6173  ve an option Has
│ │ │ │ -00023850: 6846 6f72 6d20 7768 6963 680a 7265 7475  hForm which.retu
│ │ │ │ -00023860: 726e 7320 6164 6469 7469 6f6e 616c 2069  rns additional i
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│ │ │ │ -00023890: 6473 2064 7572 696e 6720 7468 6569 7220  ds during their 
│ │ │ │ -000238a0: 7374 616e 6461 7264 0a6f 7065 7261 7469  standard.operati
│ │ │ │ -000238b0: 6f6e 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  on...+----------
│ │ │ │ +00023500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a  ---------------.
│ │ │ │ +00023510: 0a54 6865 2073 6f75 7263 6520 6f66 2074  .The source of t
│ │ │ │ +00023520: 6869 7320 646f 6375 6d65 6e74 2069 7320  his document is 
│ │ │ │ +00023530: 696e 0a2f 6275 696c 642f 7265 7072 6f64  in./build/reprod
│ │ │ │ +00023540: 7563 6962 6c65 2d70 6174 682f 6d61 6361  ucible-path/maca
│ │ │ │ +00023550: 756c 6179 322d 312e 3235 2e31 312b 6473  ulay2-1.25.11+ds
│ │ │ │ +00023560: 2f4d 322f 4d61 6361 756c 6179 322f 7061  /M2/Macaulay2/pa
│ │ │ │ +00023570: 636b 6167 6573 2f0a 4368 6172 6163 7465  ckages/.Characte
│ │ │ │ +00023580: 7269 7374 6963 436c 6173 7365 732e 6d32  risticClasses.m2
│ │ │ │ +00023590: 3a32 3035 303a 302e 0a1f 0a46 696c 653a  :2050:0....File:
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│ │ │ │ +000235b0: 6c61 7373 6573 2e69 6e66 6f2c 204e 6f64  lasses.info, Nod
│ │ │ │ +000235c0: 653a 204f 7574 7075 742c 204e 6578 743a  e: Output, Next:
│ │ │ │ +000235d0: 2070 726f 6261 6269 6c69 7374 6963 2061   probabilistic a
│ │ │ │ +000235e0: 6c67 6f72 6974 686d 2c20 5072 6576 3a20  lgorithm, Prev: 
│ │ │ │ +000235f0: 4d75 6c74 6950 726f 6a43 6f6f 7264 5269  MultiProjCoordRi
│ │ │ │ +00023600: 6e67 2c20 5570 3a20 546f 700a 0a4f 7574  ng, Up: Top..Out
│ │ │ │ +00023610: 7075 740a 2a2a 2a2a 2a2a 0a0a 4465 7363  put.******..Desc
│ │ │ │ +00023620: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
│ │ │ │ +00023630: 3d3d 3d0a 0a54 6865 206f 7074 696f 6e20  ===..The option 
│ │ │ │ +00023640: 4f75 7470 7574 2069 7320 6f6e 6c79 2075  Output is only u
│ │ │ │ +00023650: 7365 6420 6279 2074 6865 2063 6f6d 6d61  sed by the comma
│ │ │ │ +00023660: 6e64 7320 2a6e 6f74 6520 4353 4d3a 2043  nds *note CSM: C
│ │ │ │ +00023670: 534d 2c2c 202a 6e6f 7465 2053 6567 7265  SM,, *note Segre
│ │ │ │ +00023680: 3a0a 5365 6772 652c 2c20 2a6e 6f74 6520  :.Segre,, *note 
│ │ │ │ +00023690: 4368 6572 6e3a 2043 6865 726e 2c20 616e  Chern: Chern, an
│ │ │ │ +000236a0: 6420 2a6e 6f74 6520 4575 6c65 723a 2045  d *note Euler: E
│ │ │ │ +000236b0: 756c 6572 2c20 746f 2073 7065 6369 6679  uler, to specify
│ │ │ │ +000236c0: 2074 6865 2074 7970 6520 6f66 0a6f 7574   the type of.out
│ │ │ │ +000236d0: 7075 7420 746f 2062 6520 7265 7475 726e  put to be return
│ │ │ │ +000236e0: 6564 2074 6f20 7468 6520 7573 6564 2e20  ed to the used. 
│ │ │ │ +000236f0: 5468 6973 206f 7074 696f 6e20 7769 6c6c  This option will
│ │ │ │ +00023700: 2062 6520 6967 6e6f 7265 6420 7768 656e   be ignored when
│ │ │ │ +00023710: 2075 7365 6420 7769 7468 0a2a 6e6f 7465   used with.*note
│ │ │ │ +00023720: 2043 6f6d 704d 6574 686f 643a 2043 6f6d   CompMethod: Com
│ │ │ │ +00023730: 704d 6574 686f 642c 2050 6e52 6573 6964  pMethod, PnResid
│ │ │ │ +00023740: 7561 6c20 6f72 2062 6572 7469 6e69 2e20  ual or bertini. 
│ │ │ │ +00023750: 5468 6520 6f70 7469 6f6e 2077 696c 6c20  The option will 
│ │ │ │ +00023760: 616c 736f 2062 650a 6967 6e6f 7265 2077  also be.ignore w
│ │ │ │ +00023770: 6865 6e20 2a6e 6f74 6520 4d65 7468 6f64  hen *note Method
│ │ │ │ +00023780: 3a20 4d65 7468 6f64 2c3d 3e44 6972 6563  : Method,=>Direc
│ │ │ │ +00023790: 7443 6f6d 706c 6574 6549 6e74 2069 7320  tCompleteInt is 
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│ │ │ │ +000237c0: 2074 6865 7365 206d 6574 686f 6473 2069   these methods i
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│ │ │ │ +000237e0: 6e74 2077 6869 6368 2077 696c 6c20 7265  nt which will re
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│ │ │ │ +00023800: 6f66 2074 6865 2061 7070 726f 7072 6961  of the appropria
│ │ │ │ +00023810: 7465 2043 686f 7720 7269 6e67 2e20 416c  te Chow ring. Al
│ │ │ │ +00023820: 6c20 6d65 7468 6f64 7320 616c 736f 2068  l methods also h
│ │ │ │ +00023830: 6176 6520 616e 206f 7074 696f 6e20 4861  ave an option Ha
│ │ │ │ +00023840: 7368 466f 726d 2077 6869 6368 0a72 6574  shForm which.ret
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│ │ │ │ +00023880: 6f64 7320 6475 7269 6e67 2074 6865 6972  ods during their
│ │ │ │ +00023890: 2073 7461 6e64 6172 640a 6f70 6572 6174   standard.operat
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│ │ │ │ +000238b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000238c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000238d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000238e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000238f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023900: 2d2d 2d2b 0a7c 6931 203a 2052 203d 205a  ---+.|i1 : R = Z
│ │ │ │ -00023910: 5a2f 3332 3734 395b 785f 302e 2e78 5f36  Z/32749[x_0..x_6
│ │ │ │ -00023920: 5d20 2020 2020 2020 2020 2020 2020 2020  ]               
│ │ │ │ +000238f0: 2d2d 2d2d 2b0a 7c69 3120 3a20 5220 3d20  ----+.|i1 : R = 
│ │ │ │ +00023900: 5a5a 2f33 3237 3439 5b78 5f30 2e2e 785f  ZZ/32749[x_0..x_
│ │ │ │ +00023910: 365d 2020 2020 2020 2020 2020 2020 2020  6]              
│ │ │ │ +00023920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023950: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00023940: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00023950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000239a0: 2020 207c 0a7c 6f31 203d 2052 2020 2020     |.|o1 = R    
│ │ │ │ +00023990: 2020 2020 7c0a 7c6f 3120 3d20 5220 2020      |.|o1 = R   
│ │ │ │ +000239a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000239b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000239c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000239d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000239e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000239f0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000239e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000239f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023a40: 2020 207c 0a7c 6f31 203a 2050 6f6c 796e     |.|o1 : Polyn
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│ │ │ │ +00023a30: 2020 2020 7c0a 7c6f 3120 3a20 506f 6c79      |.|o1 : Poly
│ │ │ │ +00023a40: 6e6f 6d69 616c 5269 6e67 2020 2020 2020  nomialRing      
│ │ │ │ +00023a50: 2020 2020 2020 2020 2020 2020 2020 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 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00023a80: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00023a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023ae0: 2d2d 2d2b 0a7c 6932 203a 2041 3d43 686f  ---+.|i2 : A=Cho
│ │ │ │ -00023af0: 7752 696e 6728 5229 2020 2020 2020 2020  wRing(R)        
│ │ │ │ +00023ad0: 2d2d 2d2d 2b0a 7c69 3220 3a20 413d 4368  ----+.|i2 : A=Ch
│ │ │ │ +00023ae0: 6f77 5269 6e67 2852 2920 2020 2020 2020  owRing(R)       
│ │ │ │ +00023af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023b30: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00023b20: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00023b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 207c 0a7c 6f32 203d 2041 2020 2020     |.|o2 = A    
│ │ │ │ +00023b70: 2020 2020 7c0a 7c6f 3220 3d20 4120 2020      |.|o2 = A   
│ │ │ │ +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 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023bd0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00023bc0: 2020 2020 7c0a 7c20 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 2020 2020                  
│ │ │ │  00023c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023c20: 2020 207c 0a7c 6f32 203a 2051 756f 7469     |.|o2 : Quoti
│ │ │ │ -00023c30: 656e 7452 696e 6720 2020 2020 2020 2020  entRing         
│ │ │ │ +00023c10: 2020 2020 7c0a 7c6f 3220 3a20 5175 6f74      |.|o2 : Quot
│ │ │ │ +00023c20: 6965 6e74 5269 6e67 2020 2020 2020 2020  ientRing        
│ │ │ │ +00023c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023c70: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00023c60: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00023c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023cc0: 2d2d 2d2b 0a7c 6933 203a 2049 3d69 6465  ---+.|i3 : I=ide
│ │ │ │ -00023cd0: 616c 2872 616e 646f 6d28 322c 5229 2c52  al(random(2,R),R
│ │ │ │ -00023ce0: 5f30 2a52 5f31 2a52 5f36 2d52 5f30 5e33  _0*R_1*R_6-R_0^3
│ │ │ │ -00023cf0: 293b 2020 2020 2020 2020 2020 2020 2020  );              
│ │ │ │ -00023d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023d10: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00023cb0: 2d2d 2d2d 2b0a 7c69 3320 3a20 493d 6964  ----+.|i3 : I=id
│ │ │ │ +00023cc0: 6561 6c28 7261 6e64 6f6d 2832 2c52 292c  eal(random(2,R),
│ │ │ │ +00023cd0: 525f 302a 525f 312a 525f 362d 525f 305e  R_0*R_1*R_6-R_0^
│ │ │ │ +00023ce0: 3329 3b20 2020 2020 2020 2020 2020 2020  3);             
│ │ │ │ +00023cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023d00: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00023d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023d20: 2020 2020 2020 2020 2020 2020 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 207c 0a7c 6f33 203a 2049 6465 616c     |.|o3 : Ideal
│ │ │ │ -00023d70: 206f 6620 5220 2020 2020 2020 2020 2020   of R           
│ │ │ │ +00023d50: 2020 2020 7c0a 7c6f 3320 3a20 4964 6561      |.|o3 : Idea
│ │ │ │ +00023d60: 6c20 6f66 2052 2020 2020 2020 2020 2020  l of R          
│ │ │ │ +00023d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023db0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00023da0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00023db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023e00: 2d2d 2d2b 0a7c 6934 203a 2063 736d 3d43  ---+.|i4 : csm=C
│ │ │ │ -00023e10: 534d 2841 2c49 2c4f 7574 7075 743d 3e48  SM(A,I,Output=>H
│ │ │ │ -00023e20: 6173 6846 6f72 6d29 2020 2020 2020 2020  ashForm)        
│ │ │ │ +00023df0: 2d2d 2d2d 2b0a 7c69 3420 3a20 6373 6d3d  ----+.|i4 : csm=
│ │ │ │ +00023e00: 4353 4d28 412c 492c 4f75 7470 7574 3d3e  CSM(A,I,Output=>
│ │ │ │ +00023e10: 4861 7368 466f 726d 2920 2020 2020 2020  HashForm)       
│ │ │ │ +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 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00023e40: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00023e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023e60: 2020 2020 2020 2020 2020 2020 2020 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 207c 0a7c 6f34 203d 204d 7574 6162     |.|o4 = Mutab
│ │ │ │ -00023eb0: 6c65 4861 7368 5461 626c 657b 2e2e 2e34  leHashTable{...4
│ │ │ │ -00023ec0: 2e2e 2e7d 2020 2020 2020 2020 2020 2020  ...}            
│ │ │ │ +00023e90: 2020 2020 7c0a 7c6f 3420 3d20 4d75 7461      |.|o4 = Muta
│ │ │ │ +00023ea0: 626c 6548 6173 6854 6162 6c65 7b2e 2e2e  bleHashTable{...
│ │ │ │ +00023eb0: 342e 2e2e 7d20 2020 2020 2020 2020 2020  4...}           
│ │ │ │ +00023ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023ef0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00023ee0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00023ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023f40: 2020 207c 0a7c 6f34 203a 204d 7574 6162     |.|o4 : Mutab
│ │ │ │ -00023f50: 6c65 4861 7368 5461 626c 6520 2020 2020  leHashTable     
│ │ │ │ +00023f30: 2020 2020 7c0a 7c6f 3420 3a20 4d75 7461      |.|o4 : Muta
│ │ │ │ +00023f40: 626c 6548 6173 6854 6162 6c65 2020 2020  bleHashTable    
│ │ │ │ +00023f50: 2020 2020 2020 2020 2020 2020 2020 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 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00023f80: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00023f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023fe0: 2d2d 2d2b 0a7c 6935 203a 2070 6565 6b20  ---+.|i5 : peek 
│ │ │ │ -00023ff0: 6373 6d20 2020 2020 2020 2020 2020 2020  csm             
│ │ │ │ +00023fd0: 2d2d 2d2d 2b0a 7c69 3520 3a20 7065 656b  ----+.|i5 : peek
│ │ │ │ +00023fe0: 2063 736d 2020 2020 2020 2020 2020 2020   csm            
│ │ │ │ +00023ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024030: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00024020: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024080: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00024090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000240a0: 2020 2020 2020 2020 3620 2020 2020 2035          6      5
│ │ │ │ -000240b0: 2020 2020 2020 3420 2020 2020 2033 2020        4      3  
│ │ │ │ -000240c0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -000240d0: 2020 207c 0a7c 6f35 203d 204d 7574 6162     |.|o5 = Mutab
│ │ │ │ -000240e0: 6c65 4861 7368 5461 626c 657b 7b30 2c20  leHashTable{{0, 
│ │ │ │ -000240f0: 317d 203d 3e20 3268 2020 2b20 3233 6820  1} => 2h  + 23h 
│ │ │ │ -00024100: 202b 2033 3268 2020 2b20 3333 6820 202b   + 32h  + 33h  +
│ │ │ │ -00024110: 2031 3868 2020 2b20 3568 207d 2020 2020   18h  + 5h }    
│ │ │ │ -00024120: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00024130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024140: 2020 2020 2020 2020 3120 2020 2020 2031          1      1
│ │ │ │ -00024150: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ -00024160: 2020 2020 3120 2020 2020 3120 2020 2020      1     1     
│ │ │ │ -00024170: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00024180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024190: 2020 2020 2020 3620 2020 2020 2035 2020        6      5  
│ │ │ │ -000241a0: 2020 2020 3420 2020 2020 2033 2020 2020      4      3    
│ │ │ │ -000241b0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -000241c0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -000241d0: 2020 2020 2020 2020 2020 2020 4353 4d20              CSM 
│ │ │ │ -000241e0: 3d3e 2031 3068 2020 2b20 3132 6820 202b  => 10h  + 12h  +
│ │ │ │ -000241f0: 2032 3268 2020 2b20 3136 6820 202b 2036   22h  + 16h  + 6
│ │ │ │ -00024200: 6820 2020 2020 2020 2020 2020 2020 2020  h               
│ │ │ │ -00024210: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00024220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024230: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ -00024240: 2020 2020 3120 2020 2020 2031 2020 2020      1      1    
│ │ │ │ -00024250: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │ -00024260: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00024270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024280: 2020 2020 2036 2020 2020 2020 3520 2020       6      5   
│ │ │ │ -00024290: 2020 2034 2020 2020 2020 3320 2020 2020     4      3     
│ │ │ │ -000242a0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -000242b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -000242c0: 2020 2020 2020 2020 2020 2020 7b30 7d20              {0} 
│ │ │ │ -000242d0: 3d3e 2036 6820 202b 2031 3868 2020 2b20  => 6h  + 18h  + 
│ │ │ │ -000242e0: 3236 6820 202b 2032 3268 2020 2b20 3130  26h  + 22h  + 10
│ │ │ │ -000242f0: 6820 202b 2032 6820 2020 2020 2020 2020  h  + 2h         
│ │ │ │ -00024300: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00024310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024320: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ -00024330: 2020 2031 2020 2020 2020 3120 2020 2020     1      1     
│ │ │ │ -00024340: 2031 2020 2020 2031 2020 2020 2020 2020   1     1        
│ │ │ │ -00024350: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00024360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024370: 2020 2020 2036 2020 2020 2020 3520 2020       6      5   
│ │ │ │ -00024380: 2020 2034 2020 2020 2020 3320 2020 2020     4      3     
│ │ │ │ -00024390: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -000243a0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -000243b0: 2020 2020 2020 2020 2020 2020 7b31 7d20              {1} 
│ │ │ │ -000243c0: 3d3e 2036 6820 202b 2031 3768 2020 2b20  => 6h  + 17h  + 
│ │ │ │ -000243d0: 3238 6820 202b 2032 3768 2020 2b20 3134  28h  + 27h  + 14
│ │ │ │ -000243e0: 6820 202b 2033 6820 2020 2020 2020 2020  h  + 3h         
│ │ │ │ -000243f0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00024400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024410: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ -00024420: 2020 2031 2020 2020 2020 3120 2020 2020     1      1     
│ │ │ │ -00024430: 2031 2020 2020 2031 2020 2020 2020 2020   1     1        
│ │ │ │ -00024440: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00024070: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024090: 2020 2020 2020 2020 2036 2020 2020 2020           6      
│ │ │ │ +000240a0: 3520 2020 2020 2034 2020 2020 2020 3320  5      4      3 
│ │ │ │ +000240b0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +000240c0: 2020 2020 7c0a 7c6f 3520 3d20 4d75 7461      |.|o5 = Muta
│ │ │ │ +000240d0: 626c 6548 6173 6854 6162 6c65 7b7b 302c  bleHashTable{{0,
│ │ │ │ +000240e0: 2031 7d20 3d3e 2032 6820 202b 2032 3368   1} => 2h  + 23h
│ │ │ │ +000240f0: 2020 2b20 3332 6820 202b 2033 3368 2020    + 32h  + 33h  
│ │ │ │ +00024100: 2b20 3138 6820 202b 2035 6820 7d20 2020  + 18h  + 5h }   
│ │ │ │ +00024110: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024130: 2020 2020 2020 2020 2031 2020 2020 2020           1      
│ │ │ │ +00024140: 3120 2020 2020 2031 2020 2020 2020 3120  1      1      1 
│ │ │ │ +00024150: 2020 2020 2031 2020 2020 2031 2020 2020       1     1    
│ │ │ │ +00024160: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024180: 2020 2020 2020 2036 2020 2020 2020 3520         6      5 
│ │ │ │ +00024190: 2020 2020 2034 2020 2020 2020 3320 2020       4      3   
│ │ │ │ +000241a0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +000241b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000241c0: 2020 2020 2020 2020 2020 2020 2043 534d               CSM
│ │ │ │ +000241d0: 203d 3e20 3130 6820 202b 2031 3268 2020   => 10h  + 12h  
│ │ │ │ +000241e0: 2b20 3232 6820 202b 2031 3668 2020 2b20  + 22h  + 16h  + 
│ │ │ │ +000241f0: 3668 2020 2020 2020 2020 2020 2020 2020  6h              
│ │ │ │ +00024200: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024220: 2020 2020 2020 2031 2020 2020 2020 3120         1      1 
│ │ │ │ +00024230: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ +00024240: 2020 3120 2020 2020 2020 2020 2020 2020    1             
│ │ │ │ +00024250: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024270: 2020 2020 2020 3620 2020 2020 2035 2020        6      5  
│ │ │ │ +00024280: 2020 2020 3420 2020 2020 2033 2020 2020      4      3    
│ │ │ │ +00024290: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +000242a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000242b0: 2020 2020 2020 2020 2020 2020 207b 307d               {0}
│ │ │ │ +000242c0: 203d 3e20 3668 2020 2b20 3138 6820 202b   => 6h  + 18h  +
│ │ │ │ +000242d0: 2032 3668 2020 2b20 3232 6820 202b 2031   26h  + 22h  + 1
│ │ │ │ +000242e0: 3068 2020 2b20 3268 2020 2020 2020 2020  0h  + 2h        
│ │ │ │ +000242f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024310: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ +00024320: 2020 2020 3120 2020 2020 2031 2020 2020      1      1    
│ │ │ │ +00024330: 2020 3120 2020 2020 3120 2020 2020 2020    1     1       
│ │ │ │ +00024340: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024360: 2020 2020 2020 3620 2020 2020 2035 2020        6      5  
│ │ │ │ +00024370: 2020 2020 3420 2020 2020 2033 2020 2020      4      3    
│ │ │ │ +00024380: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +00024390: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000243a0: 2020 2020 2020 2020 2020 2020 207b 317d               {1}
│ │ │ │ +000243b0: 203d 3e20 3668 2020 2b20 3137 6820 202b   => 6h  + 17h  +
│ │ │ │ +000243c0: 2032 3868 2020 2b20 3237 6820 202b 2031   28h  + 27h  + 1
│ │ │ │ +000243d0: 3468 2020 2b20 3368 2020 2020 2020 2020  4h  + 3h        
│ │ │ │ +000243e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000243f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024400: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ +00024410: 2020 2020 3120 2020 2020 2031 2020 2020      1      1    
│ │ │ │ +00024420: 2020 3120 2020 2020 3120 2020 2020 2020    1     1       
│ │ │ │ +00024430: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00024440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024490: 2d2d 2d2b 0a7c 6936 203a 2043 534d 2841  ---+.|i6 : CSM(A
│ │ │ │ -000244a0: 2c69 6465 616c 2049 5f30 293d 3d63 736d  ,ideal I_0)==csm
│ │ │ │ -000244b0: 237b 307d 2020 2020 2020 2020 2020 2020  #{0}            
│ │ │ │ +00024480: 2d2d 2d2d 2b0a 7c69 3620 3a20 4353 4d28  ----+.|i6 : CSM(
│ │ │ │ +00024490: 412c 6964 6561 6c20 495f 3029 3d3d 6373  A,ideal I_0)==cs
│ │ │ │ +000244a0: 6d23 7b30 7d20 2020 2020 2020 2020 2020  m#{0}           
│ │ │ │ +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 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000244d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000244e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000244f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024530: 2020 207c 0a7c 6f36 203d 2074 7275 6520     |.|o6 = true 
│ │ │ │ +00024520: 2020 2020 7c0a 7c6f 3620 3d20 7472 7565      |.|o6 = true
│ │ │ │ +00024530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024580: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00024570: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00024580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000245a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000245b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000245c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000245d0: 2d2d 2d2b 0a7c 6937 203a 2043 534d 2841  ---+.|i7 : CSM(A
│ │ │ │ -000245e0: 2c69 6465 616c 2849 5f30 2a49 5f31 2929  ,ideal(I_0*I_1))
│ │ │ │ -000245f0: 3d3d 6373 6d23 7b30 2c31 7d20 2020 2020  ==csm#{0,1}     
│ │ │ │ +000245c0: 2d2d 2d2d 2b0a 7c69 3720 3a20 4353 4d28  ----+.|i7 : CSM(
│ │ │ │ +000245d0: 412c 6964 6561 6c28 495f 302a 495f 3129  A,ideal(I_0*I_1)
│ │ │ │ +000245e0: 293d 3d63 736d 237b 302c 317d 2020 2020  )==csm#{0,1}    
│ │ │ │ +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 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00024610: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024670: 2020 207c 0a7c 6f37 203d 2074 7275 6520     |.|o7 = true 
│ │ │ │ +00024660: 2020 2020 7c0a 7c6f 3720 3d20 7472 7565      |.|o7 = true
│ │ │ │ +00024670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000246a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000246b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000246c0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +000246b0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +000246c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000246d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000246e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000246f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024710: 2d2d 2d2b 0a7c 6938 203a 2063 3d43 6865  ---+.|i8 : c=Che
│ │ │ │ -00024720: 726e 2820 492c 204f 7574 7075 743d 3e48  rn( I, Output=>H
│ │ │ │ -00024730: 6173 6846 6f72 6d29 2020 2020 2020 2020  ashForm)        
│ │ │ │ +00024700: 2d2d 2d2d 2b0a 7c69 3820 3a20 633d 4368  ----+.|i8 : c=Ch
│ │ │ │ +00024710: 6572 6e28 2049 2c20 4f75 7470 7574 3d3e  ern( I, Output=>
│ │ │ │ +00024720: 4861 7368 466f 726d 2920 2020 2020 2020  HashForm)       
│ │ │ │ +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 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00024750: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000247a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000247b0: 2020 207c 0a7c 6f38 203d 204d 7574 6162     |.|o8 = Mutab
│ │ │ │ -000247c0: 6c65 4861 7368 5461 626c 657b 2e2e 2e36  leHashTable{...6
│ │ │ │ -000247d0: 2e2e 2e7d 2020 2020 2020 2020 2020 2020  ...}            
│ │ │ │ +000247a0: 2020 2020 7c0a 7c6f 3820 3d20 4d75 7461      |.|o8 = Muta
│ │ │ │ +000247b0: 626c 6548 6173 6854 6162 6c65 7b2e 2e2e  bleHashTable{...
│ │ │ │ +000247c0: 362e 2e2e 7d20 2020 2020 2020 2020 2020  6...}           
│ │ │ │ +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 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000247f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024810: 2020 2020 2020 2020 2020 2020 2020 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 207c 0a7c 6f38 203a 204d 7574 6162     |.|o8 : Mutab
│ │ │ │ -00024860: 6c65 4861 7368 5461 626c 6520 2020 2020  leHashTable     
│ │ │ │ +00024840: 2020 2020 7c0a 7c6f 3820 3a20 4d75 7461      |.|o8 : Muta
│ │ │ │ +00024850: 626c 6548 6173 6854 6162 6c65 2020 2020  bleHashTable    
│ │ │ │ +00024860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000248a0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00024890: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +000248a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000248b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000248c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000248d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000248e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000248f0: 2d2d 2d2b 0a7c 6939 203a 2070 6565 6b20  ---+.|i9 : peek 
│ │ │ │ -00024900: 6320 2020 2020 2020 2020 2020 2020 2020  c               
│ │ │ │ +000248e0: 2d2d 2d2d 2b0a 7c69 3920 3a20 7065 656b  ----+.|i9 : peek
│ │ │ │ +000248f0: 2063 2020 2020 2020 2020 2020 2020 2020   c              
│ │ │ │ +00024900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00024930: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024950: 2020 2020 2020 2020 2020 2020 2020 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 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00024980: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000249a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000249b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000249c0: 2020 3220 2020 2020 2033 2020 2020 2020    2      3      
│ │ │ │ -000249d0: 3420 2020 2020 2020 3520 2020 2020 2020  4       5       
│ │ │ │ -000249e0: 3620 207c 0a7c 6f39 203d 204d 7574 6162  6  |.|o9 = Mutab
│ │ │ │ -000249f0: 6c65 4861 7368 5461 626c 657b 5365 6772  leHashTable{Segr
│ │ │ │ -00024a00: 654c 6973 7420 3d3e 207b 302c 2030 2c20  eList => {0, 0, 
│ │ │ │ -00024a10: 3668 202c 202d 3330 6820 2c20 3131 3468  6h , -30h , 114h
│ │ │ │ -00024a20: 202c 202d 3339 3068 202c 2031 3236 3668   , -390h , 1266h
│ │ │ │ -00024a30: 207d 7d7c 0a7c 2020 2020 2020 2020 2020   }}|.|          
│ │ │ │ +000249b0: 2020 2032 2020 2020 2020 3320 2020 2020     2      3     
│ │ │ │ +000249c0: 2034 2020 2020 2020 2035 2020 2020 2020   4       5      
│ │ │ │ +000249d0: 2036 2020 7c0a 7c6f 3920 3d20 4d75 7461   6  |.|o9 = Muta
│ │ │ │ +000249e0: 626c 6548 6173 6854 6162 6c65 7b53 6567  bleHashTable{Seg
│ │ │ │ +000249f0: 7265 4c69 7374 203d 3e20 7b30 2c20 302c  reList => {0, 0,
│ │ │ │ +00024a00: 2036 6820 2c20 2d33 3068 202c 2031 3134   6h , -30h , 114
│ │ │ │ +00024a10: 6820 2c20 2d33 3930 6820 2c20 3132 3636  h , -390h , 1266
│ │ │ │ +00024a20: 6820 7d7d 7c0a 7c20 2020 2020 2020 2020  h }}|.|         
│ │ │ │ +00024a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024a60: 2020 3120 2020 2020 2031 2020 2020 2020    1      1      
│ │ │ │ -00024a70: 3120 2020 2020 2020 3120 2020 2020 2020  1       1       
│ │ │ │ -00024a80: 3120 207c 0a7c 2020 2020 2020 2020 2020  1  |.|          
│ │ │ │ +00024a50: 2020 2031 2020 2020 2020 3120 2020 2020     1      1     
│ │ │ │ +00024a60: 2031 2020 2020 2020 2031 2020 2020 2020   1       1      
│ │ │ │ +00024a70: 2031 2020 7c0a 7c20 2020 2020 2020 2020   1  |.|         
│ │ │ │ +00024a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024ab0: 3220 2020 2033 2020 2020 3420 2020 2035  2    3    4    5
│ │ │ │ -00024ac0: 2020 2020 3620 2020 2020 2020 2020 2020      6           
│ │ │ │ -00024ad0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00024ae0: 2020 2020 2020 2020 2020 2020 476c 6973              Glis
│ │ │ │ -00024af0: 7420 3d3e 207b 312c 2033 6820 2c20 3368  t => {1, 3h , 3h
│ │ │ │ -00024b00: 202c 2033 6820 2c20 3368 202c 2033 6820   , 3h , 3h , 3h 
│ │ │ │ -00024b10: 2c20 3368 207d 2020 2020 2020 2020 2020  , 3h }          
│ │ │ │ -00024b20: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00024b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024b40: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ -00024b50: 3120 2020 2031 2020 2020 3120 2020 2031  1    1    1    1
│ │ │ │ -00024b60: 2020 2020 3120 2020 2020 2020 2020 2020      1           
│ │ │ │ -00024b70: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00024b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024b90: 2020 2020 2020 2020 2020 3620 2020 2020            6     
│ │ │ │ -00024ba0: 2020 3520 2020 2020 2020 3420 2020 2020    5       4     
│ │ │ │ -00024bb0: 2033 2020 2020 2032 2020 2020 2020 2020   3     2        
│ │ │ │ -00024bc0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00024bd0: 2020 2020 2020 2020 2020 2020 5365 6772              Segr
│ │ │ │ -00024be0: 6520 3d3e 2031 3236 3668 2020 2d20 3339  e => 1266h  - 39
│ │ │ │ -00024bf0: 3068 2020 2b20 3131 3468 2020 2d20 3330  0h  + 114h  - 30
│ │ │ │ -00024c00: 6820 202b 2036 6820 2020 2020 2020 2020  h  + 6h         
│ │ │ │ -00024c10: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00024c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024c30: 2020 2020 2020 2020 2020 3120 2020 2020            1     
│ │ │ │ -00024c40: 2020 3120 2020 2020 2020 3120 2020 2020    1       1     
│ │ │ │ -00024c50: 2031 2020 2020 2031 2020 2020 2020 2020   1     1        
│ │ │ │ -00024c60: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00024c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024c80: 2020 2020 2020 2020 3620 2020 2020 2035          6      5
│ │ │ │ -00024c90: 2020 2020 2020 3420 2020 2020 2033 2020        4      3  
│ │ │ │ -00024ca0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -00024cb0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00024cc0: 2020 2020 2020 2020 2020 2020 4368 6572              Cher
│ │ │ │ -00024cd0: 6e20 3d3e 2039 3068 2020 2d20 3132 6820  n => 90h  - 12h 
│ │ │ │ -00024ce0: 202b 2033 3068 2020 2b20 3132 6820 202b   + 30h  + 12h  +
│ │ │ │ -00024cf0: 2036 6820 2020 2020 2020 2020 2020 2020   6h             
│ │ │ │ -00024d00: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00024d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024d20: 2020 2020 2020 2020 3120 2020 2020 2031          1      1
│ │ │ │ -00024d30: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ -00024d40: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │ -00024d50: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00024d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024d70: 2020 2020 2036 2020 2020 2020 3520 2020       6      5   
│ │ │ │ -00024d80: 2020 2034 2020 2020 2020 3320 2020 2020     4      3     
│ │ │ │ -00024d90: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -00024da0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00024db0: 2020 2020 2020 2020 2020 2020 4346 203d              CF =
│ │ │ │ -00024dc0: 3e20 3930 6820 202d 2031 3268 2020 2b20  > 90h  - 12h  + 
│ │ │ │ -00024dd0: 3330 6820 202b 2031 3268 2020 2b20 3668  30h  + 12h  + 6h
│ │ │ │ -00024de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024df0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00024e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024e10: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ -00024e20: 2020 2031 2020 2020 2020 3120 2020 2020     1      1     
│ │ │ │ -00024e30: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ -00024e40: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00024e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024e60: 2020 2036 2020 2020 2035 2020 2020 2034     6     5     4
│ │ │ │ -00024e70: 2020 2020 2033 2020 2020 2032 2020 2020       3     2    
│ │ │ │ -00024e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024e90: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00024ea0: 2020 2020 2020 2020 2020 2020 4720 3d3e              G =>
│ │ │ │ -00024eb0: 2033 6820 202b 2033 6820 202b 2033 6820   3h  + 3h  + 3h 
│ │ │ │ -00024ec0: 202b 2033 6820 202b 2033 6820 202b 2033   + 3h  + 3h  + 3
│ │ │ │ -00024ed0: 6820 202b 2031 2020 2020 2020 2020 2020  h  + 1          
│ │ │ │ -00024ee0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00024ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024f00: 2020 2031 2020 2020 2031 2020 2020 2031     1     1     1
│ │ │ │ -00024f10: 2020 2020 2031 2020 2020 2031 2020 2020       1     1    
│ │ │ │ -00024f20: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │ -00024f30: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00024aa0: 2032 2020 2020 3320 2020 2034 2020 2020   2    3    4    
│ │ │ │ +00024ab0: 3520 2020 2036 2020 2020 2020 2020 2020  5    6          
│ │ │ │ +00024ac0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024ad0: 2020 2020 2020 2020 2020 2020 2047 6c69               Gli
│ │ │ │ +00024ae0: 7374 203d 3e20 7b31 2c20 3368 202c 2033  st => {1, 3h , 3
│ │ │ │ +00024af0: 6820 2c20 3368 202c 2033 6820 2c20 3368  h , 3h , 3h , 3h
│ │ │ │ +00024b00: 202c 2033 6820 7d20 2020 2020 2020 2020   , 3h }         
│ │ │ │ +00024b10: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024b30: 2020 2020 2020 2020 2020 2020 3120 2020              1   
│ │ │ │ +00024b40: 2031 2020 2020 3120 2020 2031 2020 2020   1    1    1    
│ │ │ │ +00024b50: 3120 2020 2031 2020 2020 2020 2020 2020  1    1          
│ │ │ │ +00024b60: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024b80: 2020 2020 2020 2020 2020 2036 2020 2020             6    
│ │ │ │ +00024b90: 2020 2035 2020 2020 2020 2034 2020 2020     5       4    
│ │ │ │ +00024ba0: 2020 3320 2020 2020 3220 2020 2020 2020    3     2       
│ │ │ │ +00024bb0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024bc0: 2020 2020 2020 2020 2020 2020 2053 6567               Seg
│ │ │ │ +00024bd0: 7265 203d 3e20 3132 3636 6820 202d 2033  re => 1266h  - 3
│ │ │ │ +00024be0: 3930 6820 202b 2031 3134 6820 202d 2033  90h  + 114h  - 3
│ │ │ │ +00024bf0: 3068 2020 2b20 3668 2020 2020 2020 2020  0h  + 6h        
│ │ │ │ +00024c00: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024c20: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ +00024c30: 2020 2031 2020 2020 2020 2031 2020 2020     1       1    
│ │ │ │ +00024c40: 2020 3120 2020 2020 3120 2020 2020 2020    1     1       
│ │ │ │ +00024c50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024c70: 2020 2020 2020 2020 2036 2020 2020 2020           6      
│ │ │ │ +00024c80: 3520 2020 2020 2034 2020 2020 2020 3320  5      4      3 
│ │ │ │ +00024c90: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +00024ca0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024cb0: 2020 2020 2020 2020 2020 2020 2043 6865               Che
│ │ │ │ +00024cc0: 726e 203d 3e20 3930 6820 202d 2031 3268  rn => 90h  - 12h
│ │ │ │ +00024cd0: 2020 2b20 3330 6820 202b 2031 3268 2020    + 30h  + 12h  
│ │ │ │ +00024ce0: 2b20 3668 2020 2020 2020 2020 2020 2020  + 6h            
│ │ │ │ +00024cf0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024d10: 2020 2020 2020 2020 2031 2020 2020 2020           1      
│ │ │ │ +00024d20: 3120 2020 2020 2031 2020 2020 2020 3120  1      1      1 
│ │ │ │ +00024d30: 2020 2020 3120 2020 2020 2020 2020 2020      1           
│ │ │ │ +00024d40: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024d60: 2020 2020 2020 3620 2020 2020 2035 2020        6      5  
│ │ │ │ +00024d70: 2020 2020 3420 2020 2020 2033 2020 2020      4      3    
│ │ │ │ +00024d80: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +00024d90: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024da0: 2020 2020 2020 2020 2020 2020 2043 4620               CF 
│ │ │ │ +00024db0: 3d3e 2039 3068 2020 2d20 3132 6820 202b  => 90h  - 12h  +
│ │ │ │ +00024dc0: 2033 3068 2020 2b20 3132 6820 202b 2036   30h  + 12h  + 6
│ │ │ │ +00024dd0: 6820 2020 2020 2020 2020 2020 2020 2020  h               
│ │ │ │ +00024de0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024e00: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ +00024e10: 2020 2020 3120 2020 2020 2031 2020 2020      1      1    
│ │ │ │ +00024e20: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │ +00024e30: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024e50: 2020 2020 3620 2020 2020 3520 2020 2020      6     5     
│ │ │ │ +00024e60: 3420 2020 2020 3320 2020 2020 3220 2020  4     3     2   
│ │ │ │ +00024e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024e80: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024e90: 2020 2020 2020 2020 2020 2020 2047 203d               G =
│ │ │ │ +00024ea0: 3e20 3368 2020 2b20 3368 2020 2b20 3368  > 3h  + 3h  + 3h
│ │ │ │ +00024eb0: 2020 2b20 3368 2020 2b20 3368 2020 2b20    + 3h  + 3h  + 
│ │ │ │ +00024ec0: 3368 2020 2b20 3120 2020 2020 2020 2020  3h  + 1         
│ │ │ │ +00024ed0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024ef0: 2020 2020 3120 2020 2020 3120 2020 2020      1     1     
│ │ │ │ +00024f00: 3120 2020 2020 3120 2020 2020 3120 2020  1     1     1   
│ │ │ │ +00024f10: 2020 3120 2020 2020 2020 2020 2020 2020    1             
│ │ │ │ +00024f20: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00024f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024f80: 2d2d 2d2b 0a7c 6931 3020 3a20 7365 673d  ---+.|i10 : seg=
│ │ │ │ -00024f90: 5365 6772 6528 2049 2c20 4f75 7470 7574  Segre( I, Output
│ │ │ │ -00024fa0: 3d3e 4861 7368 466f 726d 2920 2020 2020  =>HashForm)     
│ │ │ │ +00024f70: 2d2d 2d2d 2b0a 7c69 3130 203a 2073 6567  ----+.|i10 : seg
│ │ │ │ +00024f80: 3d53 6567 7265 2820 492c 204f 7574 7075  =Segre( I, Outpu
│ │ │ │ +00024f90: 743d 3e48 6173 6846 6f72 6d29 2020 2020  t=>HashForm)    
│ │ │ │ +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 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00024fc0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024fe0: 2020 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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025020: 2020 207c 0a7c 6f31 3020 3d20 4d75 7461     |.|o10 = Muta
│ │ │ │ -00025030: 626c 6548 6173 6854 6162 6c65 7b2e 2e2e  bleHashTable{...
│ │ │ │ -00025040: 342e 2e2e 7d20 2020 2020 2020 2020 2020  4...}           
│ │ │ │ +00025010: 2020 2020 7c0a 7c6f 3130 203d 204d 7574      |.|o10 = Mut
│ │ │ │ +00025020: 6162 6c65 4861 7368 5461 626c 657b 2e2e  ableHashTable{..
│ │ │ │ +00025030: 2e34 2e2e 2e7d 2020 2020 2020 2020 2020  .4...}          
│ │ │ │ +00025040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025070: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00025060: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00025070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000250a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000250b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000250c0: 2020 207c 0a7c 6f31 3020 3a20 4d75 7461     |.|o10 : Muta
│ │ │ │ -000250d0: 626c 6548 6173 6854 6162 6c65 2020 2020  bleHashTable    
│ │ │ │ +000250b0: 2020 2020 7c0a 7c6f 3130 203a 204d 7574      |.|o10 : Mut
│ │ │ │ +000250c0: 6162 6c65 4861 7368 5461 626c 6520 2020  ableHashTable   
│ │ │ │ +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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025110: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00025100: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00025110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025160: 2d2d 2d2b 0a7c 6931 3120 3a20 7065 656b  ---+.|i11 : peek
│ │ │ │ -00025170: 2073 6567 2020 2020 2020 2020 2020 2020   seg            
│ │ │ │ +00025150: 2d2d 2d2d 2b0a 7c69 3131 203a 2070 6565  ----+.|i11 : pee
│ │ │ │ +00025160: 6b20 7365 6720 2020 2020 2020 2020 2020  k seg           
│ │ │ │ +00025170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000251a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000251b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000251a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000251b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025200: 2020 207c 0a7c 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 2032 2020 2020 2020 3320 2020 2020     2      3     
│ │ │ │ -00025240: 2034 2020 2020 2020 2035 2020 2020 2020   4       5      
│ │ │ │ -00025250: 2020 207c 0a7c 6f31 3120 3d20 4d75 7461     |.|o11 = Muta
│ │ │ │ -00025260: 626c 6548 6173 6854 6162 6c65 7b53 6567  bleHashTable{Seg
│ │ │ │ -00025270: 7265 4c69 7374 203d 3e20 7b30 2c20 302c  reList => {0, 0,
│ │ │ │ -00025280: 2036 6820 2c20 2d33 3068 202c 2031 3134   6h , -30h , 114
│ │ │ │ -00025290: 6820 2c20 2d33 3930 6820 2c20 2020 2020  h , -390h ,     
│ │ │ │ -000252a0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00025220: 2020 2020 3220 2020 2020 2033 2020 2020      2      3    
│ │ │ │ +00025230: 2020 3420 2020 2020 2020 3520 2020 2020    4       5     
│ │ │ │ +00025240: 2020 2020 7c0a 7c6f 3131 203d 204d 7574      |.|o11 = Mut
│ │ │ │ +00025250: 6162 6c65 4861 7368 5461 626c 657b 5365  ableHashTable{Se
│ │ │ │ +00025260: 6772 654c 6973 7420 3d3e 207b 302c 2030  greList => {0, 0
│ │ │ │ +00025270: 2c20 3668 202c 202d 3330 6820 2c20 3131  , 6h , -30h , 11
│ │ │ │ +00025280: 3468 202c 202d 3339 3068 202c 2020 2020  4h , -390h ,    
│ │ │ │ +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 2031 2020 2020 2020 3120 2020 2020     1      1     
│ │ │ │ -000252e0: 2031 2020 2020 2020 2031 2020 2020 2020   1       1      
│ │ │ │ -000252f0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000252c0: 2020 2020 3120 2020 2020 2031 2020 2020      1      1    
│ │ │ │ +000252d0: 2020 3120 2020 2020 2020 3120 2020 2020    1       1     
│ │ │ │ +000252e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000252f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025320: 2032 2020 2020 3320 2020 2034 2020 2020   2    3    4    
│ │ │ │ -00025330: 3520 2020 2036 2020 2020 2020 2020 2020  5    6          
│ │ │ │ -00025340: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00025350: 2020 2020 2020 2020 2020 2020 2047 6c69               Gli
│ │ │ │ -00025360: 7374 203d 3e20 7b31 2c20 3368 202c 2033  st => {1, 3h , 3
│ │ │ │ -00025370: 6820 2c20 3368 202c 2033 6820 2c20 3368  h , 3h , 3h , 3h
│ │ │ │ -00025380: 202c 2033 6820 7d20 2020 2020 2020 2020   , 3h }         
│ │ │ │ -00025390: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -000253a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000253b0: 2020 2020 2020 2020 2020 2020 3120 2020              1   
│ │ │ │ -000253c0: 2031 2020 2020 3120 2020 2031 2020 2020   1    1    1    
│ │ │ │ -000253d0: 3120 2020 2031 2020 2020 2020 2020 2020  1    1          
│ │ │ │ -000253e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -000253f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025400: 2020 2020 2020 2020 2020 2036 2020 2020             6    
│ │ │ │ -00025410: 2020 2035 2020 2020 2020 2034 2020 2020     5       4    
│ │ │ │ -00025420: 2020 3320 2020 2020 3220 2020 2020 2020    3     2       
│ │ │ │ -00025430: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00025440: 2020 2020 2020 2020 2020 2020 2053 6567               Seg
│ │ │ │ -00025450: 7265 203d 3e20 3132 3636 6820 202d 2033  re => 1266h  - 3
│ │ │ │ -00025460: 3930 6820 202b 2031 3134 6820 202d 2033  90h  + 114h  - 3
│ │ │ │ -00025470: 3068 2020 2b20 3668 2020 2020 2020 2020  0h  + 6h        
│ │ │ │ -00025480: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00025490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000254a0: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ -000254b0: 2020 2031 2020 2020 2020 2031 2020 2020     1       1    
│ │ │ │ -000254c0: 2020 3120 2020 2020 3120 2020 2020 2020    1     1       
│ │ │ │ -000254d0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -000254e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000254f0: 2020 2020 3620 2020 2020 3520 2020 2020      6     5     
│ │ │ │ -00025500: 3420 2020 2020 3320 2020 2020 3220 2020  4     3     2   
│ │ │ │ -00025510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025520: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00025530: 2020 2020 2020 2020 2020 2020 2047 203d               G =
│ │ │ │ -00025540: 3e20 3368 2020 2b20 3368 2020 2b20 3368  > 3h  + 3h  + 3h
│ │ │ │ -00025550: 2020 2b20 3368 2020 2b20 3368 2020 2b20    + 3h  + 3h  + 
│ │ │ │ -00025560: 3368 2020 2b20 3120 2020 2020 2020 2020  3h  + 1         
│ │ │ │ -00025570: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00025580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025590: 2020 2020 3120 2020 2020 3120 2020 2020      1     1     
│ │ │ │ -000255a0: 3120 2020 2020 3120 2020 2020 3120 2020  1     1     1   
│ │ │ │ -000255b0: 2020 3120 2020 2020 2020 2020 2020 2020    1             
│ │ │ │ -000255c0: 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d     |.|----------
│ │ │ │ +00025310: 2020 3220 2020 2033 2020 2020 3420 2020    2    3    4   
│ │ │ │ +00025320: 2035 2020 2020 3620 2020 2020 2020 2020   5    6         
│ │ │ │ +00025330: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00025340: 2020 2020 2020 2020 2020 2020 2020 476c                Gl
│ │ │ │ +00025350: 6973 7420 3d3e 207b 312c 2033 6820 2c20  ist => {1, 3h , 
│ │ │ │ +00025360: 3368 202c 2033 6820 2c20 3368 202c 2033  3h , 3h , 3h , 3
│ │ │ │ +00025370: 6820 2c20 3368 207d 2020 2020 2020 2020  h , 3h }        
│ │ │ │ +00025380: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00025390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000253a0: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │ +000253b0: 2020 3120 2020 2031 2020 2020 3120 2020    1    1    1   
│ │ │ │ +000253c0: 2031 2020 2020 3120 2020 2020 2020 2020   1    1         
│ │ │ │ +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 3620 2020              6   
│ │ │ │ +00025400: 2020 2020 3520 2020 2020 2020 3420 2020      5       4   
│ │ │ │ +00025410: 2020 2033 2020 2020 2032 2020 2020 2020     3     2      
│ │ │ │ +00025420: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00025430: 2020 2020 2020 2020 2020 2020 2020 5365                Se
│ │ │ │ +00025440: 6772 6520 3d3e 2031 3236 3668 2020 2d20  gre => 1266h  - 
│ │ │ │ +00025450: 3339 3068 2020 2b20 3131 3468 2020 2d20  390h  + 114h  - 
│ │ │ │ +00025460: 3330 6820 202b 2036 6820 2020 2020 2020  30h  + 6h       
│ │ │ │ +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 3120 2020              1   
│ │ │ │ +000254a0: 2020 2020 3120 2020 2020 2020 3120 2020      1       1   
│ │ │ │ +000254b0: 2020 2031 2020 2020 2031 2020 2020 2020     1     1      
│ │ │ │ +000254c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000254d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000254e0: 2020 2020 2036 2020 2020 2035 2020 2020       6     5    
│ │ │ │ +000254f0: 2034 2020 2020 2033 2020 2020 2032 2020   4     3     2  
│ │ │ │ +00025500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025510: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00025520: 2020 2020 2020 2020 2020 2020 2020 4720                G 
│ │ │ │ +00025530: 3d3e 2033 6820 202b 2033 6820 202b 2033  => 3h  + 3h  + 3
│ │ │ │ +00025540: 6820 202b 2033 6820 202b 2033 6820 202b  h  + 3h  + 3h  +
│ │ │ │ +00025550: 2033 6820 202b 2031 2020 2020 2020 2020   3h  + 1        
│ │ │ │ +00025560: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00025570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025580: 2020 2020 2031 2020 2020 2031 2020 2020       1     1    
│ │ │ │ +00025590: 2031 2020 2020 2031 2020 2020 2031 2020   1     1     1  
│ │ │ │ +000255a0: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │ +000255b0: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
│ │ │ │ +000255c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000255d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000255e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000255f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025610: 2d2d 2d7c 0a7c 2020 2020 2036 2020 2020  ---|.|     6    
│ │ │ │ +00025600: 2d2d 2d2d 7c0a 7c20 2020 2020 3620 2020  ----|.|     6   
│ │ │ │ +00025610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025660: 2020 207c 0a7c 3132 3636 6820 7d7d 2020     |.|1266h }}  
│ │ │ │ +00025650: 2020 2020 7c0a 7c31 3236 3668 207d 7d20      |.|1266h }} 
│ │ │ │ +00025660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000256a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000256b0: 2020 207c 0a7c 2020 2020 2031 2020 2020     |.|     1    
│ │ │ │ +000256a0: 2020 2020 7c0a 7c20 2020 2020 3120 2020      |.|     1   
│ │ │ │ +000256b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025700: 2020 207c 0a2b 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: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025750: 2d2d 2d2b 0a7c 6931 3220 3a20 6575 3d45  ---+.|i12 : eu=E
│ │ │ │ -00025760: 756c 6572 2820 492c 204f 7574 7075 743d  uler( I, Output=
│ │ │ │ -00025770: 3e48 6173 6846 6f72 6d29 2020 2020 2020  >HashForm)      
│ │ │ │ +00025740: 2d2d 2d2d 2b0a 7c69 3132 203a 2065 753d  ----+.|i12 : eu=
│ │ │ │ +00025750: 4575 6c65 7228 2049 2c20 4f75 7470 7574  Euler( I, Output
│ │ │ │ +00025760: 3d3e 4861 7368 466f 726d 2920 2020 2020  =>HashForm)     
│ │ │ │ +00025770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000257a0: 2020 207c 0a7c 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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000257f0: 2020 207c 0a7c 6f31 3220 3d20 4d75 7461     |.|o12 = Muta
│ │ │ │ -00025800: 626c 6548 6173 6854 6162 6c65 7b2e 2e2e  bleHashTable{...
│ │ │ │ -00025810: 352e 2e2e 7d20 2020 2020 2020 2020 2020  5...}           
│ │ │ │ +000257e0: 2020 2020 7c0a 7c6f 3132 203d 204d 7574      |.|o12 = Mut
│ │ │ │ +000257f0: 6162 6c65 4861 7368 5461 626c 657b 2e2e  ableHashTable{..
│ │ │ │ +00025800: 2e35 2e2e 2e7d 2020 2020 2020 2020 2020  .5...}          
│ │ │ │ +00025810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025840: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00025830: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00025840: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00025860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025880: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000258a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000258c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00025900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00025930: 6b20 6575 2020 2020 2020 2020 2020 2020  k eu            
│ │ │ │ +00025940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025980: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00025970: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00025980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000259a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000259b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000259c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00025a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025a10: 2020 2020 2020 2020 2020 2020 7d20 2020              }   
│ │ │ │ -00025a20: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00025a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025a40: 2020 2020 2020 2020 2036 2020 2020 2020           6      
│ │ │ │ -00025a50: 3520 2020 2020 2034 2020 2020 2020 3320  5      4      3 
│ │ │ │ -00025a60: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -00025a70: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00025a80: 2020 2020 2020 2020 2020 2020 207b 302c               {0,
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│ │ │ │ -00025aa0: 2020 2b20 3332 6820 202b 2033 3368 2020    + 32h  + 33h  
│ │ │ │ -00025ab0: 2b20 3138 6820 202b 2035 6820 2020 2020  + 18h  + 5h     
│ │ │ │ -00025ac0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00025ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025ae0: 2020 2020 2020 2020 2031 2020 2020 2020           1      
│ │ │ │ -00025af0: 3120 2020 2020 2031 2020 2020 2020 3120  1      1      1 
│ │ │ │ -00025b00: 2020 2020 2031 2020 2020 2031 2020 2020       1     1    
│ │ │ │ -00025b10: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00025b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025b30: 2020 2020 2020 2036 2020 2020 2020 3520         6      5 
│ │ │ │ -00025b40: 2020 2020 2034 2020 2020 2020 3320 2020       4      3   
│ │ │ │ -00025b50: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -00025b60: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00025b70: 2020 2020 2020 2020 2020 2020 2043 534d               CSM
│ │ │ │ -00025b80: 203d 3e20 3130 6820 202b 2031 3268 2020   => 10h  + 12h  
│ │ │ │ -00025b90: 2b20 3232 6820 202b 2031 3668 2020 2b20  + 22h  + 16h  + 
│ │ │ │ -00025ba0: 3668 2020 2020 2020 2020 2020 2020 2020  6h              
│ │ │ │ -00025bb0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00025bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025bd0: 2020 2020 2020 2031 2020 2020 2020 3120         1      1 
│ │ │ │ -00025be0: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ -00025bf0: 2020 3120 2020 2020 2020 2020 2020 2020    1             
│ │ │ │ -00025c00: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00025c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025c20: 2020 2020 2020 3620 2020 2020 2035 2020        6      5  
│ │ │ │ -00025c30: 2020 2020 3420 2020 2020 2033 2020 2020      4      3    
│ │ │ │ -00025c40: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -00025c50: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00025c60: 2020 2020 2020 2020 2020 2020 207b 307d               {0}
│ │ │ │ -00025c70: 203d 3e20 3668 2020 2b20 3138 6820 202b   => 6h  + 18h  +
│ │ │ │ -00025c80: 2032 3668 2020 2b20 3232 6820 202b 2031   26h  + 22h  + 1
│ │ │ │ -00025c90: 3068 2020 2b20 3268 2020 2020 2020 2020  0h  + 2h        
│ │ │ │ -00025ca0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00025cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025cc0: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ -00025cd0: 2020 2020 3120 2020 2020 2031 2020 2020      1      1    
│ │ │ │ -00025ce0: 2020 3120 2020 2020 3120 2020 2020 2020    1     1       
│ │ │ │ -00025cf0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00025d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025d10: 2020 2020 2020 3620 2020 2020 2035 2020        6      5  
│ │ │ │ -00025d20: 2020 2020 3420 2020 2020 2033 2020 2020      4      3    
│ │ │ │ -00025d30: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -00025d40: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00025d50: 2020 2020 2020 2020 2020 2020 207b 317d               {1}
│ │ │ │ -00025d60: 203d 3e20 3668 2020 2b20 3137 6820 202b   => 6h  + 17h  +
│ │ │ │ -00025d70: 2032 3868 2020 2b20 3237 6820 202b 2031   28h  + 27h  + 1
│ │ │ │ -00025d80: 3468 2020 2b20 3368 2020 2020 2020 2020  4h  + 3h        
│ │ │ │ -00025d90: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00025da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025db0: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ -00025dc0: 2020 2020 3120 2020 2020 2031 2020 2020      1      1    
│ │ │ │ -00025dd0: 2020 3120 2020 2020 3120 2020 2020 2020    1     1       
│ │ │ │ -00025de0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
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│ │ │ │ +000259d0: 6162 6c65 4861 7368 5461 626c 657b 4575  ableHashTable{Eu
│ │ │ │ +000259e0: 6c65 7220 3d3e 2031 3020 2020 2020 2020  ler => 10       
│ │ │ │ +000259f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025a00: 2020 2020 2020 2020 2020 2020 207d 2020               }  
│ │ │ │ +00025a10: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00025a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025a30: 2020 2020 2020 2020 2020 3620 2020 2020            6     
│ │ │ │ +00025a40: 2035 2020 2020 2020 3420 2020 2020 2033   5      4      3
│ │ │ │ +00025a50: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +00025a60: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00025a70: 2020 2020 2020 2020 2020 2020 2020 7b30                {0
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│ │ │ │ +00025a90: 6820 202b 2033 3268 2020 2b20 3333 6820  h  + 32h  + 33h 
│ │ │ │ +00025aa0: 202b 2031 3868 2020 2b20 3568 2020 2020   + 18h  + 5h    
│ │ │ │ +00025ab0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00025ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00025ae0: 2031 2020 2020 2020 3120 2020 2020 2031   1      1      1
│ │ │ │ +00025af0: 2020 2020 2020 3120 2020 2020 3120 2020        1     1   
│ │ │ │ +00025b00: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00025b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025b20: 2020 2020 2020 2020 3620 2020 2020 2035          6      5
│ │ │ │ +00025b30: 2020 2020 2020 3420 2020 2020 2033 2020        4      3  
│ │ │ │ +00025b40: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00025b50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00025b60: 2020 2020 2020 2020 2020 2020 2020 4353                CS
│ │ │ │ +00025b70: 4d20 3d3e 2031 3068 2020 2b20 3132 6820  M => 10h  + 12h 
│ │ │ │ +00025b80: 202b 2032 3268 2020 2b20 3136 6820 202b   + 22h  + 16h  +
│ │ │ │ +00025b90: 2036 6820 2020 2020 2020 2020 2020 2020   6h             
│ │ │ │ +00025ba0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00025bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025bc0: 2020 2020 2020 2020 3120 2020 2020 2031          1      1
│ │ │ │ +00025bd0: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ +00025be0: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │ +00025bf0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00025c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025c10: 2020 2020 2020 2036 2020 2020 2020 3520         6      5 
│ │ │ │ +00025c20: 2020 2020 2034 2020 2020 2020 3320 2020       4      3   
│ │ │ │ +00025c30: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00025c40: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00025c50: 2020 2020 2020 2020 2020 2020 2020 7b30                {0
│ │ │ │ +00025c60: 7d20 3d3e 2036 6820 202b 2031 3868 2020  } => 6h  + 18h  
│ │ │ │ +00025c70: 2b20 3236 6820 202b 2032 3268 2020 2b20  + 26h  + 22h  + 
│ │ │ │ +00025c80: 3130 6820 202b 2032 6820 2020 2020 2020  10h  + 2h       
│ │ │ │ +00025c90: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00025ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025cb0: 2020 2020 2020 2031 2020 2020 2020 3120         1      1 
│ │ │ │ +00025cc0: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ +00025cd0: 2020 2031 2020 2020 2031 2020 2020 2020     1     1      
│ │ │ │ +00025ce0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00025cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025d00: 2020 2020 2020 2036 2020 2020 2020 3520         6      5 
│ │ │ │ +00025d10: 2020 2020 2034 2020 2020 2020 3320 2020       4      3   
│ │ │ │ +00025d20: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00025d30: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00025d40: 2020 2020 2020 2020 2020 2020 2020 7b31                {1
│ │ │ │ +00025d50: 7d20 3d3e 2036 6820 202b 2031 3768 2020  } => 6h  + 17h  
│ │ │ │ +00025d60: 2b20 3238 6820 202b 2032 3768 2020 2b20  + 28h  + 27h  + 
│ │ │ │ +00025d70: 3134 6820 202b 2033 6820 2020 2020 2020  14h  + 3h       
│ │ │ │ +00025d80: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00025d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025da0: 2020 2020 2020 2031 2020 2020 2020 3120         1      1 
│ │ │ │ +00025db0: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ +00025dc0: 2020 2031 2020 2020 2031 2020 2020 2020     1     1      
│ │ │ │ +00025dd0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00025de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00025f70: 5768 656e 2075 7369 6e67 2074 6865 202a  When using the *
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│ │ │ │ -00026310: 5365 6772 6520 636c 6173 7365 7320 636f  Segre classes co
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│ │ │ │ -00026330: 7920 7769 6c6c 2062 6520 7468 6f73 6520  y will be those 
│ │ │ │ -00026340: 6f66 2074 6865 2072 6164 6963 616c 206f  of the radical o
│ │ │ │ -00026350: 6620 616e 2069 6465 616c 2064 6566 696e  f an ideal defin
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│ │ │ │ -00026380: 6f64 7563 7420 6f66 2073 6f6d 6520 7375  oduct of some su
│ │ │ │ -00026390: 6273 6574 206f 6620 7468 6520 6765 6e65  bset of the gene
│ │ │ │ -000263a0: 7261 746f 7273 2e20 5765 2069 6c6c 7573  rators. We illus
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│ │ │ │ -000263c0: 616e 0a65 7861 6d70 6c65 2062 656c 6f77  an.example below
│ │ │ │ -000263d0: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...+------------
│ │ │ │ +00025e20: 2d2d 2d2d 2b0a 0a54 6865 204d 7574 6162  ----+..The Mutab
│ │ │ │ +00025e30: 6c65 4861 7368 5461 626c 6520 7265 7475  leHashTable retu
│ │ │ │ +00025e40: 726e 6564 2077 6974 6820 7468 6520 6f70  rned with the op
│ │ │ │ +00025e50: 7469 6f6e 204f 7574 7075 743d 3e48 6173  tion Output=>Has
│ │ │ │ +00025e60: 6846 6f72 6d20 636f 6e74 6169 6e73 0a64  hForm contains.d
│ │ │ │ +00025e70: 6966 6665 7265 6e74 2069 6e66 6f72 6d61  ifferent informa
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│ │ │ │ +00025eb0: 6564 2e0a 4164 6469 7469 6f6e 616c 6c79  ed..Additionally
│ │ │ │ +00025ec0: 2069 6620 7468 6520 6f70 7469 6f6e 202a   if the option *
│ │ │ │ +00025ed0: 6e6f 7465 2049 6e70 7574 4973 536d 6f6f  note InputIsSmoo
│ │ │ │ +00025ee0: 7468 3a20 496e 7075 7449 7353 6d6f 6f74  th: InputIsSmoot
│ │ │ │ +00025ef0: 682c 2069 7320 7573 6564 2074 6865 6e20  h, is used then 
│ │ │ │ +00025f00: 7468 650a 6861 7368 2074 6162 6c65 2072  the.hash table r
│ │ │ │ +00025f10: 6574 7572 6e65 6420 6279 2074 6865 206d  eturned by the m
│ │ │ │ +00025f20: 6574 686f 6473 2045 756c 6572 2061 6e64  ethods Euler and
│ │ │ │ +00025f30: 2043 534d 2077 696c 6c20 6265 2074 6865   CSM will be the
│ │ │ │ +00025f40: 2073 616d 6520 6173 2074 6861 740a 7265   same as that.re
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│ │ │ │ +00025f60: 2057 6865 6e20 7573 696e 6720 7468 6520   When using the 
│ │ │ │ +00025f70: 2a6e 6f74 6520 4353 4d3a 2043 534d 2c20  *note CSM: CSM, 
│ │ │ │ +00025f80: 2063 6f6d 6d61 6e64 2069 6e20 7468 6520   command in the 
│ │ │ │ +00025f90: 6465 6661 756c 740a 636f 6e66 6967 7572  default.configur
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│ │ │ │ +00025fb0: 2a6e 6f74 6520 4d65 7468 6f64 3a20 4d65  *note Method: Me
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│ │ │ │ +00025fd0: 4578 636c 7573 696f 6e2c 202a 6e6f 7465  Exclusion, *note
│ │ │ │ +00025fe0: 0a43 6f6d 704d 6574 686f 643a 2043 6f6d  .CompMethod: Com
│ │ │ │ +00025ff0: 704d 6574 686f 642c 3d3e 5072 6f6a 6563  pMethod,=>Projec
│ │ │ │ +00026000: 7469 7665 4465 6772 6565 2920 7468 6572  tiveDegree) ther
│ │ │ │ +00026010: 6520 6973 2074 6865 2061 6464 6974 696f  e is the additio
│ │ │ │ +00026020: 6e61 6c20 6f70 7469 6f6e 2074 6f0a 7365  nal option to.se
│ │ │ │ +00026030: 7420 4f75 7470 7574 3d3e 4861 7368 466f  t Output=>HashFo
│ │ │ │ +00026040: 726d 584c 2e20 5468 6973 2072 6574 7572  rmXL. This retur
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│ │ │ │ +00026060: 2069 6e66 6f72 6d61 7469 6f6e 2074 6861   information tha
│ │ │ │ +00026070: 740a 4f75 7470 7574 3d3e 4861 7368 466f  t.Output=>HashFo
│ │ │ │ +00026080: 726d 2077 6f75 6c64 2066 6f72 2074 6869  rm would for thi
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│ │ │ │ +000260a0: 7769 7468 2074 6865 2061 6464 6974 696f  with the additio
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│ │ │ │ +000260d0: 5365 6772 6520 636c 6173 7365 7320 6f66  Segre classes of
│ │ │ │ +000260e0: 2073 696e 6775 6c61 7269 7479 2073 7562   singularity sub
│ │ │ │ +000260f0: 7363 6865 6d65 7320 6765 6e65 7261 7465  schemes generate
│ │ │ │ +00026100: 6420 6279 2074 6865 0a68 7970 6572 7375  d by the.hypersu
│ │ │ │ +00026110: 7266 6163 6573 2063 6f6e 7369 6465 7265  rfaces considere
│ │ │ │ +00026120: 6420 696e 2074 6865 2069 6e63 6c75 7369  d in the inclusi
│ │ │ │ +00026130: 6f6e 2f65 7863 6c75 7369 6f6e 2070 726f  on/exclusion pro
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│ │ │ │ +00026150: 696e 0a66 696e 6469 6e67 2074 6865 2043  in.finding the C
│ │ │ │ +00026160: 534d 2063 6c61 7373 206f 6620 616c 6c20  SM class of all 
│ │ │ │ +00026170: 6879 7065 7273 7572 6661 6365 7320 6765  hypersurfaces ge
│ │ │ │ +00026180: 6e65 7261 7465 6420 6279 2074 616b 696e  nerated by takin
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│ │ │ │ +000261a0: 6f6d 6520 7375 6273 6574 7320 6f66 2067  ome subsets of g
│ │ │ │ +000261b0: 656e 6572 6174 6f72 7320 6f66 2074 6865  enerators of the
│ │ │ │ +000261c0: 2069 6e70 7574 2069 6465 616c 2e20 4e6f   input ideal. No
│ │ │ │ +000261d0: 7465 2074 6861 742c 2073 696e 6365 2074  te that, since t
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│ │ │ │ +000261f0: 6120 7375 6273 6368 656d 6520 6571 7561  a subscheme equa
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│ │ │ │ +00026210: 206f 6620 6974 7320 7265 6475 6365 6420   of its reduced 
│ │ │ │ +00026220: 7363 6865 6d65 2c20 6f72 2065 7175 6976  scheme, or equiv
│ │ │ │ +00026230: 616c 656e 746c 7920 666f 720a 7573 2074  alently for.us t
│ │ │ │ +00026240: 6865 2043 534d 2063 6c61 7373 2063 6f72  he CSM class cor
│ │ │ │ +00026250: 7265 7370 6f6e 6469 6e67 2074 6f20 616e  responding to an
│ │ │ │ +00026260: 2069 6465 616c 2049 2065 7175 616c 7320   ideal I equals 
│ │ │ │ +00026270: 7468 6520 4353 4d20 636c 6173 7320 6f66  the CSM class of
│ │ │ │ +00026280: 2074 6865 0a72 6164 6963 616c 206f 6620   the.radical of 
│ │ │ │ +00026290: 492c 2074 6865 6e20 696e 7465 726e 616c  I, then internal
│ │ │ │ +000262a0: 6c79 2077 6520 616c 7761 7973 2077 6f72  ly we always wor
│ │ │ │ +000262b0: 6b20 7769 7468 2072 6164 6963 616c 2069  k with radical i
│ │ │ │ +000262c0: 6465 616c 7320 2866 6f72 0a65 6666 6963  deals (for.effic
│ │ │ │ +000262d0: 6965 6e63 7920 7265 6173 6f6e 7329 2e20  iency reasons). 
│ │ │ │ +000262e0: 4865 6e63 6520 7468 6520 7072 6f6a 6563  Hence the projec
│ │ │ │ +000262f0: 7469 7665 2064 6567 7265 6573 2061 6e64  tive degrees and
│ │ │ │ +00026300: 2053 6567 7265 2063 6c61 7373 6573 2063   Segre classes c
│ │ │ │ +00026310: 6f6d 7075 7465 640a 696e 7465 726e 616c  omputed.internal
│ │ │ │ +00026320: 6c79 2077 696c 6c20 6265 2074 686f 7365  ly will be those
│ │ │ │ +00026330: 206f 6620 7468 6520 7261 6469 6361 6c20   of the radical 
│ │ │ │ +00026340: 6f66 2061 6e20 6964 6561 6c20 6465 6669  of an ideal defi
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│ │ │ │ +00026360: 6961 6c0a 7768 6963 6820 6973 2061 2070  ial.which is a p
│ │ │ │ +00026370: 726f 6475 6374 206f 6620 736f 6d65 2073  roduct of some s
│ │ │ │ +00026380: 7562 7365 7420 6f66 2074 6865 2067 656e  ubset of the gen
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│ │ │ │ +000263a0: 7374 7261 7465 2074 6869 7320 7769 7468  strate this with
│ │ │ │ +000263b0: 2061 6e0a 6578 616d 706c 6520 6265 6c6f   an.example belo
│ │ │ │ +000263c0: 772e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  w...+-----------
│ │ │ │ +000263d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000263e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000263f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026420: 2d2b 0a7c 6931 3420 3a20 6373 6d58 4c68  -+.|i14 : csmXLh
│ │ │ │ -00026430: 6173 683d 4353 4d28 412c 492c 4f75 7470  ash=CSM(A,I,Outp
│ │ │ │ -00026440: 7574 3d3e 4861 7368 466f 726d 584c 2920  ut=>HashFormXL) 
│ │ │ │ +00026410: 2d2d 2b0a 7c69 3134 203a 2063 736d 584c  --+.|i14 : csmXL
│ │ │ │ +00026420: 6861 7368 3d43 534d 2841 2c49 2c4f 7574  hash=CSM(A,I,Out
│ │ │ │ +00026430: 7075 743d 3e48 6173 6846 6f72 6d58 4c29  put=>HashFormXL)
│ │ │ │ +00026440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026470: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026460: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00026470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000264a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000264b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000264c0: 207c 0a7c 6f31 3420 3d20 4d75 7461 626c   |.|o14 = Mutabl
│ │ │ │ -000264d0: 6548 6173 6854 6162 6c65 7b2e 2e2e 3130  eHashTable{...10
│ │ │ │ -000264e0: 2e2e 2e7d 2020 2020 2020 2020 2020 2020  ...}            
│ │ │ │ +000264b0: 2020 7c0a 7c6f 3134 203d 204d 7574 6162    |.|o14 = Mutab
│ │ │ │ +000264c0: 6c65 4861 7368 5461 626c 657b 2e2e 2e31  leHashTable{...1
│ │ │ │ +000264d0: 302e 2e2e 7d20 2020 2020 2020 2020 2020  0...}           
│ │ │ │ +000264e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000264f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026510: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026500: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00026510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026560: 207c 0a7c 6f31 3420 3a20 4d75 7461 626c   |.|o14 : Mutabl
│ │ │ │ -00026570: 6548 6173 6854 6162 6c65 2020 2020 2020  eHashTable      
│ │ │ │ +00026550: 2020 7c0a 7c6f 3134 203a 204d 7574 6162    |.|o14 : Mutab
│ │ │ │ +00026560: 6c65 4861 7368 5461 626c 6520 2020 2020  leHashTable     
│ │ │ │ +00026570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000265a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000265b0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +000265a0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +000265b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000265c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000265d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000265e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000265f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026600: 2d2b 0a7c 6931 3520 3a20 7065 656b 2063  -+.|i15 : peek c
│ │ │ │ -00026610: 736d 584c 6861 7368 2020 2020 2020 2020  smXLhash        
│ │ │ │ +000265f0: 2d2d 2b0a 7c69 3135 203a 2070 6565 6b20  --+.|i15 : peek 
│ │ │ │ +00026600: 6373 6d58 4c68 6173 6820 2020 2020 2020  csmXLhash       
│ │ │ │ +00026610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026650: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026640: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00026650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000266a0: 207c 0a7c 6f31 3520 3d20 4d75 7461 626c   |.|o15 = Mutabl
│ │ │ │ -000266b0: 6548 6173 6854 6162 6c65 7b47 284a 6163  eHashTable{G(Jac
│ │ │ │ -000266c0: 6f62 6961 6e29 7b30 7d20 3d3e 2030 2020  obian){0} => 0  
│ │ │ │ +00026690: 2020 7c0a 7c6f 3135 203d 204d 7574 6162    |.|o15 = Mutab
│ │ │ │ +000266a0: 6c65 4861 7368 5461 626c 657b 4728 4a61  leHashTable{G(Ja
│ │ │ │ +000266b0: 636f 6269 616e 297b 307d 203d 3e20 3020  cobian){0} => 0 
│ │ │ │ +000266c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000266d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000266e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000266f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00026700: 2020 2020 2020 2020 2020 2053 6567 7265             Segre
│ │ │ │ -00026710: 284a 6163 6f62 6961 6e29 7b30 7d20 3d3e  (Jacobian){0} =>
│ │ │ │ -00026720: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ -00026730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026740: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000266e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000266f0: 2020 2020 2020 2020 2020 2020 5365 6772              Segr
│ │ │ │ +00026700: 6528 4a61 636f 6269 616e 297b 307d 203d  e(Jacobian){0} =
│ │ │ │ +00026710: 3e20 3020 2020 2020 2020 2020 2020 2020  > 0             
│ │ │ │ +00026720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026730: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00026740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026770: 2020 2020 2020 2020 3620 2020 2020 2020          6       
│ │ │ │ -00026780: 3520 2020 2020 2020 3420 2020 2020 2020  5       4       
│ │ │ │ -00026790: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000267a0: 2020 2020 2020 2020 2020 2053 6567 7265             Segre
│ │ │ │ -000267b0: 284a 6163 6f62 6961 6e29 7b30 2c20 317d  (Jacobian){0, 1}
│ │ │ │ -000267c0: 203d 3e20 3339 3068 2020 2d20 3338 3668   => 390h  - 386h
│ │ │ │ -000267d0: 2020 2b20 3135 3068 2020 2d20 2020 2020    + 150h  -     
│ │ │ │ -000267e0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026760: 2020 2020 2020 2020 2036 2020 2020 2020           6      
│ │ │ │ +00026770: 2035 2020 2020 2020 2034 2020 2020 2020   5       4      
│ │ │ │ +00026780: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00026790: 2020 2020 2020 2020 2020 2020 5365 6772              Segr
│ │ │ │ +000267a0: 6528 4a61 636f 6269 616e 297b 302c 2031  e(Jacobian){0, 1
│ │ │ │ +000267b0: 7d20 3d3e 2033 3930 6820 202d 2033 3836  } => 390h  - 386
│ │ │ │ +000267c0: 6820 202b 2031 3530 6820 202d 2020 2020  h  + 150h  -    
│ │ │ │ +000267d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000267e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000267f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026810: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ -00026820: 3120 2020 2020 2020 3120 2020 2020 2020  1       1       
│ │ │ │ -00026830: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026800: 2020 2020 2020 2020 2031 2020 2020 2020           1      
│ │ │ │ +00026810: 2031 2020 2020 2020 2031 2020 2020 2020   1       1      
│ │ │ │ +00026820: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00026830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026860: 2020 2020 2020 2036 2020 2020 2020 3520         6      5 
│ │ │ │ -00026870: 2020 2020 3320 2020 2020 3220 2020 2020      3     2     
│ │ │ │ -00026880: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00026890: 2020 2020 2020 2020 2020 2053 6567 7265             Segre
│ │ │ │ -000268a0: 284a 6163 6f62 6961 6e29 7b31 7d20 3d3e  (Jacobian){1} =>
│ │ │ │ -000268b0: 202d 2031 3630 6820 202b 2033 3268 2020   - 160h  + 32h  
│ │ │ │ -000268c0: 2d20 3468 2020 2b20 3268 2020 2020 2020  - 4h  + 2h      
│ │ │ │ -000268d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026850: 2020 2020 2020 2020 3620 2020 2020 2035          6      5
│ │ │ │ +00026860: 2020 2020 2033 2020 2020 2032 2020 2020       3     2    
│ │ │ │ +00026870: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00026880: 2020 2020 2020 2020 2020 2020 5365 6772              Segr
│ │ │ │ +00026890: 6528 4a61 636f 6269 616e 297b 317d 203d  e(Jacobian){1} =
│ │ │ │ +000268a0: 3e20 2d20 3136 3068 2020 2b20 3332 6820  > - 160h  + 32h 
│ │ │ │ +000268b0: 202d 2034 6820 202b 2032 6820 2020 2020   - 4h  + 2h     
│ │ │ │ +000268c0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000268d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000268e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000268f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026900: 2020 2020 2020 2031 2020 2020 2020 3120         1      1 
│ │ │ │ -00026910: 2020 2020 3120 2020 2020 3120 2020 2020      1     1     
│ │ │ │ -00026920: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000268f0: 2020 2020 2020 2020 3120 2020 2020 2031          1      1
│ │ │ │ +00026900: 2020 2020 2031 2020 2020 2031 2020 2020       1     1    
│ │ │ │ +00026910: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00026920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026950: 2020 2036 2020 2020 2020 3520 2020 2020     6      5     
│ │ │ │ -00026960: 2034 2020 2020 2020 3320 2020 2020 2020   4      3       
│ │ │ │ -00026970: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00026980: 2020 2020 2020 2020 2020 2047 284a 6163             G(Jac
│ │ │ │ -00026990: 6f62 6961 6e29 7b30 2c20 317d 203d 3e20  obian){0, 1} => 
│ │ │ │ -000269a0: 3130 6820 202b 2031 3068 2020 2b20 3130  10h  + 10h  + 10
│ │ │ │ -000269b0: 6820 202b 2031 3068 2020 2b20 2020 2020  h  + 10h  +     
│ │ │ │ -000269c0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026940: 2020 2020 3620 2020 2020 2035 2020 2020      6      5    
│ │ │ │ +00026950: 2020 3420 2020 2020 2033 2020 2020 2020    4      3      
│ │ │ │ +00026960: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00026970: 2020 2020 2020 2020 2020 2020 4728 4a61              G(Ja
│ │ │ │ +00026980: 636f 6269 616e 297b 302c 2031 7d20 3d3e  cobian){0, 1} =>
│ │ │ │ +00026990: 2031 3068 2020 2b20 3130 6820 202b 2031   10h  + 10h  + 1
│ │ │ │ +000269a0: 3068 2020 2b20 3130 6820 202b 2020 2020  0h  + 10h  +    
│ │ │ │ +000269b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000269c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000269d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000269e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000269f0: 2020 2031 2020 2020 2020 3120 2020 2020     1      1     
│ │ │ │ -00026a00: 2031 2020 2020 2020 3120 2020 2020 2020   1      1       
│ │ │ │ -00026a10: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000269e0: 2020 2020 3120 2020 2020 2031 2020 2020      1      1    
│ │ │ │ +000269f0: 2020 3120 2020 2020 2031 2020 2020 2020    1      1      
│ │ │ │ +00026a00: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00026a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026a30: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +00026a30: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  00026a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026a60: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00026a70: 2020 2020 2020 2020 2020 2047 284a 6163             G(Jac
│ │ │ │ -00026a80: 6f62 6961 6e29 7b31 7d20 3d3e 2032 6820  obian){1} => 2h 
│ │ │ │ -00026a90: 202b 2032 6820 202b 2031 2020 2020 2020   + 2h  + 1      
│ │ │ │ -00026aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026ab0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026a50: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00026a60: 2020 2020 2020 2020 2020 2020 4728 4a61              G(Ja
│ │ │ │ +00026a70: 636f 6269 616e 297b 317d 203d 3e20 3268  cobian){1} => 2h
│ │ │ │ +00026a80: 2020 2b20 3268 2020 2b20 3120 2020 2020    + 2h  + 1     
│ │ │ │ +00026a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026aa0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00026ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026ad0: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ -00026ae0: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ -00026af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026b00: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00026b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026b20: 2020 2020 2020 2036 2020 2020 2020 3520         6      5 
│ │ │ │ -00026b30: 2020 2020 2034 2020 2020 2020 3320 2020       4      3   
│ │ │ │ -00026b40: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -00026b50: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00026b60: 2020 2020 2020 2020 2020 207b 302c 2031             {0, 1
│ │ │ │ -00026b70: 7d20 3d3e 2032 6820 202b 2032 3368 2020  } => 2h  + 23h  
│ │ │ │ -00026b80: 2b20 3332 6820 202b 2033 3368 2020 2b20  + 32h  + 33h  + 
│ │ │ │ -00026b90: 3138 6820 202b 2035 6820 2020 2020 2020  18h  + 5h       
│ │ │ │ -00026ba0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00026bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026bc0: 2020 2020 2020 2031 2020 2020 2020 3120         1      1 
│ │ │ │ -00026bd0: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ -00026be0: 2020 2031 2020 2020 2031 2020 2020 2020     1     1      
│ │ │ │ -00026bf0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00026c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026c10: 2020 2020 2036 2020 2020 2020 3520 2020       6      5   
│ │ │ │ -00026c20: 2020 2034 2020 2020 2020 3320 2020 2020     4      3     
│ │ │ │ -00026c30: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -00026c40: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00026c50: 2020 2020 2020 2020 2020 2043 534d 203d             CSM =
│ │ │ │ -00026c60: 3e20 3130 6820 202b 2031 3268 2020 2b20  > 10h  + 12h  + 
│ │ │ │ -00026c70: 3232 6820 202b 2031 3668 2020 2b20 3668  22h  + 16h  + 6h
│ │ │ │ -00026c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026c90: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00026ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026cb0: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ -00026cc0: 2020 2031 2020 2020 2020 3120 2020 2020     1      1     
│ │ │ │ -00026cd0: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ -00026ce0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00026cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026d00: 2020 2020 3620 2020 2020 2035 2020 2020      6      5    
│ │ │ │ -00026d10: 2020 3420 2020 2020 2033 2020 2020 2020    4      3      
│ │ │ │ -00026d20: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -00026d30: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00026d40: 2020 2020 2020 2020 2020 207b 307d 203d             {0} =
│ │ │ │ -00026d50: 3e20 3668 2020 2b20 3138 6820 202b 2032  > 6h  + 18h  + 2
│ │ │ │ -00026d60: 3668 2020 2b20 3232 6820 202b 2031 3068  6h  + 22h  + 10h
│ │ │ │ -00026d70: 2020 2b20 3268 2020 2020 2020 2020 2020    + 2h          
│ │ │ │ -00026d80: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00026d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026da0: 2020 2020 3120 2020 2020 2031 2020 2020      1      1    
│ │ │ │ -00026db0: 2020 3120 2020 2020 2031 2020 2020 2020    1      1      
│ │ │ │ -00026dc0: 3120 2020 2020 3120 2020 2020 2020 2020  1     1         
│ │ │ │ -00026dd0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00026de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026df0: 2020 2020 3620 2020 2020 2035 2020 2020      6      5    
│ │ │ │ -00026e00: 2020 3420 2020 2020 2033 2020 2020 2020    4      3      
│ │ │ │ -00026e10: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -00026e20: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00026e30: 2020 2020 2020 2020 2020 207b 317d 203d             {1} =
│ │ │ │ -00026e40: 3e20 3668 2020 2b20 3137 6820 202b 2032  > 6h  + 17h  + 2
│ │ │ │ -00026e50: 3868 2020 2b20 3237 6820 202b 2031 3468  8h  + 27h  + 14h
│ │ │ │ -00026e60: 2020 2b20 3368 2020 2020 2020 2020 2020    + 3h          
│ │ │ │ -00026e70: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00026e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026e90: 2020 2020 3120 2020 2020 2031 2020 2020      1      1    
│ │ │ │ -00026ea0: 2020 3120 2020 2020 2031 2020 2020 2020    1      1      
│ │ │ │ -00026eb0: 3120 2020 2020 3120 2020 2020 2020 2020  1     1         
│ │ │ │ -00026ec0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │ +00026ad0: 3120 2020 2020 3120 2020 2020 2020 2020  1     1         
│ │ │ │ +00026ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026af0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00026b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026b10: 2020 2020 2020 2020 3620 2020 2020 2035          6      5
│ │ │ │ +00026b20: 2020 2020 2020 3420 2020 2020 2033 2020        4      3  
│ │ │ │ +00026b30: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +00026b40: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00026b50: 2020 2020 2020 2020 2020 2020 7b30 2c20              {0, 
│ │ │ │ +00026b60: 317d 203d 3e20 3268 2020 2b20 3233 6820  1} => 2h  + 23h 
│ │ │ │ +00026b70: 202b 2033 3268 2020 2b20 3333 6820 202b   + 32h  + 33h  +
│ │ │ │ +00026b80: 2031 3868 2020 2b20 3568 2020 2020 2020   18h  + 5h      
│ │ │ │ +00026b90: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00026ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026bb0: 2020 2020 2020 2020 3120 2020 2020 2031          1      1
│ │ │ │ +00026bc0: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ +00026bd0: 2020 2020 3120 2020 2020 3120 2020 2020      1     1     
│ │ │ │ +00026be0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00026bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026c00: 2020 2020 2020 3620 2020 2020 2035 2020        6      5  
│ │ │ │ +00026c10: 2020 2020 3420 2020 2020 2033 2020 2020      4      3    
│ │ │ │ +00026c20: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +00026c30: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00026c40: 2020 2020 2020 2020 2020 2020 4353 4d20              CSM 
│ │ │ │ +00026c50: 3d3e 2031 3068 2020 2b20 3132 6820 202b  => 10h  + 12h  +
│ │ │ │ +00026c60: 2032 3268 2020 2b20 3136 6820 202b 2036   22h  + 16h  + 6
│ │ │ │ +00026c70: 6820 2020 2020 2020 2020 2020 2020 2020  h               
│ │ │ │ +00026c80: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00026c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026ca0: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ +00026cb0: 2020 2020 3120 2020 2020 2031 2020 2020      1      1    
│ │ │ │ +00026cc0: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │ +00026cd0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00026ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026cf0: 2020 2020 2036 2020 2020 2020 3520 2020       6      5   
│ │ │ │ +00026d00: 2020 2034 2020 2020 2020 3320 2020 2020     4      3     
│ │ │ │ +00026d10: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +00026d20: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00026d30: 2020 2020 2020 2020 2020 2020 7b30 7d20              {0} 
│ │ │ │ +00026d40: 3d3e 2036 6820 202b 2031 3868 2020 2b20  => 6h  + 18h  + 
│ │ │ │ +00026d50: 3236 6820 202b 2032 3268 2020 2b20 3130  26h  + 22h  + 10
│ │ │ │ +00026d60: 6820 202b 2032 6820 2020 2020 2020 2020  h  + 2h         
│ │ │ │ +00026d70: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00026d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026d90: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ +00026da0: 2020 2031 2020 2020 2020 3120 2020 2020     1      1     
│ │ │ │ +00026db0: 2031 2020 2020 2031 2020 2020 2020 2020   1     1        
│ │ │ │ +00026dc0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00026dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026de0: 2020 2020 2036 2020 2020 2020 3520 2020       6      5   
│ │ │ │ +00026df0: 2020 2034 2020 2020 2020 3320 2020 2020     4      3     
│ │ │ │ +00026e00: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +00026e10: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00026e20: 2020 2020 2020 2020 2020 2020 7b31 7d20              {1} 
│ │ │ │ +00026e30: 3d3e 2036 6820 202b 2031 3768 2020 2b20  => 6h  + 17h  + 
│ │ │ │ +00026e40: 3238 6820 202b 2032 3768 2020 2b20 3134  28h  + 27h  + 14
│ │ │ │ +00026e50: 6820 202b 2033 6820 2020 2020 2020 2020  h  + 3h         
│ │ │ │ +00026e60: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00026e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026e80: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ +00026e90: 2020 2031 2020 2020 2020 3120 2020 2020     1      1     
│ │ │ │ +00026ea0: 2031 2020 2020 2031 2020 2020 2020 2020   1     1        
│ │ │ │ +00026eb0: 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.|-----------
│ │ │ │ +00026ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026f10: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ -00026f20: 207d 2020 2020 2020 2020 2020 2020 2020   }              
│ │ │ │ +00026f00: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020  --|.|           
│ │ │ │ +00026f10: 2020 7d20 2020 2020 2020 2020 2020 2020    }             
│ │ │ │ +00026f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026f60: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026f50: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00026f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026fb0: 207c 0a7c 2020 2033 2020 2020 2032 2020   |.|   3     2  
│ │ │ │ +00026fa0: 2020 7c0a 7c20 2020 3320 2020 2020 3220    |.|   3     2 
│ │ │ │ +00026fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027000: 207c 0a7c 3432 6820 202b 2038 6820 2020   |.|42h  + 8h   
│ │ │ │ +00026ff0: 2020 7c0a 7c34 3268 2020 2b20 3868 2020    |.|42h  + 8h  
│ │ │ │ +00027000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027050: 207c 0a7c 2020 2031 2020 2020 2031 2020   |.|   1     1  
│ │ │ │ +00027040: 2020 7c0a 7c20 2020 3120 2020 2020 3120    |.|   1     1 
│ │ │ │ +00027050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000270a0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00027090: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000270a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000270b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000270c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000270d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000270e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000270f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000270e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000270f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027110: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00027130: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00027140: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00027160: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00027180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027190: 207c 0a7c 2020 3220 2020 2020 2020 2020   |.|  2         
│ │ │ │ +00027180: 2020 7c0a 7c20 2032 2020 2020 2020 2020    |.|  2        
│ │ │ │ +00027190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000271a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000271b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000271c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000271d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000271e0: 207c 0a7c 3868 2020 2b20 3468 2020 2b20   |.|8h  + 4h  + 
│ │ │ │ -000271f0: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +000271d0: 2020 7c0a 7c38 6820 202b 2034 6820 202b    |.|8h  + 4h  +
│ │ │ │ +000271e0: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │ +000271f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027230: 207c 0a7c 2020 3120 2020 2020 3120 2020   |.|  1     1   
│ │ │ │ +00027220: 2020 7c0a 7c20 2031 2020 2020 2031 2020    |.|  1     1  
│ │ │ │ +00027230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027260: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00027280: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00027270: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00027280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000272a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000272b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000272c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000272d0: 2d2b 0a7c 6931 3620 3a20 4b3d 6964 6561  -+.|i16 : K=idea
│ │ │ │ -000272e0: 6c20 495f 302a 495f 313b 2020 2020 2020  l I_0*I_1;      
│ │ │ │ +000272c0: 2d2d 2b0a 7c69 3136 203a 204b 3d69 6465  --+.|i16 : K=ide
│ │ │ │ +000272d0: 616c 2049 5f30 2a49 5f31 3b20 2020 2020  al I_0*I_1;     
│ │ │ │ +000272e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000272f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027300: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00027330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027370: 207c 0a7c 6f31 3620 3a20 4964 6561 6c20   |.|o16 : Ideal 
│ │ │ │ -00027380: 6f66 2052 2020 2020 2020 2020 2020 2020  of R            
│ │ │ │ +00027360: 2020 7c0a 7c6f 3136 203a 2049 6465 616c    |.|o16 : Ideal
│ │ │ │ +00027370: 206f 6620 5220 2020 2020 2020 2020 2020   of R           
│ │ │ │ +00027380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000273a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000273b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000273c0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +000273b0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +000273c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000273d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000273e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000273f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027410: 2d2b 0a7c 6931 3720 3a20 4353 4d28 412c  -+.|i17 : CSM(A,
│ │ │ │ -00027420: 7261 6469 6361 6c20 4b29 3d3d 4353 4d28  radical K)==CSM(
│ │ │ │ -00027430: 412c 4b29 2020 2020 2020 2020 2020 2020  A,K)            
│ │ │ │ +00027400: 2d2d 2b0a 7c69 3137 203a 2043 534d 2841  --+.|i17 : CSM(A
│ │ │ │ +00027410: 2c72 6164 6963 616c 204b 293d 3d43 534d  ,radical K)==CSM
│ │ │ │ +00027420: 2841 2c4b 2920 2020 2020 2020 2020 2020  (A,K)           
│ │ │ │ +00027430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027460: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00027450: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00027460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000274a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000274b0: 207c 0a7c 6f31 3720 3d20 7472 7565 2020   |.|o17 = true  
│ │ │ │ +000274a0: 2020 7c0a 7c6f 3137 203d 2074 7275 6520    |.|o17 = true 
│ │ │ │ +000274b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000274c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000274d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000274e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000274f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027500: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +000274f0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00027500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027550: 2d2b 0a7c 6931 3820 3a20 4a3d 6964 6561  -+.|i18 : J=idea
│ │ │ │ -00027560: 6c20 6a61 636f 6269 616e 2072 6164 6963  l jacobian radic
│ │ │ │ -00027570: 616c 204b 3b20 2020 2020 2020 2020 2020  al K;           
│ │ │ │ +00027540: 2d2d 2b0a 7c69 3138 203a 204a 3d69 6465  --+.|i18 : J=ide
│ │ │ │ +00027550: 616c 206a 6163 6f62 6961 6e20 7261 6469  al jacobian radi
│ │ │ │ +00027560: 6361 6c20 4b3b 2020 2020 2020 2020 2020  cal K;          
│ │ │ │ +00027570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000275a0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00027590: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000275a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000275b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000275c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000275d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000275e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000275f0: 207c 0a7c 6f31 3820 3a20 4964 6561 6c20   |.|o18 : Ideal 
│ │ │ │ -00027600: 6f66 2052 2020 2020 2020 2020 2020 2020  of R            
│ │ │ │ +000275e0: 2020 7c0a 7c6f 3138 203a 2049 6465 616c    |.|o18 : Ideal
│ │ │ │ +000275f0: 206f 6620 5220 2020 2020 2020 2020 2020   of R           
│ │ │ │ +00027600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027640: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00027630: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00027640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027690: 2d2b 0a7c 6931 3920 3a20 7365 674a 3d53  -+.|i19 : segJ=S
│ │ │ │ -000276a0: 6567 7265 2841 2c4a 2c4f 7574 7075 743d  egre(A,J,Output=
│ │ │ │ -000276b0: 3e48 6173 6846 6f72 6d29 2020 2020 2020  >HashForm)      
│ │ │ │ +00027680: 2d2d 2b0a 7c69 3139 203a 2073 6567 4a3d  --+.|i19 : segJ=
│ │ │ │ +00027690: 5365 6772 6528 412c 4a2c 4f75 7470 7574  Segre(A,J,Output
│ │ │ │ +000276a0: 3d3e 4861 7368 466f 726d 2920 2020 2020  =>HashForm)     
│ │ │ │ +000276b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000276c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000276d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000276e0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000276d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000276e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000276f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027730: 207c 0a7c 6f31 3920 3d20 4d75 7461 626c   |.|o19 = Mutabl
│ │ │ │ -00027740: 6548 6173 6854 6162 6c65 7b2e 2e2e 342e  eHashTable{...4.
│ │ │ │ -00027750: 2e2e 7d20 2020 2020 2020 2020 2020 2020  ..}             
│ │ │ │ +00027720: 2020 7c0a 7c6f 3139 203d 204d 7574 6162    |.|o19 = Mutab
│ │ │ │ +00027730: 6c65 4861 7368 5461 626c 657b 2e2e 2e34  leHashTable{...4
│ │ │ │ +00027740: 2e2e 2e7d 2020 2020 2020 2020 2020 2020  ...}            
│ │ │ │ +00027750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027780: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00027770: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00027780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000277a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000277b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000277c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000277d0: 207c 0a7c 6f31 3920 3a20 4d75 7461 626c   |.|o19 : Mutabl
│ │ │ │ -000277e0: 6548 6173 6854 6162 6c65 2020 2020 2020  eHashTable      
│ │ │ │ +000277c0: 2020 7c0a 7c6f 3139 203a 204d 7574 6162    |.|o19 : Mutab
│ │ │ │ +000277d0: 6c65 4861 7368 5461 626c 6520 2020 2020  leHashTable     
│ │ │ │ +000277e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000277f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027820: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00027810: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00027820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027870: 2d2b 0a7c 6932 3020 3a20 6373 6d58 4c68  -+.|i20 : csmXLh
│ │ │ │ -00027880: 6173 6823 2822 4728 4a61 636f 6269 616e  ash#("G(Jacobian
│ │ │ │ -00027890: 2922 7c74 6f53 7472 696e 6728 7b30 2c31  )"|toString({0,1
│ │ │ │ -000278a0: 7d29 293d 3d73 6567 4a23 2247 2220 2020  }))==segJ#"G"   
│ │ │ │ -000278b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000278c0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00027860: 2d2d 2b0a 7c69 3230 203a 2063 736d 584c  --+.|i20 : csmXL
│ │ │ │ +00027870: 6861 7368 2328 2247 284a 6163 6f62 6961  hash#("G(Jacobia
│ │ │ │ +00027880: 6e29 227c 746f 5374 7269 6e67 287b 302c  n)"|toString({0,
│ │ │ │ +00027890: 317d 2929 3d3d 7365 674a 2322 4722 2020  1}))==segJ#"G"  
│ │ │ │ +000278a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000278b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000278c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000278d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000278e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000278f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027910: 207c 0a7c 6f32 3020 3d20 7472 7565 2020   |.|o20 = true  
│ │ │ │ +00027900: 2020 7c0a 7c6f 3230 203d 2074 7275 6520    |.|o20 = true 
│ │ │ │ +00027910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027960: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00027950: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00027960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000279a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000279b0: 2d2b 0a7c 6932 3120 3a20 6373 6d58 4c68  -+.|i21 : csmXLh
│ │ │ │ -000279c0: 6173 6823 2822 5365 6772 6528 4a61 636f  ash#("Segre(Jaco
│ │ │ │ -000279d0: 6269 616e 2922 7c74 6f53 7472 696e 6728  bian)"|toString(
│ │ │ │ -000279e0: 7b30 2c31 7d29 293d 3d73 6567 4a23 2253  {0,1}))==segJ#"S
│ │ │ │ -000279f0: 6567 7265 2220 2020 2020 2020 2020 2020  egre"           
│ │ │ │ -00027a00: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000279a0: 2d2d 2b0a 7c69 3231 203a 2063 736d 584c  --+.|i21 : csmXL
│ │ │ │ +000279b0: 6861 7368 2328 2253 6567 7265 284a 6163  hash#("Segre(Jac
│ │ │ │ +000279c0: 6f62 6961 6e29 227c 746f 5374 7269 6e67  obian)"|toString
│ │ │ │ +000279d0: 287b 302c 317d 2929 3d3d 7365 674a 2322  ({0,1}))==segJ#"
│ │ │ │ +000279e0: 5365 6772 6522 2020 2020 2020 2020 2020  Segre"          
│ │ │ │ +000279f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00027a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027a50: 207c 0a7c 6f32 3120 3d20 7472 7565 2020   |.|o21 = true  
│ │ │ │ +00027a40: 2020 7c0a 7c6f 3231 203d 2074 7275 6520    |.|o21 = true 
│ │ │ │ +00027a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027aa0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00027a90: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00027aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027af0: 2d2b 0a0a 4675 6e63 7469 6f6e 7320 7769  -+..Functions wi
│ │ │ │ -00027b00: 7468 206f 7074 696f 6e61 6c20 6172 6775  th optional argu
│ │ │ │ -00027b10: 6d65 6e74 206e 616d 6564 204f 7574 7075  ment named Outpu
│ │ │ │ -00027b20: 743a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  t:.=============
│ │ │ │ +00027ae0: 2d2d 2b0a 0a46 756e 6374 696f 6e73 2077  --+..Functions w
│ │ │ │ +00027af0: 6974 6820 6f70 7469 6f6e 616c 2061 7267  ith optional arg
│ │ │ │ +00027b00: 756d 656e 7420 6e61 6d65 6420 4f75 7470  ument named Outp
│ │ │ │ +00027b10: 7574 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ut:.============
│ │ │ │ +00027b20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │  00027b30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00027b40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00027b50: 3d0a 0a20 202a 2022 4368 6572 6e28 2e2e  =..  * "Chern(..
│ │ │ │ -00027b60: 2e2c 4f75 7470 7574 3d3e 2e2e 2e29 2220  .,Output=>...)" 
│ │ │ │ -00027b70: 2d2d 2073 6565 202a 6e6f 7465 2043 6865  -- see *note Che
│ │ │ │ -00027b80: 726e 3a20 4368 6572 6e2c 202d 2d20 5468  rn: Chern, -- Th
│ │ │ │ -00027b90: 6520 4368 6572 6e20 636c 6173 730a 2020  e Chern class.  
│ │ │ │ -00027ba0: 2a20 2243 534d 282e 2e2e 2c4f 7574 7075  * "CSM(...,Outpu
│ │ │ │ -00027bb0: 743d 3e2e 2e2e 2922 202d 2d20 7365 6520  t=>...)" -- see 
│ │ │ │ -00027bc0: 2a6e 6f74 6520 4353 4d3a 2043 534d 2c20  *note CSM: CSM, 
│ │ │ │ -00027bd0: 2d2d 2054 6865 0a20 2020 2043 6865 726e  -- The.    Chern
│ │ │ │ -00027be0: 2d53 6368 7761 7274 7a2d 4d61 6350 6865  -Schwartz-MacPhe
│ │ │ │ -00027bf0: 7273 6f6e 2063 6c61 7373 0a20 202a 2022  rson class.  * "
│ │ │ │ -00027c00: 4575 6c65 7228 2e2e 2e2c 4f75 7470 7574  Euler(...,Output
│ │ │ │ -00027c10: 3d3e 2e2e 2e29 2220 2d2d 2073 6565 202a  =>...)" -- see *
│ │ │ │ -00027c20: 6e6f 7465 2045 756c 6572 3a20 4575 6c65  note Euler: Eule
│ │ │ │ -00027c30: 722c 202d 2d20 5468 6520 4575 6c65 720a  r, -- The Euler.
│ │ │ │ -00027c40: 2020 2020 4368 6172 6163 7465 7269 7374      Characterist
│ │ │ │ -00027c50: 6963 0a20 202a 2022 5365 6772 6528 2e2e  ic.  * "Segre(..
│ │ │ │ -00027c60: 2e2c 4f75 7470 7574 3d3e 2e2e 2e29 2220  .,Output=>...)" 
│ │ │ │ -00027c70: 2d2d 2073 6565 202a 6e6f 7465 2053 6567  -- see *note Seg
│ │ │ │ -00027c80: 7265 3a20 5365 6772 652c 202d 2d20 5468  re: Segre, -- Th
│ │ │ │ -00027c90: 6520 5365 6772 6520 636c 6173 7320 6f66  e Segre class of
│ │ │ │ -00027ca0: 2061 0a20 2020 2073 7562 7363 6865 6d65   a.    subscheme
│ │ │ │ -00027cb0: 0a0a 466f 7220 7468 6520 7072 6f67 7261  ..For the progra
│ │ │ │ -00027cc0: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  mmer.===========
│ │ │ │ -00027cd0: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a  =======..The obj
│ │ │ │ -00027ce0: 6563 7420 2a6e 6f74 6520 4f75 7470 7574  ect *note Output
│ │ │ │ -00027cf0: 3a20 4f75 7470 7574 2c20 6973 2061 202a  : Output, is a *
│ │ │ │ -00027d00: 6e6f 7465 2073 796d 626f 6c3a 2028 4d61  note symbol: (Ma
│ │ │ │ -00027d10: 6361 756c 6179 3244 6f63 2953 796d 626f  caulay2Doc)Symbo
│ │ │ │ -00027d20: 6c2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d  l,...-----------
│ │ │ │ +00027b40: 3d3d 0a0a 2020 2a20 2243 6865 726e 282e  ==..  * "Chern(.
│ │ │ │ +00027b50: 2e2e 2c4f 7574 7075 743d 3e2e 2e2e 2922  ..,Output=>...)"
│ │ │ │ +00027b60: 202d 2d20 7365 6520 2a6e 6f74 6520 4368   -- see *note Ch
│ │ │ │ +00027b70: 6572 6e3a 2043 6865 726e 2c20 2d2d 2054  ern: Chern, -- T
│ │ │ │ +00027b80: 6865 2043 6865 726e 2063 6c61 7373 0a20  he Chern class. 
│ │ │ │ +00027b90: 202a 2022 4353 4d28 2e2e 2e2c 4f75 7470   * "CSM(...,Outp
│ │ │ │ +00027ba0: 7574 3d3e 2e2e 2e29 2220 2d2d 2073 6565  ut=>...)" -- see
│ │ │ │ +00027bb0: 202a 6e6f 7465 2043 534d 3a20 4353 4d2c   *note CSM: CSM,
│ │ │ │ +00027bc0: 202d 2d20 5468 650a 2020 2020 4368 6572   -- The.    Cher
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│ │ │ │ +000282d0: 6620 7365 7665 7261 6c20 6469 6666 6572  f several differ
│ │ │ │ +000282e0: 656e 7420 6578 616d 706c 6573 2e0a 0a57  ent examples...W
│ │ │ │ +000282f0: 6520 696c 6c75 7374 7261 7465 2074 6865  e illustrate the
│ │ │ │ +00028300: 2070 726f 6261 6269 6c69 7374 6963 2062   probabilistic b
│ │ │ │ +00028310: 6568 6176 696f 7572 2077 6974 6820 616e  ehaviour with an
│ │ │ │ +00028320: 2065 7861 6d70 6c65 2077 6865 7265 2074   example where t
│ │ │ │ +00028330: 6865 2063 686f 7365 6e0a 7261 6e64 6f6d  he chosen.random
│ │ │ │ +00028340: 2073 6565 6420 6c65 6164 7320 746f 2061   seed leads to a
│ │ │ │ +00028350: 2077 726f 6e67 2072 6573 756c 7420 696e   wrong result in
│ │ │ │ +00028360: 2074 6865 2066 6972 7374 2063 616c 6375   the first calcu
│ │ │ │ +00028370: 6c61 7469 6f6e 2e0a 0a2b 2d2d 2d2d 2d2d  lation...+------
│ │ │ │ +00028380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000283a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000283b0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a  ---------+.|i1 :
│ │ │ │ -000283c0: 2073 6574 5261 6e64 6f6d 5365 6564 2031   setRandomSeed 1
│ │ │ │ -000283d0: 3231 3b20 2020 2020 2020 2020 2020 2020  21;             
│ │ │ │ -000283e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000283f0: 2d2d 2073 6574 7469 6e67 2072 616e 646f  -- setting rando
│ │ │ │ -00028400: 6d20 7365 6564 2074 6f20 3132 3120 2020  m seed to 121   
│ │ │ │ -00028410: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00028420: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000283a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120  ----------+.|i1 
│ │ │ │ +000283b0: 3a20 7365 7452 616e 646f 6d53 6565 6420  : setRandomSeed 
│ │ │ │ +000283c0: 3132 313b 2020 2020 2020 2020 2020 2020  121;            
│ │ │ │ +000283d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000283e0: 202d 2d20 7365 7474 696e 6720 7261 6e64   -- setting rand
│ │ │ │ +000283f0: 6f6d 2073 6565 6420 746f 2031 3231 2020  om seed to 121  
│ │ │ │ +00028400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028410: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00028420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028450: 2d2d 2b0a 7c69 3220 3a20 5220 3d20 5151  --+.|i2 : R = QQ
│ │ │ │ -00028460: 5b78 2c79 2c7a 2c77 5d20 2020 2020 2020  [x,y,z,w]       
│ │ │ │ -00028470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028480: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00028440: 2d2d 2d2b 0a7c 6932 203a 2052 203d 2051  ---+.|i2 : R = Q
│ │ │ │ +00028450: 515b 782c 792c 7a2c 775d 2020 2020 2020  Q[x,y,z,w]      
│ │ │ │ +00028460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028470: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00028480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000284a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000284b0: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20          |.|o2 = 
│ │ │ │ -000284c0: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │ -000284d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000284e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000284a0: 2020 2020 2020 2020 207c 0a7c 6f32 203d           |.|o2 =
│ │ │ │ +000284b0: 2052 2020 2020 2020 2020 2020 2020 2020   R              
│ │ │ │ +000284c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000284d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000284e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000284f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028510: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00028520: 7c6f 3220 3a20 506f 6c79 6e6f 6d69 616c  |o2 : Polynomial
│ │ │ │ -00028530: 5269 6e67 2020 2020 2020 2020 2020 2020  Ring            
│ │ │ │ -00028540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028550: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00028500: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00028510: 0a7c 6f32 203a 2050 6f6c 796e 6f6d 6961  .|o2 : Polynomia
│ │ │ │ +00028520: 6c52 696e 6720 2020 2020 2020 2020 2020  lRing           
│ │ │ │ +00028530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028540: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00028550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028580: 2d2d 2d2d 2b0a 7c69 3320 3a20 4920 3d20  ----+.|i3 : I = 
│ │ │ │ -00028590: 6d69 6e6f 7273 2832 2c6d 6174 7269 787b  minors(2,matrix{
│ │ │ │ -000285a0: 7b78 2c79 2c7a 7d2c 7b79 2c7a 2c77 7d7d  {x,y,z},{y,z,w}}
│ │ │ │ -000285b0: 2920 2020 2020 207c 0a7c 2020 2020 2020  )      |.|      
│ │ │ │ +00028570: 2d2d 2d2d 2d2b 0a7c 6933 203a 2049 203d  -----+.|i3 : I =
│ │ │ │ +00028580: 206d 696e 6f72 7328 322c 6d61 7472 6978   minors(2,matrix
│ │ │ │ +00028590: 7b7b 782c 792c 7a7d 2c7b 792c 7a2c 777d  {{x,y,z},{y,z,w}
│ │ │ │ +000285a0: 7d29 2020 2020 2020 7c0a 7c20 2020 2020  })      |.|     
│ │ │ │ +000285b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000285c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000285d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000285e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -000285f0: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ -00028600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028610: 2020 2020 2032 2020 2020 2020 207c 0a7c       2       |.|
│ │ │ │ -00028620: 6f33 203d 2069 6465 616c 2028 2d20 7920  o3 = ideal (- y 
│ │ │ │ -00028630: 202b 2078 2a7a 2c20 2d20 792a 7a20 2b20   + x*z, - y*z + 
│ │ │ │ -00028640: 782a 772c 202d 207a 2020 2b20 792a 7729  x*w, - z  + y*w)
│ │ │ │ -00028650: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000285d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000285e0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +000285f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028600: 2020 2020 2020 3220 2020 2020 2020 7c0a        2       |.
│ │ │ │ +00028610: 7c6f 3320 3d20 6964 6561 6c20 282d 2079  |o3 = ideal (- y
│ │ │ │ +00028620: 2020 2b20 782a 7a2c 202d 2079 2a7a 202b    + x*z, - y*z +
│ │ │ │ +00028630: 2078 2a77 2c20 2d20 7a20 202b 2079 2a77   x*w, - z  + y*w
│ │ │ │ +00028640: 297c 0a7c 2020 2020 2020 2020 2020 2020  )|.|            
│ │ │ │ +00028650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028680: 2020 207c 0a7c 6f33 203a 2049 6465 616c     |.|o3 : Ideal
│ │ │ │ -00028690: 206f 6620 5220 2020 2020 2020 2020 2020   of R           
│ │ │ │ -000286a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000286b0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00028670: 2020 2020 7c0a 7c6f 3320 3a20 4964 6561      |.|o3 : Idea
│ │ │ │ +00028680: 6c20 6f66 2052 2020 2020 2020 2020 2020  l of R          
│ │ │ │ +00028690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000286a0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +000286b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000286c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000286d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000286e0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a  ---------+.|i4 :
│ │ │ │ -000286f0: 2043 6865 726e 2028 492c 436f 6d70 4d65   Chern (I,CompMe
│ │ │ │ -00028700: 7468 6f64 3d3e 506e 5265 7369 6475 616c  thod=>PnResidual
│ │ │ │ -00028710: 2920 2020 2020 2020 2020 2020 7c0a 7c20  )           |.| 
│ │ │ │ +000286d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420  ----------+.|i4 
│ │ │ │ +000286e0: 3a20 4368 6572 6e20 2849 2c43 6f6d 704d  : Chern (I,CompM
│ │ │ │ +000286f0: 6574 686f 643d 3e50 6e52 6573 6964 7561  ethod=>PnResidua
│ │ │ │ +00028700: 6c29 2020 2020 2020 2020 2020 207c 0a7c  l)           |.|
│ │ │ │ +00028710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028740: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00028750: 0a7c 2020 2020 2020 2033 2020 2020 2032  .|       3     2
│ │ │ │ +00028740: 7c0a 7c20 2020 2020 2020 3320 2020 2020  |.|       3     
│ │ │ │ +00028750: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  00028760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028780: 2020 7c0a 7c6f 3420 3d20 3248 2020 2b20    |.|o4 = 2H  + 
│ │ │ │ -00028790: 3348 2020 2020 2020 2020 2020 2020 2020  3H              
│ │ │ │ -000287a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000287b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00028770: 2020 207c 0a7c 6f34 203d 2032 4820 202b     |.|o4 = 2H  +
│ │ │ │ +00028780: 2033 4820 2020 2020 2020 2020 2020 2020   3H             
│ │ │ │ +00028790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000287a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000287b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000287c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000287d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000287e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000287f0: 5a5a 5b48 5d20 2020 2020 2020 2020 2020  ZZ[H]           
│ │ │ │ -00028800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028810: 2020 2020 2020 2020 2020 207c 0a7c 6f34             |.|o4
│ │ │ │ -00028820: 203a 202d 2d2d 2d2d 2020 2020 2020 2020   : -----        
│ │ │ │ -00028830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028840: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00028850: 7c20 2020 2020 2020 2034 2020 2020 2020  |        4      
│ │ │ │ +000287d0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000287e0: 205a 5a5b 485d 2020 2020 2020 2020 2020   ZZ[H]          
│ │ │ │ +000287f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028800: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00028810: 3420 3a20 2d2d 2d2d 2d20 2020 2020 2020  4 : -----       
│ │ │ │ +00028820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028830: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00028840: 0a7c 2020 2020 2020 2020 3420 2020 2020  .|        4     
│ │ │ │ +00028850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028880: 207c 0a7c 2020 2020 2020 2048 2020 2020   |.|       H    
│ │ │ │ +00028870: 2020 7c0a 7c20 2020 2020 2020 4820 2020    |.|       H   
│ │ │ │ +00028880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000288a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000288b0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +000288a0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +000288b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000288c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000288d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000288e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2043  -------+.|i5 : C
│ │ │ │ -000288f0: 6865 726e 2028 492c 436f 6d70 4d65 7468  hern (I,CompMeth
│ │ │ │ -00028900: 6f64 3d3e 506e 5265 7369 6475 616c 2920  od=>PnResidual) 
│ │ │ │ -00028910: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000288d0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20  --------+.|i5 : 
│ │ │ │ +000288e0: 4368 6572 6e20 2849 2c43 6f6d 704d 6574  Chern (I,CompMet
│ │ │ │ +000288f0: 686f 643d 3e50 6e52 6573 6964 7561 6c29  hod=>PnResidual)
│ │ │ │ +00028900: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00028910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028940: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00028950: 2020 2020 2020 2033 2020 2020 2032 2020         3     2  
│ │ │ │ +00028930: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00028940: 7c20 2020 2020 2020 3320 2020 2020 3220  |       3     2 
│ │ │ │ +00028950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028980: 7c0a 7c6f 3520 3d20 3248 2020 2b20 3348  |.|o5 = 2H  + 3H
│ │ │ │ +00028970: 207c 0a7c 6f35 203d 2032 4820 202b 2033   |.|o5 = 2H  + 3
│ │ │ │ +00028980: 4820 2020 2020 2020 2020 2020 2020 2020  H               
│ │ │ │  00028990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000289a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000289b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000289a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000289b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000289c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000289d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000289e0: 2020 2020 2020 7c0a 7c20 2020 2020 5a5a        |.|     ZZ
│ │ │ │ -000289f0: 5b48 5d20 2020 2020 2020 2020 2020 2020  [H]             
│ │ │ │ -00028a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028a10: 2020 2020 2020 2020 207c 0a7c 6f35 203a           |.|o5 :
│ │ │ │ -00028a20: 202d 2d2d 2d2d 2020 2020 2020 2020 2020   -----          
│ │ │ │ -00028a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028a40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00028a50: 2020 2020 2020 2034 2020 2020 2020 2020         4        
│ │ │ │ +000289d0: 2020 2020 2020 207c 0a7c 2020 2020 205a         |.|     Z
│ │ │ │ +000289e0: 5a5b 485d 2020 2020 2020 2020 2020 2020  Z[H]            
│ │ │ │ +000289f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028a00: 2020 2020 2020 2020 2020 7c0a 7c6f 3520            |.|o5 
│ │ │ │ +00028a10: 3a20 2d2d 2d2d 2d20 2020 2020 2020 2020  : -----         
│ │ │ │ +00028a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028a30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00028a40: 2020 2020 2020 2020 3420 2020 2020 2020          4       
│ │ │ │ +00028a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028a70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00028a80: 0a7c 2020 2020 2020 2048 2020 2020 2020  .|       H      
│ │ │ │ +00028a70: 7c0a 7c20 2020 2020 2020 4820 2020 2020  |.|       H     
│ │ │ │ +00028a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028ab0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00028aa0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00028ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028ae0: 2d2d 2d2d 2d2b 0a7c 6936 203a 2043 6865  -----+.|i6 : Che
│ │ │ │ -00028af0: 726e 2028 492c 436f 6d70 4d65 7468 6f64  rn (I,CompMethod
│ │ │ │ -00028b00: 3d3e 506e 5265 7369 6475 616c 2920 2020  =>PnResidual)   
│ │ │ │ -00028b10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00028ad0: 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 4368  ------+.|i6 : Ch
│ │ │ │ +00028ae0: 6572 6e20 2849 2c43 6f6d 704d 6574 686f  ern (I,CompMetho
│ │ │ │ +00028af0: 643d 3e50 6e52 6573 6964 7561 6c29 2020  d=>PnResidual)  
│ │ │ │ +00028b00: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00028b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028b40: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00028b50: 2020 2020 2033 2020 2020 2032 2020 2020       3     2    
│ │ │ │ -00028b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028b70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00028b80: 7c6f 3620 3d20 3248 2020 2b20 3348 2020  |o6 = 2H  + 3H  
│ │ │ │ +00028b30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00028b40: 2020 2020 2020 3320 2020 2020 3220 2020        3     2   
│ │ │ │ +00028b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028b60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00028b70: 0a7c 6f36 203d 2032 4820 202b 2033 4820  .|o6 = 2H  + 3H 
│ │ │ │ +00028b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028bb0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00028ba0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00028bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028be0: 2020 2020 7c0a 7c20 2020 2020 5a5a 5b48      |.|     ZZ[H
│ │ │ │ -00028bf0: 5d20 2020 2020 2020 2020 2020 2020 2020  ]               
│ │ │ │ -00028c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028c10: 2020 2020 2020 207c 0a7c 6f36 203a 202d         |.|o6 : -
│ │ │ │ -00028c20: 2d2d 2d2d 2020 2020 2020 2020 2020 2020  ----            
│ │ │ │ -00028c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00029680: 206d 6179 2062 6520 7573 6564 2066 6f72   may be used for
│ │ │ │ -00029690: 0a20 2020 2020 2020 2073 7562 7363 6865  .        subsche
│ │ │ │ -000296a0: 6d65 7320 6f66 2061 6e79 2061 7070 6c69  mes of any appli
│ │ │ │ -000296b0: 6361 626c 6520 746f 7269 6320 7661 7269  cable toric vari
│ │ │ │ -000296c0: 6574 7920 2874 6869 7320 6d61 7920 6265  ety (this may be
│ │ │ │ -000296d0: 2063 6865 636b 6564 2075 7369 6e67 0a20   checked using. 
│ │ │ │ -000296e0: 2020 2020 2020 2074 6865 202a 6e6f 7465         the *note
│ │ │ │ -000296f0: 2043 6865 636b 546f 7269 6356 6172 6965   CheckToricVarie
│ │ │ │ -00029700: 7479 5661 6c69 643a 2043 6865 636b 546f  tyValid: CheckTo
│ │ │ │ -00029710: 7269 6356 6172 6965 7479 5661 6c69 642c  ricVarietyValid,
│ │ │ │ -00029720: 2063 6f6d 6d61 6e64 290a 2020 2020 2020   command).      
│ │ │ │ -00029730: 2a20 436f 6d70 4d65 7468 6f64 2028 6d69  * CompMethod (mi
│ │ │ │ -00029740: 7373 696e 6720 646f 6375 6d65 6e74 6174  ssing documentat
│ │ │ │ -00029750: 696f 6e29 203d 3e20 2e2e 2e2c 2064 6566  ion) => ..., def
│ │ │ │ -00029760: 6175 6c74 2076 616c 7565 0a20 2020 2020  ault value.     
│ │ │ │ -00029770: 2020 2050 726f 6a65 6374 6976 6544 6567     ProjectiveDeg
│ │ │ │ -00029780: 7265 652c 2050 6e52 6573 6964 7561 6c2c  ree, PnResidual,
│ │ │ │ -00029790: 2074 6869 7320 616c 676f 7269 7468 6d20   this algorithm 
│ │ │ │ -000297a0: 6d61 7920 6265 2075 7365 6420 666f 7220  may be used for 
│ │ │ │ -000297b0: 7375 6273 6368 656d 6573 0a20 2020 2020  subschemes.     
│ │ │ │ -000297c0: 2020 206f 6620 5c50 505e 6e20 6f6e 6c79     of \PP^n only
│ │ │ │ -000297d0: 0a20 2020 2020 202a 204f 7574 7075 7420  .      * Output 
│ │ │ │ -000297e0: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420  => ..., default 
│ │ │ │ -000297f0: 7661 6c75 6520 4368 6f77 5269 6e67 456c  value ChowRingEl
│ │ │ │ -00029800: 656d 656e 742c 2043 686f 7752 696e 6745  ement, ChowRingE
│ │ │ │ -00029810: 6c65 6d65 6e74 2c20 7265 7475 726e 730a  lement, returns.
│ │ │ │ -00029820: 2020 2020 2020 2020 6120 5269 6e67 456c          a RingEl
│ │ │ │ -00029830: 656d 656e 7420 696e 2074 6865 2043 686f  ement in the Cho
│ │ │ │ -00029840: 7720 7269 6e67 206f 6620 7468 6520 6170  w ring of the ap
│ │ │ │ -00029850: 7072 6f70 7269 6174 6520 616d 6269 656e  propriate ambien
│ │ │ │ -00029860: 7420 7370 6163 650a 2020 2020 2020 2a20  t space.      * 
│ │ │ │ -00029870: 4f75 7470 7574 203d 3e20 2e2e 2e2c 2064  Output => ..., d
│ │ │ │ -00029880: 6566 6175 6c74 2076 616c 7565 2043 686f  efault value Cho
│ │ │ │ -00029890: 7752 696e 6745 6c65 6d65 6e74 2c20 4861  wRingElement, Ha
│ │ │ │ -000298a0: 7368 466f 726d 2c20 4861 7368 466f 726d  shForm, HashForm
│ │ │ │ -000298b0: 0a20 2020 2020 2020 2072 6574 7572 6e73  .        returns
│ │ │ │ -000298c0: 2061 204d 7574 6162 6c65 4861 7368 5461   a MutableHashTa
│ │ │ │ -000298d0: 626c 6520 636f 6e74 6169 6e69 6e67 2074  ble containing t
│ │ │ │ -000298e0: 6865 2066 6f6c 6c6f 7769 6e67 206b 6579  he following key
│ │ │ │ -000298f0: 733a 2022 4722 2028 7468 650a 2020 2020  s: "G" (the.    
│ │ │ │ -00029900: 2020 2020 706f 6c79 6e6f 6d69 616c 2077      polynomial w
│ │ │ │ -00029910: 6974 6820 636f 6566 6669 6369 656e 7473  ith coefficients
│ │ │ │ -00029920: 206f 6620 7468 6520 6879 7065 7270 6c61   of the hyperpla
│ │ │ │ -00029930: 6e65 2063 6c61 7373 6573 2072 6570 7265  ne classes repre
│ │ │ │ -00029940: 7365 6e74 696e 6720 7468 650a 2020 2020  senting the.    
│ │ │ │ -00029950: 2020 2020 7072 6f6a 6563 7469 7665 2064      projective d
│ │ │ │ -00029960: 6567 7265 6573 292c 2022 476c 6973 7422  egrees), "Glist"
│ │ │ │ -00029970: 2028 7468 6520 6c69 7374 2066 6f72 6d20   (the list form 
│ │ │ │ -00029980: 6f66 2022 4722 2920 2c20 2253 6567 7265  of "G") , "Segre
│ │ │ │ -00029990: 2220 2874 6865 0a20 2020 2020 2020 2074  " (the.        t
│ │ │ │ -000299a0: 6f74 616c 2053 6567 7265 2063 6c61 7373  otal Segre class
│ │ │ │ -000299b0: 206f 6620 7468 6520 696e 7075 7429 2c22   of the input),"
│ │ │ │ -000299c0: 5365 6772 654c 6973 7422 2028 7468 6520  SegreList" (the 
│ │ │ │ -000299d0: 6c69 7374 2066 6f72 6d20 6f66 2022 5365  list form of "Se
│ │ │ │ -000299e0: 6772 6522 290a 2020 2a20 4f75 7470 7574  gre").  * Output
│ │ │ │ -000299f0: 733a 0a20 2020 2020 202a 2061 202a 6e6f  s:.      * a *no
│ │ │ │ -00029a00: 7465 2072 696e 6720 656c 656d 656e 743a  te ring element:
│ │ │ │ -00029a10: 2028 4d61 6361 756c 6179 3244 6f63 2952   (Macaulay2Doc)R
│ │ │ │ -00029a20: 696e 6745 6c65 6d65 6e74 2c2c 2074 6865  ingElement,, the
│ │ │ │ -00029a30: 2070 7573 6866 6f72 7761 7264 206f 660a   pushforward of.
│ │ │ │ -00029a40: 2020 2020 2020 2020 7468 6520 746f 7461          the tota
│ │ │ │ -00029a50: 6c20 5365 6772 6520 636c 6173 7320 6f66  l Segre class of
│ │ │ │ -00029a60: 2074 6865 2073 6368 656d 6520 5620 6465   the scheme V de
│ │ │ │ -00029a70: 6669 6e65 6420 6279 2074 6865 2069 6e70  fined by the inp
│ │ │ │ -00029a80: 7574 2069 6465 616c 2074 6f20 7468 650a  ut ideal to the.
│ │ │ │ -00029a90: 2020 2020 2020 2020 6170 7072 6f70 7269          appropri
│ │ │ │ -00029aa0: 6174 6520 4368 6f77 2072 696e 670a 0a44  ate Chow ring..D
│ │ │ │ -00029ab0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ -00029ac0: 3d3d 3d3d 3d3d 0a0a 466f 7220 6120 7375  ======..For a su
│ │ │ │ -00029ad0: 6273 6368 656d 6520 5620 6f66 2061 6e20  bscheme V of an 
│ │ │ │ -00029ae0: 6170 706c 6963 6162 6c65 2074 6f72 6963  applicable toric
│ │ │ │ -00029af0: 2076 6172 6965 7479 2058 2074 6869 7320   variety X this 
│ │ │ │ -00029b00: 636f 6d6d 616e 6420 636f 6d70 7574 6573  command computes
│ │ │ │ -00029b10: 2074 6865 0a70 7573 682d 666f 7277 6172   the.push-forwar
│ │ │ │ -00029b20: 6420 6f66 2074 6865 2074 6f74 616c 2053  d of the total S
│ │ │ │ -00029b30: 6567 7265 2063 6c61 7373 2073 2856 2c58  egre class s(V,X
│ │ │ │ -00029b40: 2920 6f66 2056 2069 6e20 5820 746f 2074  ) of V in X to t
│ │ │ │ -00029b50: 6865 2043 686f 7720 7269 6e67 206f 6620  he Chow ring of 
│ │ │ │ -00029b60: 582e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  X...+-----------
│ │ │ │ +00029100: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20  ***..  * Usage: 
│ │ │ │ +00029110: 0a20 2020 2020 2020 2053 6567 7265 2049  .        Segre I
│ │ │ │ +00029120: 0a20 2020 2020 2020 2053 6567 7265 2841  .        Segre(A
│ │ │ │ +00029130: 2c49 290a 2020 2020 2020 2020 5365 6772  ,I).        Segr
│ │ │ │ +00029140: 6528 582c 4a29 0a20 2020 2020 2020 2053  e(X,J).        S
│ │ │ │ +00029150: 6567 7265 2843 682c 582c 4a29 0a20 202a  egre(Ch,X,J).  *
│ │ │ │ +00029160: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ +00029170: 2049 2c20 616e 202a 6e6f 7465 2069 6465   I, an *note ide
│ │ │ │ +00029180: 616c 3a20 284d 6163 6175 6c61 7932 446f  al: (Macaulay2Do
│ │ │ │ +00029190: 6329 4964 6561 6c2c 2c20 6120 6d75 6c74  c)Ideal,, a mult
│ │ │ │ +000291a0: 692d 686f 6d6f 6765 6e65 6f75 7320 6964  i-homogeneous id
│ │ │ │ +000291b0: 6561 6c20 696e 2061 0a20 2020 2020 2020  eal in a.       
│ │ │ │ +000291c0: 2067 7261 6465 6420 706f 6c79 6e6f 6d69   graded polynomi
│ │ │ │ +000291d0: 616c 2072 696e 6720 6f76 6572 2061 2066  al ring over a f
│ │ │ │ +000291e0: 6965 6c64 2064 6566 696e 696e 6720 6120  ield defining a 
│ │ │ │ +000291f0: 636c 6f73 6564 2073 7562 7363 6865 6d65  closed subscheme
│ │ │ │ +00029200: 2056 206f 660a 2020 2020 2020 2020 5c50   V of.        \P
│ │ │ │ +00029210: 505e 7b6e 5f31 7d78 2e2e 2e78 5c50 505e  P^{n_1}x...x\PP^
│ │ │ │ +00029220: 7b6e 5f6d 7d0a 2020 2020 2020 2a20 412c  {n_m}.      * A,
│ │ │ │ +00029230: 2061 202a 6e6f 7465 2071 756f 7469 656e   a *note quotien
│ │ │ │ +00029240: 7420 7269 6e67 3a20 284d 6163 6175 6c61  t ring: (Macaula
│ │ │ │ +00029250: 7932 446f 6329 5175 6f74 6965 6e74 5269  y2Doc)QuotientRi
│ │ │ │ +00029260: 6e67 2c2c 0a20 2020 2020 2020 2041 3d5c  ng,,.        A=\
│ │ │ │ +00029270: 5a5a 5b68 5f31 2c2e 2e2e 2c68 5f6d 5d2f  ZZ[h_1,...,h_m]/
│ │ │ │ +00029280: 2868 5f31 5e7b 6e5f 312b 317d 2c2e 2e2e  (h_1^{n_1+1},...
│ │ │ │ +00029290: 2c68 5f6d 5e7b 6e5f 6d2b 317d 2920 7175  ,h_m^{n_m+1}) qu
│ │ │ │ +000292a0: 6f74 6965 6e74 2072 696e 670a 2020 2020  otient ring.    
│ │ │ │ +000292b0: 2020 2020 7265 7072 6573 656e 7469 6e67      representing
│ │ │ │ +000292c0: 2074 6865 2043 686f 7720 7269 6e67 206f   the Chow ring o
│ │ │ │ +000292d0: 6620 5c50 505e 7b6e 5f31 7d78 2e2e 2e78  f \PP^{n_1}x...x
│ │ │ │ +000292e0: 5c50 505e 7b6e 5f6d 7d2c 2074 6869 7320  \PP^{n_m}, this 
│ │ │ │ +000292f0: 7269 6e67 2073 686f 756c 640a 2020 2020  ring should.    
│ │ │ │ +00029300: 2020 2020 6265 2062 7569 6c74 2075 7369      be built usi
│ │ │ │ +00029310: 6e67 2074 6865 202a 6e6f 7465 2043 686f  ng the *note Cho
│ │ │ │ +00029320: 7752 696e 673a 2043 686f 7752 696e 672c  wRing: ChowRing,
│ │ │ │ +00029330: 2063 6f6d 6d61 6e64 0a20 2020 2020 202a   command.      *
│ │ │ │ +00029340: 204a 2c20 616e 202a 6e6f 7465 2069 6465   J, an *note ide
│ │ │ │ +00029350: 616c 3a20 284d 6163 6175 6c61 7932 446f  al: (Macaulay2Do
│ │ │ │ +00029360: 6329 4964 6561 6c2c 2c20 696e 2074 6865  c)Ideal,, in the
│ │ │ │ +00029370: 2067 7261 6465 6420 706f 6c79 6e6f 6d69   graded polynomi
│ │ │ │ +00029380: 616c 2072 696e 670a 2020 2020 2020 2020  al ring.        
│ │ │ │ +00029390: 7768 6963 6820 6973 2063 6f6f 7264 696e  which is coordin
│ │ │ │ +000293a0: 6174 6520 7269 6e67 206f 6620 7468 6520  ate ring of the 
│ │ │ │ +000293b0: 4e6f 726d 616c 2054 6f72 6963 2056 6172  Normal Toric Var
│ │ │ │ +000293c0: 6965 7479 2058 0a20 2020 2020 202a 2058  iety X.      * X
│ │ │ │ +000293d0: 2c20 6120 2a6e 6f74 6520 6e6f 726d 616c  , a *note normal
│ │ │ │ +000293e0: 2074 6f72 6963 2076 6172 6965 7479 3a0a   toric variety:.
│ │ │ │ +000293f0: 2020 2020 2020 2020 284e 6f72 6d61 6c54          (NormalT
│ │ │ │ +00029400: 6f72 6963 5661 7269 6574 6965 7329 4e6f  oricVarieties)No
│ │ │ │ +00029410: 726d 616c 546f 7269 6356 6172 6965 7479  rmalToricVariety
│ │ │ │ +00029420: 2c2c 2077 6869 6368 2069 7320 7468 6520  ,, which is the 
│ │ │ │ +00029430: 616d 6269 656e 7420 7370 6163 650a 2020  ambient space.  
│ │ │ │ +00029440: 2020 2020 2020 7768 6963 6820 636f 6e74        which cont
│ │ │ │ +00029450: 6169 6e73 2056 284a 290a 2020 2020 2020  ains V(J).      
│ │ │ │ +00029460: 2a20 4368 2c20 6120 2a6e 6f74 6520 7175  * Ch, a *note qu
│ │ │ │ +00029470: 6f74 6965 6e74 2072 696e 673a 2028 4d61  otient ring: (Ma
│ │ │ │ +00029480: 6361 756c 6179 3244 6f63 2951 756f 7469  caulay2Doc)Quoti
│ │ │ │ +00029490: 656e 7452 696e 672c 2c20 7468 6520 4368  entRing,, the Ch
│ │ │ │ +000294a0: 6f77 2072 696e 670a 2020 2020 2020 2020  ow ring.        
│ │ │ │ +000294b0: 6f66 2074 6865 2074 6f72 6963 2076 6172  of the toric var
│ │ │ │ +000294c0: 6965 7479 2058 2c20 4368 3d28 7269 6e67  iety X, Ch=(ring
│ │ │ │ +000294d0: 204a 292f 2853 522b 4c52 2920 7768 6572   J)/(SR+LR) wher
│ │ │ │ +000294e0: 6520 5352 2069 7320 7468 650a 2020 2020  e SR is the.    
│ │ │ │ +000294f0: 2020 2020 5374 616e 6c65 792d 5265 6973      Stanley-Reis
│ │ │ │ +00029500: 6e65 7220 6964 6561 6c20 6f66 2074 6865  ner ideal of the
│ │ │ │ +00029510: 2066 616e 2064 6566 696e 696e 6720 5820   fan defining X 
│ │ │ │ +00029520: 616e 6420 4c52 2069 7320 7468 6520 6c69  and LR is the li
│ │ │ │ +00029530: 6e65 6172 0a20 2020 2020 2020 2072 656c  near.        rel
│ │ │ │ +00029540: 6174 696f 6e73 2069 6465 616c 2c20 7468  ations ideal, th
│ │ │ │ +00029550: 6973 2072 696e 6720 7368 6f75 6c64 2062  is ring should b
│ │ │ │ +00029560: 6520 6275 696c 7420 7573 696e 6720 7468  e built using th
│ │ │ │ +00029570: 6520 2a6e 6f74 650a 2020 2020 2020 2020  e *note.        
│ │ │ │ +00029580: 546f 7269 6343 686f 7752 696e 673a 2054  ToricChowRing: T
│ │ │ │ +00029590: 6f72 6963 4368 6f77 5269 6e67 2c20 636f  oricChowRing, co
│ │ │ │ +000295a0: 6d6d 616e 640a 2020 2a20 2a6e 6f74 6520  mmand.  * *note 
│ │ │ │ +000295b0: 4f70 7469 6f6e 616c 2069 6e70 7574 733a  Optional inputs:
│ │ │ │ +000295c0: 2028 4d61 6361 756c 6179 3244 6f63 2975   (Macaulay2Doc)u
│ │ │ │ +000295d0: 7369 6e67 2066 756e 6374 696f 6e73 2077  sing functions w
│ │ │ │ +000295e0: 6974 6820 6f70 7469 6f6e 616c 2069 6e70  ith optional inp
│ │ │ │ +000295f0: 7574 732c 3a0a 2020 2020 2020 2a20 436f  uts,:.      * Co
│ │ │ │ +00029600: 6d70 4d65 7468 6f64 2028 6d69 7373 696e  mpMethod (missin
│ │ │ │ +00029610: 6720 646f 6375 6d65 6e74 6174 696f 6e29  g documentation)
│ │ │ │ +00029620: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74   => ..., default
│ │ │ │ +00029630: 2076 616c 7565 0a20 2020 2020 2020 2050   value.        P
│ │ │ │ +00029640: 726f 6a65 6374 6976 6544 6567 7265 652c  rojectiveDegree,
│ │ │ │ +00029650: 2050 726f 6a65 6374 6976 6544 6567 7265   ProjectiveDegre
│ │ │ │ +00029660: 652c 2074 6869 7320 616c 676f 7269 7468  e, this algorith
│ │ │ │ +00029670: 6d20 6d61 7920 6265 2075 7365 6420 666f  m may be used fo
│ │ │ │ +00029680: 720a 2020 2020 2020 2020 7375 6273 6368  r.        subsch
│ │ │ │ +00029690: 656d 6573 206f 6620 616e 7920 6170 706c  emes of any appl
│ │ │ │ +000296a0: 6963 6162 6c65 2074 6f72 6963 2076 6172  icable toric var
│ │ │ │ +000296b0: 6965 7479 2028 7468 6973 206d 6179 2062  iety (this may b
│ │ │ │ +000296c0: 6520 6368 6563 6b65 6420 7573 696e 670a  e checked using.
│ │ │ │ +000296d0: 2020 2020 2020 2020 7468 6520 2a6e 6f74          the *not
│ │ │ │ +000296e0: 6520 4368 6563 6b54 6f72 6963 5661 7269  e CheckToricVari
│ │ │ │ +000296f0: 6574 7956 616c 6964 3a20 4368 6563 6b54  etyValid: CheckT
│ │ │ │ +00029700: 6f72 6963 5661 7269 6574 7956 616c 6964  oricVarietyValid
│ │ │ │ +00029710: 2c20 636f 6d6d 616e 6429 0a20 2020 2020  , command).     
│ │ │ │ +00029720: 202a 2043 6f6d 704d 6574 686f 6420 286d   * CompMethod (m
│ │ │ │ +00029730: 6973 7369 6e67 2064 6f63 756d 656e 7461  issing documenta
│ │ │ │ +00029740: 7469 6f6e 2920 3d3e 202e 2e2e 2c20 6465  tion) => ..., de
│ │ │ │ +00029750: 6661 756c 7420 7661 6c75 650a 2020 2020  fault value.    
│ │ │ │ +00029760: 2020 2020 5072 6f6a 6563 7469 7665 4465      ProjectiveDe
│ │ │ │ +00029770: 6772 6565 2c20 506e 5265 7369 6475 616c  gree, PnResidual
│ │ │ │ +00029780: 2c20 7468 6973 2061 6c67 6f72 6974 686d  , this algorithm
│ │ │ │ +00029790: 206d 6179 2062 6520 7573 6564 2066 6f72   may be used for
│ │ │ │ +000297a0: 2073 7562 7363 6865 6d65 730a 2020 2020   subschemes.    
│ │ │ │ +000297b0: 2020 2020 6f66 205c 5050 5e6e 206f 6e6c      of \PP^n onl
│ │ │ │ +000297c0: 790a 2020 2020 2020 2a20 4f75 7470 7574  y.      * Output
│ │ │ │ +000297d0: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74   => ..., default
│ │ │ │ +000297e0: 2076 616c 7565 2043 686f 7752 696e 6745   value ChowRingE
│ │ │ │ +000297f0: 6c65 6d65 6e74 2c20 4368 6f77 5269 6e67  lement, ChowRing
│ │ │ │ +00029800: 456c 656d 656e 742c 2072 6574 7572 6e73  Element, returns
│ │ │ │ +00029810: 0a20 2020 2020 2020 2061 2052 696e 6745  .        a RingE
│ │ │ │ +00029820: 6c65 6d65 6e74 2069 6e20 7468 6520 4368  lement in the Ch
│ │ │ │ +00029830: 6f77 2072 696e 6720 6f66 2074 6865 2061  ow ring of the a
│ │ │ │ +00029840: 7070 726f 7072 6961 7465 2061 6d62 6965  ppropriate ambie
│ │ │ │ +00029850: 6e74 2073 7061 6365 0a20 2020 2020 202a  nt space.      *
│ │ │ │ +00029860: 204f 7574 7075 7420 3d3e 202e 2e2e 2c20   Output => ..., 
│ │ │ │ +00029870: 6465 6661 756c 7420 7661 6c75 6520 4368  default value Ch
│ │ │ │ +00029880: 6f77 5269 6e67 456c 656d 656e 742c 2048  owRingElement, H
│ │ │ │ +00029890: 6173 6846 6f72 6d2c 2048 6173 6846 6f72  ashForm, HashFor
│ │ │ │ +000298a0: 6d0a 2020 2020 2020 2020 7265 7475 726e  m.        return
│ │ │ │ +000298b0: 7320 6120 4d75 7461 626c 6548 6173 6854  s a MutableHashT
│ │ │ │ +000298c0: 6162 6c65 2063 6f6e 7461 696e 696e 6720  able containing 
│ │ │ │ +000298d0: 7468 6520 666f 6c6c 6f77 696e 6720 6b65  the following ke
│ │ │ │ +000298e0: 7973 3a20 2247 2220 2874 6865 0a20 2020  ys: "G" (the.   
│ │ │ │ +000298f0: 2020 2020 2070 6f6c 796e 6f6d 6961 6c20       polynomial 
│ │ │ │ +00029900: 7769 7468 2063 6f65 6666 6963 6965 6e74  with coefficient
│ │ │ │ +00029910: 7320 6f66 2074 6865 2068 7970 6572 706c  s of the hyperpl
│ │ │ │ +00029920: 616e 6520 636c 6173 7365 7320 7265 7072  ane classes repr
│ │ │ │ +00029930: 6573 656e 7469 6e67 2074 6865 0a20 2020  esenting the.   
│ │ │ │ +00029940: 2020 2020 2070 726f 6a65 6374 6976 6520       projective 
│ │ │ │ +00029950: 6465 6772 6565 7329 2c20 2247 6c69 7374  degrees), "Glist
│ │ │ │ +00029960: 2220 2874 6865 206c 6973 7420 666f 726d  " (the list form
│ │ │ │ +00029970: 206f 6620 2247 2229 202c 2022 5365 6772   of "G") , "Segr
│ │ │ │ +00029980: 6522 2028 7468 650a 2020 2020 2020 2020  e" (the.        
│ │ │ │ +00029990: 746f 7461 6c20 5365 6772 6520 636c 6173  total Segre clas
│ │ │ │ +000299a0: 7320 6f66 2074 6865 2069 6e70 7574 292c  s of the input),
│ │ │ │ +000299b0: 2253 6567 7265 4c69 7374 2220 2874 6865  "SegreList" (the
│ │ │ │ +000299c0: 206c 6973 7420 666f 726d 206f 6620 2253   list form of "S
│ │ │ │ +000299d0: 6567 7265 2229 0a20 202a 204f 7574 7075  egre").  * Outpu
│ │ │ │ +000299e0: 7473 3a0a 2020 2020 2020 2a20 6120 2a6e  ts:.      * a *n
│ │ │ │ +000299f0: 6f74 6520 7269 6e67 2065 6c65 6d65 6e74  ote ring element
│ │ │ │ +00029a00: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +00029a10: 5269 6e67 456c 656d 656e 742c 2c20 7468  RingElement,, th
│ │ │ │ +00029a20: 6520 7075 7368 666f 7277 6172 6420 6f66  e pushforward of
│ │ │ │ +00029a30: 0a20 2020 2020 2020 2074 6865 2074 6f74  .        the tot
│ │ │ │ +00029a40: 616c 2053 6567 7265 2063 6c61 7373 206f  al Segre class o
│ │ │ │ +00029a50: 6620 7468 6520 7363 6865 6d65 2056 2064  f the scheme V d
│ │ │ │ +00029a60: 6566 696e 6564 2062 7920 7468 6520 696e  efined by the in
│ │ │ │ +00029a70: 7075 7420 6964 6561 6c20 746f 2074 6865  put ideal to the
│ │ │ │ +00029a80: 0a20 2020 2020 2020 2061 7070 726f 7072  .        appropr
│ │ │ │ +00029a90: 6961 7465 2043 686f 7720 7269 6e67 0a0a  iate Chow ring..
│ │ │ │ +00029aa0: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
│ │ │ │ +00029ab0: 3d3d 3d3d 3d3d 3d0a 0a46 6f72 2061 2073  =======..For a s
│ │ │ │ +00029ac0: 7562 7363 6865 6d65 2056 206f 6620 616e  ubscheme V of an
│ │ │ │ +00029ad0: 2061 7070 6c69 6361 626c 6520 746f 7269   applicable tori
│ │ │ │ +00029ae0: 6320 7661 7269 6574 7920 5820 7468 6973  c variety X this
│ │ │ │ +00029af0: 2063 6f6d 6d61 6e64 2063 6f6d 7075 7465   command compute
│ │ │ │ +00029b00: 7320 7468 650a 7075 7368 2d66 6f72 7761  s the.push-forwa
│ │ │ │ +00029b10: 7264 206f 6620 7468 6520 746f 7461 6c20  rd of the total 
│ │ │ │ +00029b20: 5365 6772 6520 636c 6173 7320 7328 562c  Segre class s(V,
│ │ │ │ +00029b30: 5829 206f 6620 5620 696e 2058 2074 6f20  X) of V in X to 
│ │ │ │ +00029b40: 7468 6520 4368 6f77 2072 696e 6720 6f66  the Chow ring of
│ │ │ │ +00029b50: 2058 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d   X...+----------
│ │ │ │ +00029b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029b90: 2d2d 2b0a 7c69 3120 3a20 7365 7452 616e  --+.|i1 : setRan
│ │ │ │ -00029ba0: 646f 6d53 6565 6420 3732 3b20 2020 2020  domSeed 72;     
│ │ │ │ -00029bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029bc0: 2020 7c0a 7c20 2d2d 2073 6574 7469 6e67    |.| -- setting
│ │ │ │ -00029bd0: 2072 616e 646f 6d20 7365 6564 2074 6f20   random seed to 
│ │ │ │ -00029be0: 3732 2020 2020 2020 2020 2020 2020 2020  72              
│ │ │ │ -00029bf0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00029b80: 2d2d 2d2b 0a7c 6931 203a 2073 6574 5261  ---+.|i1 : setRa
│ │ │ │ +00029b90: 6e64 6f6d 5365 6564 2037 323b 2020 2020  ndomSeed 72;    
│ │ │ │ +00029ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029bb0: 2020 207c 0a7c 202d 2d20 7365 7474 696e     |.| -- settin
│ │ │ │ +00029bc0: 6720 7261 6e64 6f6d 2073 6565 6420 746f  g random seed to
│ │ │ │ +00029bd0: 2037 3220 2020 2020 2020 2020 2020 2020   72             
│ │ │ │ +00029be0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00029bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029c20: 2d2d 2b0a 7c69 3220 3a20 5220 3d20 5a5a  --+.|i2 : R = ZZ
│ │ │ │ -00029c30: 2f33 3237 3439 5b77 2c79 2c7a 5d20 2020  /32749[w,y,z]   
│ │ │ │ -00029c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029c50: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00029c10: 2d2d 2d2b 0a7c 6932 203a 2052 203d 205a  ---+.|i2 : R = Z
│ │ │ │ +00029c20: 5a2f 3332 3734 395b 772c 792c 7a5d 2020  Z/32749[w,y,z]  
│ │ │ │ +00029c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029c40: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00029c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029c80: 2020 7c0a 7c6f 3220 3d20 5220 2020 2020    |.|o2 = R     
│ │ │ │ +00029c70: 2020 207c 0a7c 6f32 203d 2052 2020 2020     |.|o2 = R    
│ │ │ │ +00029c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029cb0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00029ca0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00029cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029ce0: 2020 7c0a 7c6f 3220 3a20 506f 6c79 6e6f    |.|o2 : Polyno
│ │ │ │ -00029cf0: 6d69 616c 5269 6e67 2020 2020 2020 2020  mialRing        
│ │ │ │ -00029d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029d10: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00029cd0: 2020 207c 0a7c 6f32 203a 2050 6f6c 796e     |.|o2 : Polyn
│ │ │ │ +00029ce0: 6f6d 6961 6c52 696e 6720 2020 2020 2020  omialRing       
│ │ │ │ +00029cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029d00: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00029d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029d40: 2d2d 2b0a 7c69 3320 3a20 5365 6772 6528  --+.|i3 : Segre(
│ │ │ │ -00029d50: 6964 6561 6c28 772a 7929 2c43 6f6d 704d  ideal(w*y),CompM
│ │ │ │ -00029d60: 6574 686f 643d 3e50 6e52 6573 6964 7561  ethod=>PnResidua
│ │ │ │ -00029d70: 6c29 7c0a 7c20 2020 2020 2020 2020 2020  l)|.|           
│ │ │ │ +00029d30: 2d2d 2d2b 0a7c 6933 203a 2053 6567 7265  ---+.|i3 : Segre
│ │ │ │ +00029d40: 2869 6465 616c 2877 2a79 292c 436f 6d70  (ideal(w*y),Comp
│ │ │ │ +00029d50: 4d65 7468 6f64 3d3e 506e 5265 7369 6475  Method=>PnResidu
│ │ │ │ +00029d60: 616c 297c 0a7c 2020 2020 2020 2020 2020  al)|.|          
│ │ │ │ +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 7c0a 7c20 2020 2020 2020 2020 3220    |.|         2 
│ │ │ │ +00029d90: 2020 207c 0a7c 2020 2020 2020 2020 2032     |.|         2
│ │ │ │ +00029da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029dd0: 2020 7c0a 7c6f 3320 3d20 2d20 3448 2020    |.|o3 = - 4H  
│ │ │ │ -00029de0: 2b20 3248 2020 2020 2020 2020 2020 2020  + 2H            
│ │ │ │ -00029df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029e00: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00029dc0: 2020 207c 0a7c 6f33 203d 202d 2034 4820     |.|o3 = - 4H 
│ │ │ │ +00029dd0: 202b 2032 4820 2020 2020 2020 2020 2020   + 2H           
│ │ │ │ +00029de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029df0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00029e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029e30: 2020 7c0a 7c20 2020 2020 5a5a 5b48 5d20    |.|     ZZ[H] 
│ │ │ │ +00029e20: 2020 207c 0a7c 2020 2020 205a 5a5b 485d     |.|     ZZ[H]
│ │ │ │ +00029e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029e60: 2020 7c0a 7c6f 3320 3a20 2d2d 2d2d 2d20    |.|o3 : ----- 
│ │ │ │ +00029e50: 2020 207c 0a7c 6f33 203a 202d 2d2d 2d2d     |.|o3 : -----
│ │ │ │ +00029e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029e90: 2020 7c0a 7c20 2020 2020 2020 2033 2020    |.|        3  
│ │ │ │ +00029e80: 2020 207c 0a7c 2020 2020 2020 2020 3320     |.|        3 
│ │ │ │ +00029e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029ec0: 2020 7c0a 7c20 2020 2020 2020 4820 2020    |.|       H   
│ │ │ │ +00029eb0: 2020 207c 0a7c 2020 2020 2020 2048 2020     |.|       H  
│ │ │ │ +00029ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029ef0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00029ee0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00029ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029f20: 2d2d 2b0a 7c69 3420 3a20 413d 4368 6f77  --+.|i4 : A=Chow
│ │ │ │ -00029f30: 5269 6e67 2852 2920 2020 2020 2020 2020  Ring(R)         
│ │ │ │ -00029f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029f50: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00029f10: 2d2d 2d2b 0a7c 6934 203a 2041 3d43 686f  ---+.|i4 : A=Cho
│ │ │ │ +00029f20: 7752 696e 6728 5229 2020 2020 2020 2020  wRing(R)        
│ │ │ │ +00029f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029f40: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00029f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029f80: 2020 7c0a 7c6f 3420 3d20 4120 2020 2020    |.|o4 = A     
│ │ │ │ +00029f70: 2020 207c 0a7c 6f34 203d 2041 2020 2020     |.|o4 = A    
│ │ │ │ +00029f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029fb0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00029fa0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00029fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029fe0: 2020 7c0a 7c6f 3420 3a20 5175 6f74 6965    |.|o4 : Quotie
│ │ │ │ -00029ff0: 6e74 5269 6e67 2020 2020 2020 2020 2020  ntRing          
│ │ │ │ -0002a000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a010: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00029fd0: 2020 207c 0a7c 6f34 203a 2051 756f 7469     |.|o4 : Quoti
│ │ │ │ +00029fe0: 656e 7452 696e 6720 2020 2020 2020 2020  entRing         
│ │ │ │ +00029ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a000: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0002a010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a040: 2d2d 2b0a 7c69 3520 3a20 5365 6772 6528  --+.|i5 : Segre(
│ │ │ │ -0002a050: 412c 6964 6561 6c28 775e 322a 792c 772a  A,ideal(w^2*y,w*
│ │ │ │ -0002a060: 795e 3229 2920 2020 2020 2020 2020 2020  y^2))           
│ │ │ │ -0002a070: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0002a030: 2d2d 2d2b 0a7c 6935 203a 2053 6567 7265  ---+.|i5 : Segre
│ │ │ │ +0002a040: 2841 2c69 6465 616c 2877 5e32 2a79 2c77  (A,ideal(w^2*y,w
│ │ │ │ +0002a050: 2a79 5e32 2929 2020 2020 2020 2020 2020  *y^2))          
│ │ │ │ +0002a060: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0002a070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a0a0: 2020 7c0a 7c20 2020 2020 2020 2020 3220    |.|         2 
│ │ │ │ +0002a090: 2020 207c 0a7c 2020 2020 2020 2020 2032     |.|         2
│ │ │ │ +0002a0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a0d0: 2020 7c0a 7c6f 3520 3d20 2d20 3368 2020    |.|o5 = - 3h  
│ │ │ │ -0002a0e0: 2b20 3268 2020 2020 2020 2020 2020 2020  + 2h            
│ │ │ │ -0002a0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a100: 2020 7c0a 7c20 2020 2020 2020 2020 3120    |.|         1 
│ │ │ │ -0002a110: 2020 2020 3120 2020 2020 2020 2020 2020      1           
│ │ │ │ -0002a120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a130: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0002a0c0: 2020 207c 0a7c 6f35 203d 202d 2033 6820     |.|o5 = - 3h 
│ │ │ │ +0002a0d0: 202b 2032 6820 2020 2020 2020 2020 2020   + 2h           
│ │ │ │ +0002a0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a0f0: 2020 207c 0a7c 2020 2020 2020 2020 2031     |.|         1
│ │ │ │ +0002a100: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ +0002a110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a120: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0002a130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a160: 2020 7c0a 7c6f 3520 3a20 4120 2020 2020    |.|o5 : A     
│ │ │ │ +0002a150: 2020 207c 0a7c 6f35 203a 2041 2020 2020     |.|o5 : A    
│ │ │ │ +0002a160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a190: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0002a180: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0002a190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a1c0: 2d2d 2b0a 0a4e 6f77 2063 6f6e 7369 6465  --+..Now conside
│ │ │ │ -0002a1d0: 7220 616e 2065 7861 6d70 6c65 2069 6e20  r an example in 
│ │ │ │ -0002a1e0: 5c50 505e 3220 5c74 696d 6573 205c 5050  \PP^2 \times \PP
│ │ │ │ -0002a1f0: 5e32 2c20 6966 2077 6520 696e 7075 7420  ^2, if we input 
│ │ │ │ -0002a200: 7468 6520 4368 6f77 2072 696e 6720 4120  the Chow ring A 
│ │ │ │ -0002a210: 7468 650a 6f75 7470 7574 2077 696c 6c20  the.output will 
│ │ │ │ -0002a220: 6265 2072 6574 7572 6e65 6420 696e 2074  be returned in t
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│ │ │ │ -0002a270: 7468 6520 4368 6f77 2072 696e 6720 7573  the Chow ring us
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│ │ │ │ -0002a2a0: 2c20 636f 6d6d 616e 642e 2057 650a 6d61  , command. We.ma
│ │ │ │ -0002a2b0: 7920 616c 736f 2072 6574 7572 6e20 6120  y also return a 
│ │ │ │ -0002a2c0: 4d75 7461 626c 6548 6173 6854 6162 6c65  MutableHashTable
│ │ │ │ -0002a2d0: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...+------------
│ │ │ │ +0002a1b0: 2d2d 2d2b 0a0a 4e6f 7720 636f 6e73 6964  ---+..Now consid
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│ │ │ │ +0002a1e0: 505e 322c 2069 6620 7765 2069 6e70 7574  P^2, if we input
│ │ │ │ +0002a1f0: 2074 6865 2043 686f 7720 7269 6e67 2041   the Chow ring A
│ │ │ │ +0002a200: 2074 6865 0a6f 7574 7075 7420 7769 6c6c   the.output will
│ │ │ │ +0002a210: 2062 6520 7265 7475 726e 6564 2069 6e20   be returned in 
│ │ │ │ +0002a220: 7468 6520 7361 6d65 2072 696e 672e 2054  the same ring. T
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│ │ │ │ +0002a240: 6675 6e63 7469 6f6e 206f 6620 7468 650a  function of the.
│ │ │ │ +0002a250: 6d65 7468 6f64 7320 7765 2062 7569 6c64  methods we build
│ │ │ │ +0002a260: 2074 6865 2043 686f 7720 7269 6e67 2075   the Chow ring u
│ │ │ │ +0002a270: 7369 6e67 2074 6865 202a 6e6f 7465 2043  sing the *note C
│ │ │ │ +0002a280: 686f 7752 696e 673a 2043 686f 7752 696e  howRing: ChowRin
│ │ │ │ +0002a290: 672c 2063 6f6d 6d61 6e64 2e20 5765 0a6d  g, command. We.m
│ │ │ │ +0002a2a0: 6179 2061 6c73 6f20 7265 7475 726e 2061  ay also return a
│ │ │ │ +0002a2b0: 204d 7574 6162 6c65 4861 7368 5461 626c   MutableHashTabl
│ │ │ │ +0002a2c0: 652e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  e...+-----------
│ │ │ │ +0002a2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a320: 2d2b 0a7c 6936 203a 2052 3d4d 756c 7469  -+.|i6 : R=Multi
│ │ │ │ -0002a330: 5072 6f6a 436f 6f72 6452 696e 6728 7b32  ProjCoordRing({2
│ │ │ │ -0002a340: 2c32 7d29 2020 2020 2020 2020 2020 2020  ,2})            
│ │ │ │ +0002a310: 2d2d 2b0a 7c69 3620 3a20 523d 4d75 6c74  --+.|i6 : R=Mult
│ │ │ │ +0002a320: 6950 726f 6a43 6f6f 7264 5269 6e67 287b  iProjCoordRing({
│ │ │ │ +0002a330: 322c 327d 2920 2020 2020 2020 2020 2020  2,2})           
│ │ │ │ +0002a340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a370: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002a360: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0002a370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a3c0: 207c 0a7c 6f36 203d 2052 2020 2020 2020   |.|o6 = R      
│ │ │ │ +0002a3b0: 2020 7c0a 7c6f 3620 3d20 5220 2020 2020    |.|o6 = R     
│ │ │ │ +0002a3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a410: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002a400: 2020 7c0a 7c20 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a460: 207c 0a7c 6f36 203a 2050 6f6c 796e 6f6d   |.|o6 : Polynom
│ │ │ │ -0002a470: 6961 6c52 696e 6720 2020 2020 2020 2020  ialRing         
│ │ │ │ +0002a450: 2020 7c0a 7c6f 3620 3a20 506f 6c79 6e6f    |.|o6 : Polyno
│ │ │ │ +0002a460: 6d69 616c 5269 6e67 2020 2020 2020 2020  mialRing        
│ │ │ │ +0002a470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a4b0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0002a4a0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0002a4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a500: 2d2b 0a7c 6937 203a 2072 3d67 656e 7320  -+.|i7 : r=gens 
│ │ │ │ -0002a510: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │ +0002a4f0: 2d2d 2b0a 7c69 3720 3a20 723d 6765 6e73  --+.|i7 : r=gens
│ │ │ │ +0002a500: 2052 2020 2020 2020 2020 2020 2020 2020   R              
│ │ │ │ +0002a510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a550: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002a540: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0002a550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a5a0: 207c 0a7c 6f37 203d 207b 7820 2c20 7820   |.|o7 = {x , x 
│ │ │ │ -0002a5b0: 2c20 7820 2c20 7820 2c20 7820 2c20 7820  , x , x , x , x 
│ │ │ │ -0002a5c0: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +0002a590: 2020 7c0a 7c6f 3720 3d20 7b78 202c 2078    |.|o7 = {x , x
│ │ │ │ +0002a5a0: 202c 2078 202c 2078 202c 2078 202c 2078   , x , x , x , x
│ │ │ │ +0002a5b0: 207d 2020 2020 2020 2020 2020 2020 2020   }              
│ │ │ │ +0002a5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a5f0: 207c 0a7c 2020 2020 2020 2030 2020 2031   |.|       0   1
│ │ │ │ -0002a600: 2020 2032 2020 2033 2020 2034 2020 2035     2   3   4   5
│ │ │ │ +0002a5e0: 2020 7c0a 7c20 2020 2020 2020 3020 2020    |.|       0   
│ │ │ │ +0002a5f0: 3120 2020 3220 2020 3320 2020 3420 2020  1   2   3   4   
│ │ │ │ +0002a600: 3520 2020 2020 2020 2020 2020 2020 2020  5               
│ │ │ │  0002a610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a640: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002a630: 2020 7c0a 7c20 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a690: 207c 0a7c 6f37 203a 204c 6973 7420 2020   |.|o7 : List   
│ │ │ │ +0002a680: 2020 7c0a 7c6f 3720 3a20 4c69 7374 2020    |.|o7 : List  
│ │ │ │ +0002a690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a6e0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0002a6d0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0002a6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a730: 2d2b 0a7c 6938 203a 2041 3d43 686f 7752  -+.|i8 : A=ChowR
│ │ │ │ -0002a740: 696e 6728 5229 2020 2020 2020 2020 2020  ing(R)          
│ │ │ │ +0002a720: 2d2d 2b0a 7c69 3820 3a20 413d 4368 6f77  --+.|i8 : A=Chow
│ │ │ │ +0002a730: 5269 6e67 2852 2920 2020 2020 2020 2020  Ring(R)         
│ │ │ │ +0002a740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a780: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002a770: 2020 7c0a 7c20 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a7d0: 207c 0a7c 6f38 203d 2041 2020 2020 2020   |.|o8 = A      
│ │ │ │ +0002a7c0: 2020 7c0a 7c6f 3820 3d20 4120 2020 2020    |.|o8 = A     
│ │ │ │ +0002a7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a820: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002a810: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0002a820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a870: 207c 0a7c 6f38 203a 2051 756f 7469 656e   |.|o8 : Quotien
│ │ │ │ -0002a880: 7452 696e 6720 2020 2020 2020 2020 2020  tRing           
│ │ │ │ +0002a860: 2020 7c0a 7c6f 3820 3a20 5175 6f74 6965    |.|o8 : Quotie
│ │ │ │ +0002a870: 6e74 5269 6e67 2020 2020 2020 2020 2020  ntRing          
│ │ │ │ +0002a880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a8c0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0002a8b0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0002a8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a910: 2d2b 0a7c 6939 203a 2049 3d69 6465 616c  -+.|i9 : I=ideal
│ │ │ │ -0002a920: 2872 5f30 5e32 2a72 5f33 2d72 5f34 2a72  (r_0^2*r_3-r_4*r
│ │ │ │ -0002a930: 5f31 2a72 5f32 2c72 5f32 5e32 2a72 5f35  _1*r_2,r_2^2*r_5
│ │ │ │ -0002a940: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -0002a950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a960: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002a900: 2d2d 2b0a 7c69 3920 3a20 493d 6964 6561  --+.|i9 : I=idea
│ │ │ │ +0002a910: 6c28 725f 305e 322a 725f 332d 725f 342a  l(r_0^2*r_3-r_4*
│ │ │ │ +0002a920: 725f 312a 725f 322c 725f 325e 322a 725f  r_1*r_2,r_2^2*r_
│ │ │ │ +0002a930: 3529 2020 2020 2020 2020 2020 2020 2020  5)              
│ │ │ │ +0002a940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a950: 2020 7c0a 7c20 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a9b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002a9a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0002a9b0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │  0002a9c0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -0002a9d0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +0002a9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aa00: 207c 0a7c 6f39 203d 2069 6465 616c 2028   |.|o9 = ideal (
│ │ │ │ -0002aa10: 7820 7820 202d 2078 2078 2078 202c 2078  x x  - x x x , x
│ │ │ │ -0002aa20: 2078 2029 2020 2020 2020 2020 2020 2020   x )            
│ │ │ │ +0002a9f0: 2020 7c0a 7c6f 3920 3d20 6964 6561 6c20    |.|o9 = ideal 
│ │ │ │ +0002aa00: 2878 2078 2020 2d20 7820 7820 7820 2c20  (x x  - x x x , 
│ │ │ │ +0002aa10: 7820 7820 2920 2020 2020 2020 2020 2020  x x )           
│ │ │ │ +0002aa20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002aa30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aa40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aa50: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0002aa60: 2030 2033 2020 2020 3120 3220 3420 2020   0 3    1 2 4   
│ │ │ │ -0002aa70: 3220 3520 2020 2020 2020 2020 2020 2020  2 5             
│ │ │ │ +0002aa40: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0002aa50: 2020 3020 3320 2020 2031 2032 2034 2020    0 3    1 2 4  
│ │ │ │ +0002aa60: 2032 2035 2020 2020 2020 2020 2020 2020   2 5            
│ │ │ │ +0002aa70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002aa80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aa90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aaa0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002aa90: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0002aaa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002aab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002aac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002aad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aaf0: 207c 0a7c 6f39 203a 2049 6465 616c 206f   |.|o9 : Ideal o
│ │ │ │ -0002ab00: 6620 5220 2020 2020 2020 2020 2020 2020  f R             
│ │ │ │ +0002aae0: 2020 7c0a 7c6f 3920 3a20 4964 6561 6c20    |.|o9 : Ideal 
│ │ │ │ +0002aaf0: 6f66 2052 2020 2020 2020 2020 2020 2020  of R            
│ │ │ │ +0002ab00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ab10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ab20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ab30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ab40: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0002ab30: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0002ab40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ab50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ab60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ab70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ab80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ab90: 2d2b 0a7c 6931 3020 3a20 5365 6772 6520  -+.|i10 : Segre 
│ │ │ │ -0002aba0: 4920 2020 2020 2020 2020 2020 2020 2020  I               
│ │ │ │ +0002ab80: 2d2d 2b0a 7c69 3130 203a 2053 6567 7265  --+.|i10 : Segre
│ │ │ │ +0002ab90: 2049 2020 2020 2020 2020 2020 2020 2020   I              
│ │ │ │ +0002aba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002abb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002abc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002abd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002abe0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002abd0: 2020 7c0a 7c20 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ac20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ac30: 207c 0a7c 2020 2020 2020 2020 2032 2032   |.|         2 2
│ │ │ │ -0002ac40: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -0002ac50: 2032 2020 2020 2032 2020 2020 2020 2020   2     2        
│ │ │ │ -0002ac60: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -0002ac70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ac80: 207c 0a7c 6f31 3020 3d20 3732 6820 6820   |.|o10 = 72h h 
│ │ │ │ -0002ac90: 202d 2032 3468 2068 2020 2d20 3132 6820   - 24h h  - 12h 
│ │ │ │ -0002aca0: 6820 202b 2034 6820 202b 2034 6820 6820  h  + 4h  + 4h h 
│ │ │ │ -0002acb0: 202b 2068 2020 2020 2020 2020 2020 2020   + h            
│ │ │ │ -0002acc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002acd0: 207c 0a7c 2020 2020 2020 2020 2031 2032   |.|         1 2
│ │ │ │ -0002ace0: 2020 2020 2020 3120 3220 2020 2020 2031        1 2      1
│ │ │ │ -0002acf0: 2032 2020 2020 2031 2020 2020 2031 2032   2     1     1 2
│ │ │ │ -0002ad00: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -0002ad10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ad20: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002ac20: 2020 7c0a 7c20 2020 2020 2020 2020 3220    |.|         2 
│ │ │ │ +0002ac30: 3220 2020 2020 2032 2020 2020 2020 2020  2      2        
│ │ │ │ +0002ac40: 2020 3220 2020 2020 3220 2020 2020 2020    2     2       
│ │ │ │ +0002ac50: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +0002ac60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ac70: 2020 7c0a 7c6f 3130 203d 2037 3268 2068    |.|o10 = 72h h
│ │ │ │ +0002ac80: 2020 2d20 3234 6820 6820 202d 2031 3268    - 24h h  - 12h
│ │ │ │ +0002ac90: 2068 2020 2b20 3468 2020 2b20 3468 2068   h  + 4h  + 4h h
│ │ │ │ +0002aca0: 2020 2b20 6820 2020 2020 2020 2020 2020    + h           
│ │ │ │ +0002acb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002acc0: 2020 7c0a 7c20 2020 2020 2020 2020 3120    |.|         1 
│ │ │ │ +0002acd0: 3220 2020 2020 2031 2032 2020 2020 2020  2      1 2      
│ │ │ │ +0002ace0: 3120 3220 2020 2020 3120 2020 2020 3120  1 2     1     1 
│ │ │ │ +0002acf0: 3220 2020 2032 2020 2020 2020 2020 2020  2    2          
│ │ │ │ +0002ad00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ad10: 2020 7c0a 7c20 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ad60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ad70: 207c 0a7c 2020 2020 2020 5a5a 5b68 202e   |.|      ZZ[h .
│ │ │ │ -0002ad80: 2e68 205d 2020 2020 2020 2020 2020 2020  .h ]            
│ │ │ │ +0002ad60: 2020 7c0a 7c20 2020 2020 205a 5a5b 6820    |.|      ZZ[h 
│ │ │ │ +0002ad70: 2e2e 6820 5d20 2020 2020 2020 2020 2020  ..h ]           
│ │ │ │ +0002ad80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ad90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ada0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002adb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002adc0: 207c 0a7c 2020 2020 2020 2020 2020 3120   |.|          1 
│ │ │ │ -0002add0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +0002adb0: 2020 7c0a 7c20 2020 2020 2020 2020 2031    |.|          1
│ │ │ │ +0002adc0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +0002add0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ade0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002adf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ae00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ae10: 207c 0a7c 6f31 3020 3a20 2d2d 2d2d 2d2d   |.|o10 : ------
│ │ │ │ -0002ae20: 2d2d 2d2d 2020 2020 2020 2020 2020 2020  ----            
│ │ │ │ +0002ae00: 2020 7c0a 7c6f 3130 203a 202d 2d2d 2d2d    |.|o10 : -----
│ │ │ │ +0002ae10: 2d2d 2d2d 2d20 2020 2020 2020 2020 2020  -----           
│ │ │ │ +0002ae20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ae30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ae40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ae50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ae60: 207c 0a7c 2020 2020 2020 2020 2033 2020   |.|         3  
│ │ │ │ -0002ae70: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ +0002ae50: 2020 7c0a 7c20 2020 2020 2020 2020 3320    |.|         3 
│ │ │ │ +0002ae60: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │ +0002ae70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ae80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ae90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aeb0: 207c 0a7c 2020 2020 2020 2028 6820 2c20   |.|       (h , 
│ │ │ │ -0002aec0: 6820 2920 2020 2020 2020 2020 2020 2020  h )             
│ │ │ │ +0002aea0: 2020 7c0a 7c20 2020 2020 2020 2868 202c    |.|       (h ,
│ │ │ │ +0002aeb0: 2068 2029 2020 2020 2020 2020 2020 2020   h )            
│ │ │ │ +0002aec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002aed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002aee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002af00: 207c 0a7c 2020 2020 2020 2020 2031 2020   |.|         1  
│ │ │ │ -0002af10: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +0002aef0: 2020 7c0a 7c20 2020 2020 2020 2020 3120    |.|         1 
│ │ │ │ +0002af00: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +0002af10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002af20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002af30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002af40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002af50: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0002af40: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0002af50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002af60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002af70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002af80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002af90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002afa0: 2d2b 0a7c 6931 3120 3a20 7331 3d53 6567  -+.|i11 : s1=Seg
│ │ │ │ -0002afb0: 7265 2841 2c49 2920 2020 2020 2020 2020  re(A,I)         
│ │ │ │ +0002af90: 2d2d 2b0a 7c69 3131 203a 2073 313d 5365  --+.|i11 : s1=Se
│ │ │ │ +0002afa0: 6772 6528 412c 4929 2020 2020 2020 2020  gre(A,I)        
│ │ │ │ +0002afb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002afc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002afd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002afe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aff0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002afe0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0002aff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b040: 207c 0a7c 2020 2020 2020 2020 2032 2032   |.|         2 2
│ │ │ │ -0002b050: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -0002b060: 2032 2020 2020 2032 2020 2020 2020 2020   2     2        
│ │ │ │ -0002b070: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -0002b080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b090: 207c 0a7c 6f31 3120 3d20 3732 6820 6820   |.|o11 = 72h h 
│ │ │ │ -0002b0a0: 202d 2032 3468 2068 2020 2d20 3132 6820   - 24h h  - 12h 
│ │ │ │ -0002b0b0: 6820 202b 2034 6820 202b 2034 6820 6820  h  + 4h  + 4h h 
│ │ │ │ -0002b0c0: 202b 2068 2020 2020 2020 2020 2020 2020   + h            
│ │ │ │ -0002b0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b0e0: 207c 0a7c 2020 2020 2020 2020 2031 2032   |.|         1 2
│ │ │ │ -0002b0f0: 2020 2020 2020 3120 3220 2020 2020 2031        1 2      1
│ │ │ │ -0002b100: 2032 2020 2020 2031 2020 2020 2031 2032   2     1     1 2
│ │ │ │ -0002b110: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -0002b120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b130: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002b030: 2020 7c0a 7c20 2020 2020 2020 2020 3220    |.|         2 
│ │ │ │ +0002b040: 3220 2020 2020 2032 2020 2020 2020 2020  2      2        
│ │ │ │ +0002b050: 2020 3220 2020 2020 3220 2020 2020 2020    2     2       
│ │ │ │ +0002b060: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +0002b070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b080: 2020 7c0a 7c6f 3131 203d 2037 3268 2068    |.|o11 = 72h h
│ │ │ │ +0002b090: 2020 2d20 3234 6820 6820 202d 2031 3268    - 24h h  - 12h
│ │ │ │ +0002b0a0: 2068 2020 2b20 3468 2020 2b20 3468 2068   h  + 4h  + 4h h
│ │ │ │ +0002b0b0: 2020 2b20 6820 2020 2020 2020 2020 2020    + h           
│ │ │ │ +0002b0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b0d0: 2020 7c0a 7c20 2020 2020 2020 2020 3120    |.|         1 
│ │ │ │ +0002b0e0: 3220 2020 2020 2031 2032 2020 2020 2020  2      1 2      
│ │ │ │ +0002b0f0: 3120 3220 2020 2020 3120 2020 2020 3120  1 2     1     1 
│ │ │ │ +0002b100: 3220 2020 2032 2020 2020 2020 2020 2020  2    2          
│ │ │ │ +0002b110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b120: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0002b130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b180: 207c 0a7c 6f31 3120 3a20 4120 2020 2020   |.|o11 : A     
│ │ │ │ +0002b170: 2020 7c0a 7c6f 3131 203a 2041 2020 2020    |.|o11 : A    
│ │ │ │ +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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b1d0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0002b1c0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0002b1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b220: 2d2b 0a7c 6931 3220 3a20 5365 6748 6173  -+.|i12 : SegHas
│ │ │ │ -0002b230: 683d 5365 6772 6528 412c 492c 4f75 7470  h=Segre(A,I,Outp
│ │ │ │ -0002b240: 7574 3d3e 4861 7368 466f 726d 2920 2020  ut=>HashForm)   
│ │ │ │ +0002b210: 2d2d 2b0a 7c69 3132 203a 2053 6567 4861  --+.|i12 : SegHa
│ │ │ │ +0002b220: 7368 3d53 6567 7265 2841 2c49 2c4f 7574  sh=Segre(A,I,Out
│ │ │ │ +0002b230: 7075 743d 3e48 6173 6846 6f72 6d29 2020  put=>HashForm)  
│ │ │ │ +0002b240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b270: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002b260: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0002b270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b2c0: 207c 0a7c 6f31 3220 3d20 4d75 7461 626c   |.|o12 = Mutabl
│ │ │ │ -0002b2d0: 6548 6173 6854 6162 6c65 7b2e 2e2e 342e  eHashTable{...4.
│ │ │ │ -0002b2e0: 2e2e 7d20 2020 2020 2020 2020 2020 2020  ..}             
│ │ │ │ +0002b2b0: 2020 7c0a 7c6f 3132 203d 204d 7574 6162    |.|o12 = Mutab
│ │ │ │ +0002b2c0: 6c65 4861 7368 5461 626c 657b 2e2e 2e34  leHashTable{...4
│ │ │ │ +0002b2d0: 2e2e 2e7d 2020 2020 2020 2020 2020 2020  ...}            
│ │ │ │ +0002b2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b310: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002b300: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0002b310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b360: 207c 0a7c 6f31 3220 3a20 4d75 7461 626c   |.|o12 : Mutabl
│ │ │ │ -0002b370: 6548 6173 6854 6162 6c65 2020 2020 2020  eHashTable      
│ │ │ │ +0002b350: 2020 7c0a 7c6f 3132 203a 204d 7574 6162    |.|o12 : Mutab
│ │ │ │ +0002b360: 6c65 4861 7368 5461 626c 6520 2020 2020  leHashTable     
│ │ │ │ +0002b370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b3b0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0002b3a0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0002b3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b400: 2d2b 0a7c 6931 3320 3a20 7065 656b 2053  -+.|i13 : peek S
│ │ │ │ -0002b410: 6567 4861 7368 2020 2020 2020 2020 2020  egHash          
│ │ │ │ +0002b3f0: 2d2d 2b0a 7c69 3133 203a 2070 6565 6b20  --+.|i13 : peek 
│ │ │ │ +0002b400: 5365 6748 6173 6820 2020 2020 2020 2020  SegHash         
│ │ │ │ +0002b410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b450: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002b440: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0002b450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b4a0: 207c 0a7c 6f31 3320 3d20 4d75 7461 626c   |.|o13 = Mutabl
│ │ │ │ -0002b4b0: 6548 6173 6854 6162 6c65 7b47 203d 3e20  eHashTable{G => 
│ │ │ │ -0002b4c0: 3268 2020 2b20 6820 202b 2031 2020 2020  2h  + h  + 1    
│ │ │ │ +0002b490: 2020 7c0a 7c6f 3133 203d 204d 7574 6162    |.|o13 = Mutab
│ │ │ │ +0002b4a0: 6c65 4861 7368 5461 626c 657b 4720 3d3e  leHashTable{G =>
│ │ │ │ +0002b4b0: 2032 6820 202b 2068 2020 2b20 3120 2020   2h  + h  + 1   
│ │ │ │ +0002b4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b4f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0002b500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b510: 2020 3120 2020 2032 2020 2020 2020 2020    1    2        
│ │ │ │ +0002b4e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0002b4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b500: 2020 2031 2020 2020 3220 2020 2020 2020     1    2       
│ │ │ │ +0002b510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b540: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0002b550: 2020 2020 2020 2020 2020 2047 6c69 7374             Glist
│ │ │ │ -0002b560: 203d 3e20 7b31 2c20 3268 2020 2b20 6820   => {1, 2h  + h 
│ │ │ │ -0002b570: 2c20 302c 2030 2c20 307d 2020 2020 2020  , 0, 0, 0}      
│ │ │ │ -0002b580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b590: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0002b5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b5b0: 2020 2020 2020 2020 2020 3120 2020 2032            1    2
│ │ │ │ +0002b530: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0002b540: 2020 2020 2020 2020 2020 2020 476c 6973              Glis
│ │ │ │ +0002b550: 7420 3d3e 207b 312c 2032 6820 202b 2068  t => {1, 2h  + h
│ │ │ │ +0002b560: 202c 2030 2c20 302c 2030 7d20 2020 2020   , 0, 0, 0}     
│ │ │ │ +0002b570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b580: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0002b590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b5a0: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ +0002b5b0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  0002b5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b5e0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002b5d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0002b5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b610: 2032 2020 2020 2020 2020 2020 2020 3220   2            2 
│ │ │ │ -0002b620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b630: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0002b640: 2020 2020 2020 2020 2020 2053 6567 7265             Segre
│ │ │ │ -0002b650: 4c69 7374 203d 3e20 7b30 2c20 302c 2034  List => {0, 0, 4
│ │ │ │ -0002b660: 6820 202b 2034 6820 6820 202b 2068 202c  h  + 4h h  + h ,
│ │ │ │ -0002b670: 202d 2020 2020 2020 2020 2020 2020 2020   -              
│ │ │ │ -0002b680: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002b600: 2020 3220 2020 2020 2020 2020 2020 2032    2            2
│ │ │ │ +0002b610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b620: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0002b630: 2020 2020 2020 2020 2020 2020 5365 6772              Segr
│ │ │ │ +0002b640: 654c 6973 7420 3d3e 207b 302c 2030 2c20  eList => {0, 0, 
│ │ │ │ +0002b650: 3468 2020 2b20 3468 2068 2020 2b20 6820  4h  + 4h h  + h 
│ │ │ │ +0002b660: 2c20 2d20 2020 2020 2020 2020 2020 2020  , -             
│ │ │ │ +0002b670: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0002b680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b6b0: 2031 2020 2020 2031 2032 2020 2020 3220   1     1 2    2 
│ │ │ │ -0002b6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b6d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0002b6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b6f0: 2020 2020 2020 2032 2032 2020 2020 2020         2 2      
│ │ │ │ -0002b700: 3220 2020 2020 2020 2020 2032 2020 2020  2          2    
│ │ │ │ -0002b710: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -0002b720: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0002b730: 2020 2020 2020 2020 2020 2053 6567 7265             Segre
│ │ │ │ -0002b740: 203d 3e20 3732 6820 6820 202d 2032 3468   => 72h h  - 24h
│ │ │ │ -0002b750: 2068 2020 2d20 3132 6820 6820 202b 2034   h  - 12h h  + 4
│ │ │ │ -0002b760: 6820 2020 2020 2020 2020 2020 2020 2020  h               
│ │ │ │ -0002b770: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0002b780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b790: 2020 2020 2020 2031 2032 2020 2020 2020         1 2      
│ │ │ │ -0002b7a0: 3120 3220 2020 2020 2031 2032 2020 2020  1 2      1 2    
│ │ │ │ -0002b7b0: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │ -0002b7c0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │ +0002b6a0: 2020 3120 2020 2020 3120 3220 2020 2032    1     1 2    2
│ │ │ │ +0002b6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b6c0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0002b6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b6e0: 2020 2020 2020 2020 3220 3220 2020 2020          2 2     
│ │ │ │ +0002b6f0: 2032 2020 2020 2020 2020 2020 3220 2020   2          2   
│ │ │ │ +0002b700: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +0002b710: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0002b720: 2020 2020 2020 2020 2020 2020 5365 6772              Segr
│ │ │ │ +0002b730: 6520 3d3e 2037 3268 2068 2020 2d20 3234  e => 72h h  - 24
│ │ │ │ +0002b740: 6820 6820 202d 2031 3268 2068 2020 2b20  h h  - 12h h  + 
│ │ │ │ +0002b750: 3468 2020 2020 2020 2020 2020 2020 2020  4h              
│ │ │ │ +0002b760: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0002b770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b780: 2020 2020 2020 2020 3120 3220 2020 2020          1 2     
│ │ │ │ +0002b790: 2031 2032 2020 2020 2020 3120 3220 2020   1 2      1 2   
│ │ │ │ +0002b7a0: 2020 3120 2020 2020 2020 2020 2020 2020    1             
│ │ │ │ +0002b7b0: 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.|-----------
│ │ │ │ +0002b7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0002b810: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ -0002b820: 2020 2020 2020 2020 2020 2020 7d20 2020              }   
│ │ │ │ +0002b800: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020  --|.|           
│ │ │ │ +0002b810: 2020 2020 2020 2020 2020 2020 207d 2020               }  
│ │ │ │ +0002b820: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0002b850: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0002b850: 2020 7c0a 7c20 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b8b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002b8a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0002b8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b900: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002b8f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0002b900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b950: 207c 0a7c 2020 2032 2020 2020 2020 2020   |.|   2        
│ │ │ │ -0002b960: 2020 3220 2020 2020 3220 3220 2020 2020    2     2 2     
│ │ │ │ +0002b940: 2020 7c0a 7c20 2020 3220 2020 2020 2020    |.|   2       
│ │ │ │ +0002b950: 2020 2032 2020 2020 2032 2032 2020 2020     2     2 2    
│ │ │ │ +0002b960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b970: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0002b990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b9a0: 207c 0a7c 3234 6820 6820 202d 2031 3268   |.|24h h  - 12h
│ │ │ │ -0002b9b0: 2068 202c 2037 3268 2068 207d 2020 2020   h , 72h h }    
│ │ │ │ +0002b990: 2020 7c0a 7c32 3468 2068 2020 2d20 3132    |.|24h h  - 12
│ │ │ │ +0002b9a0: 6820 6820 2c20 3732 6820 6820 7d20 2020  h h , 72h h }   
│ │ │ │ +0002b9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b9f0: 207c 0a7c 2020 2031 2032 2020 2020 2020   |.|   1 2      
│ │ │ │ -0002ba00: 3120 3220 2020 2020 3120 3220 2020 2020  1 2     1 2     
│ │ │ │ +0002b9e0: 2020 7c0a 7c20 2020 3120 3220 2020 2020    |.|   1 2     
│ │ │ │ +0002b9f0: 2031 2032 2020 2020 2031 2032 2020 2020   1 2     1 2    
│ │ │ │ +0002ba00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ba10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ba20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ba30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ba40: 207c 0a7c 2020 2020 2020 2020 2020 2032   |.|           2
│ │ │ │ +0002ba30: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0002ba40: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  0002ba50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ba60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ba70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ba80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ba90: 207c 0a7c 2b20 3468 2068 2020 2b20 6820   |.|+ 4h h  + h 
│ │ │ │ +0002ba80: 2020 7c0a 7c2b 2034 6820 6820 202b 2068    |.|+ 4h h  + h
│ │ │ │ +0002ba90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002baa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bae0: 207c 0a7c 2020 2020 3120 3220 2020 2032   |.|    1 2    2
│ │ │ │ +0002bad0: 2020 7c0a 7c20 2020 2031 2032 2020 2020    |.|    1 2    
│ │ │ │ +0002bae0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  0002baf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bb30: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0002bb20: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0002bb30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002bb40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002bb50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002bb60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002bb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002bb80: 2d2b 0a7c 6931 3420 3a20 7331 3d3d 5365  -+.|i14 : s1==Se
│ │ │ │ -0002bb90: 6748 6173 6823 2253 6567 7265 2220 2020  gHash#"Segre"   
│ │ │ │ +0002bb70: 2d2d 2b0a 7c69 3134 203a 2073 313d 3d53  --+.|i14 : s1==S
│ │ │ │ +0002bb80: 6567 4861 7368 2322 5365 6772 6522 2020  egHash#"Segre"  
│ │ │ │ +0002bb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bbd0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002bbc0: 2020 7c0a 7c20 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bc20: 207c 0a7c 6f31 3420 3d20 7472 7565 2020   |.|o14 = true  
│ │ │ │ +0002bc10: 2020 7c0a 7c6f 3134 203d 2074 7275 6520    |.|o14 = true 
│ │ │ │ +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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bc70: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0002bc60: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0002bc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002bc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002bc90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002bca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002bcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002bcc0: 2d2b 0a0a 496e 2074 6865 2063 6173 6520  -+..In the case 
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│ │ │ │ -0002bd20: 7370 6163 6573 2077 6520 6d75 7374 206c  spaces we must l
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│ │ │ │ -0002bd60: 736f 2069 6e70 7574 2074 6865 2074 6f72  so input the tor
│ │ │ │ -0002bd70: 6963 2076 6172 6965 7479 2e20 4966 2074  ic variety. If t
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│ │ │ │ -0002bd90: 2069 7320 6120 7072 6f64 7563 7420 6f66   is a product of
│ │ │ │ -0002bda0: 2070 726f 6a65 6374 6976 650a 7370 6163   projective.spac
│ │ │ │ -0002bdb0: 6520 6974 2069 7320 7265 636f 6d6d 656e  e it is recommen
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│ │ │ │ -0002bdd0: 6f72 6d20 6162 6f76 6520 7261 7468 6572  orm above rather
│ │ │ │ -0002bde0: 2074 6861 6e20 696e 7075 7474 696e 6720   than inputting 
│ │ │ │ -0002bdf0: 7468 6520 746f 7269 630a 7661 7269 6574  the toric.variet
│ │ │ │ -0002be00: 7920 666f 7220 6566 6669 6369 656e 6379  y for efficiency
│ │ │ │ -0002be10: 2072 6561 736f 6e73 2e0a 0a2b 2d2d 2d2d   reasons...+----
│ │ │ │ +0002bcb0: 2d2d 2b0a 0a49 6e20 7468 6520 6361 7365  --+..In the case
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│ │ │ │ +0002bd00: 6374 0a6f 6620 7072 6f6a 6563 7469 7665  ct.of projective
│ │ │ │ +0002bd10: 2073 7061 6365 7320 7765 206d 7573 7420   spaces we must 
│ │ │ │ +0002bd20: 6c6f 6164 2074 6865 204e 6f72 6d61 6c54  load the NormalT
│ │ │ │ +0002bd30: 6f72 6963 5661 7269 6574 6965 7320 7061  oricVarieties pa
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│ │ │ │ +0002bd50: 6c73 6f20 696e 7075 7420 7468 6520 746f  lso input the to
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│ │ │ │ +0002bd70: 7468 6520 746f 7269 6320 7661 7269 6574  the toric variet
│ │ │ │ +0002bd80: 7920 6973 2061 2070 726f 6475 6374 206f  y is a product o
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│ │ │ │ +0002bda0: 6365 2069 7420 6973 2072 6563 6f6d 6d65  ce it is recomme
│ │ │ │ +0002bdb0: 6e64 6564 2074 6f20 7573 6520 7468 6520  nded to use the 
│ │ │ │ +0002bdc0: 666f 726d 2061 626f 7665 2072 6174 6865  form above rathe
│ │ │ │ +0002bdd0: 7220 7468 616e 2069 6e70 7574 7469 6e67  r than inputting
│ │ │ │ +0002bde0: 2074 6865 2074 6f72 6963 0a76 6172 6965   the toric.varie
│ │ │ │ +0002bdf0: 7479 2066 6f72 2065 6666 6963 6965 6e63  ty for efficienc
│ │ │ │ +0002be00: 7920 7265 6173 6f6e 732e 0a0a 2b2d 2d2d  y reasons...+---
│ │ │ │ +0002be10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002be20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002be30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002be40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002be50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002be60: 2d2d 2d2d 2b0a 7c69 3135 203a 206e 6565  ----+.|i15 : nee
│ │ │ │ -0002be70: 6473 5061 636b 6167 6520 224e 6f72 6d61  dsPackage "Norma
│ │ │ │ -0002be80: 6c54 6f72 6963 5661 7269 6574 6965 7322  lToricVarieties"
│ │ │ │ +0002be50: 2d2d 2d2d 2d2b 0a7c 6931 3520 3a20 6e65  -----+.|i15 : ne
│ │ │ │ +0002be60: 6564 7350 6163 6b61 6765 2022 4e6f 726d  edsPackage "Norm
│ │ │ │ +0002be70: 616c 546f 7269 6356 6172 6965 7469 6573  alToricVarieties
│ │ │ │ +0002be80: 2220 2020 2020 2020 2020 2020 2020 2020  "               
│ │ │ │  0002be90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bea0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002beb0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002bea0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0002beb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bef0: 2020 2020 2020 2020 2020 7c0a 7c6f 3135            |.|o15
│ │ │ │ -0002bf00: 203d 204e 6f72 6d61 6c54 6f72 6963 5661   = NormalToricVa
│ │ │ │ -0002bf10: 7269 6574 6965 7320 2020 2020 2020 2020  rieties         
│ │ │ │ +0002bee0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0002bef0: 3520 3d20 4e6f 726d 616c 546f 7269 6356  5 = NormalToricV
│ │ │ │ +0002bf00: 6172 6965 7469 6573 2020 2020 2020 2020  arieties        
│ │ │ │ +0002bf10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bf20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bf30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bf40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002bf30: 2020 2020 2020 7c0a 7c20 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bf80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bf90: 7c0a 7c6f 3135 203a 2050 6163 6b61 6765  |.|o15 : Package
│ │ │ │ +0002bf80: 207c 0a7c 6f31 3520 3a20 5061 636b 6167   |.|o15 : Packag
│ │ │ │ +0002bf90: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │  0002bfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bfc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bfd0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0002bfc0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0002bfd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002bfe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002bff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c020: 2d2d 2d2d 2d2d 2b0a 7c69 3136 203a 2052  ------+.|i16 : R
│ │ │ │ -0002c030: 686f 203d 207b 7b31 2c30 2c30 7d2c 7b30  ho = {{1,0,0},{0
│ │ │ │ -0002c040: 2c31 2c30 7d2c 7b30 2c30 2c31 7d2c 7b2d  ,1,0},{0,0,1},{-
│ │ │ │ -0002c050: 312c 2d31 2c30 7d2c 7b30 2c30 2c2d 317d  1,-1,0},{0,0,-1}
│ │ │ │ -0002c060: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ -0002c070: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002c010: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3620 3a20  -------+.|i16 : 
│ │ │ │ +0002c020: 5268 6f20 3d20 7b7b 312c 302c 307d 2c7b  Rho = {{1,0,0},{
│ │ │ │ +0002c030: 302c 312c 307d 2c7b 302c 302c 317d 2c7b  0,1,0},{0,0,1},{
│ │ │ │ +0002c040: 2d31 2c2d 312c 307d 2c7b 302c 302c 2d31  -1,-1,0},{0,0,-1
│ │ │ │ +0002c050: 7d7d 2020 2020 2020 2020 2020 2020 2020  }}              
│ │ │ │ +0002c060: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0002c070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c0b0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0002c0c0: 3136 203d 207b 7b31 2c20 302c 2030 7d2c  16 = {{1, 0, 0},
│ │ │ │ -0002c0d0: 207b 302c 2031 2c20 307d 2c20 7b30 2c20   {0, 1, 0}, {0, 
│ │ │ │ -0002c0e0: 302c 2031 7d2c 207b 2d31 2c20 2d31 2c20  0, 1}, {-1, -1, 
│ │ │ │ -0002c0f0: 307d 2c20 7b30 2c20 302c 202d 317d 7d20  0}, {0, 0, -1}} 
│ │ │ │ -0002c100: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002c0a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002c0b0: 6f31 3620 3d20 7b7b 312c 2030 2c20 307d  o16 = {{1, 0, 0}
│ │ │ │ +0002c0c0: 2c20 7b30 2c20 312c 2030 7d2c 207b 302c  , {0, 1, 0}, {0,
│ │ │ │ +0002c0d0: 2030 2c20 317d 2c20 7b2d 312c 202d 312c   0, 1}, {-1, -1,
│ │ │ │ +0002c0e0: 2030 7d2c 207b 302c 2030 2c20 2d31 7d7d   0}, {0, 0, -1}}
│ │ │ │ +0002c0f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002c100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c150: 2020 7c0a 7c6f 3136 203a 204c 6973 7420    |.|o16 : List 
│ │ │ │ +0002c140: 2020 207c 0a7c 6f31 3620 3a20 4c69 7374     |.|o16 : List
│ │ │ │ +0002c150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 207c 0a2b               |.+
│ │ │ │ +0002c180: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002c190: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  0002c1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c1e0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3137 203a  --------+.|i17 :
│ │ │ │ -0002c1f0: 2053 6967 6d61 203d 207b 7b30 2c31 2c32   Sigma = {{0,1,2
│ │ │ │ -0002c200: 7d2c 7b31 2c32 2c33 7d2c 7b30 2c32 2c33  },{1,2,3},{0,2,3
│ │ │ │ -0002c210: 7d2c 7b30 2c31 2c34 7d2c 7b31 2c33 2c34  },{0,1,4},{1,3,4
│ │ │ │ -0002c220: 7d2c 7b30 2c33 2c34 7d7d 2020 2020 2020  },{0,3,4}}      
│ │ │ │ -0002c230: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0002c1d0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3720  ---------+.|i17 
│ │ │ │ +0002c1e0: 3a20 5369 676d 6120 3d20 7b7b 302c 312c  : Sigma = {{0,1,
│ │ │ │ +0002c1f0: 327d 2c7b 312c 322c 337d 2c7b 302c 322c  2},{1,2,3},{0,2,
│ │ │ │ +0002c200: 337d 2c7b 302c 312c 347d 2c7b 312c 332c  3},{0,1,4},{1,3,
│ │ │ │ +0002c210: 347d 2c7b 302c 332c 347d 7d20 2020 2020  4},{0,3,4}}     
│ │ │ │ +0002c220: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002c230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c270: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002c280: 7c6f 3137 203d 207b 7b30 2c20 312c 2032  |o17 = {{0, 1, 2
│ │ │ │ -0002c290: 7d2c 207b 312c 2032 2c20 337d 2c20 7b30  }, {1, 2, 3}, {0
│ │ │ │ -0002c2a0: 2c20 322c 2033 7d2c 207b 302c 2031 2c20  , 2, 3}, {0, 1, 
│ │ │ │ -0002c2b0: 347d 2c20 7b31 2c20 332c 2034 7d2c 207b  4}, {1, 3, 4}, {
│ │ │ │ -0002c2c0: 302c 2033 2c20 347d 7d7c 0a7c 2020 2020  0, 3, 4}}|.|    
│ │ │ │ +0002c260: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002c270: 0a7c 6f31 3720 3d20 7b7b 302c 2031 2c20  .|o17 = {{0, 1, 
│ │ │ │ +0002c280: 327d 2c20 7b31 2c20 322c 2033 7d2c 207b  2}, {1, 2, 3}, {
│ │ │ │ +0002c290: 302c 2032 2c20 337d 2c20 7b30 2c20 312c  0, 2, 3}, {0, 1,
│ │ │ │ +0002c2a0: 2034 7d2c 207b 312c 2033 2c20 347d 2c20   4}, {1, 3, 4}, 
│ │ │ │ +0002c2b0: 7b30 2c20 332c 2034 7d7d 7c0a 7c20 2020  {0, 3, 4}}|.|   
│ │ │ │ +0002c2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c310: 2020 2020 7c0a 7c6f 3137 203a 204c 6973      |.|o17 : Lis
│ │ │ │ -0002c320: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │ +0002c300: 2020 2020 207c 0a7c 6f31 3720 3a20 4c69       |.|o17 : Li
│ │ │ │ +0002c310: 7374 2020 2020 2020 2020 2020 2020 2020  st              
│ │ │ │ +0002c320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c350: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002c360: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0002c350: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0002c360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3138  ----------+.|i18
│ │ │ │ -0002c3b0: 203a 2058 203d 206e 6f72 6d61 6c54 6f72   : X = normalTor
│ │ │ │ -0002c3c0: 6963 5661 7269 6574 7928 5268 6f2c 5369  icVariety(Rho,Si
│ │ │ │ -0002c3d0: 676d 612c 436f 6566 6669 6369 656e 7452  gma,CoefficientR
│ │ │ │ -0002c3e0: 696e 6720 3d3e 5a5a 2f33 3237 3439 2920  ing =>ZZ/32749) 
│ │ │ │ -0002c3f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002c390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +0002c3a0: 3820 3a20 5820 3d20 6e6f 726d 616c 546f  8 : X = normalTo
│ │ │ │ +0002c3b0: 7269 6356 6172 6965 7479 2852 686f 2c53  ricVariety(Rho,S
│ │ │ │ +0002c3c0: 6967 6d61 2c43 6f65 6666 6963 6965 6e74  igma,Coefficient
│ │ │ │ +0002c3d0: 5269 6e67 203d 3e5a 5a2f 3332 3734 3929  Ring =>ZZ/32749)
│ │ │ │ +0002c3e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0002c3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c440: 7c0a 7c6f 3138 203d 2058 2020 2020 2020  |.|o18 = X      
│ │ │ │ +0002c430: 207c 0a7c 6f31 3820 3d20 5820 2020 2020   |.|o18 = X     
│ │ │ │ +0002c440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c480: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0002c470: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c4d0: 2020 2020 2020 7c0a 7c6f 3138 203a 204e        |.|o18 : N
│ │ │ │ -0002c4e0: 6f72 6d61 6c54 6f72 6963 5661 7269 6574  ormalToricVariet
│ │ │ │ -0002c4f0: 7920 2020 2020 2020 2020 2020 2020 2020  y               
│ │ │ │ +0002c4c0: 2020 2020 2020 207c 0a7c 6f31 3820 3a20         |.|o18 : 
│ │ │ │ +0002c4d0: 4e6f 726d 616c 546f 7269 6356 6172 6965  NormalToricVarie
│ │ │ │ +0002c4e0: 7479 2020 2020 2020 2020 2020 2020 2020  ty              
│ │ │ │ +0002c4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c520: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0002c510: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0002c520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0002c570: 3139 203a 2043 6865 636b 546f 7269 6356  19 : CheckToricV
│ │ │ │ -0002c580: 6172 6965 7479 5661 6c69 6428 5829 2020  arietyValid(X)  
│ │ │ │ +0002c550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +0002c560: 6931 3920 3a20 4368 6563 6b54 6f72 6963  i19 : CheckToric
│ │ │ │ +0002c570: 5661 7269 6574 7956 616c 6964 2858 2920  VarietyValid(X) 
│ │ │ │ +0002c580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c5b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002c5a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002c5b0: 2020 2020 2020 2020 2020 2020 2020 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 7c0a 7c6f 3139 203d 2074 7275 6520    |.|o19 = true 
│ │ │ │ +0002c5f0: 2020 207c 0a7c 6f31 3920 3d20 7472 7565     |.|o19 = true
│ │ │ │ +0002c600: 2020 2020 2020 2020 2020 2020 2020 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 207c 0a2b               |.+
│ │ │ │ +0002c630: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002c640: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  0002c650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c690: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3230 203a  --------+.|i20 :
│ │ │ │ -0002c6a0: 2052 3d72 696e 6728 5829 2020 2020 2020   R=ring(X)      
│ │ │ │ +0002c680: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3020  ---------+.|i20 
│ │ │ │ +0002c690: 3a20 523d 7269 6e67 2858 2920 2020 2020  : R=ring(X)     
│ │ │ │ +0002c6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c6e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0002c6d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002c6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c720: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002c730: 7c6f 3230 203d 2052 2020 2020 2020 2020  |o20 = R        
│ │ │ │ +0002c710: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002c720: 0a7c 6f32 3020 3d20 5220 2020 2020 2020  .|o20 = R       
│ │ │ │ +0002c730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c770: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0002c760: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002c770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c7c0: 2020 2020 7c0a 7c6f 3230 203a 2050 6f6c      |.|o20 : Pol
│ │ │ │ -0002c7d0: 796e 6f6d 6961 6c52 696e 6720 2020 2020  ynomialRing     
│ │ │ │ +0002c7b0: 2020 2020 207c 0a7c 6f32 3020 3a20 506f       |.|o20 : Po
│ │ │ │ +0002c7c0: 6c79 6e6f 6d69 616c 5269 6e67 2020 2020  lynomialRing    
│ │ │ │ +0002c7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c800: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002c810: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0002c800: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0002c810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c850: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3231  ----------+.|i21
│ │ │ │ -0002c860: 203a 2049 3d69 6465 616c 2852 5f30 5e34   : I=ideal(R_0^4
│ │ │ │ -0002c870: 2a52 5f31 2c52 5f30 2a52 5f33 2a52 5f34  *R_1,R_0*R_3*R_4
│ │ │ │ -0002c880: 2a52 5f32 2d52 5f32 5e32 2a52 5f30 5e32  *R_2-R_2^2*R_0^2
│ │ │ │ -0002c890: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -0002c8a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002c840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +0002c850: 3120 3a20 493d 6964 6561 6c28 525f 305e  1 : I=ideal(R_0^
│ │ │ │ +0002c860: 342a 525f 312c 525f 302a 525f 332a 525f  4*R_1,R_0*R_3*R_
│ │ │ │ +0002c870: 342a 525f 322d 525f 325e 322a 525f 305e  4*R_2-R_2^2*R_0^
│ │ │ │ +0002c880: 3229 2020 2020 2020 2020 2020 2020 2020  2)              
│ │ │ │ +0002c890: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0002c8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c8f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0002c900: 2034 2020 2020 2020 2032 2032 2020 2020   4       2 2    
│ │ │ │ +0002c8e0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002c8f0: 2020 3420 2020 2020 2020 3220 3220 2020    4       2 2   
│ │ │ │ +0002c900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c930: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ -0002c940: 3120 3d20 6964 6561 6c20 2878 2078 202c  1 = ideal (x x ,
│ │ │ │ -0002c950: 202d 2078 2078 2020 2b20 7820 7820 7820   - x x  + x x x 
│ │ │ │ -0002c960: 7820 2920 2020 2020 2020 2020 2020 2020  x )             
│ │ │ │ -0002c970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c980: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0002c990: 2020 2020 2020 2030 2031 2020 2020 2030         0 1     0
│ │ │ │ -0002c9a0: 2032 2020 2020 3020 3220 3320 3420 2020   2    0 2 3 4   
│ │ │ │ +0002c920: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0002c930: 3231 203d 2069 6465 616c 2028 7820 7820  21 = ideal (x x 
│ │ │ │ +0002c940: 2c20 2d20 7820 7820 202b 2078 2078 2078  , - x x  + x x x
│ │ │ │ +0002c950: 2078 2029 2020 2020 2020 2020 2020 2020   x )            
│ │ │ │ +0002c960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c970: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002c980: 2020 2020 2020 2020 3020 3120 2020 2020          0 1     
│ │ │ │ +0002c990: 3020 3220 2020 2030 2032 2033 2034 2020  0 2    0 2 3 4  
│ │ │ │ +0002c9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c9d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002c9c0: 2020 7c0a 7c20 2020 2020 2020 2020 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 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0002ca20: 3231 203a 2049 6465 616c 206f 6620 5220  21 : Ideal of R 
│ │ │ │ +0002ca00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002ca10: 6f32 3120 3a20 4964 6561 6c20 6f66 2052  o21 : Ideal of R
│ │ │ │ +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 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002ca50: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0002ca60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ca70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ca80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ca90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002caa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002cab0: 2d2d 2b0a 7c69 3232 203a 2053 6567 7265  --+.|i22 : Segre
│ │ │ │ -0002cac0: 2858 2c49 2920 2020 2020 2020 2020 2020  (X,I)           
│ │ │ │ +0002caa0: 2d2d 2d2b 0a7c 6932 3220 3a20 5365 6772  ---+.|i22 : Segr
│ │ │ │ +0002cab0: 6528 582c 4929 2020 2020 2020 2020 2020  e(X,I)          
│ │ │ │ +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 207c 0a7c               |.|
│ │ │ │ +0002cae0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002caf0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0002cb00: 2020 2020 2020 2020 2020 2020 2020 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 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0002cb50: 2020 2020 2020 3220 2020 2020 2020 3220        2       2 
│ │ │ │ +0002cb30: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0002cb40: 2020 2020 2020 2032 2020 2020 2020 2032         2       2
│ │ │ │ +0002cb50: 2020 2020 2020 2020 2020 2020 2020 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 207c 0a7c 6f32 3220 3d20 2d20 3732     |.|o22 = - 72
│ │ │ │ -0002cba0: 7820 7820 202b 2033 7820 202b 2038 7820  x x  + 3x  + 8x 
│ │ │ │ -0002cbb0: 7820 202b 2078 2020 2020 2020 2020 2020  x  + x          
│ │ │ │ -0002cbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cbd0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002cbe0: 7c20 2020 2020 2020 2020 2020 3320 3420  |           3 4 
│ │ │ │ -0002cbf0: 2020 2020 3320 2020 2020 3320 3420 2020      3     3 4   
│ │ │ │ -0002cc00: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ -0002cc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cc20: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0002cb80: 2020 2020 7c0a 7c6f 3232 203d 202d 2037      |.|o22 = - 7
│ │ │ │ +0002cb90: 3278 2078 2020 2b20 3378 2020 2b20 3878  2x x  + 3x  + 8x
│ │ │ │ +0002cba0: 2078 2020 2b20 7820 2020 2020 2020 2020   x  + x         
│ │ │ │ +0002cbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cbc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002cbd0: 0a7c 2020 2020 2020 2020 2020 2033 2034  .|           3 4
│ │ │ │ +0002cbe0: 2020 2020 2033 2020 2020 2033 2034 2020       3     3 4  
│ │ │ │ +0002cbf0: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │ +0002cc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cc10: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002cc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0002cc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cc70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002cc80: 2020 2020 2020 2020 2020 2020 205a 5a5b               ZZ[
│ │ │ │ -0002cc90: 7820 2e2e 7820 5d20 2020 2020 2020 2020  x ..x ]         
│ │ │ │ +0002cc60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002cc70: 2020 2020 2020 2020 2020 2020 2020 5a5a                ZZ
│ │ │ │ +0002cc80: 5b78 202e 2e78 205d 2020 2020 2020 2020  [x ..x ]        
│ │ │ │ +0002cc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ccb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002ccc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0002ccd0: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ -0002cce0: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ -0002ccf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cd00: 2020 2020 2020 2020 2020 7c0a 7c6f 3232            |.|o22
│ │ │ │ -0002cd10: 203a 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   : -------------
│ │ │ │ -0002cd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002cd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2020 2020  ------------    
│ │ │ │ -0002cd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0002cd60: 2078 202c 2078 2078 2078 202c 2078 2020   x , x x x , x  
│ │ │ │ -0002cd70: 2d20 7820 2c20 7820 202d 2078 202c 2078  - x , x  - x , x
│ │ │ │ -0002cd80: 2020 2d20 7820 2920 2020 2020 2020 2020    - x )         
│ │ │ │ -0002cd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cda0: 7c0a 7c20 2020 2020 2020 2032 2034 2020  |.|        2 4  
│ │ │ │ -0002cdb0: 2030 2031 2033 2020 2030 2020 2020 3320   0 1 3   0    3 
│ │ │ │ -0002cdc0: 2020 3120 2020 2033 2020 2032 2020 2020    1    3   2    
│ │ │ │ -0002cdd0: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ -0002cde0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0002ccb0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0002ccc0: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ +0002ccd0: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ +0002cce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ccf0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +0002cd00: 3220 3a20 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  2 : ------------
│ │ │ │ +0002cd10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002cd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d20 2020  -------------   
│ │ │ │ +0002cd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cd40: 2020 2020 2020 7c0a 7c20 2020 2020 2028        |.|      (
│ │ │ │ +0002cd50: 7820 7820 2c20 7820 7820 7820 2c20 7820  x x , x x x , x 
│ │ │ │ +0002cd60: 202d 2078 202c 2078 2020 2d20 7820 2c20   - x , x  - x , 
│ │ │ │ +0002cd70: 7820 202d 2078 2029 2020 2020 2020 2020  x  - x )        
│ │ │ │ +0002cd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cd90: 207c 0a7c 2020 2020 2020 2020 3220 3420   |.|        2 4 
│ │ │ │ +0002cda0: 2020 3020 3120 3320 2020 3020 2020 2033    0 1 3   0    3
│ │ │ │ +0002cdb0: 2020 2031 2020 2020 3320 2020 3220 2020     1    3   2   
│ │ │ │ +0002cdc0: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ +0002cdd0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0002cde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002cdf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ce00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ce10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ce20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ce30: 2d2d 2d2d 2d2d 2b0a 7c69 3233 203a 2043  ------+.|i23 : C
│ │ │ │ -0002ce40: 683d 546f 7269 6343 686f 7752 696e 6728  h=ToricChowRing(
│ │ │ │ -0002ce50: 5829 2020 2020 2020 2020 2020 2020 2020  X)              
│ │ │ │ +0002ce20: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3320 3a20  -------+.|i23 : 
│ │ │ │ +0002ce30: 4368 3d54 6f72 6963 4368 6f77 5269 6e67  Ch=ToricChowRing
│ │ │ │ +0002ce40: 2858 2920 2020 2020 2020 2020 2020 2020  (X)             
│ │ │ │ +0002ce50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ce60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ce70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ce80: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002ce70: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0002ce80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ce90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ceb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cec0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0002ced0: 3233 203d 2043 6820 2020 2020 2020 2020  23 = Ch         
│ │ │ │ +0002ceb0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002cec0: 6f32 3320 3d20 4368 2020 2020 2020 2020  o23 = Ch        
│ │ │ │ +0002ced0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002cf00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002cf10: 2020 2020 2020 2020 2020 2020 2020 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 7c0a 7c6f 3233 203a 2051 756f 7469    |.|o23 : Quoti
│ │ │ │ -0002cf70: 656e 7452 696e 6720 2020 2020 2020 2020  entRing         
│ │ │ │ +0002cf50: 2020 207c 0a7c 6f32 3320 3a20 5175 6f74     |.|o23 : Quot
│ │ │ │ +0002cf60: 6965 6e74 5269 6e67 2020 2020 2020 2020  ientRing        
│ │ │ │ +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 207c 0a2b               |.+
│ │ │ │ +0002cf90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002cfa0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  0002cfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002cfc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002cfd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002cfe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002cff0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3234 203a  --------+.|i24 :
│ │ │ │ -0002d000: 2073 333d 5365 6772 6528 4368 2c58 2c49   s3=Segre(Ch,X,I
│ │ │ │ -0002d010: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +0002cfe0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3420  ---------+.|i24 
│ │ │ │ +0002cff0: 3a20 7333 3d53 6567 7265 2843 682c 582c  : s3=Segre(Ch,X,
│ │ │ │ +0002d000: 4929 2020 2020 2020 2020 2020 2020 2020  I)              
│ │ │ │ +0002d010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d040: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0002d030: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002d040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a                |.
│ │ │ │ -0002d090: 7c20 2020 2020 2020 2020 2020 3220 2020  |           2   
│ │ │ │ -0002d0a0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +0002d070: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002d080: 0a7c 2020 2020 2020 2020 2020 2032 2020  .|           2  
│ │ │ │ +0002d090: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +0002d0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d0d0: 2020 2020 2020 2020 207c 0a7c 6f32 3420           |.|o24 
│ │ │ │ -0002d0e0: 3d20 2d20 3732 7820 7820 202b 2033 7820  = - 72x x  + 3x 
│ │ │ │ -0002d0f0: 202b 2038 7820 7820 202b 2078 2020 2020   + 8x x  + x    
│ │ │ │ +0002d0c0: 2020 2020 2020 2020 2020 7c0a 7c6f 3234            |.|o24
│ │ │ │ +0002d0d0: 203d 202d 2037 3278 2078 2020 2b20 3378   = - 72x x  + 3x
│ │ │ │ +0002d0e0: 2020 2b20 3878 2078 2020 2b20 7820 2020    + 8x x  + x   
│ │ │ │ +0002d0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0002d110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d120: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002d130: 2020 3320 3420 2020 2020 3320 2020 2020    3 4     3     
│ │ │ │ -0002d140: 3320 3420 2020 2033 2020 2020 2020 2020  3 4    3        
│ │ │ │ +0002d110: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002d120: 2020 2033 2034 2020 2020 2033 2020 2020     3 4     3    
│ │ │ │ +0002d130: 2033 2034 2020 2020 3320 2020 2020 2020   3 4    3       
│ │ │ │ +0002d140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d160: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002d170: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002d160: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0002d170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d1b0: 2020 2020 2020 2020 2020 7c0a 7c6f 3234            |.|o24
│ │ │ │ -0002d1c0: 203a 2043 6820 2020 2020 2020 2020 2020   : Ch           
│ │ │ │ +0002d1a0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +0002d1b0: 3420 3a20 4368 2020 2020 2020 2020 2020  4 : Ch          
│ │ │ │ +0002d1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d200: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0002d1f0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0002d200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002d240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002d250: 2b0a 0a41 6c6c 2074 6865 2065 7861 6d70  +..All the examp
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│ │ │ │ -0002d310: 2042 6572 7469 6e69 2069 7320 202a 6e6f   Bertini is  *no
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│ │ │ │ -0002d3f0: 7769 7468 2061 2073 6d61 6c6c 2062 7574  with a small but
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│ │ │ │ -0002d450: 6963 2061 6c67 6f72 6974 686d 2c2e 0a0a  ic algorithm,...
│ │ │ │ -0002d460: 5761 7973 2074 6f20 7573 6520 5365 6772  Ways to use Segr
│ │ │ │ -0002d470: 653a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  e:.=============
│ │ │ │ -0002d480: 3d3d 3d3d 3d0a 0a20 202a 2022 5365 6772  =====..  * "Segr
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│ │ │ │ +0002d280: 4772 5c22 6f62 6e65 7220 6261 7365 732e  Gr\"obner bases.
│ │ │ │ +0002d290: 0a43 6861 6e67 696e 6720 7468 6520 6f70  .Changing the op
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│ │ │ │ +0002d2c0: 642c 2074 6f20 6265 7274 696e 6920 7769  d, to bertini wi
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│ │ │ │ +0002d300: 6420 4265 7274 696e 6920 6973 2020 2a6e  d Bertini is  *n
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│ │ │ │ +0002d360: 2069 7320 6f6e 6c79 2061 7661 696c 6162   is only availab
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│ │ │ │ +0002d3b0: 6261 6269 6c69 7374 6963 2061 6c67 6f72  babilistic algor
│ │ │ │ +0002d3c0: 6974 686d 2061 6e64 206d 6179 2067 6976  ithm and may giv
│ │ │ │ +0002d3d0: 6520 6120 7772 6f6e 670a 616e 7377 6572  e a wrong.answer
│ │ │ │ +0002d3e0: 2077 6974 6820 6120 736d 616c 6c20 6275   with a small bu
│ │ │ │ +0002d3f0: 7420 6e6f 6e7a 6572 6f20 7072 6f62 6162  t nonzero probab
│ │ │ │ +0002d400: 696c 6974 792e 2052 6561 6420 6d6f 7265  ility. Read more
│ │ │ │ +0002d410: 2075 6e64 6572 202a 6e6f 7465 0a70 726f   under *note.pro
│ │ │ │ +0002d420: 6261 6269 6c69 7374 6963 2061 6c67 6f72  babilistic algor
│ │ │ │ +0002d430: 6974 686d 3a20 7072 6f62 6162 696c 6973  ithm: probabilis
│ │ │ │ +0002d440: 7469 6320 616c 676f 7269 7468 6d2c 2e0a  tic algorithm,..
│ │ │ │ +0002d450: 0a57 6179 7320 746f 2075 7365 2053 6567  .Ways to use Seg
│ │ │ │ +0002d460: 7265 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  re:.============
│ │ │ │ +0002d470: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2253 6567  ======..  * "Seg
│ │ │ │ +0002d480: 7265 2849 6465 616c 2922 0a20 202a 2022  re(Ideal)".  * "
│ │ │ │ +0002d490: 5365 6772 6528 4964 6561 6c2c 5379 6d62  Segre(Ideal,Symb
│ │ │ │ +0002d4a0: 6f6c 2922 0a20 202a 2022 5365 6772 6528  ol)".  * "Segre(
│ │ │ │ +0002d4b0: 5175 6f74 6965 6e74 5269 6e67 2c49 6465  QuotientRing,Ide
│ │ │ │ +0002d4c0: 616c 2922 0a0a 466f 7220 7468 6520 7072  al)"..For the pr
│ │ │ │ +0002d4d0: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d  ogrammer.=======
│ │ │ │ +0002d4e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865  ===========..The
│ │ │ │ +0002d4f0: 206f 626a 6563 7420 2a6e 6f74 6520 5365   object *note Se
│ │ │ │ +0002d500: 6772 653a 2053 6567 7265 2c20 6973 2061  gre: Segre, is a
│ │ │ │ +0002d510: 202a 6e6f 7465 206d 6574 686f 6420 6675   *note method fu
│ │ │ │ +0002d520: 6e63 7469 6f6e 2077 6974 6820 6f70 7469  nction with opti
│ │ │ │ +0002d530: 6f6e 733a 0a28 4d61 6361 756c 6179 3244  ons:.(Macaulay2D
│ │ │ │ +0002d540: 6f63 294d 6574 686f 6446 756e 6374 696f  oc)MethodFunctio
│ │ │ │ +0002d550: 6e57 6974 684f 7074 696f 6e73 2c2e 0a0a  nWithOptions,...
│ │ │ │ +0002d560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002d5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002d5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a  --------------..
│ │ │ │ -0002d5c0: 5468 6520 736f 7572 6365 206f 6620 7468  The source of th
│ │ │ │ -0002d5d0: 6973 2064 6f63 756d 656e 7420 6973 2069  is document is i
│ │ │ │ -0002d5e0: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475  n./build/reprodu
│ │ │ │ -0002d5f0: 6369 626c 652d 7061 7468 2f6d 6163 6175  cible-path/macau
│ │ │ │ -0002d600: 6c61 7932 2d31 2e32 352e 3131 2b64 732f  lay2-1.25.11+ds/
│ │ │ │ -0002d610: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163  M2/Macaulay2/pac
│ │ │ │ -0002d620: 6b61 6765 732f 0a43 6861 7261 6374 6572  kages/.Character
│ │ │ │ -0002d630: 6973 7469 6343 6c61 7373 6573 2e6d 323a  isticClasses.m2:
│ │ │ │ -0002d640: 3137 3633 3a30 2e0a 1f0a 4669 6c65 3a20  1763:0....File: 
│ │ │ │ -0002d650: 4368 6172 6163 7465 7269 7374 6963 436c  CharacteristicCl
│ │ │ │ -0002d660: 6173 7365 732e 696e 666f 2c20 4e6f 6465  asses.info, Node
│ │ │ │ -0002d670: 3a20 546f 7269 6343 686f 7752 696e 672c  : ToricChowRing,
│ │ │ │ -0002d680: 2050 7265 763a 2053 6567 7265 2c20 5570   Prev: Segre, Up
│ │ │ │ -0002d690: 3a20 546f 700a 0a54 6f72 6963 4368 6f77  : Top..ToricChow
│ │ │ │ -0002d6a0: 5269 6e67 202d 2d20 436f 6d70 7574 6573  Ring -- Computes
│ │ │ │ -0002d6b0: 2074 6865 2043 686f 7720 7269 6e67 206f   the Chow ring o
│ │ │ │ -0002d6c0: 6620 6120 6e6f 726d 616c 2074 6f72 6963  f a normal toric
│ │ │ │ -0002d6d0: 2076 6172 6965 7479 0a2a 2a2a 2a2a 2a2a   variety.*******
│ │ │ │ +0002d5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a  ---------------.
│ │ │ │ +0002d5b0: 0a54 6865 2073 6f75 7263 6520 6f66 2074  .The source of t
│ │ │ │ +0002d5c0: 6869 7320 646f 6375 6d65 6e74 2069 7320  his document is 
│ │ │ │ +0002d5d0: 696e 0a2f 6275 696c 642f 7265 7072 6f64  in./build/reprod
│ │ │ │ +0002d5e0: 7563 6962 6c65 2d70 6174 682f 6d61 6361  ucible-path/maca
│ │ │ │ +0002d5f0: 756c 6179 322d 312e 3235 2e31 312b 6473  ulay2-1.25.11+ds
│ │ │ │ +0002d600: 2f4d 322f 4d61 6361 756c 6179 322f 7061  /M2/Macaulay2/pa
│ │ │ │ +0002d610: 636b 6167 6573 2f0a 4368 6172 6163 7465  ckages/.Characte
│ │ │ │ +0002d620: 7269 7374 6963 436c 6173 7365 732e 6d32  risticClasses.m2
│ │ │ │ +0002d630: 3a31 3736 333a 302e 0a1f 0a46 696c 653a  :1763:0....File:
│ │ │ │ +0002d640: 2043 6861 7261 6374 6572 6973 7469 6343   CharacteristicC
│ │ │ │ +0002d650: 6c61 7373 6573 2e69 6e66 6f2c 204e 6f64  lasses.info, Nod
│ │ │ │ +0002d660: 653a 2054 6f72 6963 4368 6f77 5269 6e67  e: ToricChowRing
│ │ │ │ +0002d670: 2c20 5072 6576 3a20 5365 6772 652c 2055  , Prev: Segre, U
│ │ │ │ +0002d680: 703a 2054 6f70 0a0a 546f 7269 6343 686f  p: Top..ToricCho
│ │ │ │ +0002d690: 7752 696e 6720 2d2d 2043 6f6d 7075 7465  wRing -- Compute
│ │ │ │ +0002d6a0: 7320 7468 6520 4368 6f77 2072 696e 6720  s the Chow ring 
│ │ │ │ +0002d6b0: 6f66 2061 206e 6f72 6d61 6c20 746f 7269  of a normal tori
│ │ │ │ +0002d6c0: 6320 7661 7269 6574 790a 2a2a 2a2a 2a2a  c variety.******
│ │ │ │ +0002d6d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002d6e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002d6f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002d700: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002d710: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ -0002d720: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ -0002d730: 546f 7269 6343 686f 7752 696e 6720 580a  ToricChowRing X.
│ │ │ │ -0002d740: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ -0002d750: 2020 2a20 522c 2061 202a 6e6f 7465 206e    * R, a *note n
│ │ │ │ -0002d760: 6f72 6d61 6c20 746f 7269 6320 7661 7269  ormal toric vari
│ │ │ │ -0002d770: 6574 793a 0a20 2020 2020 2020 2028 4e6f  ety:.        (No
│ │ │ │ -0002d780: 726d 616c 546f 7269 6356 6172 6965 7469  rmalToricVarieti
│ │ │ │ -0002d790: 6573 294e 6f72 6d61 6c54 6f72 6963 5661  es)NormalToricVa
│ │ │ │ -0002d7a0: 7269 6574 792c 2c20 4120 6e6f 726d 616c  riety,, A normal
│ │ │ │ -0002d7b0: 2074 6f72 6963 2076 6172 6965 7479 0a20   toric variety. 
│ │ │ │ -0002d7c0: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ -0002d7d0: 2020 2a20 6120 2a6e 6f74 6520 7175 6f74    * a *note quot
│ │ │ │ -0002d7e0: 6965 6e74 2072 696e 673a 2028 4d61 6361  ient ring: (Maca
│ │ │ │ -0002d7f0: 756c 6179 3244 6f63 2951 756f 7469 656e  ulay2Doc)Quotien
│ │ │ │ -0002d800: 7452 696e 672c 2c20 0a0a 4465 7363 7269  tRing,, ..Descri
│ │ │ │ -0002d810: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ -0002d820: 3d0a 0a4c 6574 2058 2062 6520 6120 746f  =..Let X be a to
│ │ │ │ -0002d830: 7269 6320 7661 7269 6574 7920 7769 7468  ric variety with
│ │ │ │ -0002d840: 2074 6f74 616c 2063 6f6f 7264 696e 6174   total coordinat
│ │ │ │ -0002d850: 6520 7269 6e67 2028 436f 7820 7269 6e67  e ring (Cox ring
│ │ │ │ -0002d860: 2920 522e 2054 6869 7320 6d65 7468 6f64  ) R. This method
│ │ │ │ -0002d870: 0a63 6f6d 7075 7465 7320 7468 6520 4368  .computes the Ch
│ │ │ │ -0002d880: 6f77 2072 696e 6720 2043 686f 7720 7269  ow ring  Chow ri
│ │ │ │ -0002d890: 6e67 2043 683d 522f 2853 522b 4c52 2920  ng Ch=R/(SR+LR) 
│ │ │ │ -0002d8a0: 6f66 2058 3b20 6865 7265 2053 5220 6973  of X; here SR is
│ │ │ │ -0002d8b0: 2074 6865 0a53 7461 6e6c 6579 2d52 6569   the.Stanley-Rei
│ │ │ │ -0002d8c0: 736e 6572 2069 6465 616c 206f 6620 7468  sner ideal of th
│ │ │ │ -0002d8d0: 6520 636f 7272 6573 706f 6e64 696e 6720  e corresponding 
│ │ │ │ -0002d8e0: 6661 6e20 616e 6420 4c52 2069 7320 7468  fan and LR is th
│ │ │ │ -0002d8f0: 6520 6964 6561 6c20 6f66 206c 696e 6561  e ideal of linea
│ │ │ │ -0002d900: 720a 7265 6c61 7469 6f6e 7320 616d 6f75  r.relations amou
│ │ │ │ -0002d910: 6e74 2074 6865 2072 6179 732e 2049 7420  nt the rays. It 
│ │ │ │ -0002d920: 6973 206e 6565 6465 6420 666f 7220 696e  is needed for in
│ │ │ │ -0002d930: 7075 7420 696e 746f 2074 6865 206d 6574  put into the met
│ │ │ │ -0002d940: 686f 6473 202a 6e6f 7465 2053 6567 7265  hods *note Segre
│ │ │ │ -0002d950: 3a0a 5365 6772 652c 2c20 2a6e 6f74 6520  :.Segre,, *note 
│ │ │ │ -0002d960: 4368 6572 6e3a 2043 6865 726e 2c20 616e  Chern: Chern, an
│ │ │ │ -0002d970: 6420 2a6e 6f74 6520 4353 4d3a 2043 534d  d *note CSM: CSM
│ │ │ │ -0002d980: 2c20 696e 2074 6865 2063 6173 6573 2077  , in the cases w
│ │ │ │ -0002d990: 6865 7265 2061 2074 6f72 6963 0a76 6172  here a toric.var
│ │ │ │ -0002d9a0: 6965 7479 2069 7320 616c 736f 2069 6e70  iety is also inp
│ │ │ │ -0002d9b0: 7574 2074 6f20 656e 7375 7265 2074 6861  ut to ensure tha
│ │ │ │ -0002d9c0: 7420 7468 6573 6520 6d65 7468 6f64 7320  t these methods 
│ │ │ │ -0002d9d0: 7265 7475 726e 2072 6573 756c 7473 2069  return results i
│ │ │ │ -0002d9e0: 6e20 7468 6520 7361 6d65 0a72 696e 672e  n the same.ring.
│ │ │ │ -0002d9f0: 2057 6520 6769 7665 2061 6e20 6578 616d   We give an exam
│ │ │ │ -0002da00: 706c 6520 6f66 2074 6865 2075 7365 206f  ple of the use o
│ │ │ │ -0002da10: 6620 7468 6973 206d 6574 686f 6420 746f  f this method to
│ │ │ │ -0002da20: 2077 6f72 6b20 7769 7468 2065 6c65 6d65   work with eleme
│ │ │ │ -0002da30: 6e74 7320 6f66 2074 6865 0a43 686f 7720  nts of the.Chow 
│ │ │ │ -0002da40: 7269 6e67 206f 6620 6120 746f 7269 6320  ring of a toric 
│ │ │ │ -0002da50: 7661 7269 6574 790a 0a2b 2d2d 2d2d 2d2d  variety..+------
│ │ │ │ +0002d700: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a  ***********..  *
│ │ │ │ +0002d710: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ +0002d720: 2054 6f72 6963 4368 6f77 5269 6e67 2058   ToricChowRing X
│ │ │ │ +0002d730: 0a20 202a 2049 6e70 7574 733a 0a20 2020  .  * Inputs:.   
│ │ │ │ +0002d740: 2020 202a 2052 2c20 6120 2a6e 6f74 6520     * R, a *note 
│ │ │ │ +0002d750: 6e6f 726d 616c 2074 6f72 6963 2076 6172  normal toric var
│ │ │ │ +0002d760: 6965 7479 3a0a 2020 2020 2020 2020 284e  iety:.        (N
│ │ │ │ +0002d770: 6f72 6d61 6c54 6f72 6963 5661 7269 6574  ormalToricVariet
│ │ │ │ +0002d780: 6965 7329 4e6f 726d 616c 546f 7269 6356  ies)NormalToricV
│ │ │ │ +0002d790: 6172 6965 7479 2c2c 2041 206e 6f72 6d61  ariety,, A norma
│ │ │ │ +0002d7a0: 6c20 746f 7269 6320 7661 7269 6574 790a  l toric variety.
│ │ │ │ +0002d7b0: 2020 2a20 4f75 7470 7574 733a 0a20 2020    * Outputs:.   
│ │ │ │ +0002d7c0: 2020 202a 2061 202a 6e6f 7465 2071 756f     * a *note quo
│ │ │ │ +0002d7d0: 7469 656e 7420 7269 6e67 3a20 284d 6163  tient ring: (Mac
│ │ │ │ +0002d7e0: 6175 6c61 7932 446f 6329 5175 6f74 6965  aulay2Doc)Quotie
│ │ │ │ +0002d7f0: 6e74 5269 6e67 2c2c 200a 0a44 6573 6372  ntRing,, ..Descr
│ │ │ │ +0002d800: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d  iption.=========
│ │ │ │ +0002d810: 3d3d 0a0a 4c65 7420 5820 6265 2061 2074  ==..Let X be a t
│ │ │ │ +0002d820: 6f72 6963 2076 6172 6965 7479 2077 6974  oric variety wit
│ │ │ │ +0002d830: 6820 746f 7461 6c20 636f 6f72 6469 6e61  h total coordina
│ │ │ │ +0002d840: 7465 2072 696e 6720 2843 6f78 2072 696e  te ring (Cox rin
│ │ │ │ +0002d850: 6729 2052 2e20 5468 6973 206d 6574 686f  g) R. This metho
│ │ │ │ +0002d860: 640a 636f 6d70 7574 6573 2074 6865 2043  d.computes the C
│ │ │ │ +0002d870: 686f 7720 7269 6e67 2020 4368 6f77 2072  how ring  Chow r
│ │ │ │ +0002d880: 696e 6720 4368 3d52 2f28 5352 2b4c 5229  ing Ch=R/(SR+LR)
│ │ │ │ +0002d890: 206f 6620 583b 2068 6572 6520 5352 2069   of X; here SR i
│ │ │ │ +0002d8a0: 7320 7468 650a 5374 616e 6c65 792d 5265  s the.Stanley-Re
│ │ │ │ +0002d8b0: 6973 6e65 7220 6964 6561 6c20 6f66 2074  isner ideal of t
│ │ │ │ +0002d8c0: 6865 2063 6f72 7265 7370 6f6e 6469 6e67  he corresponding
│ │ │ │ +0002d8d0: 2066 616e 2061 6e64 204c 5220 6973 2074   fan and LR is t
│ │ │ │ +0002d8e0: 6865 2069 6465 616c 206f 6620 6c69 6e65  he ideal of line
│ │ │ │ +0002d8f0: 6172 0a72 656c 6174 696f 6e73 2061 6d6f  ar.relations amo
│ │ │ │ +0002d900: 756e 7420 7468 6520 7261 7973 2e20 4974  unt the rays. It
│ │ │ │ +0002d910: 2069 7320 6e65 6564 6564 2066 6f72 2069   is needed for i
│ │ │ │ +0002d920: 6e70 7574 2069 6e74 6f20 7468 6520 6d65  nput into the me
│ │ │ │ +0002d930: 7468 6f64 7320 2a6e 6f74 6520 5365 6772  thods *note Segr
│ │ │ │ +0002d940: 653a 0a53 6567 7265 2c2c 202a 6e6f 7465  e:.Segre,, *note
│ │ │ │ +0002d950: 2043 6865 726e 3a20 4368 6572 6e2c 2061   Chern: Chern, a
│ │ │ │ +0002d960: 6e64 202a 6e6f 7465 2043 534d 3a20 4353  nd *note CSM: CS
│ │ │ │ +0002d970: 4d2c 2069 6e20 7468 6520 6361 7365 7320  M, in the cases 
│ │ │ │ +0002d980: 7768 6572 6520 6120 746f 7269 630a 7661  where a toric.va
│ │ │ │ +0002d990: 7269 6574 7920 6973 2061 6c73 6f20 696e  riety is also in
│ │ │ │ +0002d9a0: 7075 7420 746f 2065 6e73 7572 6520 7468  put to ensure th
│ │ │ │ +0002d9b0: 6174 2074 6865 7365 206d 6574 686f 6473  at these methods
│ │ │ │ +0002d9c0: 2072 6574 7572 6e20 7265 7375 6c74 7320   return results 
│ │ │ │ +0002d9d0: 696e 2074 6865 2073 616d 650a 7269 6e67  in the same.ring
│ │ │ │ +0002d9e0: 2e20 5765 2067 6976 6520 616e 2065 7861  . We give an exa
│ │ │ │ +0002d9f0: 6d70 6c65 206f 6620 7468 6520 7573 6520  mple of the use 
│ │ │ │ +0002da00: 6f66 2074 6869 7320 6d65 7468 6f64 2074  of this method t
│ │ │ │ +0002da10: 6f20 776f 726b 2077 6974 6820 656c 656d  o work with elem
│ │ │ │ +0002da20: 656e 7473 206f 6620 7468 650a 4368 6f77  ents of the.Chow
│ │ │ │ +0002da30: 2072 696e 6720 6f66 2061 2074 6f72 6963   ring of a toric
│ │ │ │ +0002da40: 2076 6172 6965 7479 0a0a 2b2d 2d2d 2d2d   variety..+-----
│ │ │ │ +0002da50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002da60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002da70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002da80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002da90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002daa0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 206e  -------+.|i1 : n
│ │ │ │ -0002dab0: 6565 6473 5061 636b 6167 6520 224e 6f72  eedsPackage "Nor
│ │ │ │ -0002dac0: 6d61 6c54 6f72 6963 5661 7269 6574 6965  malToricVarietie
│ │ │ │ -0002dad0: 7322 2020 2020 2020 2020 2020 2020 2020  s"              
│ │ │ │ -0002dae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002daf0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002da90: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20  --------+.|i1 : 
│ │ │ │ +0002daa0: 6e65 6564 7350 6163 6b61 6765 2022 4e6f  needsPackage "No
│ │ │ │ +0002dab0: 726d 616c 546f 7269 6356 6172 6965 7469  rmalToricVarieti
│ │ │ │ +0002dac0: 6573 2220 2020 2020 2020 2020 2020 2020  es"             
│ │ │ │ +0002dad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dae0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002daf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002db00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002db10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002db20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002db30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002db40: 2020 2020 2020 207c 0a7c 6f31 203d 204e         |.|o1 = N
│ │ │ │ -0002db50: 6f72 6d61 6c54 6f72 6963 5661 7269 6574  ormalToricVariet
│ │ │ │ -0002db60: 6965 7320 2020 2020 2020 2020 2020 2020  ies             
│ │ │ │ +0002db30: 2020 2020 2020 2020 7c0a 7c6f 3120 3d20          |.|o1 = 
│ │ │ │ +0002db40: 4e6f 726d 616c 546f 7269 6356 6172 6965  NormalToricVarie
│ │ │ │ +0002db50: 7469 6573 2020 2020 2020 2020 2020 2020  ties            
│ │ │ │ +0002db60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002db70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002db80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002db90: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002db80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002db90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dbe0: 2020 2020 2020 207c 0a7c 6f31 203a 2050         |.|o1 : P
│ │ │ │ -0002dbf0: 6163 6b61 6765 2020 2020 2020 2020 2020  ackage          
│ │ │ │ +0002dbd0: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20          |.|o1 : 
│ │ │ │ +0002dbe0: 5061 636b 6167 6520 2020 2020 2020 2020  Package         
│ │ │ │ +0002dbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dc30: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002dc20: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0002dc30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002dc40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002dc50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002dc60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002dc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002dc80: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2052  -------+.|i2 : R
│ │ │ │ -0002dc90: 686f 203d 207b 7b31 2c30 2c30 7d2c 7b30  ho = {{1,0,0},{0
│ │ │ │ -0002dca0: 2c31 2c30 7d2c 7b30 2c30 2c31 7d2c 7b2d  ,1,0},{0,0,1},{-
│ │ │ │ -0002dcb0: 312c 2d31 2c30 7d2c 7b30 2c30 2c2d 317d  1,-1,0},{0,0,-1}
│ │ │ │ -0002dcc0: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ -0002dcd0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002dc70: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20  --------+.|i2 : 
│ │ │ │ +0002dc80: 5268 6f20 3d20 7b7b 312c 302c 307d 2c7b  Rho = {{1,0,0},{
│ │ │ │ +0002dc90: 302c 312c 307d 2c7b 302c 302c 317d 2c7b  0,1,0},{0,0,1},{
│ │ │ │ +0002dca0: 2d31 2c2d 312c 307d 2c7b 302c 302c 2d31  -1,-1,0},{0,0,-1
│ │ │ │ +0002dcb0: 7d7d 2020 2020 2020 2020 2020 2020 2020  }}              
│ │ │ │ +0002dcc0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002dcd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dd20: 2020 2020 2020 207c 0a7c 6f32 203d 207b         |.|o2 = {
│ │ │ │ -0002dd30: 7b31 2c20 302c 2030 7d2c 207b 302c 2031  {1, 0, 0}, {0, 1
│ │ │ │ -0002dd40: 2c20 307d 2c20 7b30 2c20 302c 2031 7d2c  , 0}, {0, 0, 1},
│ │ │ │ -0002dd50: 207b 2d31 2c20 2d31 2c20 307d 2c20 7b30   {-1, -1, 0}, {0
│ │ │ │ -0002dd60: 2c20 302c 202d 317d 7d20 2020 2020 2020  , 0, -1}}       
│ │ │ │ -0002dd70: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002dd10: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20          |.|o2 = 
│ │ │ │ +0002dd20: 7b7b 312c 2030 2c20 307d 2c20 7b30 2c20  {{1, 0, 0}, {0, 
│ │ │ │ +0002dd30: 312c 2030 7d2c 207b 302c 2030 2c20 317d  1, 0}, {0, 0, 1}
│ │ │ │ +0002dd40: 2c20 7b2d 312c 202d 312c 2030 7d2c 207b  , {-1, -1, 0}, {
│ │ │ │ +0002dd50: 302c 2030 2c20 2d31 7d7d 2020 2020 2020  0, 0, -1}}      
│ │ │ │ +0002dd60: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002dd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ddb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ddc0: 2020 2020 2020 207c 0a7c 6f32 203a 204c         |.|o2 : L
│ │ │ │ -0002ddd0: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ +0002ddb0: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20          |.|o2 : 
│ │ │ │ +0002ddc0: 4c69 7374 2020 2020 2020 2020 2020 2020  List            
│ │ │ │ +0002ddd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ddf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002de00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002de10: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002de00: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0002de10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002de20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002de30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002de40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002de50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002de60: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2053  -------+.|i3 : S
│ │ │ │ -0002de70: 6967 6d61 203d 207b 7b30 2c31 2c32 7d2c  igma = {{0,1,2},
│ │ │ │ -0002de80: 7b31 2c32 2c33 7d2c 7b30 2c32 2c33 7d2c  {1,2,3},{0,2,3},
│ │ │ │ -0002de90: 7b30 2c31 2c34 7d2c 7b31 2c33 2c34 7d2c  {0,1,4},{1,3,4},
│ │ │ │ -0002dea0: 7b30 2c33 2c34 7d7d 2020 2020 2020 2020  {0,3,4}}        
│ │ │ │ -0002deb0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002de50: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │ +0002de60: 5369 676d 6120 3d20 7b7b 302c 312c 327d  Sigma = {{0,1,2}
│ │ │ │ +0002de70: 2c7b 312c 322c 337d 2c7b 302c 322c 337d  ,{1,2,3},{0,2,3}
│ │ │ │ +0002de80: 2c7b 302c 312c 347d 2c7b 312c 332c 347d  ,{0,1,4},{1,3,4}
│ │ │ │ +0002de90: 2c7b 302c 332c 347d 7d20 2020 2020 2020  ,{0,3,4}}       
│ │ │ │ +0002dea0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002deb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ded0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002def0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002df00: 2020 2020 2020 207c 0a7c 6f33 203d 207b         |.|o3 = {
│ │ │ │ -0002df10: 7b30 2c20 312c 2032 7d2c 207b 312c 2032  {0, 1, 2}, {1, 2
│ │ │ │ -0002df20: 2c20 337d 2c20 7b30 2c20 322c 2033 7d2c  , 3}, {0, 2, 3},
│ │ │ │ -0002df30: 207b 302c 2031 2c20 347d 2c20 7b31 2c20   {0, 1, 4}, {1, 
│ │ │ │ -0002df40: 332c 2034 7d2c 207b 302c 2033 2c20 347d  3, 4}, {0, 3, 4}
│ │ │ │ -0002df50: 7d20 2020 2020 207c 0a7c 2020 2020 2020  }      |.|      
│ │ │ │ +0002def0: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20          |.|o3 = 
│ │ │ │ +0002df00: 7b7b 302c 2031 2c20 327d 2c20 7b31 2c20  {{0, 1, 2}, {1, 
│ │ │ │ +0002df10: 322c 2033 7d2c 207b 302c 2032 2c20 337d  2, 3}, {0, 2, 3}
│ │ │ │ +0002df20: 2c20 7b30 2c20 312c 2034 7d2c 207b 312c  , {0, 1, 4}, {1,
│ │ │ │ +0002df30: 2033 2c20 347d 2c20 7b30 2c20 332c 2034   3, 4}, {0, 3, 4
│ │ │ │ +0002df40: 7d7d 2020 2020 2020 7c0a 7c20 2020 2020  }}      |.|     
│ │ │ │ +0002df50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002df60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002df70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002df80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002df90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dfa0: 2020 2020 2020 207c 0a7c 6f33 203a 204c         |.|o3 : L
│ │ │ │ -0002dfb0: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ +0002df90: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20          |.|o3 : 
│ │ │ │ +0002dfa0: 4c69 7374 2020 2020 2020 2020 2020 2020  List            
│ │ │ │ +0002dfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dfc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dff0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002dfe0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0002dff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e040: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2058  -------+.|i4 : X
│ │ │ │ -0002e050: 203d 206e 6f72 6d61 6c54 6f72 6963 5661   = normalToricVa
│ │ │ │ -0002e060: 7269 6574 7928 5268 6f2c 5369 676d 612c  riety(Rho,Sigma,
│ │ │ │ -0002e070: 436f 6566 6669 6369 656e 7452 696e 6720  CoefficientRing 
│ │ │ │ -0002e080: 3d3e 5a5a 2f33 3237 3439 2920 2020 2020  =>ZZ/32749)     
│ │ │ │ -0002e090: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002e030: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20  --------+.|i4 : 
│ │ │ │ +0002e040: 5820 3d20 6e6f 726d 616c 546f 7269 6356  X = normalToricV
│ │ │ │ +0002e050: 6172 6965 7479 2852 686f 2c53 6967 6d61  ariety(Rho,Sigma
│ │ │ │ +0002e060: 2c43 6f65 6666 6963 6965 6e74 5269 6e67  ,CoefficientRing
│ │ │ │ +0002e070: 203d 3e5a 5a2f 3332 3734 3929 2020 2020   =>ZZ/32749)    
│ │ │ │ +0002e080: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002e090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e0e0: 2020 2020 2020 207c 0a7c 6f34 203d 2058         |.|o4 = X
│ │ │ │ +0002e0d0: 2020 2020 2020 2020 7c0a 7c6f 3420 3d20          |.|o4 = 
│ │ │ │ +0002e0e0: 5820 2020 2020 2020 2020 2020 2020 2020  X               
│ │ │ │  0002e0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e130: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002e120: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002e130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e180: 2020 2020 2020 207c 0a7c 6f34 203a 204e         |.|o4 : N
│ │ │ │ -0002e190: 6f72 6d61 6c54 6f72 6963 5661 7269 6574  ormalToricVariet
│ │ │ │ -0002e1a0: 7920 2020 2020 2020 2020 2020 2020 2020  y               
│ │ │ │ +0002e170: 2020 2020 2020 2020 7c0a 7c6f 3420 3a20          |.|o4 : 
│ │ │ │ +0002e180: 4e6f 726d 616c 546f 7269 6356 6172 6965  NormalToricVarie
│ │ │ │ +0002e190: 7479 2020 2020 2020 2020 2020 2020 2020  ty              
│ │ │ │ +0002e1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e1d0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002e1c0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0002e1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e220: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2052  -------+.|i5 : R
│ │ │ │ -0002e230: 3d72 696e 6720 5820 2020 2020 2020 2020  =ring X         
│ │ │ │ +0002e210: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20  --------+.|i5 : 
│ │ │ │ +0002e220: 523d 7269 6e67 2058 2020 2020 2020 2020  R=ring X        
│ │ │ │ +0002e230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e270: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002e260: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002e270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 2020 207c 0a7c 6f35 203d 2052         |.|o5 = R
│ │ │ │ +0002e2b0: 2020 2020 2020 2020 7c0a 7c6f 3520 3d20          |.|o5 = 
│ │ │ │ +0002e2c0: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │  0002e2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e310: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002e300: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002e310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e360: 2020 2020 2020 207c 0a7c 6f35 203a 2050         |.|o5 : P
│ │ │ │ -0002e370: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020  olynomialRing   
│ │ │ │ +0002e350: 2020 2020 2020 2020 7c0a 7c6f 3520 3a20          |.|o5 : 
│ │ │ │ +0002e360: 506f 6c79 6e6f 6d69 616c 5269 6e67 2020  PolynomialRing  
│ │ │ │ +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 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002e3a0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0002e3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e400: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 2043  -------+.|i6 : C
│ │ │ │ -0002e410: 683d 546f 7269 6343 686f 7752 696e 6728  h=ToricChowRing(
│ │ │ │ -0002e420: 5829 2020 2020 2020 2020 2020 2020 2020  X)              
│ │ │ │ +0002e3f0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20  --------+.|i6 : 
│ │ │ │ +0002e400: 4368 3d54 6f72 6963 4368 6f77 5269 6e67  Ch=ToricChowRing
│ │ │ │ +0002e410: 2858 2920 2020 2020 2020 2020 2020 2020  (X)             
│ │ │ │ +0002e420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e450: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002e440: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002e450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e4a0: 2020 2020 2020 207c 0a7c 6f36 203d 2043         |.|o6 = C
│ │ │ │ -0002e4b0: 6820 2020 2020 2020 2020 2020 2020 2020  h               
│ │ │ │ +0002e490: 2020 2020 2020 2020 7c0a 7c6f 3620 3d20          |.|o6 = 
│ │ │ │ +0002e4a0: 4368 2020 2020 2020 2020 2020 2020 2020  Ch              
│ │ │ │ +0002e4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e4f0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002e4e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002e4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e540: 2020 2020 2020 207c 0a7c 6f36 203a 2051         |.|o6 : Q
│ │ │ │ -0002e550: 756f 7469 656e 7452 696e 6720 2020 2020  uotientRing     
│ │ │ │ +0002e530: 2020 2020 2020 2020 7c0a 7c6f 3620 3a20          |.|o6 : 
│ │ │ │ +0002e540: 5175 6f74 6965 6e74 5269 6e67 2020 2020  QuotientRing    
│ │ │ │ +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 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002e580: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0002e590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e5e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a 2064  -------+.|i7 : d
│ │ │ │ -0002e5f0: 6573 6372 6962 6520 4368 2020 2020 2020  escribe Ch      
│ │ │ │ +0002e5d0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20  --------+.|i7 : 
│ │ │ │ +0002e5e0: 6465 7363 7269 6265 2043 6820 2020 2020  describe Ch     
│ │ │ │ +0002e5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e630: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002e620: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002e630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e650: 2020 2020 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 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0002e690: 2020 2020 2020 2020 2020 2020 2020 205a                 Z
│ │ │ │ -0002e6a0: 5a5b 7820 2e2e 7820 5d20 2020 2020 2020  Z[x ..x ]       
│ │ │ │ +0002e670: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002e680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e690: 5a5a 5b78 202e 2e78 205d 2020 2020 2020  ZZ[x ..x ]      
│ │ │ │ +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 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0002e6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e6f0: 2020 2030 2020 2034 2020 2020 2020 2020     0   4        
│ │ │ │ +0002e6c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002e6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e6e0: 2020 2020 3020 2020 3420 2020 2020 2020      0   4       
│ │ │ │ +0002e6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e720: 2020 2020 2020 207c 0a7c 6f37 203d 202d         |.|o7 = -
│ │ │ │ +0002e710: 2020 2020 2020 2020 7c0a 7c6f 3720 3d20          |.|o7 = 
│ │ │ │ +0002e720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e750: 2d2d 2d2d 2d2d 2d2d 2020 2020 2020 2020  --------        
│ │ │ │ -0002e760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e770: 2020 2020 2020 207c 0a7c 2020 2020 2028         |.|     (
│ │ │ │ -0002e780: 7820 7820 2c20 7820 7820 7820 2c20 7820  x x , x x x , x 
│ │ │ │ -0002e790: 202d 2078 202c 2078 2020 2d20 7820 2c20   - x , x  - x , 
│ │ │ │ -0002e7a0: 7820 202d 2078 2029 2020 2020 2020 2020  x  - x )        
│ │ │ │ -0002e7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e7c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0002e7d0: 2032 2034 2020 2030 2031 2033 2020 2030   2 4   0 1 3   0
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│ │ │ │ -0002e7f0: 2032 2020 2020 3420 2020 2020 2020 2020   2    4         
│ │ │ │ -0002e800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e810: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002e740: 2d2d 2d2d 2d2d 2d2d 2d20 2020 2020 2020  ---------       
│ │ │ │ +0002e750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e760: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002e770: 2878 2078 202c 2078 2078 2078 202c 2078  (x x , x x x , x
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│ │ │ │ +0002e7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e7b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002e7c0: 2020 3220 3420 2020 3020 3120 3320 2020    2 4   0 1 3   
│ │ │ │ +0002e7d0: 3020 2020 2033 2020 2031 2020 2020 3320  0    3   1    3 
│ │ │ │ +0002e7e0: 2020 3220 2020 2034 2020 2020 2020 2020    2    4        
│ │ │ │ +0002e7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e800: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0002e810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e860: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 2072  -------+.|i8 : r
│ │ │ │ -0002e870: 3d67 656e 7320 5220 2020 2020 2020 2020  =gens R         
│ │ │ │ +0002e850: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20  --------+.|i8 : 
│ │ │ │ +0002e860: 723d 6765 6e73 2052 2020 2020 2020 2020  r=gens R        
│ │ │ │ +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 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002e8a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002e8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e900: 2020 2020 2020 207c 0a7c 6f38 203d 207b         |.|o8 = {
│ │ │ │ -0002e910: 7820 2c20 7820 2c20 7820 2c20 7820 2c20  x , x , x , x , 
│ │ │ │ -0002e920: 7820 7d20 2020 2020 2020 2020 2020 2020  x }             
│ │ │ │ +0002e8f0: 2020 2020 2020 2020 7c0a 7c6f 3820 3d20          |.|o8 = 
│ │ │ │ +0002e900: 7b78 202c 2078 202c 2078 202c 2078 202c  {x , x , x , x ,
│ │ │ │ +0002e910: 2078 207d 2020 2020 2020 2020 2020 2020   x }            
│ │ │ │ +0002e920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e950: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0002e960: 2030 2020 2031 2020 2032 2020 2033 2020   0   1   2   3  
│ │ │ │ -0002e970: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ +0002e940: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002e950: 2020 3020 2020 3120 2020 3220 2020 3320    0   1   2   3 
│ │ │ │ +0002e960: 2020 3420 2020 2020 2020 2020 2020 2020    4             
│ │ │ │ +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 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002e990: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002e9a0: 2020 2020 2020 2020 2020 2020 2020 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 2020 207c 0a7c 6f38 203a 204c         |.|o8 : L
│ │ │ │ -0002ea00: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ +0002e9e0: 2020 2020 2020 2020 7c0a 7c6f 3820 3a20          |.|o8 : 
│ │ │ │ +0002e9f0: 4c69 7374 2020 2020 2020 2020 2020 2020  List            
│ │ │ │ +0002ea00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ea10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ea20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ea30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ea40: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002ea30: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0002ea40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ea50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ea60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ea70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ea80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ea90: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2049  -------+.|i9 : I
│ │ │ │ -0002eaa0: 3d69 6465 616c 2872 616e 646f 6d28 7b31  =ideal(random({1
│ │ │ │ -0002eab0: 2c30 7d2c 5229 2920 2020 2020 2020 2020  ,0},R))         
│ │ │ │ +0002ea80: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3920 3a20  --------+.|i9 : 
│ │ │ │ +0002ea90: 493d 6964 6561 6c28 7261 6e64 6f6d 287b  I=ideal(random({
│ │ │ │ +0002eaa0: 312c 307d 2c52 2929 2020 2020 2020 2020  1,0},R))        
│ │ │ │ +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 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002ead0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002eae0: 2020 2020 2020 2020 2020 2020 2020 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 2020 207c 0a7c 6f39 203d 2069         |.|o9 = i
│ │ │ │ -0002eb40: 6465 616c 2831 3037 7820 202b 2034 3337  deal(107x  + 437
│ │ │ │ -0002eb50: 3678 2020 2d20 3633 3136 7820 2920 2020  6x  - 6316x )   
│ │ │ │ +0002eb20: 2020 2020 2020 2020 7c0a 7c6f 3920 3d20          |.|o9 = 
│ │ │ │ +0002eb30: 6964 6561 6c28 3130 3778 2020 2b20 3433  ideal(107x  + 43
│ │ │ │ +0002eb40: 3736 7820 202d 2036 3331 3678 2029 2020  76x  - 6316x )  
│ │ │ │ +0002eb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002eb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002eb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002eb80: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0002eb90: 2020 2020 2020 2020 2030 2020 2020 2020           0      
│ │ │ │ -0002eba0: 2020 3120 2020 2020 2020 2033 2020 2020    1        3    
│ │ │ │ +0002eb70: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002eb80: 2020 2020 2020 2020 2020 3020 2020 2020            0     
│ │ │ │ +0002eb90: 2020 2031 2020 2020 2020 2020 3320 2020     1        3   
│ │ │ │ +0002eba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ebb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ebc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ebd0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002ebc0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002ebd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 2020 207c 0a7c 6f39 203a 2049         |.|o9 : I
│ │ │ │ -0002ec30: 6465 616c 206f 6620 5220 2020 2020 2020  deal of R       
│ │ │ │ +0002ec10: 2020 2020 2020 2020 7c0a 7c6f 3920 3a20          |.|o9 : 
│ │ │ │ +0002ec20: 4964 6561 6c20 6f66 2052 2020 2020 2020  Ideal of R      
│ │ │ │ +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 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002ec60: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0002ec70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ec80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ec90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002eca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ecb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ecc0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20  -------+.|i10 : 
│ │ │ │ -0002ecd0: 4b3d 6964 6561 6c28 7261 6e64 6f6d 287b  K=ideal(random({
│ │ │ │ -0002ece0: 312c 317d 2c52 2929 2020 2020 2020 2020  1,1},R))        
│ │ │ │ +0002ecb0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3130 203a  --------+.|i10 :
│ │ │ │ +0002ecc0: 204b 3d69 6465 616c 2872 616e 646f 6d28   K=ideal(random(
│ │ │ │ +0002ecd0: 7b31 2c31 7d2c 5229 2920 2020 2020 2020  {1,1},R))       
│ │ │ │ +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 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002ed00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002ed10: 2020 2020 2020 2020 2020 2020 2020 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 2020 207c 0a7c 6f31 3020 3d20         |.|o10 = 
│ │ │ │ -0002ed70: 6964 6561 6c28 3331 3837 7820 7820 202d  ideal(3187x x  -
│ │ │ │ -0002ed80: 2036 3035 3378 2078 2020 2d20 3136 3039   6053x x  - 1609
│ │ │ │ -0002ed90: 3078 2078 2020 2b20 3337 3833 7820 7820  0x x  + 3783x x 
│ │ │ │ -0002eda0: 202b 2038 3537 3078 2078 2020 2b20 3834   + 8570x x  + 84
│ │ │ │ -0002edb0: 3434 7820 7820 297c 0a7c 2020 2020 2020  44x x )|.|      
│ │ │ │ -0002edc0: 2020 2020 2020 2020 2020 2030 2032 2020             0 2  
│ │ │ │ -0002edd0: 2020 2020 2020 3120 3220 2020 2020 2020        1 2       
│ │ │ │ -0002ede0: 2020 3220 3320 2020 2020 2020 2030 2034    2 3        0 4
│ │ │ │ -0002edf0: 2020 2020 2020 2020 3120 3420 2020 2020          1 4     
│ │ │ │ -0002ee00: 2020 2033 2034 207c 0a7c 2020 2020 2020     3 4 |.|      
│ │ │ │ +0002ed50: 2020 2020 2020 2020 7c0a 7c6f 3130 203d          |.|o10 =
│ │ │ │ +0002ed60: 2069 6465 616c 2833 3138 3778 2078 2020   ideal(3187x x  
│ │ │ │ +0002ed70: 2d20 3630 3533 7820 7820 202d 2031 3630  - 6053x x  - 160
│ │ │ │ +0002ed80: 3930 7820 7820 202b 2033 3738 3378 2078  90x x  + 3783x x
│ │ │ │ +0002ed90: 2020 2b20 3835 3730 7820 7820 202b 2038    + 8570x x  + 8
│ │ │ │ +0002eda0: 3434 3478 2078 2029 7c0a 7c20 2020 2020  444x x )|.|     
│ │ │ │ +0002edb0: 2020 2020 2020 2020 2020 2020 3020 3220              0 2 
│ │ │ │ +0002edc0: 2020 2020 2020 2031 2032 2020 2020 2020         1 2      
│ │ │ │ +0002edd0: 2020 2032 2033 2020 2020 2020 2020 3020     2 3        0 
│ │ │ │ +0002ede0: 3420 2020 2020 2020 2031 2034 2020 2020  4        1 4    
│ │ │ │ +0002edf0: 2020 2020 3320 3420 7c0a 7c20 2020 2020      3 4 |.|     
│ │ │ │ +0002ee00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ee10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ee20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ee30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ee40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ee50: 2020 2020 2020 207c 0a7c 6f31 3020 3a20         |.|o10 : 
│ │ │ │ -0002ee60: 4964 6561 6c20 6f66 2052 2020 2020 2020  Ideal of R      
│ │ │ │ +0002ee40: 2020 2020 2020 2020 7c0a 7c6f 3130 203a          |.|o10 :
│ │ │ │ +0002ee50: 2049 6465 616c 206f 6620 5220 2020 2020   Ideal of R     
│ │ │ │ +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 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002ee90: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0002eea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002eeb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002eec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002eed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002eee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002eef0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20  -------+.|i11 : 
│ │ │ │ -0002ef00: 633d 4368 6572 6e28 4368 2c58 2c49 2920  c=Chern(Ch,X,I) 
│ │ │ │ +0002eee0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131 203a  --------+.|i11 :
│ │ │ │ +0002eef0: 2063 3d43 6865 726e 2843 682c 582c 4929   c=Chern(Ch,X,I)
│ │ │ │ +0002ef00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ef10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ef20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ef30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ef40: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002ef30: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002ef40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0002efa0: 2020 3220 2020 2020 2020 3220 2020 2020    2       2     
│ │ │ │ +0002ef80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002ef90: 2020 2032 2020 2020 2020 2032 2020 2020     2       2    
│ │ │ │ +0002efa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002efb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002efc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002efd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002efe0: 2020 2020 2020 207c 0a7c 6f31 3120 3d20         |.|o11 = 
│ │ │ │ -0002eff0: 3478 2078 2020 2b20 3278 2020 2b20 3278  4x x  + 2x  + 2x
│ │ │ │ -0002f000: 2078 2020 2b20 7820 2020 2020 2020 2020   x  + x         
│ │ │ │ +0002efd0: 2020 2020 2020 2020 7c0a 7c6f 3131 203d          |.|o11 =
│ │ │ │ +0002efe0: 2034 7820 7820 202b 2032 7820 202b 2032   4x x  + 2x  + 2
│ │ │ │ +0002eff0: 7820 7820 202b 2078 2020 2020 2020 2020  x x  + x        
│ │ │ │ +0002f000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f030: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0002f040: 2020 3320 3420 2020 2020 3320 2020 2020    3 4     3     
│ │ │ │ -0002f050: 3320 3420 2020 2033 2020 2020 2020 2020  3 4    3        
│ │ │ │ +0002f020: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002f030: 2020 2033 2034 2020 2020 2033 2020 2020     3 4     3    
│ │ │ │ +0002f040: 2033 2034 2020 2020 3320 2020 2020 2020   3 4    3       
│ │ │ │ +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 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002f070: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002f080: 2020 2020 2020 2020 2020 2020 2020 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 2020 207c 0a7c 6f31 3120 3a20         |.|o11 : 
│ │ │ │ -0002f0e0: 4368 2020 2020 2020 2020 2020 2020 2020  Ch              
│ │ │ │ +0002f0c0: 2020 2020 2020 2020 7c0a 7c6f 3131 203a          |.|o11 :
│ │ │ │ +0002f0d0: 2043 6820 2020 2020 2020 2020 2020 2020   Ch             
│ │ │ │ +0002f0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f120: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002f110: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0002f120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002f130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002f140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002f150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f170: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220 3a20  -------+.|i12 : 
│ │ │ │ -0002f180: 733d 5365 6772 6528 4368 2c58 2c4b 2920  s=Segre(Ch,X,K) 
│ │ │ │ +0002f160: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3132 203a  --------+.|i12 :
│ │ │ │ +0002f170: 2073 3d53 6567 7265 2843 682c 582c 4b29   s=Segre(Ch,X,K)
│ │ │ │ +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 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002f1b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002f1c0: 2020 2020 2020 2020 2020 2020 2020 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 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0002f220: 2020 3220 2020 2020 2032 2020 2020 2020    2      2      
│ │ │ │ +0002f200: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002f210: 2020 2032 2020 2020 2020 3220 2020 2020     2      2     
│ │ │ │ +0002f220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f260: 2020 2020 2020 207c 0a7c 6f31 3220 3d20         |.|o12 = 
│ │ │ │ -0002f270: 3378 2078 2020 2d20 7820 202d 2032 7820  3x x  - x  - 2x 
│ │ │ │ -0002f280: 7820 202b 2078 2020 2b20 7820 2020 2020  x  + x  + x     
│ │ │ │ +0002f250: 2020 2020 2020 2020 7c0a 7c6f 3132 203d          |.|o12 =
│ │ │ │ +0002f260: 2033 7820 7820 202d 2078 2020 2d20 3278   3x x  - x  - 2x
│ │ │ │ +0002f270: 2078 2020 2b20 7820 202b 2078 2020 2020   x  + x  + x    
│ │ │ │ +0002f280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f2b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0002f2c0: 2020 3320 3420 2020 2033 2020 2020 2033    3 4    3     3
│ │ │ │ -0002f2d0: 2034 2020 2020 3320 2020 2034 2020 2020   4    3    4    
│ │ │ │ +0002f2a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002f2b0: 2020 2033 2034 2020 2020 3320 2020 2020     3 4    3     
│ │ │ │ +0002f2c0: 3320 3420 2020 2033 2020 2020 3420 2020  3 4    3    4   
│ │ │ │ +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 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002f2f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002f300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f350: 2020 2020 2020 207c 0a7c 6f31 3220 3a20         |.|o12 : 
│ │ │ │ -0002f360: 4368 2020 2020 2020 2020 2020 2020 2020  Ch              
│ │ │ │ +0002f340: 2020 2020 2020 2020 7c0a 7c6f 3132 203a          |.|o12 :
│ │ │ │ +0002f350: 2043 6820 2020 2020 2020 2020 2020 2020   Ch             
│ │ │ │ +0002f360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f3a0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002f390: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0002f3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002f3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002f3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002f3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f3f0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3320 3a20  -------+.|i13 : 
│ │ │ │ -0002f400: 732d 6320 2020 2020 2020 2020 2020 2020  s-c             
│ │ │ │ +0002f3e0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3133 203a  --------+.|i13 :
│ │ │ │ +0002f3f0: 2073 2d63 2020 2020 2020 2020 2020 2020   s-c            
│ │ │ │ +0002f400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f440: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002f430: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002f440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f490: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0002f4a0: 2020 2032 2020 2020 2020 2032 2020 2020     2       2    
│ │ │ │ +0002f480: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002f490: 2020 2020 3220 2020 2020 2020 3220 2020      2       2   
│ │ │ │ +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 2020 207c 0a7c 6f31 3320 3d20         |.|o13 = 
│ │ │ │ -0002f4f0: 2d20 7820 7820 202d 2033 7820 202d 2034  - x x  - 3x  - 4
│ │ │ │ -0002f500: 7820 7820 202b 2078 2020 2020 2020 2020  x x  + x        
│ │ │ │ +0002f4d0: 2020 2020 2020 2020 7c0a 7c6f 3133 203d          |.|o13 =
│ │ │ │ +0002f4e0: 202d 2078 2078 2020 2d20 3378 2020 2d20   - x x  - 3x  - 
│ │ │ │ +0002f4f0: 3478 2078 2020 2b20 7820 2020 2020 2020  4x x  + x       
│ │ │ │ +0002f500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f530: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0002f540: 2020 2033 2034 2020 2020 2033 2020 2020     3 4     3    
│ │ │ │ -0002f550: 2033 2034 2020 2020 3420 2020 2020 2020   3 4    4       
│ │ │ │ +0002f520: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002f530: 2020 2020 3320 3420 2020 2020 3320 2020      3 4     3   
│ │ │ │ +0002f540: 2020 3320 3420 2020 2034 2020 2020 2020    3 4    4      
│ │ │ │ +0002f550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f580: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002f570: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002f580: 2020 2020 2020 2020 2020 2020 2020 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 2020 207c 0a7c 6f31 3320 3a20         |.|o13 : 
│ │ │ │ -0002f5e0: 4368 2020 2020 2020 2020 2020 2020 2020  Ch              
│ │ │ │ +0002f5c0: 2020 2020 2020 2020 7c0a 7c6f 3133 203a          |.|o13 :
│ │ │ │ +0002f5d0: 2043 6820 2020 2020 2020 2020 2020 2020   Ch             
│ │ │ │ +0002f5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f620: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002f610: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0002f620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002f630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002f640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002f650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f670: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3420 3a20  -------+.|i14 : 
│ │ │ │ -0002f680: 732a 6320 2020 2020 2020 2020 2020 2020  s*c             
│ │ │ │ +0002f660: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3134 203a  --------+.|i14 :
│ │ │ │ +0002f670: 2073 2a63 2020 2020 2020 2020 2020 2020   s*c            
│ │ │ │ +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 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002f6b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002f6c0: 2020 2020 2020 2020 2020 2020 2020 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 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0002f720: 2020 3220 2020 2020 2032 2020 2020 2020    2      2      
│ │ │ │ +0002f700: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002f710: 2020 2032 2020 2020 2020 3220 2020 2020     2      2     
│ │ │ │ +0002f720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 2020 207c 0a7c 6f31 3420 3d20         |.|o14 = 
│ │ │ │ -0002f770: 3278 2078 2020 2b20 7820 202b 2078 2078  2x x  + x  + x x
│ │ │ │ +0002f750: 2020 2020 2020 2020 7c0a 7c6f 3134 203d          |.|o14 =
│ │ │ │ +0002f760: 2032 7820 7820 202b 2078 2020 2b20 7820   2x x  + x  + x 
│ │ │ │ +0002f770: 7820 2020 2020 2020 2020 2020 2020 2020  x               
│ │ │ │  0002f780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f7b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0002f7c0: 2020 3320 3420 2020 2033 2020 2020 3320    3 4    3    3 
│ │ │ │ -0002f7d0: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ +0002f7a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002f7b0: 2020 2033 2034 2020 2020 3320 2020 2033     3 4    3    3
│ │ │ │ +0002f7c0: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ +0002f7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f800: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002f7f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002f800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f850: 2020 2020 2020 207c 0a7c 6f31 3420 3a20         |.|o14 : 
│ │ │ │ -0002f860: 4368 2020 2020 2020 2020 2020 2020 2020  Ch              
│ │ │ │ +0002f840: 2020 2020 2020 2020 7c0a 7c6f 3134 203a          |.|o14 :
│ │ │ │ +0002f850: 2043 6820 2020 2020 2020 2020 2020 2020   Ch             
│ │ │ │ +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 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002f890: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0002f8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002f8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002f8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002f8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f8f0: 2d2d 2d2d 2d2d 2d2b 0a0a 466f 7220 7468  -------+..For th
│ │ │ │ -0002f900: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d  e programmer.===
│ │ │ │ -0002f910: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ -0002f920: 0a54 6865 206f 626a 6563 7420 2a6e 6f74  .The object *not
│ │ │ │ -0002f930: 6520 546f 7269 6343 686f 7752 696e 673a  e ToricChowRing:
│ │ │ │ -0002f940: 2054 6f72 6963 4368 6f77 5269 6e67 2c20   ToricChowRing, 
│ │ │ │ -0002f950: 6973 2061 202a 6e6f 7465 206d 6574 686f  is a *note metho
│ │ │ │ -0002f960: 6420 6675 6e63 7469 6f6e 3a0a 284d 6163  d function:.(Mac
│ │ │ │ -0002f970: 6175 6c61 7932 446f 6329 4d65 7468 6f64  aulay2Doc)Method
│ │ │ │ -0002f980: 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d  Function,...----
│ │ │ │ +0002f8e0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a46 6f72 2074  --------+..For t
│ │ │ │ +0002f8f0: 6865 2070 726f 6772 616d 6d65 720a 3d3d  he programmer.==
│ │ │ │ +0002f900: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0002f910: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f  ..The object *no
│ │ │ │ +0002f920: 7465 2054 6f72 6963 4368 6f77 5269 6e67  te ToricChowRing
│ │ │ │ +0002f930: 3a20 546f 7269 6343 686f 7752 696e 672c  : ToricChowRing,
│ │ │ │ +0002f940: 2069 7320 6120 2a6e 6f74 6520 6d65 7468   is a *note meth
│ │ │ │ +0002f950: 6f64 2066 756e 6374 696f 6e3a 0a28 4d61  od function:.(Ma
│ │ │ │ +0002f960: 6361 756c 6179 3244 6f63 294d 6574 686f  caulay2Doc)Metho
│ │ │ │ +0002f970: 6446 756e 6374 696f 6e2c 2e0a 0a2d 2d2d  dFunction,...---
│ │ │ │ +0002f980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002f990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002f9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002f9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865  -----------..The
│ │ │ │ -0002f9e0: 2073 6f75 7263 6520 6f66 2074 6869 7320   source of this 
│ │ │ │ -0002f9f0: 646f 6375 6d65 6e74 2069 7320 696e 0a2f  document is in./
│ │ │ │ -0002fa00: 6275 696c 642f 7265 7072 6f64 7563 6962  build/reproducib
│ │ │ │ -0002fa10: 6c65 2d70 6174 682f 6d61 6361 756c 6179  le-path/macaulay
│ │ │ │ -0002fa20: 322d 312e 3235 2e31 312b 6473 2f4d 322f  2-1.25.11+ds/M2/
│ │ │ │ -0002fa30: 4d61 6361 756c 6179 322f 7061 636b 6167  Macaulay2/packag
│ │ │ │ -0002fa40: 6573 2f0a 4368 6172 6163 7465 7269 7374  es/.Characterist
│ │ │ │ -0002fa50: 6963 436c 6173 7365 732e 6d32 3a31 3935  icClasses.m2:195
│ │ │ │ -0002fa60: 313a 302e 0a1f 0a54 6167 2054 6162 6c65  1:0....Tag Table
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│ │ │ │ -0002fb50: 6f64 653a 2063 6f6e 6669 6775 7269 6e67  ode: configuring
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│ │ │ │ -0002fc70: 653a 2053 6567 7265 7f31 3638 3032 310a  e: Segre.168021.
│ │ │ │ -0002fc80: 4e6f 6465 3a20 546f 7269 6343 686f 7752  Node: ToricChowR
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│ │ │ │ +0002faa0: 320a 4e6f 6465 3a20 4368 6563 6b54 6f72  2.Node: CheckTor
│ │ │ │ +0002fab0: 6963 5661 7269 6574 7956 616c 6964 7f32  icVarietyValid.2
│ │ │ │ +0002fac0: 3237 3138 0a4e 6f64 653a 2043 6865 726e  2718.Node: Chern
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│ │ │ │ +0002fc10: 5072 6f6a 436f 6f72 6452 696e 677f 3133  ProjCoordRing.13
│ │ │ │ +0002fc20: 3831 3235 0a4e 6f64 653a 204f 7574 7075  8125.Node: Outpu
│ │ │ │ +0002fc30: 747f 3134 3437 3933 0a4e 6f64 653a 2070  t.144793.Node: p
│ │ │ │ +0002fc40: 726f 6261 6269 6c69 7374 6963 2061 6c67  robabilistic alg
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│ │ │ │ +0002fc60: 6465 3a20 5365 6772 657f 3136 3830 3036  de: Segre.168006
│ │ │ │ +0002fc70: 0a4e 6f64 653a 2054 6f72 6963 4368 6f77  .Node: ToricChow
│ │ │ │ +0002fc80: 5269 6e67 7f31 3835 3931 330a 1f0a 456e  Ring.185913...En
│ │ │ │ +0002fc90: 6420 5461 6720 5461 626c 650a            d Tag Table.
│ │ ├── ./usr/share/info/Chordal.info.gz
│ │ │ ├── Chordal.info
│ │ │ │ @@ -3949,30 +3949,30 @@
│ │ │ │  0000f6c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │  0000f710: 2020 2020 2020 2020 2020 207c 0a7c 6f34             |.|o4
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│ │ │ │ +0000f730: 2063 7d20 2020 2020 2020 2020 2020 2020   c}             
│ │ │ │  0000f740: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0000f770: 2020 2020 2020 2020 2020 2020 6220 3d3e              b =>
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│ │ │ │  0000f7c0: 2020 2020 2020 2020 2020 2020 6320 3d3e              c =>
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│ │ │ │  0000f7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f800: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0000f810: 2020 2020 2020 2020 2020 2020 6420 3d3e              d =>
│ │ │ │ -0000f820: 206e 756c 6c20 2020 2020 2020 2020 2020   null           
│ │ │ │ +0000f820: 2062 2020 2020 2020 2020 2020 2020 2020   b              
│ │ │ │  0000f830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f850: 2020 2020 2020 2020 2020 207c 0a7c 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 2020 2020                  
│ │ │ │  0000f890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -4033,31 +4033,31 @@
│ │ │ │  0000fc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fc10: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0000fc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fc60: 2020 2020 2020 2020 2020 207c 0a7c 6f37             |.|o7
│ │ │ │ -0000fc70: 203d 2043 686f 7264 616c 4e65 747b 2064   = ChordalNet{ d
│ │ │ │ -0000fc80: 203d 3e20 7b20 2c20 647d 2020 2020 7d20   => { , d}    } 
│ │ │ │ +0000fc70: 203d 2043 686f 7264 616c 4e65 747b 2061   = ChordalNet{ a
│ │ │ │ +0000fc80: 203d 3e20 7b61 2c20 207d 2020 2020 7d20   => {a,  }    } 
│ │ │ │  0000fc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fcb0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0000fcc0: 2020 2020 2020 2020 2020 2020 2020 2062                 b
│ │ │ │ -0000fcd0: 203d 3e20 7b62 2c20 202c 2062 7d20 2020   => {b,  , b}   
│ │ │ │ +0000fcc0: 2020 2020 2020 2020 2020 2020 2020 2063                 c
│ │ │ │ +0000fcd0: 203d 3e20 7b20 2c20 637d 2020 2020 2020   => { , c}      
│ │ │ │  0000fce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fd00: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0000fd10: 2020 2020 2020 2020 2020 2020 2020 2061                 a
│ │ │ │ -0000fd20: 203d 3e20 7b61 2c20 207d 2020 2020 2020   => {a,  }      
│ │ │ │ +0000fd10: 2020 2020 2020 2020 2020 2020 2020 2064                 d
│ │ │ │ +0000fd20: 203d 3e20 7b20 2c20 647d 2020 2020 2020   => { , d}      
│ │ │ │  0000fd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fd50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0000fd60: 2020 2020 2020 2020 2020 2020 2020 2063                 c
│ │ │ │ -0000fd70: 203d 3e20 7b20 2c20 637d 2020 2020 2020   => { , c}      
│ │ │ │ +0000fd60: 2020 2020 2020 2020 2020 2020 2020 2062                 b
│ │ │ │ +0000fd70: 203d 3e20 7b62 2c20 202c 2062 7d20 2020   => {b,  , b}   
│ │ │ │  0000fd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fda0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0000fdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fde0: 2020 2020 2020 2020 2020 2020 2020 2020
│ │ ├── ./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 2032 2e39 3733 3836 7320 656c  | -- 2.97386s el
│ │ │ │ +00004180: 7c20 2d2d 2033 2e31 3538 3235 7320 656c  | -- 3.15825s 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 3339 3039 3736 7320 656c  | -- .390976s el
│ │ │ │ +00006930: 7c20 2d2d 202e 3532 3335 3037 7320 656c  | -- .523507s 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
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│ │ │ │  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 302e 3333 3031 7320 656c  | -- 10.3301s el
│ │ │ │ +00006b60: 7c20 2d2d 2039 2e39 3035 3437 7320 656c  | -- 9.90547s 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 3334 3435 3632 7320 656c  | -- .344562s el
│ │ │ │ +000070b0: 7c20 2d2d 202e 3531 3537 3739 7320 656c  | -- .515779s 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 3535 3638 3938 7320 656c  | -- .556898s el
│ │ │ │ -000072f0: 6170 7365 6420 2020 2020 2020 2020 2020  apsed           
│ │ │ │ +000072e0: 7c20 2d2d 202e 3437 3537 3273 2065 6c61  | -- .47572s 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 3535 3639 3332 7320 656c  | -- .556932s el
│ │ │ │ +00007330: 7c20 2d2d 202e 3437 3537 3436 7320 656c  | -- .475746s 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
│ │ │ │ @@ -4343,18 +4343,18 @@
│ │ │ │  00010f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010f80: 2d2d 2d2b 0a7c 6937 203a 2074 696d 6520  ---+.|i7 : time 
│ │ │ │  00010f90: 4720 3d20 4569 7365 6e62 7564 5368 616d  G = EisenbudSham
│ │ │ │  00010fa0: 6173 6828 6666 2c46 2c6c 656e 2920 2020  ash(ff,F,len)   
│ │ │ │  00010fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010fd0: 2020 207c 0a7c 202d 2d20 7573 6564 2036     |.| -- used 6
│ │ │ │ -00010fe0: 2e38 3231 3235 7320 2863 7075 293b 2035  .82125s (cpu); 5
│ │ │ │ -00010ff0: 2e31 3938 3933 7320 2874 6872 6561 6429  .19893s (thread)
│ │ │ │ -00011000: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │ +00010fd0: 2020 207c 0a7c 202d 2d20 7573 6564 2038     |.| -- used 8
│ │ │ │ +00010fe0: 2e30 3239 3436 7320 2863 7075 293b 2035  .02946s (cpu); 5
│ │ │ │ +00010ff0: 2e39 3231 3873 2028 7468 7265 6164 293b  .9218s (thread);
│ │ │ │ +00011000: 2030 7320 2867 6329 2020 2020 2020 2020   0s (gc)        
│ │ │ │  00011010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011020: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00011030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011040: 2020 2020 2020 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 207c 0a7c 2020 2020 202f 2020 2020     |.|     /    
│ │ │ │ @@ -4884,16 +4884,16 @@
│ │ │ │  00013130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013150: 2d2b 0a7c 6932 3020 3a20 4646 203d 2074  -+.|i20 : FF = t
│ │ │ │  00013160: 696d 6520 5368 616d 6173 6828 5231 2c46  ime Shamash(R1,F
│ │ │ │  00013170: 2c34 2920 2020 2020 2020 2020 2020 2020  ,4)             
│ │ │ │  00013180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013190: 2020 207c 0a7c 202d 2d20 7573 6564 2030     |.| -- used 0
│ │ │ │ -000131a0: 2e31 3834 3033 3373 2028 6370 7529 3b20  .184033s (cpu); 
│ │ │ │ -000131b0: 302e 3130 3730 3037 7320 2874 6872 6561  0.107007s (threa
│ │ │ │ +000131a0: 2e32 3131 3730 3673 2028 6370 7529 3b20  .211706s (cpu); 
│ │ │ │ +000131b0: 302e 3131 3833 3938 7320 2874 6872 6561  0.118398s (threa
│ │ │ │  000131c0: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │  000131d0: 2020 2020 207c 0a7c 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 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00013220: 2020 3120 2020 2020 2020 3620 2020 2020    1       6     
│ │ │ │ @@ -4925,17 +4925,17 @@
│ │ │ │  000133c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000133d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000133e0: 2d2d 2d2d 2d2b 0a7c 6932 3120 3a20 4747  -----+.|i21 : GG
│ │ │ │  000133f0: 203d 2074 696d 6520 4569 7365 6e62 7564   = time Eisenbud
│ │ │ │  00013400: 5368 616d 6173 6828 6666 2c46 2c34 2920  Shamash(ff,F,4) 
│ │ │ │  00013410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013420: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -00013430: 6564 2030 2e39 3533 3535 3473 2028 6370  ed 0.953554s (cp
│ │ │ │ -00013440: 7529 3b20 302e 3732 3432 3334 7320 2874  u); 0.724234s (t
│ │ │ │ -00013450: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │ +00013430: 6564 2031 2e31 3432 3739 7320 2863 7075  ed 1.14279s (cpu
│ │ │ │ +00013440: 293b 2030 2e38 3634 3033 3373 2028 7468  ); 0.864033s (th
│ │ │ │ +00013450: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │  00013460: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00013470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000134a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000134b0: 2020 2020 2f20 525c 3120 2020 2020 2f20      / R\1     / 
│ │ │ │  000134c0: 525c 3620 2020 2020 2f20 525c 3138 2020  R\6     / R\18  
│ │ │ │ @@ -4977,24032 +4977,24030 @@
│ │ │ │  00013700: 5468 6520 6675 6e63 7469 6f6e 2061 6c73  The function als
│ │ │ │  00013710: 6f20 6465 616c 7320 636f 7272 6563 746c  o deals correctl
│ │ │ │  00013720: 7920 7769 7468 2063 6f6d 706c 6578 6573  y with complexes
│ │ │ │  00013730: 2046 2077 6865 7265 206d 696e 2046 2069   F where min F i
│ │ │ │  00013740: 7320 6e6f 7420 303a 0a0a 2b2d 2d2d 2d2d  s not 0:..+-----
│ │ │ │  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: 2b0a 7c69 3232 203a 2047 4720 3d20 7469  +.|i22 : GG = ti
│ │ │ │ -00013790: 6d65 2045 6973 656e 6275 6453 6861 6d61  me EisenbudShama
│ │ │ │ -000137a0: 7368 2852 312c 465b 325d 2c34 2920 2020  sh(R1,F[2],4)   
│ │ │ │ -000137b0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075          |.| -- u
│ │ │ │ -000137c0: 7365 6420 302e 3934 3530 3835 7320 2863  sed 0.945085s (c
│ │ │ │ -000137d0: 7075 293b 2030 2e37 3430 3130 3173 2028  pu); 0.740101s (
│ │ │ │ -000137e0: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 0s (gc)
│ │ │ │ -000137f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00013770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00013780: 0a7c 6932 3220 3a20 4747 203d 2074 696d  .|i22 : GG = tim
│ │ │ │ +00013790: 6520 4569 7365 6e62 7564 5368 616d 6173  e EisenbudShamas
│ │ │ │ +000137a0: 6828 5231 2c46 5b32 5d2c 3429 2020 2020  h(R1,F[2],4)    
│ │ │ │ +000137b0: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365        |.| -- use
│ │ │ │ +000137c0: 6420 312e 3135 3234 3873 2028 6370 7529  d 1.15248s (cpu)
│ │ │ │ +000137d0: 3b20 302e 3838 3533 3135 7320 2874 6872  ; 0.885315s (thr
│ │ │ │ +000137e0: 6561 6429 3b20 3073 2028 6763 297c 0a7c  ead); 0s (gc)|.|
│ │ │ │ +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 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00013830: 2020 2031 2020 2020 2020 2036 2020 2020     1       6    
│ │ │ │ -00013840: 2020 2031 3820 2020 2020 2020 3338 2020     18       38  
│ │ │ │ -00013850: 2020 2020 2036 3620 2020 2020 2020 2020       66         
│ │ │ │ -00013860: 7c0a 7c6f 3232 203d 2052 3120 203c 2d2d  |.|o22 = R1  <--
│ │ │ │ -00013870: 2052 3120 203c 2d2d 2052 3120 2020 3c2d   R1  <-- R1   <-
│ │ │ │ -00013880: 2d20 5231 2020 203c 2d2d 2052 3120 2020  - R1   <-- R1   
│ │ │ │ -00013890: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00013820: 2020 2020 7c0a 7c20 2020 2020 2020 2031      |.|        1
│ │ │ │ +00013830: 2020 2020 2020 2036 2020 2020 2020 2031         6       1
│ │ │ │ +00013840: 3820 2020 2020 2020 3338 2020 2020 2020  8       38      
│ │ │ │ +00013850: 2036 3620 2020 2020 2020 207c 0a7c 6f32   66        |.|o2
│ │ │ │ +00013860: 3220 3d20 5231 2020 3c2d 2d20 5231 2020  2 = R1  <-- R1  
│ │ │ │ +00013870: 3c2d 2d20 5231 2020 203c 2d2d 2052 3120  <-- R1   <-- R1 
│ │ │ │ +00013880: 2020 3c2d 2d20 5231 2020 2020 2020 2020    <-- R1        
│ │ │ │ +00013890: 2020 7c0a 7c20 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 2020 2020                  
│ │ │ │ -000138d0: 7c0a 7c20 2020 2020 202d 3220 2020 2020  |.|      -2     
│ │ │ │ -000138e0: 202d 3120 2020 2020 2030 2020 2020 2020   -1      0      
│ │ │ │ -000138f0: 2020 3120 2020 2020 2020 2032 2020 2020    1        2    
│ │ │ │ -00013900: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000138c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000138d0: 2020 2d32 2020 2020 2020 2d31 2020 2020    -2      -1    
│ │ │ │ +000138e0: 2020 3020 2020 2020 2020 2031 2020 2020    0        1    
│ │ │ │ +000138f0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +00013900: 7c0a 7c20 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                  
│ │ │ │ -00013940: 7c0a 7c6f 3232 203a 2043 6f6d 706c 6578  |.|o22 : Complex
│ │ │ │ +00013930: 2020 2020 2020 207c 0a7c 6f32 3220 3a20         |.|o22 : 
│ │ │ │ +00013940: 436f 6d70 6c65 7820 2020 2020 2020 2020  Complex         
│ │ │ │  00013950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013970: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00013960: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013970: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00013980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000139a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000139b0: 2b0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d  +..See also.====
│ │ │ │ -000139c0: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520  ====..  * *note 
│ │ │ │ -000139d0: 6d61 6b65 486f 6d6f 746f 7069 6573 3a20  makeHomotopies: 
│ │ │ │ -000139e0: 6d61 6b65 486f 6d6f 746f 7069 6573 2c20  makeHomotopies, 
│ │ │ │ -000139f0: 2d2d 2072 6574 7572 6e73 2061 2073 7973  -- returns a sys
│ │ │ │ -00013a00: 7465 6d20 6f66 2068 6967 6865 720a 2020  tem of higher.  
│ │ │ │ -00013a10: 2020 686f 6d6f 746f 7069 6573 0a20 202a    homotopies.  *
│ │ │ │ -00013a20: 202a 6e6f 7465 2053 6861 6d61 7368 3a20   *note Shamash: 
│ │ │ │ -00013a30: 5368 616d 6173 682c 202d 2d20 436f 6d70  Shamash, -- Comp
│ │ │ │ -00013a40: 7574 6573 2074 6865 2053 6861 6d61 7368  utes the Shamash
│ │ │ │ -00013a50: 2043 6f6d 706c 6578 0a20 202a 202a 6e6f   Complex.  * *no
│ │ │ │ -00013a60: 7465 2065 7870 6f3a 2065 7870 6f2c 202d  te expo: expo, -
│ │ │ │ -00013a70: 2d20 7265 7475 726e 7320 6120 7365 7420  - returns a set 
│ │ │ │ -00013a80: 636f 7272 6573 706f 6e64 696e 6720 746f  corresponding to
│ │ │ │ -00013a90: 2074 6865 2062 6173 6973 206f 6620 6120   the basis of a 
│ │ │ │ -00013aa0: 6469 7669 6465 640a 2020 2020 706f 7765  divided.    powe
│ │ │ │ -00013ab0: 720a 0a57 6179 7320 746f 2075 7365 2045  r..Ways to use E
│ │ │ │ -00013ac0: 6973 656e 6275 6453 6861 6d61 7368 3a0a  isenbudShamash:.
│ │ │ │ +000139a0: 2d2d 2d2d 2d2b 0a0a 5365 6520 616c 736f  -----+..See also
│ │ │ │ +000139b0: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a  .========..  * *
│ │ │ │ +000139c0: 6e6f 7465 206d 616b 6548 6f6d 6f74 6f70  note makeHomotop
│ │ │ │ +000139d0: 6965 733a 206d 616b 6548 6f6d 6f74 6f70  ies: makeHomotop
│ │ │ │ +000139e0: 6965 732c 202d 2d20 7265 7475 726e 7320  ies, -- returns 
│ │ │ │ +000139f0: 6120 7379 7374 656d 206f 6620 6869 6768  a system of high
│ │ │ │ +00013a00: 6572 0a20 2020 2068 6f6d 6f74 6f70 6965  er.    homotopie
│ │ │ │ +00013a10: 730a 2020 2a20 2a6e 6f74 6520 5368 616d  s.  * *note Sham
│ │ │ │ +00013a20: 6173 683a 2053 6861 6d61 7368 2c20 2d2d  ash: Shamash, --
│ │ │ │ +00013a30: 2043 6f6d 7075 7465 7320 7468 6520 5368   Computes the Sh
│ │ │ │ +00013a40: 616d 6173 6820 436f 6d70 6c65 780a 2020  amash Complex.  
│ │ │ │ +00013a50: 2a20 2a6e 6f74 6520 6578 706f 3a20 6578  * *note expo: ex
│ │ │ │ +00013a60: 706f 2c20 2d2d 2072 6574 7572 6e73 2061  po, -- returns a
│ │ │ │ +00013a70: 2073 6574 2063 6f72 7265 7370 6f6e 6469   set correspondi
│ │ │ │ +00013a80: 6e67 2074 6f20 7468 6520 6261 7369 7320  ng to the basis 
│ │ │ │ +00013a90: 6f66 2061 2064 6976 6964 6564 0a20 2020  of a divided.   
│ │ │ │ +00013aa0: 2070 6f77 6572 0a0a 5761 7973 2074 6f20   power..Ways to 
│ │ │ │ +00013ab0: 7573 6520 4569 7365 6e62 7564 5368 616d  use EisenbudSham
│ │ │ │ +00013ac0: 6173 683a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  ash:.===========
│ │ │ │  00013ad0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00013ae0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020  ============..  
│ │ │ │ -00013af0: 2a20 2245 6973 656e 6275 6453 6861 6d61  * "EisenbudShama
│ │ │ │ -00013b00: 7368 284d 6174 7269 782c 436f 6d70 6c65  sh(Matrix,Comple
│ │ │ │ -00013b10: 782c 5a5a 2922 0a20 202a 2022 4569 7365  x,ZZ)".  * "Eise
│ │ │ │ -00013b20: 6e62 7564 5368 616d 6173 6828 5269 6e67  nbudShamash(Ring
│ │ │ │ -00013b30: 2c43 6f6d 706c 6578 2c5a 5a29 220a 0a46  ,Complex,ZZ)"..F
│ │ │ │ -00013b40: 6f72 2074 6865 2070 726f 6772 616d 6d65  or the programme
│ │ │ │ -00013b50: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  r.==============
│ │ │ │ -00013b60: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374  ====..The object
│ │ │ │ -00013b70: 202a 6e6f 7465 2045 6973 656e 6275 6453   *note EisenbudS
│ │ │ │ -00013b80: 6861 6d61 7368 3a20 4569 7365 6e62 7564  hamash: Eisenbud
│ │ │ │ -00013b90: 5368 616d 6173 682c 2069 7320 6120 2a6e  Shamash, is a *n
│ │ │ │ -00013ba0: 6f74 6520 6d65 7468 6f64 2066 756e 6374  ote method funct
│ │ │ │ -00013bb0: 696f 6e3a 0a28 4d61 6361 756c 6179 3244  ion:.(Macaulay2D
│ │ │ │ -00013bc0: 6f63 294d 6574 686f 6446 756e 6374 696f  oc)MethodFunctio
│ │ │ │ -00013bd0: 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d  n,...-----------
│ │ │ │ +00013ae0: 3d0a 0a20 202a 2022 4569 7365 6e62 7564  =..  * "Eisenbud
│ │ │ │ +00013af0: 5368 616d 6173 6828 4d61 7472 6978 2c43  Shamash(Matrix,C
│ │ │ │ +00013b00: 6f6d 706c 6578 2c5a 5a29 220a 2020 2a20  omplex,ZZ)".  * 
│ │ │ │ +00013b10: 2245 6973 656e 6275 6453 6861 6d61 7368  "EisenbudShamash
│ │ │ │ +00013b20: 2852 696e 672c 436f 6d70 6c65 782c 5a5a  (Ring,Complex,ZZ
│ │ │ │ +00013b30: 2922 0a0a 466f 7220 7468 6520 7072 6f67  )"..For the prog
│ │ │ │ +00013b40: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d  rammer.=========
│ │ │ │ +00013b50: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f  =========..The o
│ │ │ │ +00013b60: 626a 6563 7420 2a6e 6f74 6520 4569 7365  bject *note Eise
│ │ │ │ +00013b70: 6e62 7564 5368 616d 6173 683a 2045 6973  nbudShamash: Eis
│ │ │ │ +00013b80: 656e 6275 6453 6861 6d61 7368 2c20 6973  enbudShamash, is
│ │ │ │ +00013b90: 2061 202a 6e6f 7465 206d 6574 686f 6420   a *note method 
│ │ │ │ +00013ba0: 6675 6e63 7469 6f6e 3a0a 284d 6163 6175  function:.(Macau
│ │ │ │ +00013bb0: 6c61 7932 446f 6329 4d65 7468 6f64 4675  lay2Doc)MethodFu
│ │ │ │ +00013bc0: 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d  nction,...------
│ │ │ │ +00013bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013c20: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365  ----..The source
│ │ │ │ -00013c30: 206f 6620 7468 6973 2064 6f63 756d 656e   of this documen
│ │ │ │ -00013c40: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72  t is in./build/r
│ │ │ │ -00013c50: 6570 726f 6475 6369 626c 652d 7061 7468  eproducible-path
│ │ │ │ -00013c60: 2f6d 6163 6175 6c61 7932 2d31 2e32 352e  /macaulay2-1.25.
│ │ │ │ -00013c70: 3131 2b64 732f 4d32 2f4d 6163 6175 6c61  11+ds/M2/Macaula
│ │ │ │ -00013c80: 7932 2f70 6163 6b61 6765 732f 0a43 6f6d  y2/packages/.Com
│ │ │ │ -00013c90: 706c 6574 6549 6e74 6572 7365 6374 696f  pleteIntersectio
│ │ │ │ -00013ca0: 6e52 6573 6f6c 7574 696f 6e73 2e6d 323a  nResolutions.m2:
│ │ │ │ -00013cb0: 3438 3432 3a30 2e0a 1f0a 4669 6c65 3a20  4842:0....File: 
│ │ │ │ -00013cc0: 436f 6d70 6c65 7465 496e 7465 7273 6563  CompleteIntersec
│ │ │ │ -00013cd0: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e  tionResolutions.
│ │ │ │ -00013ce0: 696e 666f 2c20 4e6f 6465 3a20 4569 7365  info, Node: Eise
│ │ │ │ -00013cf0: 6e62 7564 5368 616d 6173 6854 6f74 616c  nbudShamashTotal
│ │ │ │ -00013d00: 2c20 4e65 7874 3a20 6576 656e 4578 744d  , Next: evenExtM
│ │ │ │ -00013d10: 6f64 756c 652c 2050 7265 763a 2045 6973  odule, Prev: Eis
│ │ │ │ -00013d20: 656e 6275 6453 6861 6d61 7368 2c20 5570  enbudShamash, Up
│ │ │ │ -00013d30: 3a20 546f 700a 0a45 6973 656e 6275 6453  : Top..EisenbudS
│ │ │ │ -00013d40: 6861 6d61 7368 546f 7461 6c20 2d2d 2050  hamashTotal -- P
│ │ │ │ -00013d50: 7265 6375 7273 6f72 2063 6f6d 706c 6578  recursor complex
│ │ │ │ -00013d60: 206f 6620 746f 7461 6c20 4578 740a 2a2a   of total Ext.**
│ │ │ │ +00013c10: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073  ---------..The s
│ │ │ │ +00013c20: 6f75 7263 6520 6f66 2074 6869 7320 646f  ource of this do
│ │ │ │ +00013c30: 6375 6d65 6e74 2069 7320 696e 0a2f 6275  cument is in./bu
│ │ │ │ +00013c40: 696c 642f 7265 7072 6f64 7563 6962 6c65  ild/reproducible
│ │ │ │ +00013c50: 2d70 6174 682f 6d61 6361 756c 6179 322d  -path/macaulay2-
│ │ │ │ +00013c60: 312e 3235 2e31 312b 6473 2f4d 322f 4d61  1.25.11+ds/M2/Ma
│ │ │ │ +00013c70: 6361 756c 6179 322f 7061 636b 6167 6573  caulay2/packages
│ │ │ │ +00013c80: 2f0a 436f 6d70 6c65 7465 496e 7465 7273  /.CompleteInters
│ │ │ │ +00013c90: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e  ectionResolution
│ │ │ │ +00013ca0: 732e 6d32 3a34 3834 323a 302e 0a1f 0a46  s.m2:4842:0....F
│ │ │ │ +00013cb0: 696c 653a 2043 6f6d 706c 6574 6549 6e74  ile: CompleteInt
│ │ │ │ +00013cc0: 6572 7365 6374 696f 6e52 6573 6f6c 7574  ersectionResolut
│ │ │ │ +00013cd0: 696f 6e73 2e69 6e66 6f2c 204e 6f64 653a  ions.info, Node:
│ │ │ │ +00013ce0: 2045 6973 656e 6275 6453 6861 6d61 7368   EisenbudShamash
│ │ │ │ +00013cf0: 546f 7461 6c2c 204e 6578 743a 2065 7665  Total, Next: eve
│ │ │ │ +00013d00: 6e45 7874 4d6f 6475 6c65 2c20 5072 6576  nExtModule, Prev
│ │ │ │ +00013d10: 3a20 4569 7365 6e62 7564 5368 616d 6173  : EisenbudShamas
│ │ │ │ +00013d20: 682c 2055 703a 2054 6f70 0a0a 4569 7365  h, Up: Top..Eise
│ │ │ │ +00013d30: 6e62 7564 5368 616d 6173 6854 6f74 616c  nbudShamashTotal
│ │ │ │ +00013d40: 202d 2d20 5072 6563 7572 736f 7220 636f   -- Precursor co
│ │ │ │ +00013d50: 6d70 6c65 7820 6f66 2074 6f74 616c 2045  mplex of total E
│ │ │ │ +00013d60: 7874 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  xt.*************
│ │ │ │  00013d70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00013d80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00013d90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00013da0: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a  ****..  * Usage:
│ │ │ │ -00013db0: 200a 2020 2020 2020 2020 2864 302c 6431   .        (d0,d1
│ │ │ │ -00013dc0: 2920 3d20 2045 6973 656e 6275 6453 6861  ) =  EisenbudSha
│ │ │ │ -00013dd0: 6d61 7368 546f 7461 6c20 4d0a 2020 2a20  mashTotal M.  * 
│ │ │ │ -00013de0: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ -00013df0: 4d2c 2061 202a 6e6f 7465 206d 6f64 756c  M, a *note modul
│ │ │ │ -00013e00: 653a 2028 4d61 6361 756c 6179 3244 6f63  e: (Macaulay2Doc
│ │ │ │ -00013e10: 294d 6f64 756c 652c 2c20 6f76 6572 2061  )Module,, over a
│ │ │ │ -00013e20: 2063 6f6d 706c 6574 6520 696e 7465 7273   complete inters
│ │ │ │ -00013e30: 6563 7469 6f6e 0a20 202a 202a 6e6f 7465  ection.  * *note
│ │ │ │ -00013e40: 204f 7074 696f 6e61 6c20 696e 7075 7473   Optional inputs
│ │ │ │ -00013e50: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ -00013e60: 7573 696e 6720 6675 6e63 7469 6f6e 7320  using functions 
│ │ │ │ -00013e70: 7769 7468 206f 7074 696f 6e61 6c20 696e  with optional in
│ │ │ │ -00013e80: 7075 7473 2c3a 0a20 2020 2020 202a 2043  puts,:.      * C
│ │ │ │ -00013e90: 6865 636b 203d 3e20 2e2e 2e2c 2064 6566  heck => ..., def
│ │ │ │ -00013ea0: 6175 6c74 2076 616c 7565 2066 616c 7365  ault value false
│ │ │ │ -00013eb0: 0a20 2020 2020 202a 2047 7261 6469 6e67  .      * Grading
│ │ │ │ -00013ec0: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74   => ..., default
│ │ │ │ -00013ed0: 2076 616c 7565 2032 0a20 2020 2020 202a   value 2.      *
│ │ │ │ -00013ee0: 2056 6172 6961 626c 6573 203d 3e20 2e2e   Variables => ..
│ │ │ │ -00013ef0: 2e2c 2064 6566 6175 6c74 2076 616c 7565  ., default value
│ │ │ │ -00013f00: 2073 0a20 202a 204f 7574 7075 7473 3a0a   s.  * Outputs:.
│ │ │ │ -00013f10: 2020 2020 2020 2a20 6430 2c20 6120 2a6e        * d0, a *n
│ │ │ │ -00013f20: 6f74 6520 6d61 7472 6978 3a20 284d 6163  ote matrix: (Mac
│ │ │ │ -00013f30: 6175 6c61 7932 446f 6329 4d61 7472 6978  aulay2Doc)Matrix
│ │ │ │ -00013f40: 2c2c 206d 6170 206f 6620 6672 6565 206d  ,, map of free m
│ │ │ │ -00013f50: 6f64 756c 6573 206f 7665 7220 616e 0a20  odules over an. 
│ │ │ │ -00013f60: 2020 2020 2020 2065 6e6c 6172 6765 6420         enlarged 
│ │ │ │ -00013f70: 7269 6e67 0a20 2020 2020 202a 2064 312c  ring.      * d1,
│ │ │ │ -00013f80: 2061 202a 6e6f 7465 206d 6174 7269 783a   a *note matrix:
│ │ │ │ -00013f90: 2028 4d61 6361 756c 6179 3244 6f63 294d   (Macaulay2Doc)M
│ │ │ │ -00013fa0: 6174 7269 782c 2c20 6d61 7020 6f66 2066  atrix,, map of f
│ │ │ │ -00013fb0: 7265 6520 6d6f 6475 6c65 7320 6f76 6572  ree modules over
│ │ │ │ -00013fc0: 2061 6e0a 2020 2020 2020 2020 656e 6c61   an.        enla
│ │ │ │ -00013fd0: 7267 6564 2072 696e 670a 0a44 6573 6372  rged ring..Descr
│ │ │ │ -00013fe0: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d  iption.=========
│ │ │ │ -00013ff0: 3d3d 0a0a 4173 7375 6d65 2074 6861 7420  ==..Assume that 
│ │ │ │ -00014000: 4d20 6973 2064 6566 696e 6564 206f 7665  M is defined ove
│ │ │ │ -00014010: 7220 6120 7269 6e67 206f 6620 7468 6520  r a ring of the 
│ │ │ │ -00014020: 666f 726d 2052 6261 7220 3d20 522f 2866  form Rbar = R/(f
│ │ │ │ -00014030: 5f30 2e2e 665f 7b63 2d31 7d29 2c20 610a  _0..f_{c-1}), a.
│ │ │ │ -00014040: 636f 6d70 6c65 7465 2069 6e74 6572 7365  complete interse
│ │ │ │ -00014050: 6374 696f 6e2c 2061 6e64 2074 6861 7420  ction, and that 
│ │ │ │ -00014060: 4d20 6861 7320 6120 6669 6e69 7465 2066  M has a finite f
│ │ │ │ -00014070: 7265 6520 7265 736f 6c75 7469 6f6e 2047  ree resolution G
│ │ │ │ -00014080: 206f 7665 7220 522e 2049 6e0a 7468 6973   over R. In.this
│ │ │ │ -00014090: 2063 6173 6520 4d20 6861 7320 6120 6672   case M has a fr
│ │ │ │ -000140a0: 6565 2072 6573 6f6c 7574 696f 6e20 4620  ee resolution F 
│ │ │ │ -000140b0: 6f76 6572 2052 6261 7220 7768 6f73 6520  over Rbar whose 
│ │ │ │ -000140c0: 6475 616c 2c20 465e 2a20 6973 2061 2066  dual, F^* is a f
│ │ │ │ -000140d0: 696e 6974 656c 790a 6765 6e65 7261 7465  initely.generate
│ │ │ │ -000140e0: 642c 205a 2d67 7261 6465 6420 6672 6565  d, Z-graded free
│ │ │ │ -000140f0: 206d 6f64 756c 6520 6f76 6572 2061 2072   module over a r
│ │ │ │ -00014100: 696e 6720 5362 6172 5c63 6f6e 6720 6b6b  ing Sbar\cong kk
│ │ │ │ -00014110: 5b73 5f30 2e2e 735f 7b63 2d31 7d2c 6765  [s_0..s_{c-1},ge
│ │ │ │ -00014120: 6e73 0a52 6261 725d 2c20 7768 6572 6520  ns.Rbar], where 
│ │ │ │ -00014130: 7468 6520 6465 6772 6565 7320 6f66 2074  the degrees of t
│ │ │ │ -00014140: 6865 2073 5f69 2061 7265 207b 2d32 2c20  he s_i are {-2, 
│ │ │ │ -00014150: 2d64 6567 7265 6520 665f 697d 2e20 5468  -degree f_i}. Th
│ │ │ │ -00014160: 6973 2072 6573 6f6c 7574 696f 6e20 6973  is resolution is
│ │ │ │ -00014170: 0a69 7320 636f 6e73 7472 7563 7465 6420  .is constructed 
│ │ │ │ -00014180: 6672 6f6d 2074 6865 2064 7561 6c20 6f66  from the dual of
│ │ │ │ -00014190: 2047 2c20 746f 6765 7468 6572 2077 6974   G, together wit
│ │ │ │ -000141a0: 6820 7468 6520 6475 616c 7320 6f66 2074  h the duals of t
│ │ │ │ -000141b0: 6865 2068 6967 6865 720a 686f 6d6f 746f  he higher.homoto
│ │ │ │ -000141c0: 7069 6573 206f 6e20 4720 6465 6669 6e65  pies on G define
│ │ │ │ -000141d0: 6420 6279 2045 6973 656e 6275 642e 0a0a  d by Eisenbud...
│ │ │ │ -000141e0: 5468 6520 6675 6e63 7469 6f6e 2072 6574  The function ret
│ │ │ │ -000141f0: 7572 6e73 2074 6865 2064 6966 6665 7265  urns the differe
│ │ │ │ -00014200: 6e74 6961 6c73 2064 303a 465e 2a5f 7b65  ntials d0:F^*_{e
│ │ │ │ -00014210: 7665 6e7d 205c 746f 2046 5e2a 5f7b 6f64  ven} \to F^*_{od
│ │ │ │ -00014220: 647d 2061 6e64 0a64 313a 465e 2a5f 7b6f  d} and.d1:F^*_{o
│ │ │ │ -00014230: 6464 7d5c 746f 2046 5e2a 5f7b 6576 656e  dd}\to F^*_{even
│ │ │ │ -00014240: 7d2e 0a0a 5468 6520 6d61 7073 2064 302c  }...The maps d0,
│ │ │ │ -00014250: 6431 2066 6f72 6d20 6120 6d61 7472 6978  d1 form a matrix
│ │ │ │ -00014260: 2066 6163 746f 7269 7a61 7469 6f6e 206f   factorization o
│ │ │ │ -00014270: 6620 7375 6d28 632c 2069 2d3e 735f 692a  f sum(c, i->s_i*
│ │ │ │ -00014280: 665f 6929 2e20 5468 6520 6861 7665 2074  f_i). The have t
│ │ │ │ -00014290: 6865 0a70 726f 7065 7274 7920 7468 6174  he.property that
│ │ │ │ -000142a0: 2066 6f72 2061 6e79 2052 6261 7220 6d6f   for any Rbar mo
│ │ │ │ -000142b0: 6475 6c65 204e 2c0a 0a48 485f 3120 636f  dule N,..HH_1 co
│ │ │ │ -000142c0: 6d70 6c65 7820 5c7b 6430 2a2a 4e2c 2064  mplex \{d0**N, d
│ │ │ │ -000142d0: 312a 2a4e 5c7d 203d 2045 7874 5e7b 6576  1**N\} = Ext^{ev
│ │ │ │ -000142e0: 656e 7d5f 7b52 6261 727d 284d 2c4e 290a  en}_{Rbar}(M,N).
│ │ │ │ -000142f0: 0a53 5e7b 7b31 2c30 7d7d 2a2a 4848 5f31  .S^{{1,0}}**HH_1
│ │ │ │ -00014300: 2063 6f6d 706c 6578 205c 7b53 5e7b 7b2d   complex \{S^{{-
│ │ │ │ -00014310: 322c 307d 7d2a 2a64 312a 2a4e 2c20 6430  2,0}}**d1**N, d0
│ │ │ │ -00014320: 2a2a 4e5c 7d20 3d20 4578 745e 7b6f 6464  **N\} = Ext^{odd
│ │ │ │ -00014330: 7d5f 7b52 6261 727d 284d 2c4e 290a 0a54  }_{Rbar}(M,N)..T
│ │ │ │ -00014340: 6869 7320 6973 2065 6e63 6f64 6564 2069  his is encoded i
│ │ │ │ -00014350: 6e20 7468 6520 7363 7269 7074 206e 6577  n the script new
│ │ │ │ -00014360: 4578 740a 0a4f 7074 696f 6e20 6465 6661  Ext..Option defa
│ │ │ │ -00014370: 756c 7473 3a20 4368 6563 6b3d 3e66 616c  ults: Check=>fal
│ │ │ │ -00014380: 7365 2056 6172 6961 626c 6573 3d3e 6765  se Variables=>ge
│ │ │ │ -00014390: 7453 796d 626f 6c20 2273 222c 2047 7261  tSymbol "s", Gra
│ │ │ │ -000143a0: 6469 6e67 203d 3e32 7d0a 0a49 6620 4772  ding =>2}..If Gr
│ │ │ │ -000143b0: 6164 696e 6720 3d3e 312c 2074 6865 6e20  ading =>1, then 
│ │ │ │ -000143c0: 6120 7369 6e67 6c79 2067 7261 6465 6420  a singly graded 
│ │ │ │ -000143d0: 7265 7375 6c74 2069 7320 7265 7475 726e  result is return
│ │ │ │ -000143e0: 6564 2028 6a75 7374 2066 6f72 6765 7474  ed (just forgett
│ │ │ │ -000143f0: 696e 6720 7468 650a 686f 6d6f 6c6f 6769  ing the.homologi
│ │ │ │ -00014400: 6361 6c20 6772 6164 696e 672e 290a 0a0a  cal grading.)...
│ │ │ │ -00014410: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00013d90: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055  *********..  * U
│ │ │ │ +00013da0: 7361 6765 3a20 0a20 2020 2020 2020 2028  sage: .        (
│ │ │ │ +00013db0: 6430 2c64 3129 203d 2020 4569 7365 6e62  d0,d1) =  Eisenb
│ │ │ │ +00013dc0: 7564 5368 616d 6173 6854 6f74 616c 204d  udShamashTotal M
│ │ │ │ +00013dd0: 0a20 202a 2049 6e70 7574 733a 0a20 2020  .  * Inputs:.   
│ │ │ │ +00013de0: 2020 202a 204d 2c20 6120 2a6e 6f74 6520     * M, a *note 
│ │ │ │ +00013df0: 6d6f 6475 6c65 3a20 284d 6163 6175 6c61  module: (Macaula
│ │ │ │ +00013e00: 7932 446f 6329 4d6f 6475 6c65 2c2c 206f  y2Doc)Module,, o
│ │ │ │ +00013e10: 7665 7220 6120 636f 6d70 6c65 7465 2069  ver a complete i
│ │ │ │ +00013e20: 6e74 6572 7365 6374 696f 6e0a 2020 2a20  ntersection.  * 
│ │ │ │ +00013e30: 2a6e 6f74 6520 4f70 7469 6f6e 616c 2069  *note Optional i
│ │ │ │ +00013e40: 6e70 7574 733a 2028 4d61 6361 756c 6179  nputs: (Macaulay
│ │ │ │ +00013e50: 3244 6f63 2975 7369 6e67 2066 756e 6374  2Doc)using funct
│ │ │ │ +00013e60: 696f 6e73 2077 6974 6820 6f70 7469 6f6e  ions with option
│ │ │ │ +00013e70: 616c 2069 6e70 7574 732c 3a0a 2020 2020  al inputs,:.    
│ │ │ │ +00013e80: 2020 2a20 4368 6563 6b20 3d3e 202e 2e2e    * Check => ...
│ │ │ │ +00013e90: 2c20 6465 6661 756c 7420 7661 6c75 6520  , default value 
│ │ │ │ +00013ea0: 6661 6c73 650a 2020 2020 2020 2a20 4772  false.      * Gr
│ │ │ │ +00013eb0: 6164 696e 6720 3d3e 202e 2e2e 2c20 6465  ading => ..., de
│ │ │ │ +00013ec0: 6661 756c 7420 7661 6c75 6520 320a 2020  fault value 2.  
│ │ │ │ +00013ed0: 2020 2020 2a20 5661 7269 6162 6c65 7320      * Variables 
│ │ │ │ +00013ee0: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420  => ..., default 
│ │ │ │ +00013ef0: 7661 6c75 6520 730a 2020 2a20 4f75 7470  value s.  * Outp
│ │ │ │ +00013f00: 7574 733a 0a20 2020 2020 202a 2064 302c  uts:.      * d0,
│ │ │ │ +00013f10: 2061 202a 6e6f 7465 206d 6174 7269 783a   a *note matrix:
│ │ │ │ +00013f20: 2028 4d61 6361 756c 6179 3244 6f63 294d   (Macaulay2Doc)M
│ │ │ │ +00013f30: 6174 7269 782c 2c20 6d61 7020 6f66 2066  atrix,, map of f
│ │ │ │ +00013f40: 7265 6520 6d6f 6475 6c65 7320 6f76 6572  ree modules over
│ │ │ │ +00013f50: 2061 6e0a 2020 2020 2020 2020 656e 6c61   an.        enla
│ │ │ │ +00013f60: 7267 6564 2072 696e 670a 2020 2020 2020  rged ring.      
│ │ │ │ +00013f70: 2a20 6431 2c20 6120 2a6e 6f74 6520 6d61  * d1, a *note ma
│ │ │ │ +00013f80: 7472 6978 3a20 284d 6163 6175 6c61 7932  trix: (Macaulay2
│ │ │ │ +00013f90: 446f 6329 4d61 7472 6978 2c2c 206d 6170  Doc)Matrix,, map
│ │ │ │ +00013fa0: 206f 6620 6672 6565 206d 6f64 756c 6573   of free modules
│ │ │ │ +00013fb0: 206f 7665 7220 616e 0a20 2020 2020 2020   over an.       
│ │ │ │ +00013fc0: 2065 6e6c 6172 6765 6420 7269 6e67 0a0a   enlarged ring..
│ │ │ │ +00013fd0: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
│ │ │ │ +00013fe0: 3d3d 3d3d 3d3d 3d0a 0a41 7373 756d 6520  =======..Assume 
│ │ │ │ +00013ff0: 7468 6174 204d 2069 7320 6465 6669 6e65  that M is define
│ │ │ │ +00014000: 6420 6f76 6572 2061 2072 696e 6720 6f66  d over a ring of
│ │ │ │ +00014010: 2074 6865 2066 6f72 6d20 5262 6172 203d   the form Rbar =
│ │ │ │ +00014020: 2052 2f28 665f 302e 2e66 5f7b 632d 317d   R/(f_0..f_{c-1}
│ │ │ │ +00014030: 292c 2061 0a63 6f6d 706c 6574 6520 696e  ), a.complete in
│ │ │ │ +00014040: 7465 7273 6563 7469 6f6e 2c20 616e 6420  tersection, and 
│ │ │ │ +00014050: 7468 6174 204d 2068 6173 2061 2066 696e  that M has a fin
│ │ │ │ +00014060: 6974 6520 6672 6565 2072 6573 6f6c 7574  ite free resolut
│ │ │ │ +00014070: 696f 6e20 4720 6f76 6572 2052 2e20 496e  ion G over R. In
│ │ │ │ +00014080: 0a74 6869 7320 6361 7365 204d 2068 6173  .this case M has
│ │ │ │ +00014090: 2061 2066 7265 6520 7265 736f 6c75 7469   a free resoluti
│ │ │ │ +000140a0: 6f6e 2046 206f 7665 7220 5262 6172 2077  on F over Rbar w
│ │ │ │ +000140b0: 686f 7365 2064 7561 6c2c 2046 5e2a 2069  hose dual, F^* i
│ │ │ │ +000140c0: 7320 6120 6669 6e69 7465 6c79 0a67 656e  s a finitely.gen
│ │ │ │ +000140d0: 6572 6174 6564 2c20 5a2d 6772 6164 6564  erated, Z-graded
│ │ │ │ +000140e0: 2066 7265 6520 6d6f 6475 6c65 206f 7665   free module ove
│ │ │ │ +000140f0: 7220 6120 7269 6e67 2053 6261 725c 636f  r a ring Sbar\co
│ │ │ │ +00014100: 6e67 206b 6b5b 735f 302e 2e73 5f7b 632d  ng kk[s_0..s_{c-
│ │ │ │ +00014110: 317d 2c67 656e 730a 5262 6172 5d2c 2077  1},gens.Rbar], w
│ │ │ │ +00014120: 6865 7265 2074 6865 2064 6567 7265 6573  here the degrees
│ │ │ │ +00014130: 206f 6620 7468 6520 735f 6920 6172 6520   of the s_i are 
│ │ │ │ +00014140: 7b2d 322c 202d 6465 6772 6565 2066 5f69  {-2, -degree f_i
│ │ │ │ +00014150: 7d2e 2054 6869 7320 7265 736f 6c75 7469  }. This resoluti
│ │ │ │ +00014160: 6f6e 2069 730a 6973 2063 6f6e 7374 7275  on is.is constru
│ │ │ │ +00014170: 6374 6564 2066 726f 6d20 7468 6520 6475  cted from the du
│ │ │ │ +00014180: 616c 206f 6620 472c 2074 6f67 6574 6865  al of G, togethe
│ │ │ │ +00014190: 7220 7769 7468 2074 6865 2064 7561 6c73  r with the duals
│ │ │ │ +000141a0: 206f 6620 7468 6520 6869 6768 6572 0a68   of the higher.h
│ │ │ │ +000141b0: 6f6d 6f74 6f70 6965 7320 6f6e 2047 2064  omotopies on G d
│ │ │ │ +000141c0: 6566 696e 6564 2062 7920 4569 7365 6e62  efined by Eisenb
│ │ │ │ +000141d0: 7564 2e0a 0a54 6865 2066 756e 6374 696f  ud...The functio
│ │ │ │ +000141e0: 6e20 7265 7475 726e 7320 7468 6520 6469  n returns the di
│ │ │ │ +000141f0: 6666 6572 656e 7469 616c 7320 6430 3a46  fferentials d0:F
│ │ │ │ +00014200: 5e2a 5f7b 6576 656e 7d20 5c74 6f20 465e  ^*_{even} \to F^
│ │ │ │ +00014210: 2a5f 7b6f 6464 7d20 616e 640a 6431 3a46  *_{odd} and.d1:F
│ │ │ │ +00014220: 5e2a 5f7b 6f64 647d 5c74 6f20 465e 2a5f  ^*_{odd}\to F^*_
│ │ │ │ +00014230: 7b65 7665 6e7d 2e0a 0a54 6865 206d 6170  {even}...The map
│ │ │ │ +00014240: 7320 6430 2c64 3120 666f 726d 2061 206d  s d0,d1 form a m
│ │ │ │ +00014250: 6174 7269 7820 6661 6374 6f72 697a 6174  atrix factorizat
│ │ │ │ +00014260: 696f 6e20 6f66 2073 756d 2863 2c20 692d  ion of sum(c, i-
│ │ │ │ +00014270: 3e73 5f69 2a66 5f69 292e 2054 6865 2068  >s_i*f_i). The h
│ │ │ │ +00014280: 6176 6520 7468 650a 7072 6f70 6572 7479  ave the.property
│ │ │ │ +00014290: 2074 6861 7420 666f 7220 616e 7920 5262   that for any Rb
│ │ │ │ +000142a0: 6172 206d 6f64 756c 6520 4e2c 0a0a 4848  ar module N,..HH
│ │ │ │ +000142b0: 5f31 2063 6f6d 706c 6578 205c 7b64 302a  _1 complex \{d0*
│ │ │ │ +000142c0: 2a4e 2c20 6431 2a2a 4e5c 7d20 3d20 4578  *N, d1**N\} = Ex
│ │ │ │ +000142d0: 745e 7b65 7665 6e7d 5f7b 5262 6172 7d28  t^{even}_{Rbar}(
│ │ │ │ +000142e0: 4d2c 4e29 0a0a 535e 7b7b 312c 307d 7d2a  M,N)..S^{{1,0}}*
│ │ │ │ +000142f0: 2a48 485f 3120 636f 6d70 6c65 7820 5c7b  *HH_1 complex \{
│ │ │ │ +00014300: 535e 7b7b 2d32 2c30 7d7d 2a2a 6431 2a2a  S^{{-2,0}}**d1**
│ │ │ │ +00014310: 4e2c 2064 302a 2a4e 5c7d 203d 2045 7874  N, d0**N\} = Ext
│ │ │ │ +00014320: 5e7b 6f64 647d 5f7b 5262 6172 7d28 4d2c  ^{odd}_{Rbar}(M,
│ │ │ │ +00014330: 4e29 0a0a 5468 6973 2069 7320 656e 636f  N)..This is enco
│ │ │ │ +00014340: 6465 6420 696e 2074 6865 2073 6372 6970  ded in the scrip
│ │ │ │ +00014350: 7420 6e65 7745 7874 0a0a 4f70 7469 6f6e  t newExt..Option
│ │ │ │ +00014360: 2064 6566 6175 6c74 733a 2043 6865 636b   defaults: Check
│ │ │ │ +00014370: 3d3e 6661 6c73 6520 5661 7269 6162 6c65  =>false Variable
│ │ │ │ +00014380: 733d 3e67 6574 5379 6d62 6f6c 2022 7322  s=>getSymbol "s"
│ │ │ │ +00014390: 2c20 4772 6164 696e 6720 3d3e 327d 0a0a  , Grading =>2}..
│ │ │ │ +000143a0: 4966 2047 7261 6469 6e67 203d 3e31 2c20  If Grading =>1, 
│ │ │ │ +000143b0: 7468 656e 2061 2073 696e 676c 7920 6772  then a singly gr
│ │ │ │ +000143c0: 6164 6564 2072 6573 756c 7420 6973 2072  aded result is r
│ │ │ │ +000143d0: 6574 7572 6e65 6420 286a 7573 7420 666f  eturned (just fo
│ │ │ │ +000143e0: 7267 6574 7469 6e67 2074 6865 0a68 6f6d  rgetting the.hom
│ │ │ │ +000143f0: 6f6c 6f67 6963 616c 2067 7261 6469 6e67  ological grading
│ │ │ │ +00014400: 2e29 0a0a 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d  .)....+---------
│ │ │ │ +00014410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00014460: 0a7c 6931 203a 206e 203d 2033 2020 2020  .|i1 : n = 3    
│ │ │ │ +00014450: 2d2d 2d2d 2b0a 7c69 3120 3a20 6e20 3d20  ----+.|i1 : n = 
│ │ │ │ +00014460: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │  00014470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000144a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000144b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000144a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000144b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000144c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000144d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000144e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000144f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014500: 0a7c 6f31 203d 2033 2020 2020 2020 2020  .|o1 = 3        
│ │ │ │ +000144f0: 2020 2020 7c0a 7c6f 3120 3d20 3320 2020      |.|o1 = 3   
│ │ │ │ +00014500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014540: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014550: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00014540: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00014550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000145a0: 0a7c 6932 203a 2063 203d 2032 2020 2020  .|i2 : c = 2    
│ │ │ │ +00014590: 2d2d 2d2d 2b0a 7c69 3220 3a20 6320 3d20  ----+.|i2 : c = 
│ │ │ │ +000145a0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  000145b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000145c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000145d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000145e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000145f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000145e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000145f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014630: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014640: 0a7c 6f32 203d 2032 2020 2020 2020 2020  .|o2 = 2        
│ │ │ │ +00014630: 2020 2020 7c0a 7c6f 3220 3d20 3220 2020      |.|o2 = 2   
│ │ │ │ +00014640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014680: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014690: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00014680: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00014690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000146a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000146b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000146c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000146d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000146e0: 0a7c 6933 203a 206b 6b20 3d20 5a5a 2f31  .|i3 : kk = ZZ/1
│ │ │ │ -000146f0: 3031 2020 2020 2020 2020 2020 2020 2020  01              
│ │ │ │ +000146d0: 2d2d 2d2d 2b0a 7c69 3320 3a20 6b6b 203d  ----+.|i3 : kk =
│ │ │ │ +000146e0: 205a 5a2f 3130 3120 2020 2020 2020 2020   ZZ/101         
│ │ │ │ +000146f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014720: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014730: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00014720: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00014730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014770: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014780: 0a7c 6f33 203d 206b 6b20 2020 2020 2020  .|o3 = kk       
│ │ │ │ +00014770: 2020 2020 7c0a 7c6f 3320 3d20 6b6b 2020      |.|o3 = kk  
│ │ │ │ +00014780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000147a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000147b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000147c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000147d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000147c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000147d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000147e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000147f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014810: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014820: 0a7c 6f33 203a 2051 756f 7469 656e 7452  .|o3 : QuotientR
│ │ │ │ -00014830: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │ +00014810: 2020 2020 7c0a 7c6f 3320 3a20 5175 6f74      |.|o3 : Quot
│ │ │ │ +00014820: 6965 6e74 5269 6e67 2020 2020 2020 2020  ientRing        
│ │ │ │ +00014830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014860: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014870: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00014860: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00014870: 2d2d 2d2d 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 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000148b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000148c0: 0a7c 6934 203a 2052 203d 206b 6b5b 785f  .|i4 : R = kk[x_
│ │ │ │ -000148d0: 302e 2e78 5f28 6e2d 3129 5d20 2020 2020  0..x_(n-1)]     
│ │ │ │ +000148b0: 2d2d 2d2d 2b0a 7c69 3420 3a20 5220 3d20  ----+.|i4 : R = 
│ │ │ │ +000148c0: 6b6b 5b78 5f30 2e2e 785f 286e 2d31 295d  kk[x_0..x_(n-1)]
│ │ │ │ +000148d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000148e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000148f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014900: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014910: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00014900: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00014910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014920: 2020 2020 2020 2020 2020 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 207c                 |
│ │ │ │ -00014960: 0a7c 6f34 203d 2052 2020 2020 2020 2020  .|o4 = R        
│ │ │ │ +00014950: 2020 2020 7c0a 7c6f 3420 3d20 5220 2020      |.|o4 = R   
│ │ │ │ +00014960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000149a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000149b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000149a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000149b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000149c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000149d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000149e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000149f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014a00: 0a7c 6f34 203a 2050 6f6c 796e 6f6d 6961  .|o4 : Polynomia
│ │ │ │ -00014a10: 6c52 696e 6720 2020 2020 2020 2020 2020  lRing           
│ │ │ │ +000149f0: 2020 2020 7c0a 7c6f 3420 3a20 506f 6c79      |.|o4 : Poly
│ │ │ │ +00014a00: 6e6f 6d69 616c 5269 6e67 2020 2020 2020  nomialRing      
│ │ │ │ +00014a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014a40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014a50: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00014a40: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00014a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00014aa0: 0a7c 6935 203a 2049 203d 2069 6465 616c  .|i5 : I = ideal
│ │ │ │ -00014ab0: 2878 5f30 5e32 2c20 785f 325e 3329 2020  (x_0^2, x_2^3)  
│ │ │ │ +00014a90: 2d2d 2d2d 2b0a 7c69 3520 3a20 4920 3d20  ----+.|i5 : I = 
│ │ │ │ +00014aa0: 6964 6561 6c28 785f 305e 322c 2078 5f32  ideal(x_0^2, x_2
│ │ │ │ +00014ab0: 5e33 2920 2020 2020 2020 2020 2020 2020  ^3)             
│ │ │ │  00014ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014ae0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014af0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00014ae0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00014af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014b30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014b40: 0a7c 2020 2020 2020 2020 2020 2020 2032  .|             2
│ │ │ │ -00014b50: 2020 2033 2020 2020 2020 2020 2020 2020     3            
│ │ │ │ +00014b30: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00014b40: 2020 2020 3220 2020 3320 2020 2020 2020      2   3       
│ │ │ │ +00014b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014b80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014b90: 0a7c 6f35 203d 2069 6465 616c 2028 7820  .|o5 = ideal (x 
│ │ │ │ -00014ba0: 2c20 7820 2920 2020 2020 2020 2020 2020  , x )           
│ │ │ │ +00014b80: 2020 2020 7c0a 7c6f 3520 3d20 6964 6561      |.|o5 = idea
│ │ │ │ +00014b90: 6c20 2878 202c 2078 2029 2020 2020 2020  l (x , x )      
│ │ │ │ +00014ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014bd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014be0: 0a7c 2020 2020 2020 2020 2020 2020 2030  .|             0
│ │ │ │ -00014bf0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00014bd0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00014be0: 2020 2020 3020 2020 3220 2020 2020 2020      0   2       
│ │ │ │ +00014bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014c20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014c30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00014c20: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00014c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014c70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014c80: 0a7c 6f35 203a 2049 6465 616c 206f 6620  .|o5 : Ideal of 
│ │ │ │ -00014c90: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │ +00014c70: 2020 2020 7c0a 7c6f 3520 3a20 4964 6561      |.|o5 : Idea
│ │ │ │ +00014c80: 6c20 6f66 2052 2020 2020 2020 2020 2020  l of R          
│ │ │ │ +00014c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014cc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014cd0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00014cc0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00014cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00014d20: 0a7c 6936 203a 2066 6620 3d20 6765 6e73  .|i6 : ff = gens
│ │ │ │ -00014d30: 2049 2020 2020 2020 2020 2020 2020 2020   I              
│ │ │ │ +00014d10: 2d2d 2d2d 2b0a 7c69 3620 3a20 6666 203d  ----+.|i6 : ff =
│ │ │ │ +00014d20: 2067 656e 7320 4920 2020 2020 2020 2020   gens I         
│ │ │ │ +00014d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014d60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014d70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00014d60: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00014d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014db0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014dc0: 0a7c 6f36 203d 207c 2078 5f30 5e32 2078  .|o6 = | x_0^2 x
│ │ │ │ -00014dd0: 5f32 5e33 207c 2020 2020 2020 2020 2020  _2^3 |          
│ │ │ │ +00014db0: 2020 2020 7c0a 7c6f 3620 3d20 7c20 785f      |.|o6 = | x_
│ │ │ │ +00014dc0: 305e 3220 785f 325e 3320 7c20 2020 2020  0^2 x_2^3 |     
│ │ │ │ +00014dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014e00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014e10: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00014e00: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00014e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014e50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014e60: 0a7c 2020 2020 2020 2020 2020 2020 2031  .|             1
│ │ │ │ -00014e70: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +00014e50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00014e60: 2020 2020 3120 2020 2020 2032 2020 2020      1      2    
│ │ │ │ +00014e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014ea0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014eb0: 0a7c 6f36 203a 204d 6174 7269 7820 5220  .|o6 : Matrix R 
│ │ │ │ -00014ec0: 203c 2d2d 2052 2020 2020 2020 2020 2020   <-- R          
│ │ │ │ +00014ea0: 2020 2020 7c0a 7c6f 3620 3a20 4d61 7472      |.|o6 : Matr
│ │ │ │ +00014eb0: 6978 2052 2020 3c2d 2d20 5220 2020 2020  ix R  <-- R     
│ │ │ │ +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 2020 207c                 |
│ │ │ │ -00014f00: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00014ef0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00014f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00014f50: 0a7c 6937 203a 2052 6261 7220 3d20 522f  .|i7 : Rbar = R/
│ │ │ │ -00014f60: 4920 2020 2020 2020 2020 2020 2020 2020  I               
│ │ │ │ +00014f40: 2d2d 2d2d 2b0a 7c69 3720 3a20 5262 6172  ----+.|i7 : Rbar
│ │ │ │ +00014f50: 203d 2052 2f49 2020 2020 2020 2020 2020   = R/I          
│ │ │ │ +00014f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014f90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014fa0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00014f90: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +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 2020 2020                  
│ │ │ │ -00014fe0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014ff0: 0a7c 6f37 203d 2052 6261 7220 2020 2020  .|o7 = Rbar     
│ │ │ │ +00014fe0: 2020 2020 7c0a 7c6f 3720 3d20 5262 6172      |.|o7 = Rbar
│ │ │ │ +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 2020 2020                  
│ │ │ │ -00015030: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015040: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00015030: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +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 2020 2020                  
│ │ │ │ -00015080: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015090: 0a7c 6f37 203a 2051 756f 7469 656e 7452  .|o7 : QuotientR
│ │ │ │ -000150a0: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │ +00015080: 2020 2020 7c0a 7c6f 3720 3a20 5175 6f74      |.|o7 : Quot
│ │ │ │ +00015090: 6965 6e74 5269 6e67 2020 2020 2020 2020  ientRing        
│ │ │ │ +000150a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000150b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000150c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000150d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000150e0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000150d0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +000150e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000150f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00015130: 0a7c 6938 203a 2062 6172 203d 206d 6170  .|i8 : bar = map
│ │ │ │ -00015140: 2852 6261 722c 2052 2920 2020 2020 2020  (Rbar, R)       
│ │ │ │ +00015120: 2d2d 2d2d 2b0a 7c69 3820 3a20 6261 7220  ----+.|i8 : bar 
│ │ │ │ +00015130: 3d20 6d61 7028 5262 6172 2c20 5229 2020  = map(Rbar, R)  
│ │ │ │ +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 2020 2020                  
│ │ │ │ -00015170: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015180: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00015170: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +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 2020 2020                  
│ │ │ │ -000151c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000151d0: 0a7c 6f38 203d 206d 6170 2028 5262 6172  .|o8 = map (Rbar
│ │ │ │ -000151e0: 2c20 522c 207b 7820 2c20 7820 2c20 7820  , R, {x , x , x 
│ │ │ │ -000151f0: 7d29 2020 2020 2020 2020 2020 2020 2020  })              
│ │ │ │ +000151c0: 2020 2020 7c0a 7c6f 3820 3d20 6d61 7020      |.|o8 = map 
│ │ │ │ +000151d0: 2852 6261 722c 2052 2c20 7b78 202c 2078  (Rbar, R, {x , x
│ │ │ │ +000151e0: 202c 2078 207d 2920 2020 2020 2020 2020   , x })         
│ │ │ │ +000151f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015210: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015220: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00015230: 2020 2020 2020 2030 2020 2031 2020 2032         0   1   2
│ │ │ │ +00015210: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00015220: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ +00015230: 3120 2020 3220 2020 2020 2020 2020 2020  1   2           
│ │ │ │  00015240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015260: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015270: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00015260: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00015270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000152a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000152b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000152c0: 0a7c 6f38 203a 2052 696e 674d 6170 2052  .|o8 : RingMap R
│ │ │ │ -000152d0: 6261 7220 3c2d 2d20 5220 2020 2020 2020  bar <-- R       
│ │ │ │ +000152b0: 2020 2020 7c0a 7c6f 3820 3a20 5269 6e67      |.|o8 : Ring
│ │ │ │ +000152c0: 4d61 7020 5262 6172 203c 2d2d 2052 2020  Map Rbar <-- R  
│ │ │ │ +000152d0: 2020 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 2020 207c                 |
│ │ │ │ -00015310: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00015300: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00015310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00015360: 0a7c 6939 203a 204d 6261 7220 3d20 7072  .|i9 : Mbar = pr
│ │ │ │ -00015370: 756e 6520 636f 6b65 7220 7261 6e64 6f6d  une coker random
│ │ │ │ -00015380: 2852 6261 725e 312c 2052 6261 725e 7b2d  (Rbar^1, Rbar^{-
│ │ │ │ -00015390: 327d 2920 2020 2020 2020 2020 2020 2020  2})             
│ │ │ │ -000153a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000153b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00015350: 2d2d 2d2d 2b0a 7c69 3920 3a20 4d62 6172  ----+.|i9 : Mbar
│ │ │ │ +00015360: 203d 2070 7275 6e65 2063 6f6b 6572 2072   = prune coker r
│ │ │ │ +00015370: 616e 646f 6d28 5262 6172 5e31 2c20 5262  andom(Rbar^1, Rb
│ │ │ │ +00015380: 6172 5e7b 2d32 7d29 2020 2020 2020 2020  ar^{-2})        
│ │ │ │ +00015390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000153a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000153b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000153c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000153d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000153e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000153f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015400: 0a7c 6f39 203d 2063 6f6b 6572 6e65 6c20  .|o9 = cokernel 
│ │ │ │ -00015410: 7c20 785f 3078 5f31 2b32 3478 5f31 5e32  | x_0x_1+24x_1^2
│ │ │ │ -00015420: 2b34 3978 5f30 785f 322b 3378 5f31 785f  +49x_0x_2+3x_1x_
│ │ │ │ -00015430: 322b 3578 5f32 5e32 207c 2020 2020 2020  2+5x_2^2 |      
│ │ │ │ -00015440: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015450: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000153f0: 2020 2020 7c0a 7c6f 3920 3d20 636f 6b65      |.|o9 = coke
│ │ │ │ +00015400: 726e 656c 207c 2078 5f30 785f 312b 3234  rnel | x_0x_1+24
│ │ │ │ +00015410: 785f 315e 322b 3439 785f 3078 5f32 2b33  x_1^2+49x_0x_2+3
│ │ │ │ +00015420: 785f 3178 5f32 2b35 785f 325e 3220 7c20  x_1x_2+5x_2^2 | 
│ │ │ │ +00015430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015440: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00015450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015490: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000154a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000154b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000154c0: 2020 2020 3120 2020 2020 2020 2020 2020      1           
│ │ │ │ +00015490: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000154a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000154b0: 2020 2020 2020 2020 2031 2020 2020 2020           1      
│ │ │ │ +000154c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000154d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000154e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000154f0: 0a7c 6f39 203a 2052 6261 722d 6d6f 6475  .|o9 : Rbar-modu
│ │ │ │ -00015500: 6c65 2c20 7175 6f74 6965 6e74 206f 6620  le, quotient of 
│ │ │ │ -00015510: 5262 6172 2020 2020 2020 2020 2020 2020  Rbar            
│ │ │ │ +000154e0: 2020 2020 7c0a 7c6f 3920 3a20 5262 6172      |.|o9 : Rbar
│ │ │ │ +000154f0: 2d6d 6f64 756c 652c 2071 756f 7469 656e  -module, quotien
│ │ │ │ +00015500: 7420 6f66 2052 6261 7220 2020 2020 2020  t of Rbar       
│ │ │ │ +00015510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015530: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015540: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00015530: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00015540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00015590: 0a7c 6931 3020 3a20 2864 302c 6431 2920  .|i10 : (d0,d1) 
│ │ │ │ -000155a0: 3d20 4569 7365 6e62 7564 5368 616d 6173  = EisenbudShamas
│ │ │ │ -000155b0: 6854 6f74 616c 284d 6261 722c 4772 6164  hTotal(Mbar,Grad
│ │ │ │ -000155c0: 696e 6720 3d3e 3129 2020 2020 2020 2020  ing =>1)        
│ │ │ │ -000155d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000155e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00015580: 2d2d 2d2d 2b0a 7c69 3130 203a 2028 6430  ----+.|i10 : (d0
│ │ │ │ +00015590: 2c64 3129 203d 2045 6973 656e 6275 6453  ,d1) = EisenbudS
│ │ │ │ +000155a0: 6861 6d61 7368 546f 7461 6c28 4d62 6172  hamashTotal(Mbar
│ │ │ │ +000155b0: 2c47 7261 6469 6e67 203d 3e31 2920 2020  ,Grading =>1)   
│ │ │ │ +000155c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000155d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000155e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000155f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015620: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015630: 0a7c 6f31 3020 3d20 287b 2d32 7d20 7c20  .|o10 = ({-2} | 
│ │ │ │ -00015640: 785f 305e 3220 2020 2020 2020 2020 2020  x_0^2           
│ │ │ │ -00015650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015660: 2020 2020 2020 2030 2020 2020 2020 2020         0        
│ │ │ │ -00015670: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015680: 0a7c 2020 2020 2020 207b 2d32 7d20 7c20  .|       {-2} | 
│ │ │ │ -00015690: 785f 3078 5f31 2b32 3478 5f31 5e32 2b34  x_0x_1+24x_1^2+4
│ │ │ │ -000156a0: 3978 5f30 785f 322b 3378 5f31 785f 322b  9x_0x_2+3x_1x_2+
│ │ │ │ -000156b0: 3578 5f32 5e32 2033 3073 5f30 2020 2020  5x_2^2 30s_0    
│ │ │ │ -000156c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000156d0: 0a7c 2020 2020 2020 207b 2d33 7d20 7c20  .|       {-3} | 
│ │ │ │ -000156e0: 785f 325e 3320 2020 2020 2020 2020 2020  x_2^3           
│ │ │ │ -000156f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015700: 2020 2020 2020 2030 2020 2020 2020 2020         0        
│ │ │ │ -00015710: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015720: 0a7c 2020 2020 2020 207b 2d37 7d20 7c20  .|       {-7} | 
│ │ │ │ -00015730: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ -00015740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015750: 2020 2020 2020 2078 5f32 5e33 2020 2020         x_2^3    
│ │ │ │ -00015760: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015770: 0a7c 2020 2020 2020 2d2d 2d2d 2d2d 2d2d  .|      --------
│ │ │ │ +00015620: 2020 2020 7c0a 7c6f 3130 203d 2028 7b2d      |.|o10 = ({-
│ │ │ │ +00015630: 327d 207c 2078 5f30 5e32 2020 2020 2020  2} | x_0^2      
│ │ │ │ +00015640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015650: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ +00015660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015670: 2020 2020 7c0a 7c20 2020 2020 2020 7b2d      |.|       {-
│ │ │ │ +00015680: 327d 207c 2078 5f30 785f 312b 3234 785f  2} | x_0x_1+24x_
│ │ │ │ +00015690: 315e 322b 3439 785f 3078 5f32 2b33 785f  1^2+49x_0x_2+3x_
│ │ │ │ +000156a0: 3178 5f32 2b35 785f 325e 3220 3330 735f  1x_2+5x_2^2 30s_
│ │ │ │ +000156b0: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ +000156c0: 2020 2020 7c0a 7c20 2020 2020 2020 7b2d      |.|       {-
│ │ │ │ +000156d0: 337d 207c 2078 5f32 5e33 2020 2020 2020  3} | x_2^3      
│ │ │ │ +000156e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000156f0: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ +00015700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015710: 2020 2020 7c0a 7c20 2020 2020 2020 7b2d      |.|       {-
│ │ │ │ +00015720: 377d 207c 2030 2020 2020 2020 2020 2020  7} | 0          
│ │ │ │ +00015730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015740: 2020 2020 2020 2020 2020 2020 785f 325e              x_2^
│ │ │ │ +00015750: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ +00015760: 2020 2020 7c0a 7c20 2020 2020 202d 2d2d      |.|      ---
│ │ │ │ +00015770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000157a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000157b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -000157c0: 0a7c 2020 2020 2020 2d73 5f31 2020 2020  .|      -s_1    
│ │ │ │ +000157b0: 2d2d 2d2d 7c0a 7c20 2020 2020 202d 735f  ----|.|      -s_
│ │ │ │ +000157c0: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │  000157d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000157e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000157f0: 2020 3020 2020 2020 2020 207c 2c20 7b30    0        |, {0
│ │ │ │ -00015800: 7d20 207c 2020 2020 2020 2020 2020 207c  }  |           |
│ │ │ │ -00015810: 0a7c 2020 2020 2020 3020 2020 2020 2020  .|      0       
│ │ │ │ +000157e0: 2020 2020 2020 2030 2020 2020 2020 2020         0        
│ │ │ │ +000157f0: 7c2c 207b 307d 2020 7c20 2020 2020 2020  |, {0}  |       
│ │ │ │ +00015800: 2020 2020 7c0a 7c20 2020 2020 2030 2020      |.|      0  
│ │ │ │ +00015810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015840: 2020 2d73 5f31 2020 2020 207c 2020 7b2d    -s_1     |  {-
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│ │ │ │ -00015860: 0a7c 2020 2020 2020 735f 3020 2020 2020  .|      s_0     
│ │ │ │ +00015830: 2020 2020 2020 202d 735f 3120 2020 2020         -s_1     
│ │ │ │ +00015840: 7c20 207b 2d34 7d20 7c20 2020 2020 2020  |  {-4} |       
│ │ │ │ +00015850: 2020 2020 7c0a 7c20 2020 2020 2073 5f30      |.|      s_0
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│ │ │ │  00015870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015880: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000158b0: 0a7c 2020 2020 2020 3337 785f 3078 5f31  .|      37x_0x_1
│ │ │ │ -000158c0: 2d32 3178 5f31 5e32 2d35 785f 3078 5f32  -21x_1^2-5x_0x_2
│ │ │ │ -000158d0: 2b31 3078 5f31 785f 322d 3137 785f 325e  +10x_1x_2-17x_2^
│ │ │ │ -000158e0: 3220 2d33 3778 5f30 5e32 207c 2020 7b2d  2 -37x_0^2 |  {-
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│ │ │ │ -00015900: 0a7c 2020 2020 2020 2d2d 2d2d 2d2d 2d2d  .|      --------
│ │ │ │ +00015880: 2020 2020 2020 2030 2020 2020 2020 2020         0        
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│ │ │ │ +000158a0: 2020 2020 7c0a 7c20 2020 2020 2033 3778      |.|      37x
│ │ │ │ +000158b0: 5f30 785f 312d 3231 785f 315e 322d 3578  _0x_1-21x_1^2-5x
│ │ │ │ +000158c0: 5f30 785f 322b 3130 785f 3178 5f32 2d31  _0x_2+10x_1x_2-1
│ │ │ │ +000158d0: 3778 5f32 5e32 202d 3337 785f 305e 3220  7x_2^2 -37x_0^2 
│ │ │ │ +000158e0: 7c20 207b 2d35 7d20 7c20 2020 2020 2020  |  {-5} |       
│ │ │ │ +000158f0: 2020 2020 7c0a 7c20 2020 2020 202d 2d2d      |.|      ---
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│ │ │ │  00015910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00015950: 0a7c 2020 2020 2020 735f 3020 2020 2020  .|      s_0     
│ │ │ │ +00015940: 2d2d 2d2d 7c0a 7c20 2020 2020 2073 5f30  ----|.|      s_0
│ │ │ │ +00015950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015980: 2020 3020 2020 2020 2020 2020 2020 2020    0             
│ │ │ │ -00015990: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000159a0: 0a7c 2020 2020 2020 3337 785f 3078 5f31  .|      37x_0x_1
│ │ │ │ -000159b0: 2d32 3178 5f31 5e32 2d35 785f 3078 5f32  -21x_1^2-5x_0x_2
│ │ │ │ -000159c0: 2b31 3078 5f31 785f 322d 3137 785f 325e  +10x_1x_2-17x_2^
│ │ │ │ -000159d0: 3220 2d33 3778 5f30 5e32 2020 2020 2020  2 -37x_0^2      
│ │ │ │ -000159e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000159f0: 0a7c 2020 2020 2020 2d78 5f32 5e33 2020  .|      -x_2^3  
│ │ │ │ +00015970: 2020 2020 2020 2030 2020 2020 2020 2020         0        
│ │ │ │ +00015980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015990: 2020 2020 7c0a 7c20 2020 2020 2033 3778      |.|      37x
│ │ │ │ +000159a0: 5f30 785f 312d 3231 785f 315e 322d 3578  _0x_1-21x_1^2-5x
│ │ │ │ +000159b0: 5f30 785f 322b 3130 785f 3178 5f32 2d31  _0x_2+10x_1x_2-1
│ │ │ │ +000159c0: 3778 5f32 5e32 202d 3337 785f 305e 3220  7x_2^2 -37x_0^2 
│ │ │ │ +000159d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000159e0: 2020 2020 7c0a 7c20 2020 2020 202d 785f      |.|      -x_
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│ │ │ │  00015a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015a20: 2020 3020 2020 2020 2020 2020 2020 2020    0             
│ │ │ │ -00015a30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015a40: 0a7c 2020 2020 2020 3020 2020 2020 2020  .|      0       
│ │ │ │ +00015a10: 2020 2020 2020 2030 2020 2020 2020 2020         0        
│ │ │ │ +00015a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015a30: 2020 2020 7c0a 7c20 2020 2020 2030 2020      |.|      0  
│ │ │ │ +00015a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015a70: 2020 2d78 5f32 5e33 2020 2020 2020 2020    -x_2^3        
│ │ │ │ -00015a80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015a90: 0a7c 2020 2020 2020 2d2d 2d2d 2d2d 2d2d  .|      --------
│ │ │ │ +00015a60: 2020 2020 2020 202d 785f 325e 3320 2020         -x_2^3   
│ │ │ │ +00015a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015a80: 2020 2020 7c0a 7c20 2020 2020 202d 2d2d      |.|      ---
│ │ │ │ +00015a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00015ae0: 0a7c 2020 2020 2020 735f 3120 2020 2020  .|      s_1     
│ │ │ │ +00015ad0: 2d2d 2d2d 7c0a 7c20 2020 2020 2073 5f31  ----|.|      s_1
│ │ │ │ +00015ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015b00: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ -00015b10: 2020 2020 207c 2920 2020 2020 2020 2020       |)         
│ │ │ │ -00015b20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015b30: 0a7c 2020 2020 2020 3020 2020 2020 2020  .|      0       
│ │ │ │ +00015b00: 2020 2020 3020 2020 2020 7c29 2020 2020      0     |)    
│ │ │ │ +00015b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015b20: 2020 2020 7c0a 7c20 2020 2020 2030 2020      |.|      0  
│ │ │ │ +00015b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015b50: 2020 2020 2020 2020 2020 2020 2020 2073                 s
│ │ │ │ -00015b60: 5f31 2020 207c 2020 2020 2020 2020 2020  _1   |          
│ │ │ │ -00015b70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015b80: 0a7c 2020 2020 2020 785f 305e 3220 2020  .|      x_0^2   
│ │ │ │ +00015b50: 2020 2020 735f 3120 2020 7c20 2020 2020      s_1   |     
│ │ │ │ +00015b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015b70: 2020 2020 7c0a 7c20 2020 2020 2078 5f30      |.|      x_0
│ │ │ │ +00015b80: 5e32 2020 2020 2020 2020 2020 2020 2020  ^2              
│ │ │ │  00015b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015ba0: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ -00015bb0: 2020 2020 207c 2020 2020 2020 2020 2020       |          
│ │ │ │ -00015bc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015bd0: 0a7c 2020 2020 2020 785f 3078 5f31 2b32  .|      x_0x_1+2
│ │ │ │ -00015be0: 3478 5f31 5e32 2b34 3978 5f30 785f 322b  4x_1^2+49x_0x_2+
│ │ │ │ -00015bf0: 3378 5f31 785f 322b 3578 5f32 5e32 2033  3x_1x_2+5x_2^2 3
│ │ │ │ -00015c00: 3073 5f30 207c 2020 2020 2020 2020 2020  0s_0 |          
│ │ │ │ -00015c10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015c20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00015ba0: 2020 2020 3020 2020 2020 7c20 2020 2020      0     |     
│ │ │ │ +00015bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015bc0: 2020 2020 7c0a 7c20 2020 2020 2078 5f30      |.|      x_0
│ │ │ │ +00015bd0: 785f 312b 3234 785f 315e 322b 3439 785f  x_1+24x_1^2+49x_
│ │ │ │ +00015be0: 3078 5f32 2b33 785f 3178 5f32 2b35 785f  0x_2+3x_1x_2+5x_
│ │ │ │ +00015bf0: 325e 3220 3330 735f 3020 7c20 2020 2020  2^2 30s_0 |     
│ │ │ │ +00015c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015c10: 2020 2020 7c0a 7c20 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 2020 2020                  
│ │ │ │ -00015c60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015c70: 0a7c 6f31 3020 3a20 5365 7175 656e 6365  .|o10 : Sequence
│ │ │ │ +00015c60: 2020 2020 7c0a 7c6f 3130 203a 2053 6571      |.|o10 : Seq
│ │ │ │ +00015c70: 7565 6e63 6520 2020 2020 2020 2020 2020  uence           
│ │ │ │  00015c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015cb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015cc0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00015cb0: 2020 2020 7c0a 2b2d 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 2d2d 2d2d  ----------------
│ │ │ │ -00015d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00015d10: 0a7c 6931 3120 3a20 6430 2a64 3120 2020  .|i11 : d0*d1   
│ │ │ │ +00015d00: 2d2d 2d2d 2b0a 7c69 3131 203a 2064 302a  ----+.|i11 : d0*
│ │ │ │ +00015d10: 6431 2020 2020 2020 2020 2020 2020 2020  d1              
│ │ │ │  00015d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015d50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015d60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00015d50: 2020 2020 7c0a 7c20 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 2020 2020                  
│ │ │ │ -00015da0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015db0: 0a7c 6f31 3120 3d20 7b2d 327d 207c 2073  .|o11 = {-2} | s
│ │ │ │ -00015dc0: 5f31 785f 325e 332b 735f 3078 5f30 5e32  _1x_2^3+s_0x_0^2
│ │ │ │ -00015dd0: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ -00015de0: 2020 2030 2020 2020 2020 2020 2020 2020     0            
│ │ │ │ -00015df0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015e00: 0a7c 2020 2020 2020 7b2d 327d 207c 2030  .|      {-2} | 0
│ │ │ │ -00015e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015e20: 2073 5f31 785f 325e 332b 735f 3078 5f30   s_1x_2^3+s_0x_0
│ │ │ │ -00015e30: 5e32 2030 2020 2020 2020 2020 2020 2020  ^2 0            
│ │ │ │ -00015e40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015e50: 0a7c 2020 2020 2020 7b2d 337d 207c 2030  .|      {-3} | 0
│ │ │ │ -00015e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015e70: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ -00015e80: 2020 2073 5f31 785f 325e 332b 735f 3078     s_1x_2^3+s_0x
│ │ │ │ -00015e90: 5f30 5e32 2020 2020 2020 2020 2020 207c  _0^2           |
│ │ │ │ -00015ea0: 0a7c 2020 2020 2020 7b2d 377d 207c 2030  .|      {-7} | 0
│ │ │ │ -00015eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015ec0: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ -00015ed0: 2020 2030 2020 2020 2020 2020 2020 2020     0            
│ │ │ │ -00015ee0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015ef0: 0a7c 2020 2020 2020 2d2d 2d2d 2d2d 2d2d  .|      --------
│ │ │ │ +00015da0: 2020 2020 7c0a 7c6f 3131 203d 207b 2d32      |.|o11 = {-2
│ │ │ │ +00015db0: 7d20 7c20 735f 3178 5f32 5e33 2b73 5f30  } | s_1x_2^3+s_0
│ │ │ │ +00015dc0: 785f 305e 3220 3020 2020 2020 2020 2020  x_0^2 0         
│ │ │ │ +00015dd0: 2020 2020 2020 2020 3020 2020 2020 2020          0       
│ │ │ │ +00015de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015df0: 2020 2020 7c0a 7c20 2020 2020 207b 2d32      |.|      {-2
│ │ │ │ +00015e00: 7d20 7c20 3020 2020 2020 2020 2020 2020  } | 0           
│ │ │ │ +00015e10: 2020 2020 2020 735f 3178 5f32 5e33 2b73        s_1x_2^3+s
│ │ │ │ +00015e20: 5f30 785f 305e 3220 3020 2020 2020 2020  _0x_0^2 0       
│ │ │ │ +00015e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015e40: 2020 2020 7c0a 7c20 2020 2020 207b 2d33      |.|      {-3
│ │ │ │ +00015e50: 7d20 7c20 3020 2020 2020 2020 2020 2020  } | 0           
│ │ │ │ +00015e60: 2020 2020 2020 3020 2020 2020 2020 2020        0         
│ │ │ │ +00015e70: 2020 2020 2020 2020 735f 3178 5f32 5e33          s_1x_2^3
│ │ │ │ +00015e80: 2b73 5f30 785f 305e 3220 2020 2020 2020  +s_0x_0^2       
│ │ │ │ +00015e90: 2020 2020 7c0a 7c20 2020 2020 207b 2d37      |.|      {-7
│ │ │ │ +00015ea0: 7d20 7c20 3020 2020 2020 2020 2020 2020  } | 0           
│ │ │ │ +00015eb0: 2020 2020 2020 3020 2020 2020 2020 2020        0         
│ │ │ │ +00015ec0: 2020 2020 2020 2020 3020 2020 2020 2020          0       
│ │ │ │ +00015ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015ee0: 2020 2020 7c0a 7c20 2020 2020 202d 2d2d      |.|      ---
│ │ │ │ +00015ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00015f40: 0a7c 2020 2020 2020 3020 2020 2020 2020  .|      0       
│ │ │ │ -00015f50: 2020 2020 2020 2020 2020 7c20 2020 2020            |     
│ │ │ │ +00015f30: 2d2d 2d2d 7c0a 7c20 2020 2020 2030 2020  ----|.|      0  
│ │ │ │ +00015f40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00015f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015f80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015f90: 0a7c 2020 2020 2020 3020 2020 2020 2020  .|      0       
│ │ │ │ -00015fa0: 2020 2020 2020 2020 2020 7c20 2020 2020            |     
│ │ │ │ +00015f80: 2020 2020 7c0a 7c20 2020 2020 2030 2020      |.|      0  
│ │ │ │ +00015f90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00015fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015fd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015fe0: 0a7c 2020 2020 2020 3020 2020 2020 2020  .|      0       
│ │ │ │ -00015ff0: 2020 2020 2020 2020 2020 7c20 2020 2020            |     
│ │ │ │ +00015fd0: 2020 2020 7c0a 7c20 2020 2020 2030 2020      |.|      0  
│ │ │ │ +00015fe0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00015ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016020: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016030: 0a7c 2020 2020 2020 735f 3178 5f32 5e33  .|      s_1x_2^3
│ │ │ │ -00016040: 2b73 5f30 785f 305e 3220 7c20 2020 2020  +s_0x_0^2 |     
│ │ │ │ +00016020: 2020 2020 7c0a 7c20 2020 2020 2073 5f31      |.|      s_1
│ │ │ │ +00016030: 785f 325e 332b 735f 3078 5f30 5e32 207c  x_2^3+s_0x_0^2 |
│ │ │ │ +00016040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016070: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016080: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00016070: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00016080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000160a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000160b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000160c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000160d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000160e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000160f0: 2020 2034 2020 2020 2020 2020 2020 2020     4            
│ │ │ │ -00016100: 2020 2020 2020 2020 2020 2020 2034 2020               4  
│ │ │ │ -00016110: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016120: 0a7c 6f31 3120 3a20 4d61 7472 6978 2028  .|o11 : Matrix (
│ │ │ │ -00016130: 6b6b 5b73 202e 2e73 202c 2078 202e 2e78  kk[s ..s , x ..x
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│ │ │ │ -00016160: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016170: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00016180: 2020 2020 3020 2020 3120 2020 3020 2020      0   1   0   
│ │ │ │ -00016190: 3220 2020 2020 2020 2020 2020 2020 3020  2             0 
│ │ │ │ -000161a0: 2020 3120 2020 3020 2020 3220 2020 2020    1   0   2     
│ │ │ │ -000161b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000161c0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000160c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000160d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000160e0: 2020 2020 2020 2020 3420 2020 2020 2020          4       
│ │ │ │ +000160f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016100: 2020 3420 2020 2020 2020 2020 2020 2020    4             
│ │ │ │ +00016110: 2020 2020 7c0a 7c6f 3131 203a 204d 6174      |.|o11 : Mat
│ │ │ │ +00016120: 7269 7820 286b 6b5b 7320 2e2e 7320 2c20  rix (kk[s ..s , 
│ │ │ │ +00016130: 7820 2e2e 7820 5d29 2020 3c2d 2d20 286b  x ..x ])  <-- (k
│ │ │ │ +00016140: 6b5b 7320 2e2e 7320 2c20 7820 2e2e 7820  k[s ..s , x ..x 
│ │ │ │ +00016150: 5d29 2020 2020 2020 2020 2020 2020 2020  ])              
│ │ │ │ +00016160: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00016170: 2020 2020 2020 2020 2030 2020 2031 2020           0   1  
│ │ │ │ +00016180: 2030 2020 2032 2020 2020 2020 2020 2020   0   2          
│ │ │ │ +00016190: 2020 2030 2020 2031 2020 2030 2020 2032     0   1   0   2
│ │ │ │ +000161a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000161b0: 2020 2020 7c0a 2b2d 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 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00016210: 0a7c 6931 3220 3a20 6431 2a64 3020 2020  .|i12 : d1*d0   
│ │ │ │ +00016200: 2d2d 2d2d 2b0a 7c69 3132 203a 2064 312a  ----+.|i12 : d1*
│ │ │ │ +00016210: 6430 2020 2020 2020 2020 2020 2020 2020  d0              
│ │ │ │  00016220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016250: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016260: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00016250: 2020 2020 7c0a 7c20 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 2020 2020 2020                  
│ │ │ │ -000162a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000162b0: 0a7c 6f31 3220 3d20 7b30 7d20 207c 2073  .|o12 = {0}  | s
│ │ │ │ -000162c0: 5f31 785f 325e 332b 735f 3078 5f30 5e32  _1x_2^3+s_0x_0^2
│ │ │ │ -000162d0: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ -000162e0: 2020 2030 2020 2020 2020 2020 2020 2020     0            
│ │ │ │ -000162f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016300: 0a7c 2020 2020 2020 7b2d 347d 207c 2030  .|      {-4} | 0
│ │ │ │ -00016310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016320: 2073 5f31 785f 325e 332b 735f 3078 5f30   s_1x_2^3+s_0x_0
│ │ │ │ -00016330: 5e32 2030 2020 2020 2020 2020 2020 2020  ^2 0            
│ │ │ │ -00016340: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016350: 0a7c 2020 2020 2020 7b2d 357d 207c 2030  .|      {-5} | 0
│ │ │ │ -00016360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016370: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ -00016380: 2020 2073 5f31 785f 325e 332b 735f 3078     s_1x_2^3+s_0x
│ │ │ │ -00016390: 5f30 5e32 2020 2020 2020 2020 2020 207c  _0^2           |
│ │ │ │ -000163a0: 0a7c 2020 2020 2020 7b2d 357d 207c 2030  .|      {-5} | 0
│ │ │ │ -000163b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000163c0: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ -000163d0: 2020 2030 2020 2020 2020 2020 2020 2020     0            
│ │ │ │ -000163e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000163f0: 0a7c 2020 2020 2020 2d2d 2d2d 2d2d 2d2d  .|      --------
│ │ │ │ +000162a0: 2020 2020 7c0a 7c6f 3132 203d 207b 307d      |.|o12 = {0}
│ │ │ │ +000162b0: 2020 7c20 735f 3178 5f32 5e33 2b73 5f30    | s_1x_2^3+s_0
│ │ │ │ +000162c0: 785f 305e 3220 3020 2020 2020 2020 2020  x_0^2 0         
│ │ │ │ +000162d0: 2020 2020 2020 2020 3020 2020 2020 2020          0       
│ │ │ │ +000162e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000162f0: 2020 2020 7c0a 7c20 2020 2020 207b 2d34      |.|      {-4
│ │ │ │ +00016300: 7d20 7c20 3020 2020 2020 2020 2020 2020  } | 0           
│ │ │ │ +00016310: 2020 2020 2020 735f 3178 5f32 5e33 2b73        s_1x_2^3+s
│ │ │ │ +00016320: 5f30 785f 305e 3220 3020 2020 2020 2020  _0x_0^2 0       
│ │ │ │ +00016330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016340: 2020 2020 7c0a 7c20 2020 2020 207b 2d35      |.|      {-5
│ │ │ │ +00016350: 7d20 7c20 3020 2020 2020 2020 2020 2020  } | 0           
│ │ │ │ +00016360: 2020 2020 2020 3020 2020 2020 2020 2020        0         
│ │ │ │ +00016370: 2020 2020 2020 2020 735f 3178 5f32 5e33          s_1x_2^3
│ │ │ │ +00016380: 2b73 5f30 785f 305e 3220 2020 2020 2020  +s_0x_0^2       
│ │ │ │ +00016390: 2020 2020 7c0a 7c20 2020 2020 207b 2d35      |.|      {-5
│ │ │ │ +000163a0: 7d20 7c20 3020 2020 2020 2020 2020 2020  } | 0           
│ │ │ │ +000163b0: 2020 2020 2020 3020 2020 2020 2020 2020        0         
│ │ │ │ +000163c0: 2020 2020 2020 2020 3020 2020 2020 2020          0       
│ │ │ │ +000163d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000163e0: 2020 2020 7c0a 7c20 2020 2020 202d 2d2d      |.|      ---
│ │ │ │ +000163f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00016440: 0a7c 2020 2020 2020 3020 2020 2020 2020  .|      0       
│ │ │ │ -00016450: 2020 2020 2020 2020 2020 7c20 2020 2020            |     
│ │ │ │ +00016430: 2d2d 2d2d 7c0a 7c20 2020 2020 2030 2020  ----|.|      0  
│ │ │ │ +00016440: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00016450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016480: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016490: 0a7c 2020 2020 2020 3020 2020 2020 2020  .|      0       
│ │ │ │ -000164a0: 2020 2020 2020 2020 2020 7c20 2020 2020            |     
│ │ │ │ +00016480: 2020 2020 7c0a 7c20 2020 2020 2030 2020      |.|      0  
│ │ │ │ +00016490: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000164a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000164b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000164c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000164d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000164e0: 0a7c 2020 2020 2020 3020 2020 2020 2020  .|      0       
│ │ │ │ -000164f0: 2020 2020 2020 2020 2020 7c20 2020 2020            |     
│ │ │ │ +000164d0: 2020 2020 7c0a 7c20 2020 2020 2030 2020      |.|      0  
│ │ │ │ +000164e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000164f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016520: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016530: 0a7c 2020 2020 2020 735f 3178 5f32 5e33  .|      s_1x_2^3
│ │ │ │ -00016540: 2b73 5f30 785f 305e 3220 7c20 2020 2020  +s_0x_0^2 |     
│ │ │ │ +00016520: 2020 2020 7c0a 7c20 2020 2020 2073 5f31      |.|      s_1
│ │ │ │ +00016530: 785f 325e 332b 735f 3078 5f30 5e32 207c  x_2^3+s_0x_0^2 |
│ │ │ │ +00016540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016570: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016580: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00016570: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00016580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000165a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000165b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000165c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000165d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000165e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000165f0: 2020 2034 2020 2020 2020 2020 2020 2020     4            
│ │ │ │ -00016600: 2020 2020 2020 2020 2020 2020 2034 2020               4  
│ │ │ │ -00016610: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016620: 0a7c 6f31 3220 3a20 4d61 7472 6978 2028  .|o12 : Matrix (
│ │ │ │ -00016630: 6b6b 5b73 202e 2e73 202c 2078 202e 2e78  kk[s ..s , x ..x
│ │ │ │ -00016640: 205d 2920 203c 2d2d 2028 6b6b 5b73 202e   ])  <-- (kk[s .
│ │ │ │ -00016650: 2e73 202c 2078 202e 2e78 205d 2920 2020  .s , x ..x ])   
│ │ │ │ -00016660: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016670: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00016680: 2020 2020 3020 2020 3120 2020 3020 2020      0   1   0   
│ │ │ │ -00016690: 3220 2020 2020 2020 2020 2020 2020 3020  2             0 
│ │ │ │ -000166a0: 2020 3120 2020 3020 2020 3220 2020 2020    1   0   2     
│ │ │ │ -000166b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000166c0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000165c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000165d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000165e0: 2020 2020 2020 2020 3420 2020 2020 2020          4       
│ │ │ │ +000165f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016600: 2020 3420 2020 2020 2020 2020 2020 2020    4             
│ │ │ │ +00016610: 2020 2020 7c0a 7c6f 3132 203a 204d 6174      |.|o12 : Mat
│ │ │ │ +00016620: 7269 7820 286b 6b5b 7320 2e2e 7320 2c20  rix (kk[s ..s , 
│ │ │ │ +00016630: 7820 2e2e 7820 5d29 2020 3c2d 2d20 286b  x ..x ])  <-- (k
│ │ │ │ +00016640: 6b5b 7320 2e2e 7320 2c20 7820 2e2e 7820  k[s ..s , x ..x 
│ │ │ │ +00016650: 5d29 2020 2020 2020 2020 2020 2020 2020  ])              
│ │ │ │ +00016660: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00016670: 2020 2020 2020 2020 2030 2020 2031 2020           0   1  
│ │ │ │ +00016680: 2030 2020 2032 2020 2020 2020 2020 2020   0   2          
│ │ │ │ +00016690: 2020 2030 2020 2031 2020 2030 2020 2032     0   1   0   2
│ │ │ │ +000166a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000166b0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +000166c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000166d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000166e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000166f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00016710: 0a7c 6931 3320 3a20 5320 3d20 7269 6e67  .|i13 : S = ring
│ │ │ │ -00016720: 2064 3020 2020 2020 2020 2020 2020 2020   d0             
│ │ │ │ +00016700: 2d2d 2d2d 2b0a 7c69 3133 203a 2053 203d  ----+.|i13 : S =
│ │ │ │ +00016710: 2072 696e 6720 6430 2020 2020 2020 2020   ring d0        
│ │ │ │ +00016720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016750: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016760: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00016750: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00016760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000167a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000167b0: 0a7c 6f31 3320 3d20 5320 2020 2020 2020  .|o13 = S       
│ │ │ │ +000167a0: 2020 2020 7c0a 7c6f 3133 203d 2053 2020      |.|o13 = S  
│ │ │ │ +000167b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000167c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000167d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000167e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000167f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016800: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000167f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00016800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016840: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016850: 0a7c 6f31 3320 3a20 506f 6c79 6e6f 6d69  .|o13 : Polynomi
│ │ │ │ -00016860: 616c 5269 6e67 2020 2020 2020 2020 2020  alRing          
│ │ │ │ +00016840: 2020 2020 7c0a 7c6f 3133 203a 2050 6f6c      |.|o13 : Pol
│ │ │ │ +00016850: 796e 6f6d 6961 6c52 696e 6720 2020 2020  ynomialRing     
│ │ │ │ +00016860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016890: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000168a0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00016890: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +000168a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000168b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000168c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000168d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000168e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000168f0: 0a7c 6931 3420 3a20 7068 6920 3d20 6d61  .|i14 : phi = ma
│ │ │ │ -00016900: 7028 532c 5229 2020 2020 2020 2020 2020  p(S,R)          
│ │ │ │ +000168e0: 2d2d 2d2d 2b0a 7c69 3134 203a 2070 6869  ----+.|i14 : phi
│ │ │ │ +000168f0: 203d 206d 6170 2853 2c52 2920 2020 2020   = map(S,R)     
│ │ │ │ +00016900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016930: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016940: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00016930: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00016940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016980: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016990: 0a7c 6f31 3420 3d20 6d61 7020 2853 2c20  .|o14 = map (S, 
│ │ │ │ -000169a0: 522c 207b 7820 2c20 7820 2c20 7820 7d29  R, {x , x , x })
│ │ │ │ +00016980: 2020 2020 7c0a 7c6f 3134 203d 206d 6170      |.|o14 = map
│ │ │ │ +00016990: 2028 532c 2052 2c20 7b78 202c 2078 202c   (S, R, {x , x ,
│ │ │ │ +000169a0: 2078 207d 2920 2020 2020 2020 2020 2020   x })           
│ │ │ │  000169b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000169c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000169d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000169e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000169f0: 2020 2020 2030 2020 2031 2020 2032 2020       0   1   2  
│ │ │ │ +000169d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000169e0: 2020 2020 2020 2020 2020 3020 2020 3120            0   1 
│ │ │ │ +000169f0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │  00016a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016a20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016a30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00016a20: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00016a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016a70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016a80: 0a7c 6f31 3420 3a20 5269 6e67 4d61 7020  .|o14 : RingMap 
│ │ │ │ -00016a90: 5320 3c2d 2d20 5220 2020 2020 2020 2020  S <-- R         
│ │ │ │ +00016a70: 2020 2020 7c0a 7c6f 3134 203a 2052 696e      |.|o14 : Rin
│ │ │ │ +00016a80: 674d 6170 2053 203c 2d2d 2052 2020 2020  gMap S <-- R    
│ │ │ │ +00016a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ac0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016ad0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00016ac0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00016ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00016b20: 0a7c 6931 3520 3a20 4953 203d 2070 6869  .|i15 : IS = phi
│ │ │ │ -00016b30: 2049 2020 2020 2020 2020 2020 2020 2020   I              
│ │ │ │ +00016b10: 2d2d 2d2d 2b0a 7c69 3135 203a 2049 5320  ----+.|i15 : IS 
│ │ │ │ +00016b20: 3d20 7068 6920 4920 2020 2020 2020 2020  = phi I         
│ │ │ │ +00016b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016b60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016b70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00016b60: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00016b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016bb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016bc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00016bd0: 3220 2020 3320 2020 2020 2020 2020 2020  2   3           
│ │ │ │ +00016bb0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00016bc0: 2020 2020 2032 2020 2033 2020 2020 2020       2   3      
│ │ │ │ +00016bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016c00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016c10: 0a7c 6f31 3520 3d20 6964 6561 6c20 2878  .|o15 = ideal (x
│ │ │ │ -00016c20: 202c 2078 2029 2020 2020 2020 2020 2020   , x )          
│ │ │ │ +00016c00: 2020 2020 7c0a 7c6f 3135 203d 2069 6465      |.|o15 = ide
│ │ │ │ +00016c10: 616c 2028 7820 2c20 7820 2920 2020 2020  al (x , x )     
│ │ │ │ +00016c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016c50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016c60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00016c70: 3020 2020 3220 2020 2020 2020 2020 2020  0   2           
│ │ │ │ +00016c50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00016c60: 2020 2020 2030 2020 2032 2020 2020 2020       0   2      
│ │ │ │ +00016c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ca0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016cb0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00016ca0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00016cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016cf0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016d00: 0a7c 6f31 3520 3a20 4964 6561 6c20 6f66  .|o15 : Ideal of
│ │ │ │ -00016d10: 2053 2020 2020 2020 2020 2020 2020 2020   S              
│ │ │ │ +00016cf0: 2020 2020 7c0a 7c6f 3135 203a 2049 6465      |.|o15 : Ide
│ │ │ │ +00016d00: 616c 206f 6620 5320 2020 2020 2020 2020  al of S         
│ │ │ │ +00016d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016d40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016d50: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00016d40: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00016d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00016da0: 0a7c 6931 3620 3a20 5362 6172 203d 2053  .|i16 : Sbar = S
│ │ │ │ -00016db0: 2f49 5320 2020 2020 2020 2020 2020 2020  /IS             
│ │ │ │ +00016d90: 2d2d 2d2d 2b0a 7c69 3136 203a 2053 6261  ----+.|i16 : Sba
│ │ │ │ +00016da0: 7220 3d20 532f 4953 2020 2020 2020 2020  r = S/IS        
│ │ │ │ +00016db0: 2020 2020 2020 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 207c                 |
│ │ │ │ -00016df0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00016de0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00016df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016e00: 2020 2020 2020 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 207c                 |
│ │ │ │ -00016e40: 0a7c 6f31 3620 3d20 5362 6172 2020 2020  .|o16 = Sbar    
│ │ │ │ +00016e30: 2020 2020 7c0a 7c6f 3136 203d 2053 6261      |.|o16 = Sba
│ │ │ │ +00016e40: 7220 2020 2020 2020 2020 2020 2020 2020  r               
│ │ │ │  00016e50: 2020 2020 2020 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 207c                 |
│ │ │ │ -00016e90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00016e80: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00016e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ed0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016ee0: 0a7c 6f31 3620 3a20 5175 6f74 6965 6e74  .|o16 : Quotient
│ │ │ │ -00016ef0: 5269 6e67 2020 2020 2020 2020 2020 2020  Ring            
│ │ │ │ +00016ed0: 2020 2020 7c0a 7c6f 3136 203a 2051 756f      |.|o16 : Quo
│ │ │ │ +00016ee0: 7469 656e 7452 696e 6720 2020 2020 2020  tientRing       
│ │ │ │ +00016ef0: 2020 2020 2020 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 207c                 |
│ │ │ │ -00016f30: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00016f20: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00016f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00016f80: 0a7c 6931 3720 3a20 534d 6261 7220 3d20  .|i17 : SMbar = 
│ │ │ │ -00016f90: 5362 6172 2a2a 4d62 6172 2020 2020 2020  Sbar**Mbar      
│ │ │ │ +00016f70: 2d2d 2d2d 2b0a 7c69 3137 203a 2053 4d62  ----+.|i17 : SMb
│ │ │ │ +00016f80: 6172 203d 2053 6261 722a 2a4d 6261 7220  ar = Sbar**Mbar 
│ │ │ │ +00016f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016fc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016fd0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00016fc0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00016fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017010: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00017020: 0a7c 6f31 3720 3d20 636f 6b65 726e 656c  .|o17 = cokernel
│ │ │ │ -00017030: 207c 2078 5f30 785f 312b 3234 785f 315e   | x_0x_1+24x_1^
│ │ │ │ -00017040: 322b 3439 785f 3078 5f32 2b33 785f 3178  2+49x_0x_2+3x_1x
│ │ │ │ -00017050: 5f32 2b35 785f 325e 3220 7c20 2020 2020  _2+5x_2^2 |     
│ │ │ │ -00017060: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00017070: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00017010: 2020 2020 7c0a 7c6f 3137 203d 2063 6f6b      |.|o17 = cok
│ │ │ │ +00017020: 6572 6e65 6c20 7c20 785f 3078 5f31 2b32  ernel | x_0x_1+2
│ │ │ │ +00017030: 3478 5f31 5e32 2b34 3978 5f30 785f 322b  4x_1^2+49x_0x_2+
│ │ │ │ +00017040: 3378 5f31 785f 322b 3578 5f32 5e32 207c  3x_1x_2+5x_2^2 |
│ │ │ │ +00017050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017060: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00017070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017080: 2020 2020 2020 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 207c                 |
│ │ │ │ -000170c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000170d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000170e0: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ +000170b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000170c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000170d0: 2020 2020 2020 2020 2020 3120 2020 2020            1     
│ │ │ │ +000170e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000170f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017100: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00017110: 0a7c 6f31 3720 3a20 5362 6172 2d6d 6f64  .|o17 : Sbar-mod
│ │ │ │ -00017120: 756c 652c 2071 756f 7469 656e 7420 6f66  ule, quotient of
│ │ │ │ -00017130: 2053 6261 7220 2020 2020 2020 2020 2020   Sbar           
│ │ │ │ +00017100: 2020 2020 7c0a 7c6f 3137 203a 2053 6261      |.|o17 : Sba
│ │ │ │ +00017110: 722d 6d6f 6475 6c65 2c20 7175 6f74 6965  r-module, quotie
│ │ │ │ +00017120: 6e74 206f 6620 5362 6172 2020 2020 2020  nt of Sbar      
│ │ │ │ +00017130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017150: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00017160: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00017150: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00017160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000171a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000171b0: 0a0a 486f 6d28 6430 2c53 6261 7229 2061  ..Hom(d0,Sbar) a
│ │ │ │ -000171c0: 6e64 2048 6f6d 2864 312c 5362 6172 2920  nd Hom(d1,Sbar) 
│ │ │ │ -000171d0: 746f 6765 7468 6572 2066 6f72 6d20 7468  together form th
│ │ │ │ -000171e0: 6520 7265 736f 6c75 7469 6f6e 206f 6620  e resolution of 
│ │ │ │ -000171f0: 4d62 6172 3b20 7468 7573 2074 6865 0a68  Mbar; thus the.h
│ │ │ │ -00017200: 6f6d 6f6c 6f67 7920 6f66 206f 6e65 2063  omology of one c
│ │ │ │ -00017210: 6f6d 706f 7369 7469 6f6e 2069 7320 302c  omposition is 0,
│ │ │ │ -00017220: 2077 6869 6c65 2074 6865 206f 7468 6572   while the other
│ │ │ │ -00017230: 2069 7320 4d62 6172 0a0a 2b2d 2d2d 2d2d   is Mbar..+-----
│ │ │ │ +000171a0: 2d2d 2d2d 2b0a 0a48 6f6d 2864 302c 5362  ----+..Hom(d0,Sb
│ │ │ │ +000171b0: 6172 2920 616e 6420 486f 6d28 6431 2c53  ar) and Hom(d1,S
│ │ │ │ +000171c0: 6261 7229 2074 6f67 6574 6865 7220 666f  bar) together fo
│ │ │ │ +000171d0: 726d 2074 6865 2072 6573 6f6c 7574 696f  rm the resolutio
│ │ │ │ +000171e0: 6e20 6f66 204d 6261 723b 2074 6875 7320  n of Mbar; thus 
│ │ │ │ +000171f0: 7468 650a 686f 6d6f 6c6f 6779 206f 6620  the.homology of 
│ │ │ │ +00017200: 6f6e 6520 636f 6d70 6f73 6974 696f 6e20  one composition 
│ │ │ │ +00017210: 6973 2030 2c20 7768 696c 6520 7468 6520  is 0, while the 
│ │ │ │ +00017220: 6f74 6865 7220 6973 204d 6261 720a 0a2b  other is Mbar..+
│ │ │ │ +00017230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017280: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3138 203a  --------+.|i18 :
│ │ │ │ -00017290: 2070 7275 6e65 2048 485f 3120 636f 6d70   prune HH_1 comp
│ │ │ │ -000172a0: 6c65 787b 6475 616c 2028 5362 6172 2a2a  lex{dual (Sbar**
│ │ │ │ -000172b0: 6430 292c 2064 7561 6c28 5362 6172 2a2a  d0), dual(Sbar**
│ │ │ │ -000172c0: 6431 297d 203d 3d20 3020 2020 2020 2020  d1)} == 0       
│ │ │ │ -000172d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00017270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00017280: 6931 3820 3a20 7072 756e 6520 4848 5f31  i18 : prune HH_1
│ │ │ │ +00017290: 2063 6f6d 706c 6578 7b64 7561 6c20 2853   complex{dual (S
│ │ │ │ +000172a0: 6261 722a 2a64 3029 2c20 6475 616c 2853  bar**d0), dual(S
│ │ │ │ +000172b0: 6261 722a 2a64 3129 7d20 3d3d 2030 2020  bar**d1)} == 0  
│ │ │ │ +000172c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000172d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000172e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000172f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017320: 2020 2020 2020 2020 7c0a 7c6f 3138 203d          |.|o18 =
│ │ │ │ -00017330: 2074 7275 6520 2020 2020 2020 2020 2020   true           
│ │ │ │ +00017310: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00017320: 6f31 3820 3d20 7472 7565 2020 2020 2020  o18 = true      
│ │ │ │ +00017330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017370: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00017360: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00017370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000173a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000173b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000173c0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3139 203a  --------+.|i19 :
│ │ │ │ -000173d0: 204d 6261 7227 203d 2053 6261 725e 312f   Mbar' = Sbar^1/
│ │ │ │ -000173e0: 2853 6261 725f 302c 2053 6261 725f 3129  (Sbar_0, Sbar_1)
│ │ │ │ -000173f0: 2a2a 534d 6261 7220 2020 2020 2020 2020  **SMbar         
│ │ │ │ -00017400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017410: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000173b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +000173c0: 6931 3920 3a20 4d62 6172 2720 3d20 5362  i19 : Mbar' = Sb
│ │ │ │ +000173d0: 6172 5e31 2f28 5362 6172 5f30 2c20 5362  ar^1/(Sbar_0, Sb
│ │ │ │ +000173e0: 6172 5f31 292a 2a53 4d62 6172 2020 2020  ar_1)**SMbar    
│ │ │ │ +000173f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017400: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00017410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017460: 2020 2020 2020 2020 7c0a 7c6f 3139 203d          |.|o19 =
│ │ │ │ -00017470: 2063 6f6b 6572 6e65 6c20 7c20 785f 3078   cokernel | x_0x
│ │ │ │ -00017480: 5f31 2b32 3478 5f31 5e32 2b34 3978 5f30  _1+24x_1^2+49x_0
│ │ │ │ -00017490: 785f 322b 3378 5f31 785f 322b 3578 5f32  x_2+3x_1x_2+5x_2
│ │ │ │ -000174a0: 5e32 2073 5f30 2073 5f31 207c 2020 2020  ^2 s_0 s_1 |    
│ │ │ │ -000174b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00017450: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00017460: 6f31 3920 3d20 636f 6b65 726e 656c 207c  o19 = cokernel |
│ │ │ │ +00017470: 2078 5f30 785f 312b 3234 785f 315e 322b   x_0x_1+24x_1^2+
│ │ │ │ +00017480: 3439 785f 3078 5f32 2b33 785f 3178 5f32  49x_0x_2+3x_1x_2
│ │ │ │ +00017490: 2b35 785f 325e 3220 735f 3020 735f 3120  +5x_2^2 s_0 s_1 
│ │ │ │ +000174a0: 7c20 2020 2020 2020 2020 2020 207c 0a7c  |            |.|
│ │ │ │ +000174b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000174c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000174d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000174e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000174f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017500: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000174f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00017500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017520: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ +00017520: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │  00017530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017550: 2020 2020 2020 2020 7c0a 7c6f 3139 203a          |.|o19 :
│ │ │ │ -00017560: 2053 6261 722d 6d6f 6475 6c65 2c20 7175   Sbar-module, qu
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│ │ │ │ -00017d80: 2020 2020 2020 6765 6e73 2069 6e20 6465        gens in de
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│ │ │ │ -00017de0: 4966 2074 6865 206f 7074 696f 6e61 6c20  If the optional 
│ │ │ │ -00017df0: 6172 6775 6d65 6e74 204f 7574 5269 6e67  argument OutRing
│ │ │ │ -00017e00: 0a3d 3e20 5420 6973 2067 6976 656e 2c20  .=> T is given, 
│ │ │ │ -00017e10: 616e 6420 636c 6173 7320 5420 3d3d 3d20  and class T === 
│ │ │ │ -00017e20: 506f 6c79 6e6f 6d69 616c 5269 6e67 2c20  PolynomialRing, 
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│ │ │ │ -00017e40: 7769 6c6c 2062 6520 6120 6d6f 6475 6c65  will be a module
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│ │ │ │ +00017c00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a  **************..
│ │ │ │ +00017c10: 2020 2a20 5573 6167 653a 200a 2020 2020    * Usage: .    
│ │ │ │ +00017c20: 2020 2020 4520 3d20 6576 656e 4578 744d      E = evenExtM
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│ │ │ │ +00017c40: 7473 3a0a 2020 2020 2020 2a20 4d2c 2061  ts:.      * M, a
│ │ │ │ +00017c50: 202a 6e6f 7465 206d 6f64 756c 653a 2028   *note module: (
│ │ │ │ +00017c60: 4d61 6361 756c 6179 3244 6f63 294d 6f64  Macaulay2Doc)Mod
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│ │ │ │ +00017ca0: 2020 2a20 2a6e 6f74 6520 4f70 7469 6f6e    * *note Option
│ │ │ │ +00017cb0: 616c 2069 6e70 7574 733a 2028 4d61 6361  al inputs: (Maca
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│ │ │ │ +00017ce0: 7469 6f6e 616c 2069 6e70 7574 732c 3a0a  tional inputs,:.
│ │ │ │ +00017cf0: 2020 2020 2020 2a20 4f75 7452 696e 6720        * OutRing 
│ │ │ │ +00017d00: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420  => ..., default 
│ │ │ │ +00017d10: 7661 6c75 6520 300a 2020 2a20 4f75 7470  value 0.  * Outp
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│ │ │ │ +00017d30: 6120 2a6e 6f74 6520 6d6f 6475 6c65 3a20  a *note module: 
│ │ │ │ +00017d40: 284d 6163 6175 6c61 7932 446f 6329 4d6f  (Macaulay2Doc)Mo
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│ │ │ │ +00017d90: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ +00017da0: 3d3d 3d3d 0a0a 4578 7472 6163 7473 2074  ====..Extracts t
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│ │ │ │ +00017dc0: 6172 7420 6672 6f6d 2045 7874 4d6f 6475  art from ExtModu
│ │ │ │ +00017dd0: 6c65 204d 2049 6620 7468 6520 6f70 7469  le M If the opti
│ │ │ │ +00017de0: 6f6e 616c 2061 7267 756d 656e 7420 4f75  onal argument Ou
│ │ │ │ +00017df0: 7452 696e 670a 3d3e 2054 2069 7320 6769  tRing.=> T is gi
│ │ │ │ +00017e00: 7665 6e2c 2061 6e64 2063 6c61 7373 2054  ven, and class T
│ │ │ │ +00017e10: 203d 3d3d 2050 6f6c 796e 6f6d 6961 6c52   === PolynomialR
│ │ │ │ +00017e20: 696e 672c 2074 6865 6e20 7468 6520 6f75  ing, then the ou
│ │ │ │ +00017e30: 7470 7574 2077 696c 6c20 6265 2061 206d  tput will be a m
│ │ │ │ +00017e40: 6f64 756c 650a 6f76 6572 2054 2e0a 0a2b  odule.over T...+
│ │ │ │ +00017e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017e90: 2d2d 2d2d 2b0a 7c69 3120 3a20 6b6b 3d20  ----+.|i1 : kk= 
│ │ │ │ -00017ea0: 5a5a 2f31 3031 2020 2020 2020 2020 2020  ZZ/101          
│ │ │ │ +00017e80: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a  ---------+.|i1 :
│ │ │ │ +00017e90: 206b 6b3d 205a 5a2f 3130 3120 2020 2020   kk= ZZ/101     
│ │ │ │ +00017ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017ed0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00017ec0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00017ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017f00: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00017f10: 3120 3d20 6b6b 2020 2020 2020 2020 2020  1 = kk          
│ │ │ │ +00017f00: 207c 0a7c 6f31 203d 206b 6b20 2020 2020   |.|o1 = kk     
│ │ │ │ +00017f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017f40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00017f30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00017f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017f80: 2020 2020 7c0a 7c6f 3120 3a20 5175 6f74      |.|o1 : Quot
│ │ │ │ -00017f90: 6965 6e74 5269 6e67 2020 2020 2020 2020  ientRing        
│ │ │ │ +00017f70: 2020 2020 2020 2020 207c 0a7c 6f31 203a           |.|o1 :
│ │ │ │ +00017f80: 2051 756f 7469 656e 7452 696e 6720 2020   QuotientRing   
│ │ │ │ +00017f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017fc0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00017fb0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00017fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00018000: 3220 3a20 5320 3d20 6b6b 5b78 2c79 2c7a  2 : S = kk[x,y,z
│ │ │ │ -00018010: 5d20 2020 2020 2020 2020 2020 2020 2020  ]               
│ │ │ │ -00018020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018030: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00017ff0: 2d2b 0a7c 6932 203a 2053 203d 206b 6b5b  -+.|i2 : S = kk[
│ │ │ │ +00018000: 782c 792c 7a5d 2020 2020 2020 2020 2020  x,y,z]          
│ │ │ │ +00018010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018020: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00018030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018070: 2020 2020 7c0a 7c6f 3220 3d20 5320 2020      |.|o2 = S   
│ │ │ │ +00018060: 2020 2020 2020 2020 207c 0a7c 6f32 203d           |.|o2 =
│ │ │ │ +00018070: 2053 2020 2020 2020 2020 2020 2020 2020   S              
│ │ │ │  00018080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000180a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000180b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000180a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000180b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000180c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000180d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000180e0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -000180f0: 3220 3a20 506f 6c79 6e6f 6d69 616c 5269  2 : PolynomialRi
│ │ │ │ -00018100: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │ -00018110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018120: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +000180e0: 207c 0a7c 6f32 203a 2050 6f6c 796e 6f6d   |.|o2 : Polynom
│ │ │ │ +000180f0: 6961 6c52 696e 6720 2020 2020 2020 2020  ialRing         
│ │ │ │ +00018100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018110: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00018120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018160: 2d2d 2d2d 2b0a 7c69 3320 3a20 4932 203d  ----+.|i3 : I2 =
│ │ │ │ -00018170: 2069 6465 616c 2278 332c 797a 2220 2020   ideal"x3,yz"   
│ │ │ │ +00018150: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a  ---------+.|i3 :
│ │ │ │ +00018160: 2049 3220 3d20 6964 6561 6c22 7833 2c79   I2 = ideal"x3,y
│ │ │ │ +00018170: 7a22 2020 2020 2020 2020 2020 2020 2020  z"              
│ │ │ │  00018180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000181a0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00018190: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000181a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000181b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000181c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000181d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000181e0: 2020 2020 2020 2020 2020 2020 3320 2020              3   
│ │ │ │ +000181d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000181e0: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │  000181f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018210: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20          |.|o3 = 
│ │ │ │ -00018220: 6964 6561 6c20 2878 202c 2079 2a7a 2920  ideal (x , y*z) 
│ │ │ │ +00018200: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00018210: 6f33 203d 2069 6465 616c 2028 7820 2c20  o3 = ideal (x , 
│ │ │ │ +00018220: 792a 7a29 2020 2020 2020 2020 2020 2020  y*z)            
│ │ │ │  00018230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018250: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00018240: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00018250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018290: 7c0a 7c6f 3320 3a20 4964 6561 6c20 6f66  |.|o3 : Ideal of
│ │ │ │ -000182a0: 2053 2020 2020 2020 2020 2020 2020 2020   S              
│ │ │ │ +00018280: 2020 2020 207c 0a7c 6f33 203a 2049 6465       |.|o3 : Ide
│ │ │ │ +00018290: 616c 206f 6620 5320 2020 2020 2020 2020  al of S         
│ │ │ │ +000182a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000182b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000182c0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +000182c0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  000182d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000182e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000182f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018300: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20  --------+.|i4 : 
│ │ │ │ -00018310: 5232 203d 2053 2f49 3220 2020 2020 2020  R2 = S/I2       
│ │ │ │ +000182f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00018300: 6934 203a 2052 3220 3d20 532f 4932 2020  i4 : R2 = S/I2  
│ │ │ │ +00018310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018340: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00018330: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00018340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018380: 7c0a 7c6f 3420 3d20 5232 2020 2020 2020  |.|o4 = R2      
│ │ │ │ +00018370: 2020 2020 207c 0a7c 6f34 203d 2052 3220       |.|o4 = R2 
│ │ │ │ +00018380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000183a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000183b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000183b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  000183c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000183d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000183e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000183f0: 2020 2020 2020 2020 7c0a 7c6f 3420 3a20          |.|o4 : 
│ │ │ │ -00018400: 5175 6f74 6965 6e74 5269 6e67 2020 2020  QuotientRing    
│ │ │ │ +000183e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000183f0: 6f34 203a 2051 756f 7469 656e 7452 696e  o4 : QuotientRin
│ │ │ │ +00018400: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │  00018410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018430: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00018420: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00018430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018470: 2b0a 7c69 3520 3a20 4d32 203d 2052 325e  +.|i5 : M2 = R2^
│ │ │ │ -00018480: 312f 6964 6561 6c22 7832 2c79 2c7a 2220  1/ideal"x2,y,z" 
│ │ │ │ +00018460: 2d2d 2d2d 2d2b 0a7c 6935 203a 204d 3220  -----+.|i5 : M2 
│ │ │ │ +00018470: 3d20 5232 5e31 2f69 6465 616c 2278 322c  = R2^1/ideal"x2,
│ │ │ │ +00018480: 792c 7a22 2020 2020 2020 2020 2020 2020  y,z"            
│ │ │ │  00018490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000184a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000184a0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  000184b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000184c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000184d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000184e0: 2020 2020 2020 2020 7c0a 7c6f 3520 3d20          |.|o5 = 
│ │ │ │ -000184f0: 636f 6b65 726e 656c 207c 2078 3220 7920  cokernel | x2 y 
│ │ │ │ -00018500: 7a20 7c20 2020 2020 2020 2020 2020 2020  z |             
│ │ │ │ -00018510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018520: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000184d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000184e0: 6f35 203d 2063 6f6b 6572 6e65 6c20 7c20  o5 = cokernel | 
│ │ │ │ +000184f0: 7832 2079 207a 207c 2020 2020 2020 2020  x2 y z |        
│ │ │ │ +00018500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018510: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00018520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018560: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00018570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018580: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │ -00018590: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -000185a0: 3520 3a20 5232 2d6d 6f64 756c 652c 2071  5 : R2-module, q
│ │ │ │ -000185b0: 756f 7469 656e 7420 6f66 2052 3220 2020  uotient of R2   
│ │ │ │ -000185c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000185d0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00018550: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00018560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018570: 2020 2020 2020 3120 2020 2020 2020 2020        1         
│ │ │ │ +00018580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018590: 207c 0a7c 6f35 203a 2052 322d 6d6f 6475   |.|o5 : R2-modu
│ │ │ │ +000185a0: 6c65 2c20 7175 6f74 6965 6e74 206f 6620  le, quotient of 
│ │ │ │ +000185b0: 5232 2020 2020 2020 2020 2020 2020 2020  R2              
│ │ │ │ +000185c0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +000185d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000185e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000185f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018610: 2d2d 2d2d 2b0a 7c69 3620 3a20 6265 7474  ----+.|i6 : bett
│ │ │ │ -00018620: 6920 6672 6565 5265 736f 6c75 7469 6f6e  i freeResolution
│ │ │ │ -00018630: 2028 4d32 2c20 4c65 6e67 7468 4c69 6d69   (M2, LengthLimi
│ │ │ │ -00018640: 7420 3d3e 3130 2920 2020 2020 2020 2020  t =>10)         
│ │ │ │ -00018650: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00018600: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a  ---------+.|i6 :
│ │ │ │ +00018610: 2062 6574 7469 2066 7265 6552 6573 6f6c   betti freeResol
│ │ │ │ +00018620: 7574 696f 6e20 284d 322c 204c 656e 6774  ution (M2, Lengt
│ │ │ │ +00018630: 684c 696d 6974 203d 3e31 3029 2020 2020  hLimit =>10)    
│ │ │ │ +00018640: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00018650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018680: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00018690: 2020 2020 2020 2020 2020 2030 2031 2032             0 1 2
│ │ │ │ -000186a0: 2033 2034 2020 3520 2036 2020 3720 2038   3 4  5  6  7  8
│ │ │ │ -000186b0: 2020 3920 3130 2020 2020 2020 2020 2020    9 10          
│ │ │ │ -000186c0: 2020 2020 2020 2020 7c0a 7c6f 3620 3d20          |.|o6 = 
│ │ │ │ -000186d0: 746f 7461 6c3a 2031 2033 2035 2037 2039  total: 1 3 5 7 9
│ │ │ │ -000186e0: 2031 3120 3133 2031 3520 3137 2031 3920   11 13 15 17 19 
│ │ │ │ -000186f0: 3231 2020 2020 2020 2020 2020 2020 2020  21              
│ │ │ │ -00018700: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00018710: 303a 2031 2032 2032 2032 2032 2020 3220  0: 1 2 2 2 2  2 
│ │ │ │ -00018720: 2032 2020 3220 2032 2020 3220 2032 2020   2  2  2  2  2  
│ │ │ │ -00018730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018740: 7c0a 7c20 2020 2020 2020 2020 313a 202e  |.|         1: .
│ │ │ │ -00018750: 2031 2033 2034 2034 2020 3420 2034 2020   1 3 4 4  4  4  
│ │ │ │ -00018760: 3420 2034 2020 3420 2034 2020 2020 2020  4  4  4  4      
│ │ │ │ -00018770: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00018780: 2020 2020 2020 2020 323a 202e 202e 202e          2: . . .
│ │ │ │ -00018790: 2031 2033 2020 3420 2034 2020 3420 2034   1 3  4  4  4  4
│ │ │ │ -000187a0: 2020 3420 2034 2020 2020 2020 2020 2020    4  4          
│ │ │ │ -000187b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000187c0: 2020 2020 333a 202e 202e 202e 202e 202e      3: . . . . .
│ │ │ │ -000187d0: 2020 3120 2033 2020 3420 2034 2020 3420    1  3  4  4  4 
│ │ │ │ -000187e0: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ -000187f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00018800: 343a 202e 202e 202e 202e 202e 2020 2e20  4: . . . . .  . 
│ │ │ │ -00018810: 202e 2020 3120 2033 2020 3420 2034 2020   .  1  3  4  4  
│ │ │ │ -00018820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018830: 7c0a 7c20 2020 2020 2020 2020 353a 202e  |.|         5: .
│ │ │ │ -00018840: 202e 202e 202e 202e 2020 2e20 202e 2020   . . . .  .  .  
│ │ │ │ -00018850: 2e20 202e 2020 3120 2033 2020 2020 2020  .  .  1  3      
│ │ │ │ -00018860: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00018680: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00018690: 3020 3120 3220 3320 3420 2035 2020 3620  0 1 2 3 4  5  6 
│ │ │ │ +000186a0: 2037 2020 3820 2039 2031 3020 2020 2020   7  8  9 10     
│ │ │ │ +000186b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000186c0: 6f36 203d 2074 6f74 616c 3a20 3120 3320  o6 = total: 1 3 
│ │ │ │ +000186d0: 3520 3720 3920 3131 2031 3320 3135 2031  5 7 9 11 13 15 1
│ │ │ │ +000186e0: 3720 3139 2032 3120 2020 2020 2020 2020  7 19 21         
│ │ │ │ +000186f0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00018700: 2020 2020 2030 3a20 3120 3220 3220 3220       0: 1 2 2 2 
│ │ │ │ +00018710: 3220 2032 2020 3220 2032 2020 3220 2032  2  2  2  2  2  2
│ │ │ │ +00018720: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +00018730: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00018740: 2031 3a20 2e20 3120 3320 3420 3420 2034   1: . 1 3 4 4  4
│ │ │ │ +00018750: 2020 3420 2034 2020 3420 2034 2020 3420    4  4  4  4  4 
│ │ │ │ +00018760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018770: 207c 0a7c 2020 2020 2020 2020 2032 3a20   |.|         2: 
│ │ │ │ +00018780: 2e20 2e20 2e20 3120 3320 2034 2020 3420  . . . 1 3  4  4 
│ │ │ │ +00018790: 2034 2020 3420 2034 2020 3420 2020 2020   4  4  4  4     
│ │ │ │ +000187a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000187b0: 2020 2020 2020 2020 2033 3a20 2e20 2e20           3: . . 
│ │ │ │ +000187c0: 2e20 2e20 2e20 2031 2020 3320 2034 2020  . . .  1  3  4  
│ │ │ │ +000187d0: 3420 2034 2020 3420 2020 2020 2020 2020  4  4  4         
│ │ │ │ +000187e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000187f0: 2020 2020 2034 3a20 2e20 2e20 2e20 2e20       4: . . . . 
│ │ │ │ +00018800: 2e20 202e 2020 2e20 2031 2020 3320 2034  .  .  .  1  3  4
│ │ │ │ +00018810: 2020 3420 2020 2020 2020 2020 2020 2020    4             
│ │ │ │ +00018820: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00018830: 2035 3a20 2e20 2e20 2e20 2e20 2e20 202e   5: . . . . .  .
│ │ │ │ +00018840: 2020 2e20 202e 2020 2e20 2031 2020 3320    .  .  .  1  3 
│ │ │ │ +00018850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018860: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00018870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000188a0: 2020 2020 2020 2020 7c0a 7c6f 3620 3a20          |.|o6 : 
│ │ │ │ -000188b0: 4265 7474 6954 616c 6c79 2020 2020 2020  BettiTally      
│ │ │ │ +00018890: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000188a0: 6f36 203a 2042 6574 7469 5461 6c6c 7920  o6 : BettiTally 
│ │ │ │ +000188b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000188c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000188d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000188e0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +000188d0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +000188e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000188f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018920: 2b0a 7c69 3720 3a20 4520 3d20 4578 744d  +.|i7 : E = ExtM
│ │ │ │ -00018930: 6f64 756c 6520 4d32 2020 2020 2020 2020  odule M2        
│ │ │ │ +00018910: 2d2d 2d2d 2d2b 0a7c 6937 203a 2045 203d  -----+.|i7 : E =
│ │ │ │ +00018920: 2045 7874 4d6f 6475 6c65 204d 3220 2020   ExtModule M2   
│ │ │ │ +00018930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018950: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00018950: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00018960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018990: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000189a0: 2020 2020 2020 2020 2020 2020 3820 2020              8   
│ │ │ │ +00018980: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00018990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000189a0: 2038 2020 2020 2020 2020 2020 2020 2020   8              
│ │ │ │  000189b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000189c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000189d0: 2020 2020 7c0a 7c6f 3720 3d20 286b 6b5b      |.|o7 = (kk[
│ │ │ │ -000189e0: 5820 2e2e 5820 5d29 2020 2020 2020 2020  X ..X ])        
│ │ │ │ +000189c0: 2020 2020 2020 2020 207c 0a7c 6f37 203d           |.|o7 =
│ │ │ │ +000189d0: 2028 6b6b 5b58 202e 2e58 205d 2920 2020   (kk[X ..X ])   
│ │ │ │ +000189e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000189f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018a10: 7c0a 7c20 2020 2020 2020 2020 2030 2020  |.|          0  
│ │ │ │ -00018a20: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │ +00018a00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00018a10: 2020 3020 2020 3120 2020 2020 2020 2020    0   1         
│ │ │ │ +00018a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018a40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00018a40: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00018a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018a80: 2020 2020 2020 2020 7c0a 7c6f 3720 3a20          |.|o7 : 
│ │ │ │ -00018a90: 6b6b 5b58 202e 2e58 205d 2d6d 6f64 756c  kk[X ..X ]-modul
│ │ │ │ -00018aa0: 652c 2066 7265 652c 2064 6567 7265 6573  e, free, degrees
│ │ │ │ -00018ab0: 207b 302e 2e31 2c20 323a 312c 2033 3a32   {0..1, 2:1, 3:2
│ │ │ │ -00018ac0: 2c20 337d 7c0a 7c20 2020 2020 2020 2020  , 3}|.|         
│ │ │ │ -00018ad0: 3020 2020 3120 2020 2020 2020 2020 2020  0   1           
│ │ │ │ +00018a70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00018a80: 6f37 203a 206b 6b5b 5820 2e2e 5820 5d2d  o7 : kk[X ..X ]-
│ │ │ │ +00018a90: 6d6f 6475 6c65 2c20 6672 6565 2c20 6465  module, free, de
│ │ │ │ +00018aa0: 6772 6565 7320 7b30 2e2e 312c 2032 3a31  grees {0..1, 2:1
│ │ │ │ +00018ab0: 2c20 333a 322c 2033 7d7c 0a7c 2020 2020  , 3:2, 3}|.|    
│ │ │ │ +00018ac0: 2020 2020 2030 2020 2031 2020 2020 2020       0   1      
│ │ │ │ +00018ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018b00: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00018af0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00018b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00018b40: 3820 3a20 6170 706c 7928 746f 4c69 7374  8 : apply(toList
│ │ │ │ -00018b50: 2830 2e2e 3130 292c 2069 2d3e 6869 6c62  (0..10), i->hilb
│ │ │ │ -00018b60: 6572 7446 756e 6374 696f 6e28 692c 2045  ertFunction(i, E
│ │ │ │ -00018b70: 2929 2020 2020 2020 7c0a 7c20 2020 2020  ))      |.|     
│ │ │ │ +00018b30: 2d2b 0a7c 6938 203a 2061 7070 6c79 2874  -+.|i8 : apply(t
│ │ │ │ +00018b40: 6f4c 6973 7428 302e 2e31 3029 2c20 692d  oList(0..10), i-
│ │ │ │ +00018b50: 3e68 696c 6265 7274 4675 6e63 7469 6f6e  >hilbertFunction
│ │ │ │ +00018b60: 2869 2c20 4529 2920 2020 2020 207c 0a7c  (i, E))      |.|
│ │ │ │ +00018b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018bb0: 2020 2020 7c0a 7c6f 3820 3d20 7b31 2c20      |.|o8 = {1, 
│ │ │ │ -00018bc0: 332c 2035 2c20 372c 2039 2c20 3131 2c20  3, 5, 7, 9, 11, 
│ │ │ │ -00018bd0: 3133 2c20 3135 2c20 3137 2c20 3139 2c20  13, 15, 17, 19, 
│ │ │ │ -00018be0: 3231 7d20 2020 2020 2020 2020 2020 2020  21}             
│ │ │ │ -00018bf0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00018ba0: 2020 2020 2020 2020 207c 0a7c 6f38 203d           |.|o8 =
│ │ │ │ +00018bb0: 207b 312c 2033 2c20 352c 2037 2c20 392c   {1, 3, 5, 7, 9,
│ │ │ │ +00018bc0: 2031 312c 2031 332c 2031 352c 2031 372c   11, 13, 15, 17,
│ │ │ │ +00018bd0: 2031 392c 2032 317d 2020 2020 2020 2020   19, 21}        
│ │ │ │ +00018be0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00018bf0: 2020 2020 2020 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 7c0a 7c6f              |.|o
│ │ │ │ -00018c30: 3820 3a20 4c69 7374 2020 2020 2020 2020  8 : List        
│ │ │ │ +00018c20: 207c 0a7c 6f38 203a 204c 6973 7420 2020   |.|o8 : List   
│ │ │ │ +00018c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018c60: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00018c50: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00018c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018ca0: 2d2d 2d2d 2b0a 7c69 3920 3a20 4565 7665  ----+.|i9 : Eeve
│ │ │ │ -00018cb0: 6e20 3d20 6576 656e 4578 744d 6f64 756c  n = evenExtModul
│ │ │ │ -00018cc0: 6520 4d32 2020 2020 2020 2020 2020 2020  e M2            
│ │ │ │ -00018cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018ce0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00018c90: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a  ---------+.|i9 :
│ │ │ │ +00018ca0: 2045 6576 656e 203d 2065 7665 6e45 7874   Eeven = evenExt
│ │ │ │ +00018cb0: 4d6f 6475 6c65 204d 3220 2020 2020 2020  Module M2       
│ │ │ │ +00018cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018cd0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00018ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018d10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00018d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018d30: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ -00018d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018d50: 2020 2020 2020 2020 7c0a 7c6f 3920 3d20          |.|o9 = 
│ │ │ │ -00018d60: 286b 6b5b 5820 2e2e 5820 5d29 2020 2020  (kk[X ..X ])    
│ │ │ │ +00018d10: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00018d20: 2020 2020 2034 2020 2020 2020 2020 2020       4          
│ │ │ │ +00018d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018d40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00018d50: 6f39 203d 2028 6b6b 5b58 202e 2e58 205d  o9 = (kk[X ..X ]
│ │ │ │ +00018d60: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  00018d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018d90: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00018da0: 2030 2020 2031 2020 2020 2020 2020 2020   0   1          
│ │ │ │ +00018d80: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00018d90: 2020 2020 2020 3020 2020 3120 2020 2020        0   1     
│ │ │ │ +00018da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018dd0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00018dc0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00018dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018e00: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00018e10: 3920 3a20 6b6b 5b58 202e 2e58 205d 2d6d  9 : kk[X ..X ]-m
│ │ │ │ -00018e20: 6f64 756c 652c 2066 7265 652c 2064 6567  odule, free, deg
│ │ │ │ -00018e30: 7265 6573 207b 302e 2e31 2c20 323a 317d  rees {0..1, 2:1}
│ │ │ │ -00018e40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00018e50: 2020 2020 3020 2020 3120 2020 2020 2020      0   1       
│ │ │ │ +00018e00: 207c 0a7c 6f39 203a 206b 6b5b 5820 2e2e   |.|o9 : kk[X ..
│ │ │ │ +00018e10: 5820 5d2d 6d6f 6475 6c65 2c20 6672 6565  X ]-module, free
│ │ │ │ +00018e20: 2c20 6465 6772 6565 7320 7b30 2e2e 312c  , degrees {0..1,
│ │ │ │ +00018e30: 2032 3a31 7d20 2020 2020 2020 207c 0a7c   2:1}        |.|
│ │ │ │ +00018e40: 2020 2020 2020 2020 2030 2020 2031 2020           0   1  
│ │ │ │ +00018e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018e80: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00018e70: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00018e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018ec0: 2b0a 7c69 3130 203a 2061 7070 6c79 2874  +.|i10 : apply(t
│ │ │ │ -00018ed0: 6f4c 6973 7428 302e 2e35 292c 2069 2d3e  oList(0..5), i->
│ │ │ │ -00018ee0: 6869 6c62 6572 7446 756e 6374 696f 6e28  hilbertFunction(
│ │ │ │ -00018ef0: 692c 2045 6576 656e 2929 2020 7c0a 7c20  i, Eeven))  |.| 
│ │ │ │ +00018eb0: 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20 6170  -----+.|i10 : ap
│ │ │ │ +00018ec0: 706c 7928 746f 4c69 7374 2830 2e2e 3529  ply(toList(0..5)
│ │ │ │ +00018ed0: 2c20 692d 3e68 696c 6265 7274 4675 6e63  , i->hilbertFunc
│ │ │ │ +00018ee0: 7469 6f6e 2869 2c20 4565 7665 6e29 2920  tion(i, Eeven)) 
│ │ │ │ +00018ef0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00018f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018f30: 2020 2020 2020 2020 7c0a 7c6f 3130 203d          |.|o10 =
│ │ │ │ -00018f40: 207b 312c 2035 2c20 392c 2031 332c 2031   {1, 5, 9, 13, 1
│ │ │ │ -00018f50: 372c 2032 317d 2020 2020 2020 2020 2020  7, 21}          
│ │ │ │ -00018f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018f70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00018f20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00018f30: 6f31 3020 3d20 7b31 2c20 352c 2039 2c20  o10 = {1, 5, 9, 
│ │ │ │ +00018f40: 3133 2c20 3137 2c20 3231 7d20 2020 2020  13, 17, 21}     
│ │ │ │ +00018f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018f60: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00018f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018fb0: 7c0a 7c6f 3130 203a 204c 6973 7420 2020  |.|o10 : List   
│ │ │ │ +00018fa0: 2020 2020 207c 0a7c 6f31 3020 3a20 4c69       |.|o10 : Li
│ │ │ │ +00018fb0: 7374 2020 2020 2020 2020 2020 2020 2020  st              
│ │ │ │  00018fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018fe0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00018fe0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  00018ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019020: 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061  --------+..See a
│ │ │ │ -00019030: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020  lso.========..  
│ │ │ │ -00019040: 2a20 2a6e 6f74 6520 4578 744d 6f64 756c  * *note ExtModul
│ │ │ │ -00019050: 653a 2045 7874 4d6f 6475 6c65 2c20 2d2d  e: ExtModule, --
│ │ │ │ -00019060: 2045 7874 5e2a 284d 2c6b 2920 6f76 6572   Ext^*(M,k) over
│ │ │ │ -00019070: 2061 2063 6f6d 706c 6574 6520 696e 7465   a complete inte
│ │ │ │ -00019080: 7273 6563 7469 6f6e 2061 730a 2020 2020  rsection as.    
│ │ │ │ -00019090: 6d6f 6475 6c65 206f 7665 7220 4349 206f  module over CI o
│ │ │ │ -000190a0: 7065 7261 746f 7220 7269 6e67 0a20 202a  perator ring.  *
│ │ │ │ -000190b0: 202a 6e6f 7465 206f 6464 4578 744d 6f64   *note oddExtMod
│ │ │ │ -000190c0: 756c 653a 206f 6464 4578 744d 6f64 756c  ule: oddExtModul
│ │ │ │ -000190d0: 652c 202d 2d20 6f64 6420 7061 7274 206f  e, -- odd part o
│ │ │ │ -000190e0: 6620 4578 745e 2a28 4d2c 6b29 206f 7665  f Ext^*(M,k) ove
│ │ │ │ -000190f0: 7220 6120 636f 6d70 6c65 7465 0a20 2020  r a complete.   
│ │ │ │ -00019100: 2069 6e74 6572 7365 6374 696f 6e20 6173   intersection as
│ │ │ │ -00019110: 206d 6f64 756c 6520 6f76 6572 2043 4920   module over CI 
│ │ │ │ -00019120: 6f70 6572 6174 6f72 2072 696e 670a 2020  operator ring.  
│ │ │ │ -00019130: 2a20 2a6e 6f74 6520 4f75 7452 696e 673a  * *note OutRing:
│ │ │ │ -00019140: 204f 7574 5269 6e67 2c20 2d2d 204f 7074   OutRing, -- Opt
│ │ │ │ -00019150: 696f 6e20 616c 6c6f 7769 6e67 2073 7065  ion allowing spe
│ │ │ │ -00019160: 6369 6669 6361 7469 6f6e 206f 6620 7468  cification of th
│ │ │ │ -00019170: 6520 7269 6e67 206f 7665 720a 2020 2020  e ring over.    
│ │ │ │ -00019180: 7768 6963 6820 7468 6520 6f75 7470 7574  which the output
│ │ │ │ -00019190: 2069 7320 6465 6669 6e65 640a 0a57 6179   is defined..Way
│ │ │ │ -000191a0: 7320 746f 2075 7365 2065 7665 6e45 7874  s to use evenExt
│ │ │ │ -000191b0: 4d6f 6475 6c65 3a0a 3d3d 3d3d 3d3d 3d3d  Module:.========
│ │ │ │ -000191c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000191d0: 3d3d 0a0a 2020 2a20 2265 7665 6e45 7874  ==..  * "evenExt
│ │ │ │ -000191e0: 4d6f 6475 6c65 284d 6f64 756c 6529 220a  Module(Module)".
│ │ │ │ -000191f0: 0a46 6f72 2074 6865 2070 726f 6772 616d  .For the program
│ │ │ │ -00019200: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  mer.============
│ │ │ │ -00019210: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65  ======..The obje
│ │ │ │ -00019220: 6374 202a 6e6f 7465 2065 7665 6e45 7874  ct *note evenExt
│ │ │ │ -00019230: 4d6f 6475 6c65 3a20 6576 656e 4578 744d  Module: evenExtM
│ │ │ │ -00019240: 6f64 756c 652c 2069 7320 6120 2a6e 6f74  odule, is a *not
│ │ │ │ -00019250: 6520 6d65 7468 6f64 2066 756e 6374 696f  e method functio
│ │ │ │ -00019260: 6e20 7769 7468 0a6f 7074 696f 6e73 3a20  n with.options: 
│ │ │ │ -00019270: 284d 6163 6175 6c61 7932 446f 6329 4d65  (Macaulay2Doc)Me
│ │ │ │ -00019280: 7468 6f64 4675 6e63 7469 6f6e 5769 7468  thodFunctionWith
│ │ │ │ -00019290: 4f70 7469 6f6e 732c 2e0a 0a2d 2d2d 2d2d  Options,...-----
│ │ │ │ +00019010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ +00019020: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d  See also.=======
│ │ │ │ +00019030: 3d0a 0a20 202a 202a 6e6f 7465 2045 7874  =..  * *note Ext
│ │ │ │ +00019040: 4d6f 6475 6c65 3a20 4578 744d 6f64 756c  Module: ExtModul
│ │ │ │ +00019050: 652c 202d 2d20 4578 745e 2a28 4d2c 6b29  e, -- Ext^*(M,k)
│ │ │ │ +00019060: 206f 7665 7220 6120 636f 6d70 6c65 7465   over a complete
│ │ │ │ +00019070: 2069 6e74 6572 7365 6374 696f 6e20 6173   intersection as
│ │ │ │ +00019080: 0a20 2020 206d 6f64 756c 6520 6f76 6572  .    module over
│ │ │ │ +00019090: 2043 4920 6f70 6572 6174 6f72 2072 696e   CI operator rin
│ │ │ │ +000190a0: 670a 2020 2a20 2a6e 6f74 6520 6f64 6445  g.  * *note oddE
│ │ │ │ +000190b0: 7874 4d6f 6475 6c65 3a20 6f64 6445 7874  xtModule: oddExt
│ │ │ │ +000190c0: 4d6f 6475 6c65 2c20 2d2d 206f 6464 2070  Module, -- odd p
│ │ │ │ +000190d0: 6172 7420 6f66 2045 7874 5e2a 284d 2c6b  art of Ext^*(M,k
│ │ │ │ +000190e0: 2920 6f76 6572 2061 2063 6f6d 706c 6574  ) over a complet
│ │ │ │ +000190f0: 650a 2020 2020 696e 7465 7273 6563 7469  e.    intersecti
│ │ │ │ +00019100: 6f6e 2061 7320 6d6f 6475 6c65 206f 7665  on as module ove
│ │ │ │ +00019110: 7220 4349 206f 7065 7261 746f 7220 7269  r CI operator ri
│ │ │ │ +00019120: 6e67 0a20 202a 202a 6e6f 7465 204f 7574  ng.  * *note Out
│ │ │ │ +00019130: 5269 6e67 3a20 4f75 7452 696e 672c 202d  Ring: OutRing, -
│ │ │ │ +00019140: 2d20 4f70 7469 6f6e 2061 6c6c 6f77 696e  - Option allowin
│ │ │ │ +00019150: 6720 7370 6563 6966 6963 6174 696f 6e20  g specification 
│ │ │ │ +00019160: 6f66 2074 6865 2072 696e 6720 6f76 6572  of the ring over
│ │ │ │ +00019170: 0a20 2020 2077 6869 6368 2074 6865 206f  .    which the o
│ │ │ │ +00019180: 7574 7075 7420 6973 2064 6566 696e 6564  utput is defined
│ │ │ │ +00019190: 0a0a 5761 7973 2074 6f20 7573 6520 6576  ..Ways to use ev
│ │ │ │ +000191a0: 656e 4578 744d 6f64 756c 653a 0a3d 3d3d  enExtModule:.===
│ │ │ │ +000191b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +000191c0: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 6576  =======..  * "ev
│ │ │ │ +000191d0: 656e 4578 744d 6f64 756c 6528 4d6f 6475  enExtModule(Modu
│ │ │ │ +000191e0: 6c65 2922 0a0a 466f 7220 7468 6520 7072  le)"..For the pr
│ │ │ │ +000191f0: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d  ogrammer.=======
│ │ │ │ +00019200: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865  ===========..The
│ │ │ │ +00019210: 206f 626a 6563 7420 2a6e 6f74 6520 6576   object *note ev
│ │ │ │ +00019220: 656e 4578 744d 6f64 756c 653a 2065 7665  enExtModule: eve
│ │ │ │ +00019230: 6e45 7874 4d6f 6475 6c65 2c20 6973 2061  nExtModule, is a
│ │ │ │ +00019240: 202a 6e6f 7465 206d 6574 686f 6420 6675   *note method fu
│ │ │ │ +00019250: 6e63 7469 6f6e 2077 6974 680a 6f70 7469  nction with.opti
│ │ │ │ +00019260: 6f6e 733a 2028 4d61 6361 756c 6179 3244  ons: (Macaulay2D
│ │ │ │ +00019270: 6f63 294d 6574 686f 6446 756e 6374 696f  oc)MethodFunctio
│ │ │ │ +00019280: 6e57 6974 684f 7074 696f 6e73 2c2e 0a0a  nWithOptions,...
│ │ │ │ +00019290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000192a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000192b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000192c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000192d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000192e0: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520  ----------..The 
│ │ │ │ -000192f0: 736f 7572 6365 206f 6620 7468 6973 2064  source of this d
│ │ │ │ -00019300: 6f63 756d 656e 7420 6973 2069 6e0a 2f62  ocument is in./b
│ │ │ │ -00019310: 7569 6c64 2f72 6570 726f 6475 6369 626c  uild/reproducibl
│ │ │ │ -00019320: 652d 7061 7468 2f6d 6163 6175 6c61 7932  e-path/macaulay2
│ │ │ │ -00019330: 2d31 2e32 352e 3131 2b64 732f 4d32 2f4d  -1.25.11+ds/M2/M
│ │ │ │ -00019340: 6163 6175 6c61 7932 2f70 6163 6b61 6765  acaulay2/package
│ │ │ │ -00019350: 732f 0a43 6f6d 706c 6574 6549 6e74 6572  s/.CompleteInter
│ │ │ │ -00019360: 7365 6374 696f 6e52 6573 6f6c 7574 696f  sectionResolutio
│ │ │ │ -00019370: 6e73 2e6d 323a 3336 3433 3a30 2e0a 1f0a  ns.m2:3643:0....
│ │ │ │ -00019380: 4669 6c65 3a20 436f 6d70 6c65 7465 496e  File: CompleteIn
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│ │ │ │ -000193b0: 3a20 6578 706f 2c20 4e65 7874 3a20 6578  : expo, Next: ex
│ │ │ │ -000193c0: 7465 7269 6f72 4578 744d 6f64 756c 652c  teriorExtModule,
│ │ │ │ -000193d0: 2050 7265 763a 2065 7665 6e45 7874 4d6f   Prev: evenExtMo
│ │ │ │ -000193e0: 6475 6c65 2c20 5570 3a20 546f 700a 0a65  dule, Up: Top..e
│ │ │ │ -000193f0: 7870 6f20 2d2d 2072 6574 7572 6e73 2061  xpo -- returns a
│ │ │ │ -00019400: 2073 6574 2063 6f72 7265 7370 6f6e 6469   set correspondi
│ │ │ │ -00019410: 6e67 2074 6f20 7468 6520 6261 7369 7320  ng to the basis 
│ │ │ │ -00019420: 6f66 2061 2064 6976 6964 6564 2070 6f77  of a divided pow
│ │ │ │ -00019430: 6572 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  er.*************
│ │ │ │ +000192d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a  ---------------.
│ │ │ │ +000192e0: 0a54 6865 2073 6f75 7263 6520 6f66 2074  .The source of t
│ │ │ │ +000192f0: 6869 7320 646f 6375 6d65 6e74 2069 7320  his document is 
│ │ │ │ +00019300: 696e 0a2f 6275 696c 642f 7265 7072 6f64  in./build/reprod
│ │ │ │ +00019310: 7563 6962 6c65 2d70 6174 682f 6d61 6361  ucible-path/maca
│ │ │ │ +00019320: 756c 6179 322d 312e 3235 2e31 312b 6473  ulay2-1.25.11+ds
│ │ │ │ +00019330: 2f4d 322f 4d61 6361 756c 6179 322f 7061  /M2/Macaulay2/pa
│ │ │ │ +00019340: 636b 6167 6573 2f0a 436f 6d70 6c65 7465  ckages/.Complete
│ │ │ │ +00019350: 496e 7465 7273 6563 7469 6f6e 5265 736f  IntersectionReso
│ │ │ │ +00019360: 6c75 7469 6f6e 732e 6d32 3a33 3634 333a  lutions.m2:3643:
│ │ │ │ +00019370: 302e 0a1f 0a46 696c 653a 2043 6f6d 706c  0....File: Compl
│ │ │ │ +00019380: 6574 6549 6e74 6572 7365 6374 696f 6e52  eteIntersectionR
│ │ │ │ +00019390: 6573 6f6c 7574 696f 6e73 2e69 6e66 6f2c  esolutions.info,
│ │ │ │ +000193a0: 204e 6f64 653a 2065 7870 6f2c 204e 6578   Node: expo, Nex
│ │ │ │ +000193b0: 743a 2065 7874 6572 696f 7245 7874 4d6f  t: exteriorExtMo
│ │ │ │ +000193c0: 6475 6c65 2c20 5072 6576 3a20 6576 656e  dule, Prev: even
│ │ │ │ +000193d0: 4578 744d 6f64 756c 652c 2055 703a 2054  ExtModule, Up: T
│ │ │ │ +000193e0: 6f70 0a0a 6578 706f 202d 2d20 7265 7475  op..expo -- retu
│ │ │ │ +000193f0: 726e 7320 6120 7365 7420 636f 7272 6573  rns a set corres
│ │ │ │ +00019400: 706f 6e64 696e 6720 746f 2074 6865 2062  ponding to the b
│ │ │ │ +00019410: 6173 6973 206f 6620 6120 6469 7669 6465  asis of a divide
│ │ │ │ +00019420: 6420 706f 7765 720a 2a2a 2a2a 2a2a 2a2a  d power.********
│ │ │ │ +00019430: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00019440: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00019450: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00019460: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00019470: 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 6167  ******..  * Usag
│ │ │ │ -00019480: 653a 200a 2020 2020 2020 2020 4220 3d20  e: .        B = 
│ │ │ │ -00019490: 6578 706f 2863 2c4e 290a 2020 2020 2020  expo(c,N).      
│ │ │ │ -000194a0: 2020 4220 3d20 6578 706f 2863 2c4c 290a    B = expo(c,L).
│ │ │ │ -000194b0: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ -000194c0: 2020 2a20 4e2c 2061 6e20 2a6e 6f74 6520    * N, an *note 
│ │ │ │ -000194d0: 696e 7465 6765 723a 2028 4d61 6361 756c  integer: (Macaul
│ │ │ │ -000194e0: 6179 3244 6f63 295a 5a2c 2c20 0a20 2020  ay2Doc)ZZ,, .   
│ │ │ │ -000194f0: 2020 202a 2063 2c20 616e 202a 6e6f 7465     * c, an *note
│ │ │ │ -00019500: 2069 6e74 6567 6572 3a20 284d 6163 6175   integer: (Macau
│ │ │ │ -00019510: 6c61 7932 446f 6329 5a5a 2c2c 200a 2020  lay2Doc)ZZ,, .  
│ │ │ │ -00019520: 2020 2020 2a20 4c2c 2061 202a 6e6f 7465      * L, a *note
│ │ │ │ -00019530: 206c 6973 743a 2028 4d61 6361 756c 6179   list: (Macaulay
│ │ │ │ -00019540: 3244 6f63 294c 6973 742c 2c20 6f66 2063  2Doc)List,, of c
│ │ │ │ -00019550: 206e 6f6e 2d6e 6567 6174 6976 6520 696e   non-negative in
│ │ │ │ -00019560: 7465 6765 7273 0a20 202a 204f 7574 7075  tegers.  * Outpu
│ │ │ │ -00019570: 7473 3a0a 2020 2020 2020 2a20 422c 2061  ts:.      * B, a
│ │ │ │ -00019580: 202a 6e6f 7465 206c 6973 743a 2028 4d61   *note list: (Ma
│ │ │ │ -00019590: 6361 756c 6179 3244 6f63 294c 6973 742c  caulay2Doc)List,
│ │ │ │ -000195a0: 2c20 7061 7274 6974 696f 6e73 2077 6974  , partitions wit
│ │ │ │ -000195b0: 6820 6320 6e6f 6e2d 6e65 6761 7469 7665  h c non-negative
│ │ │ │ -000195c0: 0a20 2020 2020 2020 2070 6172 7473 0a0a  .        parts..
│ │ │ │ -000195d0: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
│ │ │ │ -000195e0: 3d3d 3d3d 3d3d 3d0a 0a54 6865 2066 6f72  =======..The for
│ │ │ │ -000195f0: 6d20 6578 706f 2863 2c4e 2920 7265 7475  m expo(c,N) retu
│ │ │ │ -00019600: 726e 7320 7061 7274 6974 696f 6e73 206f  rns partitions o
│ │ │ │ -00019610: 6620 4e20 7769 7468 2063 206e 6f6e 2d6e  f N with c non-n
│ │ │ │ -00019620: 6567 6174 6976 6520 7061 7274 732e 2054  egative parts. T
│ │ │ │ -00019630: 6865 2066 6f72 6d0a 6578 706f 2863 2c20  he form.expo(c, 
│ │ │ │ -00019640: 4c29 2072 6574 7572 6e73 2070 6172 7469  L) returns parti
│ │ │ │ -00019650: 7469 6f6e 7320 7769 7468 206e 6f6e 2d6e  tions with non-n
│ │ │ │ -00019660: 6567 6174 6976 6520 7061 7274 7320 7468  egative parts th
│ │ │ │ -00019670: 6174 2061 7265 2063 6f6d 706f 6e65 6e74  at are component
│ │ │ │ -00019680: 7769 7365 203c 3d0a 4c20 2861 6e64 2061  wise <=.L (and a
│ │ │ │ -00019690: 6e79 2073 756d 203c 3d20 7375 6d20 4c29  ny sum <= sum L)
│ │ │ │ -000196a0: 2e0a 0a54 6865 206c 6973 7420 6578 706f  ...The list expo
│ │ │ │ -000196b0: 2863 2c4e 2920 206d 6179 2062 6520 7468  (c,N)  may be th
│ │ │ │ -000196c0: 6f75 6768 7420 6f66 2061 7320 7468 6520  ought of as the 
│ │ │ │ -000196d0: 6c69 7374 206f 6620 6578 706f 6e65 6e74  list of exponent
│ │ │ │ -000196e0: 2076 6563 746f 7273 206f 6620 7468 650a   vectors of the.
│ │ │ │ -000196f0: 6d6f 6e6f 6d69 616c 7320 6f66 2064 6567  monomials of deg
│ │ │ │ -00019700: 7265 6520 4e20 696e 2063 2076 6172 6961  ree N in c varia
│ │ │ │ -00019710: 626c 6573 2e20 5468 6973 2069 7320 7573  bles. This is us
│ │ │ │ -00019720: 6564 2069 6e20 7468 6520 636f 6e73 7472  ed in the constr
│ │ │ │ -00019730: 7563 7469 6f6e 206f 6620 7468 650a 4569  uction of the.Ei
│ │ │ │ -00019740: 7365 6e62 7564 2d53 6861 6d61 7368 2072  senbud-Shamash r
│ │ │ │ -00019750: 6573 6f6c 7574 696f 6e2e 0a0a 5468 6520  esolution...The 
│ │ │ │ -00019760: 6c69 7374 2065 7870 6f28 632c 204c 292c  list expo(c, L),
│ │ │ │ -00019770: 206f 6e20 7468 6520 6f74 6865 7220 6861   on the other ha
│ │ │ │ -00019780: 6e64 2c20 6d61 7920 6265 2074 686f 7567  nd, may be thoug
│ │ │ │ -00019790: 6874 206f 6620 6173 2074 6865 206c 6973  ht of as the lis
│ │ │ │ -000197a0: 7420 6f66 0a64 6976 6973 6f72 7320 6f66  t of.divisors of
│ │ │ │ -000197b0: 2065 5e4c 203d 2065 5f30 5e7b 4c5f 307d   e^L = e_0^{L_0}
│ │ │ │ -000197c0: 202e 2e2e 2065 5f63 5e7b 4c5f 637d 2e20   ... e_c^{L_c}. 
│ │ │ │ -000197d0: 5468 6973 2069 7320 7573 6564 2069 6e20  This is used in 
│ │ │ │ -000197e0: 7468 6520 636f 6e73 7472 7563 7469 6f6e  the construction
│ │ │ │ -000197f0: 206f 660a 7468 6520 6869 6768 6572 2068   of.the higher h
│ │ │ │ -00019800: 6f6d 6f74 6f70 6965 7320 6f6e 2061 2063  omotopies on a c
│ │ │ │ -00019810: 6f6d 706c 6578 2e0a 0a2b 2d2d 2d2d 2d2d  omplex...+------
│ │ │ │ +00019460: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a  ***********..  *
│ │ │ │ +00019470: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ +00019480: 2042 203d 2065 7870 6f28 632c 4e29 0a20   B = expo(c,N). 
│ │ │ │ +00019490: 2020 2020 2020 2042 203d 2065 7870 6f28         B = expo(
│ │ │ │ +000194a0: 632c 4c29 0a20 202a 2049 6e70 7574 733a  c,L).  * Inputs:
│ │ │ │ +000194b0: 0a20 2020 2020 202a 204e 2c20 616e 202a  .      * N, an *
│ │ │ │ +000194c0: 6e6f 7465 2069 6e74 6567 6572 3a20 284d  note integer: (M
│ │ │ │ +000194d0: 6163 6175 6c61 7932 446f 6329 5a5a 2c2c  acaulay2Doc)ZZ,,
│ │ │ │ +000194e0: 200a 2020 2020 2020 2a20 632c 2061 6e20   .      * c, an 
│ │ │ │ +000194f0: 2a6e 6f74 6520 696e 7465 6765 723a 2028  *note integer: (
│ │ │ │ +00019500: 4d61 6361 756c 6179 3244 6f63 295a 5a2c  Macaulay2Doc)ZZ,
│ │ │ │ +00019510: 2c20 0a20 2020 2020 202a 204c 2c20 6120  , .      * L, a 
│ │ │ │ +00019520: 2a6e 6f74 6520 6c69 7374 3a20 284d 6163  *note list: (Mac
│ │ │ │ +00019530: 6175 6c61 7932 446f 6329 4c69 7374 2c2c  aulay2Doc)List,,
│ │ │ │ +00019540: 206f 6620 6320 6e6f 6e2d 6e65 6761 7469   of c non-negati
│ │ │ │ +00019550: 7665 2069 6e74 6567 6572 730a 2020 2a20  ve integers.  * 
│ │ │ │ +00019560: 4f75 7470 7574 733a 0a20 2020 2020 202a  Outputs:.      *
│ │ │ │ +00019570: 2042 2c20 6120 2a6e 6f74 6520 6c69 7374   B, a *note list
│ │ │ │ +00019580: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +00019590: 4c69 7374 2c2c 2070 6172 7469 7469 6f6e  List,, partition
│ │ │ │ +000195a0: 7320 7769 7468 2063 206e 6f6e 2d6e 6567  s with c non-neg
│ │ │ │ +000195b0: 6174 6976 650a 2020 2020 2020 2020 7061  ative.        pa
│ │ │ │ +000195c0: 7274 730a 0a44 6573 6372 6970 7469 6f6e  rts..Description
│ │ │ │ +000195d0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  .===========..Th
│ │ │ │ +000195e0: 6520 666f 726d 2065 7870 6f28 632c 4e29  e form expo(c,N)
│ │ │ │ +000195f0: 2072 6574 7572 6e73 2070 6172 7469 7469   returns partiti
│ │ │ │ +00019600: 6f6e 7320 6f66 204e 2077 6974 6820 6320  ons of N with c 
│ │ │ │ +00019610: 6e6f 6e2d 6e65 6761 7469 7665 2070 6172  non-negative par
│ │ │ │ +00019620: 7473 2e20 5468 6520 666f 726d 0a65 7870  ts. The form.exp
│ │ │ │ +00019630: 6f28 632c 204c 2920 7265 7475 726e 7320  o(c, L) returns 
│ │ │ │ +00019640: 7061 7274 6974 696f 6e73 2077 6974 6820  partitions with 
│ │ │ │ +00019650: 6e6f 6e2d 6e65 6761 7469 7665 2070 6172  non-negative par
│ │ │ │ +00019660: 7473 2074 6861 7420 6172 6520 636f 6d70  ts that are comp
│ │ │ │ +00019670: 6f6e 656e 7477 6973 6520 3c3d 0a4c 2028  onentwise <=.L (
│ │ │ │ +00019680: 616e 6420 616e 7920 7375 6d20 3c3d 2073  and any sum <= s
│ │ │ │ +00019690: 756d 204c 292e 0a0a 5468 6520 6c69 7374  um L)...The list
│ │ │ │ +000196a0: 2065 7870 6f28 632c 4e29 2020 6d61 7920   expo(c,N)  may 
│ │ │ │ +000196b0: 6265 2074 686f 7567 6874 206f 6620 6173  be thought of as
│ │ │ │ +000196c0: 2074 6865 206c 6973 7420 6f66 2065 7870   the list of exp
│ │ │ │ +000196d0: 6f6e 656e 7420 7665 6374 6f72 7320 6f66  onent vectors of
│ │ │ │ +000196e0: 2074 6865 0a6d 6f6e 6f6d 6961 6c73 206f   the.monomials o
│ │ │ │ +000196f0: 6620 6465 6772 6565 204e 2069 6e20 6320  f degree N in c 
│ │ │ │ +00019700: 7661 7269 6162 6c65 732e 2054 6869 7320  variables. This 
│ │ │ │ +00019710: 6973 2075 7365 6420 696e 2074 6865 2063  is used in the c
│ │ │ │ +00019720: 6f6e 7374 7275 6374 696f 6e20 6f66 2074  onstruction of t
│ │ │ │ +00019730: 6865 0a45 6973 656e 6275 642d 5368 616d  he.Eisenbud-Sham
│ │ │ │ +00019740: 6173 6820 7265 736f 6c75 7469 6f6e 2e0a  ash resolution..
│ │ │ │ +00019750: 0a54 6865 206c 6973 7420 6578 706f 2863  .The list expo(c
│ │ │ │ +00019760: 2c20 4c29 2c20 6f6e 2074 6865 206f 7468  , L), on the oth
│ │ │ │ +00019770: 6572 2068 616e 642c 206d 6179 2062 6520  er hand, may be 
│ │ │ │ +00019780: 7468 6f75 6768 7420 6f66 2061 7320 7468  thought of as th
│ │ │ │ +00019790: 6520 6c69 7374 206f 660a 6469 7669 736f  e list of.diviso
│ │ │ │ +000197a0: 7273 206f 6620 655e 4c20 3d20 655f 305e  rs of e^L = e_0^
│ │ │ │ +000197b0: 7b4c 5f30 7d20 2e2e 2e20 655f 635e 7b4c  {L_0} ... e_c^{L
│ │ │ │ +000197c0: 5f63 7d2e 2054 6869 7320 6973 2075 7365  _c}. This is use
│ │ │ │ +000197d0: 6420 696e 2074 6865 2063 6f6e 7374 7275  d in the constru
│ │ │ │ +000197e0: 6374 696f 6e20 6f66 0a74 6865 2068 6967  ction of.the hig
│ │ │ │ +000197f0: 6865 7220 686f 6d6f 746f 7069 6573 206f  her homotopies o
│ │ │ │ +00019800: 6e20 6120 636f 6d70 6c65 782e 0a0a 2b2d  n a complex...+-
│ │ │ │ +00019810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019860: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2065  -------+.|i1 : e
│ │ │ │ -00019870: 7870 6f28 332c 3529 2020 2020 2020 2020  xpo(3,5)        
│ │ │ │ +00019850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00019860: 3120 3a20 6578 706f 2833 2c35 2920 2020  1 : expo(3,5)   
│ │ │ │ +00019870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000198a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000198b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000198a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000198b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000198c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000198d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000198e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000198f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019900: 2020 2020 2020 207c 0a7c 6f31 203d 207b         |.|o1 = {
│ │ │ │ -00019910: 7b35 2c20 302c 2030 7d2c 207b 342c 2031  {5, 0, 0}, {4, 1
│ │ │ │ -00019920: 2c20 307d 2c20 7b34 2c20 302c 2031 7d2c  , 0}, {4, 0, 1},
│ │ │ │ -00019930: 207b 332c 2032 2c20 307d 2c20 7b33 2c20   {3, 2, 0}, {3, 
│ │ │ │ -00019940: 312c 2031 7d2c 207b 332c 2030 2c20 327d  1, 1}, {3, 0, 2}
│ │ │ │ -00019950: 2c20 7b32 2c20 207c 0a7c 2020 2020 202d  , {2,  |.|     -
│ │ │ │ +000198f0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00019900: 3120 3d20 7b7b 352c 2030 2c20 307d 2c20  1 = {{5, 0, 0}, 
│ │ │ │ +00019910: 7b34 2c20 312c 2030 7d2c 207b 342c 2030  {4, 1, 0}, {4, 0
│ │ │ │ +00019920: 2c20 317d 2c20 7b33 2c20 322c 2030 7d2c  , 1}, {3, 2, 0},
│ │ │ │ +00019930: 207b 332c 2031 2c20 317d 2c20 7b33 2c20   {3, 1, 1}, {3, 
│ │ │ │ +00019940: 302c 2032 7d2c 207b 322c 2020 7c0a 7c20  0, 2}, {2,  |.| 
│ │ │ │ +00019950: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d      ------------
│ │ │ │  00019960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000199a0: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2033  -------|.|     3
│ │ │ │ -000199b0: 2c20 307d 2c20 7b32 2c20 322c 2031 7d2c  , 0}, {2, 2, 1},
│ │ │ │ -000199c0: 207b 322c 2031 2c20 327d 2c20 7b32 2c20   {2, 1, 2}, {2, 
│ │ │ │ -000199d0: 302c 2033 7d2c 207b 312c 2034 2c20 307d  0, 3}, {1, 4, 0}
│ │ │ │ -000199e0: 2c20 7b31 2c20 332c 2031 7d2c 207b 312c  , {1, 3, 1}, {1,
│ │ │ │ -000199f0: 2032 2c20 327d 2c7c 0a7c 2020 2020 202d   2, 2},|.|     -
│ │ │ │ +00019990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +000199a0: 2020 2020 332c 2030 7d2c 207b 322c 2032      3, 0}, {2, 2
│ │ │ │ +000199b0: 2c20 317d 2c20 7b32 2c20 312c 2032 7d2c  , 1}, {2, 1, 2},
│ │ │ │ +000199c0: 207b 322c 2030 2c20 337d 2c20 7b31 2c20   {2, 0, 3}, {1, 
│ │ │ │ +000199d0: 342c 2030 7d2c 207b 312c 2033 2c20 317d  4, 0}, {1, 3, 1}
│ │ │ │ +000199e0: 2c20 7b31 2c20 322c 2032 7d2c 7c0a 7c20  , {1, 2, 2},|.| 
│ │ │ │ +000199f0: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d      ------------
│ │ │ │  00019a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019a40: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 207b  -------|.|     {
│ │ │ │ -00019a50: 312c 2031 2c20 337d 2c20 7b31 2c20 302c  1, 1, 3}, {1, 0,
│ │ │ │ -00019a60: 2034 7d2c 207b 302c 2035 2c20 307d 2c20   4}, {0, 5, 0}, 
│ │ │ │ -00019a70: 7b30 2c20 342c 2031 7d2c 207b 302c 2033  {0, 4, 1}, {0, 3
│ │ │ │ -00019a80: 2c20 327d 2c20 7b30 2c20 322c 2033 7d2c  , 2}, {0, 2, 3},
│ │ │ │ -00019a90: 207b 302c 2031 2c7c 0a7c 2020 2020 202d   {0, 1,|.|     -
│ │ │ │ +00019a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +00019a40: 2020 2020 7b31 2c20 312c 2033 7d2c 207b      {1, 1, 3}, {
│ │ │ │ +00019a50: 312c 2030 2c20 347d 2c20 7b30 2c20 352c  1, 0, 4}, {0, 5,
│ │ │ │ +00019a60: 2030 7d2c 207b 302c 2034 2c20 317d 2c20   0}, {0, 4, 1}, 
│ │ │ │ +00019a70: 7b30 2c20 332c 2032 7d2c 207b 302c 2032  {0, 3, 2}, {0, 2
│ │ │ │ +00019a80: 2c20 337d 2c20 7b30 2c20 312c 7c0a 7c20  , 3}, {0, 1,|.| 
│ │ │ │ +00019a90: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d      ------------
│ │ │ │  00019aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019ae0: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2034  -------|.|     4
│ │ │ │ -00019af0: 7d2c 207b 302c 2030 2c20 357d 7d20 2020  }, {0, 0, 5}}   
│ │ │ │ +00019ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +00019ae0: 2020 2020 347d 2c20 7b30 2c20 302c 2035      4}, {0, 0, 5
│ │ │ │ +00019af0: 7d7d 2020 2020 2020 2020 2020 2020 2020  }}              
│ │ │ │  00019b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019b30: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00019b20: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00019b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019b80: 2020 2020 2020 207c 0a7c 6f31 203a 204c         |.|o1 : L
│ │ │ │ -00019b90: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ +00019b70: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00019b80: 3120 3a20 4c69 7374 2020 2020 2020 2020  1 : List        
│ │ │ │ +00019b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019bd0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00019bc0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00019bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019c20: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2065  -------+.|i2 : e
│ │ │ │ -00019c30: 7870 6f28 332c 207b 332c 322c 317d 2920  xpo(3, {3,2,1}) 
│ │ │ │ +00019c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00019c20: 3220 3a20 6578 706f 2833 2c20 7b33 2c32  2 : expo(3, {3,2
│ │ │ │ +00019c30: 2c31 7d29 2020 2020 2020 2020 2020 2020  ,1})            
│ │ │ │  00019c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019c70: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00019c60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00019c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019cc0: 2020 2020 2020 207c 0a7c 6f32 203d 207b         |.|o2 = {
│ │ │ │ -00019cd0: 7b30 2c20 302c 2030 7d2c 207b 312c 2030  {0, 0, 0}, {1, 0
│ │ │ │ -00019ce0: 2c20 307d 2c20 7b30 2c20 312c 2030 7d2c  , 0}, {0, 1, 0},
│ │ │ │ -00019cf0: 207b 302c 2030 2c20 317d 2c20 7b32 2c20   {0, 0, 1}, {2, 
│ │ │ │ -00019d00: 302c 2030 7d2c 207b 312c 2031 2c20 307d  0, 0}, {1, 1, 0}
│ │ │ │ -00019d10: 2c20 7b31 2c20 207c 0a7c 2020 2020 202d  , {1,  |.|     -
│ │ │ │ +00019cb0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00019cc0: 3220 3d20 7b7b 302c 2030 2c20 307d 2c20  2 = {{0, 0, 0}, 
│ │ │ │ +00019cd0: 7b31 2c20 302c 2030 7d2c 207b 302c 2031  {1, 0, 0}, {0, 1
│ │ │ │ +00019ce0: 2c20 307d 2c20 7b30 2c20 302c 2031 7d2c  , 0}, {0, 0, 1},
│ │ │ │ +00019cf0: 207b 322c 2030 2c20 307d 2c20 7b31 2c20   {2, 0, 0}, {1, 
│ │ │ │ +00019d00: 312c 2030 7d2c 207b 312c 2020 7c0a 7c20  1, 0}, {1,  |.| 
│ │ │ │ +00019d10: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d      ------------
│ │ │ │  00019d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00019e40: 2c20 317d 2c20 7b31 2c20 322c 7c0a 7c20  , 1}, {1, 2,|.| 
│ │ │ │ +00019e50: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d      ------------
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│ │ │ │  00019e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019ea0: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2031  -------|.|     1
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│ │ │ │ -00019ef0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
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│ │ │ │ +00019ea0: 2020 2020 317d 2c20 7b33 2c20 322c 2030      1}, {3, 2, 0
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│ │ │ │ +00019ee0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
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│ │ │ │  00019f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019f40: 2020 2020 2020 207c 0a7c 6f32 203a 204c         |.|o2 : L
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│ │ │ │ +00019f40: 3220 3a20 4c69 7374 2020 2020 2020 2020  2 : List        
│ │ │ │ +00019f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00019fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0001a100: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d  e programmer.===
│ │ │ │ -0001a110: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
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│ │ │ │ -0001a160: 6c61 7932 446f 6329 4d65 7468 6f64 4675  lay2Doc)MethodFu
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│ │ │ │ +00019fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53  ------------+..S
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│ │ │ │ +0001a000: 6e62 7564 5368 616d 6173 683a 2045 6973  nbudShamash: Eis
│ │ │ │ +0001a010: 656e 6275 6453 6861 6d61 7368 2c20 2d2d  enbudShamash, --
│ │ │ │ +0001a020: 2043 6f6d 7075 7465 7320 7468 6520 4569   Computes the Ei
│ │ │ │ +0001a030: 7365 6e62 7564 2d53 6861 6d61 7368 0a20  senbud-Shamash. 
│ │ │ │ +0001a040: 2020 2043 6f6d 706c 6578 0a20 202a 202a     Complex.  * *
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│ │ │ │ +0001a060: 6965 733a 206d 616b 6548 6f6d 6f74 6f70  ies: makeHomotop
│ │ │ │ +0001a070: 6965 732c 202d 2d20 7265 7475 726e 7320  ies, -- returns 
│ │ │ │ +0001a080: 6120 7379 7374 656d 206f 6620 6869 6768  a system of high
│ │ │ │ +0001a090: 6572 0a20 2020 2068 6f6d 6f74 6f70 6965  er.    homotopie
│ │ │ │ +0001a0a0: 730a 0a57 6179 7320 746f 2075 7365 2065  s..Ways to use e
│ │ │ │ +0001a0b0: 7870 6f3a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  xpo:.===========
│ │ │ │ +0001a0c0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2265 7870  ======..  * "exp
│ │ │ │ +0001a0d0: 6f28 5a5a 2c4c 6973 7429 220a 2020 2a20  o(ZZ,List)".  * 
│ │ │ │ +0001a0e0: 2265 7870 6f28 5a5a 2c5a 5a29 220a 0a46  "expo(ZZ,ZZ)"..F
│ │ │ │ +0001a0f0: 6f72 2074 6865 2070 726f 6772 616d 6d65  or the programme
│ │ │ │ +0001a100: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  r.==============
│ │ │ │ +0001a110: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374  ====..The object
│ │ │ │ +0001a120: 202a 6e6f 7465 2065 7870 6f3a 2065 7870   *note expo: exp
│ │ │ │ +0001a130: 6f2c 2069 7320 6120 2a6e 6f74 6520 6d65  o, is a *note me
│ │ │ │ +0001a140: 7468 6f64 2066 756e 6374 696f 6e3a 0a28  thod function:.(
│ │ │ │ +0001a150: 4d61 6361 756c 6179 3244 6f63 294d 6574  Macaulay2Doc)Met
│ │ │ │ +0001a160: 686f 6446 756e 6374 696f 6e2c 2e0a 0a2d  hodFunction,...-
│ │ │ │ +0001a170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a1c0: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073  ---------..The s
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│ │ │ │ -0001a200: 2d70 6174 682f 6d61 6361 756c 6179 322d  -path/macaulay2-
│ │ │ │ -0001a210: 312e 3235 2e31 312b 6473 2f4d 322f 4d61  1.25.11+ds/M2/Ma
│ │ │ │ -0001a220: 6361 756c 6179 322f 7061 636b 6167 6573  caulay2/packages
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│ │ │ │ -0001a250: 732e 6d32 3a35 3038 363a 302e 0a1f 0a46  s.m2:5086:0....F
│ │ │ │ -0001a260: 696c 653a 2043 6f6d 706c 6574 6549 6e74  ile: CompleteInt
│ │ │ │ -0001a270: 6572 7365 6374 696f 6e52 6573 6f6c 7574  ersectionResolut
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│ │ │ │ -0001a2b0: 6f72 486f 6d6f 6c6f 6779 4d6f 6475 6c65  orHomologyModule
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│ │ │ │ -0001a2d0: 3a20 546f 700a 0a65 7874 6572 696f 7245  : Top..exteriorE
│ │ │ │ -0001a2e0: 7874 4d6f 6475 6c65 202d 2d20 4578 7428  xtModule -- Ext(
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│ │ │ │ -0001a310: 7220 616e 2065 7874 6572 696f 7220 616c  r an exterior al
│ │ │ │ -0001a320: 6765 6272 610a 2a2a 2a2a 2a2a 2a2a 2a2a  gebra.**********
│ │ │ │ +0001a1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a  --------------..
│ │ │ │ +0001a1c0: 5468 6520 736f 7572 6365 206f 6620 7468  The source of th
│ │ │ │ +0001a1d0: 6973 2064 6f63 756d 656e 7420 6973 2069  is document is i
│ │ │ │ +0001a1e0: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475  n./build/reprodu
│ │ │ │ +0001a1f0: 6369 626c 652d 7061 7468 2f6d 6163 6175  cible-path/macau
│ │ │ │ +0001a200: 6c61 7932 2d31 2e32 352e 3131 2b64 732f  lay2-1.25.11+ds/
│ │ │ │ +0001a210: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163  M2/Macaulay2/pac
│ │ │ │ +0001a220: 6b61 6765 732f 0a43 6f6d 706c 6574 6549  kages/.CompleteI
│ │ │ │ +0001a230: 6e74 6572 7365 6374 696f 6e52 6573 6f6c  ntersectionResol
│ │ │ │ +0001a240: 7574 696f 6e73 2e6d 323a 3530 3836 3a30  utions.m2:5086:0
│ │ │ │ +0001a250: 2e0a 1f0a 4669 6c65 3a20 436f 6d70 6c65  ....File: Comple
│ │ │ │ +0001a260: 7465 496e 7465 7273 6563 7469 6f6e 5265  teIntersectionRe
│ │ │ │ +0001a270: 736f 6c75 7469 6f6e 732e 696e 666f 2c20  solutions.info, 
│ │ │ │ +0001a280: 4e6f 6465 3a20 6578 7465 7269 6f72 4578  Node: exteriorEx
│ │ │ │ +0001a290: 744d 6f64 756c 652c 204e 6578 743a 2065  tModule, Next: e
│ │ │ │ +0001a2a0: 7874 6572 696f 7248 6f6d 6f6c 6f67 794d  xteriorHomologyM
│ │ │ │ +0001a2b0: 6f64 756c 652c 2050 7265 763a 2065 7870  odule, Prev: exp
│ │ │ │ +0001a2c0: 6f2c 2055 703a 2054 6f70 0a0a 6578 7465  o, Up: Top..exte
│ │ │ │ +0001a2d0: 7269 6f72 4578 744d 6f64 756c 6520 2d2d  riorExtModule --
│ │ │ │ +0001a2e0: 2045 7874 284d 2c6b 2920 6f72 2045 7874   Ext(M,k) or Ext
│ │ │ │ +0001a2f0: 284d 2c4e 2920 6173 2061 206d 6f64 756c  (M,N) as a modul
│ │ │ │ +0001a300: 6520 6f76 6572 2061 6e20 6578 7465 7269  e over an exteri
│ │ │ │ +0001a310: 6f72 2061 6c67 6562 7261 0a2a 2a2a 2a2a  or algebra.*****
│ │ │ │ +0001a320: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001a330: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001a340: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001a350: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001a360: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001a370: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a  ****..  * Usage:
│ │ │ │ -0001a380: 200a 2020 2020 2020 2020 4520 3d20 6578   .        E = ex
│ │ │ │ -0001a390: 7465 7269 6f72 4578 744d 6f64 756c 6528  teriorExtModule(
│ │ │ │ -0001a3a0: 662c 4d29 0a20 202a 2049 6e70 7574 733a  f,M).  * Inputs:
│ │ │ │ -0001a3b0: 0a20 2020 2020 202a 2066 2c20 6120 2a6e  .      * f, a *n
│ │ │ │ -0001a3c0: 6f74 6520 6d61 7472 6978 3a20 284d 6163  ote matrix: (Mac
│ │ │ │ -0001a3d0: 6175 6c61 7932 446f 6329 4d61 7472 6978  aulay2Doc)Matrix
│ │ │ │ -0001a3e0: 2c2c 2031 2078 2063 2c20 656e 7472 6965  ,, 1 x c, entrie
│ │ │ │ -0001a3f0: 7320 6d75 7374 2062 650a 2020 2020 2020  s must be.      
│ │ │ │ -0001a400: 2020 686f 6d6f 746f 7069 6320 746f 2030    homotopic to 0
│ │ │ │ -0001a410: 206f 6e20 460a 2020 2020 2020 2a20 4d2c   on F.      * M,
│ │ │ │ -0001a420: 2061 202a 6e6f 7465 206d 6f64 756c 653a   a *note module:
│ │ │ │ -0001a430: 2028 4d61 6361 756c 6179 3244 6f63 294d   (Macaulay2Doc)M
│ │ │ │ -0001a440: 6f64 756c 652c 2c20 616e 6e69 6869 6c61  odule,, annihila
│ │ │ │ -0001a450: 7465 6420 6279 2074 6865 2065 6c65 6d65  ted by the eleme
│ │ │ │ -0001a460: 6e74 730a 2020 2020 2020 2020 6f66 2066  nts.        of f
│ │ │ │ -0001a470: 660a 2020 2020 2020 2a20 4e2c 2061 202a  f.      * N, a *
│ │ │ │ -0001a480: 6e6f 7465 206d 6f64 756c 653a 2028 4d61  note module: (Ma
│ │ │ │ -0001a490: 6361 756c 6179 3244 6f63 294d 6f64 756c  caulay2Doc)Modul
│ │ │ │ -0001a4a0: 652c 2c20 616e 6e69 6869 6c61 7465 6420  e,, annihilated 
│ │ │ │ -0001a4b0: 6279 2074 6865 2065 6c65 6d65 6e74 730a  by the elements.
│ │ │ │ -0001a4c0: 2020 2020 2020 2020 6f66 2066 660a 2020          of ff.  
│ │ │ │ -0001a4d0: 2a20 4f75 7470 7574 733a 0a20 2020 2020  * Outputs:.     
│ │ │ │ -0001a4e0: 202a 2045 2c20 6120 2a6e 6f74 6520 6d6f   * E, a *note mo
│ │ │ │ -0001a4f0: 6475 6c65 3a20 284d 6163 6175 6c61 7932  dule: (Macaulay2
│ │ │ │ -0001a500: 446f 6329 4d6f 6475 6c65 2c2c 204d 6f64  Doc)Module,, Mod
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│ │ │ │ -0001a560: 2066 0a0a 4465 7363 7269 7074 696f 6e0a   f..Description.
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│ │ │ │ -0001a580: 4d2c 4e20 6172 6520 532d 6d6f 6475 6c65  M,N are S-module
│ │ │ │ -0001a590: 7320 616e 6e69 6869 6c61 7465 6420 6279  s annihilated by
│ │ │ │ -0001a5a0: 2074 6865 2065 6c65 6d65 6e74 7320 6f66   the elements of
│ │ │ │ -0001a5b0: 2074 6865 206d 6174 7269 7820 6666 203d   the matrix ff =
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│ │ │ │ -0001a5d0: 206b 2069 7320 7468 6520 7265 7369 6475   k is the residu
│ │ │ │ -0001a5e0: 6520 6669 656c 6420 6f66 2053 2c20 7468  e field of S, th
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│ │ │ │ -0001a600: 7465 7269 6f72 4578 744d 6f64 756c 6528  teriorExtModule(
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│ │ │ │ -0001a620: 5f53 284d 2c20 6b29 2061 7320 6120 6d6f  _S(M, k) as a mo
│ │ │ │ -0001a630: 6475 6c65 206f 7665 7220 616e 2065 7874  dule over an ext
│ │ │ │ -0001a640: 6572 696f 7220 616c 6765 6272 6120 4520  erior algebra E 
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│ │ │ │ -0001a670: 6861 7665 2064 6567 7265 6520 312e 2049  have degree 1. I
│ │ │ │ -0001a680: 7420 6973 2063 6f6d 7075 7465 6420 6173  t is computed as
│ │ │ │ -0001a690: 2074 6865 2045 2d64 7561 6c20 6f66 2065   the E-dual of e
│ │ │ │ -0001a6a0: 7874 6572 696f 7254 6f72 4d6f 6475 6c65  xteriorTorModule
│ │ │ │ -0001a6b0: 2e0a 0a54 6865 2073 6372 6970 7420 6578  ...The script ex
│ │ │ │ -0001a6c0: 7465 7269 6f72 546f 724d 6f64 756c 6528  teriorTorModule(
│ │ │ │ -0001a6d0: 662c 4d2c 4e29 2072 6574 7572 6e73 2045  f,M,N) returns E
│ │ │ │ -0001a6e0: 7874 5f53 284d 2c4e 2920 6173 2061 206d  xt_S(M,N) as a m
│ │ │ │ -0001a6f0: 6f64 756c 6520 6f76 6572 2061 0a62 6967  odule over a.big
│ │ │ │ -0001a700: 7261 6465 6420 7269 6e67 2053 4520 3d20  raded ring SE = 
│ │ │ │ -0001a710: 533c 655f 312c 2e2e 2c65 5f63 3e2c 2077  S, w
│ │ │ │ -0001a720: 6865 7265 2074 6865 2065 5f69 2068 6176  here the e_i hav
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│ │ │ │ -0001a740: 7d2c 2077 6865 7265 2064 5f69 0a69 7320  }, where d_i.is 
│ │ │ │ -0001a750: 7468 6520 6465 6772 6565 206f 6620 665f  the degree of f_
│ │ │ │ -0001a760: 692e 2054 6865 206d 6f64 756c 6520 7374  i. The module st
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│ │ │ │ -0001a780: 6572 2063 6173 652c 2069 7320 6465 6669  er case, is defi
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│ │ │ │ -0001a7b0: 6920 6f6e 2074 6865 2072 6573 6f6c 7574  i on the resolut
│ │ │ │ -0001a7c0: 696f 6e20 6f66 204d 2c20 636f 6d70 7574  ion of M, comput
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│ │ │ │ -0001a7e0: 0a6d 616b 6548 6f6d 6f74 6f70 6965 7331  .makeHomotopies1
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│ │ │ │ -0001a800: 7320 6d61 6b65 4d6f 6475 6c65 2074 6f20  s makeModule to 
│ │ │ │ -0001a810: 636f 6d70 7574 6520 6120 286e 6f6e 2d6d  compute a (non-m
│ │ │ │ -0001a820: 696e 696d 616c 290a 7072 6573 656e 7461  inimal).presenta
│ │ │ │ -0001a830: 7469 6f6e 206f 6620 7468 6973 206d 6f64  tion of this mod
│ │ │ │ -0001a840: 756c 652e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d  ule...+---------
│ │ │ │ +0001a360: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055  *********..  * U
│ │ │ │ +0001a370: 7361 6765 3a20 0a20 2020 2020 2020 2045  sage: .        E
│ │ │ │ +0001a380: 203d 2065 7874 6572 696f 7245 7874 4d6f   = exteriorExtMo
│ │ │ │ +0001a390: 6475 6c65 2866 2c4d 290a 2020 2a20 496e  dule(f,M).  * In
│ │ │ │ +0001a3a0: 7075 7473 3a0a 2020 2020 2020 2a20 662c  puts:.      * f,
│ │ │ │ +0001a3b0: 2061 202a 6e6f 7465 206d 6174 7269 783a   a *note matrix:
│ │ │ │ +0001a3c0: 2028 4d61 6361 756c 6179 3244 6f63 294d   (Macaulay2Doc)M
│ │ │ │ +0001a3d0: 6174 7269 782c 2c20 3120 7820 632c 2065  atrix,, 1 x c, e
│ │ │ │ +0001a3e0: 6e74 7269 6573 206d 7573 7420 6265 0a20  ntries must be. 
│ │ │ │ +0001a3f0: 2020 2020 2020 2068 6f6d 6f74 6f70 6963         homotopic
│ │ │ │ +0001a400: 2074 6f20 3020 6f6e 2046 0a20 2020 2020   to 0 on F.     
│ │ │ │ +0001a410: 202a 204d 2c20 6120 2a6e 6f74 6520 6d6f   * M, a *note mo
│ │ │ │ +0001a420: 6475 6c65 3a20 284d 6163 6175 6c61 7932  dule: (Macaulay2
│ │ │ │ +0001a430: 446f 6329 4d6f 6475 6c65 2c2c 2061 6e6e  Doc)Module,, ann
│ │ │ │ +0001a440: 6968 696c 6174 6564 2062 7920 7468 6520  ihilated by the 
│ │ │ │ +0001a450: 656c 656d 656e 7473 0a20 2020 2020 2020  elements.       
│ │ │ │ +0001a460: 206f 6620 6666 0a20 2020 2020 202a 204e   of ff.      * N
│ │ │ │ +0001a470: 2c20 6120 2a6e 6f74 6520 6d6f 6475 6c65  , a *note module
│ │ │ │ +0001a480: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +0001a490: 4d6f 6475 6c65 2c2c 2061 6e6e 6968 696c  Module,, annihil
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│ │ │ │ +0001a4c0: 6666 0a20 202a 204f 7574 7075 7473 3a0a  ff.  * Outputs:.
│ │ │ │ +0001a4d0: 2020 2020 2020 2a20 452c 2061 202a 6e6f        * E, a *no
│ │ │ │ +0001a4e0: 7465 206d 6f64 756c 653a 2028 4d61 6361  te module: (Maca
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│ │ │ │ +0001a500: 2c20 4d6f 6475 6c65 206f 7665 7220 616e  , Module over an
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│ │ │ │ +0001a560: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  tion.===========
│ │ │ │ +0001a570: 0a0a 4966 204d 2c4e 2061 7265 2053 2d6d  ..If M,N are S-m
│ │ │ │ +0001a580: 6f64 756c 6573 2061 6e6e 6968 696c 6174  odules annihilat
│ │ │ │ +0001a590: 6564 2062 7920 7468 6520 656c 656d 656e  ed by the elemen
│ │ │ │ +0001a5a0: 7473 206f 6620 7468 6520 6d61 7472 6978  ts of the matrix
│ │ │ │ +0001a5b0: 2066 6620 3d20 2866 5f31 2e2e 665f 6329   ff = (f_1..f_c)
│ │ │ │ +0001a5c0: 2c0a 616e 6420 6b20 6973 2074 6865 2072  ,.and k is the r
│ │ │ │ +0001a5d0: 6573 6964 7565 2066 6965 6c64 206f 6620  esidue field of 
│ │ │ │ +0001a5e0: 532c 2074 6865 6e20 7468 6520 7363 7269  S, then the scri
│ │ │ │ +0001a5f0: 7074 2065 7874 6572 696f 7245 7874 4d6f  pt exteriorExtMo
│ │ │ │ +0001a600: 6475 6c65 2866 2c4d 2920 7265 7475 726e  dule(f,M) return
│ │ │ │ +0001a610: 730a 4578 745f 5328 4d2c 206b 2920 6173  s.Ext_S(M, k) as
│ │ │ │ +0001a620: 2061 206d 6f64 756c 6520 6f76 6572 2061   a module over a
│ │ │ │ +0001a630: 6e20 6578 7465 7269 6f72 2061 6c67 6562  n exterior algeb
│ │ │ │ +0001a640: 7261 2045 203d 206b 3c65 5f31 2c2e 2e2e  ra E = k, where the
│ │ │ │ +0001a660: 0a65 5f69 2068 6176 6520 6465 6772 6565  .e_i have degree
│ │ │ │ +0001a670: 2031 2e20 4974 2069 7320 636f 6d70 7574   1. It is comput
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│ │ │ │ +0001a690: 206f 6620 6578 7465 7269 6f72 546f 724d   of exteriorTorM
│ │ │ │ +0001a6a0: 6f64 756c 652e 0a0a 5468 6520 7363 7269  odule...The scri
│ │ │ │ +0001a6b0: 7074 2065 7874 6572 696f 7254 6f72 4d6f  pt exteriorTorMo
│ │ │ │ +0001a6c0: 6475 6c65 2866 2c4d 2c4e 2920 7265 7475  dule(f,M,N) retu
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│ │ │ │ +0001a6f0: 610a 6269 6772 6164 6564 2072 696e 6720  a.bigraded ring 
│ │ │ │ +0001a700: 5345 203d 2053 3c65 5f31 2c2e 2e2c 655f  SE = S, where the e_
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│ │ │ │ +0001a750: 6f66 2066 5f69 2e20 5468 6520 6d6f 6475  of f_i. The modu
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│ │ │ │ +0001a840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0001a880: 6b20 3d20 5a5a 2f31 3031 2020 2020 2020  k = ZZ/101      
│ │ │ │ +0001a890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a8c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0001a8b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001a8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a8f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001a900: 7c6f 3120 3d20 6b6b 2020 2020 2020 2020  |o1 = kk        
│ │ │ │ +0001a8f0: 2020 207c 0a7c 6f31 203d 206b 6b20 2020     |.|o1 = kk   
│ │ │ │ +0001a900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a930: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001a930: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001a940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a970: 2020 2020 2020 2020 2020 7c0a 7c6f 3120            |.|o1 
│ │ │ │ -0001a980: 3a20 5175 6f74 6965 6e74 5269 6e67 2020  : QuotientRing  
│ │ │ │ +0001a960: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │ +0001a980: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │  0001a990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a9b0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0001a9a0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0001a9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a9f0: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 5320  ------+.|i2 : S 
│ │ │ │ -0001aa00: 3d20 6b6b 5b61 2c62 2c63 5d20 2020 2020  = kk[a,b,c]     
│ │ │ │ +0001a9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +0001a9f0: 203a 2053 203d 206b 6b5b 612c 622c 635d   : S = kk[a,b,c]
│ │ │ │ +0001aa00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001aa10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aa20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aa30: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001aa20: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aa70: 2020 7c0a 7c6f 3220 3d20 5320 2020 2020    |.|o2 = S     
│ │ │ │ +0001aa60: 2020 2020 2020 207c 0a7c 6f32 203d 2053         |.|o2 = S
│ │ │ │ +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 2020                  
│ │ │ │ -0001aaa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aab0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0001aaa0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001aab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001aac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001aad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aae0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001aaf0: 7c6f 3220 3a20 506f 6c79 6e6f 6d69 616c  |o2 : Polynomial
│ │ │ │ -0001ab00: 5269 6e67 2020 2020 2020 2020 2020 2020  Ring            
│ │ │ │ +0001aae0: 2020 207c 0a7c 6f32 203a 2050 6f6c 796e     |.|o2 : Polyn
│ │ │ │ +0001aaf0: 6f6d 6961 6c52 696e 6720 2020 2020 2020  omialRing       
│ │ │ │ +0001ab00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ab10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ab20: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0001ab20: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0001ab30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ab40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ab50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ab60: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320  ----------+.|i3 
│ │ │ │ -0001ab70: 3a20 6620 3d20 6d61 7472 6978 2261 342c  : f = matrix"a4,
│ │ │ │ -0001ab80: 6234 2c63 3422 2020 2020 2020 2020 2020  b4,c4"          
│ │ │ │ -0001ab90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aba0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001ab50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0001ab60: 0a7c 6933 203a 2066 203d 206d 6174 7269  .|i3 : f = matri
│ │ │ │ +0001ab70: 7822 6134 2c62 342c 6334 2220 2020 2020  x"a4,b4,c4"     
│ │ │ │ +0001ab80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ab90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001aba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001abb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001abc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001abd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001abe0: 2020 2020 2020 7c0a 7c6f 3320 3d20 7c20        |.|o3 = | 
│ │ │ │ -0001abf0: 6134 2062 3420 6334 207c 2020 2020 2020  a4 b4 c4 |      
│ │ │ │ +0001abd0: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ +0001abe0: 203d 207c 2061 3420 6234 2063 3420 7c20   = | a4 b4 c4 | 
│ │ │ │ +0001abf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ac00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ac10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ac20: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001ac10: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001ac20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ac30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ac40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ac50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ac60: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0001ac70: 2020 3120 2020 2020 2033 2020 2020 2020    1      3      
│ │ │ │ +0001ac50: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001ac60: 2020 2020 2020 2031 2020 2020 2020 3320         1      3 
│ │ │ │ +0001ac70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ac80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ac90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aca0: 7c0a 7c6f 3320 3a20 4d61 7472 6978 2053  |.|o3 : Matrix S
│ │ │ │ -0001acb0: 2020 3c2d 2d20 5320 2020 2020 2020 2020    <-- S         
│ │ │ │ +0001ac90: 2020 2020 207c 0a7c 6f33 203a 204d 6174       |.|o3 : Mat
│ │ │ │ +0001aca0: 7269 7820 5320 203c 2d2d 2053 2020 2020  rix S  <-- S    
│ │ │ │ +0001acb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001acc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001acd0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001ace0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0001acd0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0001ace0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001acf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ad00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ad10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0001ad20: 3420 3a20 5220 3d20 532f 6964 6561 6c20  4 : R = S/ideal 
│ │ │ │ -0001ad30: 6620 2020 2020 2020 2020 2020 2020 2020  f               
│ │ │ │ -0001ad40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ad50: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001ad10: 2d2b 0a7c 6934 203a 2052 203d 2053 2f69  -+.|i4 : R = S/i
│ │ │ │ +0001ad20: 6465 616c 2066 2020 2020 2020 2020 2020  deal f          
│ │ │ │ +0001ad30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ad40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001ad50: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0001ad60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ad70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ad80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ad90: 2020 2020 2020 2020 7c0a 7c6f 3420 3d20          |.|o4 = 
│ │ │ │ -0001ada0: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │ +0001ad80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001ad90: 6f34 203d 2052 2020 2020 2020 2020 2020  o4 = R          
│ │ │ │ +0001ada0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001adb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001adc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001add0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001adc0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001add0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ade0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001adf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ae00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ae10: 2020 2020 7c0a 7c6f 3420 3a20 5175 6f74      |.|o4 : Quot
│ │ │ │ -0001ae20: 6965 6e74 5269 6e67 2020 2020 2020 2020  ientRing        
│ │ │ │ +0001ae00: 2020 2020 2020 2020 207c 0a7c 6f34 203a           |.|o4 :
│ │ │ │ +0001ae10: 2051 756f 7469 656e 7452 696e 6720 2020   QuotientRing   
│ │ │ │ +0001ae20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ae30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ae40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ae50: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0001ae40: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0001ae50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ae60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ae70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ae80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ae90: 2b0a 7c69 3520 3a20 7020 3d20 6d61 7028  +.|i5 : p = map(
│ │ │ │ -0001aea0: 522c 5329 2020 2020 2020 2020 2020 2020  R,S)            
│ │ │ │ +0001ae80: 2d2d 2d2d 2d2b 0a7c 6935 203a 2070 203d  -----+.|i5 : p =
│ │ │ │ +0001ae90: 206d 6170 2852 2c53 2920 2020 2020 2020   map(R,S)       
│ │ │ │ +0001aea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001aeb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aec0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001aed0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0001aec0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0001aed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001aee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001aef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001af00: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0001af10: 3520 3d20 6d61 7020 2852 2c20 532c 207b  5 = map (R, S, {
│ │ │ │ -0001af20: 612c 2062 2c20 637d 2920 2020 2020 2020  a, b, c})       
│ │ │ │ -0001af30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001af40: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001af00: 207c 0a7c 6f35 203d 206d 6170 2028 522c   |.|o5 = map (R,
│ │ │ │ +0001af10: 2053 2c20 7b61 2c20 622c 2063 7d29 2020   S, {a, b, c})  
│ │ │ │ +0001af20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001af30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001af40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0001af50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001af60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001af70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001af80: 2020 2020 2020 2020 7c0a 7c6f 3520 3a20          |.|o5 : 
│ │ │ │ -0001af90: 5269 6e67 4d61 7020 5220 3c2d 2d20 5320  RingMap R <-- S 
│ │ │ │ +0001af70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001af80: 6f35 203a 2052 696e 674d 6170 2052 203c  o5 : RingMap R <
│ │ │ │ +0001af90: 2d2d 2053 2020 2020 2020 2020 2020 2020  -- S            
│ │ │ │  0001afa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001afb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001afc0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0001afb0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0001afc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001afd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001afe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001aff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b000: 2d2d 2d2d 2b0a 7c69 3620 3a20 4d20 3d20  ----+.|i6 : M = 
│ │ │ │ -0001b010: 636f 6b65 7220 6d61 7028 525e 322c 2052  coker map(R^2, R
│ │ │ │ -0001b020: 5e7b 333a 2d31 7d2c 207b 7b61 2c62 2c63  ^{3:-1}, {{a,b,c
│ │ │ │ -0001b030: 7d2c 7b62 2c63 2c61 7d7d 2920 2020 2020  },{b,c,a}})     
│ │ │ │ -0001b040: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001aff0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a  ---------+.|i6 :
│ │ │ │ +0001b000: 204d 203d 2063 6f6b 6572 206d 6170 2852   M = coker map(R
│ │ │ │ +0001b010: 5e32 2c20 525e 7b33 3a2d 317d 2c20 7b7b  ^2, R^{3:-1}, {{
│ │ │ │ +0001b020: 612c 622c 637d 2c7b 622c 632c 617d 7d29  a,b,c},{b,c,a}})
│ │ │ │ +0001b030: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001b040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b080: 7c0a 7c6f 3620 3d20 636f 6b65 726e 656c  |.|o6 = cokernel
│ │ │ │ -0001b090: 207c 2061 2062 2063 207c 2020 2020 2020   | a b c |      
│ │ │ │ +0001b070: 2020 2020 207c 0a7c 6f36 203d 2063 6f6b       |.|o6 = cok
│ │ │ │ +0001b080: 6572 6e65 6c20 7c20 6120 6220 6320 7c20  ernel | a b c | 
│ │ │ │ +0001b090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b0b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001b0c0: 7c20 2020 2020 2020 2020 2020 2020 207c  |              |
│ │ │ │ -0001b0d0: 2062 2063 2061 207c 2020 2020 2020 2020   b c a |        
│ │ │ │ +0001b0b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0001b0c0: 2020 2020 7c20 6220 6320 6120 7c20 2020      | b c a |   
│ │ │ │ +0001b0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b0f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001b0f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001b100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b130: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0001b140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b150: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ -0001b160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b170: 2020 2020 2020 2020 7c0a 7c6f 3620 3a20          |.|o6 : 
│ │ │ │ -0001b180: 522d 6d6f 6475 6c65 2c20 7175 6f74 6965  R-module, quotie
│ │ │ │ -0001b190: 6e74 206f 6620 5220 2020 2020 2020 2020  nt of R         
│ │ │ │ -0001b1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b1b0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0001b120: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001b130: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0001b140: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +0001b150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b160: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001b170: 6f36 203a 2052 2d6d 6f64 756c 652c 2071  o6 : R-module, q
│ │ │ │ +0001b180: 756f 7469 656e 7420 6f66 2052 2020 2020  uotient of R    
│ │ │ │ +0001b190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b1a0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0001b1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b1f0: 2d2d 2d2d 2b0a 7c69 3720 3a20 6265 7474  ----+.|i7 : bett
│ │ │ │ -0001b200: 6920 2846 4620 3d66 7265 6552 6573 6f6c  i (FF =freeResol
│ │ │ │ -0001b210: 7574 696f 6e28 204d 2c20 4c65 6e67 7468  ution( M, Length
│ │ │ │ -0001b220: 4c69 6d69 7420 3d3e 3629 2920 2020 2020  Limit =>6))     
│ │ │ │ -0001b230: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001b1e0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a  ---------+.|i7 :
│ │ │ │ +0001b1f0: 2062 6574 7469 2028 4646 203d 6672 6565   betti (FF =free
│ │ │ │ +0001b200: 5265 736f 6c75 7469 6f6e 2820 4d2c 204c  Resolution( M, L
│ │ │ │ +0001b210: 656e 6774 684c 696d 6974 203d 3e36 2929  engthLimit =>6))
│ │ │ │ +0001b220: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001b230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b270: 7c0a 7c20 2020 2020 2020 2020 2020 2030  |.|            0
│ │ │ │ -0001b280: 2031 2032 2033 2034 2020 3520 2036 2020   1 2 3 4  5  6  
│ │ │ │ +0001b260: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001b270: 2020 2020 3020 3120 3220 3320 3420 2035      0 1 2 3 4  5
│ │ │ │ +0001b280: 2020 3620 2020 2020 2020 2020 2020 2020    6             
│ │ │ │  0001b290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b2a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001b2b0: 7c6f 3720 3d20 746f 7461 6c3a 2032 2033  |o7 = total: 2 3
│ │ │ │ -0001b2c0: 2034 2036 2039 2031 3320 3138 2020 2020   4 6 9 13 18    
│ │ │ │ +0001b2a0: 2020 207c 0a7c 6f37 203d 2074 6f74 616c     |.|o7 = total
│ │ │ │ +0001b2b0: 3a20 3220 3320 3420 3620 3920 3133 2031  : 2 3 4 6 9 13 1
│ │ │ │ +0001b2c0: 3820 2020 2020 2020 2020 2020 2020 2020  8               
│ │ │ │  0001b2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b2e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001b2f0: 2020 2020 2020 2020 303a 2032 2033 202e          0: 2 3 .
│ │ │ │ -0001b300: 202e 202e 2020 2e20 202e 2020 2020 2020   . .  .  .      
│ │ │ │ -0001b310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b320: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0001b330: 2020 2020 2020 313a 202e 202e 2031 202e        1: . . 1 .
│ │ │ │ -0001b340: 202e 2020 2e20 202e 2020 2020 2020 2020   .  .  .        
│ │ │ │ -0001b350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b360: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0001b370: 2020 2020 323a 202e 202e 2033 2033 202e      2: . . 3 3 .
│ │ │ │ -0001b380: 2020 2e20 202e 2020 2020 2020 2020 2020    .  .          
│ │ │ │ -0001b390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b3a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001b3b0: 2020 333a 202e 202e 202e 2033 2033 2020    3: . . . 3 3  
│ │ │ │ -0001b3c0: 2e20 202e 2020 2020 2020 2020 2020 2020  .  .            
│ │ │ │ -0001b3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b3e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001b3f0: 343a 202e 202e 202e 202e 2033 2020 3320  4: . . . . 3  3 
│ │ │ │ -0001b400: 202e 2020 2020 2020 2020 2020 2020 2020   .              
│ │ │ │ -0001b410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b420: 2020 7c0a 7c20 2020 2020 2020 2020 353a    |.|         5:
│ │ │ │ -0001b430: 202e 202e 202e 202e 2033 2020 3920 2036   . . . . 3  9  6
│ │ │ │ +0001b2e0: 207c 0a7c 2020 2020 2020 2020 2030 3a20   |.|         0: 
│ │ │ │ +0001b2f0: 3220 3320 2e20 2e20 2e20 202e 2020 2e20  2 3 . . .  .  . 
│ │ │ │ +0001b300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b310: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001b320: 0a7c 2020 2020 2020 2020 2031 3a20 2e20  .|         1: . 
│ │ │ │ +0001b330: 2e20 3120 2e20 2e20 202e 2020 2e20 2020  . 1 . .  .  .   
│ │ │ │ +0001b340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b350: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001b360: 2020 2020 2020 2020 2032 3a20 2e20 2e20           2: . . 
│ │ │ │ +0001b370: 3320 3320 2e20 202e 2020 2e20 2020 2020  3 3 .  .  .     
│ │ │ │ +0001b380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b390: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001b3a0: 2020 2020 2020 2033 3a20 2e20 2e20 2e20         3: . . . 
│ │ │ │ +0001b3b0: 3320 3320 202e 2020 2e20 2020 2020 2020  3 3  .  .       
│ │ │ │ +0001b3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b3d0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001b3e0: 2020 2020 2034 3a20 2e20 2e20 2e20 2e20       4: . . . . 
│ │ │ │ +0001b3f0: 3320 2033 2020 2e20 2020 2020 2020 2020  3  3  .         
│ │ │ │ +0001b400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b410: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001b420: 2020 2035 3a20 2e20 2e20 2e20 2e20 3320     5: . . . . 3 
│ │ │ │ +0001b430: 2039 2020 3620 2020 2020 2020 2020 2020   9  6           
│ │ │ │  0001b440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b460: 7c0a 7c20 2020 2020 2020 2020 363a 202e  |.|         6: .
│ │ │ │ -0001b470: 202e 202e 202e 202e 2020 2e20 2033 2020   . . . .  .  3  
│ │ │ │ +0001b450: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001b460: 2036 3a20 2e20 2e20 2e20 2e20 2e20 202e   6: . . . . .  .
│ │ │ │ +0001b470: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │  0001b480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b490: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001b4a0: 7c20 2020 2020 2020 2020 373a 202e 202e  |         7: . .
│ │ │ │ -0001b4b0: 202e 202e 202e 2020 3120 2039 2020 2020   . . .  1  9    
│ │ │ │ +0001b490: 2020 207c 0a7c 2020 2020 2020 2020 2037     |.|         7
│ │ │ │ +0001b4a0: 3a20 2e20 2e20 2e20 2e20 2e20 2031 2020  : . . . . .  1  
│ │ │ │ +0001b4b0: 3920 2020 2020 2020 2020 2020 2020 2020  9               
│ │ │ │  0001b4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b4d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001b4d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001b4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b510: 2020 2020 2020 2020 2020 7c0a 7c6f 3720            |.|o7 
│ │ │ │ -0001b520: 3a20 4265 7474 6954 616c 6c79 2020 2020  : BettiTally    
│ │ │ │ +0001b500: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001b510: 0a7c 6f37 203a 2042 6574 7469 5461 6c6c  .|o7 : BettiTall
│ │ │ │ +0001b520: 7920 2020 2020 2020 2020 2020 2020 2020  y               
│ │ │ │  0001b530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b550: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0001b540: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0001b550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b590: 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20 4d53  ------+.|i8 : MS
│ │ │ │ -0001b5a0: 203d 2070 7275 6e65 2070 7573 6846 6f72   = prune pushFor
│ │ │ │ -0001b5b0: 7761 7264 2870 2c20 636f 6b65 7220 4646  ward(p, coker FF
│ │ │ │ -0001b5c0: 2e64 645f 3629 3b20 2020 2020 2020 2020  .dd_6);         
│ │ │ │ -0001b5d0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0001b580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938  -----------+.|i8
│ │ │ │ +0001b590: 203a 204d 5320 3d20 7072 756e 6520 7075   : MS = prune pu
│ │ │ │ +0001b5a0: 7368 466f 7277 6172 6428 702c 2063 6f6b  shForward(p, cok
│ │ │ │ +0001b5b0: 6572 2046 462e 6464 5f36 293b 2020 2020  er FF.dd_6);    
│ │ │ │ +0001b5c0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0001b5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b610: 2d2d 2b0a 7c69 3920 3a20 7265 7346 6c64  --+.|i9 : resFld
│ │ │ │ -0001b620: 203a 3d20 7075 7368 466f 7277 6172 6428   := pushForward(
│ │ │ │ -0001b630: 702c 2063 6f6b 6572 2076 6172 7320 5229  p, coker vars R)
│ │ │ │ -0001b640: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ -0001b650: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0001b600: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2072  -------+.|i9 : r
│ │ │ │ +0001b610: 6573 466c 6420 3a3d 2070 7573 6846 6f72  esFld := pushFor
│ │ │ │ +0001b620: 7761 7264 2870 2c20 636f 6b65 7220 7661  ward(p, coker va
│ │ │ │ +0001b630: 7273 2052 293b 2020 2020 2020 2020 2020  rs R);          
│ │ │ │ +0001b640: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001b650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0001b690: 7c69 3130 203a 2054 203d 2065 7874 6572  |i10 : T = exter
│ │ │ │ -0001b6a0: 696f 7254 6f72 4d6f 6475 6c65 2866 2c4d  iorTorModule(f,M
│ │ │ │ -0001b6b0: 5329 3b20 2020 2020 2020 2020 2020 2020  S);             
│ │ │ │ -0001b6c0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0001b680: 2d2d 2d2b 0a7c 6931 3020 3a20 5420 3d20  ---+.|i10 : T = 
│ │ │ │ +0001b690: 6578 7465 7269 6f72 546f 724d 6f64 756c  exteriorTorModul
│ │ │ │ +0001b6a0: 6528 662c 4d53 293b 2020 2020 2020 2020  e(f,MS);        
│ │ │ │ +0001b6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b6c0: 207c 0a2b 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 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b700: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131  ----------+.|i11
│ │ │ │ -0001b710: 203a 2045 203d 2065 7874 6572 696f 7245   : E = exteriorE
│ │ │ │ -0001b720: 7874 4d6f 6475 6c65 2866 2c4d 5329 3b20  xtModule(f,MS); 
│ │ │ │ -0001b730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b740: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0001b6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0001b700: 0a7c 6931 3120 3a20 4520 3d20 6578 7465  .|i11 : E = exte
│ │ │ │ +0001b710: 7269 6f72 4578 744d 6f64 756c 6528 662c  riorExtModule(f,
│ │ │ │ +0001b720: 4d53 293b 2020 2020 2020 2020 2020 2020  MS);            
│ │ │ │ +0001b730: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0001b740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b780: 2d2d 2d2d 2d2d 2b0a 7c69 3132 203a 2068  ------+.|i12 : h
│ │ │ │ -0001b790: 6628 2d34 2e2e 302c 4529 2020 2020 2020  f(-4..0,E)      
│ │ │ │ +0001b770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +0001b780: 3220 3a20 6866 282d 342e 2e30 2c45 2920  2 : hf(-4..0,E) 
│ │ │ │ +0001b790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b7c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001b7b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001b7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b800: 2020 7c0a 7c6f 3132 203d 207b 302c 2039    |.|o12 = {0, 9
│ │ │ │ -0001b810: 2c20 3239 2c20 3333 2c20 3133 7d20 2020  , 29, 33, 13}   
│ │ │ │ +0001b7f0: 2020 2020 2020 207c 0a7c 6f31 3220 3d20         |.|o12 = 
│ │ │ │ +0001b800: 7b30 2c20 392c 2032 392c 2033 332c 2031  {0, 9, 29, 33, 1
│ │ │ │ +0001b810: 337d 2020 2020 2020 2020 2020 2020 2020  3}              
│ │ │ │  0001b820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b840: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0001b830: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001b840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b870: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001b880: 7c6f 3132 203a 204c 6973 7420 2020 2020  |o12 : List     
│ │ │ │ +0001b870: 2020 207c 0a7c 6f31 3220 3a20 4c69 7374     |.|o12 : List
│ │ │ │ +0001b880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b8b0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0001b8b0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0001b8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3133  ----------+.|i13
│ │ │ │ -0001b900: 203a 2062 6574 7469 2066 7265 6552 6573   : betti freeRes
│ │ │ │ -0001b910: 6f6c 7574 696f 6e20 4d53 2020 2020 2020  olution MS      
│ │ │ │ -0001b920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b930: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001b8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0001b8f0: 0a7c 6931 3320 3a20 6265 7474 6920 6672  .|i13 : betti fr
│ │ │ │ +0001b900: 6565 5265 736f 6c75 7469 6f6e 204d 5320  eeResolution MS 
│ │ │ │ +0001b910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b920: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001b930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b970: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001b980: 2020 2020 2020 2030 2020 3120 2032 2033         0  1  2 3
│ │ │ │ +0001b960: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001b970: 2020 2020 2020 2020 2020 2020 3020 2031              0  1
│ │ │ │ +0001b980: 2020 3220 3320 2020 2020 2020 2020 2020    2 3           
│ │ │ │  0001b990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b9b0: 2020 2020 7c0a 7c6f 3133 203d 2074 6f74      |.|o13 = tot
│ │ │ │ -0001b9c0: 616c 3a20 3133 2033 3320 3239 2039 2020  al: 13 33 29 9  
│ │ │ │ +0001b9a0: 2020 2020 2020 2020 207c 0a7c 6f31 3320           |.|o13 
│ │ │ │ +0001b9b0: 3d20 746f 7461 6c3a 2031 3320 3333 2032  = total: 13 33 2
│ │ │ │ +0001b9c0: 3920 3920 2020 2020 2020 2020 2020 2020  9 9             
│ │ │ │  0001b9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b9f0: 2020 7c0a 7c20 2020 2020 2020 2020 2039    |.|          9
│ │ │ │ -0001ba00: 3a20 2033 2020 2e20 202e 202e 2020 2020  :  3  .  . .    
│ │ │ │ +0001b9e0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001b9f0: 2020 2020 393a 2020 3320 202e 2020 2e20      9:  3  .  . 
│ │ │ │ +0001ba00: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │  0001ba10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ba20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ba30: 7c0a 7c20 2020 2020 2020 2020 3130 3a20  |.|         10: 
│ │ │ │ -0001ba40: 2039 2020 3620 202e 202e 2020 2020 2020   9  6  . .      
│ │ │ │ +0001ba20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001ba30: 2031 303a 2020 3920 2036 2020 2e20 2e20   10:  9  6  . . 
│ │ │ │ +0001ba40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ba50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ba60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001ba70: 7c20 2020 2020 2020 2020 3131 3a20 202e  |         11:  .
│ │ │ │ -0001ba80: 2020 3320 202e 202e 2020 2020 2020 2020    3  . .        
│ │ │ │ +0001ba60: 2020 207c 0a7c 2020 2020 2020 2020 2031     |.|         1
│ │ │ │ +0001ba70: 313a 2020 2e20 2033 2020 2e20 2e20 2020  1:  .  3  . .   
│ │ │ │ +0001ba80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ba90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001baa0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001bab0: 2020 2020 2020 2020 3132 3a20 2031 2031          12:  1 1
│ │ │ │ -0001bac0: 3520 202e 202e 2020 2020 2020 2020 2020  5  . .          
│ │ │ │ -0001bad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bae0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0001baf0: 2020 2020 2020 3133 3a20 202e 2020 3920        13:  .  9 
│ │ │ │ -0001bb00: 2038 202e 2020 2020 2020 2020 2020 2020   8 .            
│ │ │ │ -0001bb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bb20: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0001bb30: 2020 2020 3134 3a20 202e 2020 2e20 2036      14:  .  .  6
│ │ │ │ -0001bb40: 202e 2020 2020 2020 2020 2020 2020 2020   .              
│ │ │ │ -0001bb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bb60: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001bb70: 2020 3135 3a20 202e 2020 2e20 3132 202e    15:  .  . 12 .
│ │ │ │ +0001baa0: 207c 0a7c 2020 2020 2020 2020 2031 323a   |.|         12:
│ │ │ │ +0001bab0: 2020 3120 3135 2020 2e20 2e20 2020 2020    1 15  . .     
│ │ │ │ +0001bac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bad0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001bae0: 0a7c 2020 2020 2020 2020 2031 333a 2020  .|         13:  
│ │ │ │ +0001baf0: 2e20 2039 2020 3820 2e20 2020 2020 2020  .  9  8 .       
│ │ │ │ +0001bb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bb10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001bb20: 2020 2020 2020 2020 2031 343a 2020 2e20           14:  . 
│ │ │ │ +0001bb30: 202e 2020 3620 2e20 2020 2020 2020 2020   .  6 .         
│ │ │ │ +0001bb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bb50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001bb60: 2020 2020 2020 2031 353a 2020 2e20 202e         15:  .  .
│ │ │ │ +0001bb70: 2031 3220 2e20 2020 2020 2020 2020 2020   12 .           
│ │ │ │  0001bb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bba0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001bbb0: 3136 3a20 202e 2020 2e20 2033 2033 2020  16:  .  .  3 3  
│ │ │ │ +0001bb90: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001bba0: 2020 2020 2031 363a 2020 2e20 202e 2020       16:  .  .  
│ │ │ │ +0001bbb0: 3320 3320 2020 2020 2020 2020 2020 2020  3 3             
│ │ │ │  0001bbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bbe0: 2020 7c0a 7c20 2020 2020 2020 2020 3137    |.|         17
│ │ │ │ -0001bbf0: 3a20 202e 2020 2e20 202e 2033 2020 2020  :  .  .  . 3    
│ │ │ │ +0001bbd0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001bbe0: 2020 2031 373a 2020 2e20 202e 2020 2e20     17:  .  .  . 
│ │ │ │ +0001bbf0: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │  0001bc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bc20: 7c0a 7c20 2020 2020 2020 2020 3138 3a20  |.|         18: 
│ │ │ │ -0001bc30: 202e 2020 2e20 202e 2033 2020 2020 2020   .  .  . 3      
│ │ │ │ +0001bc10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001bc20: 2031 383a 2020 2e20 202e 2020 2e20 3320   18:  .  .  . 3 
│ │ │ │ +0001bc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bc50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001bc60: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0001bc50: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0001bc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bc90: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0001bca0: 3133 203a 2042 6574 7469 5461 6c6c 7920  13 : BettiTally 
│ │ │ │ +0001bc90: 207c 0a7c 6f31 3320 3a20 4265 7474 6954   |.|o13 : BettiT
│ │ │ │ +0001bca0: 616c 6c79 2020 2020 2020 2020 2020 2020  ally            
│ │ │ │  0001bcb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bcd0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0001bcc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001bcd0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  0001bce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001bcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bd10: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3134 203a  --------+.|i14 :
│ │ │ │ -0001bd20: 2062 6574 7469 2066 7265 6552 6573 6f6c   betti freeResol
│ │ │ │ -0001bd30: 7574 696f 6e20 2850 4520 3d20 7072 756e  ution (PE = prun
│ │ │ │ -0001bd40: 6520 452c 204c 656e 6774 684c 696d 6974  e E, LengthLimit
│ │ │ │ -0001bd50: 203d 3e20 3629 7c0a 7c20 2020 2020 2020   => 6)|.|       
│ │ │ │ +0001bd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +0001bd10: 6931 3420 3a20 6265 7474 6920 6672 6565  i14 : betti free
│ │ │ │ +0001bd20: 5265 736f 6c75 7469 6f6e 2028 5045 203d  Resolution (PE =
│ │ │ │ +0001bd30: 2070 7275 6e65 2045 2c20 4c65 6e67 7468   prune E, Length
│ │ │ │ +0001bd40: 4c69 6d69 7420 3d3e 2036 297c 0a7c 2020  Limit => 6)|.|  
│ │ │ │ +0001bd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bd90: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001bda0: 2020 2020 2030 2020 3120 2032 2020 3320       0  1  2  3 
│ │ │ │ -0001bdb0: 2034 2020 2035 2020 2036 2020 2020 2020   4   5   6      
│ │ │ │ -0001bdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bdd0: 2020 7c0a 7c6f 3134 203d 2074 6f74 616c    |.|o14 = total
│ │ │ │ -0001bde0: 3a20 3136 2031 3320 3235 2034 3920 3831  : 16 13 25 49 81
│ │ │ │ -0001bdf0: 2031 3231 2031 3639 2020 2020 2020 2020   121 169        
│ │ │ │ -0001be00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001be10: 7c0a 7c20 2020 2020 2020 2020 2d33 3a20  |.|         -3: 
│ │ │ │ -0001be20: 2039 2020 3420 2033 2020 3320 2033 2020   9  4  3  3  3  
│ │ │ │ -0001be30: 2033 2020 2033 2020 2020 2020 2020 2020   3   3          
│ │ │ │ -0001be40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001be50: 7c20 2020 2020 2020 2020 2d32 3a20 2036  |         -2:  6
│ │ │ │ -0001be60: 2020 3320 202e 2020 2e20 202e 2020 202e    3  .  .  .   .
│ │ │ │ -0001be70: 2020 202e 2020 2020 2020 2020 2020 2020     .            
│ │ │ │ -0001be80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001be90: 2020 2020 2020 2020 2d31 3a20 202e 2020          -1:  .  
│ │ │ │ -0001bea0: 2e20 2037 2031 3820 3333 2020 3532 2020  .  7 18 33  52  
│ │ │ │ -0001beb0: 3735 2020 2020 2020 2020 2020 2020 2020  75              
│ │ │ │ -0001bec0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0001bed0: 2020 2020 2020 2030 3a20 2031 2020 3620         0:  1  6 
│ │ │ │ -0001bee0: 3135 2032 3820 3435 2020 3636 2020 3931  15 28 45  66  91
│ │ │ │ -0001bef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bf00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001bd80: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001bd90: 2020 2020 2020 2020 2020 3020 2031 2020            0  1  
│ │ │ │ +0001bda0: 3220 2033 2020 3420 2020 3520 2020 3620  2  3  4   5   6 
│ │ │ │ +0001bdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bdc0: 2020 2020 2020 207c 0a7c 6f31 3420 3d20         |.|o14 = 
│ │ │ │ +0001bdd0: 746f 7461 6c3a 2031 3620 3133 2032 3520  total: 16 13 25 
│ │ │ │ +0001bde0: 3439 2038 3120 3132 3120 3136 3920 2020  49 81 121 169   
│ │ │ │ +0001bdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001be00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001be10: 202d 333a 2020 3920 2034 2020 3320 2033   -3:  9  4  3  3
│ │ │ │ +0001be20: 2020 3320 2020 3320 2020 3320 2020 2020    3   3   3     
│ │ │ │ +0001be30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001be40: 2020 207c 0a7c 2020 2020 2020 2020 202d     |.|         -
│ │ │ │ +0001be50: 323a 2020 3620 2033 2020 2e20 202e 2020  2:  6  3  .  .  
│ │ │ │ +0001be60: 2e20 2020 2e20 2020 2e20 2020 2020 2020  .   .   .       
│ │ │ │ +0001be70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001be80: 207c 0a7c 2020 2020 2020 2020 202d 313a   |.|         -1:
│ │ │ │ +0001be90: 2020 2e20 202e 2020 3720 3138 2033 3320    .  .  7 18 33 
│ │ │ │ +0001bea0: 2035 3220 2037 3520 2020 2020 2020 2020   52  75         
│ │ │ │ +0001beb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001bec0: 0a7c 2020 2020 2020 2020 2020 303a 2020  .|          0:  
│ │ │ │ +0001bed0: 3120 2036 2031 3520 3238 2034 3520 2036  1  6 15 28 45  6
│ │ │ │ +0001bee0: 3620 2039 3120 2020 2020 2020 2020 2020  6  91           
│ │ │ │ +0001bef0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001bf00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bf10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bf20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bf30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bf40: 2020 2020 2020 7c0a 7c6f 3134 203a 2042        |.|o14 : B
│ │ │ │ -0001bf50: 6574 7469 5461 6c6c 7920 2020 2020 2020  ettiTally       
│ │ │ │ +0001bf30: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0001bf40: 3420 3a20 4265 7474 6954 616c 6c79 2020  4 : BettiTally  
│ │ │ │ +0001bf50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bf60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bf70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bf80: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0001bf70: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0001bf80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001bf90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001bfa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bfc0: 2d2d 2b0a 7c69 3135 203a 2062 6574 7469  --+.|i15 : betti
│ │ │ │ -0001bfd0: 2066 7265 6552 6573 6f6c 7574 696f 6e20   freeResolution 
│ │ │ │ -0001bfe0: 2850 5420 3d20 7072 756e 6520 542c 204c  (PT = prune T, L
│ │ │ │ -0001bff0: 656e 6774 684c 696d 6974 203d 3e20 3629  engthLimit => 6)
│ │ │ │ -0001c000: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0001bfb0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3520 3a20  -------+.|i15 : 
│ │ │ │ +0001bfc0: 6265 7474 6920 6672 6565 5265 736f 6c75  betti freeResolu
│ │ │ │ +0001bfd0: 7469 6f6e 2028 5054 203d 2070 7275 6e65  tion (PT = prune
│ │ │ │ +0001bfe0: 2054 2c20 4c65 6e67 7468 4c69 6d69 7420   T, LengthLimit 
│ │ │ │ +0001bff0: 3d3e 2036 297c 0a7c 2020 2020 2020 2020  => 6)|.|        
│ │ │ │ +0001c000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c030: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001c040: 7c20 2020 2020 2020 2020 2020 2020 2030  |              0
│ │ │ │ -0001c050: 2020 3120 2032 2020 2033 2020 2034 2020    1  2   3   4  
│ │ │ │ -0001c060: 2035 2020 2036 2020 2020 2020 2020 2020   5   6          
│ │ │ │ -0001c070: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0001c080: 3135 203d 2074 6f74 616c 3a20 3331 2035  15 = total: 31 5
│ │ │ │ -0001c090: 3520 3837 2031 3237 2031 3735 2032 3331  5 87 127 175 231
│ │ │ │ -0001c0a0: 2032 3935 2020 2020 2020 2020 2020 2020   295            
│ │ │ │ -0001c0b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0001c0c0: 2020 2020 2020 2030 3a20 3133 2032 3420         0: 13 24 
│ │ │ │ -0001c0d0: 3339 2020 3538 2020 3831 2031 3038 2031  39  58  81 108 1
│ │ │ │ -0001c0e0: 3339 2020 2020 2020 2020 2020 2020 2020  39              
│ │ │ │ -0001c0f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0001c100: 2020 2020 2031 3a20 3138 2033 3120 3438       1: 18 31 48
│ │ │ │ -0001c110: 2020 3639 2020 3934 2031 3233 2031 3536    69  94 123 156
│ │ │ │ -0001c120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c130: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001c030: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0001c040: 2020 2020 3020 2031 2020 3220 2020 3320      0  1  2   3 
│ │ │ │ +0001c050: 2020 3420 2020 3520 2020 3620 2020 2020    4   5   6     
│ │ │ │ +0001c060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c070: 207c 0a7c 6f31 3520 3d20 746f 7461 6c3a   |.|o15 = total:
│ │ │ │ +0001c080: 2033 3120 3535 2038 3720 3132 3720 3137   31 55 87 127 17
│ │ │ │ +0001c090: 3520 3233 3120 3239 3520 2020 2020 2020  5 231 295       
│ │ │ │ +0001c0a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001c0b0: 0a7c 2020 2020 2020 2020 2020 303a 2031  .|          0: 1
│ │ │ │ +0001c0c0: 3320 3234 2033 3920 2035 3820 2038 3120  3 24 39  58  81 
│ │ │ │ +0001c0d0: 3130 3820 3133 3920 2020 2020 2020 2020  108 139         
│ │ │ │ +0001c0e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001c0f0: 2020 2020 2020 2020 2020 313a 2031 3820            1: 18 
│ │ │ │ +0001c100: 3331 2034 3820 2036 3920 2039 3420 3132  31 48  69  94 12
│ │ │ │ +0001c110: 3320 3135 3620 2020 2020 2020 2020 2020  3 156           
│ │ │ │ +0001c120: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001c130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c170: 2020 2020 7c0a 7c6f 3135 203a 2042 6574      |.|o15 : Bet
│ │ │ │ -0001c180: 7469 5461 6c6c 7920 2020 2020 2020 2020  tiTally         
│ │ │ │ +0001c160: 2020 2020 2020 2020 207c 0a7c 6f31 3520           |.|o15 
│ │ │ │ +0001c170: 3a20 4265 7474 6954 616c 6c79 2020 2020  : BettiTally    
│ │ │ │ +0001c180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c1b0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0001c1a0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0001c1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c1f0: 2b0a 7c69 3136 203a 2045 3120 3d20 7072  +.|i16 : E1 = pr
│ │ │ │ -0001c200: 756e 6520 6578 7465 7269 6f72 4578 744d  une exteriorExtM
│ │ │ │ -0001c210: 6f64 756c 6528 662c 204d 532c 2072 6573  odule(f, MS, res
│ │ │ │ -0001c220: 466c 6429 3b20 2020 2020 2020 2020 7c0a  Fld);         |.
│ │ │ │ -0001c230: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0001c1e0: 2d2d 2d2d 2d2b 0a7c 6931 3620 3a20 4531  -----+.|i16 : E1
│ │ │ │ +0001c1f0: 203d 2070 7275 6e65 2065 7874 6572 696f   = prune exterio
│ │ │ │ +0001c200: 7245 7874 4d6f 6475 6c65 2866 2c20 4d53  rExtModule(f, MS
│ │ │ │ +0001c210: 2c20 7265 7346 6c64 293b 2020 2020 2020  , resFld);      
│ │ │ │ +0001c220: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0001c230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0001c270: 3137 203a 2072 696e 6720 4531 2020 2020  17 : ring E1    
│ │ │ │ +0001c260: 2d2b 0a7c 6931 3720 3a20 7269 6e67 2045  -+.|i17 : ring E
│ │ │ │ +0001c270: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │  0001c280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c2a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001c290: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001c2a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0001c2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c2e0: 2020 2020 2020 2020 7c0a 7c6f 3137 203d          |.|o17 =
│ │ │ │ -0001c2f0: 206b 6b5b 5820 2e2e 5820 2c20 6520 2e2e   kk[X ..X , e ..
│ │ │ │ -0001c300: 6520 5d20 2020 2020 2020 2020 2020 2020  e ]             
│ │ │ │ -0001c310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c320: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001c330: 2020 2030 2020 2032 2020 2030 2020 2032     0   2   0   2
│ │ │ │ +0001c2d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001c2e0: 6f31 3720 3d20 6b6b 5b58 202e 2e58 202c  o17 = kk[X ..X ,
│ │ │ │ +0001c2f0: 2065 202e 2e65 205d 2020 2020 2020 2020   e ..e ]        
│ │ │ │ +0001c300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c310: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001c320: 2020 2020 2020 2020 3020 2020 3220 2020          0   2   
│ │ │ │ +0001c330: 3020 2020 3220 2020 2020 2020 2020 2020  0   2           
│ │ │ │  0001c340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c360: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001c350: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001c360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c3a0: 2020 7c0a 7c6f 3137 203a 2050 6f6c 796e    |.|o17 : Polyn
│ │ │ │ -0001c3b0: 6f6d 6961 6c52 696e 672c 2033 2073 6b65  omialRing, 3 ske
│ │ │ │ -0001c3c0: 7720 636f 6d6d 7574 6174 6976 6520 7661  w commutative va
│ │ │ │ -0001c3d0: 7269 6162 6c65 2873 2920 2020 2020 2020  riable(s)       
│ │ │ │ -0001c3e0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0001c390: 2020 2020 2020 207c 0a7c 6f31 3720 3a20         |.|o17 : 
│ │ │ │ +0001c3a0: 506f 6c79 6e6f 6d69 616c 5269 6e67 2c20  PolynomialRing, 
│ │ │ │ +0001c3b0: 3320 736b 6577 2063 6f6d 6d75 7461 7469  3 skew commutati
│ │ │ │ +0001c3c0: 7665 2076 6172 6961 626c 6528 7329 2020  ve variable(s)  
│ │ │ │ +0001c3d0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001c3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0001c420: 7c69 3138 203a 2065 7852 696e 6720 3d20  |i18 : exRing = 
│ │ │ │ -0001c430: 6b6b 5b65 5f30 2c65 5f31 2c65 5f32 2c20  kk[e_0,e_1,e_2, 
│ │ │ │ -0001c440: 536b 6577 436f 6d6d 7574 6174 6976 6520  SkewCommutative 
│ │ │ │ -0001c450: 3d3e 7472 7565 5d20 2020 2020 7c0a 7c20  =>true]     |.| 
│ │ │ │ +0001c410: 2d2d 2d2b 0a7c 6931 3820 3a20 6578 5269  ---+.|i18 : exRi
│ │ │ │ +0001c420: 6e67 203d 206b 6b5b 655f 302c 655f 312c  ng = kk[e_0,e_1,
│ │ │ │ +0001c430: 655f 322c 2053 6b65 7743 6f6d 6d75 7461  e_2, SkewCommuta
│ │ │ │ +0001c440: 7469 7665 203d 3e74 7275 655d 2020 2020  tive =>true]    
│ │ │ │ +0001c450: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001c460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c490: 2020 2020 2020 2020 2020 7c0a 7c6f 3138            |.|o18
│ │ │ │ -0001c4a0: 203d 2065 7852 696e 6720 2020 2020 2020   = exRing       
│ │ │ │ +0001c480: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001c490: 0a7c 6f31 3820 3d20 6578 5269 6e67 2020  .|o18 = exRing  
│ │ │ │ +0001c4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c4d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001c4c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +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 3138 203a 2050        |.|o18 : P
│ │ │ │ -0001c520: 6f6c 796e 6f6d 6961 6c52 696e 672c 2033  olynomialRing, 3
│ │ │ │ -0001c530: 2073 6b65 7720 636f 6d6d 7574 6174 6976   skew commutativ
│ │ │ │ -0001c540: 6520 7661 7269 6162 6c65 2873 2920 2020  e variable(s)   
│ │ │ │ -0001c550: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0001c500: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0001c510: 3820 3a20 506f 6c79 6e6f 6d69 616c 5269  8 : PolynomialRi
│ │ │ │ +0001c520: 6e67 2c20 3320 736b 6577 2063 6f6d 6d75  ng, 3 skew commu
│ │ │ │ +0001c530: 7461 7469 7665 2076 6172 6961 626c 6528  tative variable(
│ │ │ │ +0001c540: 7329 2020 2020 2020 207c 0a2b 2d2d 2d2d  s)       |.+----
│ │ │ │ +0001c550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c590: 2d2d 2b0a 0a57 6520 6361 6e20 616c 736f  --+..We can also
│ │ │ │ -0001c5a0: 2063 6f6e 7374 7275 6374 2074 6865 2065   construct the e
│ │ │ │ -0001c5b0: 7874 6572 696f 7245 7874 4d6f 6475 6c65  xteriorExtModule
│ │ │ │ -0001c5c0: 2061 7320 6120 6269 6772 6164 6564 206d   as a bigraded m
│ │ │ │ -0001c5d0: 6f64 756c 652c 206f 7665 7220 6120 7269  odule, over a ri
│ │ │ │ -0001c5e0: 6e67 0a53 4520 7468 6174 2068 6173 2062  ng.SE that has b
│ │ │ │ -0001c5f0: 6f74 6820 706f 6c79 6e6f 6d69 616c 2076  oth polynomial v
│ │ │ │ -0001c600: 6172 6961 626c 6573 206c 696b 6520 5320  ariables like S 
│ │ │ │ -0001c610: 616e 6420 6578 7465 7269 6f72 2076 6172  and exterior var
│ │ │ │ -0001c620: 6961 626c 6573 206c 696b 6520 452e 2054  iables like E. T
│ │ │ │ -0001c630: 6865 0a70 6f6c 796e 6f6d 6961 6c20 7661  he.polynomial va
│ │ │ │ -0001c640: 7269 6162 6c65 7320 6861 7665 2064 6567  riables have deg
│ │ │ │ -0001c650: 7265 6573 207b 312c 307d 2e20 5468 6520  rees {1,0}. The 
│ │ │ │ -0001c660: 6578 7465 7269 6f72 2076 6172 6961 626c  exterior variabl
│ │ │ │ -0001c670: 6573 2068 6176 6520 6465 6772 6565 730a  es have degrees.
│ │ │ │ -0001c680: 7b64 6567 2066 665f 692c 2031 7d2e 0a0a  {deg ff_i, 1}...
│ │ │ │ -0001c690: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0001c580: 2d2d 2d2d 2d2d 2d2b 0a0a 5765 2063 616e  -------+..We can
│ │ │ │ +0001c590: 2061 6c73 6f20 636f 6e73 7472 7563 7420   also construct 
│ │ │ │ +0001c5a0: 7468 6520 6578 7465 7269 6f72 4578 744d  the exteriorExtM
│ │ │ │ +0001c5b0: 6f64 756c 6520 6173 2061 2062 6967 7261  odule as a bigra
│ │ │ │ +0001c5c0: 6465 6420 6d6f 6475 6c65 2c20 6f76 6572  ded module, over
│ │ │ │ +0001c5d0: 2061 2072 696e 670a 5345 2074 6861 7420   a ring.SE that 
│ │ │ │ +0001c5e0: 6861 7320 626f 7468 2070 6f6c 796e 6f6d  has both polynom
│ │ │ │ +0001c5f0: 6961 6c20 7661 7269 6162 6c65 7320 6c69  ial variables li
│ │ │ │ +0001c600: 6b65 2053 2061 6e64 2065 7874 6572 696f  ke S and exterio
│ │ │ │ +0001c610: 7220 7661 7269 6162 6c65 7320 6c69 6b65  r variables like
│ │ │ │ +0001c620: 2045 2e20 5468 650a 706f 6c79 6e6f 6d69   E. The.polynomi
│ │ │ │ +0001c630: 616c 2076 6172 6961 626c 6573 2068 6176  al variables hav
│ │ │ │ +0001c640: 6520 6465 6772 6565 7320 7b31 2c30 7d2e  e degrees {1,0}.
│ │ │ │ +0001c650: 2054 6865 2065 7874 6572 696f 7220 7661   The exterior va
│ │ │ │ +0001c660: 7269 6162 6c65 7320 6861 7665 2064 6567  riables have deg
│ │ │ │ +0001c670: 7265 6573 0a7b 6465 6720 6666 5f69 2c20  rees.{deg ff_i, 
│ │ │ │ +0001c680: 317d 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  1}...+----------
│ │ │ │ +0001c690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c6c0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3920 3a20  -------+.|i19 : 
│ │ │ │ -0001c6d0: 4531 203d 2070 7275 6e65 2065 7874 6572  E1 = prune exter
│ │ │ │ -0001c6e0: 696f 7245 7874 4d6f 6475 6c65 2866 2c20  iorExtModule(f, 
│ │ │ │ -0001c6f0: 4d53 2c20 7265 7346 6c64 293b 2020 2020  MS, resFld);    
│ │ │ │ -0001c700: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0001c6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0001c6c0: 3139 203a 2045 3120 3d20 7072 756e 6520  19 : E1 = prune 
│ │ │ │ +0001c6d0: 6578 7465 7269 6f72 4578 744d 6f64 756c  exteriorExtModul
│ │ │ │ +0001c6e0: 6528 662c 204d 532c 2072 6573 466c 6429  e(f, MS, resFld)
│ │ │ │ +0001c6f0: 3b20 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d  ;    |.+--------
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│ │ │ │  0001c710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c730: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3020  ---------+.|i20 
│ │ │ │ -0001c740: 3a20 7269 6e67 2045 3120 2020 2020 2020  : ring E1       
│ │ │ │ +0001c720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0001c730: 7c69 3230 203a 2072 696e 6720 4531 2020  |i20 : ring E1  
│ │ │ │ +0001c740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c770: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001c760: 2020 2020 2020 207c 0a7c 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 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c7a0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ -0001c7b0: 3020 3d20 6b6b 5b58 202e 2e58 202c 2065  0 = kk[X ..X , e
│ │ │ │ -0001c7c0: 202e 2e65 205d 2020 2020 2020 2020 2020   ..e ]          
│ │ │ │ -0001c7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c7e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001c7f0: 2030 2020 2032 2020 2030 2020 2032 2020   0   2   0   2  
│ │ │ │ +0001c7a0: 7c0a 7c6f 3230 203d 206b 6b5b 5820 2e2e  |.|o20 = kk[X ..
│ │ │ │ +0001c7b0: 5820 2c20 6520 2e2e 6520 5d20 2020 2020  X , e ..e ]     
│ │ │ │ +0001c7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c7d0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001c7e0: 2020 2020 2020 3020 2020 3220 2020 3020        0   2   0 
│ │ │ │ +0001c7f0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │  0001c800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c810: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001c810: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0001c820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c850: 2020 2020 2020 7c0a 7c6f 3230 203a 2050        |.|o20 : P
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│ │ │ │ -0001c890: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0001c840: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +0001c850: 3020 3a20 506f 6c79 6e6f 6d69 616c 5269  0 : PolynomialRi
│ │ │ │ +0001c860: 6e67 2c20 3320 736b 6577 2063 6f6d 6d75  ng, 3 skew commu
│ │ │ │ +0001c870: 7461 7469 7665 2076 6172 6961 626c 6528  tative variable(
│ │ │ │ +0001c880: 7329 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d  s)  |.+---------
│ │ │ │ +0001c890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c8c0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3231 203a  --------+.|i21 :
│ │ │ │ -0001c8d0: 2065 7852 696e 6720 3d20 6b6b 5b65 5f30   exRing = kk[e_0
│ │ │ │ -0001c8e0: 2c65 5f31 2c65 5f32 2c20 536b 6577 436f  ,e_1,e_2, SkewCo
│ │ │ │ -0001c8f0: 6d6d 7574 6174 6976 6520 3d3e 7472 7565  mmutative =>true
│ │ │ │ -0001c900: 5d7c 0a7c 2020 2020 2020 2020 2020 2020  ]|.|            
│ │ │ │ +0001c8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +0001c8c0: 6932 3120 3a20 6578 5269 6e67 203d 206b  i21 : exRing = k
│ │ │ │ +0001c8d0: 6b5b 655f 302c 655f 312c 655f 322c 2053  k[e_0,e_1,e_2, S
│ │ │ │ +0001c8e0: 6b65 7743 6f6d 6d75 7461 7469 7665 203d  kewCommutative =
│ │ │ │ +0001c8f0: 3e74 7275 655d 7c0a 7c20 2020 2020 2020  >true]|.|       
│ │ │ │ +0001c900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c930: 2020 2020 2020 2020 2020 7c0a 7c6f 3231            |.|o21
│ │ │ │ -0001c940: 203d 2065 7852 696e 6720 2020 2020 2020   = exRing       
│ │ │ │ +0001c920: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001c930: 0a7c 6f32 3120 3d20 6578 5269 6e67 2020  .|o21 = exRing  
│ │ │ │ +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 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0001c960: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001c970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c9a0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0001c9b0: 3231 203a 2050 6f6c 796e 6f6d 6961 6c52  21 : PolynomialR
│ │ │ │ -0001c9c0: 696e 672c 2033 2073 6b65 7720 636f 6d6d  ing, 3 skew comm
│ │ │ │ -0001c9d0: 7574 6174 6976 6520 7661 7269 6162 6c65  utative variable
│ │ │ │ -0001c9e0: 2873 2920 207c 0a2b 2d2d 2d2d 2d2d 2d2d  (s)  |.+--------
│ │ │ │ +0001c9a0: 207c 0a7c 6f32 3120 3a20 506f 6c79 6e6f   |.|o21 : Polyno
│ │ │ │ +0001c9b0: 6d69 616c 5269 6e67 2c20 3320 736b 6577  mialRing, 3 skew
│ │ │ │ +0001c9c0: 2063 6f6d 6d75 7461 7469 7665 2076 6172   commutative var
│ │ │ │ +0001c9d0: 6961 626c 6528 7329 2020 7c0a 2b2d 2d2d  iable(s)  |.+---
│ │ │ │ +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 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0001ca20: 0a54 6f20 7365 6520 7468 6174 2074 6869  .To see that thi
│ │ │ │ -0001ca30: 7320 6973 2072 6561 6c6c 7920 7468 6520  s is really the 
│ │ │ │ -0001ca40: 7361 6d65 206d 6f64 756c 652c 2077 6974  same module, wit
│ │ │ │ -0001ca50: 6820 6120 6d6f 7265 2063 6f6d 706c 6578  h a more complex
│ │ │ │ -0001ca60: 2067 7261 6469 6e67 2c20 7765 2063 616e   grading, we can
│ │ │ │ -0001ca70: 0a62 7269 6e67 2069 7420 6f76 6572 2074  .bring it over t
│ │ │ │ -0001ca80: 6f20 6120 7075 7265 2065 7874 6572 696f  o a pure exterio
│ │ │ │ -0001ca90: 7220 616c 6765 6272 612e 204e 6f74 6520  r algebra. Note 
│ │ │ │ -0001caa0: 7468 6174 2074 6865 206e 6563 6573 7361  that the necessa
│ │ │ │ -0001cab0: 7279 206d 6170 206f 6620 7269 6e67 730a  ry map of rings.
│ │ │ │ -0001cac0: 6d75 7374 2063 6f6e 7461 696e 2061 2044  must contain a D
│ │ │ │ -0001cad0: 6567 7265 654d 6170 206f 7074 696f 6e2e  egreeMap option.
│ │ │ │ -0001cae0: 2049 6e20 6765 6e65 7261 6c20 7765 2063   In general we c
│ │ │ │ -0001caf0: 6f75 6c64 206f 6e6c 7920 7461 6b65 2074  ould only take t
│ │ │ │ -0001cb00: 6865 2064 6567 7265 6573 206f 660a 7468  he degrees of.th
│ │ │ │ -0001cb10: 6520 6765 6e65 7261 746f 7273 206f 6620  e generators of 
│ │ │ │ -0001cb20: 7468 6520 6578 7465 7269 6f72 2061 6c67  the exterior alg
│ │ │ │ -0001cb30: 6562 7261 2074 6f20 6265 2074 6865 2067  ebra to be the g
│ │ │ │ -0001cb40: 6364 206f 6620 2074 6865 2064 6567 2066  cd of  the deg f
│ │ │ │ -0001cb50: 665f 6920 3b20 696e 2074 6865 0a65 7861  f_i ; in the.exa
│ │ │ │ -0001cb60: 6d70 6c65 2061 626f 7665 2074 6869 7320  mple above this 
│ │ │ │ -0001cb70: 6973 2031 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  is 1...+--------
│ │ │ │ +0001ca10: 2d2d 2d2b 0a0a 546f 2073 6565 2074 6861  ---+..To see tha
│ │ │ │ +0001ca20: 7420 7468 6973 2069 7320 7265 616c 6c79  t this is really
│ │ │ │ +0001ca30: 2074 6865 2073 616d 6520 6d6f 6475 6c65   the same module
│ │ │ │ +0001ca40: 2c20 7769 7468 2061 206d 6f72 6520 636f  , with a more co
│ │ │ │ +0001ca50: 6d70 6c65 7820 6772 6164 696e 672c 2077  mplex grading, w
│ │ │ │ +0001ca60: 6520 6361 6e0a 6272 696e 6720 6974 206f  e can.bring it o
│ │ │ │ +0001ca70: 7665 7220 746f 2061 2070 7572 6520 6578  ver to a pure ex
│ │ │ │ +0001ca80: 7465 7269 6f72 2061 6c67 6562 7261 2e20  terior algebra. 
│ │ │ │ +0001ca90: 4e6f 7465 2074 6861 7420 7468 6520 6e65  Note that the ne
│ │ │ │ +0001caa0: 6365 7373 6172 7920 6d61 7020 6f66 2072  cessary map of r
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│ │ │ │ +0001cad0: 7469 6f6e 2e20 496e 2067 656e 6572 616c  tion. In general
│ │ │ │ +0001cae0: 2077 6520 636f 756c 6420 6f6e 6c79 2074   we could only t
│ │ │ │ +0001caf0: 616b 6520 7468 6520 6465 6772 6565 7320  ake the degrees 
│ │ │ │ +0001cb00: 6f66 0a74 6865 2067 656e 6572 6174 6f72  of.the generator
│ │ │ │ +0001cb10: 7320 6f66 2074 6865 2065 7874 6572 696f  s of the exterio
│ │ │ │ +0001cb20: 7220 616c 6765 6272 6120 746f 2062 6520  r algebra to be 
│ │ │ │ +0001cb30: 7468 6520 6763 6420 6f66 2020 7468 6520  the gcd of  the 
│ │ │ │ +0001cb40: 6465 6720 6666 5f69 203b 2069 6e20 7468  deg ff_i ; in th
│ │ │ │ +0001cb50: 650a 6578 616d 706c 6520 6162 6f76 6520  e.example above 
│ │ │ │ +0001cb60: 7468 6973 2069 7320 312e 0a0a 2b2d 2d2d  this is 1...+---
│ │ │ │ +0001cb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001cb80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001cb90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001cba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cbb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cbc0: 2d2d 2b0a 7c69 3232 203a 2071 203d 206d  --+.|i22 : q = m
│ │ │ │ -0001cbd0: 6170 2865 7852 696e 672c 2072 696e 6720  ap(exRing, ring 
│ │ │ │ -0001cbe0: 4531 2c20 7b33 3a30 2c65 5f30 2c65 5f31  E1, {3:0,e_0,e_1
│ │ │ │ -0001cbf0: 2c65 5f32 7d2c 2044 6567 7265 654d 6170  ,e_2}, DegreeMap
│ │ │ │ -0001cc00: 203d 3e20 6420 2d3e 207b 645f 317d 297c   => d -> {d_1})|
│ │ │ │ -0001cc10: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0001cbb0: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3220 3a20  -------+.|i22 : 
│ │ │ │ +0001cbc0: 7120 3d20 6d61 7028 6578 5269 6e67 2c20  q = map(exRing, 
│ │ │ │ +0001cbd0: 7269 6e67 2045 312c 207b 333a 302c 655f  ring E1, {3:0,e_
│ │ │ │ +0001cbe0: 302c 655f 312c 655f 327d 2c20 4465 6772  0,e_1,e_2}, Degr
│ │ │ │ +0001cbf0: 6565 4d61 7020 3d3e 2064 202d 3e20 7b64  eeMap => d -> {d
│ │ │ │ +0001cc00: 5f31 7d29 7c0a 7c20 2020 2020 2020 2020  _1})|.|         
│ │ │ │ +0001cc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cc50: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0001cc60: 3232 203d 206d 6170 2028 6578 5269 6e67  22 = map (exRing
│ │ │ │ -0001cc70: 2c20 6b6b 5b58 202e 2e58 202c 2065 202e  , kk[X ..X , e .
│ │ │ │ -0001cc80: 2e65 205d 2c20 7b30 2c20 302c 2030 2c20  .e ], {0, 0, 0, 
│ │ │ │ -0001cc90: 6520 2c20 6520 2c20 6520 7d29 2020 2020  e , e , e })    
│ │ │ │ -0001cca0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0001ccb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ccc0: 2020 2030 2020 2032 2020 2030 2020 2032     0   2   0   2
│ │ │ │ -0001ccd0: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -0001cce0: 2020 3120 2020 3220 2020 2020 2020 2020    1   2         
│ │ │ │ -0001ccf0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001cc50: 207c 0a7c 6f32 3220 3d20 6d61 7020 2865   |.|o22 = map (e
│ │ │ │ +0001cc60: 7852 696e 672c 206b 6b5b 5820 2e2e 5820  xRing, kk[X ..X 
│ │ │ │ +0001cc70: 2c20 6520 2e2e 6520 5d2c 207b 302c 2030  , e ..e ], {0, 0
│ │ │ │ +0001cc80: 2c20 302c 2065 202c 2065 202c 2065 207d  , 0, e , e , e }
│ │ │ │ +0001cc90: 2920 2020 2020 2020 2020 2020 2020 7c0a  )             |.
│ │ │ │ +0001cca0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0001ccb0: 2020 2020 2020 2020 3020 2020 3220 2020          0   2   
│ │ │ │ +0001ccc0: 3020 2020 3220 2020 2020 2020 2020 2020  0   2           
│ │ │ │ +0001ccd0: 2020 2030 2020 2031 2020 2032 2020 2020     0   1   2    
│ │ │ │ +0001cce0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001ccf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cd40: 2020 207c 0a7c 6f32 3220 3a20 5269 6e67     |.|o22 : Ring
│ │ │ │ -0001cd50: 4d61 7020 6578 5269 6e67 203c 2d2d 206b  Map exRing <-- k
│ │ │ │ -0001cd60: 6b5b 5820 2e2e 5820 2c20 6520 2e2e 6520  k[X ..X , e ..e 
│ │ │ │ -0001cd70: 5d20 2020 2020 2020 2020 2020 2020 2020  ]               
│ │ │ │ -0001cd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cd90: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0001cda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cdb0: 3020 2020 3220 2020 3020 2020 3220 2020  0   2   0   2   
│ │ │ │ +0001cd30: 2020 2020 2020 2020 7c0a 7c6f 3232 203a          |.|o22 :
│ │ │ │ +0001cd40: 2052 696e 674d 6170 2065 7852 696e 6720   RingMap exRing 
│ │ │ │ +0001cd50: 3c2d 2d20 6b6b 5b58 202e 2e58 202c 2065  <-- kk[X ..X , e
│ │ │ │ +0001cd60: 202e 2e65 205d 2020 2020 2020 2020 2020   ..e ]          
│ │ │ │ +0001cd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cd80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001cd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cda0: 2020 2020 2030 2020 2032 2020 2030 2020       0   2   0  
│ │ │ │ +0001cdb0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │  0001cdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cdd0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0001cdd0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  0001cde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001cdf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ce00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ce10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ce20: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3233  ----------+.|i23
│ │ │ │ -0001ce30: 203a 2045 3220 3d20 636f 6b65 7220 7120   : E2 = coker q 
│ │ │ │ -0001ce40: 7072 6573 656e 7461 7469 6f6e 2045 313b  presentation E1;
│ │ │ │ +0001ce10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0001ce20: 0a7c 6932 3320 3a20 4532 203d 2063 6f6b  .|i23 : E2 = cok
│ │ │ │ +0001ce30: 6572 2071 2070 7265 7365 6e74 6174 696f  er q presentatio
│ │ │ │ +0001ce40: 6e20 4531 3b20 2020 2020 2020 2020 2020  n E1;           
│ │ │ │  0001ce50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ce60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ce70: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0001ce60: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0001ce70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ce80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ce90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001cea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ceb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cec0: 2d2d 2d2d 2b0a 7c69 3234 203a 2068 6628  ----+.|i24 : hf(
│ │ │ │ -0001ced0: 2d35 2e2e 352c 4532 2920 3d3d 2068 6628  -5..5,E2) == hf(
│ │ │ │ -0001cee0: 2d35 2e2e 352c 4529 2020 2020 2020 2020  -5..5,E)        
│ │ │ │ +0001ceb0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3420  ---------+.|i24 
│ │ │ │ +0001cec0: 3a20 6866 282d 352e 2e35 2c45 3229 203d  : hf(-5..5,E2) =
│ │ │ │ +0001ced0: 3d20 6866 282d 352e 2e35 2c45 2920 2020  = hf(-5..5,E)   
│ │ │ │ +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: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0001cf00: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001cf10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cf20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cf30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cf40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cf50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
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│ │ │ │  0001d3e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001d3f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001d400: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001d410: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ +0001d4b0: 2065 6c65 6d65 6e74 7320 7468 6174 2061   elements that a
│ │ │ │ +0001d4c0: 7265 0a20 2020 2020 2020 2068 6f6d 6f74  re.        homot
│ │ │ │ +0001d4d0: 6f70 6963 2074 6f20 3020 6f6e 2043 0a20  opic to 0 on C. 
│ │ │ │ +0001d4e0: 2020 2020 202a 2043 2c20 6120 2a6e 6f74       * C, a *not
│ │ │ │ +0001d4f0: 6520 636f 6d70 6c65 783a 2028 436f 6d70  e complex: (Comp
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│ │ │ │ +0001d510: 0a20 202a 204f 7574 7075 7473 3a0a 2020  .  * Outputs:.  
│ │ │ │ +0001d520: 2020 2020 2a20 4d2c 2061 202a 6e6f 7465      * M, a *note
│ │ │ │ +0001d530: 206d 6f64 756c 653a 2028 4d61 6361 756c   module: (Macaul
│ │ │ │ +0001d540: 6179 3244 6f63 294d 6f64 756c 652c 2c20  ay2Doc)Module,, 
│ │ │ │ +0001d550: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d  ..Description.==
│ │ │ │ +0001d560: 3d3d 3d3d 3d3d 3d3d 3d0a 0a41 7373 756d  =========..Assum
│ │ │ │ +0001d570: 696e 6720 7468 6174 2074 6865 2065 6c65  ing that the ele
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│ │ │ │ +0001d590: 206d 6174 7269 7820 6666 2061 7265 206e   matrix ff are n
│ │ │ │ +0001d5a0: 756c 6c2d 686f 6d6f 746f 7069 6320 6f6e  ull-homotopic on
│ │ │ │ +0001d5b0: 2043 2c20 7468 650a 7363 7269 7074 2072   C, the.script r
│ │ │ │ +0001d5c0: 6574 7572 6e73 2074 6865 2064 6972 6563  eturns the direc
│ │ │ │ +0001d5d0: 7420 7375 6d20 6f66 2074 6865 2068 6f6d  t sum of the hom
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│ │ │ │ +0001d5f0: 6d6f 6475 6c65 206f 7665 7220 6120 6e65  module over a ne
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│ │ │ │ +0001d610: 6e67 206f 6620 7269 6e67 2043 2077 6974  ng of ring C wit
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│ │ │ │ +0001d630: 6961 626c 6573 2061 646a 6f69 6e65 642e  iables adjoined.
│ │ │ │ +0001d640: 2054 6865 2073 6372 6970 7420 6973 2074   The script is t
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│ │ │ │ +0001d670: 4d6f 6475 6c65 0a0a 5365 6520 616c 736f  Module..See also
│ │ │ │ +0001d680: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a  .========..  * *
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│ │ │ │ +0001d6b0: 546f 724d 6f64 756c 652c 202d 2d20 546f  TorModule, -- To
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│ │ │ │ +0001d6f0: 6772 6164 6564 2061 6c67 6562 7261 0a20  graded algebra. 
│ │ │ │ +0001d700: 202a 202a 6e6f 7465 206d 616b 6548 6f6d   * *note makeHom
│ │ │ │ +0001d710: 6f74 6f70 6965 734f 6e48 6f6d 6f6c 6f67  otopiesOnHomolog
│ │ │ │ +0001d720: 793a 206d 616b 6548 6f6d 6f74 6f70 6965  y: makeHomotopie
│ │ │ │ +0001d730: 734f 6e48 6f6d 6f6c 6f67 792c 202d 2d20  sOnHomology, -- 
│ │ │ │ +0001d740: 486f 6d6f 6c6f 6779 206f 6620 610a 2020  Homology of a.  
│ │ │ │ +0001d750: 2020 636f 6d70 6c65 7820 6173 2065 7874    complex as ext
│ │ │ │ +0001d760: 6572 696f 7220 6d6f 6475 6c65 0a0a 5761  erior module..Wa
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│ │ │ │ +0001d780: 6f72 486f 6d6f 6c6f 6779 4d6f 6475 6c65  orHomologyModule
│ │ │ │ +0001d790: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  :.==============
│ │ │ │  0001d7a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001d7b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ -0001d7d0: 6f6d 6f6c 6f67 794d 6f64 756c 6528 4d61  omologyModule(Ma
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│ │ │ │ -0001d7f0: 466f 7220 7468 6520 7072 6f67 7261 6d6d  For the programm
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│ │ │ │ -0001d810: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563  =====..The objec
│ │ │ │ -0001d820: 7420 2a6e 6f74 6520 6578 7465 7269 6f72  t *note exterior
│ │ │ │ -0001d830: 486f 6d6f 6c6f 6779 4d6f 6475 6c65 3a20  HomologyModule: 
│ │ │ │ -0001d840: 6578 7465 7269 6f72 486f 6d6f 6c6f 6779  exteriorHomology
│ │ │ │ -0001d850: 4d6f 6475 6c65 2c20 6973 2061 202a 6e6f  Module, is a *no
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│ │ │ │ +0001d7f0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d  grammer.========
│ │ │ │ +0001d800: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520  ==========..The 
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│ │ │ │ +0001d820: 6572 696f 7248 6f6d 6f6c 6f67 794d 6f64  eriorHomologyMod
│ │ │ │ +0001d830: 756c 653a 2065 7874 6572 696f 7248 6f6d  ule: exteriorHom
│ │ │ │ +0001d840: 6f6c 6f67 794d 6f64 756c 652c 2069 7320  ologyModule, is 
│ │ │ │ +0001d850: 6120 2a6e 6f74 650a 6d65 7468 6f64 2066  a *note.method f
│ │ │ │ +0001d860: 756e 6374 696f 6e3a 2028 4d61 6361 756c  unction: (Macaul
│ │ │ │ +0001d870: 6179 3244 6f63 294d 6574 686f 6446 756e  ay2Doc)MethodFun
│ │ │ │ +0001d880: 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d  ction,...-------
│ │ │ │ +0001d890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d8e0: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520  ---..The source 
│ │ │ │ -0001d8f0: 6f66 2074 6869 7320 646f 6375 6d65 6e74  of this document
│ │ │ │ -0001d900: 2069 7320 696e 0a2f 6275 696c 642f 7265   is in./build/re
│ │ │ │ -0001d910: 7072 6f64 7563 6962 6c65 2d70 6174 682f  producible-path/
│ │ │ │ -0001d920: 6d61 6361 756c 6179 322d 312e 3235 2e31  macaulay2-1.25.1
│ │ │ │ -0001d930: 312b 6473 2f4d 322f 4d61 6361 756c 6179  1+ds/M2/Macaulay
│ │ │ │ -0001d940: 322f 7061 636b 6167 6573 2f0a 436f 6d70  2/packages/.Comp
│ │ │ │ -0001d950: 6c65 7465 496e 7465 7273 6563 7469 6f6e  leteIntersection
│ │ │ │ -0001d960: 5265 736f 6c75 7469 6f6e 732e 6d32 3a32  Resolutions.m2:2
│ │ │ │ -0001d970: 3738 353a 302e 0a1f 0a46 696c 653a 2043  785:0....File: C
│ │ │ │ -0001d980: 6f6d 706c 6574 6549 6e74 6572 7365 6374  ompleteIntersect
│ │ │ │ -0001d990: 696f 6e52 6573 6f6c 7574 696f 6e73 2e69  ionResolutions.i
│ │ │ │ -0001d9a0: 6e66 6f2c 204e 6f64 653a 2065 7874 6572  nfo, Node: exter
│ │ │ │ -0001d9b0: 696f 7254 6f72 4d6f 6475 6c65 2c20 4e65  iorTorModule, Ne
│ │ │ │ -0001d9c0: 7874 3a20 6578 7449 734f 6e65 506f 6c79  xt: extIsOnePoly
│ │ │ │ -0001d9d0: 6e6f 6d69 616c 2c20 5072 6576 3a20 6578  nomial, Prev: ex
│ │ │ │ -0001d9e0: 7465 7269 6f72 486f 6d6f 6c6f 6779 4d6f  teriorHomologyMo
│ │ │ │ -0001d9f0: 6475 6c65 2c20 5570 3a20 546f 700a 0a65  dule, Up: Top..e
│ │ │ │ -0001da00: 7874 6572 696f 7254 6f72 4d6f 6475 6c65  xteriorTorModule
│ │ │ │ -0001da10: 202d 2d20 546f 7220 6173 2061 206d 6f64   -- Tor as a mod
│ │ │ │ -0001da20: 756c 6520 6f76 6572 2061 6e20 6578 7465  ule over an exte
│ │ │ │ -0001da30: 7269 6f72 2061 6c67 6562 7261 206f 7220  rior algebra or 
│ │ │ │ -0001da40: 6269 6772 6164 6564 2061 6c67 6562 7261  bigraded algebra
│ │ │ │ -0001da50: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  .***************
│ │ │ │ +0001d8d0: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f  --------..The so
│ │ │ │ +0001d8e0: 7572 6365 206f 6620 7468 6973 2064 6f63  urce of this doc
│ │ │ │ +0001d8f0: 756d 656e 7420 6973 2069 6e0a 2f62 7569  ument is in./bui
│ │ │ │ +0001d900: 6c64 2f72 6570 726f 6475 6369 626c 652d  ld/reproducible-
│ │ │ │ +0001d910: 7061 7468 2f6d 6163 6175 6c61 7932 2d31  path/macaulay2-1
│ │ │ │ +0001d920: 2e32 352e 3131 2b64 732f 4d32 2f4d 6163  .25.11+ds/M2/Mac
│ │ │ │ +0001d930: 6175 6c61 7932 2f70 6163 6b61 6765 732f  aulay2/packages/
│ │ │ │ +0001d940: 0a43 6f6d 706c 6574 6549 6e74 6572 7365  .CompleteInterse
│ │ │ │ +0001d950: 6374 696f 6e52 6573 6f6c 7574 696f 6e73  ctionResolutions
│ │ │ │ +0001d960: 2e6d 323a 3237 3835 3a30 2e0a 1f0a 4669  .m2:2785:0....Fi
│ │ │ │ +0001d970: 6c65 3a20 436f 6d70 6c65 7465 496e 7465  le: CompleteInte
│ │ │ │ +0001d980: 7273 6563 7469 6f6e 5265 736f 6c75 7469  rsectionResoluti
│ │ │ │ +0001d990: 6f6e 732e 696e 666f 2c20 4e6f 6465 3a20  ons.info, Node: 
│ │ │ │ +0001d9a0: 6578 7465 7269 6f72 546f 724d 6f64 756c  exteriorTorModul
│ │ │ │ +0001d9b0: 652c 204e 6578 743a 2065 7874 4973 4f6e  e, Next: extIsOn
│ │ │ │ +0001d9c0: 6550 6f6c 796e 6f6d 6961 6c2c 2050 7265  ePolynomial, Pre
│ │ │ │ +0001d9d0: 763a 2065 7874 6572 696f 7248 6f6d 6f6c  v: exteriorHomol
│ │ │ │ +0001d9e0: 6f67 794d 6f64 756c 652c 2055 703a 2054  ogyModule, Up: T
│ │ │ │ +0001d9f0: 6f70 0a0a 6578 7465 7269 6f72 546f 724d  op..exteriorTorM
│ │ │ │ +0001da00: 6f64 756c 6520 2d2d 2054 6f72 2061 7320  odule -- Tor as 
│ │ │ │ +0001da10: 6120 6d6f 6475 6c65 206f 7665 7220 616e  a module over an
│ │ │ │ +0001da20: 2065 7874 6572 696f 7220 616c 6765 6272   exterior algebr
│ │ │ │ +0001da30: 6120 6f72 2062 6967 7261 6465 6420 616c  a or bigraded al
│ │ │ │ +0001da40: 6765 6272 610a 2a2a 2a2a 2a2a 2a2a 2a2a  gebra.**********
│ │ │ │ +0001da50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001da60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001da70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001da80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001da90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001daa0: 2a2a 0a0a 2020 2a20 5573 6167 653a 200a  **..  * Usage: .
│ │ │ │ -0001dab0: 2020 2020 2020 2020 5420 3d20 6578 7465          T = exte
│ │ │ │ -0001dac0: 7269 6f72 546f 724d 6f64 756c 6528 662c  riorTorModule(f,
│ │ │ │ -0001dad0: 4629 0a20 2020 2020 2020 2054 203d 2065  F).        T = e
│ │ │ │ -0001dae0: 7874 6572 696f 7254 6f72 4d6f 6475 6c65  xteriorTorModule
│ │ │ │ -0001daf0: 2866 2c4d 2c4e 290a 2020 2a20 496e 7075  (f,M,N).  * Inpu
│ │ │ │ -0001db00: 7473 3a0a 2020 2020 2020 2a20 662c 2061  ts:.      * f, a
│ │ │ │ -0001db10: 202a 6e6f 7465 206d 6174 7269 783a 2028   *note matrix: (
│ │ │ │ -0001db20: 4d61 6361 756c 6179 3244 6f63 294d 6174  Macaulay2Doc)Mat
│ │ │ │ -0001db30: 7269 782c 2c20 3120 7820 632c 2065 6e74  rix,, 1 x c, ent
│ │ │ │ -0001db40: 7269 6573 206d 7573 7420 6265 0a20 2020  ries must be.   
│ │ │ │ -0001db50: 2020 2020 2068 6f6d 6f74 6f70 6963 2074       homotopic t
│ │ │ │ -0001db60: 6f20 3020 6f6e 2046 0a20 2020 2020 202a  o 0 on F.      *
│ │ │ │ -0001db70: 204d 2c20 6120 2a6e 6f74 6520 6d6f 6475   M, a *note modu
│ │ │ │ -0001db80: 6c65 3a20 284d 6163 6175 6c61 7932 446f  le: (Macaulay2Do
│ │ │ │ -0001db90: 6329 4d6f 6475 6c65 2c2c 2053 2d6d 6f64  c)Module,, S-mod
│ │ │ │ -0001dba0: 756c 6520 616e 6e69 6869 6c61 7465 6420  ule annihilated 
│ │ │ │ -0001dbb0: 6279 2069 6465 616c 0a20 2020 2020 2020  by ideal.       
│ │ │ │ -0001dbc0: 2066 0a20 2020 2020 202a 204e 2c20 6120   f.      * N, a 
│ │ │ │ -0001dbd0: 2a6e 6f74 6520 6d6f 6475 6c65 3a20 284d  *note module: (M
│ │ │ │ -0001dbe0: 6163 6175 6c61 7932 446f 6329 4d6f 6475  acaulay2Doc)Modu
│ │ │ │ -0001dbf0: 6c65 2c2c 2053 2d6d 6f64 756c 6520 616e  le,, S-module an
│ │ │ │ -0001dc00: 6e69 6869 6c61 7465 6420 6279 2069 6465  nihilated by ide
│ │ │ │ -0001dc10: 616c 0a20 2020 2020 2020 2066 0a20 202a  al.        f.  *
│ │ │ │ -0001dc20: 204f 7574 7075 7473 3a0a 2020 2020 2020   Outputs:.      
│ │ │ │ -0001dc30: 2a20 542c 2061 202a 6e6f 7465 206d 6f64  * T, a *note mod
│ │ │ │ -0001dc40: 756c 653a 2028 4d61 6361 756c 6179 3244  ule: (Macaulay2D
│ │ │ │ -0001dc50: 6f63 294d 6f64 756c 652c 2c20 546f 725e  oc)Module,, Tor^
│ │ │ │ -0001dc60: 5328 4d2c 4e29 2061 7320 6120 4d6f 6475  S(M,N) as a Modu
│ │ │ │ -0001dc70: 6c65 206f 7665 720a 2020 2020 2020 2020  le over.        
│ │ │ │ -0001dc80: 616e 2065 7874 6572 696f 7220 616c 6765  an exterior alge
│ │ │ │ -0001dc90: 6272 610a 0a44 6573 6372 6970 7469 6f6e  bra..Description
│ │ │ │ -0001dca0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 4966  .===========..If
│ │ │ │ -0001dcb0: 204d 2c4e 2061 7265 2053 2d6d 6f64 756c   M,N are S-modul
│ │ │ │ -0001dcc0: 6573 2061 6e6e 6968 696c 6174 6564 2062  es annihilated b
│ │ │ │ -0001dcd0: 7920 7468 6520 656c 656d 656e 7473 206f  y the elements o
│ │ │ │ -0001dce0: 6620 7468 6520 6d61 7472 6978 2066 6620  f the matrix ff 
│ │ │ │ -0001dcf0: 3d20 2866 5f31 2e2e 665f 6329 2c0a 616e  = (f_1..f_c),.an
│ │ │ │ -0001dd00: 6420 6b20 6973 2074 6865 2072 6573 6964  d k is the resid
│ │ │ │ -0001dd10: 7565 2066 6965 6c64 206f 6620 532c 2074  ue field of S, t
│ │ │ │ -0001dd20: 6865 6e20 7468 6520 7363 7269 7074 2065  hen the script e
│ │ │ │ -0001dd30: 7874 6572 696f 7254 6f72 4d6f 6475 6c65  xteriorTorModule
│ │ │ │ -0001dd40: 2866 2c4d 2920 7265 7475 726e 730a 546f  (f,M) returns.To
│ │ │ │ -0001dd50: 725e 5328 4d2c 206b 2920 6173 2061 206d  r^S(M, k) as a m
│ │ │ │ -0001dd60: 6f64 756c 6520 6f76 6572 2061 6e20 6578  odule over an ex
│ │ │ │ -0001dd70: 7465 7269 6f72 2061 6c67 6562 7261 206b  terior algebra k
│ │ │ │ -0001dd80: 3c65 5f31 2c2e 2e2e 2c65 5f63 3e2c 2077  , w
│ │ │ │ -0001dd90: 6865 7265 2074 6865 2065 5f69 0a68 6176  here the e_i.hav
│ │ │ │ -0001dda0: 6520 6465 6772 6565 2031 2c20 7768 696c  e degree 1, whil
│ │ │ │ -0001ddb0: 6520 6578 7465 7269 6f72 546f 724d 6f64  e exteriorTorMod
│ │ │ │ -0001ddc0: 756c 6528 662c 4d2c 4e29 2072 6574 7572  ule(f,M,N) retur
│ │ │ │ -0001ddd0: 6e73 2054 6f72 5e53 284d 2c4e 2920 6173  ns Tor^S(M,N) as
│ │ │ │ -0001dde0: 2061 206d 6f64 756c 650a 6f76 6572 2061   a module.over a
│ │ │ │ -0001ddf0: 2062 6967 7261 6465 6420 7269 6e67 2053   bigraded ring S
│ │ │ │ -0001de00: 4520 3d20 533c 655f 312c 2e2e 2c65 5f63  E = S, where the e_i
│ │ │ │ -0001de20: 2068 6176 6520 6465 6772 6565 7320 7b64   have degrees {d
│ │ │ │ -0001de30: 5f69 2c31 7d2c 0a77 6865 7265 2064 5f69  _i,1},.where d_i
│ │ │ │ -0001de40: 2069 7320 7468 6520 6465 6772 6565 206f   is the degree o
│ │ │ │ -0001de50: 6620 665f 692e 2054 6865 206d 6f64 756c  f f_i. The modul
│ │ │ │ -0001de60: 6520 7374 7275 6374 7572 652c 2069 6e20  e structure, in 
│ │ │ │ -0001de70: 6569 7468 6572 2063 6173 652c 2069 730a  either case, is.
│ │ │ │ -0001de80: 6465 6669 6e65 6420 6279 2074 6865 2068  defined by the h
│ │ │ │ -0001de90: 6f6d 6f74 6f70 6965 7320 666f 7220 7468  omotopies for th
│ │ │ │ -0001dea0: 6520 665f 6920 6f6e 2074 6865 2072 6573  e f_i on the res
│ │ │ │ -0001deb0: 6f6c 7574 696f 6e20 6f66 204d 2c20 636f  olution of M, co
│ │ │ │ -0001dec0: 6d70 7574 6564 2062 7920 7468 650a 7363  mputed by the.sc
│ │ │ │ -0001ded0: 7269 7074 206d 616b 6548 6f6d 6f74 6f70  ript makeHomotop
│ │ │ │ -0001dee0: 6965 7331 2e0a 0a54 6865 2073 6372 6970  ies1...The scrip
│ │ │ │ -0001def0: 7473 2063 616c 6c20 6d61 6b65 4d6f 6475  ts call makeModu
│ │ │ │ -0001df00: 6c65 2074 6f20 636f 6d70 7574 6520 6120  le to compute a 
│ │ │ │ -0001df10: 286e 6f6e 2d6d 696e 696d 616c 2920 7072  (non-minimal) pr
│ │ │ │ -0001df20: 6573 656e 7461 7469 6f6e 206f 6620 7468  esentation of th
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│ │ │ │ -0001df40: 2074 6865 2064 6573 6372 6970 7469 6f6e   the description
│ │ │ │ -0001df50: 2062 7920 6d61 7472 6978 2066 6163 746f   by matrix facto
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│ │ │ │ -0001df80: 6120 6d6f 6475 6c65 0a6f 7665 7220 616e  a module.over an
│ │ │ │ -0001df90: 2065 7874 6572 696f 7220 616c 6765 6272   exterior algebr
│ │ │ │ -0001dfa0: 6122 206f 6620 4569 7365 6e62 7564 2c20  a" of Eisenbud, 
│ │ │ │ -0001dfb0: 5065 6576 6120 616e 6420 5363 6872 6579  Peeva and Schrey
│ │ │ │ -0001dfc0: 6572 2069 7420 666f 6c6c 6f77 7320 7468  er it follows th
│ │ │ │ -0001dfd0: 6174 2077 6865 6e0a 4d20 6973 2061 2068  at when.M is a h
│ │ │ │ -0001dfe0: 6967 6820 7379 7a79 6779 2061 6e64 2046  igh syzygy and F
│ │ │ │ -0001dff0: 2069 7320 6974 7320 7265 736f 6c75 7469   is its resoluti
│ │ │ │ -0001e000: 6f6e 2c20 7468 656e 2074 6865 2070 7265  on, then the pre
│ │ │ │ -0001e010: 7365 6e74 6174 696f 6e20 6f66 0a54 6f72  sentation of.Tor
│ │ │ │ -0001e020: 284d 2c53 5e31 2f6d 6d29 2061 6c77 6179  (M,S^1/mm) alway
│ │ │ │ -0001e030: 7320 6861 7320 6765 6e65 7261 746f 7273  s has generators
│ │ │ │ -0001e040: 2069 6e20 6465 6772 6565 7320 302c 312c   in degrees 0,1,
│ │ │ │ -0001e050: 2063 6f72 7265 7370 6f6e 6469 6e67 2074   corresponding t
│ │ │ │ -0001e060: 6f20 7468 650a 7461 7267 6574 7320 616e  o the.targets an
│ │ │ │ -0001e070: 6420 736f 7572 6365 7320 6f66 2074 6865  d sources of the
│ │ │ │ -0001e080: 2073 7461 636b 206f 6620 6d61 7073 2042   stack of maps B
│ │ │ │ -0001e090: 2869 292c 2061 6e64 2074 6861 7420 7468  (i), and that th
│ │ │ │ -0001e0a0: 6520 7265 736f 6c75 7469 6f6e 2069 730a  e resolution is.
│ │ │ │ -0001e0b0: 636f 6d70 6f6e 656e 7477 6973 6520 6c69  componentwise li
│ │ │ │ -0001e0c0: 6e65 6172 2069 6e20 6120 7375 6974 6162  near in a suitab
│ │ │ │ -0001e0d0: 6c65 2073 656e 7365 2e20 496e 2074 6865  le sense. In the
│ │ │ │ -0001e0e0: 2066 6f6c 6c6f 7769 6e67 2065 7861 6d70   following examp
│ │ │ │ -0001e0f0: 6c65 2c20 7468 6573 6520 6661 6374 730a  le, these facts.
│ │ │ │ -0001e100: 6172 6520 7665 7269 6669 6564 2e20 5468  are verified. Th
│ │ │ │ -0001e110: 6520 546f 7220 6d6f 6475 6c65 2064 6f65  e Tor module doe
│ │ │ │ -0001e120: 7320 4e4f 5420 7370 6c69 7420 696e 746f  s NOT split into
│ │ │ │ -0001e130: 2074 6865 2064 6972 6563 7420 7375 6d20   the direct sum 
│ │ │ │ -0001e140: 6f66 2074 6865 0a73 7562 6d6f 6475 6c65  of the.submodule
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│ │ │ │ -0001e170: 686f 7765 7665 722e 0a0a 0a0a 2b2d 2d2d  however.....+---
│ │ │ │ +0001da90: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 7361  *******..  * Usa
│ │ │ │ +0001daa0: 6765 3a20 0a20 2020 2020 2020 2054 203d  ge: .        T =
│ │ │ │ +0001dab0: 2065 7874 6572 696f 7254 6f72 4d6f 6475   exteriorTorModu
│ │ │ │ +0001dac0: 6c65 2866 2c46 290a 2020 2020 2020 2020  le(f,F).        
│ │ │ │ +0001dad0: 5420 3d20 6578 7465 7269 6f72 546f 724d  T = exteriorTorM
│ │ │ │ +0001dae0: 6f64 756c 6528 662c 4d2c 4e29 0a20 202a  odule(f,M,N).  *
│ │ │ │ +0001daf0: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ +0001db00: 2066 2c20 6120 2a6e 6f74 6520 6d61 7472   f, a *note matr
│ │ │ │ +0001db10: 6978 3a20 284d 6163 6175 6c61 7932 446f  ix: (Macaulay2Do
│ │ │ │ +0001db20: 6329 4d61 7472 6978 2c2c 2031 2078 2063  c)Matrix,, 1 x c
│ │ │ │ +0001db30: 2c20 656e 7472 6965 7320 6d75 7374 2062  , entries must b
│ │ │ │ +0001db40: 650a 2020 2020 2020 2020 686f 6d6f 746f  e.        homoto
│ │ │ │ +0001db50: 7069 6320 746f 2030 206f 6e20 460a 2020  pic to 0 on F.  
│ │ │ │ +0001db60: 2020 2020 2a20 4d2c 2061 202a 6e6f 7465      * M, a *note
│ │ │ │ +0001db70: 206d 6f64 756c 653a 2028 4d61 6361 756c   module: (Macaul
│ │ │ │ +0001db80: 6179 3244 6f63 294d 6f64 756c 652c 2c20  ay2Doc)Module,, 
│ │ │ │ +0001db90: 532d 6d6f 6475 6c65 2061 6e6e 6968 696c  S-module annihil
│ │ │ │ +0001dba0: 6174 6564 2062 7920 6964 6561 6c0a 2020  ated by ideal.  
│ │ │ │ +0001dbb0: 2020 2020 2020 660a 2020 2020 2020 2a20        f.      * 
│ │ │ │ +0001dbc0: 4e2c 2061 202a 6e6f 7465 206d 6f64 756c  N, a *note modul
│ │ │ │ +0001dbd0: 653a 2028 4d61 6361 756c 6179 3244 6f63  e: (Macaulay2Doc
│ │ │ │ +0001dbe0: 294d 6f64 756c 652c 2c20 532d 6d6f 6475  )Module,, S-modu
│ │ │ │ +0001dbf0: 6c65 2061 6e6e 6968 696c 6174 6564 2062  le annihilated b
│ │ │ │ +0001dc00: 7920 6964 6561 6c0a 2020 2020 2020 2020  y ideal.        
│ │ │ │ +0001dc10: 660a 2020 2a20 4f75 7470 7574 733a 0a20  f.  * Outputs:. 
│ │ │ │ +0001dc20: 2020 2020 202a 2054 2c20 6120 2a6e 6f74       * T, a *not
│ │ │ │ +0001dc30: 6520 6d6f 6475 6c65 3a20 284d 6163 6175  e module: (Macau
│ │ │ │ +0001dc40: 6c61 7932 446f 6329 4d6f 6475 6c65 2c2c  lay2Doc)Module,,
│ │ │ │ +0001dc50: 2054 6f72 5e53 284d 2c4e 2920 6173 2061   Tor^S(M,N) as a
│ │ │ │ +0001dc60: 204d 6f64 756c 6520 6f76 6572 0a20 2020   Module over.   
│ │ │ │ +0001dc70: 2020 2020 2061 6e20 6578 7465 7269 6f72       an exterior
│ │ │ │ +0001dc80: 2061 6c67 6562 7261 0a0a 4465 7363 7269   algebra..Descri
│ │ │ │ +0001dc90: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ +0001dca0: 3d0a 0a49 6620 4d2c 4e20 6172 6520 532d  =..If M,N are S-
│ │ │ │ +0001dcb0: 6d6f 6475 6c65 7320 616e 6e69 6869 6c61  modules annihila
│ │ │ │ +0001dcc0: 7465 6420 6279 2074 6865 2065 6c65 6d65  ted by the eleme
│ │ │ │ +0001dcd0: 6e74 7320 6f66 2074 6865 206d 6174 7269  nts of the matri
│ │ │ │ +0001dce0: 7820 6666 203d 2028 665f 312e 2e66 5f63  x ff = (f_1..f_c
│ │ │ │ +0001dcf0: 292c 0a61 6e64 206b 2069 7320 7468 6520  ),.and k is the 
│ │ │ │ +0001dd00: 7265 7369 6475 6520 6669 656c 6420 6f66  residue field of
│ │ │ │ +0001dd10: 2053 2c20 7468 656e 2074 6865 2073 6372   S, then the scr
│ │ │ │ +0001dd20: 6970 7420 6578 7465 7269 6f72 546f 724d  ipt exteriorTorM
│ │ │ │ +0001dd30: 6f64 756c 6528 662c 4d29 2072 6574 7572  odule(f,M) retur
│ │ │ │ +0001dd40: 6e73 0a54 6f72 5e53 284d 2c20 6b29 2061  ns.Tor^S(M, k) a
│ │ │ │ +0001dd50: 7320 6120 6d6f 6475 6c65 206f 7665 7220  s a module over 
│ │ │ │ +0001dd60: 616e 2065 7874 6572 696f 7220 616c 6765  an exterior alge
│ │ │ │ +0001dd70: 6272 6120 6b3c 655f 312c 2e2e 2e2c 655f  bra k, where the e_
│ │ │ │ +0001dd90: 690a 6861 7665 2064 6567 7265 6520 312c  i.have degree 1,
│ │ │ │ +0001dda0: 2077 6869 6c65 2065 7874 6572 696f 7254   while exteriorT
│ │ │ │ +0001ddb0: 6f72 4d6f 6475 6c65 2866 2c4d 2c4e 2920  orModule(f,M,N) 
│ │ │ │ +0001ddc0: 7265 7475 726e 7320 546f 725e 5328 4d2c  returns Tor^S(M,
│ │ │ │ +0001ddd0: 4e29 2061 7320 6120 6d6f 6475 6c65 0a6f  N) as a module.o
│ │ │ │ +0001dde0: 7665 7220 6120 6269 6772 6164 6564 2072  ver a bigraded r
│ │ │ │ +0001ddf0: 696e 6720 5345 203d 2053 3c65 5f31 2c2e  ing SE = S, where th
│ │ │ │ +0001de10: 6520 655f 6920 6861 7665 2064 6567 7265  e e_i have degre
│ │ │ │ +0001de20: 6573 207b 645f 692c 317d 2c0a 7768 6572  es {d_i,1},.wher
│ │ │ │ +0001de30: 6520 645f 6920 6973 2074 6865 2064 6567  e d_i is the deg
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│ │ │ │ +0001de50: 6d6f 6475 6c65 2073 7472 7563 7475 7265  module structure
│ │ │ │ +0001de60: 2c20 696e 2065 6974 6865 7220 6361 7365  , in either case
│ │ │ │ +0001de70: 2c20 6973 0a64 6566 696e 6564 2062 7920  , is.defined by 
│ │ │ │ +0001de80: 7468 6520 686f 6d6f 746f 7069 6573 2066  the homotopies f
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│ │ │ │ +0001deb0: 4d2c 2063 6f6d 7075 7465 6420 6279 2074  M, computed by t
│ │ │ │ +0001dec0: 6865 0a73 6372 6970 7420 6d61 6b65 486f  he.script makeHo
│ │ │ │ +0001ded0: 6d6f 746f 7069 6573 312e 0a0a 5468 6520  motopies1...The 
│ │ │ │ +0001dee0: 7363 7269 7074 7320 6361 6c6c 206d 616b  scripts call mak
│ │ │ │ +0001def0: 654d 6f64 756c 6520 746f 2063 6f6d 7075  eModule to compu
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│ │ │ │ +0001df10: 6c29 2070 7265 7365 6e74 6174 696f 6e20  l) presentation 
│ │ │ │ +0001df20: 6f66 2074 6869 730a 6d6f 6475 6c65 2e0a  of this.module..
│ │ │ │ +0001df30: 0a46 726f 6d20 7468 6520 6465 7363 7269  .From the descri
│ │ │ │ +0001df40: 7074 696f 6e20 6279 206d 6174 7269 7820  ption by matrix 
│ │ │ │ +0001df50: 6661 6374 6f72 697a 6174 696f 6e73 2061  factorizations a
│ │ │ │ +0001df60: 6e64 2074 6865 2070 6170 6572 2022 546f  nd the paper "To
│ │ │ │ +0001df70: 7220 6173 2061 206d 6f64 756c 650a 6f76  r as a module.ov
│ │ │ │ +0001df80: 6572 2061 6e20 6578 7465 7269 6f72 2061  er an exterior a
│ │ │ │ +0001df90: 6c67 6562 7261 2220 6f66 2045 6973 656e  lgebra" of Eisen
│ │ │ │ +0001dfa0: 6275 642c 2050 6565 7661 2061 6e64 2053  bud, Peeva and S
│ │ │ │ +0001dfb0: 6368 7265 7965 7220 6974 2066 6f6c 6c6f  chreyer it follo
│ │ │ │ +0001dfc0: 7773 2074 6861 7420 7768 656e 0a4d 2069  ws that when.M i
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│ │ │ │ +0001dfe0: 616e 6420 4620 6973 2069 7473 2072 6573  and F is its res
│ │ │ │ +0001dff0: 6f6c 7574 696f 6e2c 2074 6865 6e20 7468  olution, then th
│ │ │ │ +0001e000: 6520 7072 6573 656e 7461 7469 6f6e 206f  e presentation o
│ │ │ │ +0001e010: 660a 546f 7228 4d2c 535e 312f 6d6d 2920  f.Tor(M,S^1/mm) 
│ │ │ │ +0001e020: 616c 7761 7973 2068 6173 2067 656e 6572  always has gener
│ │ │ │ +0001e030: 6174 6f72 7320 696e 2064 6567 7265 6573  ators in degrees
│ │ │ │ +0001e040: 2030 2c31 2c20 636f 7272 6573 706f 6e64   0,1, correspond
│ │ │ │ +0001e050: 696e 6720 746f 2074 6865 0a74 6172 6765  ing to the.targe
│ │ │ │ +0001e060: 7473 2061 6e64 2073 6f75 7263 6573 206f  ts and sources o
│ │ │ │ +0001e070: 6620 7468 6520 7374 6163 6b20 6f66 206d  f the stack of m
│ │ │ │ +0001e080: 6170 7320 4228 6929 2c20 616e 6420 7468  aps B(i), and th
│ │ │ │ +0001e090: 6174 2074 6865 2072 6573 6f6c 7574 696f  at the resolutio
│ │ │ │ +0001e0a0: 6e20 6973 0a63 6f6d 706f 6e65 6e74 7769  n is.componentwi
│ │ │ │ +0001e0b0: 7365 206c 696e 6561 7220 696e 2061 2073  se linear in a s
│ │ │ │ +0001e0c0: 7569 7461 626c 6520 7365 6e73 652e 2049  uitable sense. I
│ │ │ │ +0001e0d0: 6e20 7468 6520 666f 6c6c 6f77 696e 6720  n the following 
│ │ │ │ +0001e0e0: 6578 616d 706c 652c 2074 6865 7365 2066  example, these f
│ │ │ │ +0001e0f0: 6163 7473 0a61 7265 2076 6572 6966 6965  acts.are verifie
│ │ │ │ +0001e100: 642e 2054 6865 2054 6f72 206d 6f64 756c  d. The Tor modul
│ │ │ │ +0001e110: 6520 646f 6573 204e 4f54 2073 706c 6974  e does NOT split
│ │ │ │ +0001e120: 2069 6e74 6f20 7468 6520 6469 7265 6374   into the direct
│ │ │ │ +0001e130: 2073 756d 206f 6620 7468 650a 7375 626d   sum of the.subm
│ │ │ │ +0001e140: 6f64 756c 6573 2067 656e 6572 6174 6564  odules generated
│ │ │ │ +0001e150: 2069 6e20 6465 6772 6565 7320 3020 616e   in degrees 0 an
│ │ │ │ +0001e160: 6420 312c 2068 6f77 6576 6572 2e0a 0a0a  d 1, however....
│ │ │ │ +0001e170: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  0001e180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e1c0: 2b0a 7c69 3120 3a20 6b6b 203d 205a 5a2f  +.|i1 : kk = ZZ/
│ │ │ │ -0001e1d0: 3130 3120 2020 2020 2020 2020 2020 2020  101             
│ │ │ │ +0001e1b0: 2d2d 2d2d 2d2b 0a7c 6931 203a 206b 6b20  -----+.|i1 : kk 
│ │ │ │ +0001e1c0: 3d20 5a5a 2f31 3031 2020 2020 2020 2020  = ZZ/101        
│ │ │ │ +0001e1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e200: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001e1f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001e200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e240: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0001e250: 3120 3d20 6b6b 2020 2020 2020 2020 2020  1 = kk          
│ │ │ │ +0001e240: 207c 0a7c 6f31 203d 206b 6b20 2020 2020   |.|o1 = kk     
│ │ │ │ +0001e250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e290: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001e280: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001e290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e2d0: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20          |.|o1 : 
│ │ │ │ -0001e2e0: 5175 6f74 6965 6e74 5269 6e67 2020 2020  QuotientRing    
│ │ │ │ +0001e2c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001e2d0: 6f31 203a 2051 756f 7469 656e 7452 696e  o1 : QuotientRin
│ │ │ │ +0001e2e0: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │  0001e2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e310: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001e320: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0001e310: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0001e320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e360: 2d2d 2d2d 2b0a 7c69 3220 3a20 5320 3d20  ----+.|i2 : S = 
│ │ │ │ -0001e370: 6b6b 5b61 2c62 2c63 5d20 2020 2020 2020  kk[a,b,c]       
│ │ │ │ +0001e350: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a  ---------+.|i2 :
│ │ │ │ +0001e360: 2053 203d 206b 6b5b 612c 622c 635d 2020   S = kk[a,b,c]  
│ │ │ │ +0001e370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e3a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001e390: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001e3a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0001e3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e3f0: 7c0a 7c6f 3220 3d20 5320 2020 2020 2020  |.|o2 = S       
│ │ │ │ +0001e3e0: 2020 2020 207c 0a7c 6f32 203d 2053 2020       |.|o2 = S  
│ │ │ │ +0001e3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e430: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001e420: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001e430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e470: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0001e480: 3220 3a20 506f 6c79 6e6f 6d69 616c 5269  2 : PolynomialRi
│ │ │ │ -0001e490: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │ +0001e470: 207c 0a7c 6f32 203a 2050 6f6c 796e 6f6d   |.|o2 : Polynom
│ │ │ │ +0001e480: 6961 6c52 696e 6720 2020 2020 2020 2020  ialRing         
│ │ │ │ +0001e490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e4c0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0001e4b0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0001e4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e500: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │ -0001e510: 6620 3d20 6d61 7472 6978 2261 342c 6234  f = matrix"a4,b4
│ │ │ │ -0001e520: 2c63 3422 2020 2020 2020 2020 2020 2020  ,c4"            
│ │ │ │ +0001e4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +0001e500: 6933 203a 2066 203d 206d 6174 7269 7822  i3 : f = matrix"
│ │ │ │ +0001e510: 6134 2c62 342c 6334 2220 2020 2020 2020  a4,b4,c4"       
│ │ │ │ +0001e520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e540: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001e550: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0001e540: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0001e550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e590: 2020 2020 7c0a 7c6f 3320 3d20 7c20 6134      |.|o3 = | a4
│ │ │ │ -0001e5a0: 2062 3420 6334 207c 2020 2020 2020 2020   b4 c4 |        
│ │ │ │ +0001e580: 2020 2020 2020 2020 207c 0a7c 6f33 203d           |.|o3 =
│ │ │ │ +0001e590: 207c 2061 3420 6234 2063 3420 7c20 2020   | a4 b4 c4 |   
│ │ │ │ +0001e5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e5d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001e5c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001e5d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0001e5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e620: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0001e630: 3120 2020 2020 2033 2020 2020 2020 2020  1      3        
│ │ │ │ +0001e610: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001e620: 2020 2020 2031 2020 2020 2020 3320 2020       1      3   
│ │ │ │ +0001e630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e660: 2020 2020 2020 7c0a 7c6f 3320 3a20 4d61        |.|o3 : Ma
│ │ │ │ -0001e670: 7472 6978 2053 2020 3c2d 2d20 5320 2020  trix S  <-- S   
│ │ │ │ +0001e650: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ +0001e660: 203a 204d 6174 7269 7820 5320 203c 2d2d   : Matrix S  <--
│ │ │ │ +0001e670: 2053 2020 2020 2020 2020 2020 2020 2020   S              
│ │ │ │  0001e680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e6a0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0001e6a0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0001e6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e6f0: 2d2d 2b0a 7c69 3420 3a20 5220 3d20 532f  --+.|i4 : R = S/
│ │ │ │ -0001e700: 6964 6561 6c20 6620 2020 2020 2020 2020  ideal f         
│ │ │ │ +0001e6e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2052  -------+.|i4 : R
│ │ │ │ +0001e6f0: 203d 2053 2f69 6465 616c 2066 2020 2020   = S/ideal f    
│ │ │ │ +0001e700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e730: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001e720: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001e730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e770: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001e780: 7c6f 3420 3d20 5220 2020 2020 2020 2020  |o4 = R         
│ │ │ │ +0001e770: 2020 207c 0a7c 6f34 203d 2052 2020 2020     |.|o4 = R    
│ │ │ │ +0001e780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e7c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001e7b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001e7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e800: 2020 2020 2020 2020 2020 7c0a 7c6f 3420            |.|o4 
│ │ │ │ -0001e810: 3a20 5175 6f74 6965 6e74 5269 6e67 2020  : QuotientRing  
│ │ │ │ +0001e7f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001e800: 0a7c 6f34 203a 2051 756f 7469 656e 7452  .|o4 : QuotientR
│ │ │ │ +0001e810: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │  0001e820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e850: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0001e840: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001e850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e890: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 7020  ------+.|i5 : p 
│ │ │ │ -0001e8a0: 3d20 6d61 7028 522c 5329 2020 2020 2020  = map(R,S)      
│ │ │ │ +0001e880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935  -----------+.|i5
│ │ │ │ +0001e890: 203a 2070 203d 206d 6170 2852 2c53 2920   : p = map(R,S) 
│ │ │ │ +0001e8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e8d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001e8d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001e8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e920: 2020 7c0a 7c6f 3520 3d20 6d61 7020 2852    |.|o5 = map (R
│ │ │ │ -0001e930: 2c20 532c 207b 612c 2062 2c20 637d 2920  , S, {a, b, c}) 
│ │ │ │ +0001e910: 2020 2020 2020 207c 0a7c 6f35 203d 206d         |.|o5 = m
│ │ │ │ +0001e920: 6170 2028 522c 2053 2c20 7b61 2c20 622c  ap (R, S, {a, b,
│ │ │ │ +0001e930: 2063 7d29 2020 2020 2020 2020 2020 2020   c})            
│ │ │ │  0001e940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e960: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001e950: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001e960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e9a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001e9b0: 7c6f 3520 3a20 5269 6e67 4d61 7020 5220  |o5 : RingMap R 
│ │ │ │ -0001e9c0: 3c2d 2d20 5320 2020 2020 2020 2020 2020  <-- S           
│ │ │ │ +0001e9a0: 2020 207c 0a7c 6f35 203a 2052 696e 674d     |.|o5 : RingM
│ │ │ │ +0001e9b0: 6170 2052 203c 2d2d 2053 2020 2020 2020  ap R <-- S      
│ │ │ │ +0001e9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e9f0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0001e9e0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0001e9f0: 2d2d 2d2d 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 2b0a 7c69 3620  ----------+.|i6 
│ │ │ │ -0001ea40: 3a20 4d20 3d20 636f 6b65 7220 6d61 7028  : M = coker map(
│ │ │ │ -0001ea50: 525e 322c 2052 5e7b 333a 2d31 7d2c 207b  R^2, R^{3:-1}, {
│ │ │ │ -0001ea60: 7b61 2c62 2c63 7d2c 7b62 2c63 2c61 7d7d  {a,b,c},{b,c,a}}
│ │ │ │ -0001ea70: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -0001ea80: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0001ea20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0001ea30: 0a7c 6936 203a 204d 203d 2063 6f6b 6572  .|i6 : M = coker
│ │ │ │ +0001ea40: 206d 6170 2852 5e32 2c20 525e 7b33 3a2d   map(R^2, R^{3:-
│ │ │ │ +0001ea50: 317d 2c20 7b7b 612c 622c 637d 2c7b 622c  1}, {{a,b,c},{b,
│ │ │ │ +0001ea60: 632c 617d 7d29 2020 2020 2020 2020 2020  c,a}})          
│ │ │ │ +0001ea70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001ea80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ea90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001eaa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001eab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001eac0: 2020 2020 2020 7c0a 7c6f 3620 3d20 636f        |.|o6 = co
│ │ │ │ -0001ead0: 6b65 726e 656c 207c 2061 2062 2063 207c  kernel | a b c |
│ │ │ │ +0001eab0: 2020 2020 2020 2020 2020 207c 0a7c 6f36             |.|o6
│ │ │ │ +0001eac0: 203d 2063 6f6b 6572 6e65 6c20 7c20 6120   = cokernel | a 
│ │ │ │ +0001ead0: 6220 6320 7c20 2020 2020 2020 2020 2020  b c |           
│ │ │ │  0001eae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001eaf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001eb00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001eb10: 2020 2020 2020 2020 2020 2020 207c 2062               | b
│ │ │ │ -0001eb20: 2063 2061 207c 2020 2020 2020 2020 2020   c a |          
│ │ │ │ +0001eb00: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0001eb10: 2020 7c20 6220 6320 6120 7c20 2020 2020    | b c a |     
│ │ │ │ +0001eb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001eb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001eb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001eb50: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001eb40: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001eb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001eb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001eb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001eb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001eb90: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0001eba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ebb0: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +0001eb80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001eb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001eba0: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +0001ebb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ebc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ebd0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001ebe0: 7c6f 3620 3a20 522d 6d6f 6475 6c65 2c20  |o6 : R-module, 
│ │ │ │ -0001ebf0: 7175 6f74 6965 6e74 206f 6620 5220 2020  quotient of R   
│ │ │ │ +0001ebd0: 2020 207c 0a7c 6f36 203a 2052 2d6d 6f64     |.|o6 : R-mod
│ │ │ │ +0001ebe0: 756c 652c 2071 756f 7469 656e 7420 6f66  ule, quotient of
│ │ │ │ +0001ebf0: 2052 2020 2020 2020 2020 2020 2020 2020   R              
│ │ │ │  0001ec00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ec10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ec20: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0001ec10: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0001ec20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ec30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ec40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ec50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ec60: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720  ----------+.|i7 
│ │ │ │ -0001ec70: 3a20 6265 7474 6920 2846 4620 3d66 7265  : betti (FF =fre
│ │ │ │ -0001ec80: 6552 6573 6f6c 7574 696f 6e28 204d 2c20  eResolution( M, 
│ │ │ │ -0001ec90: 4c65 6e67 7468 4c69 6d69 7420 3d3e 3629  LengthLimit =>6)
│ │ │ │ -0001eca0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -0001ecb0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0001ec50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0001ec60: 0a7c 6937 203a 2062 6574 7469 2028 4646  .|i7 : betti (FF
│ │ │ │ +0001ec70: 203d 6672 6565 5265 736f 6c75 7469 6f6e   =freeResolution
│ │ │ │ +0001ec80: 2820 4d2c 204c 656e 6774 684c 696d 6974  ( M, LengthLimit
│ │ │ │ +0001ec90: 203d 3e36 2929 2020 2020 2020 2020 2020   =>6))          
│ │ │ │ +0001eca0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001ecb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ecc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ecd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ece0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ecf0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001ed00: 2020 2020 2030 2031 2032 2033 2034 2020       0 1 2 3 4  
│ │ │ │ -0001ed10: 3520 2036 2020 2020 2020 2020 2020 2020  5  6            
│ │ │ │ +0001ece0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001ecf0: 2020 2020 2020 2020 2020 3020 3120 3220            0 1 2 
│ │ │ │ +0001ed00: 3320 3420 2035 2020 3620 2020 2020 2020  3 4  5  6       
│ │ │ │ +0001ed10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ed20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ed30: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0001ed40: 3720 3d20 746f 7461 6c3a 2032 2033 2034  7 = total: 2 3 4
│ │ │ │ -0001ed50: 2036 2039 2031 3320 3138 2020 2020 2020   6 9 13 18      
│ │ │ │ +0001ed30: 207c 0a7c 6f37 203d 2074 6f74 616c 3a20   |.|o7 = total: 
│ │ │ │ +0001ed40: 3220 3320 3420 3620 3920 3133 2031 3820  2 3 4 6 9 13 18 
│ │ │ │ +0001ed50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ed60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ed70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ed80: 2020 7c0a 7c20 2020 2020 2020 2020 303a    |.|         0:
│ │ │ │ -0001ed90: 2032 2033 202e 202e 202e 2020 2e20 202e   2 3 . . .  .  .
│ │ │ │ +0001ed70: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001ed80: 2020 2030 3a20 3220 3320 2e20 2e20 2e20     0: 2 3 . . . 
│ │ │ │ +0001ed90: 202e 2020 2e20 2020 2020 2020 2020 2020   .  .           
│ │ │ │  0001eda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001edb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001edc0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0001edd0: 2020 2020 313a 202e 202e 2031 202e 202e      1: . . 1 . .
│ │ │ │ -0001ede0: 2020 2e20 202e 2020 2020 2020 2020 2020    .  .          
│ │ │ │ +0001edb0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001edc0: 2020 2020 2020 2020 2031 3a20 2e20 2e20           1: . . 
│ │ │ │ +0001edd0: 3120 2e20 2e20 202e 2020 2e20 2020 2020  1 . .  .  .     
│ │ │ │ +0001ede0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001edf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ee00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001ee10: 7c20 2020 2020 2020 2020 323a 202e 202e  |         2: . .
│ │ │ │ -0001ee20: 2033 2033 202e 2020 2e20 202e 2020 2020   3 3 .  .  .    
│ │ │ │ +0001ee00: 2020 207c 0a7c 2020 2020 2020 2020 2032     |.|         2
│ │ │ │ +0001ee10: 3a20 2e20 2e20 3320 3320 2e20 202e 2020  : . . 3 3 .  .  
│ │ │ │ +0001ee20: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │  0001ee30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ee40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ee50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001ee60: 333a 202e 202e 202e 2033 2033 2020 2e20  3: . . . 3 3  . 
│ │ │ │ -0001ee70: 202e 2020 2020 2020 2020 2020 2020 2020   .              
│ │ │ │ -0001ee80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ee90: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0001eea0: 2020 2020 2020 343a 202e 202e 202e 202e        4: . . . .
│ │ │ │ -0001eeb0: 2033 2020 3320 202e 2020 2020 2020 2020   3  3  .        
│ │ │ │ +0001ee40: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001ee50: 2020 2020 2033 3a20 2e20 2e20 2e20 3320       3: . . . 3 
│ │ │ │ +0001ee60: 3320 202e 2020 2e20 2020 2020 2020 2020  3  .  .         
│ │ │ │ +0001ee70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ee80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001ee90: 0a7c 2020 2020 2020 2020 2034 3a20 2e20  .|         4: . 
│ │ │ │ +0001eea0: 2e20 2e20 2e20 3320 2033 2020 2e20 2020  . . . 3  3  .   
│ │ │ │ +0001eeb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001eec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001eed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001eee0: 7c0a 7c20 2020 2020 2020 2020 353a 202e  |.|         5: .
│ │ │ │ -0001eef0: 202e 202e 202e 2033 2020 3920 2036 2020   . . . 3  9  6  
│ │ │ │ +0001eed0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001eee0: 2035 3a20 2e20 2e20 2e20 2e20 3320 2039   5: . . . . 3  9
│ │ │ │ +0001eef0: 2020 3620 2020 2020 2020 2020 2020 2020    6             
│ │ │ │  0001ef00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ef10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ef20: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001ef30: 2020 363a 202e 202e 202e 202e 202e 2020    6: . . . . .  
│ │ │ │ -0001ef40: 2e20 2033 2020 2020 2020 2020 2020 2020  .  3            
│ │ │ │ +0001ef10: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001ef20: 2020 2020 2020 2036 3a20 2e20 2e20 2e20         6: . . . 
│ │ │ │ +0001ef30: 2e20 2e20 202e 2020 3320 2020 2020 2020  . .  .  3       
│ │ │ │ +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 7c0a 7c20              |.| 
│ │ │ │ -0001ef70: 2020 2020 2020 2020 373a 202e 202e 202e          7: . . .
│ │ │ │ -0001ef80: 202e 202e 2020 3120 2039 2020 2020 2020   . .  1  9      
│ │ │ │ +0001ef60: 207c 0a7c 2020 2020 2020 2020 2037 3a20   |.|         7: 
│ │ │ │ +0001ef70: 2e20 2e20 2e20 2e20 2e20 2031 2020 3920  . . . . .  1  9 
│ │ │ │ +0001ef80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ef90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001efa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001efb0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001efa0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001efb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001efc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001efd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001efe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001eff0: 2020 2020 2020 2020 7c0a 7c6f 3720 3a20          |.|o7 : 
│ │ │ │ -0001f000: 4265 7474 6954 616c 6c79 2020 2020 2020  BettiTally      
│ │ │ │ +0001efe0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001eff0: 6f37 203a 2042 6574 7469 5461 6c6c 7920  o7 : BettiTally 
│ │ │ │ +0001f000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f030: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001f040: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0001f030: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0001f040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f080: 2d2d 2d2d 2b0a 7c69 3820 3a20 4d53 203d  ----+.|i8 : MS =
│ │ │ │ -0001f090: 2070 7275 6e65 2070 7573 6846 6f72 7761   prune pushForwa
│ │ │ │ -0001f0a0: 7264 2870 2c20 636f 6b65 7220 4646 2e64  rd(p, coker FF.d
│ │ │ │ -0001f0b0: 645f 3629 3b20 2020 2020 2020 2020 2020  d_6);           
│ │ │ │ -0001f0c0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0001f070: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a  ---------+.|i8 :
│ │ │ │ +0001f080: 204d 5320 3d20 7072 756e 6520 7075 7368   MS = prune push
│ │ │ │ +0001f090: 466f 7277 6172 6428 702c 2063 6f6b 6572  Forward(p, coker
│ │ │ │ +0001f0a0: 2046 462e 6464 5f36 293b 2020 2020 2020   FF.dd_6);      
│ │ │ │ +0001f0b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001f0c0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  0001f0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f110: 2b0a 7c69 3920 3a20 5420 3d20 6578 7465  +.|i9 : T = exte
│ │ │ │ -0001f120: 7269 6f72 546f 724d 6f64 756c 6528 662c  riorTorModule(f,
│ │ │ │ -0001f130: 4d53 293b 2020 2020 2020 2020 2020 2020  MS);            
│ │ │ │ -0001f140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f150: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0001f100: 2d2d 2d2d 2d2b 0a7c 6939 203a 2054 203d  -----+.|i9 : T =
│ │ │ │ +0001f110: 2065 7874 6572 696f 7254 6f72 4d6f 6475   exteriorTorModu
│ │ │ │ +0001f120: 6c65 2866 2c4d 5329 3b20 2020 2020 2020  le(f,MS);       
│ │ │ │ +0001f130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f140: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0001f150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0001f1a0: 3130 203a 2062 6574 7469 2054 2020 2020  10 : betti T    
│ │ │ │ +0001f190: 2d2b 0a7c 6931 3020 3a20 6265 7474 6920  -+.|i10 : betti 
│ │ │ │ +0001f1a0: 5420 2020 2020 2020 2020 2020 2020 2020  T               
│ │ │ │  0001f1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f1e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001f1d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001f1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f220: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0001f230: 2020 2020 2020 2020 2030 2020 2031 2020           0   1  
│ │ │ │ +0001f210: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001f220: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ +0001f230: 2020 3120 2020 2020 2020 2020 2020 2020    1             
│ │ │ │  0001f240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f260: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001f270: 7c6f 3130 203d 2074 6f74 616c 3a20 3834  |o10 = total: 84
│ │ │ │ -0001f280: 2032 3532 2020 2020 2020 2020 2020 2020   252            
│ │ │ │ +0001f260: 2020 207c 0a7c 6f31 3020 3d20 746f 7461     |.|o10 = tota
│ │ │ │ +0001f270: 6c3a 2038 3420 3235 3220 2020 2020 2020  l: 84 252       
│ │ │ │ +0001f280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f2b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001f2c0: 2030 3a20 3133 2020 3339 2020 2020 2020   0: 13  39      
│ │ │ │ +0001f2a0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001f2b0: 2020 2020 2020 303a 2031 3320 2033 3920        0: 13  39 
│ │ │ │ +0001f2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f2f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0001f300: 2020 2020 2020 2031 3a20 3333 2020 3939         1: 33  99
│ │ │ │ +0001f2e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001f2f0: 0a7c 2020 2020 2020 2020 2020 313a 2033  .|          1: 3
│ │ │ │ +0001f300: 3320 2039 3920 2020 2020 2020 2020 2020  3  99           
│ │ │ │  0001f310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f340: 7c0a 7c20 2020 2020 2020 2020 2032 3a20  |.|          2: 
│ │ │ │ -0001f350: 3239 2020 3837 2020 2020 2020 2020 2020  29  87          
│ │ │ │ +0001f330: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001f340: 2020 323a 2032 3920 2038 3720 2020 2020    2: 29  87     
│ │ │ │ +0001f350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f380: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001f390: 2020 2033 3a20 2039 2020 3237 2020 2020     3:  9  27    
│ │ │ │ +0001f370: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001f380: 2020 2020 2020 2020 333a 2020 3920 2032          3:  9  2
│ │ │ │ +0001f390: 3720 2020 2020 2020 2020 2020 2020 2020  7               
│ │ │ │  0001f3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f3c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001f3c0: 207c 0a7c 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f410: 2020 7c0a 7c6f 3130 203a 2042 6574 7469    |.|o10 : Betti
│ │ │ │ -0001f420: 5461 6c6c 7920 2020 2020 2020 2020 2020  Tally           
│ │ │ │ +0001f400: 2020 2020 2020 207c 0a7c 6f31 3020 3a20         |.|o10 : 
│ │ │ │ +0001f410: 4265 7474 6954 616c 6c79 2020 2020 2020  BettiTally      
│ │ │ │ +0001f420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f450: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0001f440: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0001f450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0001f4a0: 7c69 3131 203a 2062 6574 7469 2066 7265  |i11 : betti fre
│ │ │ │ -0001f4b0: 6552 6573 6f6c 7574 696f 6e20 2850 5420  eResolution (PT 
│ │ │ │ -0001f4c0: 3d20 7072 756e 6520 542c 204c 656e 6774  = prune T, Lengt
│ │ │ │ -0001f4d0: 684c 696d 6974 203d 3e20 3429 2020 2020  hLimit => 4)    
│ │ │ │ -0001f4e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001f490: 2d2d 2d2b 0a7c 6931 3120 3a20 6265 7474  ---+.|i11 : bett
│ │ │ │ +0001f4a0: 6920 6672 6565 5265 736f 6c75 7469 6f6e  i freeResolution
│ │ │ │ +0001f4b0: 2028 5054 203d 2070 7275 6e65 2054 2c20   (PT = prune T, 
│ │ │ │ +0001f4c0: 4c65 6e67 7468 4c69 6d69 7420 3d3e 2034  LengthLimit => 4
│ │ │ │ +0001f4d0: 2920 2020 2020 2020 207c 0a7c 2020 2020  )        |.|    
│ │ │ │ +0001f4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f520: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0001f530: 2020 2020 2020 2020 2020 2030 2020 3120             0  1 
│ │ │ │ -0001f540: 2032 2020 2033 2020 2034 2020 2020 2020   2   3   4      
│ │ │ │ +0001f510: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001f520: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0001f530: 3020 2031 2020 3220 2020 3320 2020 3420  0  1  2   3   4 
│ │ │ │ +0001f540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f570: 7c0a 7c6f 3131 203d 2074 6f74 616c 3a20  |.|o11 = total: 
│ │ │ │ -0001f580: 3331 2035 3520 3837 2031 3237 2031 3735  31 55 87 127 175
│ │ │ │ +0001f560: 2020 2020 207c 0a7c 6f31 3120 3d20 746f       |.|o11 = to
│ │ │ │ +0001f570: 7461 6c3a 2033 3120 3535 2038 3720 3132  tal: 31 55 87 12
│ │ │ │ +0001f580: 3720 3137 3520 2020 2020 2020 2020 2020  7 175           
│ │ │ │  0001f590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f5b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001f5c0: 2020 2030 3a20 3133 2032 3420 3339 2020     0: 13 24 39  
│ │ │ │ -0001f5d0: 3538 2020 3831 2020 2020 2020 2020 2020  58  81          
│ │ │ │ +0001f5a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001f5b0: 2020 2020 2020 2020 303a 2031 3320 3234          0: 13 24
│ │ │ │ +0001f5c0: 2033 3920 2035 3820 2038 3120 2020 2020   39  58  81     
│ │ │ │ +0001f5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f5f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001f600: 2020 2020 2020 2020 2031 3a20 3138 2033           1: 18 3
│ │ │ │ -0001f610: 3120 3438 2020 3639 2020 3934 2020 2020  1 48  69  94    
│ │ │ │ +0001f5f0: 207c 0a7c 2020 2020 2020 2020 2020 313a   |.|          1:
│ │ │ │ +0001f600: 2031 3820 3331 2034 3820 2036 3920 2039   18 31 48  69  9
│ │ │ │ +0001f610: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │  0001f620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f640: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001f630: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001f640: 2020 2020 2020 2020 2020 2020 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 2020 2020 7c0a 7c6f 3131 203a          |.|o11 :
│ │ │ │ -0001f690: 2042 6574 7469 5461 6c6c 7920 2020 2020   BettiTally     
│ │ │ │ +0001f670: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001f680: 6f31 3120 3a20 4265 7474 6954 616c 6c79  o11 : BettiTally
│ │ │ │ +0001f690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f6c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001f6d0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0001f6c0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0001f6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f710: 2d2d 2d2d 2b0a 7c69 3132 203a 2061 6e6e  ----+.|i12 : ann
│ │ │ │ -0001f720: 2050 5420 2020 2020 2020 2020 2020 2020   PT             
│ │ │ │ +0001f700: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220  ---------+.|i12 
│ │ │ │ +0001f710: 3a20 616e 6e20 5054 2020 2020 2020 2020  : ann PT        
│ │ │ │ +0001f720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f750: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001f740: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001f750: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0001f760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f7a0: 7c0a 7c6f 3132 203d 2069 6465 616c 2865  |.|o12 = ideal(e
│ │ │ │ -0001f7b0: 2065 2065 2029 2020 2020 2020 2020 2020   e e )          
│ │ │ │ +0001f790: 2020 2020 207c 0a7c 6f31 3220 3d20 6964       |.|o12 = id
│ │ │ │ +0001f7a0: 6561 6c28 6520 6520 6520 2920 2020 2020  eal(e e e )     
│ │ │ │ +0001f7b0: 2020 2020 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 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001f7f0: 2020 2020 2020 3020 3120 3220 2020 2020        0 1 2     
│ │ │ │ +0001f7d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001f7e0: 2020 2020 2020 2020 2020 2030 2031 2032             0 1 2
│ │ │ │ +0001f7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f820: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001f820: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001f830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f870: 2020 7c0a 7c6f 3132 203a 2049 6465 616c    |.|o12 : Ideal
│ │ │ │ -0001f880: 206f 6620 6b6b 5b65 202e 2e65 205d 2020   of kk[e ..e ]  
│ │ │ │ +0001f860: 2020 2020 2020 207c 0a7c 6f31 3220 3a20         |.|o12 : 
│ │ │ │ +0001f870: 4964 6561 6c20 6f66 206b 6b5b 6520 2e2e  Ideal of kk[e ..
│ │ │ │ +0001f880: 6520 5d20 2020 2020 2020 2020 2020 2020  e ]             
│ │ │ │  0001f890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f8b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0001f8c0: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -0001f8d0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +0001f8a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001f8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f8c0: 2020 2030 2020 2032 2020 2020 2020 2020     0   2        
│ │ │ │ +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 7c0a                |.
│ │ │ │ -0001f900: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0001f8f0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0001f900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f940: 2d2d 2d2d 2b0a 7c69 3133 203a 2050 5430  ----+.|i13 : PT0
│ │ │ │ -0001f950: 203d 2069 6d61 6765 2028 696e 6475 6365   = image (induce
│ │ │ │ -0001f960: 644d 6170 2850 542c 636f 7665 7220 5054  dMap(PT,cover PT
│ │ │ │ -0001f970: 292a 2028 2863 6f76 6572 2050 5429 5f7b  )* ((cover PT)_{
│ │ │ │ -0001f980: 302e 2e31 327d 2929 3b20 7c0a 2b2d 2d2d  0..12})); |.+---
│ │ │ │ +0001f930: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3320  ---------+.|i13 
│ │ │ │ +0001f940: 3a20 5054 3020 3d20 696d 6167 6520 2869  : PT0 = image (i
│ │ │ │ +0001f950: 6e64 7563 6564 4d61 7028 5054 2c63 6f76  nducedMap(PT,cov
│ │ │ │ +0001f960: 6572 2050 5429 2a20 2828 636f 7665 7220  er PT)* ((cover 
│ │ │ │ +0001f970: 5054 295f 7b30 2e2e 3132 7d29 293b 207c  PT)_{0..12})); |
│ │ │ │ +0001f980: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  0001f990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f9d0: 2b0a 7c69 3134 203a 2050 5431 203d 2069  +.|i14 : PT1 = i
│ │ │ │ -0001f9e0: 6d61 6765 2028 696e 6475 6365 644d 6170  mage (inducedMap
│ │ │ │ -0001f9f0: 2850 542c 636f 7665 7220 5054 292a 2028  (PT,cover PT)* (
│ │ │ │ -0001fa00: 2863 6f76 6572 2050 5429 5f7b 3133 2e2e  (cover PT)_{13..
│ │ │ │ -0001fa10: 3330 7d29 293b 7c0a 2b2d 2d2d 2d2d 2d2d  30}));|.+-------
│ │ │ │ +0001f9c0: 2d2d 2d2d 2d2b 0a7c 6931 3420 3a20 5054  -----+.|i14 : PT
│ │ │ │ +0001f9d0: 3120 3d20 696d 6167 6520 2869 6e64 7563  1 = image (induc
│ │ │ │ +0001f9e0: 6564 4d61 7028 5054 2c63 6f76 6572 2050  edMap(PT,cover P
│ │ │ │ +0001f9f0: 5429 2a20 2828 636f 7665 7220 5054 295f  T)* ((cover PT)_
│ │ │ │ +0001fa00: 7b31 332e 2e33 307d 2929 3b7c 0a2b 2d2d  {13..30}));|.+--
│ │ │ │ +0001fa10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001fa20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001fa30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001fa40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001fa50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0001fa60: 3135 203a 2062 6574 7469 2066 7265 6552  15 : betti freeR
│ │ │ │ -0001fa70: 6573 6f6c 7574 696f 6e28 7072 756e 6520  esolution(prune 
│ │ │ │ -0001fa80: 5054 302c 204c 656e 6774 684c 696d 6974  PT0, LengthLimit
│ │ │ │ -0001fa90: 203d 3e20 3429 2020 2020 2020 2020 2020   => 4)          
│ │ │ │ -0001faa0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001fa50: 2d2b 0a7c 6931 3520 3a20 6265 7474 6920  -+.|i15 : betti 
│ │ │ │ +0001fa60: 6672 6565 5265 736f 6c75 7469 6f6e 2870  freeResolution(p
│ │ │ │ +0001fa70: 7275 6e65 2050 5430 2c20 4c65 6e67 7468  rune PT0, Length
│ │ │ │ +0001fa80: 4c69 6d69 7420 3d3e 2034 2920 2020 2020  Limit => 4)     
│ │ │ │ +0001fa90: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001faa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fae0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0001faf0: 2020 2020 2020 2020 2030 2020 3120 2032           0  1  2
│ │ │ │ -0001fb00: 2020 3320 2034 2020 2020 2020 2020 2020    3  4          
│ │ │ │ +0001fad0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001fae0: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ +0001faf0: 2031 2020 3220 2033 2020 3420 2020 2020   1  2  3  4     
│ │ │ │ +0001fb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fb20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001fb30: 7c6f 3135 203d 2074 6f74 616c 3a20 3133  |o15 = total: 13
│ │ │ │ -0001fb40: 2032 3420 3339 2035 3820 3831 2020 2020   24 39 58 81    
│ │ │ │ +0001fb20: 2020 207c 0a7c 6f31 3520 3d20 746f 7461     |.|o15 = tota
│ │ │ │ +0001fb30: 6c3a 2031 3320 3234 2033 3920 3538 2038  l: 13 24 39 58 8
│ │ │ │ +0001fb40: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │  0001fb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fb70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001fb80: 2030 3a20 3133 2032 3420 3339 2035 3820   0: 13 24 39 58 
│ │ │ │ -0001fb90: 3831 2020 2020 2020 2020 2020 2020 2020  81              
│ │ │ │ -0001fba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fbb0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001fb60: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001fb70: 2020 2020 2020 303a 2031 3320 3234 2033        0: 13 24 3
│ │ │ │ +0001fb80: 3920 3538 2038 3120 2020 2020 2020 2020  9 58 81         
│ │ │ │ +0001fb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fba0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001fbb0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0001fbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fc00: 7c0a 7c6f 3135 203a 2042 6574 7469 5461  |.|o15 : BettiTa
│ │ │ │ -0001fc10: 6c6c 7920 2020 2020 2020 2020 2020 2020  lly             
│ │ │ │ +0001fbf0: 2020 2020 207c 0a7c 6f31 3520 3a20 4265       |.|o15 : Be
│ │ │ │ +0001fc00: 7474 6954 616c 6c79 2020 2020 2020 2020  ttiTally        
│ │ │ │ +0001fc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fc40: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0001fc30: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0001fc40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001fc50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001fc60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001fc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001fc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0001fc90: 3136 203a 2062 6574 7469 2066 7265 6552  16 : betti freeR
│ │ │ │ -0001fca0: 6573 6f6c 7574 696f 6e28 7072 756e 6520  esolution(prune 
│ │ │ │ -0001fcb0: 5054 312c 204c 656e 6774 684c 696d 6974  PT1, LengthLimit
│ │ │ │ -0001fcc0: 203d 3e20 3429 2020 2020 2020 2020 2020   => 4)          
│ │ │ │ -0001fcd0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001fc80: 2d2b 0a7c 6931 3620 3a20 6265 7474 6920  -+.|i16 : betti 
│ │ │ │ +0001fc90: 6672 6565 5265 736f 6c75 7469 6f6e 2870  freeResolution(p
│ │ │ │ +0001fca0: 7275 6e65 2050 5431 2c20 4c65 6e67 7468  rune PT1, Length
│ │ │ │ +0001fcb0: 4c69 6d69 7420 3d3e 2034 2920 2020 2020  Limit => 4)     
│ │ │ │ +0001fcc0: 2020 2020 2020 207c 0a7c 2020 2020 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 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0001fd20: 2020 2020 2020 2020 2030 2020 3120 2032           0  1  2
│ │ │ │ -0001fd30: 2020 3320 2034 2020 2020 2020 2020 2020    3  4          
│ │ │ │ +0001fd00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001fd10: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ +0001fd20: 2031 2020 3220 2033 2020 3420 2020 2020   1  2  3  4     
│ │ │ │ +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 3136 203d 2074 6f74 616c 3a20 3138  |o16 = total: 18
│ │ │ │ -0001fd70: 2032 3820 3339 2035 3120 3634 2020 2020   28 39 51 64    
│ │ │ │ +0001fd50: 2020 207c 0a7c 6f31 3620 3d20 746f 7461     |.|o16 = tota
│ │ │ │ +0001fd60: 6c3a 2031 3820 3238 2033 3920 3531 2036  l: 18 28 39 51 6
│ │ │ │ +0001fd70: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │  0001fd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fda0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001fdb0: 2031 3a20 3138 2032 3820 3339 2035 3120   1: 18 28 39 51 
│ │ │ │ -0001fdc0: 3634 2020 2020 2020 2020 2020 2020 2020  64              
│ │ │ │ -0001fdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fde0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001fd90: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001fda0: 2020 2020 2020 313a 2031 3820 3238 2033        1: 18 28 3
│ │ │ │ +0001fdb0: 3920 3531 2036 3420 2020 2020 2020 2020  9 51 64         
│ │ │ │ +0001fdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fdd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001fde0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0001fdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fe00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fe10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fe20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fe30: 7c0a 7c6f 3136 203a 2042 6574 7469 5461  |.|o16 : BettiTa
│ │ │ │ -0001fe40: 6c6c 7920 2020 2020 2020 2020 2020 2020  lly             
│ │ │ │ +0001fe20: 2020 2020 207c 0a7c 6f31 3620 3a20 4265       |.|o16 : Be
│ │ │ │ +0001fe30: 7474 6954 616c 6c79 2020 2020 2020 2020  ttiTally        
│ │ │ │ +0001fe40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fe50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fe60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fe70: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0001fe60: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0001fe70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001fe80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001fe90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001fea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001feb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0001fec0: 3137 203a 2062 6574 7469 2066 7265 6552  17 : betti freeR
│ │ │ │ -0001fed0: 6573 6f6c 7574 696f 6e28 7072 756e 6520  esolution(prune 
│ │ │ │ -0001fee0: 5054 2c20 4c65 6e67 7468 4c69 6d69 7420  PT, LengthLimit 
│ │ │ │ -0001fef0: 3d3e 2034 2920 2020 2020 2020 2020 2020  => 4)           
│ │ │ │ -0001ff00: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001feb0: 2d2b 0a7c 6931 3720 3a20 6265 7474 6920  -+.|i17 : betti 
│ │ │ │ +0001fec0: 6672 6565 5265 736f 6c75 7469 6f6e 2870  freeResolution(p
│ │ │ │ +0001fed0: 7275 6e65 2050 542c 204c 656e 6774 684c  rune PT, LengthL
│ │ │ │ +0001fee0: 696d 6974 203d 3e20 3429 2020 2020 2020  imit => 4)      
│ │ │ │ +0001fef0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001ff00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ff10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ff20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ff30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ff40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0001ff50: 2020 2020 2020 2020 2030 2020 3120 2032           0  1  2
│ │ │ │ -0001ff60: 2020 2033 2020 2034 2020 2020 2020 2020     3   4        
│ │ │ │ +0001ff30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001ff40: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ +0001ff50: 2031 2020 3220 2020 3320 2020 3420 2020   1  2   3   4   
│ │ │ │ +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 7c0a                |.
│ │ │ │ -0001ff90: 7c6f 3137 203d 2074 6f74 616c 3a20 3331  |o17 = total: 31
│ │ │ │ -0001ffa0: 2035 3520 3837 2031 3237 2031 3735 2020   55 87 127 175  
│ │ │ │ +0001ff80: 2020 207c 0a7c 6f31 3720 3d20 746f 7461     |.|o17 = tota
│ │ │ │ +0001ff90: 6c3a 2033 3120 3535 2038 3720 3132 3720  l: 31 55 87 127 
│ │ │ │ +0001ffa0: 3137 3520 2020 2020 2020 2020 2020 2020  175             
│ │ │ │  0001ffb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ffc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ffd0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001ffe0: 2030 3a20 3133 2032 3420 3339 2020 3538   0: 13 24 39  58
│ │ │ │ -0001fff0: 2020 3831 2020 2020 2020 2020 2020 2020    81            
│ │ │ │ -00020000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020010: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00020020: 2020 2020 2020 2031 3a20 3138 2033 3120         1: 18 31 
│ │ │ │ -00020030: 3438 2020 3639 2020 3934 2020 2020 2020  48  69  94      
│ │ │ │ +0001ffc0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001ffd0: 2020 2020 2020 303a 2031 3320 3234 2033        0: 13 24 3
│ │ │ │ +0001ffe0: 3920 2035 3820 2038 3120 2020 2020 2020  9  58  81       
│ │ │ │ +0001fff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020000: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00020010: 0a7c 2020 2020 2020 2020 2020 313a 2031  .|          1: 1
│ │ │ │ +00020020: 3820 3331 2034 3820 2036 3920 2039 3420  8 31 48  69  94 
│ │ │ │ +00020030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020060: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00020050: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00020060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000200a0: 2020 2020 2020 7c0a 7c6f 3137 203a 2042        |.|o17 : B
│ │ │ │ -000200b0: 6574 7469 5461 6c6c 7920 2020 2020 2020  ettiTally       
│ │ │ │ +00020090: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +000200a0: 3720 3a20 4265 7474 6954 616c 6c79 2020  7 : BettiTally  
│ │ │ │ +000200b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000200c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000200d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000200e0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +000200e0: 207c 0a2b 2d2d 2d2d 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 2d2d  ----------------
│ │ │ │ -00020120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020130: 2d2d 2b0a 0a53 6565 2061 6c73 6f0a 3d3d  --+..See also.==
│ │ │ │ -00020140: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74  ======..  * *not
│ │ │ │ -00020150: 6520 6d61 6b65 4d6f 6475 6c65 3a20 6d61  e makeModule: ma
│ │ │ │ -00020160: 6b65 4d6f 6475 6c65 2c20 2d2d 206d 616b  keModule, -- mak
│ │ │ │ -00020170: 6573 2061 204d 6f64 756c 6520 6f75 7420  es a Module out 
│ │ │ │ -00020180: 6f66 2061 2063 6f6c 6c65 6374 696f 6e20  of a collection 
│ │ │ │ -00020190: 6f66 0a20 2020 206d 6f64 756c 6573 2061  of.    modules a
│ │ │ │ -000201a0: 6e64 206d 6170 730a 0a57 6179 7320 746f  nd maps..Ways to
│ │ │ │ -000201b0: 2075 7365 2065 7874 6572 696f 7254 6f72   use exteriorTor
│ │ │ │ -000201c0: 4d6f 6475 6c65 3a0a 3d3d 3d3d 3d3d 3d3d  Module:.========
│ │ │ │ -000201d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000201e0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2265 7874  ======..  * "ext
│ │ │ │ -000201f0: 6572 696f 7254 6f72 4d6f 6475 6c65 284d  eriorTorModule(M
│ │ │ │ -00020200: 6174 7269 782c 4d6f 6475 6c65 2922 0a20  atrix,Module)". 
│ │ │ │ -00020210: 202a 2022 6578 7465 7269 6f72 546f 724d   * "exteriorTorM
│ │ │ │ -00020220: 6f64 756c 6528 4d61 7472 6978 2c4d 6f64  odule(Matrix,Mod
│ │ │ │ -00020230: 756c 652c 4d6f 6475 6c65 2922 0a0a 466f  ule,Module)"..Fo
│ │ │ │ -00020240: 7220 7468 6520 7072 6f67 7261 6d6d 6572  r the programmer
│ │ │ │ -00020250: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ -00020260: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420  ===..The object 
│ │ │ │ -00020270: 2a6e 6f74 6520 6578 7465 7269 6f72 546f  *note exteriorTo
│ │ │ │ -00020280: 724d 6f64 756c 653a 2065 7874 6572 696f  rModule: exterio
│ │ │ │ -00020290: 7254 6f72 4d6f 6475 6c65 2c20 6973 2061  rTorModule, is a
│ │ │ │ -000202a0: 202a 6e6f 7465 206d 6574 686f 640a 6675   *note method.fu
│ │ │ │ -000202b0: 6e63 7469 6f6e 3a20 284d 6163 6175 6c61  nction: (Macaula
│ │ │ │ -000202c0: 7932 446f 6329 4d65 7468 6f64 4675 6e63  y2Doc)MethodFunc
│ │ │ │ -000202d0: 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d  tion,...--------
│ │ │ │ +00020120: 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 616c  -------+..See al
│ │ │ │ +00020130: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  so.========..  *
│ │ │ │ +00020140: 202a 6e6f 7465 206d 616b 654d 6f64 756c   *note makeModul
│ │ │ │ +00020150: 653a 206d 616b 654d 6f64 756c 652c 202d  e: makeModule, -
│ │ │ │ +00020160: 2d20 6d61 6b65 7320 6120 4d6f 6475 6c65  - makes a Module
│ │ │ │ +00020170: 206f 7574 206f 6620 6120 636f 6c6c 6563   out of a collec
│ │ │ │ +00020180: 7469 6f6e 206f 660a 2020 2020 6d6f 6475  tion of.    modu
│ │ │ │ +00020190: 6c65 7320 616e 6420 6d61 7073 0a0a 5761  les and maps..Wa
│ │ │ │ +000201a0: 7973 2074 6f20 7573 6520 6578 7465 7269  ys to use exteri
│ │ │ │ +000201b0: 6f72 546f 724d 6f64 756c 653a 0a3d 3d3d  orTorModule:.===
│ │ │ │ +000201c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +000201d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ +000201e0: 2022 6578 7465 7269 6f72 546f 724d 6f64   "exteriorTorMod
│ │ │ │ +000201f0: 756c 6528 4d61 7472 6978 2c4d 6f64 756c  ule(Matrix,Modul
│ │ │ │ +00020200: 6529 220a 2020 2a20 2265 7874 6572 696f  e)".  * "exterio
│ │ │ │ +00020210: 7254 6f72 4d6f 6475 6c65 284d 6174 7269  rTorModule(Matri
│ │ │ │ +00020220: 782c 4d6f 6475 6c65 2c4d 6f64 756c 6529  x,Module,Module)
│ │ │ │ +00020230: 220a 0a46 6f72 2074 6865 2070 726f 6772  "..For the progr
│ │ │ │ +00020240: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d  ammer.==========
│ │ │ │ +00020250: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62  ========..The ob
│ │ │ │ +00020260: 6a65 6374 202a 6e6f 7465 2065 7874 6572  ject *note exter
│ │ │ │ +00020270: 696f 7254 6f72 4d6f 6475 6c65 3a20 6578  iorTorModule: ex
│ │ │ │ +00020280: 7465 7269 6f72 546f 724d 6f64 756c 652c  teriorTorModule,
│ │ │ │ +00020290: 2069 7320 6120 2a6e 6f74 6520 6d65 7468   is a *note meth
│ │ │ │ +000202a0: 6f64 0a66 756e 6374 696f 6e3a 2028 4d61  od.function: (Ma
│ │ │ │ +000202b0: 6361 756c 6179 3244 6f63 294d 6574 686f  caulay2Doc)Metho
│ │ │ │ +000202c0: 6446 756e 6374 696f 6e2c 2e0a 0a2d 2d2d  dFunction,...---
│ │ │ │ +000202d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000202e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000202f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020320: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75  -------..The sou
│ │ │ │ -00020330: 7263 6520 6f66 2074 6869 7320 646f 6375  rce of this docu
│ │ │ │ -00020340: 6d65 6e74 2069 7320 696e 0a2f 6275 696c  ment is in./buil
│ │ │ │ -00020350: 642f 7265 7072 6f64 7563 6962 6c65 2d70  d/reproducible-p
│ │ │ │ -00020360: 6174 682f 6d61 6361 756c 6179 322d 312e  ath/macaulay2-1.
│ │ │ │ -00020370: 3235 2e31 312b 6473 2f4d 322f 4d61 6361  25.11+ds/M2/Maca
│ │ │ │ -00020380: 756c 6179 322f 7061 636b 6167 6573 2f0a  ulay2/packages/.
│ │ │ │ -00020390: 436f 6d70 6c65 7465 496e 7465 7273 6563  CompleteIntersec
│ │ │ │ -000203a0: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e  tionResolutions.
│ │ │ │ -000203b0: 6d32 3a34 3138 323a 302e 0a1f 0a46 696c  m2:4182:0....Fil
│ │ │ │ -000203c0: 653a 2043 6f6d 706c 6574 6549 6e74 6572  e: CompleteInter
│ │ │ │ -000203d0: 7365 6374 696f 6e52 6573 6f6c 7574 696f  sectionResolutio
│ │ │ │ -000203e0: 6e73 2e69 6e66 6f2c 204e 6f64 653a 2065  ns.info, Node: e
│ │ │ │ -000203f0: 7874 4973 4f6e 6550 6f6c 796e 6f6d 6961  xtIsOnePolynomia
│ │ │ │ -00020400: 6c2c 204e 6578 743a 2045 7874 4d6f 6475  l, Next: ExtModu
│ │ │ │ -00020410: 6c65 2c20 5072 6576 3a20 6578 7465 7269  le, Prev: exteri
│ │ │ │ -00020420: 6f72 546f 724d 6f64 756c 652c 2055 703a  orTorModule, Up:
│ │ │ │ -00020430: 2054 6f70 0a0a 6578 7449 734f 6e65 506f   Top..extIsOnePo
│ │ │ │ -00020440: 6c79 6e6f 6d69 616c 202d 2d20 6368 6563  lynomial -- chec
│ │ │ │ -00020450: 6b20 7768 6574 6865 7220 7468 6520 4869  k whether the Hi
│ │ │ │ -00020460: 6c62 6572 7420 6675 6e63 7469 6f6e 206f  lbert function o
│ │ │ │ -00020470: 6620 4578 7428 4d2c 6b29 2069 7320 6f6e  f Ext(M,k) is on
│ │ │ │ -00020480: 6520 706f 6c79 6e6f 6d69 616c 0a2a 2a2a  e polynomial.***
│ │ │ │ +00020310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468  ------------..Th
│ │ │ │ +00020320: 6520 736f 7572 6365 206f 6620 7468 6973  e source of this
│ │ │ │ +00020330: 2064 6f63 756d 656e 7420 6973 2069 6e0a   document is in.
│ │ │ │ +00020340: 2f62 7569 6c64 2f72 6570 726f 6475 6369  /build/reproduci
│ │ │ │ +00020350: 626c 652d 7061 7468 2f6d 6163 6175 6c61  ble-path/macaula
│ │ │ │ +00020360: 7932 2d31 2e32 352e 3131 2b64 732f 4d32  y2-1.25.11+ds/M2
│ │ │ │ +00020370: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61  /Macaulay2/packa
│ │ │ │ +00020380: 6765 732f 0a43 6f6d 706c 6574 6549 6e74  ges/.CompleteInt
│ │ │ │ +00020390: 6572 7365 6374 696f 6e52 6573 6f6c 7574  ersectionResolut
│ │ │ │ +000203a0: 696f 6e73 2e6d 323a 3431 3832 3a30 2e0a  ions.m2:4182:0..
│ │ │ │ +000203b0: 1f0a 4669 6c65 3a20 436f 6d70 6c65 7465  ..File: Complete
│ │ │ │ +000203c0: 496e 7465 7273 6563 7469 6f6e 5265 736f  IntersectionReso
│ │ │ │ +000203d0: 6c75 7469 6f6e 732e 696e 666f 2c20 4e6f  lutions.info, No
│ │ │ │ +000203e0: 6465 3a20 6578 7449 734f 6e65 506f 6c79  de: extIsOnePoly
│ │ │ │ +000203f0: 6e6f 6d69 616c 2c20 4e65 7874 3a20 4578  nomial, Next: Ex
│ │ │ │ +00020400: 744d 6f64 756c 652c 2050 7265 763a 2065  tModule, Prev: e
│ │ │ │ +00020410: 7874 6572 696f 7254 6f72 4d6f 6475 6c65  xteriorTorModule
│ │ │ │ +00020420: 2c20 5570 3a20 546f 700a 0a65 7874 4973  , Up: Top..extIs
│ │ │ │ +00020430: 4f6e 6550 6f6c 796e 6f6d 6961 6c20 2d2d  OnePolynomial --
│ │ │ │ +00020440: 2063 6865 636b 2077 6865 7468 6572 2074   check whether t
│ │ │ │ +00020450: 6865 2048 696c 6265 7274 2066 756e 6374  he Hilbert funct
│ │ │ │ +00020460: 696f 6e20 6f66 2045 7874 284d 2c6b 2920  ion of Ext(M,k) 
│ │ │ │ +00020470: 6973 206f 6e65 2070 6f6c 796e 6f6d 6961  is one polynomia
│ │ │ │ +00020480: 6c0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  l.**************
│ │ │ │  00020490: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000204a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000204b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000204c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000204d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000204e0: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20  ***..  * Usage: 
│ │ │ │ -000204f0: 0a20 2020 2020 2020 2028 702c 7429 203d  .        (p,t) =
│ │ │ │ -00020500: 2065 7874 4973 4f6e 6550 6f6c 796e 6f6d   extIsOnePolynom
│ │ │ │ -00020510: 6961 6c20 4d0a 2020 2a20 496e 7075 7473  ial M.  * Inputs
│ │ │ │ -00020520: 3a0a 2020 2020 2020 2a20 4d2c 2061 202a  :.      * M, a *
│ │ │ │ -00020530: 6e6f 7465 206d 6f64 756c 653a 2028 4d61  note module: (Ma
│ │ │ │ -00020540: 6361 756c 6179 3244 6f63 294d 6f64 756c  caulay2Doc)Modul
│ │ │ │ -00020550: 652c 2c20 6d6f 6475 6c65 206f 7665 7220  e,, module over 
│ │ │ │ -00020560: 6120 636f 6d70 6c65 7465 0a20 2020 2020  a complete.     
│ │ │ │ -00020570: 2020 2069 6e74 6572 7365 6374 696f 6e0a     intersection.
│ │ │ │ -00020580: 2020 2a20 4f75 7470 7574 733a 0a20 2020    * Outputs:.   
│ │ │ │ -00020590: 2020 202a 2070 2c20 6120 2a6e 6f74 6520     * p, a *note 
│ │ │ │ -000205a0: 7269 6e67 2065 6c65 6d65 6e74 3a20 284d  ring element: (M
│ │ │ │ -000205b0: 6163 6175 6c61 7932 446f 6329 5269 6e67  acaulay2Doc)Ring
│ │ │ │ -000205c0: 456c 656d 656e 742c 2c20 7028 7a29 3d70  Element,, p(z)=p
│ │ │ │ -000205d0: 6528 7a2f 3229 2c0a 2020 2020 2020 2020  e(z/2),.        
│ │ │ │ -000205e0: 7768 6572 6520 7065 2069 7320 7468 6520  where pe is the 
│ │ │ │ -000205f0: 4869 6c62 6572 7420 706f 6c79 206f 6620  Hilbert poly of 
│ │ │ │ -00020600: 4578 745e 7b65 7665 6e7d 284d 2c6b 290a  Ext^{even}(M,k).
│ │ │ │ -00020610: 2020 2020 2020 2a20 742c 2061 202a 6e6f        * t, a *no
│ │ │ │ -00020620: 7465 2042 6f6f 6c65 616e 2076 616c 7565  te Boolean value
│ │ │ │ -00020630: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ -00020640: 426f 6f6c 6561 6e2c 2c20 7472 7565 2069  Boolean,, true i
│ │ │ │ -00020650: 6620 7468 6520 6576 656e 2061 6e64 0a20  f the even and. 
│ │ │ │ -00020660: 2020 2020 2020 206f 6464 2070 6f6c 796e         odd polyn
│ │ │ │ -00020670: 6f6d 6961 6c73 206d 6174 6368 2074 6f20  omials match to 
│ │ │ │ -00020680: 666f 726d 206f 6e65 2070 6f6c 796e 6f6d  form one polynom
│ │ │ │ -00020690: 6961 6c0a 0a44 6573 6372 6970 7469 6f6e  ial..Description
│ │ │ │ -000206a0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 436f  .===========..Co
│ │ │ │ -000206b0: 6d70 7574 6573 2074 6865 2048 696c 6265  mputes the Hilbe
│ │ │ │ -000206c0: 7274 2070 6f6c 796e 6f6d 6961 6c73 2070  rt polynomials p
│ │ │ │ -000206d0: 6528 7a29 2c20 706f 287a 2920 6f66 2065  e(z), po(z) of e
│ │ │ │ -000206e0: 7665 6e45 7874 4d6f 6475 6c65 2061 6e64  venExtModule and
│ │ │ │ -000206f0: 0a6f 6464 4578 744d 6f64 756c 652e 2049  .oddExtModule. I
│ │ │ │ -00020700: 7420 7265 7475 726e 7320 7065 287a 2f32  t returns pe(z/2
│ │ │ │ -00020710: 292c 2061 6e64 2063 6f6d 7061 7265 7320  ), and compares 
│ │ │ │ -00020720: 746f 2073 6565 2077 6865 7468 6572 2074  to see whether t
│ │ │ │ -00020730: 6869 7320 6973 2065 7175 616c 2074 6f0a  his is equal to.
│ │ │ │ -00020740: 706f 287a 2f32 2d31 2f32 292e 2041 7672  po(z/2-1/2). Avr
│ │ │ │ -00020750: 616d 6f76 2c20 5365 6365 6c65 616e 7520  amov, Seceleanu 
│ │ │ │ -00020760: 616e 6420 5a68 656e 6720 6861 7665 2070  and Zheng have p
│ │ │ │ -00020770: 726f 7665 6e20 7468 6174 2069 6620 7468  roven that if th
│ │ │ │ -00020780: 6520 6964 6561 6c20 6f66 0a71 7561 6472  e ideal of.quadr
│ │ │ │ -00020790: 6174 6963 206c 6561 6469 6e67 2066 6f72  atic leading for
│ │ │ │ -000207a0: 6d73 206f 6620 6120 636f 6d70 6c65 7465  ms of a complete
│ │ │ │ -000207b0: 2069 6e74 6572 7365 6374 696f 6e20 6f66   intersection of
│ │ │ │ -000207c0: 2063 6f64 696d 656e 7369 6f6e 2063 2067   codimension c g
│ │ │ │ -000207d0: 656e 6572 6174 6520 616e 0a69 6465 616c  enerate an.ideal
│ │ │ │ -000207e0: 206f 6620 636f 6469 6d65 6e73 696f 6e20   of codimension 
│ │ │ │ -000207f0: 6174 206c 6561 7374 2063 2d31 2c20 7468  at least c-1, th
│ │ │ │ -00020800: 656e 2074 6865 2042 6574 7469 206e 756d  en the Betti num
│ │ │ │ -00020810: 6265 7273 206f 6620 616e 7920 6d6f 6475  bers of any modu
│ │ │ │ -00020820: 6c65 2067 726f 772c 0a65 7665 6e74 7561  le grow,.eventua
│ │ │ │ -00020830: 6c6c 792c 2061 7320 6120 7369 6e67 6c65  lly, as a single
│ │ │ │ -00020840: 2070 6f6c 796e 6f6d 6961 6c20 2869 6e73   polynomial (ins
│ │ │ │ -00020850: 7465 6164 206f 6620 7265 7175 6972 696e  tead of requirin
│ │ │ │ -00020860: 6720 7365 7061 7261 7465 2070 6f6c 796e  g separate polyn
│ │ │ │ -00020870: 6f6d 6961 6c73 0a66 6f72 2065 7665 6e20  omials.for even 
│ │ │ │ -00020880: 616e 6420 6f64 6420 7465 726d 732e 2920  and odd terms.) 
│ │ │ │ -00020890: 5468 6973 2073 6372 6970 7420 6368 6563  This script chec
│ │ │ │ -000208a0: 6b73 2074 6865 2072 6573 756c 7420 696e  ks the result in
│ │ │ │ -000208b0: 2074 6865 2068 6f6d 6f67 656e 656f 7573   the homogeneous
│ │ │ │ -000208c0: 2063 6173 650a 2869 6e20 7768 6963 6820   case.(in which 
│ │ │ │ -000208d0: 6361 7365 2074 6865 2063 6f6e 6469 7469  case the conditi
│ │ │ │ -000208e0: 6f6e 2069 7320 6e65 6365 7373 6172 7920  on is necessary 
│ │ │ │ -000208f0: 616e 6420 7375 6666 6963 6965 6e74 2e29  and sufficient.)
│ │ │ │ -00020900: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..+-------------
│ │ │ │ +000204d0: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573  ********..  * Us
│ │ │ │ +000204e0: 6167 653a 200a 2020 2020 2020 2020 2870  age: .        (p
│ │ │ │ +000204f0: 2c74 2920 3d20 6578 7449 734f 6e65 506f  ,t) = extIsOnePo
│ │ │ │ +00020500: 6c79 6e6f 6d69 616c 204d 0a20 202a 2049  lynomial M.  * I
│ │ │ │ +00020510: 6e70 7574 733a 0a20 2020 2020 202a 204d  nputs:.      * M
│ │ │ │ +00020520: 2c20 6120 2a6e 6f74 6520 6d6f 6475 6c65  , a *note module
│ │ │ │ +00020530: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +00020540: 4d6f 6475 6c65 2c2c 206d 6f64 756c 6520  Module,, module 
│ │ │ │ +00020550: 6f76 6572 2061 2063 6f6d 706c 6574 650a  over a complete.
│ │ │ │ +00020560: 2020 2020 2020 2020 696e 7465 7273 6563          intersec
│ │ │ │ +00020570: 7469 6f6e 0a20 202a 204f 7574 7075 7473  tion.  * Outputs
│ │ │ │ +00020580: 3a0a 2020 2020 2020 2a20 702c 2061 202a  :.      * p, a *
│ │ │ │ +00020590: 6e6f 7465 2072 696e 6720 656c 656d 656e  note ring elemen
│ │ │ │ +000205a0: 743a 2028 4d61 6361 756c 6179 3244 6f63  t: (Macaulay2Doc
│ │ │ │ +000205b0: 2952 696e 6745 6c65 6d65 6e74 2c2c 2070  )RingElement,, p
│ │ │ │ +000205c0: 287a 293d 7065 287a 2f32 292c 0a20 2020  (z)=pe(z/2),.   
│ │ │ │ +000205d0: 2020 2020 2077 6865 7265 2070 6520 6973       where pe is
│ │ │ │ +000205e0: 2074 6865 2048 696c 6265 7274 2070 6f6c   the Hilbert pol
│ │ │ │ +000205f0: 7920 6f66 2045 7874 5e7b 6576 656e 7d28  y of Ext^{even}(
│ │ │ │ +00020600: 4d2c 6b29 0a20 2020 2020 202a 2074 2c20  M,k).      * t, 
│ │ │ │ +00020610: 6120 2a6e 6f74 6520 426f 6f6c 6561 6e20  a *note Boolean 
│ │ │ │ +00020620: 7661 6c75 653a 2028 4d61 6361 756c 6179  value: (Macaulay
│ │ │ │ +00020630: 3244 6f63 2942 6f6f 6c65 616e 2c2c 2074  2Doc)Boolean,, t
│ │ │ │ +00020640: 7275 6520 6966 2074 6865 2065 7665 6e20  rue if the even 
│ │ │ │ +00020650: 616e 640a 2020 2020 2020 2020 6f64 6420  and.        odd 
│ │ │ │ +00020660: 706f 6c79 6e6f 6d69 616c 7320 6d61 7463  polynomials matc
│ │ │ │ +00020670: 6820 746f 2066 6f72 6d20 6f6e 6520 706f  h to form one po
│ │ │ │ +00020680: 6c79 6e6f 6d69 616c 0a0a 4465 7363 7269  lynomial..Descri
│ │ │ │ +00020690: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ +000206a0: 3d0a 0a43 6f6d 7075 7465 7320 7468 6520  =..Computes the 
│ │ │ │ +000206b0: 4869 6c62 6572 7420 706f 6c79 6e6f 6d69  Hilbert polynomi
│ │ │ │ +000206c0: 616c 7320 7065 287a 292c 2070 6f28 7a29  als pe(z), po(z)
│ │ │ │ +000206d0: 206f 6620 6576 656e 4578 744d 6f64 756c   of evenExtModul
│ │ │ │ +000206e0: 6520 616e 640a 6f64 6445 7874 4d6f 6475  e and.oddExtModu
│ │ │ │ +000206f0: 6c65 2e20 4974 2072 6574 7572 6e73 2070  le. It returns p
│ │ │ │ +00020700: 6528 7a2f 3229 2c20 616e 6420 636f 6d70  e(z/2), and comp
│ │ │ │ +00020710: 6172 6573 2074 6f20 7365 6520 7768 6574  ares to see whet
│ │ │ │ +00020720: 6865 7220 7468 6973 2069 7320 6571 7561  her this is equa
│ │ │ │ +00020730: 6c20 746f 0a70 6f28 7a2f 322d 312f 3229  l to.po(z/2-1/2)
│ │ │ │ +00020740: 2e20 4176 7261 6d6f 762c 2053 6563 656c  . Avramov, Secel
│ │ │ │ +00020750: 6561 6e75 2061 6e64 205a 6865 6e67 2068  eanu and Zheng h
│ │ │ │ +00020760: 6176 6520 7072 6f76 656e 2074 6861 7420  ave proven that 
│ │ │ │ +00020770: 6966 2074 6865 2069 6465 616c 206f 660a  if the ideal of.
│ │ │ │ +00020780: 7175 6164 7261 7469 6320 6c65 6164 696e  quadratic leadin
│ │ │ │ +00020790: 6720 666f 726d 7320 6f66 2061 2063 6f6d  g forms of a com
│ │ │ │ +000207a0: 706c 6574 6520 696e 7465 7273 6563 7469  plete intersecti
│ │ │ │ +000207b0: 6f6e 206f 6620 636f 6469 6d65 6e73 696f  on of codimensio
│ │ │ │ +000207c0: 6e20 6320 6765 6e65 7261 7465 2061 6e0a  n c generate an.
│ │ │ │ +000207d0: 6964 6561 6c20 6f66 2063 6f64 696d 656e  ideal of codimen
│ │ │ │ +000207e0: 7369 6f6e 2061 7420 6c65 6173 7420 632d  sion at least c-
│ │ │ │ +000207f0: 312c 2074 6865 6e20 7468 6520 4265 7474  1, then the Bett
│ │ │ │ +00020800: 6920 6e75 6d62 6572 7320 6f66 2061 6e79  i numbers of any
│ │ │ │ +00020810: 206d 6f64 756c 6520 6772 6f77 2c0a 6576   module grow,.ev
│ │ │ │ +00020820: 656e 7475 616c 6c79 2c20 6173 2061 2073  entually, as a s
│ │ │ │ +00020830: 696e 676c 6520 706f 6c79 6e6f 6d69 616c  ingle polynomial
│ │ │ │ +00020840: 2028 696e 7374 6561 6420 6f66 2072 6571   (instead of req
│ │ │ │ +00020850: 7569 7269 6e67 2073 6570 6172 6174 6520  uiring separate 
│ │ │ │ +00020860: 706f 6c79 6e6f 6d69 616c 730a 666f 7220  polynomials.for 
│ │ │ │ +00020870: 6576 656e 2061 6e64 206f 6464 2074 6572  even and odd ter
│ │ │ │ +00020880: 6d73 2e29 2054 6869 7320 7363 7269 7074  ms.) This script
│ │ │ │ +00020890: 2063 6865 636b 7320 7468 6520 7265 7375   checks the resu
│ │ │ │ +000208a0: 6c74 2069 6e20 7468 6520 686f 6d6f 6765  lt in the homoge
│ │ │ │ +000208b0: 6e65 6f75 7320 6361 7365 0a28 696e 2077  neous case.(in w
│ │ │ │ +000208c0: 6869 6368 2063 6173 6520 7468 6520 636f  hich case the co
│ │ │ │ +000208d0: 6e64 6974 696f 6e20 6973 206e 6563 6573  ndition is neces
│ │ │ │ +000208e0: 7361 7279 2061 6e64 2073 7566 6669 6369  sary and suffici
│ │ │ │ +000208f0: 656e 742e 290a 0a2b 2d2d 2d2d 2d2d 2d2d  ent.)..+--------
│ │ │ │ +00020900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -00020940: 203a 2052 313d 5a5a 2f31 3031 5b61 2c62   : R1=ZZ/101[a,b
│ │ │ │ -00020950: 2c63 5d2f 6964 6561 6c28 615e 322c 625e  ,c]/ideal(a^2,b^
│ │ │ │ -00020960: 322c 635e 3529 2020 2020 2020 2020 2020  2,c^5)          
│ │ │ │ -00020970: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00020930: 2b0a 7c69 3120 3a20 5231 3d5a 5a2f 3130  +.|i1 : R1=ZZ/10
│ │ │ │ +00020940: 315b 612c 622c 635d 2f69 6465 616c 2861  1[a,b,c]/ideal(a
│ │ │ │ +00020950: 5e32 2c62 5e32 2c63 5e35 2920 2020 2020  ^2,b^2,c^5)     
│ │ │ │ +00020960: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00020970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000209a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000209b0: 207c 0a7c 6f31 203d 2052 3120 2020 2020   |.|o1 = R1     
│ │ │ │ +000209a0: 2020 2020 2020 7c0a 7c6f 3120 3d20 5231        |.|o1 = R1
│ │ │ │ +000209b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000209c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000209d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000209e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000209e0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  000209f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020a20: 2020 2020 2020 207c 0a7c 6f31 203a 2051         |.|o1 : Q
│ │ │ │ -00020a30: 756f 7469 656e 7452 696e 6720 2020 2020  uotientRing     
│ │ │ │ +00020a10: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00020a20: 3120 3a20 5175 6f74 6965 6e74 5269 6e67  1 : QuotientRing
│ │ │ │ +00020a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020a60: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00020a50: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00020a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00020aa0: 6932 203a 2052 323d 5a5a 2f31 3031 5b61  i2 : R2=ZZ/101[a
│ │ │ │ -00020ab0: 2c62 2c63 5d2f 6964 6561 6c28 615e 332c  ,b,c]/ideal(a^3,
│ │ │ │ -00020ac0: 625e 3329 2020 2020 2020 2020 2020 2020  b^3)            
│ │ │ │ -00020ad0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00020a90: 2d2d 2b0a 7c69 3220 3a20 5232 3d5a 5a2f  --+.|i2 : R2=ZZ/
│ │ │ │ +00020aa0: 3130 315b 612c 622c 635d 2f69 6465 616c  101[a,b,c]/ideal
│ │ │ │ +00020ab0: 2861 5e33 2c62 5e33 2920 2020 2020 2020  (a^3,b^3)       
│ │ │ │ +00020ac0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00020ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020b10: 2020 207c 0a7c 6f32 203d 2052 3220 2020     |.|o2 = R2   
│ │ │ │ +00020b00: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20          |.|o2 = 
│ │ │ │ +00020b10: 5232 2020 2020 2020 2020 2020 2020 2020  R2              
│ │ │ │  00020b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020b40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00020b50: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00020b40: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00020b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020b80: 2020 2020 2020 2020 207c 0a7c 6f32 203a           |.|o2 :
│ │ │ │ -00020b90: 2051 756f 7469 656e 7452 696e 6720 2020   QuotientRing   
│ │ │ │ +00020b70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00020b80: 7c6f 3220 3a20 5175 6f74 6965 6e74 5269  |o2 : QuotientRi
│ │ │ │ +00020b90: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │  00020ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020bc0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00020bb0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00020bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00020c00: 0a7c 6933 203a 2065 7874 4973 4f6e 6550  .|i3 : extIsOneP
│ │ │ │ -00020c10: 6f6c 796e 6f6d 6961 6c20 636f 6b65 7220  olynomial coker 
│ │ │ │ -00020c20: 7261 6e64 6f6d 2852 315e 7b30 2c31 7d2c  random(R1^{0,1},
│ │ │ │ -00020c30: 5231 5e7b 333a 2d31 7d29 7c0a 7c20 2020  R1^{3:-1})|.|   
│ │ │ │ +00020bf0: 2d2d 2d2d 2b0a 7c69 3320 3a20 6578 7449  ----+.|i3 : extI
│ │ │ │ +00020c00: 734f 6e65 506f 6c79 6e6f 6d69 616c 2063  sOnePolynomial c
│ │ │ │ +00020c10: 6f6b 6572 2072 616e 646f 6d28 5231 5e7b  oker random(R1^{
│ │ │ │ +00020c20: 302c 317d 2c52 315e 7b33 3a2d 317d 297c  0,1},R1^{3:-1})|
│ │ │ │ +00020c30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00020c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020c70: 2020 2020 207c 0a7c 2020 2020 2020 3120       |.|      1 
│ │ │ │ -00020c80: 3220 2020 3120 2020 2020 2020 2020 2020  2   1           
│ │ │ │ +00020c60: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00020c70: 2020 2031 2032 2020 2031 2020 2020 2020     1 2   1      
│ │ │ │ +00020c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020cb0: 7c0a 7c6f 3320 3d20 282d 7a20 202d 202d  |.|o3 = (-z  - -
│ │ │ │ -00020cc0: 7a20 2b20 332c 2074 7275 6529 2020 2020  z + 3, true)    
│ │ │ │ +00020ca0: 2020 2020 207c 0a7c 6f33 203d 2028 2d7a       |.|o3 = (-z
│ │ │ │ +00020cb0: 2020 2d20 2d7a 202b 2033 2c20 7472 7565    - -z + 3, true
│ │ │ │ +00020cc0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  00020cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020ce0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00020cf0: 2020 2020 3220 2020 2020 3220 2020 2020      2     2     
│ │ │ │ +00020ce0: 7c0a 7c20 2020 2020 2032 2020 2020 2032  |.|      2     2
│ │ │ │ +00020cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020d20: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00020d10: 2020 2020 2020 2020 2020 207c 0a7c 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020d60: 207c 0a7c 6f33 203a 2053 6571 7565 6e63   |.|o3 : Sequenc
│ │ │ │ -00020d70: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │ +00020d50: 2020 2020 2020 7c0a 7c6f 3320 3a20 5365        |.|o3 : Se
│ │ │ │ +00020d60: 7175 656e 6365 2020 2020 2020 2020 2020  quence          
│ │ │ │ +00020d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020d90: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00020d90: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  00020da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020dd0: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2065  -------+.|i4 : e
│ │ │ │ -00020de0: 7874 4973 4f6e 6550 6f6c 796e 6f6d 6961  xtIsOnePolynomia
│ │ │ │ -00020df0: 6c20 636f 6b65 7220 7261 6e64 6f6d 2852  l coker random(R
│ │ │ │ -00020e00: 325e 7b30 2c31 7d2c 5232 5e7b 333a 2d31  2^{0,1},R2^{3:-1
│ │ │ │ -00020e10: 7d29 7c0a 7c20 2020 2020 2020 2020 2020  })|.|           
│ │ │ │ +00020dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00020dd0: 3420 3a20 6578 7449 734f 6e65 506f 6c79  4 : extIsOnePoly
│ │ │ │ +00020de0: 6e6f 6d69 616c 2063 6f6b 6572 2072 616e  nomial coker ran
│ │ │ │ +00020df0: 646f 6d28 5232 5e7b 302c 317d 2c52 325e  dom(R2^{0,1},R2^
│ │ │ │ +00020e00: 7b33 3a2d 317d 297c 0a7c 2020 2020 2020  {3:-1})|.|      
│ │ │ │ +00020e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020e40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00020e50: 6f34 203d 2028 337a 202d 2032 2c20 6661  o4 = (3z - 2, fa
│ │ │ │ -00020e60: 6c73 6529 2020 2020 2020 2020 2020 2020  lse)            
│ │ │ │ -00020e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020e80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00020e40: 2020 7c0a 7c6f 3420 3d20 2833 7a20 2d20    |.|o4 = (3z - 
│ │ │ │ +00020e50: 322c 2066 616c 7365 2920 2020 2020 2020  2, false)       
│ │ │ │ +00020e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020e70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00020e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020ec0: 2020 207c 0a7c 6f34 203a 2053 6571 7565     |.|o4 : Seque
│ │ │ │ -00020ed0: 6e63 6520 2020 2020 2020 2020 2020 2020  nce             
│ │ │ │ +00020eb0: 2020 2020 2020 2020 7c0a 7c6f 3420 3a20          |.|o4 : 
│ │ │ │ +00020ec0: 5365 7175 656e 6365 2020 2020 2020 2020  Sequence        
│ │ │ │ +00020ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020ef0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00020f00: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00020ef0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00020f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020f30: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520  ---------+..See 
│ │ │ │ -00020f40: 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20  also.========.. 
│ │ │ │ -00020f50: 202a 202a 6e6f 7465 2065 7665 6e45 7874   * *note evenExt
│ │ │ │ -00020f60: 4d6f 6475 6c65 3a20 6576 656e 4578 744d  Module: evenExtM
│ │ │ │ -00020f70: 6f64 756c 652c 202d 2d20 6576 656e 2070  odule, -- even p
│ │ │ │ -00020f80: 6172 7420 6f66 2045 7874 5e2a 284d 2c6b  art of Ext^*(M,k
│ │ │ │ -00020f90: 2920 6f76 6572 2061 0a20 2020 2063 6f6d  ) over a.    com
│ │ │ │ -00020fa0: 706c 6574 6520 696e 7465 7273 6563 7469  plete intersecti
│ │ │ │ -00020fb0: 6f6e 2061 7320 6d6f 6475 6c65 206f 7665  on as module ove
│ │ │ │ -00020fc0: 7220 4349 206f 7065 7261 746f 7220 7269  r CI operator ri
│ │ │ │ -00020fd0: 6e67 0a20 202a 202a 6e6f 7465 206f 6464  ng.  * *note odd
│ │ │ │ -00020fe0: 4578 744d 6f64 756c 653a 206f 6464 4578  ExtModule: oddEx
│ │ │ │ -00020ff0: 744d 6f64 756c 652c 202d 2d20 6f64 6420  tModule, -- odd 
│ │ │ │ -00021000: 7061 7274 206f 6620 4578 745e 2a28 4d2c  part of Ext^*(M,
│ │ │ │ -00021010: 6b29 206f 7665 7220 6120 636f 6d70 6c65  k) over a comple
│ │ │ │ -00021020: 7465 0a20 2020 2069 6e74 6572 7365 6374  te.    intersect
│ │ │ │ -00021030: 696f 6e20 6173 206d 6f64 756c 6520 6f76  ion as module ov
│ │ │ │ -00021040: 6572 2043 4920 6f70 6572 6174 6f72 2072  er CI operator r
│ │ │ │ -00021050: 696e 670a 0a57 6179 7320 746f 2075 7365  ing..Ways to use
│ │ │ │ -00021060: 2065 7874 4973 4f6e 6550 6f6c 796e 6f6d   extIsOnePolynom
│ │ │ │ -00021070: 6961 6c3a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  ial:.===========
│ │ │ │ -00021080: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00021090: 3d3d 3d3d 0a0a 2020 2a20 2265 7874 4973  ====..  * "extIs
│ │ │ │ -000210a0: 4f6e 6550 6f6c 796e 6f6d 6961 6c28 4d6f  OnePolynomial(Mo
│ │ │ │ -000210b0: 6475 6c65 2922 0a0a 466f 7220 7468 6520  dule)"..For the 
│ │ │ │ -000210c0: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d  programmer.=====
│ │ │ │ -000210d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  =============..T
│ │ │ │ -000210e0: 6865 206f 626a 6563 7420 2a6e 6f74 6520  he object *note 
│ │ │ │ -000210f0: 6578 7449 734f 6e65 506f 6c79 6e6f 6d69  extIsOnePolynomi
│ │ │ │ -00021100: 616c 3a20 6578 7449 734f 6e65 506f 6c79  al: extIsOnePoly
│ │ │ │ -00021110: 6e6f 6d69 616c 2c20 6973 2061 202a 6e6f  nomial, is a *no
│ │ │ │ -00021120: 7465 206d 6574 686f 640a 6675 6e63 7469  te method.functi
│ │ │ │ -00021130: 6f6e 3a20 284d 6163 6175 6c61 7932 446f  on: (Macaulay2Do
│ │ │ │ -00021140: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e  c)MethodFunction
│ │ │ │ -00021150: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ,...------------
│ │ │ │ +00020f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00020f30: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d  .See also.======
│ │ │ │ +00020f40: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6576  ==..  * *note ev
│ │ │ │ +00020f50: 656e 4578 744d 6f64 756c 653a 2065 7665  enExtModule: eve
│ │ │ │ +00020f60: 6e45 7874 4d6f 6475 6c65 2c20 2d2d 2065  nExtModule, -- e
│ │ │ │ +00020f70: 7665 6e20 7061 7274 206f 6620 4578 745e  ven part of Ext^
│ │ │ │ +00020f80: 2a28 4d2c 6b29 206f 7665 7220 610a 2020  *(M,k) over a.  
│ │ │ │ +00020f90: 2020 636f 6d70 6c65 7465 2069 6e74 6572    complete inter
│ │ │ │ +00020fa0: 7365 6374 696f 6e20 6173 206d 6f64 756c  section as modul
│ │ │ │ +00020fb0: 6520 6f76 6572 2043 4920 6f70 6572 6174  e over CI operat
│ │ │ │ +00020fc0: 6f72 2072 696e 670a 2020 2a20 2a6e 6f74  or ring.  * *not
│ │ │ │ +00020fd0: 6520 6f64 6445 7874 4d6f 6475 6c65 3a20  e oddExtModule: 
│ │ │ │ +00020fe0: 6f64 6445 7874 4d6f 6475 6c65 2c20 2d2d  oddExtModule, --
│ │ │ │ +00020ff0: 206f 6464 2070 6172 7420 6f66 2045 7874   odd part of Ext
│ │ │ │ +00021000: 5e2a 284d 2c6b 2920 6f76 6572 2061 2063  ^*(M,k) over a c
│ │ │ │ +00021010: 6f6d 706c 6574 650a 2020 2020 696e 7465  omplete.    inte
│ │ │ │ +00021020: 7273 6563 7469 6f6e 2061 7320 6d6f 6475  rsection as modu
│ │ │ │ +00021030: 6c65 206f 7665 7220 4349 206f 7065 7261  le over CI opera
│ │ │ │ +00021040: 746f 7220 7269 6e67 0a0a 5761 7973 2074  tor ring..Ways t
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│ │ │ │ +00021300: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00021310: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00021320: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00021330: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00021340: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00021350: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 7361  *******..  * Usa
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│ │ │ │ +00021360: 2020 4520 3d20 4578 744d 6f64 756c 6520    E = ExtModule 
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│ │ │ │ +00021380: 2020 2020 2a20 4d2c 2061 202a 6e6f 7465      * M, a *note
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│ │ │ │ +000213c0: 696e 7465 7273 6563 7469 6f6e 0a20 2020  intersection.   
│ │ │ │ +000213d0: 2020 2020 2072 696e 670a 2020 2a20 4f75       ring.  * Ou
│ │ │ │ +000213e0: 7470 7574 733a 0a20 2020 2020 202a 2045  tputs:.      * E
│ │ │ │ +000213f0: 2c20 6120 2a6e 6f74 6520 6d6f 6475 6c65  , a *note module
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│ │ │ │ +00021410: 4d6f 6475 6c65 2c2c 206f 7665 7220 6120  Module,, over a 
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│ │ │ │ +000214b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000214c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000214d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000214e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -000214f0: 203a 206b 6b3d 205a 5a2f 3130 3120 2020   : kk= ZZ/101   
│ │ │ │ +000214e0: 2b0a 7c69 3120 3a20 6b6b 3d20 5a5a 2f31  +.|i1 : kk= ZZ/1
│ │ │ │ +000214f0: 3031 2020 2020 2020 2020 2020 2020 2020  01              
│ │ │ │  00021500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021520: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00021510: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00021520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021560: 2020 2020 207c 0a7c 6f31 203d 206b 6b20       |.|o1 = kk 
│ │ │ │ +00021550: 2020 2020 2020 2020 2020 7c0a 7c6f 3120            |.|o1 
│ │ │ │ +00021560: 3d20 6b6b 2020 2020 2020 2020 2020 2020  = kk            
│ │ │ │  00021570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000215a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00021590: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000215a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000215b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000215c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000215d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000215e0: 0a7c 6f31 203a 2051 756f 7469 656e 7452  .|o1 : QuotientR
│ │ │ │ -000215f0: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │ +000215d0: 2020 2020 7c0a 7c6f 3120 3a20 5175 6f74      |.|o1 : Quot
│ │ │ │ +000215e0: 6965 6e74 5269 6e67 2020 2020 2020 2020  ientRing        
│ │ │ │ +000215f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021610: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00021610: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  00021620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021650: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a  ---------+.|i2 :
│ │ │ │ -00021660: 2053 203d 206b 6b5b 782c 792c 7a5d 2020   S = kk[x,y,z]  
│ │ │ │ +00021640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00021650: 7c69 3220 3a20 5320 3d20 6b6b 5b78 2c79  |i2 : S = kk[x,y
│ │ │ │ +00021660: 2c7a 5d20 2020 2020 2020 2020 2020 2020  ,z]             
│ │ │ │  00021670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021690: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00021680: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00021690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000216a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000216b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000216c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000216d0: 2020 207c 0a7c 6f32 203d 2053 2020 2020     |.|o2 = S    
│ │ │ │ +000216c0: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20          |.|o2 = 
│ │ │ │ +000216d0: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │  000216e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000216f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021710: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00021700: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +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 207c 0a7c               |.|
│ │ │ │ -00021750: 6f32 203a 2050 6f6c 796e 6f6d 6961 6c52  o2 : PolynomialR
│ │ │ │ -00021760: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │ -00021770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021780: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00021740: 2020 7c0a 7c6f 3220 3a20 506f 6c79 6e6f    |.|o2 : Polyno
│ │ │ │ +00021750: 6d69 616c 5269 6e67 2020 2020 2020 2020  mialRing        
│ │ │ │ +00021760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021770: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00021780: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00021790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000217a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000217b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000217c0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2049  -------+.|i3 : I
│ │ │ │ -000217d0: 3120 3d20 6964 6561 6c20 2278 3379 2220  1 = ideal "x3y" 
│ │ │ │ +000217b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +000217c0: 3320 3a20 4931 203d 2069 6465 616c 2022  3 : I1 = ideal "
│ │ │ │ +000217d0: 7833 7922 2020 2020 2020 2020 2020 2020  x3y"            
│ │ │ │  000217e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000217f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021800: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000217f0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00021800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021840: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00021850: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ +00021830: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00021840: 2020 2020 2033 2020 2020 2020 2020 2020       3          
│ │ │ │ +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 7c0a                |.
│ │ │ │ -00021880: 7c6f 3320 3d20 6964 6561 6c28 7820 7929  |o3 = ideal(x y)
│ │ │ │ +00021870: 2020 207c 0a7c 6f33 203d 2069 6465 616c     |.|o3 = ideal
│ │ │ │ +00021880: 2878 2079 2920 2020 2020 2020 2020 2020  (x y)           
│ │ │ │  00021890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000218a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000218b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000218b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000218c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000218d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000218e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000218f0: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20          |.|o3 : 
│ │ │ │ -00021900: 4964 6561 6c20 6f66 2053 2020 2020 2020  Ideal of S      
│ │ │ │ +000218e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000218f0: 6f33 203a 2049 6465 616c 206f 6620 5320  o3 : Ideal of S 
│ │ │ │ +00021900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021930: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00021920: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00021930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021970: 2d2d 2b0a 7c69 3420 3a20 5231 203d 2053  --+.|i4 : R1 = S
│ │ │ │ -00021980: 2f49 3120 2020 2020 2020 2020 2020 2020  /I1             
│ │ │ │ +00021960: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2052  -------+.|i4 : R
│ │ │ │ +00021970: 3120 3d20 532f 4931 2020 2020 2020 2020  1 = S/I1        
│ │ │ │ +00021980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000219a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000219b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000219a0: 2020 2020 7c0a 7c20 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000219e0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -000219f0: 3420 3d20 5231 2020 2020 2020 2020 2020  4 = R1          
│ │ │ │ +000219e0: 207c 0a7c 6f34 203d 2052 3120 2020 2020   |.|o4 = R1     
│ │ │ │ +000219f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021a20: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00021a10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00021a20: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00021a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021a60: 2020 2020 2020 7c0a 7c6f 3420 3a20 5175        |.|o4 : Qu
│ │ │ │ -00021a70: 6f74 6965 6e74 5269 6e67 2020 2020 2020  otientRing      
│ │ │ │ +00021a50: 2020 2020 2020 2020 2020 207c 0a7c 6f34             |.|o4
│ │ │ │ +00021a60: 203a 2051 756f 7469 656e 7452 696e 6720   : QuotientRing 
│ │ │ │ +00021a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021aa0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00021a90: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00021aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021ae0: 2b0a 7c69 3520 3a20 4d31 203d 2052 315e  +.|i5 : M1 = R1^
│ │ │ │ -00021af0: 312f 6964 6561 6c28 785e 3229 2020 2020  1/ideal(x^2)    
│ │ │ │ +00021ad0: 2d2d 2d2d 2d2b 0a7c 6935 203a 204d 3120  -----+.|i5 : M1 
│ │ │ │ +00021ae0: 3d20 5231 5e31 2f69 6465 616c 2878 5e32  = R1^1/ideal(x^2
│ │ │ │ +00021af0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  00021b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021b10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00021b10: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00021b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021b50: 2020 2020 2020 2020 2020 7c0a 7c6f 3520            |.|o5 
│ │ │ │ -00021b60: 3d20 636f 6b65 726e 656c 207c 2078 3220  = cokernel | x2 
│ │ │ │ -00021b70: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00021b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021b90: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00021b40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00021b50: 0a7c 6f35 203d 2063 6f6b 6572 6e65 6c20  .|o5 = cokernel 
│ │ │ │ +00021b60: 7c20 7832 207c 2020 2020 2020 2020 2020  | x2 |          
│ │ │ │ +00021b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021b80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00021b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021bd0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00021be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021bf0: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ -00021c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021c10: 207c 0a7c 6f35 203a 2052 312d 6d6f 6475   |.|o5 : R1-modu
│ │ │ │ -00021c20: 6c65 2c20 7175 6f74 6965 6e74 206f 6620  le, quotient of 
│ │ │ │ -00021c30: 5231 2020 2020 2020 2020 2020 2020 2020  R1              
│ │ │ │ -00021c40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00021c50: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00021bc0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00021bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021be0: 2020 2020 2020 2020 2020 3120 2020 2020            1     
│ │ │ │ +00021bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021c00: 2020 2020 2020 7c0a 7c6f 3520 3a20 5231        |.|o5 : R1
│ │ │ │ +00021c10: 2d6d 6f64 756c 652c 2071 756f 7469 656e  -module, quotien
│ │ │ │ +00021c20: 7420 6f66 2052 3120 2020 2020 2020 2020  t of R1         
│ │ │ │ +00021c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021c40: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00021c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936  -----------+.|i6
│ │ │ │ -00021c90: 203a 2062 6574 7469 2066 7265 6552 6573   : betti freeRes
│ │ │ │ -00021ca0: 6f6c 7574 696f 6e20 284d 312c 204c 656e  olution (M1, Len
│ │ │ │ -00021cb0: 6774 684c 696d 6974 203d 3e35 2920 2020  gthLimit =>5)   
│ │ │ │ -00021cc0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00021c80: 2b0a 7c69 3620 3a20 6265 7474 6920 6672  +.|i6 : betti fr
│ │ │ │ +00021c90: 6565 5265 736f 6c75 7469 6f6e 2028 4d31  eeResolution (M1
│ │ │ │ +00021ca0: 2c20 4c65 6e67 7468 4c69 6d69 7420 3d3e  , LengthLimit =>
│ │ │ │ +00021cb0: 3529 2020 2020 2020 2020 2020 207c 0a7c  5)           |.|
│ │ │ │ +00021cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021d00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00021d10: 2020 2020 3020 3120 3220 3320 3420 3520      0 1 2 3 4 5 
│ │ │ │ +00021cf0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00021d00: 2020 2020 2020 2020 2030 2031 2032 2033           0 1 2 3
│ │ │ │ +00021d10: 2034 2035 2020 2020 2020 2020 2020 2020   4 5            
│ │ │ │  00021d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021d40: 2020 7c0a 7c6f 3620 3d20 746f 7461 6c3a    |.|o6 = total:
│ │ │ │ -00021d50: 2031 2031 2031 2031 2031 2031 2020 2020   1 1 1 1 1 1    
│ │ │ │ +00021d30: 2020 2020 2020 207c 0a7c 6f36 203d 2074         |.|o6 = t
│ │ │ │ +00021d40: 6f74 616c 3a20 3120 3120 3120 3120 3120  otal: 1 1 1 1 1 
│ │ │ │ +00021d50: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │  00021d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021d70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00021d80: 0a7c 2020 2020 2020 2020 2030 3a20 3120  .|         0: 1 
│ │ │ │ -00021d90: 2e20 2e20 2e20 2e20 2e20 2020 2020 2020  . . . . .       
│ │ │ │ +00021d70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00021d80: 303a 2031 202e 202e 202e 202e 202e 2020  0: 1 . . . . .  
│ │ │ │ +00021d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021db0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00021dc0: 2020 2020 2020 2020 313a 202e 2031 202e          1: . 1 .
│ │ │ │ -00021dd0: 202e 202e 202e 2020 2020 2020 2020 2020   . . .          
│ │ │ │ -00021de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021df0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00021e00: 2020 2020 2032 3a20 2e20 2e20 3120 2e20       2: . . 1 . 
│ │ │ │ -00021e10: 2e20 2e20 2020 2020 2020 2020 2020 2020  . .             
│ │ │ │ -00021e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021e30: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00021e40: 2020 333a 202e 202e 202e 2031 202e 202e    3: . . . 1 . .
│ │ │ │ +00021db0: 207c 0a7c 2020 2020 2020 2020 2031 3a20   |.|         1: 
│ │ │ │ +00021dc0: 2e20 3120 2e20 2e20 2e20 2e20 2020 2020  . 1 . . . .     
│ │ │ │ +00021dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021de0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00021df0: 7c20 2020 2020 2020 2020 323a 202e 202e  |         2: . .
│ │ │ │ +00021e00: 2031 202e 202e 202e 2020 2020 2020 2020   1 . . .        
│ │ │ │ +00021e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021e20: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00021e30: 2020 2020 2020 2033 3a20 2e20 2e20 2e20         3: . . . 
│ │ │ │ +00021e40: 3120 2e20 2e20 2020 2020 2020 2020 2020  1 . .           
│ │ │ │  00021e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021e70: 2020 207c 0a7c 2020 2020 2020 2020 2034     |.|         4
│ │ │ │ -00021e80: 3a20 2e20 2e20 2e20 2e20 3120 2e20 2020  : . . . . 1 .   
│ │ │ │ +00021e60: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00021e70: 2020 2020 343a 202e 202e 202e 202e 2031      4: . . . . 1
│ │ │ │ +00021e80: 202e 2020 2020 2020 2020 2020 2020 2020   .              
│ │ │ │  00021e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021eb0: 7c0a 7c20 2020 2020 2020 2020 353a 202e  |.|         5: .
│ │ │ │ -00021ec0: 202e 202e 202e 202e 2031 2020 2020 2020   . . . . 1      
│ │ │ │ +00021ea0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00021eb0: 2035 3a20 2e20 2e20 2e20 2e20 2e20 3120   5: . . . . . 1 
│ │ │ │ +00021ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021ee0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00021ee0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00021ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021f20: 2020 2020 2020 2020 2020 7c0a 7c6f 3620            |.|o6 
│ │ │ │ -00021f30: 3a20 4265 7474 6954 616c 6c79 2020 2020  : BettiTally    
│ │ │ │ +00021f10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00021f20: 0a7c 6f36 203a 2042 6574 7469 5461 6c6c  .|o6 : BettiTall
│ │ │ │ +00021f30: 7920 2020 2020 2020 2020 2020 2020 2020  y               
│ │ │ │  00021f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021f60: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00021f50: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00021f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021fa0: 2d2d 2d2d 2b0a 7c69 3720 3a20 4520 3d20  ----+.|i7 : E = 
│ │ │ │ -00021fb0: 4578 744d 6f64 756c 6520 4d31 2020 2020  ExtModule M1    
│ │ │ │ +00021f90: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a  ---------+.|i7 :
│ │ │ │ +00021fa0: 2045 203d 2045 7874 4d6f 6475 6c65 204d   E = ExtModule M
│ │ │ │ +00021fb0: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │  00021fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021fe0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00021fd0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00021fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022010: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00022020: 7c20 2020 2020 2020 2020 2020 2020 3220  |             2 
│ │ │ │ +00022010: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00022020: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │  00022030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022050: 2020 2020 2020 2020 2020 207c 0a7c 6f37             |.|o7
│ │ │ │ -00022060: 203d 2028 6b6b 5b58 205d 2920 2020 2020   = (kk[X ])     
│ │ │ │ +00022050: 7c0a 7c6f 3720 3d20 286b 6b5b 5820 5d29  |.|o7 = (kk[X ])
│ │ │ │ +00022060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022090: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000220a0: 2020 2020 2030 2020 2020 2020 2020 2020       0          
│ │ │ │ +00022080: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022090: 2020 2020 2020 2020 2020 3020 2020 2020            0     
│ │ │ │ +000220a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000220b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000220c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000220d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000220c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000220d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000220e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000220f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022110: 2020 7c0a 7c6f 3720 3a20 6b6b 5b58 205d    |.|o7 : kk[X ]
│ │ │ │ -00022120: 2d6d 6f64 756c 652c 2066 7265 652c 2064  -module, free, d
│ │ │ │ -00022130: 6567 7265 6573 207b 302e 2e31 7d20 2020  egrees {0..1}   
│ │ │ │ -00022140: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022150: 0a7c 2020 2020 2020 2020 2030 2020 2020  .|         0    
│ │ │ │ +00022100: 2020 2020 2020 207c 0a7c 6f37 203a 206b         |.|o7 : k
│ │ │ │ +00022110: 6b5b 5820 5d2d 6d6f 6475 6c65 2c20 6672  k[X ]-module, fr
│ │ │ │ +00022120: 6565 2c20 6465 6772 6565 7320 7b30 2e2e  ee, degrees {0..
│ │ │ │ +00022130: 317d 2020 2020 2020 2020 2020 2020 2020  1}              
│ │ │ │ +00022140: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00022150: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │  00022160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022180: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00022180: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  00022190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000221a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000221b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000221c0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a  ---------+.|i8 :
│ │ │ │ -000221d0: 2061 7070 6c79 2874 6f4c 6973 7428 302e   apply(toList(0.
│ │ │ │ -000221e0: 2e31 3029 2c20 692d 3e68 696c 6265 7274  .10), i->hilbert
│ │ │ │ -000221f0: 4675 6e63 7469 6f6e 2869 2c20 4529 2920  Function(i, E)) 
│ │ │ │ -00022200: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000221b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000221c0: 7c69 3820 3a20 6170 706c 7928 746f 4c69  |i8 : apply(toLi
│ │ │ │ +000221d0: 7374 2830 2e2e 3130 292c 2069 2d3e 6869  st(0..10), i->hi
│ │ │ │ +000221e0: 6c62 6572 7446 756e 6374 696f 6e28 692c  lbertFunction(i,
│ │ │ │ +000221f0: 2045 2929 2020 2020 2020 207c 0a7c 2020   E))       |.|  
│ │ │ │ +00022200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022240: 2020 207c 0a7c 6f38 203d 207b 312c 2031     |.|o8 = {1, 1
│ │ │ │ -00022250: 2c20 312c 2031 2c20 312c 2031 2c20 312c  , 1, 1, 1, 1, 1,
│ │ │ │ -00022260: 2031 2c20 312c 2031 2c20 317d 2020 2020   1, 1, 1, 1}    
│ │ │ │ -00022270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022280: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00022230: 2020 2020 2020 2020 7c0a 7c6f 3820 3d20          |.|o8 = 
│ │ │ │ +00022240: 7b31 2c20 312c 2031 2c20 312c 2031 2c20  {1, 1, 1, 1, 1, 
│ │ │ │ +00022250: 312c 2031 2c20 312c 2031 2c20 312c 2031  1, 1, 1, 1, 1, 1
│ │ │ │ +00022260: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +00022270: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00022280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000222a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000222b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000222c0: 6f38 203a 204c 6973 7420 2020 2020 2020  o8 : List       
│ │ │ │ +000222b0: 2020 7c0a 7c6f 3820 3a20 4c69 7374 2020    |.|o8 : List  
│ │ │ │ +000222c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000222d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000222e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000222f0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +000222e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000222f0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00022300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022330: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2045  -------+.|i9 : E
│ │ │ │ -00022340: 6576 656e 203d 2065 7665 6e45 7874 4d6f  even = evenExtMo
│ │ │ │ -00022350: 6475 6c65 284d 3129 2020 2020 2020 2020  dule(M1)        
│ │ │ │ -00022360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022370: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00022320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00022330: 3920 3a20 4565 7665 6e20 3d20 6576 656e  9 : Eeven = even
│ │ │ │ +00022340: 4578 744d 6f64 756c 6528 4d31 2920 2020  ExtModule(M1)   
│ │ │ │ +00022350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022360: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00022370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000223a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000223b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000223c0: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │ +000223a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000223b0: 2020 2020 2020 3120 2020 2020 2020 2020        1         
│ │ │ │ +000223c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000223d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000223e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000223f0: 7c6f 3920 3d20 286b 6b5b 5820 5d29 2020  |o9 = (kk[X ])  
│ │ │ │ +000223e0: 2020 207c 0a7c 6f39 203d 2028 6b6b 5b58     |.|o9 = (kk[X
│ │ │ │ +000223f0: 205d 2920 2020 2020 2020 2020 2020 2020   ])             
│ │ │ │  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 0a7c 2020             |.|  
│ │ │ │ -00022430: 2020 2020 2020 2020 3020 2020 2020 2020          0       
│ │ │ │ +00022420: 7c0a 7c20 2020 2020 2020 2020 2030 2020  |.|          0  
│ │ │ │ +00022430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022460: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00022450: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000224a0: 2020 2020 207c 0a7c 6f39 203a 206b 6b5b       |.|o9 : kk[
│ │ │ │ -000224b0: 5820 5d2d 6d6f 6475 6c65 2c20 6672 6565  X ]-module, free
│ │ │ │ +00022490: 2020 2020 2020 2020 2020 7c0a 7c6f 3920            |.|o9 
│ │ │ │ +000224a0: 3a20 6b6b 5b58 205d 2d6d 6f64 756c 652c  : kk[X ]-module,
│ │ │ │ +000224b0: 2066 7265 6520 2020 2020 2020 2020 2020   free           
│ │ │ │  000224c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000224d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000224e0: 2020 7c0a 7c20 2020 2020 2020 2020 3020    |.|         0 
│ │ │ │ +000224d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000224e0: 2020 2030 2020 2020 2020 2020 2020 2020     0            
│ │ │ │  000224f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022510: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022520: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00022510: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00022520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00022560: 3130 203a 2061 7070 6c79 2874 6f4c 6973  10 : apply(toLis
│ │ │ │ -00022570: 7428 302e 2e35 292c 2069 2d3e 6869 6c62  t(0..5), i->hilb
│ │ │ │ -00022580: 6572 7446 756e 6374 696f 6e28 692c 2045  ertFunction(i, E
│ │ │ │ -00022590: 6576 656e 2929 2020 207c 0a7c 2020 2020  even))   |.|    
│ │ │ │ +00022550: 2d2b 0a7c 6931 3020 3a20 6170 706c 7928  -+.|i10 : apply(
│ │ │ │ +00022560: 746f 4c69 7374 2830 2e2e 3529 2c20 692d  toList(0..5), i-
│ │ │ │ +00022570: 3e68 696c 6265 7274 4675 6e63 7469 6f6e  >hilbertFunction
│ │ │ │ +00022580: 2869 2c20 4565 7665 6e29 2920 2020 7c0a  (i, Eeven))   |.
│ │ │ │ +00022590: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000225a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000225b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000225c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000225d0: 2020 2020 2020 7c0a 7c6f 3130 203d 207b        |.|o10 = {
│ │ │ │ -000225e0: 312c 2031 2c20 312c 2031 2c20 312c 2031  1, 1, 1, 1, 1, 1
│ │ │ │ -000225f0: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ -00022600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022610: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000225c0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +000225d0: 3020 3d20 7b31 2c20 312c 2031 2c20 312c  0 = {1, 1, 1, 1,
│ │ │ │ +000225e0: 2031 2c20 317d 2020 2020 2020 2020 2020   1, 1}          
│ │ │ │ +000225f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022600: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00022610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022650: 7c0a 7c6f 3130 203a 204c 6973 7420 2020  |.|o10 : List   
│ │ │ │ +00022640: 2020 2020 207c 0a7c 6f31 3020 3a20 4c69       |.|o10 : Li
│ │ │ │ +00022650: 7374 2020 2020 2020 2020 2020 2020 2020  st              
│ │ │ │  00022660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022680: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00022680: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00022690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000226a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000226b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000226c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131  ----------+.|i11
│ │ │ │ -000226d0: 203a 2045 6f64 6420 3d20 6f64 6445 7874   : Eodd = oddExt
│ │ │ │ -000226e0: 4d6f 6475 6c65 284d 3129 2020 2020 2020  Module(M1)      
│ │ │ │ -000226f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022700: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000226b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +000226c0: 0a7c 6931 3120 3a20 456f 6464 203d 206f  .|i11 : Eodd = o
│ │ │ │ +000226d0: 6464 4578 744d 6f64 756c 6528 4d31 2920  ddExtModule(M1) 
│ │ │ │ +000226e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000226f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00022700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022740: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00022750: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ +00022730: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00022740: 2020 2020 2020 2020 2020 3120 2020 2020            1     
│ │ │ │ +00022750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022780: 207c 0a7c 6f31 3120 3d20 286b 6b5b 5820   |.|o11 = (kk[X 
│ │ │ │ -00022790: 5d29 2020 2020 2020 2020 2020 2020 2020  ])              
│ │ │ │ +00022770: 2020 2020 2020 7c0a 7c6f 3131 203d 2028        |.|o11 = (
│ │ │ │ +00022780: 6b6b 5b58 205d 2920 2020 2020 2020 2020  kk[X ])         
│ │ │ │ +00022790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000227a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000227b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000227c0: 7c20 2020 2020 2020 2020 2020 3020 2020  |           0   
│ │ │ │ +000227b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000227c0: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │  000227d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000227e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000227f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000227f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00022800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022830: 2020 2020 2020 2020 7c0a 7c6f 3131 203a          |.|o11 :
│ │ │ │ -00022840: 206b 6b5b 5820 5d2d 6d6f 6475 6c65 2c20   kk[X ]-module, 
│ │ │ │ -00022850: 6672 6565 2020 2020 2020 2020 2020 2020  free            
│ │ │ │ -00022860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022870: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00022880: 2020 3020 2020 2020 2020 2020 2020 2020    0             
│ │ │ │ +00022820: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022830: 6f31 3120 3a20 6b6b 5b58 205d 2d6d 6f64  o11 : kk[X ]-mod
│ │ │ │ +00022840: 756c 652c 2066 7265 6520 2020 2020 2020  ule, free       
│ │ │ │ +00022850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022860: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00022870: 2020 2020 2020 2030 2020 2020 2020 2020         0        
│ │ │ │ +00022880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000228a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000228b0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +000228a0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +000228b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000228c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000228d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000228e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000228f0: 0a7c 6931 3220 3a20 6170 706c 7928 746f  .|i12 : apply(to
│ │ │ │ -00022900: 4c69 7374 2830 2e2e 3529 2c20 692d 3e68  List(0..5), i->h
│ │ │ │ -00022910: 696c 6265 7274 4675 6e63 7469 6f6e 2869  ilbertFunction(i
│ │ │ │ -00022920: 2c20 456f 6464 2929 2020 2020 7c0a 7c20  , Eodd))    |.| 
│ │ │ │ +000228e0: 2d2d 2d2d 2b0a 7c69 3132 203a 2061 7070  ----+.|i12 : app
│ │ │ │ +000228f0: 6c79 2874 6f4c 6973 7428 302e 2e35 292c  ly(toList(0..5),
│ │ │ │ +00022900: 2069 2d3e 6869 6c62 6572 7446 756e 6374   i->hilbertFunct
│ │ │ │ +00022910: 696f 6e28 692c 2045 6f64 6429 2920 2020  ion(i, Eodd))   
│ │ │ │ +00022920: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00022930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022960: 2020 2020 2020 2020 207c 0a7c 6f31 3220           |.|o12 
│ │ │ │ -00022970: 3d20 7b31 2c20 312c 2031 2c20 312c 2031  = {1, 1, 1, 1, 1
│ │ │ │ -00022980: 2c20 317d 2020 2020 2020 2020 2020 2020  , 1}            
│ │ │ │ -00022990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000229a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00022950: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00022960: 7c6f 3132 203d 207b 312c 2031 2c20 312c  |o12 = {1, 1, 1,
│ │ │ │ +00022970: 2031 2c20 312c 2031 7d20 2020 2020 2020   1, 1, 1}       
│ │ │ │ +00022980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022990: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000229a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000229b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000229c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000229d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000229e0: 2020 207c 0a7c 6f31 3220 3a20 4c69 7374     |.|o12 : List
│ │ │ │ +000229d0: 2020 2020 2020 2020 7c0a 7c6f 3132 203a          |.|o12 :
│ │ │ │ +000229e0: 204c 6973 7420 2020 2020 2020 2020 2020   List           
│ │ │ │  000229f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022a20: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00022a10: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00022a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00022a60: 6931 3320 3a20 7573 6520 5320 2020 2020  i13 : use S     
│ │ │ │ +00022a50: 2d2d 2b0a 7c69 3133 203a 2075 7365 2053  --+.|i13 : use S
│ │ │ │ +00022a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022a90: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00022a80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00022a90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00022aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022ad0: 2020 2020 2020 207c 0a7c 6f31 3320 3d20         |.|o13 = 
│ │ │ │ -00022ae0: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │ +00022ac0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00022ad0: 3133 203d 2053 2020 2020 2020 2020 2020  13 = S          
│ │ │ │ +00022ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022b10: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00022b00: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00022b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022b50: 207c 0a7c 6f31 3320 3a20 506f 6c79 6e6f   |.|o13 : Polyno
│ │ │ │ -00022b60: 6d69 616c 5269 6e67 2020 2020 2020 2020  mialRing        
│ │ │ │ +00022b40: 2020 2020 2020 7c0a 7c6f 3133 203a 2050        |.|o13 : P
│ │ │ │ +00022b50: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020  olynomialRing   
│ │ │ │ +00022b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022b80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00022b90: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00022b80: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00022b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -00022bd0: 3420 3a20 4932 203d 2069 6465 616c 2278  4 : I2 = ideal"x
│ │ │ │ -00022be0: 332c 797a 2220 2020 2020 2020 2020 2020  3,yz"           
│ │ │ │ -00022bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022c00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00022bc0: 2b0a 7c69 3134 203a 2049 3220 3d20 6964  +.|i14 : I2 = id
│ │ │ │ +00022bd0: 6561 6c22 7833 2c79 7a22 2020 2020 2020  eal"x3,yz"      
│ │ │ │ +00022be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022bf0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022c40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00022c50: 2020 2020 2020 3320 2020 2020 2020 2020        3         
│ │ │ │ +00022c30: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00022c40: 2020 2020 2020 2020 2020 2033 2020 2020             3    
│ │ │ │ +00022c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022c80: 2020 7c0a 7c6f 3134 203d 2069 6465 616c    |.|o14 = ideal
│ │ │ │ -00022c90: 2028 7820 2c20 792a 7a29 2020 2020 2020   (x , y*z)      
│ │ │ │ +00022c70: 2020 2020 2020 207c 0a7c 6f31 3420 3d20         |.|o14 = 
│ │ │ │ +00022c80: 6964 6561 6c20 2878 202c 2079 2a7a 2920  ideal (x , y*z) 
│ │ │ │ +00022c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022cb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022cc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00022cb0: 2020 2020 7c0a 7c20 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022cf0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00022d00: 3134 203a 2049 6465 616c 206f 6620 5320  14 : Ideal of S 
│ │ │ │ +00022cf0: 207c 0a7c 6f31 3420 3a20 4964 6561 6c20   |.|o14 : Ideal 
│ │ │ │ +00022d00: 6f66 2053 2020 2020 2020 2020 2020 2020  of S            
│ │ │ │  00022d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022d30: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00022d20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00022d30: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00022d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022d70: 2d2d 2d2d 2d2d 2b0a 7c69 3135 203a 2052  ------+.|i15 : R
│ │ │ │ -00022d80: 3220 3d20 532f 4932 2020 2020 2020 2020  2 = S/I2        
│ │ │ │ +00022d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +00022d70: 3520 3a20 5232 203d 2053 2f49 3220 2020  5 : R2 = S/I2   
│ │ │ │ +00022d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022db0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00022da0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00022db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022df0: 7c0a 7c6f 3135 203d 2052 3220 2020 2020  |.|o15 = R2     
│ │ │ │ +00022de0: 2020 2020 207c 0a7c 6f31 3520 3d20 5232       |.|o15 = R2
│ │ │ │ +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 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022e20: 2020 7c0a 7c20 2020 2020 2020 2020 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 7c0a 7c6f 3135            |.|o15
│ │ │ │ -00022e70: 203a 2051 756f 7469 656e 7452 696e 6720   : QuotientRing 
│ │ │ │ +00022e50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00022e60: 0a7c 6f31 3520 3a20 5175 6f74 6965 6e74  .|o15 : Quotient
│ │ │ │ +00022e70: 5269 6e67 2020 2020 2020 2020 2020 2020  Ring            
│ │ │ │  00022e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022ea0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00022e90: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00022ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022ee0: 2d2d 2d2d 2b0a 7c69 3136 203a 204d 3220  ----+.|i16 : M2 
│ │ │ │ -00022ef0: 3d20 5232 5e31 2f69 6465 616c 2278 322c  = R2^1/ideal"x2,
│ │ │ │ -00022f00: 792c 7a22 2020 2020 2020 2020 2020 2020  y,z"            
│ │ │ │ -00022f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022f20: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00022ed0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3620  ---------+.|i16 
│ │ │ │ +00022ee0: 3a20 4d32 203d 2052 325e 312f 6964 6561  : M2 = R2^1/idea
│ │ │ │ +00022ef0: 6c22 7832 2c79 2c7a 2220 2020 2020 2020  l"x2,y,z"       
│ │ │ │ +00022f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022f10: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00022f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a                |.
│ │ │ │ -00022f60: 7c6f 3136 203d 2063 6f6b 6572 6e65 6c20  |o16 = cokernel 
│ │ │ │ -00022f70: 7c20 7832 2079 207a 207c 2020 2020 2020  | x2 y z |      
│ │ │ │ +00022f50: 2020 207c 0a7c 6f31 3620 3d20 636f 6b65     |.|o16 = coke
│ │ │ │ +00022f60: 726e 656c 207c 2078 3220 7920 7a20 7c20  rnel | x2 y z | 
│ │ │ │ +00022f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022f90: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00022f90: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00022fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022fd0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00022fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022ff0: 2020 2020 2020 2020 2020 3120 2020 2020            1     
│ │ │ │ -00023000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023010: 2020 2020 207c 0a7c 6f31 3620 3a20 5232       |.|o16 : R2
│ │ │ │ -00023020: 2d6d 6f64 756c 652c 2071 756f 7469 656e  -module, quotien
│ │ │ │ -00023030: 7420 6f66 2052 3220 2020 2020 2020 2020  t of R2         
│ │ │ │ -00023040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023050: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00022fc0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022fe0: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ +00022ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023000: 2020 2020 2020 2020 2020 7c0a 7c6f 3136            |.|o16
│ │ │ │ +00023010: 203a 2052 322d 6d6f 6475 6c65 2c20 7175   : R2-module, qu
│ │ │ │ +00023020: 6f74 6965 6e74 206f 6620 5232 2020 2020  otient of R2    
│ │ │ │ +00023030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023040: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00023050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00023090: 0a7c 6931 3720 3a20 6265 7474 6920 6672  .|i17 : betti fr
│ │ │ │ -000230a0: 6565 5265 736f 6c75 7469 6f6e 2028 4d32  eeResolution (M2
│ │ │ │ -000230b0: 2c20 4c65 6e67 7468 4c69 6d69 7420 3d3e  , LengthLimit =>
│ │ │ │ -000230c0: 3130 2920 2020 2020 2020 2020 7c0a 7c20  10)         |.| 
│ │ │ │ +00023080: 2d2d 2d2d 2b0a 7c69 3137 203a 2062 6574  ----+.|i17 : bet
│ │ │ │ +00023090: 7469 2066 7265 6552 6573 6f6c 7574 696f  ti freeResolutio
│ │ │ │ +000230a0: 6e20 284d 322c 204c 656e 6774 684c 696d  n (M2, LengthLim
│ │ │ │ +000230b0: 6974 203d 3e31 3029 2020 2020 2020 2020  it =>10)        
│ │ │ │ +000230c0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  000230d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000230e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000230f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023100: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00023110: 2020 2020 2020 2020 2030 2031 2032 2033           0 1 2 3
│ │ │ │ -00023120: 2034 2020 3520 2036 2020 3720 2038 2020   4  5  6  7  8  
│ │ │ │ -00023130: 3920 3130 2020 2020 2020 2020 2020 2020  9 10            
│ │ │ │ -00023140: 2020 2020 2020 7c0a 7c6f 3137 203d 2074        |.|o17 = t
│ │ │ │ -00023150: 6f74 616c 3a20 3120 3320 3520 3720 3920  otal: 1 3 5 7 9 
│ │ │ │ -00023160: 3131 2031 3320 3135 2031 3720 3139 2032  11 13 15 17 19 2
│ │ │ │ -00023170: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ -00023180: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00023190: 303a 2031 2032 2032 2032 2032 2020 3220  0: 1 2 2 2 2  2 
│ │ │ │ -000231a0: 2032 2020 3220 2032 2020 3220 2032 2020   2  2  2  2  2  
│ │ │ │ -000231b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000231c0: 7c0a 7c20 2020 2020 2020 2020 2031 3a20  |.|          1: 
│ │ │ │ -000231d0: 2e20 3120 3320 3420 3420 2034 2020 3420  . 1 3 4 4  4  4 
│ │ │ │ -000231e0: 2034 2020 3420 2034 2020 3420 2020 2020   4  4  4  4     
│ │ │ │ -000231f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00023200: 2020 2020 2020 2020 2020 323a 202e 202e            2: . .
│ │ │ │ -00023210: 202e 2031 2033 2020 3420 2034 2020 3420   . 1 3  4  4  4 
│ │ │ │ -00023220: 2034 2020 3420 2034 2020 2020 2020 2020   4  4  4        
│ │ │ │ -00023230: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00023240: 2020 2020 2020 2033 3a20 2e20 2e20 2e20         3: . . . 
│ │ │ │ -00023250: 2e20 2e20 2031 2020 3320 2034 2020 3420  . .  1  3  4  4 
│ │ │ │ -00023260: 2034 2020 3420 2020 2020 2020 2020 2020   4  4           
│ │ │ │ -00023270: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00023280: 2020 2020 343a 202e 202e 202e 202e 202e      4: . . . . .
│ │ │ │ -00023290: 2020 2e20 202e 2020 3120 2033 2020 3420    .  .  1  3  4 
│ │ │ │ -000232a0: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ -000232b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000232c0: 2035 3a20 2e20 2e20 2e20 2e20 2e20 202e   5: . . . . .  .
│ │ │ │ -000232d0: 2020 2e20 202e 2020 2e20 2031 2020 3320    .  .  .  1  3 
│ │ │ │ -000232e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000232f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000230f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00023100: 7c20 2020 2020 2020 2020 2020 2020 3020  |             0 
│ │ │ │ +00023110: 3120 3220 3320 3420 2035 2020 3620 2037  1 2 3 4  5  6  7
│ │ │ │ +00023120: 2020 3820 2039 2031 3020 2020 2020 2020    8  9 10       
│ │ │ │ +00023130: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +00023140: 3720 3d20 746f 7461 6c3a 2031 2033 2035  7 = total: 1 3 5
│ │ │ │ +00023150: 2037 2039 2031 3120 3133 2031 3520 3137   7 9 11 13 15 17
│ │ │ │ +00023160: 2031 3920 3231 2020 2020 2020 2020 2020   19 21          
│ │ │ │ +00023170: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00023180: 2020 2020 2030 3a20 3120 3220 3220 3220       0: 1 2 2 2 
│ │ │ │ +00023190: 3220 2032 2020 3220 2032 2020 3220 2032  2  2  2  2  2  2
│ │ │ │ +000231a0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +000231b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000231c0: 2020 313a 202e 2031 2033 2034 2034 2020    1: . 1 3 4 4  
│ │ │ │ +000231d0: 3420 2034 2020 3420 2034 2020 3420 2034  4  4  4  4  4  4
│ │ │ │ +000231e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000231f0: 2020 7c0a 7c20 2020 2020 2020 2020 2032    |.|          2
│ │ │ │ +00023200: 3a20 2e20 2e20 2e20 3120 3320 2034 2020  : . . . 1 3  4  
│ │ │ │ +00023210: 3420 2034 2020 3420 2034 2020 3420 2020  4  4  4  4  4   
│ │ │ │ +00023220: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023230: 0a7c 2020 2020 2020 2020 2020 333a 202e  .|          3: .
│ │ │ │ +00023240: 202e 202e 202e 202e 2020 3120 2033 2020   . . . .  1  3  
│ │ │ │ +00023250: 3420 2034 2020 3420 2034 2020 2020 2020  4  4  4  4      
│ │ │ │ +00023260: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00023270: 2020 2020 2020 2020 2034 3a20 2e20 2e20           4: . . 
│ │ │ │ +00023280: 2e20 2e20 2e20 202e 2020 2e20 2031 2020  . . .  .  .  1  
│ │ │ │ +00023290: 3320 2034 2020 3420 2020 2020 2020 2020  3  4  4         
│ │ │ │ +000232a0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000232b0: 2020 2020 2020 353a 202e 202e 202e 202e        5: . . . .
│ │ │ │ +000232c0: 202e 2020 2e20 202e 2020 2e20 202e 2020   .  .  .  .  .  
│ │ │ │ +000232d0: 3120 2033 2020 2020 2020 2020 2020 2020  1  3            
│ │ │ │ +000232e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000232f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023320: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00023330: 7c6f 3137 203a 2042 6574 7469 5461 6c6c  |o17 : BettiTall
│ │ │ │ -00023340: 7920 2020 2020 2020 2020 2020 2020 2020  y               
│ │ │ │ +00023320: 2020 207c 0a7c 6f31 3720 3a20 4265 7474     |.|o17 : Bett
│ │ │ │ +00023330: 6954 616c 6c79 2020 2020 2020 2020 2020  iTally          
│ │ │ │ +00023340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023360: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00023360: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00023370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000233a0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3138 203a  --------+.|i18 :
│ │ │ │ -000233b0: 2045 203d 2045 7874 4d6f 6475 6c65 204d   E = ExtModule M
│ │ │ │ -000233c0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -000233d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000233e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00023390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +000233a0: 6931 3820 3a20 4520 3d20 4578 744d 6f64  i18 : E = ExtMod
│ │ │ │ +000233b0: 756c 6520 4d32 2020 2020 2020 2020 2020  ule M2          
│ │ │ │ +000233c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000233d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000233e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000233f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023420: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00023430: 2020 2020 2020 2038 2020 2020 2020 2020         8        
│ │ │ │ +00023410: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00023420: 2020 2020 2020 2020 2020 2020 3820 2020              8   
│ │ │ │ +00023430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023450: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023460: 0a7c 6f31 3820 3d20 286b 6b5b 5820 2e2e  .|o18 = (kk[X ..
│ │ │ │ -00023470: 5820 5d29 2020 2020 2020 2020 2020 2020  X ])            
│ │ │ │ +00023450: 2020 2020 7c0a 7c6f 3138 203d 2028 6b6b      |.|o18 = (kk
│ │ │ │ +00023460: 5b58 202e 2e58 205d 2920 2020 2020 2020  [X ..X ])       
│ │ │ │ +00023470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023490: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000234a0: 2020 2020 2020 2020 2020 3020 2020 3120            0   1 
│ │ │ │ +00023490: 207c 0a7c 2020 2020 2020 2020 2020 2030   |.|           0
│ │ │ │ +000234a0: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │  000234b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000234c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000234d0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000234c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000234d0: 7c20 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023510: 2020 2020 2020 7c0a 7c6f 3138 203a 206b        |.|o18 : k
│ │ │ │ -00023520: 6b5b 5820 2e2e 5820 5d2d 6d6f 6475 6c65  k[X ..X ]-module
│ │ │ │ -00023530: 2c20 6672 6565 2c20 6465 6772 6565 7320  , free, degrees 
│ │ │ │ -00023540: 7b30 2e2e 312c 2032 3a31 2c20 333a 322c  {0..1, 2:1, 3:2,
│ │ │ │ -00023550: 2033 7d7c 0a7c 2020 2020 2020 2020 2020   3}|.|          
│ │ │ │ -00023560: 3020 2020 3120 2020 2020 2020 2020 2020  0   1           
│ │ │ │ +00023500: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +00023510: 3820 3a20 6b6b 5b58 202e 2e58 205d 2d6d  8 : kk[X ..X ]-m
│ │ │ │ +00023520: 6f64 756c 652c 2066 7265 652c 2064 6567  odule, free, deg
│ │ │ │ +00023530: 7265 6573 207b 302e 2e31 2c20 323a 312c  rees {0..1, 2:1,
│ │ │ │ +00023540: 2033 3a32 2c20 337d 7c0a 7c20 2020 2020   3:2, 3}|.|     
│ │ │ │ +00023550: 2020 2020 2030 2020 2031 2020 2020 2020       0   1      
│ │ │ │ +00023560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023590: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00023580: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00023590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000235a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000235b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000235c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -000235d0: 6931 3920 3a20 6170 706c 7928 746f 4c69  i19 : apply(toLi
│ │ │ │ -000235e0: 7374 2830 2e2e 3130 292c 2069 2d3e 6869  st(0..10), i->hi
│ │ │ │ -000235f0: 6c62 6572 7446 756e 6374 696f 6e28 692c  lbertFunction(i,
│ │ │ │ -00023600: 2045 2929 2020 2020 2020 7c0a 7c20 2020   E))      |.|   
│ │ │ │ +000235c0: 2d2d 2b0a 7c69 3139 203a 2061 7070 6c79  --+.|i19 : apply
│ │ │ │ +000235d0: 2874 6f4c 6973 7428 302e 2e31 3029 2c20  (toList(0..10), 
│ │ │ │ +000235e0: 692d 3e68 696c 6265 7274 4675 6e63 7469  i->hilbertFuncti
│ │ │ │ +000235f0: 6f6e 2869 2c20 4529 2920 2020 2020 207c  on(i, E))      |
│ │ │ │ +00023600: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00023610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023640: 2020 2020 2020 207c 0a7c 6f31 3920 3d20         |.|o19 = 
│ │ │ │ -00023650: 7b31 2c20 332c 2035 2c20 372c 2039 2c20  {1, 3, 5, 7, 9, 
│ │ │ │ -00023660: 3131 2c20 3133 2c20 3135 2c20 3137 2c20  11, 13, 15, 17, 
│ │ │ │ -00023670: 3139 2c20 3231 7d20 2020 2020 2020 2020  19, 21}         
│ │ │ │ -00023680: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00023630: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00023640: 3139 203d 207b 312c 2033 2c20 352c 2037  19 = {1, 3, 5, 7
│ │ │ │ +00023650: 2c20 392c 2031 312c 2031 332c 2031 352c  , 9, 11, 13, 15,
│ │ │ │ +00023660: 2031 372c 2031 392c 2032 317d 2020 2020   17, 19, 21}    
│ │ │ │ +00023670: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00023680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000236a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000236b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000236c0: 207c 0a7c 6f31 3920 3a20 4c69 7374 2020   |.|o19 : List  
│ │ │ │ +000236b0: 2020 2020 2020 7c0a 7c6f 3139 203a 204c        |.|o19 : L
│ │ │ │ +000236c0: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │  000236d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000236e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000236f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00023700: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +000236f0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00023700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ -00023740: 3020 3a20 4565 7665 6e20 3d20 6576 656e  0 : Eeven = even
│ │ │ │ -00023750: 4578 744d 6f64 756c 6520 4d32 2020 2020  ExtModule M2    
│ │ │ │ -00023760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023770: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00023730: 2b0a 7c69 3230 203a 2045 6576 656e 203d  +.|i20 : Eeven =
│ │ │ │ +00023740: 2065 7665 6e45 7874 4d6f 6475 6c65 204d   evenExtModule M
│ │ │ │ +00023750: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00023760: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00023770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000237a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000237b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000237c0: 2020 2020 2020 2020 2020 3420 2020 2020            4     
│ │ │ │ +000237a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000237b0: 2020 2020 2020 2020 2020 2020 2020 2034                 4
│ │ │ │ +000237c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000237d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000237e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000237f0: 2020 7c0a 7c6f 3230 203d 2028 6b6b 5b58    |.|o20 = (kk[X
│ │ │ │ -00023800: 202e 2e58 205d 2920 2020 2020 2020 2020   ..X ])         
│ │ │ │ +000237e0: 2020 2020 2020 207c 0a7c 6f32 3020 3d20         |.|o20 = 
│ │ │ │ +000237f0: 286b 6b5b 5820 2e2e 5820 5d29 2020 2020  (kk[X ..X ])    
│ │ │ │ +00023800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023820: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023830: 0a7c 2020 2020 2020 2020 2020 2030 2020  .|           0  
│ │ │ │ -00023840: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │ +00023820: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00023830: 2020 3020 2020 3120 2020 2020 2020 2020    0   1         
│ │ │ │ +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 7c0a 7c20              |.| 
│ │ │ │ +00023860: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00023870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000238a0: 2020 2020 2020 2020 207c 0a7c 6f32 3020           |.|o20 
│ │ │ │ -000238b0: 3a20 6b6b 5b58 202e 2e58 205d 2d6d 6f64  : kk[X ..X ]-mod
│ │ │ │ -000238c0: 756c 652c 2066 7265 652c 2064 6567 7265  ule, free, degre
│ │ │ │ -000238d0: 6573 207b 302e 2e31 2c20 323a 317d 2020  es {0..1, 2:1}  
│ │ │ │ -000238e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000238f0: 2020 2030 2020 2031 2020 2020 2020 2020     0   1        
│ │ │ │ +00023890: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000238a0: 7c6f 3230 203a 206b 6b5b 5820 2e2e 5820  |o20 : kk[X ..X 
│ │ │ │ +000238b0: 5d2d 6d6f 6475 6c65 2c20 6672 6565 2c20  ]-module, free, 
│ │ │ │ +000238c0: 6465 6772 6565 7320 7b30 2e2e 312c 2032  degrees {0..1, 2
│ │ │ │ +000238d0: 3a31 7d20 2020 2020 2020 207c 0a7c 2020  :1}        |.|  
│ │ │ │ +000238e0: 2020 2020 2020 2020 3020 2020 3120 2020          0   1   
│ │ │ │ +000238f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023920: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00023910: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00023920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023960: 2b0a 7c69 3231 203a 2061 7070 6c79 2874  +.|i21 : apply(t
│ │ │ │ -00023970: 6f4c 6973 7428 302e 2e35 292c 2069 2d3e  oList(0..5), i->
│ │ │ │ -00023980: 6869 6c62 6572 7446 756e 6374 696f 6e28  hilbertFunction(
│ │ │ │ -00023990: 692c 2045 6576 656e 2929 2020 207c 0a7c  i, Eeven))   |.|
│ │ │ │ +00023950: 2d2d 2d2d 2d2b 0a7c 6932 3120 3a20 6170  -----+.|i21 : ap
│ │ │ │ +00023960: 706c 7928 746f 4c69 7374 2830 2e2e 3529  ply(toList(0..5)
│ │ │ │ +00023970: 2c20 692d 3e68 696c 6265 7274 4675 6e63  , i->hilbertFunc
│ │ │ │ +00023980: 7469 6f6e 2869 2c20 4565 7665 6e29 2920  tion(i, Eeven)) 
│ │ │ │ +00023990: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000239a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000239b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000239c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000239d0: 2020 2020 2020 2020 2020 7c0a 7c6f 3231            |.|o21
│ │ │ │ -000239e0: 203d 207b 312c 2035 2c20 392c 2031 332c   = {1, 5, 9, 13,
│ │ │ │ -000239f0: 2031 372c 2032 317d 2020 2020 2020 2020   17, 21}        
│ │ │ │ -00023a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023a10: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000239c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000239d0: 0a7c 6f32 3120 3d20 7b31 2c20 352c 2039  .|o21 = {1, 5, 9
│ │ │ │ +000239e0: 2c20 3133 2c20 3137 2c20 3231 7d20 2020  , 13, 17, 21}   
│ │ │ │ +000239f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023a00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +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 7c6f 3231 203a 204c 6973      |.|o21 : Lis
│ │ │ │ -00023a60: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │ +00023a40: 2020 2020 2020 2020 207c 0a7c 6f32 3120           |.|o21 
│ │ │ │ +00023a50: 3a20 4c69 7374 2020 2020 2020 2020 2020  : List          
│ │ │ │ +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: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00023a80: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00023a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  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 2b0a  --------------+.
│ │ │ │ -00023ad0: 7c69 3232 203a 2045 6f64 6420 3d20 6f64  |i22 : Eodd = od
│ │ │ │ -00023ae0: 6445 7874 4d6f 6475 6c65 204d 3220 2020  dExtModule M2   
│ │ │ │ +00023ac0: 2d2d 2d2b 0a7c 6932 3220 3a20 456f 6464  ---+.|i22 : Eodd
│ │ │ │ +00023ad0: 203d 206f 6464 4578 744d 6f64 756c 6520   = oddExtModule 
│ │ │ │ +00023ae0: 4d32 2020 2020 2020 2020 2020 2020 2020  M2              
│ │ │ │  00023af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023b00: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00023b00: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00023b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023b40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00023b50: 2020 2020 2020 2020 2020 2020 2034 2020               4  
│ │ │ │ +00023b30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00023b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023b50: 2020 3420 2020 2020 2020 2020 2020 2020    4             
│ │ │ │  00023b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023b80: 2020 2020 207c 0a7c 6f32 3220 3d20 286b       |.|o22 = (k
│ │ │ │ -00023b90: 6b5b 5820 2e2e 5820 5d29 2020 2020 2020  k[X ..X ])      
│ │ │ │ +00023b70: 2020 2020 2020 2020 2020 7c0a 7c6f 3232            |.|o22
│ │ │ │ +00023b80: 203d 2028 6b6b 5b58 202e 2e58 205d 2920   = (kk[X ..X ]) 
│ │ │ │ +00023b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023bc0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00023bd0: 3020 2020 3120 2020 2020 2020 2020 2020  0   1           
│ │ │ │ +00023bb0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00023bc0: 2020 2020 2030 2020 2031 2020 2020 2020       0   1      
│ │ │ │ +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 2020 207c                 |
│ │ │ │ -00023c00: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00023bf0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00023c00: 2020 2020 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 7c0a 7c6f              |.|o
│ │ │ │ -00023c40: 3232 203a 206b 6b5b 5820 2e2e 5820 5d2d  22 : kk[X ..X ]-
│ │ │ │ -00023c50: 6d6f 6475 6c65 2c20 6672 6565 2c20 6465  module, free, de
│ │ │ │ -00023c60: 6772 6565 7320 7b33 3a30 2c20 317d 2020  grees {3:0, 1}  
│ │ │ │ -00023c70: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00023c80: 2020 2020 2020 3020 2020 3120 2020 2020        0   1     
│ │ │ │ +00023c30: 207c 0a7c 6f32 3220 3a20 6b6b 5b58 202e   |.|o22 : kk[X .
│ │ │ │ +00023c40: 2e58 205d 2d6d 6f64 756c 652c 2066 7265  .X ]-module, fre
│ │ │ │ +00023c50: 652c 2064 6567 7265 6573 207b 333a 302c  e, degrees {3:0,
│ │ │ │ +00023c60: 2031 7d20 2020 2020 2020 2020 2020 7c0a   1}           |.
│ │ │ │ +00023c70: 7c20 2020 2020 2020 2020 2030 2020 2031  |          0   1
│ │ │ │ +00023c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023cb0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00023ca0: 2020 2020 2020 2020 2020 207c 0a2b 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 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023cf0: 2d2d 2d2b 0a7c 6932 3320 3a20 6170 706c  ---+.|i23 : appl
│ │ │ │ -00023d00: 7928 746f 4c69 7374 2830 2e2e 3529 2c20  y(toList(0..5), 
│ │ │ │ -00023d10: 692d 3e68 696c 6265 7274 4675 6e63 7469  i->hilbertFuncti
│ │ │ │ -00023d20: 6f6e 2869 2c20 456f 6464 2929 2020 2020  on(i, Eodd))    
│ │ │ │ -00023d30: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00023ce0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3233 203a  --------+.|i23 :
│ │ │ │ +00023cf0: 2061 7070 6c79 2874 6f4c 6973 7428 302e   apply(toList(0.
│ │ │ │ +00023d00: 2e35 292c 2069 2d3e 6869 6c62 6572 7446  .5), i->hilbertF
│ │ │ │ +00023d10: 756e 6374 696f 6e28 692c 2045 6f64 6429  unction(i, Eodd)
│ │ │ │ +00023d20: 2920 2020 207c 0a7c 2020 2020 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 207c 0a7c               |.|
│ │ │ │ -00023d70: 6f32 3320 3d20 7b33 2c20 372c 2031 312c  o23 = {3, 7, 11,
│ │ │ │ -00023d80: 2031 352c 2031 392c 2032 337d 2020 2020   15, 19, 23}    
│ │ │ │ -00023d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023da0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00023d60: 2020 7c0a 7c6f 3233 203d 207b 332c 2037    |.|o23 = {3, 7
│ │ │ │ +00023d70: 2c20 3131 2c20 3135 2c20 3139 2c20 3233  , 11, 15, 19, 23
│ │ │ │ +00023d80: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +00023d90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023da0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00023db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023de0: 2020 2020 2020 207c 0a7c 6f32 3320 3a20         |.|o23 : 
│ │ │ │ -00023df0: 4c69 7374 2020 2020 2020 2020 2020 2020  List            
│ │ │ │ +00023dd0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00023de0: 3233 203a 204c 6973 7420 2020 2020 2020  23 : List       
│ │ │ │ +00023df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023e20: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00023e10: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00023e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023e60: 2d2b 0a0a 5365 6520 616c 736f 0a3d 3d3d  -+..See also.===
│ │ │ │ -00023e70: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
│ │ │ │ -00023e80: 2065 7665 6e45 7874 4d6f 6475 6c65 3a20   evenExtModule: 
│ │ │ │ -00023e90: 6576 656e 4578 744d 6f64 756c 652c 202d  evenExtModule, -
│ │ │ │ -00023ea0: 2d20 6576 656e 2070 6172 7420 6f66 2045  - even part of E
│ │ │ │ -00023eb0: 7874 5e2a 284d 2c6b 2920 6f76 6572 2061  xt^*(M,k) over a
│ │ │ │ -00023ec0: 0a20 2020 2063 6f6d 706c 6574 6520 696e  .    complete in
│ │ │ │ -00023ed0: 7465 7273 6563 7469 6f6e 2061 7320 6d6f  tersection as mo
│ │ │ │ -00023ee0: 6475 6c65 206f 7665 7220 4349 206f 7065  dule over CI ope
│ │ │ │ -00023ef0: 7261 746f 7220 7269 6e67 0a20 202a 202a  rator ring.  * *
│ │ │ │ -00023f00: 6e6f 7465 206f 6464 4578 744d 6f64 756c  note oddExtModul
│ │ │ │ -00023f10: 653a 206f 6464 4578 744d 6f64 756c 652c  e: oddExtModule,
│ │ │ │ -00023f20: 202d 2d20 6f64 6420 7061 7274 206f 6620   -- odd part of 
│ │ │ │ -00023f30: 4578 745e 2a28 4d2c 6b29 206f 7665 7220  Ext^*(M,k) over 
│ │ │ │ -00023f40: 6120 636f 6d70 6c65 7465 0a20 2020 2069  a complete.    i
│ │ │ │ -00023f50: 6e74 6572 7365 6374 696f 6e20 6173 206d  ntersection as m
│ │ │ │ -00023f60: 6f64 756c 6520 6f76 6572 2043 4920 6f70  odule over CI op
│ │ │ │ -00023f70: 6572 6174 6f72 2072 696e 670a 0a57 6179  erator ring..Way
│ │ │ │ -00023f80: 7320 746f 2075 7365 2045 7874 4d6f 6475  s to use ExtModu
│ │ │ │ -00023f90: 6c65 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  le:.============
│ │ │ │ -00023fa0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ -00023fb0: 2245 7874 4d6f 6475 6c65 284d 6f64 756c  "ExtModule(Modul
│ │ │ │ -00023fc0: 6529 220a 0a46 6f72 2074 6865 2070 726f  e)"..For the pro
│ │ │ │ -00023fd0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d  grammer.========
│ │ │ │ -00023fe0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520  ==========..The 
│ │ │ │ -00023ff0: 6f62 6a65 6374 202a 6e6f 7465 2045 7874  object *note Ext
│ │ │ │ -00024000: 4d6f 6475 6c65 3a20 4578 744d 6f64 756c  Module: ExtModul
│ │ │ │ -00024010: 652c 2069 7320 6120 2a6e 6f74 6520 6d65  e, is a *note me
│ │ │ │ -00024020: 7468 6f64 2066 756e 6374 696f 6e3a 0a28  thod function:.(
│ │ │ │ -00024030: 4d61 6361 756c 6179 3244 6f63 294d 6574  Macaulay2Doc)Met
│ │ │ │ -00024040: 686f 6446 756e 6374 696f 6e2c 2e0a 0a2d  hodFunction,...-
│ │ │ │ +00023e50: 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061 6c73  ------+..See als
│ │ │ │ +00023e60: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  o.========..  * 
│ │ │ │ +00023e70: 2a6e 6f74 6520 6576 656e 4578 744d 6f64  *note evenExtMod
│ │ │ │ +00023e80: 756c 653a 2065 7665 6e45 7874 4d6f 6475  ule: evenExtModu
│ │ │ │ +00023e90: 6c65 2c20 2d2d 2065 7665 6e20 7061 7274  le, -- even part
│ │ │ │ +00023ea0: 206f 6620 4578 745e 2a28 4d2c 6b29 206f   of Ext^*(M,k) o
│ │ │ │ +00023eb0: 7665 7220 610a 2020 2020 636f 6d70 6c65  ver a.    comple
│ │ │ │ +00023ec0: 7465 2069 6e74 6572 7365 6374 696f 6e20  te intersection 
│ │ │ │ +00023ed0: 6173 206d 6f64 756c 6520 6f76 6572 2043  as module over C
│ │ │ │ +00023ee0: 4920 6f70 6572 6174 6f72 2072 696e 670a  I operator ring.
│ │ │ │ +00023ef0: 2020 2a20 2a6e 6f74 6520 6f64 6445 7874    * *note oddExt
│ │ │ │ +00023f00: 4d6f 6475 6c65 3a20 6f64 6445 7874 4d6f  Module: oddExtMo
│ │ │ │ +00023f10: 6475 6c65 2c20 2d2d 206f 6464 2070 6172  dule, -- odd par
│ │ │ │ +00023f20: 7420 6f66 2045 7874 5e2a 284d 2c6b 2920  t of Ext^*(M,k) 
│ │ │ │ +00023f30: 6f76 6572 2061 2063 6f6d 706c 6574 650a  over a complete.
│ │ │ │ +00023f40: 2020 2020 696e 7465 7273 6563 7469 6f6e      intersection
│ │ │ │ +00023f50: 2061 7320 6d6f 6475 6c65 206f 7665 7220   as module over 
│ │ │ │ +00023f60: 4349 206f 7065 7261 746f 7220 7269 6e67  CI operator ring
│ │ │ │ +00023f70: 0a0a 5761 7973 2074 6f20 7573 6520 4578  ..Ways to use Ex
│ │ │ │ +00023f80: 744d 6f64 756c 653a 0a3d 3d3d 3d3d 3d3d  tModule:.=======
│ │ │ │ +00023f90: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ +00023fa0: 0a20 202a 2022 4578 744d 6f64 756c 6528  .  * "ExtModule(
│ │ │ │ +00023fb0: 4d6f 6475 6c65 2922 0a0a 466f 7220 7468  Module)"..For th
│ │ │ │ +00023fc0: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d  e programmer.===
│ │ │ │ +00023fd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ +00023fe0: 0a54 6865 206f 626a 6563 7420 2a6e 6f74  .The object *not
│ │ │ │ +00023ff0: 6520 4578 744d 6f64 756c 653a 2045 7874  e ExtModule: Ext
│ │ │ │ +00024000: 4d6f 6475 6c65 2c20 6973 2061 202a 6e6f  Module, is a *no
│ │ │ │ +00024010: 7465 206d 6574 686f 6420 6675 6e63 7469  te method functi
│ │ │ │ +00024020: 6f6e 3a0a 284d 6163 6175 6c61 7932 446f  on:.(Macaulay2Do
│ │ │ │ +00024030: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e  c)MethodFunction
│ │ │ │ +00024040: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ,...------------
│ │ │ │  00024050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a  --------------..
│ │ │ │ -000240a0: 5468 6520 736f 7572 6365 206f 6620 7468  The source of th
│ │ │ │ -000240b0: 6973 2064 6f63 756d 656e 7420 6973 2069  is document is i
│ │ │ │ -000240c0: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475  n./build/reprodu
│ │ │ │ -000240d0: 6369 626c 652d 7061 7468 2f6d 6163 6175  cible-path/macau
│ │ │ │ -000240e0: 6c61 7932 2d31 2e32 352e 3131 2b64 732f  lay2-1.25.11+ds/
│ │ │ │ -000240f0: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163  M2/Macaulay2/pac
│ │ │ │ -00024100: 6b61 6765 732f 0a43 6f6d 706c 6574 6549  kages/.CompleteI
│ │ │ │ -00024110: 6e74 6572 7365 6374 696f 6e52 6573 6f6c  ntersectionResol
│ │ │ │ -00024120: 7574 696f 6e73 2e6d 323a 3335 3936 3a30  utions.m2:3596:0
│ │ │ │ -00024130: 2e0a 1f0a 4669 6c65 3a20 436f 6d70 6c65  ....File: Comple
│ │ │ │ -00024140: 7465 496e 7465 7273 6563 7469 6f6e 5265  teIntersectionRe
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│ │ │ │ -00024160: 4e6f 6465 3a20 4578 744d 6f64 756c 6544  Node: ExtModuleD
│ │ │ │ -00024170: 6174 612c 204e 6578 743a 2065 7874 5673  ata, Next: extVs
│ │ │ │ -00024180: 436f 686f 6d6f 6c6f 6779 2c20 5072 6576  Cohomology, Prev
│ │ │ │ -00024190: 3a20 4578 744d 6f64 756c 652c 2055 703a  : ExtModule, Up:
│ │ │ │ -000241a0: 2054 6f70 0a0a 4578 744d 6f64 756c 6544   Top..ExtModuleD
│ │ │ │ -000241b0: 6174 6120 2d2d 2045 7665 6e20 616e 6420  ata -- Even and 
│ │ │ │ -000241c0: 6f64 6420 4578 7420 6d6f 6475 6c65 7320  odd Ext modules 
│ │ │ │ -000241d0: 616e 6420 7468 6569 7220 7265 6775 6c61  and their regula
│ │ │ │ -000241e0: 7269 7479 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a  rity.***********
│ │ │ │ +00024090: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520  ---..The source 
│ │ │ │ +000240a0: 6f66 2074 6869 7320 646f 6375 6d65 6e74  of this document
│ │ │ │ +000240b0: 2069 7320 696e 0a2f 6275 696c 642f 7265   is in./build/re
│ │ │ │ +000240c0: 7072 6f64 7563 6962 6c65 2d70 6174 682f  producible-path/
│ │ │ │ +000240d0: 6d61 6361 756c 6179 322d 312e 3235 2e31  macaulay2-1.25.1
│ │ │ │ +000240e0: 312b 6473 2f4d 322f 4d61 6361 756c 6179  1+ds/M2/Macaulay
│ │ │ │ +000240f0: 322f 7061 636b 6167 6573 2f0a 436f 6d70  2/packages/.Comp
│ │ │ │ +00024100: 6c65 7465 496e 7465 7273 6563 7469 6f6e  leteIntersection
│ │ │ │ +00024110: 5265 736f 6c75 7469 6f6e 732e 6d32 3a33  Resolutions.m2:3
│ │ │ │ +00024120: 3539 363a 302e 0a1f 0a46 696c 653a 2043  596:0....File: C
│ │ │ │ +00024130: 6f6d 706c 6574 6549 6e74 6572 7365 6374  ompleteIntersect
│ │ │ │ +00024140: 696f 6e52 6573 6f6c 7574 696f 6e73 2e69  ionResolutions.i
│ │ │ │ +00024150: 6e66 6f2c 204e 6f64 653a 2045 7874 4d6f  nfo, Node: ExtMo
│ │ │ │ +00024160: 6475 6c65 4461 7461 2c20 4e65 7874 3a20  duleData, Next: 
│ │ │ │ +00024170: 6578 7456 7343 6f68 6f6d 6f6c 6f67 792c  extVsCohomology,
│ │ │ │ +00024180: 2050 7265 763a 2045 7874 4d6f 6475 6c65   Prev: ExtModule
│ │ │ │ +00024190: 2c20 5570 3a20 546f 700a 0a45 7874 4d6f  , Up: Top..ExtMo
│ │ │ │ +000241a0: 6475 6c65 4461 7461 202d 2d20 4576 656e  duleData -- Even
│ │ │ │ +000241b0: 2061 6e64 206f 6464 2045 7874 206d 6f64   and odd Ext mod
│ │ │ │ +000241c0: 756c 6573 2061 6e64 2074 6865 6972 2072  ules and their r
│ │ │ │ +000241d0: 6567 756c 6172 6974 790a 2a2a 2a2a 2a2a  egularity.******
│ │ │ │ +000241e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000241f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00024200: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00024210: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00024220: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20  ***..  * Usage: 
│ │ │ │ -00024230: 0a20 2020 2020 2020 204c 203d 2045 7874  .        L = Ext
│ │ │ │ -00024240: 4d6f 6475 6c65 4461 7461 204d 0a20 202a  ModuleData M.  *
│ │ │ │ -00024250: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ -00024260: 204d 2c20 6120 2a6e 6f74 6520 6d6f 6475   M, a *note modu
│ │ │ │ -00024270: 6c65 3a20 284d 6163 6175 6c61 7932 446f  le: (Macaulay2Do
│ │ │ │ -00024280: 6329 4d6f 6475 6c65 2c2c 204d 6f64 756c  c)Module,, Modul
│ │ │ │ -00024290: 6520 6f76 6572 2061 2063 6f6d 706c 6574  e over a complet
│ │ │ │ -000242a0: 650a 2020 2020 2020 2020 696e 7465 7273  e.        inters
│ │ │ │ -000242b0: 6563 7469 6f6e 2053 0a20 202a 204f 7574  ection S.  * Out
│ │ │ │ -000242c0: 7075 7473 3a0a 2020 2020 2020 2a20 4c2c  puts:.      * L,
│ │ │ │ -000242d0: 2061 202a 6e6f 7465 206c 6973 743a 2028   a *note list: (
│ │ │ │ -000242e0: 4d61 6361 756c 6179 3244 6f63 294c 6973  Macaulay2Doc)Lis
│ │ │ │ -000242f0: 742c 2c20 4c20 3d20 5c7b 6576 656e 4578  t,, L = \{evenEx
│ │ │ │ -00024300: 744d 6f64 756c 652c 0a20 2020 2020 2020  tModule,.       
│ │ │ │ -00024310: 206f 6464 4578 744d 6f64 756c 652c 2072   oddExtModule, r
│ │ │ │ -00024320: 6567 302c 2072 6567 315c 7d0a 0a44 6573  eg0, reg1\}..Des
│ │ │ │ -00024330: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ -00024340: 3d3d 3d3d 0a0a 5375 7070 6f73 6520 7468  ====..Suppose th
│ │ │ │ -00024350: 6174 204d 2069 7320 6120 6d6f 6475 6c65  at M is a module
│ │ │ │ -00024360: 206f 7665 7220 6120 636f 6d70 6c65 7465   over a complete
│ │ │ │ -00024370: 2069 6e74 6572 7365 6374 696f 6e20 5220   intersection R 
│ │ │ │ -00024380: 736f 2074 6861 740a 0a45 203a 3d20 4578  so that..E := Ex
│ │ │ │ -00024390: 744d 6f64 756c 6520 4d0a 0a69 7320 6120  tModule M..is a 
│ │ │ │ -000243a0: 6d6f 6475 6c65 2067 656e 6572 6174 6564  module generated
│ │ │ │ -000243b0: 2069 6e20 6465 6772 6565 7320 3e3d 3020   in degrees >=0 
│ │ │ │ -000243c0: 6f76 6572 2061 2070 6f6c 796e 6f6d 6961  over a polynomia
│ │ │ │ -000243d0: 6c20 7269 6e67 2054 2720 6765 6e65 7261  l ring T' genera
│ │ │ │ -000243e0: 7465 6420 696e 0a64 6567 7265 6520 322c  ted in.degree 2,
│ │ │ │ -000243f0: 2061 6e64 0a0a 4530 203a 3d20 6576 656e   and..E0 := even
│ │ │ │ -00024400: 4578 744d 6f64 756c 6520 4d20 616e 6420  ExtModule M and 
│ │ │ │ -00024410: 4531 203a 3d20 6f64 6445 7874 4d6f 6475  E1 := oddExtModu
│ │ │ │ -00024420: 6c65 204d 0a0a 6172 6520 6d6f 6475 6c65  le M..are module
│ │ │ │ -00024430: 7320 6765 6e65 7261 7465 6420 696e 2064  s generated in d
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│ │ │ │ -00024450: 6120 706f 6c79 6e6f 6d69 616c 2072 696e  a polynomial rin
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│ │ │ │ -00024470: 6f72 730a 696e 2064 6567 7265 6520 312e  ors.in degree 1.
│ │ │ │ -00024480: 0a0a 5468 6520 7363 7269 7074 2072 6574  ..The script ret
│ │ │ │ -00024490: 7572 6e73 0a0a 4c20 3d20 5c7b 4530 2c45  urns..L = \{E0,E
│ │ │ │ -000244a0: 312c 2072 6567 756c 6172 6974 7920 4530  1, regularity E0
│ │ │ │ -000244b0: 2c20 7265 6775 6c61 7269 7479 2045 315c  , regularity E1\
│ │ │ │ -000244c0: 7d0a 0a61 6e64 2070 7269 6e74 7320 6120  }..and prints a 
│ │ │ │ -000244d0: 6d65 7373 6167 6520 6966 207c 7265 6730  message if |reg0
│ │ │ │ -000244e0: 2d72 6567 317c 3e31 2e0a 0a49 6620 7765  -reg1|>1...If we
│ │ │ │ -000244f0: 2073 6574 2072 203d 206d 6178 2832 2a72   set r = max(2*r
│ │ │ │ -00024500: 6567 302c 2031 2b32 2a72 6567 3129 2c20  eg0, 1+2*reg1), 
│ │ │ │ -00024510: 616e 6420 4620 6973 2061 2072 6573 6f6c  and F is a resol
│ │ │ │ -00024520: 7574 696f 6e20 6f66 204d 2c20 7468 656e  ution of M, then
│ │ │ │ -00024530: 2063 6f6b 6572 0a46 2e64 645f 7b28 722b   coker.F.dd_{(r+
│ │ │ │ -00024540: 3129 7d20 6973 2074 6865 2066 6972 7374  1)} is the first
│ │ │ │ -00024550: 2073 7a79 6779 206d 6f64 756c 6520 6f66   szygy module of
│ │ │ │ -00024560: 204d 2073 7563 6820 7468 6174 2072 6567   M such that reg
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│ │ │ │ -00024580: 6f64 756c 650a 4d20 3d30 2041 4e44 2072  odule.M =0 AND r
│ │ │ │ -00024590: 6567 756c 6172 6974 7920 6f64 6445 7874  egularity oddExt
│ │ │ │ -000245a0: 4d6f 6475 6c65 204d 203d 300a 0a57 6520  Module M =0..We 
│ │ │ │ -000245b0: 6861 7665 2062 6565 6e20 7573 696e 6720  have been using 
│ │ │ │ -000245c0: 7265 6775 6c61 7269 7479 2045 7874 4d6f  regularity ExtMo
│ │ │ │ -000245d0: 6475 6c65 204d 2061 7320 6120 7375 6273  dule M as a subs
│ │ │ │ -000245e0: 7469 7475 7465 2066 6f72 2072 2c20 6275  titute for r, bu
│ │ │ │ -000245f0: 7420 7468 6174 2773 206e 6f74 0a61 6c77  t that's not.alw
│ │ │ │ -00024600: 6179 7320 7468 6520 7361 6d65 2e0a 0a54  ays the same...T
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│ │ │ │ -00024670: 726f 6475 6365 6420 7769 7468 2073 6574  roduced with set
│ │ │ │ -00024680: 5261 6e64 6f6d 5365 6564 2030 2053 203d  RandomSeed 0 S =
│ │ │ │ -00024690: 205a 5a2f 3130 315b 612c 622c 632c 645d   ZZ/101[a,b,c,d]
│ │ │ │ -000246a0: 2066 660a 3d6d 6174 7269 7822 6134 2c62   ff.=matrix"a4,b
│ │ │ │ -000246b0: 342c 6334 2c64 3422 2052 203d 2053 2f69  4,c4,d4" R = S/i
│ │ │ │ -000246c0: 6465 616c 2066 6620 4e20 3d20 636f 6b65  deal ff N = coke
│ │ │ │ -000246d0: 7220 7261 6e64 6f6d 2852 5e7b 302c 317d  r random(R^{0,1}
│ │ │ │ -000246e0: 2c20 525e 7b20 2d31 2c2d 322c 2d33 2c2d  , R^{ -1,-2,-3,-
│ │ │ │ -000246f0: 347d 290a 2d2d 6769 7665 7320 7265 6720  4}).--gives reg 
│ │ │ │ -00024700: 4578 745e 6576 656e 203d 2034 2c20 7265  Ext^even = 4, re
│ │ │ │ -00024710: 6720 4578 745e 6f64 6420 3d20 3320 4c20  g Ext^odd = 3 L 
│ │ │ │ -00024720: 3d20 4578 744d 6f64 756c 6544 6174 6120  = ExtModuleData 
│ │ │ │ -00024730: 4e3b 2062 7574 2074 616b 6573 2073 6f6d  N; but takes som
│ │ │ │ -00024740: 650a 7469 6d65 2074 6f20 636f 6d70 7574  e.time to comput
│ │ │ │ -00024750: 652e 0a0a 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d  e.....+---------
│ │ │ │ +00024210: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573  ********..  * Us
│ │ │ │ +00024220: 6167 653a 200a 2020 2020 2020 2020 4c20  age: .        L 
│ │ │ │ +00024230: 3d20 4578 744d 6f64 756c 6544 6174 6120  = ExtModuleData 
│ │ │ │ +00024240: 4d0a 2020 2a20 496e 7075 7473 3a0a 2020  M.  * Inputs:.  
│ │ │ │ +00024250: 2020 2020 2a20 4d2c 2061 202a 6e6f 7465      * M, a *note
│ │ │ │ +00024260: 206d 6f64 756c 653a 2028 4d61 6361 756c   module: (Macaul
│ │ │ │ +00024270: 6179 3244 6f63 294d 6f64 756c 652c 2c20  ay2Doc)Module,, 
│ │ │ │ +00024280: 4d6f 6475 6c65 206f 7665 7220 6120 636f  Module over a co
│ │ │ │ +00024290: 6d70 6c65 7465 0a20 2020 2020 2020 2069  mplete.        i
│ │ │ │ +000242a0: 6e74 6572 7365 6374 696f 6e20 530a 2020  ntersection S.  
│ │ │ │ +000242b0: 2a20 4f75 7470 7574 733a 0a20 2020 2020  * Outputs:.     
│ │ │ │ +000242c0: 202a 204c 2c20 6120 2a6e 6f74 6520 6c69   * L, a *note li
│ │ │ │ +000242d0: 7374 3a20 284d 6163 6175 6c61 7932 446f  st: (Macaulay2Do
│ │ │ │ +000242e0: 6329 4c69 7374 2c2c 204c 203d 205c 7b65  c)List,, L = \{e
│ │ │ │ +000242f0: 7665 6e45 7874 4d6f 6475 6c65 2c0a 2020  venExtModule,.  
│ │ │ │ +00024300: 2020 2020 2020 6f64 6445 7874 4d6f 6475        oddExtModu
│ │ │ │ +00024310: 6c65 2c20 7265 6730 2c20 7265 6731 5c7d  le, reg0, reg1\}
│ │ │ │ +00024320: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d  ..Description.==
│ │ │ │ +00024330: 3d3d 3d3d 3d3d 3d3d 3d0a 0a53 7570 706f  =========..Suppo
│ │ │ │ +00024340: 7365 2074 6861 7420 4d20 6973 2061 206d  se that M is a m
│ │ │ │ +00024350: 6f64 756c 6520 6f76 6572 2061 2063 6f6d  odule over a com
│ │ │ │ +00024360: 706c 6574 6520 696e 7465 7273 6563 7469  plete intersecti
│ │ │ │ +00024370: 6f6e 2052 2073 6f20 7468 6174 0a0a 4520  on R so that..E 
│ │ │ │ +00024380: 3a3d 2045 7874 4d6f 6475 6c65 204d 0a0a  := ExtModule M..
│ │ │ │ +00024390: 6973 2061 206d 6f64 756c 6520 6765 6e65  is a module gene
│ │ │ │ +000243a0: 7261 7465 6420 696e 2064 6567 7265 6573  rated in degrees
│ │ │ │ +000243b0: 203e 3d30 206f 7665 7220 6120 706f 6c79   >=0 over a poly
│ │ │ │ +000243c0: 6e6f 6d69 616c 2072 696e 6720 5427 2067  nomial ring T' g
│ │ │ │ +000243d0: 656e 6572 6174 6564 2069 6e0a 6465 6772  enerated in.degr
│ │ │ │ +000243e0: 6565 2032 2c20 616e 640a 0a45 3020 3a3d  ee 2, and..E0 :=
│ │ │ │ +000243f0: 2065 7665 6e45 7874 4d6f 6475 6c65 204d   evenExtModule M
│ │ │ │ +00024400: 2061 6e64 2045 3120 3a3d 206f 6464 4578   and E1 := oddEx
│ │ │ │ +00024410: 744d 6f64 756c 6520 4d0a 0a61 7265 206d  tModule M..are m
│ │ │ │ +00024420: 6f64 756c 6573 2067 656e 6572 6174 6564  odules generated
│ │ │ │ +00024430: 2069 6e20 6465 6772 6565 203e 3d20 3020   in degree >= 0 
│ │ │ │ +00024440: 6f76 6572 2061 2070 6f6c 796e 6f6d 6961  over a polynomia
│ │ │ │ +00024450: 6c20 7269 6e67 2054 2077 6974 6820 6765  l ring T with ge
│ │ │ │ +00024460: 6e65 7261 746f 7273 0a69 6e20 6465 6772  nerators.in degr
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│ │ │ │ +00024480: 7420 7265 7475 726e 730a 0a4c 203d 205c  t returns..L = \
│ │ │ │ +00024490: 7b45 302c 4531 2c20 7265 6775 6c61 7269  {E0,E1, regulari
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│ │ │ │ +000244b0: 7920 4531 5c7d 0a0a 616e 6420 7072 696e  y E1\}..and prin
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│ │ │ │ +000244d0: 7c72 6567 302d 7265 6731 7c3e 312e 0a0a  |reg0-reg1|>1...
│ │ │ │ +000244e0: 4966 2077 6520 7365 7420 7220 3d20 6d61  If we set r = ma
│ │ │ │ +000244f0: 7828 322a 7265 6730 2c20 312b 322a 7265  x(2*reg0, 1+2*re
│ │ │ │ +00024500: 6731 292c 2061 6e64 2046 2069 7320 6120  g1), and F is a 
│ │ │ │ +00024510: 7265 736f 6c75 7469 6f6e 206f 6620 4d2c  resolution of M,
│ │ │ │ +00024520: 2074 6865 6e20 636f 6b65 720a 462e 6464   then coker.F.dd
│ │ │ │ +00024530: 5f7b 2872 2b31 297d 2069 7320 7468 6520  _{(r+1)} is the 
│ │ │ │ +00024540: 6669 7273 7420 737a 7967 7920 6d6f 6475  first szygy modu
│ │ │ │ +00024550: 6c65 206f 6620 4d20 7375 6368 2074 6861  le of M such tha
│ │ │ │ +00024560: 7420 7265 6775 6c61 7269 7479 2065 7665  t regularity eve
│ │ │ │ +00024570: 6e45 7874 4d6f 6475 6c65 0a4d 203d 3020  nExtModule.M =0 
│ │ │ │ +00024580: 414e 4420 7265 6775 6c61 7269 7479 206f  AND regularity o
│ │ │ │ +00024590: 6464 4578 744d 6f64 756c 6520 4d20 3d30  ddExtModule M =0
│ │ │ │ +000245a0: 0a0a 5765 2068 6176 6520 6265 656e 2075  ..We have been u
│ │ │ │ +000245b0: 7369 6e67 2072 6567 756c 6172 6974 7920  sing regularity 
│ │ │ │ +000245c0: 4578 744d 6f64 756c 6520 4d20 6173 2061  ExtModule M as a
│ │ │ │ +000245d0: 2073 7562 7374 6974 7574 6520 666f 7220   substitute for 
│ │ │ │ +000245e0: 722c 2062 7574 2074 6861 7427 7320 6e6f  r, but that's no
│ │ │ │ +000245f0: 740a 616c 7761 7973 2074 6865 2073 616d  t.always the sam
│ │ │ │ +00024600: 652e 0a0a 5468 6520 7265 6775 6c61 7269  e...The regulari
│ │ │ │ +00024610: 7469 6573 206f 6620 7468 6520 6576 656e  ties of the even
│ │ │ │ +00024620: 2061 6e64 206f 6464 2045 7874 206d 6f64   and odd Ext mod
│ │ │ │ +00024630: 756c 6573 202a 6361 6e2a 2064 6966 6665  ules *can* diffe
│ │ │ │ +00024640: 7220 6279 206d 6f72 6520 7468 616e 2031  r by more than 1
│ │ │ │ +00024650: 2e0a 416e 2065 7861 6d70 6c65 2063 616e  ..An example can
│ │ │ │ +00024660: 2062 6520 7072 6f64 7563 6564 2077 6974   be produced wit
│ │ │ │ +00024670: 6820 7365 7452 616e 646f 6d53 6565 6420  h setRandomSeed 
│ │ │ │ +00024680: 3020 5320 3d20 5a5a 2f31 3031 5b61 2c62  0 S = ZZ/101[a,b
│ │ │ │ +00024690: 2c63 2c64 5d20 6666 0a3d 6d61 7472 6978  ,c,d] ff.=matrix
│ │ │ │ +000246a0: 2261 342c 6234 2c63 342c 6434 2220 5220  "a4,b4,c4,d4" R 
│ │ │ │ +000246b0: 3d20 532f 6964 6561 6c20 6666 204e 203d  = S/ideal ff N =
│ │ │ │ +000246c0: 2063 6f6b 6572 2072 616e 646f 6d28 525e   coker random(R^
│ │ │ │ +000246d0: 7b30 2c31 7d2c 2052 5e7b 202d 312c 2d32  {0,1}, R^{ -1,-2
│ │ │ │ +000246e0: 2c2d 332c 2d34 7d29 0a2d 2d67 6976 6573  ,-3,-4}).--gives
│ │ │ │ +000246f0: 2072 6567 2045 7874 5e65 7665 6e20 3d20   reg Ext^even = 
│ │ │ │ +00024700: 342c 2072 6567 2045 7874 5e6f 6464 203d  4, reg Ext^odd =
│ │ │ │ +00024710: 2033 204c 203d 2045 7874 4d6f 6475 6c65   3 L = ExtModule
│ │ │ │ +00024720: 4461 7461 204e 3b20 6275 7420 7461 6b65  Data N; but take
│ │ │ │ +00024730: 7320 736f 6d65 0a74 696d 6520 746f 2063  s some.time to c
│ │ │ │ +00024740: 6f6d 7075 7465 2e0a 0a0a 0a2b 2d2d 2d2d  ompute.....+----
│ │ │ │ +00024750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -00024790: 203a 2073 6574 5261 6e64 6f6d 5365 6564   : setRandomSeed
│ │ │ │ -000247a0: 2031 3030 2020 2020 2020 2020 2020 2020   100            
│ │ │ │ -000247b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000247c0: 2020 7c0a 7c20 2d2d 2073 6574 7469 6e67    |.| -- setting
│ │ │ │ -000247d0: 2072 616e 646f 6d20 7365 6564 2074 6f20   random seed to 
│ │ │ │ -000247e0: 3130 3020 2020 2020 2020 2020 2020 2020  100             
│ │ │ │ -000247f0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00024780: 2b0a 7c69 3120 3a20 7365 7452 616e 646f  +.|i1 : setRando
│ │ │ │ +00024790: 6d53 6565 6420 3130 3020 2020 2020 2020  mSeed 100       
│ │ │ │ +000247a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000247b0: 2020 2020 2020 207c 0a7c 202d 2d20 7365         |.| -- se
│ │ │ │ +000247c0: 7474 696e 6720 7261 6e64 6f6d 2073 6565  tting random see
│ │ │ │ +000247d0: 6420 746f 2031 3030 2020 2020 2020 2020  d to 100        
│ │ │ │ +000247e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000247f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00024800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024830: 7c0a 7c6f 3120 3d20 3130 3020 2020 2020  |.|o1 = 100     
│ │ │ │ +00024820: 2020 2020 207c 0a7c 6f31 203d 2031 3030       |.|o1 = 100
│ │ │ │ +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 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00024850: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00024860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -000248a0: 7c69 3220 3a20 5320 3d20 5a5a 2f31 3031  |i2 : S = ZZ/101
│ │ │ │ -000248b0: 5b61 2c62 2c63 2c64 5d3b 2020 2020 2020  [a,b,c,d];      
│ │ │ │ -000248c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000248d0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00024890: 2d2d 2d2b 0a7c 6932 203a 2053 203d 205a  ---+.|i2 : S = Z
│ │ │ │ +000248a0: 5a2f 3130 315b 612c 622c 632c 645d 3b20  Z/101[a,b,c,d]; 
│ │ │ │ +000248b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000248c0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +000248d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000248e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000248f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00024910: 3320 3a20 6620 3d20 6d61 7028 535e 312c  3 : f = map(S^1,
│ │ │ │ -00024920: 2053 5e34 2c20 2869 2c6a 2920 2d3e 2053   S^4, (i,j) -> S
│ │ │ │ -00024930: 5f6a 5e33 2920 2020 2020 2020 2020 2020  _j^3)           
│ │ │ │ -00024940: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00024900: 2d2b 0a7c 6933 203a 2066 203d 206d 6170  -+.|i3 : f = map
│ │ │ │ +00024910: 2853 5e31 2c20 535e 342c 2028 692c 6a29  (S^1, S^4, (i,j)
│ │ │ │ +00024920: 202d 3e20 535f 6a5e 3329 2020 2020 2020   -> S_j^3)      
│ │ │ │ +00024930: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00024940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024970: 2020 2020 2020 2020 2020 7c0a 7c6f 3320            |.|o3 
│ │ │ │ -00024980: 3d20 7c20 6133 2062 3320 6333 2064 3320  = | a3 b3 c3 d3 
│ │ │ │ -00024990: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -000249a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000249b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00024960: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00024970: 0a7c 6f33 203d 207c 2061 3320 6233 2063  .|o3 = | a3 b3 c
│ │ │ │ +00024980: 3320 6433 207c 2020 2020 2020 2020 2020  3 d3 |          
│ │ │ │ +00024990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000249a0: 2020 2020 2020 7c0a 7c20 2020 2020 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 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000249f0: 2020 2020 2020 2020 3120 2020 2020 2034          1      4
│ │ │ │ +000249d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000249e0: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │ +000249f0: 2020 2020 3420 2020 2020 2020 2020 2020      4           
│ │ │ │  00024a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024a10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00024a20: 0a7c 6f33 203a 204d 6174 7269 7820 5320  .|o3 : Matrix S 
│ │ │ │ -00024a30: 203c 2d2d 2053 2020 2020 2020 2020 2020   <-- S          
│ │ │ │ -00024a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024a50: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00024a10: 2020 2020 7c0a 7c6f 3320 3a20 4d61 7472      |.|o3 : Matr
│ │ │ │ +00024a20: 6978 2053 2020 3c2d 2d20 5320 2020 2020  ix S  <-- S     
│ │ │ │ +00024a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024a40: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00024a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00024a90: 6934 203a 2052 203d 2053 2f69 6465 616c  i4 : R = S/ideal
│ │ │ │ -00024aa0: 2066 3b20 2020 2020 2020 2020 2020 2020   f;             
│ │ │ │ -00024ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024ac0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00024a80: 2d2d 2b0a 7c69 3420 3a20 5220 3d20 532f  --+.|i4 : R = S/
│ │ │ │ +00024a90: 6964 6561 6c20 663b 2020 2020 2020 2020  ideal f;        
│ │ │ │ +00024aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024ab0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00024ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935  -----------+.|i5
│ │ │ │ -00024b00: 203a 204d 203d 2052 5e31 2f69 6465 616c   : M = R^1/ideal
│ │ │ │ -00024b10: 2261 6232 2b63 6432 223b 2020 2020 2020  "ab2+cd2";      
│ │ │ │ -00024b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024b30: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00024af0: 2b0a 7c69 3520 3a20 4d20 3d20 525e 312f  +.|i5 : M = R^1/
│ │ │ │ +00024b00: 6964 6561 6c22 6162 322b 6364 3222 3b20  ideal"ab2+cd2"; 
│ │ │ │ +00024b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024b20: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00024b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024b60: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a  ---------+.|i6 :
│ │ │ │ -00024b70: 2062 6574 7469 2028 4620 3d20 6672 6565   betti (F = free
│ │ │ │ -00024b80: 5265 736f 6c75 7469 6f6e 284d 2c20 4c65  Resolution(M, Le
│ │ │ │ -00024b90: 6e67 7468 4c69 6d69 7420 3d3e 2035 2929  ngthLimit => 5))
│ │ │ │ -00024ba0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00024b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00024b60: 7c69 3620 3a20 6265 7474 6920 2846 203d  |i6 : betti (F =
│ │ │ │ +00024b70: 2066 7265 6552 6573 6f6c 7574 696f 6e28   freeResolution(
│ │ │ │ +00024b80: 4d2c 204c 656e 6774 684c 696d 6974 203d  M, LengthLimit =
│ │ │ │ +00024b90: 3e20 3529 297c 0a7c 2020 2020 2020 2020  > 5))|.|        
│ │ │ │ +00024ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024bd0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00024be0: 2020 2020 2020 3020 3120 3220 2033 2020        0 1 2  3  
│ │ │ │ -00024bf0: 3420 2035 2020 2020 2020 2020 2020 2020  4  5            
│ │ │ │ -00024c00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00024c10: 7c6f 3620 3d20 746f 7461 6c3a 2031 2031  |o6 = total: 1 1
│ │ │ │ -00024c20: 2035 2031 3620 3335 2036 3420 2020 2020   5 16 35 64     
│ │ │ │ -00024c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024c40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00024c50: 2030 3a20 3120 2e20 2e20 202e 2020 2e20   0: 1 . .  .  . 
│ │ │ │ -00024c60: 202e 2020 2020 2020 2020 2020 2020 2020   .              
│ │ │ │ -00024c70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00024c80: 2020 2020 2020 2020 313a 202e 202e 202e          1: . . .
│ │ │ │ -00024c90: 2020 2e20 202e 2020 2e20 2020 2020 2020    .  .  .       
│ │ │ │ -00024ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024cb0: 2020 207c 0a7c 2020 2020 2020 2020 2032     |.|         2
│ │ │ │ -00024cc0: 3a20 2e20 3120 2e20 202e 2020 2e20 202e  : . 1 .  .  .  .
│ │ │ │ -00024cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024ce0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00024cf0: 2020 2020 2020 333a 202e 202e 2031 2020        3: . . 1  
│ │ │ │ -00024d00: 2e20 202e 2020 2e20 2020 2020 2020 2020  .  .  .         
│ │ │ │ -00024d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024d20: 207c 0a7c 2020 2020 2020 2020 2034 3a20   |.|         4: 
│ │ │ │ -00024d30: 2e20 2e20 3320 2038 2020 3520 202e 2020  . . 3  8  5  .  
│ │ │ │ -00024d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024d50: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00024d60: 2020 2020 353a 202e 202e 2031 2020 3820      5: . . 1  8 
│ │ │ │ -00024d70: 3235 2033 3220 2020 2020 2020 2020 2020  25 32           
│ │ │ │ -00024d80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00024d90: 0a7c 2020 2020 2020 2020 2036 3a20 2e20  .|         6: . 
│ │ │ │ -00024da0: 2e20 2e20 202e 2020 3520 3332 2020 2020  . .  .  5 32    
│ │ │ │ -00024db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024dc0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00024bc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00024bd0: 2020 2020 2020 2020 2020 2030 2031 2032             0 1 2
│ │ │ │ +00024be0: 2020 3320 2034 2020 3520 2020 2020 2020    3  4  5       
│ │ │ │ +00024bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024c00: 2020 207c 0a7c 6f36 203d 2074 6f74 616c     |.|o6 = total
│ │ │ │ +00024c10: 3a20 3120 3120 3520 3136 2033 3520 3634  : 1 1 5 16 35 64
│ │ │ │ +00024c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024c30: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00024c40: 2020 2020 2020 303a 2031 202e 202e 2020        0: 1 . .  
│ │ │ │ +00024c50: 2e20 202e 2020 2e20 2020 2020 2020 2020  .  .  .         
│ │ │ │ +00024c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024c70: 207c 0a7c 2020 2020 2020 2020 2031 3a20   |.|         1: 
│ │ │ │ +00024c80: 2e20 2e20 2e20 202e 2020 2e20 202e 2020  . . .  .  .  .  
│ │ │ │ +00024c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024ca0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00024cb0: 2020 2020 323a 202e 2031 202e 2020 2e20      2: . 1 .  . 
│ │ │ │ +00024cc0: 202e 2020 2e20 2020 2020 2020 2020 2020   .  .           
│ │ │ │ +00024cd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00024ce0: 0a7c 2020 2020 2020 2020 2033 3a20 2e20  .|         3: . 
│ │ │ │ +00024cf0: 2e20 3120 202e 2020 2e20 202e 2020 2020  . 1  .  .  .    
│ │ │ │ +00024d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024d10: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00024d20: 2020 343a 202e 202e 2033 2020 3820 2035    4: . . 3  8  5
│ │ │ │ +00024d30: 2020 2e20 2020 2020 2020 2020 2020 2020    .             
│ │ │ │ +00024d40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00024d50: 2020 2020 2020 2020 2035 3a20 2e20 2e20           5: . . 
│ │ │ │ +00024d60: 3120 2038 2032 3520 3332 2020 2020 2020  1  8 25 32      
│ │ │ │ +00024d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024d80: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024d90: 363a 202e 202e 202e 2020 2e20 2035 2033  6: . . .  .  5 3
│ │ │ │ +00024da0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00024db0: 2020 2020 2020 2020 2020 207c 0a7c 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 207c 0a7c               |.|
│ │ │ │ -00024e00: 6f36 203a 2042 6574 7469 5461 6c6c 7920  o6 : BettiTally 
│ │ │ │ +00024df0: 2020 7c0a 7c6f 3620 3a20 4265 7474 6954    |.|o6 : BettiT
│ │ │ │ +00024e00: 616c 6c79 2020 2020 2020 2020 2020 2020  ally            
│ │ │ │  00024e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024e30: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00024e20: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00024e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937  -----------+.|i7
│ │ │ │ -00024e70: 203a 2045 203d 2045 7874 4d6f 6475 6c65   : E = ExtModule
│ │ │ │ -00024e80: 4461 7461 204d 3b20 2020 2020 2020 2020  Data M;         
│ │ │ │ -00024e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024ea0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00024e60: 2b0a 7c69 3720 3a20 4520 3d20 4578 744d  +.|i7 : E = ExtM
│ │ │ │ +00024e70: 6f64 756c 6544 6174 6120 4d3b 2020 2020  oduleData M;    
│ │ │ │ +00024e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024e90: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00024ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024ed0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a  ---------+.|i8 :
│ │ │ │ -00024ee0: 2045 5f32 2020 2020 2020 2020 2020 2020   E_2            
│ │ │ │ +00024ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00024ed0: 7c69 3820 3a20 455f 3220 2020 2020 2020  |i8 : E_2       
│ │ │ │ +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: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00024f00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00024f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024f40: 2020 2020 2020 207c 0a7c 6f38 203d 2032         |.|o8 = 2
│ │ │ │ +00024f30: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00024f40: 3820 3d20 3220 2020 2020 2020 2020 2020  8 = 2           
│ │ │ │  00024f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024f70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00024f80: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00024f70: 2020 207c 0a2b 2d2d 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 2d2b 0a7c 6939 203a 2045 5f33  -----+.|i9 : E_3
│ │ │ │ +00024fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3920  ----------+.|i9 
│ │ │ │ +00024fb0: 3a20 455f 3320 2020 2020 2020 2020 2020  : E_3           
│ │ │ │  00024fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024fe0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00024fe0: 207c 0a7c 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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025020: 2020 207c 0a7c 6f39 203d 2031 2020 2020     |.|o9 = 1    
│ │ │ │ +00025010: 2020 2020 2020 2020 7c0a 7c6f 3920 3d20          |.|o9 = 
│ │ │ │ +00025020: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │  00025030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025050: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00025040: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00025050: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00025060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025090: 2d2b 0a7c 6931 3020 3a20 7220 3d20 6d61  -+.|i10 : r = ma
│ │ │ │ -000250a0: 7828 322a 455f 322c 322a 455f 332b 3129  x(2*E_2,2*E_3+1)
│ │ │ │ -000250b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000250c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00025080: 2d2d 2d2d 2d2d 2b0a 7c69 3130 203a 2072  ------+.|i10 : r
│ │ │ │ +00025090: 203d 206d 6178 2832 2a45 5f32 2c32 2a45   = max(2*E_2,2*E
│ │ │ │ +000250a0: 5f33 2b31 2920 2020 2020 2020 2020 2020  _3+1)           
│ │ │ │ +000250b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +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 207c                 |
│ │ │ │ -00025100: 0a7c 6f31 3020 3d20 3420 2020 2020 2020  .|o10 = 4       
│ │ │ │ +000250f0: 2020 2020 7c0a 7c6f 3130 203d 2034 2020      |.|o10 = 4  
│ │ │ │ +00025100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025130: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00025120: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00025130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00025170: 6931 3120 3a20 4572 203d 2045 7874 4d6f  i11 : Er = ExtMo
│ │ │ │ -00025180: 6475 6c65 4461 7461 2063 6f6b 6572 2046  duleData coker F
│ │ │ │ -00025190: 2e64 645f 723b 2020 2020 2020 2020 2020  .dd_r;          
│ │ │ │ -000251a0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00025160: 2d2d 2b0a 7c69 3131 203a 2045 7220 3d20  --+.|i11 : Er = 
│ │ │ │ +00025170: 4578 744d 6f64 756c 6544 6174 6120 636f  ExtModuleData co
│ │ │ │ +00025180: 6b65 7220 462e 6464 5f72 3b20 2020 2020  ker F.dd_r;     
│ │ │ │ +00025190: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +000251a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000251b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000251c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000251d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -000251e0: 3220 3a20 7265 6775 6c61 7269 7479 2045  2 : regularity E
│ │ │ │ -000251f0: 725f 3020 2020 2020 2020 2020 2020 2020  r_0             
│ │ │ │ -00025200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025210: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000251d0: 2b0a 7c69 3132 203a 2072 6567 756c 6172  +.|i12 : regular
│ │ │ │ +000251e0: 6974 7920 4572 5f30 2020 2020 2020 2020  ity Er_0        
│ │ │ │ +000251f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025200: 2020 2020 2020 207c 0a7c 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: 2020 2020 2020 2020 207c 0a7c 6f31 3220           |.|o12 
│ │ │ │ -00025250: 3d20 3020 2020 2020 2020 2020 2020 2020  = 0             
│ │ │ │ +00025230: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00025240: 7c6f 3132 203d 2030 2020 2020 2020 2020  |o12 = 0        
│ │ │ │ +00025250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025280: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00025270: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00025280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000252a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000252b0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3320 3a20  -------+.|i13 : 
│ │ │ │ -000252c0: 7265 6775 6c61 7269 7479 2045 725f 3120  regularity Er_1 
│ │ │ │ +000252a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +000252b0: 3133 203a 2072 6567 756c 6172 6974 7920  13 : regularity 
│ │ │ │ +000252c0: 4572 5f31 2020 2020 2020 2020 2020 2020  Er_1            
│ │ │ │  000252d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000252e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000252f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000252e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000252f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025320: 2020 2020 207c 0a7c 6f31 3320 3d20 3020       |.|o13 = 0 
│ │ │ │ +00025310: 2020 2020 2020 2020 2020 7c0a 7c6f 3133            |.|o13
│ │ │ │ +00025320: 203d 2030 2020 2020 2020 2020 2020 2020   = 0            
│ │ │ │  00025330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025350: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00025350: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  00025360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025390: 2d2d 2d2b 0a7c 6931 3420 3a20 7265 6775  ---+.|i14 : regu
│ │ │ │ -000253a0: 6c61 7269 7479 2065 7665 6e45 7874 4d6f  larity evenExtMo
│ │ │ │ -000253b0: 6475 6c65 2863 6f6b 6572 2046 2e64 645f  dule(coker F.dd_
│ │ │ │ -000253c0: 2872 2d31 2929 2020 2020 7c0a 7c20 2020  (r-1))    |.|   
│ │ │ │ +00025380: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3134 203a  --------+.|i14 :
│ │ │ │ +00025390: 2072 6567 756c 6172 6974 7920 6576 656e   regularity even
│ │ │ │ +000253a0: 4578 744d 6f64 756c 6528 636f 6b65 7220  ExtModule(coker 
│ │ │ │ +000253b0: 462e 6464 5f28 722d 3129 2920 2020 207c  F.dd_(r-1))    |
│ │ │ │ +000253c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000253d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000253e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000253f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025400: 207c 0a7c 6f31 3420 3d20 3120 2020 2020   |.|o14 = 1     
│ │ │ │ +000253f0: 2020 2020 2020 7c0a 7c6f 3134 203d 2031        |.|o14 = 1
│ │ │ │ +00025400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025430: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00025420: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00025430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00025470: 0a7c 6931 3520 3a20 6666 203d 2066 2a72  .|i15 : ff = f*r
│ │ │ │ -00025480: 616e 646f 6d28 736f 7572 6365 2066 2c20  andom(source f, 
│ │ │ │ -00025490: 736f 7572 6365 2066 293b 2020 2020 2020  source f);      
│ │ │ │ -000254a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00025460: 2d2d 2d2d 2b0a 7c69 3135 203a 2066 6620  ----+.|i15 : ff 
│ │ │ │ +00025470: 3d20 662a 7261 6e64 6f6d 2873 6f75 7263  = f*random(sourc
│ │ │ │ +00025480: 6520 662c 2073 6f75 7263 6520 6629 3b20  e f, source f); 
│ │ │ │ +00025490: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000254a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000254b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000254c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000254d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000254e0: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ -000254f0: 2020 2020 2034 2020 2020 2020 2020 2020       4          
│ │ │ │ -00025500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025510: 2020 2020 7c0a 7c6f 3135 203a 204d 6174      |.|o15 : Mat
│ │ │ │ -00025520: 7269 7820 5320 203c 2d2d 2053 2020 2020  rix S  <-- S    
│ │ │ │ +000254d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000254e0: 2020 2031 2020 2020 2020 3420 2020 2020     1      4     
│ │ │ │ +000254f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025500: 2020 2020 2020 2020 207c 0a7c 6f31 3520           |.|o15 
│ │ │ │ +00025510: 3a20 4d61 7472 6978 2053 2020 3c2d 2d20  : Matrix S  <-- 
│ │ │ │ +00025520: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │  00025530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025540: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00025540: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00025550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025580: 2d2d 2b0a 7c69 3136 203a 206d 6174 7269  --+.|i16 : matri
│ │ │ │ -00025590: 7846 6163 746f 7269 7a61 7469 6f6e 2866  xFactorization(f
│ │ │ │ -000255a0: 662c 2063 6f6b 6572 2046 2e64 645f 2872  f, coker F.dd_(r
│ │ │ │ -000255b0: 2b31 2929 3b20 2020 207c 0a2b 2d2d 2d2d  +1));    |.+----
│ │ │ │ +00025570: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3620 3a20  -------+.|i16 : 
│ │ │ │ +00025580: 6d61 7472 6978 4661 6374 6f72 697a 6174  matrixFactorizat
│ │ │ │ +00025590: 696f 6e28 6666 2c20 636f 6b65 7220 462e  ion(ff, coker F.
│ │ │ │ +000255a0: 6464 5f28 722b 3129 293b 2020 2020 7c0a  dd_(r+1));    |.
│ │ │ │ +000255b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  000255c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000255d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000255e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000255f0: 2b0a 0a54 6869 7320 7375 6363 6565 6473  +..This succeeds
│ │ │ │ -00025600: 2c20 6275 7420 7765 2063 6f75 6c64 2067  , but we could g
│ │ │ │ -00025610: 6574 2061 6e20 6572 726f 7220 6672 6f6d  et an error from
│ │ │ │ -00025620: 0a0a 6d61 7472 6978 4661 6374 6f72 697a  ..matrixFactoriz
│ │ │ │ -00025630: 6174 696f 6e28 6666 2c20 636f 6b65 7220  ation(ff, coker 
│ │ │ │ -00025640: 462e 6464 5f72 290a 0a69 6620 6f6e 6520  F.dd_r)..if one 
│ │ │ │ -00025650: 6f66 2074 6865 2043 4920 6f70 6572 6174  of the CI operat
│ │ │ │ -00025660: 6f72 7320 7765 7265 206e 6f74 2073 7572  ors were not sur
│ │ │ │ -00025670: 6a65 6374 6976 652e 0a0a 4361 7665 6174  jective...Caveat
│ │ │ │ -00025680: 0a3d 3d3d 3d3d 3d0a 0a45 7874 4d6f 6475  .======..ExtModu
│ │ │ │ -00025690: 6c65 2063 7265 6174 6573 2061 2072 696e  le creates a rin
│ │ │ │ -000256a0: 6720 696e 7369 6465 2074 6865 2073 6372  g inside the scr
│ │ │ │ -000256b0: 6970 742c 2073 6f20 6966 2069 7427 7320  ipt, so if it's 
│ │ │ │ -000256c0: 7275 6e20 7477 6963 6520 796f 7520 6765  run twice you ge
│ │ │ │ -000256d0: 740a 6d6f 6475 6c65 7320 6f76 6572 2064  t.modules over d
│ │ │ │ -000256e0: 6966 6665 7265 6e74 2072 696e 6773 2e20  ifferent rings. 
│ │ │ │ -000256f0: 5468 6973 2073 686f 756c 6420 6265 2063  This should be c
│ │ │ │ -00025700: 6861 6e67 6564 2e0a 0a53 6565 2061 6c73  hanged...See als
│ │ │ │ -00025710: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  o.========..  * 
│ │ │ │ -00025720: 2a6e 6f74 6520 4578 744d 6f64 756c 653a  *note ExtModule:
│ │ │ │ -00025730: 2045 7874 4d6f 6475 6c65 2c20 2d2d 2045   ExtModule, -- E
│ │ │ │ -00025740: 7874 5e2a 284d 2c6b 2920 6f76 6572 2061  xt^*(M,k) over a
│ │ │ │ -00025750: 2063 6f6d 706c 6574 6520 696e 7465 7273   complete inters
│ │ │ │ -00025760: 6563 7469 6f6e 2061 730a 2020 2020 6d6f  ection as.    mo
│ │ │ │ -00025770: 6475 6c65 206f 7665 7220 4349 206f 7065  dule over CI ope
│ │ │ │ -00025780: 7261 746f 7220 7269 6e67 0a20 202a 202a  rator ring.  * *
│ │ │ │ -00025790: 6e6f 7465 2065 7665 6e45 7874 4d6f 6475  note evenExtModu
│ │ │ │ -000257a0: 6c65 3a20 6576 656e 4578 744d 6f64 756c  le: evenExtModul
│ │ │ │ -000257b0: 652c 202d 2d20 6576 656e 2070 6172 7420  e, -- even part 
│ │ │ │ -000257c0: 6f66 2045 7874 5e2a 284d 2c6b 2920 6f76  of Ext^*(M,k) ov
│ │ │ │ -000257d0: 6572 2061 0a20 2020 2063 6f6d 706c 6574  er a.    complet
│ │ │ │ -000257e0: 6520 696e 7465 7273 6563 7469 6f6e 2061  e intersection a
│ │ │ │ -000257f0: 7320 6d6f 6475 6c65 206f 7665 7220 4349  s module over CI
│ │ │ │ -00025800: 206f 7065 7261 746f 7220 7269 6e67 0a20   operator ring. 
│ │ │ │ -00025810: 202a 202a 6e6f 7465 206f 6464 4578 744d   * *note oddExtM
│ │ │ │ -00025820: 6f64 756c 653a 206f 6464 4578 744d 6f64  odule: oddExtMod
│ │ │ │ -00025830: 756c 652c 202d 2d20 6f64 6420 7061 7274  ule, -- odd part
│ │ │ │ -00025840: 206f 6620 4578 745e 2a28 4d2c 6b29 206f   of Ext^*(M,k) o
│ │ │ │ -00025850: 7665 7220 6120 636f 6d70 6c65 7465 0a20  ver a complete. 
│ │ │ │ -00025860: 2020 2069 6e74 6572 7365 6374 696f 6e20     intersection 
│ │ │ │ -00025870: 6173 206d 6f64 756c 6520 6f76 6572 2043  as module over C
│ │ │ │ -00025880: 4920 6f70 6572 6174 6f72 2072 696e 670a  I operator ring.
│ │ │ │ -00025890: 0a57 6179 7320 746f 2075 7365 2045 7874  .Ways to use Ext
│ │ │ │ -000258a0: 4d6f 6475 6c65 4461 7461 3a0a 3d3d 3d3d  ModuleData:.====
│ │ │ │ -000258b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000258c0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2245 7874  ======..  * "Ext
│ │ │ │ -000258d0: 4d6f 6475 6c65 4461 7461 284d 6f64 756c  ModuleData(Modul
│ │ │ │ -000258e0: 6529 220a 0a46 6f72 2074 6865 2070 726f  e)"..For the pro
│ │ │ │ -000258f0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d  grammer.========
│ │ │ │ -00025900: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520  ==========..The 
│ │ │ │ -00025910: 6f62 6a65 6374 202a 6e6f 7465 2045 7874  object *note Ext
│ │ │ │ -00025920: 4d6f 6475 6c65 4461 7461 3a20 4578 744d  ModuleData: ExtM
│ │ │ │ -00025930: 6f64 756c 6544 6174 612c 2069 7320 6120  oduleData, is a 
│ │ │ │ -00025940: 2a6e 6f74 6520 6d65 7468 6f64 2066 756e  *note method fun
│ │ │ │ -00025950: 6374 696f 6e3a 0a28 4d61 6361 756c 6179  ction:.(Macaulay
│ │ │ │ -00025960: 3244 6f63 294d 6574 686f 6446 756e 6374  2Doc)MethodFunct
│ │ │ │ -00025970: 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d  ion,...---------
│ │ │ │ +000255e0: 2d2d 2d2d 2d2b 0a0a 5468 6973 2073 7563  -----+..This suc
│ │ │ │ +000255f0: 6365 6564 732c 2062 7574 2077 6520 636f  ceeds, but we co
│ │ │ │ +00025600: 756c 6420 6765 7420 616e 2065 7272 6f72  uld get an error
│ │ │ │ +00025610: 2066 726f 6d0a 0a6d 6174 7269 7846 6163   from..matrixFac
│ │ │ │ +00025620: 746f 7269 7a61 7469 6f6e 2866 662c 2063  torization(ff, c
│ │ │ │ +00025630: 6f6b 6572 2046 2e64 645f 7229 0a0a 6966  oker F.dd_r)..if
│ │ │ │ +00025640: 206f 6e65 206f 6620 7468 6520 4349 206f   one of the CI o
│ │ │ │ +00025650: 7065 7261 746f 7273 2077 6572 6520 6e6f  perators were no
│ │ │ │ +00025660: 7420 7375 726a 6563 7469 7665 2e0a 0a43  t surjective...C
│ │ │ │ +00025670: 6176 6561 740a 3d3d 3d3d 3d3d 0a0a 4578  aveat.======..Ex
│ │ │ │ +00025680: 744d 6f64 756c 6520 6372 6561 7465 7320  tModule creates 
│ │ │ │ +00025690: 6120 7269 6e67 2069 6e73 6964 6520 7468  a ring inside th
│ │ │ │ +000256a0: 6520 7363 7269 7074 2c20 736f 2069 6620  e script, so if 
│ │ │ │ +000256b0: 6974 2773 2072 756e 2074 7769 6365 2079  it's run twice y
│ │ │ │ +000256c0: 6f75 2067 6574 0a6d 6f64 756c 6573 206f  ou get.modules o
│ │ │ │ +000256d0: 7665 7220 6469 6666 6572 656e 7420 7269  ver different ri
│ │ │ │ +000256e0: 6e67 732e 2054 6869 7320 7368 6f75 6c64  ngs. This should
│ │ │ │ +000256f0: 2062 6520 6368 616e 6765 642e 0a0a 5365   be changed...Se
│ │ │ │ +00025700: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a  e also.========.
│ │ │ │ +00025710: 0a20 202a 202a 6e6f 7465 2045 7874 4d6f  .  * *note ExtMo
│ │ │ │ +00025720: 6475 6c65 3a20 4578 744d 6f64 756c 652c  dule: ExtModule,
│ │ │ │ +00025730: 202d 2d20 4578 745e 2a28 4d2c 6b29 206f   -- Ext^*(M,k) o
│ │ │ │ +00025740: 7665 7220 6120 636f 6d70 6c65 7465 2069  ver a complete i
│ │ │ │ +00025750: 6e74 6572 7365 6374 696f 6e20 6173 0a20  ntersection as. 
│ │ │ │ +00025760: 2020 206d 6f64 756c 6520 6f76 6572 2043     module over C
│ │ │ │ +00025770: 4920 6f70 6572 6174 6f72 2072 696e 670a  I operator ring.
│ │ │ │ +00025780: 2020 2a20 2a6e 6f74 6520 6576 656e 4578    * *note evenEx
│ │ │ │ +00025790: 744d 6f64 756c 653a 2065 7665 6e45 7874  tModule: evenExt
│ │ │ │ +000257a0: 4d6f 6475 6c65 2c20 2d2d 2065 7665 6e20  Module, -- even 
│ │ │ │ +000257b0: 7061 7274 206f 6620 4578 745e 2a28 4d2c  part of Ext^*(M,
│ │ │ │ +000257c0: 6b29 206f 7665 7220 610a 2020 2020 636f  k) over a.    co
│ │ │ │ +000257d0: 6d70 6c65 7465 2069 6e74 6572 7365 6374  mplete intersect
│ │ │ │ +000257e0: 696f 6e20 6173 206d 6f64 756c 6520 6f76  ion as module ov
│ │ │ │ +000257f0: 6572 2043 4920 6f70 6572 6174 6f72 2072  er CI operator r
│ │ │ │ +00025800: 696e 670a 2020 2a20 2a6e 6f74 6520 6f64  ing.  * *note od
│ │ │ │ +00025810: 6445 7874 4d6f 6475 6c65 3a20 6f64 6445  dExtModule: oddE
│ │ │ │ +00025820: 7874 4d6f 6475 6c65 2c20 2d2d 206f 6464  xtModule, -- odd
│ │ │ │ +00025830: 2070 6172 7420 6f66 2045 7874 5e2a 284d   part of Ext^*(M
│ │ │ │ +00025840: 2c6b 2920 6f76 6572 2061 2063 6f6d 706c  ,k) over a compl
│ │ │ │ +00025850: 6574 650a 2020 2020 696e 7465 7273 6563  ete.    intersec
│ │ │ │ +00025860: 7469 6f6e 2061 7320 6d6f 6475 6c65 206f  tion as module o
│ │ │ │ +00025870: 7665 7220 4349 206f 7065 7261 746f 7220  ver CI operator 
│ │ │ │ +00025880: 7269 6e67 0a0a 5761 7973 2074 6f20 7573  ring..Ways to us
│ │ │ │ +00025890: 6520 4578 744d 6f64 756c 6544 6174 613a  e ExtModuleData:
│ │ │ │ +000258a0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ +000258b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ +000258c0: 2022 4578 744d 6f64 756c 6544 6174 6128   "ExtModuleData(
│ │ │ │ +000258d0: 4d6f 6475 6c65 2922 0a0a 466f 7220 7468  Module)"..For th
│ │ │ │ +000258e0: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d  e programmer.===
│ │ │ │ +000258f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ +00025900: 0a54 6865 206f 626a 6563 7420 2a6e 6f74  .The object *not
│ │ │ │ +00025910: 6520 4578 744d 6f64 756c 6544 6174 613a  e ExtModuleData:
│ │ │ │ +00025920: 2045 7874 4d6f 6475 6c65 4461 7461 2c20   ExtModuleData, 
│ │ │ │ +00025930: 6973 2061 202a 6e6f 7465 206d 6574 686f  is a *note metho
│ │ │ │ +00025940: 6420 6675 6e63 7469 6f6e 3a0a 284d 6163  d function:.(Mac
│ │ │ │ +00025950: 6175 6c61 7932 446f 6329 4d65 7468 6f64  aulay2Doc)Method
│ │ │ │ +00025960: 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d  Function,...----
│ │ │ │ +00025970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000259a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000259b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000259c0: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572  ------..The sour
│ │ │ │ -000259d0: 6365 206f 6620 7468 6973 2064 6f63 756d  ce of this docum
│ │ │ │ -000259e0: 656e 7420 6973 2069 6e0a 2f62 7569 6c64  ent is in./build
│ │ │ │ -000259f0: 2f72 6570 726f 6475 6369 626c 652d 7061  /reproducible-pa
│ │ │ │ -00025a00: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32  th/macaulay2-1.2
│ │ │ │ -00025a10: 352e 3131 2b64 732f 4d32 2f4d 6163 6175  5.11+ds/M2/Macau
│ │ │ │ -00025a20: 6c61 7932 2f70 6163 6b61 6765 732f 0a43  lay2/packages/.C
│ │ │ │ -00025a30: 6f6d 706c 6574 6549 6e74 6572 7365 6374  ompleteIntersect
│ │ │ │ -00025a40: 696f 6e52 6573 6f6c 7574 696f 6e73 2e6d  ionResolutions.m
│ │ │ │ -00025a50: 323a 3334 3433 3a30 2e0a 1f0a 4669 6c65  2:3443:0....File
│ │ │ │ -00025a60: 3a20 436f 6d70 6c65 7465 496e 7465 7273  : CompleteInters
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│ │ │ │ -00025a80: 732e 696e 666f 2c20 4e6f 6465 3a20 6578  s.info, Node: ex
│ │ │ │ -00025a90: 7456 7343 6f68 6f6d 6f6c 6f67 792c 204e  tVsCohomology, N
│ │ │ │ -00025aa0: 6578 743a 2066 696e 6974 6542 6574 7469  ext: finiteBetti
│ │ │ │ -00025ab0: 4e75 6d62 6572 732c 2050 7265 763a 2045  Numbers, Prev: E
│ │ │ │ -00025ac0: 7874 4d6f 6475 6c65 4461 7461 2c20 5570  xtModuleData, Up
│ │ │ │ -00025ad0: 3a20 546f 700a 0a65 7874 5673 436f 686f  : Top..extVsCoho
│ │ │ │ -00025ae0: 6d6f 6c6f 6779 202d 2d20 636f 6d70 6172  mology -- compar
│ │ │ │ -00025af0: 6573 2045 7874 5f53 284d 2c6b 2920 6173  es Ext_S(M,k) as
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│ │ │ │ -00025b10: 2077 6974 6820 636f 6820 7461 626c 6520   with coh table 
│ │ │ │ -00025b20: 6f66 2073 6865 6166 2045 7874 5f52 284d  of sheaf Ext_R(M
│ │ │ │ -00025b30: 2c6b 290a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ,k).************
│ │ │ │ +000259b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865  -----------..The
│ │ │ │ +000259c0: 2073 6f75 7263 6520 6f66 2074 6869 7320   source of this 
│ │ │ │ +000259d0: 646f 6375 6d65 6e74 2069 7320 696e 0a2f  document is in./
│ │ │ │ +000259e0: 6275 696c 642f 7265 7072 6f64 7563 6962  build/reproducib
│ │ │ │ +000259f0: 6c65 2d70 6174 682f 6d61 6361 756c 6179  le-path/macaulay
│ │ │ │ +00025a00: 322d 312e 3235 2e31 312b 6473 2f4d 322f  2-1.25.11+ds/M2/
│ │ │ │ +00025a10: 4d61 6361 756c 6179 322f 7061 636b 6167  Macaulay2/packag
│ │ │ │ +00025a20: 6573 2f0a 436f 6d70 6c65 7465 496e 7465  es/.CompleteInte
│ │ │ │ +00025a30: 7273 6563 7469 6f6e 5265 736f 6c75 7469  rsectionResoluti
│ │ │ │ +00025a40: 6f6e 732e 6d32 3a33 3434 333a 302e 0a1f  ons.m2:3443:0...
│ │ │ │ +00025a50: 0a46 696c 653a 2043 6f6d 706c 6574 6549  .File: CompleteI
│ │ │ │ +00025a60: 6e74 6572 7365 6374 696f 6e52 6573 6f6c  ntersectionResol
│ │ │ │ +00025a70: 7574 696f 6e73 2e69 6e66 6f2c 204e 6f64  utions.info, Nod
│ │ │ │ +00025a80: 653a 2065 7874 5673 436f 686f 6d6f 6c6f  e: extVsCohomolo
│ │ │ │ +00025a90: 6779 2c20 4e65 7874 3a20 6669 6e69 7465  gy, Next: finite
│ │ │ │ +00025aa0: 4265 7474 694e 756d 6265 7273 2c20 5072  BettiNumbers, Pr
│ │ │ │ +00025ab0: 6576 3a20 4578 744d 6f64 756c 6544 6174  ev: ExtModuleDat
│ │ │ │ +00025ac0: 612c 2055 703a 2054 6f70 0a0a 6578 7456  a, Up: Top..extV
│ │ │ │ +00025ad0: 7343 6f68 6f6d 6f6c 6f67 7920 2d2d 2063  sCohomology -- c
│ │ │ │ +00025ae0: 6f6d 7061 7265 7320 4578 745f 5328 4d2c  ompares Ext_S(M,
│ │ │ │ +00025af0: 6b29 2061 7320 6578 7465 7269 6f72 206d  k) as exterior m
│ │ │ │ +00025b00: 6f64 756c 6520 7769 7468 2063 6f68 2074  odule with coh t
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│ │ │ │ +00025b20: 745f 5228 4d2c 6b29 0a2a 2a2a 2a2a 2a2a  t_R(M,k).*******
│ │ │ │ +00025b30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00025b40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00025b50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00025b60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00025b70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00025b80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00025b90: 0a0a 2020 2a20 5573 6167 653a 200a 2020  ..  * Usage: .  
│ │ │ │ -00025ba0: 2020 2020 2020 2845 2c54 2920 3d20 6578        (E,T) = ex
│ │ │ │ -00025bb0: 7456 7343 6f68 6f6d 6f6c 6f67 7928 6666  tVsCohomology(ff
│ │ │ │ -00025bc0: 2c4e 290a 2020 2a20 496e 7075 7473 3a0a  ,N).  * Inputs:.
│ │ │ │ -00025bd0: 2020 2020 2020 2a20 6666 2c20 6120 2a6e        * ff, a *n
│ │ │ │ -00025be0: 6f74 6520 6d61 7472 6978 3a20 284d 6163  ote matrix: (Mac
│ │ │ │ -00025bf0: 6175 6c61 7932 446f 6329 4d61 7472 6978  aulay2Doc)Matrix
│ │ │ │ -00025c00: 2c2c 2072 6567 756c 6172 2073 6571 7565  ,, regular seque
│ │ │ │ -00025c10: 6e63 6520 696e 2061 0a20 2020 2020 2020  nce in a.       
│ │ │ │ -00025c20: 2072 6567 756c 6172 2072 696e 6720 530a   regular ring S.
│ │ │ │ -00025c30: 2020 2020 2020 2a20 4e2c 2061 202a 6e6f        * N, a *no
│ │ │ │ -00025c40: 7465 206d 6f64 756c 653a 2028 4d61 6361  te module: (Maca
│ │ │ │ -00025c50: 756c 6179 3244 6f63 294d 6f64 756c 652c  ulay2Doc)Module,
│ │ │ │ -00025c60: 2c20 6772 6164 6564 206d 6f64 756c 6520  , graded module 
│ │ │ │ -00025c70: 6f76 6572 2052 203d 0a20 2020 2020 2020  over R =.       
│ │ │ │ -00025c80: 2053 2f69 6465 616c 2866 6629 2028 7573   S/ideal(ff) (us
│ │ │ │ -00025c90: 7561 6c6c 7920 6120 6869 6768 2073 797a  ually a high syz
│ │ │ │ -00025ca0: 7967 7929 0a20 202a 204f 7574 7075 7473  ygy).  * Outputs
│ │ │ │ -00025cb0: 3a0a 2020 2020 2020 2a20 452c 2061 202a  :.      * E, a *
│ │ │ │ -00025cc0: 6e6f 7465 206d 6f64 756c 653a 2028 4d61  note module: (Ma
│ │ │ │ -00025cd0: 6361 756c 6179 3244 6f63 294d 6f64 756c  caulay2Doc)Modul
│ │ │ │ -00025ce0: 652c 2c20 0a20 2020 2020 202a 2054 2c20  e,, .      * T, 
│ │ │ │ -00025cf0: 6120 2a6e 6f74 6520 6d6f 6475 6c65 3a20  a *note module: 
│ │ │ │ -00025d00: 284d 6163 6175 6c61 7932 446f 6329 4d6f  (Macaulay2Doc)Mo
│ │ │ │ -00025d10: 6475 6c65 2c2c 2045 7874 2061 6e64 2054  dule,, Ext and T
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│ │ │ │ -00025d30: 2020 2020 2020 206d 6f64 756c 6573 0a0a         modules..
│ │ │ │ -00025d40: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
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│ │ │ │ -00025d70: 696e 696e 6720 6120 7265 6775 6c61 7220  ining a regular 
│ │ │ │ -00025d80: 7365 7175 656e 6365 2069 6e20 6120 706f  sequence in a po
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│ │ │ │ -00025da0: 6f76 6572 206b 2c0a 7365 7420 5220 3d20  over k,.set R = 
│ │ │ │ -00025db0: 532f 2869 6465 616c 2066 6629 2e20 4966  S/(ideal ff). If
│ │ │ │ -00025dc0: 204e 2069 7320 6120 6772 6164 6564 2052   N is a graded R
│ │ │ │ -00025dd0: 2d6d 6f64 756c 652c 2061 6e64 204d 2069  -module, and M i
│ │ │ │ -00025de0: 7320 7468 6520 6d6f 6475 6c65 204e 2072  s the module N r
│ │ │ │ -00025df0: 6567 6172 6465 640a 6173 2061 6e20 532d  egarded.as an S-
│ │ │ │ -00025e00: 6d6f 6475 6c65 2c20 7468 6520 7363 7269  module, the scri
│ │ │ │ -00025e10: 7074 2072 6574 7572 6e73 2045 203d 2045  pt returns E = E
│ │ │ │ -00025e20: 7874 5f53 284d 2c6b 2920 616e 6420 5420  xt_S(M,k) and T 
│ │ │ │ -00025e30: 3d20 546f 725e 5328 4d2c 6b29 2061 7320  = Tor^S(M,k) as 
│ │ │ │ -00025e40: 6d6f 6475 6c65 730a 6f76 6572 2061 6e20  modules.over an 
│ │ │ │ -00025e50: 6578 7465 7269 6f72 2061 6c67 6562 7261  exterior algebra
│ │ │ │ -00025e60: 2e0a 0a54 6865 2073 6372 6970 7420 7072  ...The script pr
│ │ │ │ -00025e70: 696e 7473 2074 6865 2054 6174 6520 7265  ints the Tate re
│ │ │ │ -00025e80: 736f 6c75 7469 6f6e 206f 6620 453b 2061  solution of E; a
│ │ │ │ -00025e90: 6e64 2074 6865 2063 6f68 6f6d 6f6c 6f67  nd the cohomolog
│ │ │ │ -00025ea0: 7920 7461 626c 6520 6f66 2074 6865 0a73  y table of the.s
│ │ │ │ -00025eb0: 6865 6166 2061 7373 6f63 6961 7465 6420  heaf associated 
│ │ │ │ -00025ec0: 746f 2045 7874 5f52 284e 2c6b 2920 6f76  to Ext_R(N,k) ov
│ │ │ │ -00025ed0: 6572 2074 6865 2072 696e 6720 6f66 2043  er the ring of C
│ │ │ │ -00025ee0: 4920 6f70 6572 6174 6f72 732c 2077 6869  I operators, whi
│ │ │ │ -00025ef0: 6368 2069 7320 610a 706f 6c79 6e6f 6d69  ch is a.polynomi
│ │ │ │ -00025f00: 616c 2072 696e 6720 6f76 6572 206b 206f  al ring over k o
│ │ │ │ -00025f10: 6e20 6320 7661 7269 6162 6c65 732e 0a0a  n c variables...
│ │ │ │ -00025f20: 5468 6520 6f75 7470 7574 2063 616e 2062  The output can b
│ │ │ │ -00025f30: 6520 7573 6564 2074 6f20 2873 6f6d 6574  e used to (somet
│ │ │ │ -00025f40: 696d 6573 2920 6368 6563 6b20 7768 6574  imes) check whet
│ │ │ │ -00025f50: 6865 7220 7468 6520 7375 626d 6f64 756c  her the submodul
│ │ │ │ -00025f60: 6520 6f66 2045 7874 5f53 284d 2c6b 290a  e of Ext_S(M,k).
│ │ │ │ -00025f70: 6765 6e65 7261 7465 6420 696e 2064 6567  generated in deg
│ │ │ │ -00025f80: 7265 6520 3020 7370 6c69 7473 2028 6173  ree 0 splits (as
│ │ │ │ -00025f90: 2061 6e20 6578 7465 7269 6f72 206d 6f64   an exterior mod
│ │ │ │ -00025fa0: 756c 650a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  ule..+----------
│ │ │ │ +00025b80: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765  *****..  * Usage
│ │ │ │ +00025b90: 3a20 0a20 2020 2020 2020 2028 452c 5429  : .        (E,T)
│ │ │ │ +00025ba0: 203d 2065 7874 5673 436f 686f 6d6f 6c6f   = extVsCohomolo
│ │ │ │ +00025bb0: 6779 2866 662c 4e29 0a20 202a 2049 6e70  gy(ff,N).  * Inp
│ │ │ │ +00025bc0: 7574 733a 0a20 2020 2020 202a 2066 662c  uts:.      * ff,
│ │ │ │ +00025bd0: 2061 202a 6e6f 7465 206d 6174 7269 783a   a *note matrix:
│ │ │ │ +00025be0: 2028 4d61 6361 756c 6179 3244 6f63 294d   (Macaulay2Doc)M
│ │ │ │ +00025bf0: 6174 7269 782c 2c20 7265 6775 6c61 7220  atrix,, regular 
│ │ │ │ +00025c00: 7365 7175 656e 6365 2069 6e20 610a 2020  sequence in a.  
│ │ │ │ +00025c10: 2020 2020 2020 7265 6775 6c61 7220 7269        regular ri
│ │ │ │ +00025c20: 6e67 2053 0a20 2020 2020 202a 204e 2c20  ng S.      * N, 
│ │ │ │ +00025c30: 6120 2a6e 6f74 6520 6d6f 6475 6c65 3a20  a *note module: 
│ │ │ │ +00025c40: 284d 6163 6175 6c61 7932 446f 6329 4d6f  (Macaulay2Doc)Mo
│ │ │ │ +00025c50: 6475 6c65 2c2c 2067 7261 6465 6420 6d6f  dule,, graded mo
│ │ │ │ +00025c60: 6475 6c65 206f 7665 7220 5220 3d0a 2020  dule over R =.  
│ │ │ │ +00025c70: 2020 2020 2020 532f 6964 6561 6c28 6666        S/ideal(ff
│ │ │ │ +00025c80: 2920 2875 7375 616c 6c79 2061 2068 6967  ) (usually a hig
│ │ │ │ +00025c90: 6820 7379 7a79 6779 290a 2020 2a20 4f75  h syzygy).  * Ou
│ │ │ │ +00025ca0: 7470 7574 733a 0a20 2020 2020 202a 2045  tputs:.      * E
│ │ │ │ +00025cb0: 2c20 6120 2a6e 6f74 6520 6d6f 6475 6c65  , a *note module
│ │ │ │ +00025cc0: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +00025cd0: 4d6f 6475 6c65 2c2c 200a 2020 2020 2020  Module,, .      
│ │ │ │ +00025ce0: 2a20 542c 2061 202a 6e6f 7465 206d 6f64  * T, a *note mod
│ │ │ │ +00025cf0: 756c 653a 2028 4d61 6361 756c 6179 3244  ule: (Macaulay2D
│ │ │ │ +00025d00: 6f63 294d 6f64 756c 652c 2c20 4578 7420  oc)Module,, Ext 
│ │ │ │ +00025d10: 616e 6420 546f 7220 6173 2065 7874 6572  and Tor as exter
│ │ │ │ +00025d20: 696f 720a 2020 2020 2020 2020 6d6f 6475  ior.        modu
│ │ │ │ +00025d30: 6c65 730a 0a44 6573 6372 6970 7469 6f6e  les..Description
│ │ │ │ +00025d40: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 4769  .===========..Gi
│ │ │ │ +00025d50: 7665 6e20 6120 6d61 7472 6978 2066 6620  ven a matrix ff 
│ │ │ │ +00025d60: 636f 6e74 6169 6e69 6e67 2061 2072 6567  containing a reg
│ │ │ │ +00025d70: 756c 6172 2073 6571 7565 6e63 6520 696e  ular sequence in
│ │ │ │ +00025d80: 2061 2070 6f6c 796e 6f6d 6961 6c20 7269   a polynomial ri
│ │ │ │ +00025d90: 6e67 2053 206f 7665 7220 6b2c 0a73 6574  ng S over k,.set
│ │ │ │ +00025da0: 2052 203d 2053 2f28 6964 6561 6c20 6666   R = S/(ideal ff
│ │ │ │ +00025db0: 292e 2049 6620 4e20 6973 2061 2067 7261  ). If N is a gra
│ │ │ │ +00025dc0: 6465 6420 522d 6d6f 6475 6c65 2c20 616e  ded R-module, an
│ │ │ │ +00025dd0: 6420 4d20 6973 2074 6865 206d 6f64 756c  d M is the modul
│ │ │ │ +00025de0: 6520 4e20 7265 6761 7264 6564 0a61 7320  e N regarded.as 
│ │ │ │ +00025df0: 616e 2053 2d6d 6f64 756c 652c 2074 6865  an S-module, the
│ │ │ │ +00025e00: 2073 6372 6970 7420 7265 7475 726e 7320   script returns 
│ │ │ │ +00025e10: 4520 3d20 4578 745f 5328 4d2c 6b29 2061  E = Ext_S(M,k) a
│ │ │ │ +00025e20: 6e64 2054 203d 2054 6f72 5e53 284d 2c6b  nd T = Tor^S(M,k
│ │ │ │ +00025e30: 2920 6173 206d 6f64 756c 6573 0a6f 7665  ) as modules.ove
│ │ │ │ +00025e40: 7220 616e 2065 7874 6572 696f 7220 616c  r an exterior al
│ │ │ │ +00025e50: 6765 6272 612e 0a0a 5468 6520 7363 7269  gebra...The scri
│ │ │ │ +00025e60: 7074 2070 7269 6e74 7320 7468 6520 5461  pt prints the Ta
│ │ │ │ +00025e70: 7465 2072 6573 6f6c 7574 696f 6e20 6f66  te resolution of
│ │ │ │ +00025e80: 2045 3b20 616e 6420 7468 6520 636f 686f   E; and the coho
│ │ │ │ +00025e90: 6d6f 6c6f 6779 2074 6162 6c65 206f 6620  mology table of 
│ │ │ │ +00025ea0: 7468 650a 7368 6561 6620 6173 736f 6369  the.sheaf associ
│ │ │ │ +00025eb0: 6174 6564 2074 6f20 4578 745f 5228 4e2c  ated to Ext_R(N,
│ │ │ │ +00025ec0: 6b29 206f 7665 7220 7468 6520 7269 6e67  k) over the ring
│ │ │ │ +00025ed0: 206f 6620 4349 206f 7065 7261 746f 7273   of CI operators
│ │ │ │ +00025ee0: 2c20 7768 6963 6820 6973 2061 0a70 6f6c  , which is a.pol
│ │ │ │ +00025ef0: 796e 6f6d 6961 6c20 7269 6e67 206f 7665  ynomial ring ove
│ │ │ │ +00025f00: 7220 6b20 6f6e 2063 2076 6172 6961 626c  r k on c variabl
│ │ │ │ +00025f10: 6573 2e0a 0a54 6865 206f 7574 7075 7420  es...The output 
│ │ │ │ +00025f20: 6361 6e20 6265 2075 7365 6420 746f 2028  can be used to (
│ │ │ │ +00025f30: 736f 6d65 7469 6d65 7329 2063 6865 636b  sometimes) check
│ │ │ │ +00025f40: 2077 6865 7468 6572 2074 6865 2073 7562   whether the sub
│ │ │ │ +00025f50: 6d6f 6475 6c65 206f 6620 4578 745f 5328  module of Ext_S(
│ │ │ │ +00025f60: 4d2c 6b29 0a67 656e 6572 6174 6564 2069  M,k).generated i
│ │ │ │ +00025f70: 6e20 6465 6772 6565 2030 2073 706c 6974  n degree 0 split
│ │ │ │ +00025f80: 7320 2861 7320 616e 2065 7874 6572 696f  s (as an exterio
│ │ │ │ +00025f90: 7220 6d6f 6475 6c65 0a0a 2b2d 2d2d 2d2d  r module..+-----
│ │ │ │ +00025fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025fe0: 2d2d 2d2b 0a7c 6931 203a 2053 203d 205a  ---+.|i1 : S = Z
│ │ │ │ -00025ff0: 5a2f 3130 315b 612c 622c 635d 2020 2020  Z/101[a,b,c]    
│ │ │ │ +00025fd0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20  --------+.|i1 : 
│ │ │ │ +00025fe0: 5320 3d20 5a5a 2f31 3031 5b61 2c62 2c63  S = ZZ/101[a,b,c
│ │ │ │ +00025ff0: 5d20 2020 2020 2020 2020 2020 2020 2020  ]               
│ │ │ │  00026000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026020: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00026010: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00026020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026060: 2020 207c 0a7c 6f31 203d 2053 2020 2020     |.|o1 = S    
│ │ │ │ +00026050: 2020 2020 2020 2020 7c0a 7c6f 3120 3d20          |.|o1 = 
│ │ │ │ +00026060: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │  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 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00026090: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000260a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000260b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000260c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000260d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000260e0: 2020 207c 0a7c 6f31 203a 2050 6f6c 796e     |.|o1 : Polyn
│ │ │ │ -000260f0: 6f6d 6961 6c52 696e 6720 2020 2020 2020  omialRing       
│ │ │ │ +000260d0: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20          |.|o1 : 
│ │ │ │ +000260e0: 506f 6c79 6e6f 6d69 616c 5269 6e67 2020  PolynomialRing  
│ │ │ │ +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 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00026110: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00026120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026160: 2d2d 2d2b 0a7c 6932 203a 2066 6620 3d20  ---+.|i2 : ff = 
│ │ │ │ -00026170: 6d61 7472 6978 2022 6132 2c62 322c 6332  matrix "a2,b2,c2
│ │ │ │ -00026180: 2220 2020 2020 2020 2020 2020 2020 2020  "               
│ │ │ │ -00026190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000261a0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00026150: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20  --------+.|i2 : 
│ │ │ │ +00026160: 6666 203d 206d 6174 7269 7820 2261 322c  ff = matrix "a2,
│ │ │ │ +00026170: 6232 2c63 3222 2020 2020 2020 2020 2020  b2,c2"          
│ │ │ │ +00026180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026190: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000261a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000261b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000261c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000261d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000261e0: 2020 207c 0a7c 6f32 203d 207c 2061 3220     |.|o2 = | a2 
│ │ │ │ -000261f0: 6232 2063 3220 7c20 2020 2020 2020 2020  b2 c2 |         
│ │ │ │ +000261d0: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20          |.|o2 = 
│ │ │ │ +000261e0: 7c20 6132 2062 3220 6332 207c 2020 2020  | a2 b2 c2 |    
│ │ │ │ +000261f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026220: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00026210: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00026220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026260: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00026270: 2020 2031 2020 2020 2020 3320 2020 2020     1      3     
│ │ │ │ +00026250: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00026260: 2020 2020 2020 2020 3120 2020 2020 2033          1      3
│ │ │ │ +00026270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000262a0: 2020 207c 0a7c 6f32 203a 204d 6174 7269     |.|o2 : Matri
│ │ │ │ -000262b0: 7820 5320 203c 2d2d 2053 2020 2020 2020  x S  <-- S      
│ │ │ │ +00026290: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20          |.|o2 : 
│ │ │ │ +000262a0: 4d61 7472 6978 2053 2020 3c2d 2d20 5320  Matrix S  <-- S 
│ │ │ │ +000262b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000262c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000262d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000262e0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +000262d0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +000262e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000262f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026320: 2d2d 2d2b 0a7c 6933 203a 2052 203d 2053  ---+.|i3 : R = S
│ │ │ │ -00026330: 2f28 6964 6561 6c20 6666 2920 2020 2020  /(ideal ff)     
│ │ │ │ +00026310: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │ +00026320: 5220 3d20 532f 2869 6465 616c 2066 6629  R = S/(ideal ff)
│ │ │ │ +00026330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026360: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00026350: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00026360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000263a0: 2020 207c 0a7c 6f33 203d 2052 2020 2020     |.|o3 = R    
│ │ │ │ +00026390: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20          |.|o3 = 
│ │ │ │ +000263a0: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │  000263b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000263c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000263d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000263e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000263d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000263e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000263f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026420: 2020 207c 0a7c 6f33 203a 2051 756f 7469     |.|o3 : Quoti
│ │ │ │ -00026430: 656e 7452 696e 6720 2020 2020 2020 2020  entRing         
│ │ │ │ +00026410: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20          |.|o3 : 
│ │ │ │ +00026420: 5175 6f74 6965 6e74 5269 6e67 2020 2020  QuotientRing    
│ │ │ │ +00026430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026460: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00026450: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00026460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000264a0: 2d2d 2d2b 0a7c 6934 203a 204e 203d 2068  ---+.|i4 : N = h
│ │ │ │ -000264b0: 6967 6853 797a 7967 7928 525e 312f 6964  ighSyzygy(R^1/id
│ │ │ │ -000264c0: 6561 6c28 612a 622c 6329 2920 2020 2020  eal(a*b,c))     
│ │ │ │ -000264d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000264e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00026490: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20  --------+.|i4 : 
│ │ │ │ +000264a0: 4e20 3d20 6869 6768 5379 7a79 6779 2852  N = highSyzygy(R
│ │ │ │ +000264b0: 5e31 2f69 6465 616c 2861 2a62 2c63 2929  ^1/ideal(a*b,c))
│ │ │ │ +000264c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000264d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000264e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000264f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026520: 2020 207c 0a7c 6f34 203d 2063 6f6b 6572     |.|o4 = coker
│ │ │ │ -00026530: 6e65 6c20 7b34 7d20 7c20 6320 2d61 6220  nel {4} | c -ab 
│ │ │ │ -00026540: 3020 3020 3020 2030 2020 3020 2030 2030  0 0 0  0  0  0 0
│ │ │ │ -00026550: 2030 2030 2020 3020 3020 2030 2020 3020   0 0  0 0  0  0 
│ │ │ │ -00026560: 3020 7c7c 0a7c 2020 2020 2020 2020 2020  0 ||.|          
│ │ │ │ -00026570: 2020 2020 7b35 7d20 7c20 3020 6320 2020      {5} | 0 c   
│ │ │ │ -00026580: 6220 6120 3020 2030 2020 3020 2030 2030  b a 0  0  0  0 0
│ │ │ │ -00026590: 2030 2030 2020 3020 3020 2030 2020 3020   0 0  0 0  0  0 
│ │ │ │ -000265a0: 3020 7c7c 0a7c 2020 2020 2020 2020 2020  0 ||.|          
│ │ │ │ -000265b0: 2020 2020 7b35 7d20 7c20 3020 3020 2020      {5} | 0 0   
│ │ │ │ -000265c0: 6320 3020 2d62 2061 2020 3020 2030 2030  c 0 -b a  0  0 0
│ │ │ │ -000265d0: 2030 2030 2020 3020 3020 2030 2020 3020   0 0  0 0  0  0 
│ │ │ │ -000265e0: 3020 7c7c 0a7c 2020 2020 2020 2020 2020  0 ||.|          
│ │ │ │ -000265f0: 2020 2020 7b35 7d20 7c20 3020 3020 2020      {5} | 0 0   
│ │ │ │ -00026600: 3020 6320 3020 202d 6220 2d61 2030 2030  0 c 0  -b -a 0 0
│ │ │ │ -00026610: 2030 2030 2020 3020 3020 2030 2020 3020   0 0  0 0  0  0 
│ │ │ │ -00026620: 3020 7c7c 0a7c 2020 2020 2020 2020 2020  0 ||.|          
│ │ │ │ -00026630: 2020 2020 7b35 7d20 7c20 3020 3020 2020      {5} | 0 0   
│ │ │ │ -00026640: 3020 3020 6320 2030 2020 3020 2062 2061  0 0 c  0  0  b a
│ │ │ │ -00026650: 2030 2030 2020 3020 3020 2030 2020 3020   0 0  0 0  0  0 
│ │ │ │ -00026660: 3020 7c7c 0a7c 2020 2020 2020 2020 2020  0 ||.|          
│ │ │ │ -00026670: 2020 2020 7b35 7d20 7c20 3020 3020 2020      {5} | 0 0   
│ │ │ │ -00026680: 3020 3020 3020 2063 2020 3020 2030 2062  0 0 0  c  0  0 b
│ │ │ │ -00026690: 2030 2030 2020 3020 2d61 2030 2020 3020   0 0  0 -a 0  0 
│ │ │ │ -000266a0: 3020 7c7c 0a7c 2020 2020 2020 2020 2020  0 ||.|          
│ │ │ │ -000266b0: 2020 2020 7b35 7d20 7c20 3020 3020 2020      {5} | 0 0   
│ │ │ │ -000266c0: 3020 3020 3020 2030 2020 6320 2030 2030  0 0 0  0  c  0 0
│ │ │ │ -000266d0: 2030 2030 2020 3020 6220 2030 2020 6120   0 0  0 b  0  a 
│ │ │ │ -000266e0: 3020 7c7c 0a7c 2020 2020 2020 2020 2020  0 ||.|          
│ │ │ │ -000266f0: 2020 2020 7b35 7d20 7c20 3020 3020 2020      {5} | 0 0   
│ │ │ │ -00026700: 3020 3020 3020 2030 2020 3020 2063 2030  0 0 0  0  0  c 0
│ │ │ │ -00026710: 2062 202d 6120 3020 3020 2030 2020 3020   b -a 0 0  0  0 
│ │ │ │ -00026720: 3020 7c7c 0a7c 2020 2020 2020 2020 2020  0 ||.|          
│ │ │ │ -00026730: 2020 2020 7b35 7d20 7c20 3020 3020 2020      {5} | 0 0   
│ │ │ │ -00026740: 3020 3020 3020 2030 2020 3020 2030 2063  0 0 0  0  0  0 c
│ │ │ │ -00026750: 2030 2062 2020 6120 3020 2030 2020 3020   0 b  a 0  0  0 
│ │ │ │ -00026760: 3020 7c7c 0a7c 2020 2020 2020 2020 2020  0 ||.|          
│ │ │ │ -00026770: 2020 2020 7b35 7d20 7c20 3020 3020 2020      {5} | 0 0   
│ │ │ │ -00026780: 3020 3020 3020 2030 2020 3020 2030 2030  0 0 0  0  0  0 0
│ │ │ │ -00026790: 2030 2030 2020 6220 6320 202d 6120 3020   0 0  b c  -a 0 
│ │ │ │ -000267a0: 3020 7c7c 0a7c 2020 2020 2020 2020 2020  0 ||.|          
│ │ │ │ -000267b0: 2020 2020 7b35 7d20 7c20 3020 3020 2020      {5} | 0 0   
│ │ │ │ -000267c0: 3020 3020 3020 2030 2020 3020 2030 2030  0 0 0  0  0  0 0
│ │ │ │ -000267d0: 2030 2030 2020 3020 3020 2062 2020 6320   0 0  0 0  b  c 
│ │ │ │ -000267e0: 6120 7c7c 0a7c 2020 2020 2020 2020 2020  a ||.|          
│ │ │ │ +00026510: 2020 2020 2020 2020 7c0a 7c6f 3420 3d20          |.|o4 = 
│ │ │ │ +00026520: 636f 6b65 726e 656c 207b 347d 207c 2063  cokernel {4} | c
│ │ │ │ +00026530: 202d 6162 2030 2030 2030 2020 3020 2030   -ab 0 0 0  0  0
│ │ │ │ +00026540: 2020 3020 3020 3020 3020 2030 2030 2020    0 0 0 0  0 0  
│ │ │ │ +00026550: 3020 2030 2030 207c 7c0a 7c20 2020 2020  0  0 0 ||.|     
│ │ │ │ +00026560: 2020 2020 2020 2020 207b 357d 207c 2030           {5} | 0
│ │ │ │ +00026570: 2063 2020 2062 2061 2030 2020 3020 2030   c   b a 0  0  0
│ │ │ │ +00026580: 2020 3020 3020 3020 3020 2030 2030 2020    0 0 0 0  0 0  
│ │ │ │ +00026590: 3020 2030 2030 207c 7c0a 7c20 2020 2020  0  0 0 ||.|     
│ │ │ │ +000265a0: 2020 2020 2020 2020 207b 357d 207c 2030           {5} | 0
│ │ │ │ +000265b0: 2030 2020 2063 2030 202d 6220 6120 2030   0   c 0 -b a  0
│ │ │ │ +000265c0: 2020 3020 3020 3020 3020 2030 2030 2020    0 0 0 0  0 0  
│ │ │ │ +000265d0: 3020 2030 2030 207c 7c0a 7c20 2020 2020  0  0 0 ||.|     
│ │ │ │ +000265e0: 2020 2020 2020 2020 207b 357d 207c 2030           {5} | 0
│ │ │ │ +000265f0: 2030 2020 2030 2063 2030 2020 2d62 202d   0   0 c 0  -b -
│ │ │ │ +00026600: 6120 3020 3020 3020 3020 2030 2030 2020  a 0 0 0 0  0 0  
│ │ │ │ +00026610: 3020 2030 2030 207c 7c0a 7c20 2020 2020  0  0 0 ||.|     
│ │ │ │ +00026620: 2020 2020 2020 2020 207b 357d 207c 2030           {5} | 0
│ │ │ │ +00026630: 2030 2020 2030 2030 2063 2020 3020 2030   0   0 0 c  0  0
│ │ │ │ +00026640: 2020 6220 6120 3020 3020 2030 2030 2020    b a 0 0  0 0  
│ │ │ │ +00026650: 3020 2030 2030 207c 7c0a 7c20 2020 2020  0  0 0 ||.|     
│ │ │ │ +00026660: 2020 2020 2020 2020 207b 357d 207c 2030           {5} | 0
│ │ │ │ +00026670: 2030 2020 2030 2030 2030 2020 6320 2030   0   0 0 0  c  0
│ │ │ │ +00026680: 2020 3020 6220 3020 3020 2030 202d 6120    0 b 0 0  0 -a 
│ │ │ │ +00026690: 3020 2030 2030 207c 7c0a 7c20 2020 2020  0  0 0 ||.|     
│ │ │ │ +000266a0: 2020 2020 2020 2020 207b 357d 207c 2030           {5} | 0
│ │ │ │ +000266b0: 2030 2020 2030 2030 2030 2020 3020 2063   0   0 0 0  0  c
│ │ │ │ +000266c0: 2020 3020 3020 3020 3020 2030 2062 2020    0 0 0 0  0 b  
│ │ │ │ +000266d0: 3020 2061 2030 207c 7c0a 7c20 2020 2020  0  a 0 ||.|     
│ │ │ │ +000266e0: 2020 2020 2020 2020 207b 357d 207c 2030           {5} | 0
│ │ │ │ +000266f0: 2030 2020 2030 2030 2030 2020 3020 2030   0   0 0 0  0  0
│ │ │ │ +00026700: 2020 6320 3020 6220 2d61 2030 2030 2020    c 0 b -a 0 0  
│ │ │ │ +00026710: 3020 2030 2030 207c 7c0a 7c20 2020 2020  0  0 0 ||.|     
│ │ │ │ +00026720: 2020 2020 2020 2020 207b 357d 207c 2030           {5} | 0
│ │ │ │ +00026730: 2030 2020 2030 2030 2030 2020 3020 2030   0   0 0 0  0  0
│ │ │ │ +00026740: 2020 3020 6320 3020 6220 2061 2030 2020    0 c 0 b  a 0  
│ │ │ │ +00026750: 3020 2030 2030 207c 7c0a 7c20 2020 2020  0  0 0 ||.|     
│ │ │ │ +00026760: 2020 2020 2020 2020 207b 357d 207c 2030           {5} | 0
│ │ │ │ +00026770: 2030 2020 2030 2030 2030 2020 3020 2030   0   0 0 0  0  0
│ │ │ │ +00026780: 2020 3020 3020 3020 3020 2062 2063 2020    0 0 0 0  b c  
│ │ │ │ +00026790: 2d61 2030 2030 207c 7c0a 7c20 2020 2020  -a 0 0 ||.|     
│ │ │ │ +000267a0: 2020 2020 2020 2020 207b 357d 207c 2030           {5} | 0
│ │ │ │ +000267b0: 2030 2020 2030 2030 2030 2020 3020 2030   0   0 0 0  0  0
│ │ │ │ +000267c0: 2020 3020 3020 3020 3020 2030 2030 2020    0 0 0 0  0 0  
│ │ │ │ +000267d0: 6220 2063 2061 207c 7c0a 7c20 2020 2020  b  c a ||.|     
│ │ │ │ +000267e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000267f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026820: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00026830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026840: 2020 3131 2020 2020 2020 2020 2020 2020    11            
│ │ │ │ -00026850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026860: 2020 207c 0a7c 6f34 203a 2052 2d6d 6f64     |.|o4 : R-mod
│ │ │ │ -00026870: 756c 652c 2071 756f 7469 656e 7420 6f66  ule, quotient of
│ │ │ │ -00026880: 2052 2020 2020 2020 2020 2020 2020 2020   R              
│ │ │ │ -00026890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000268a0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00026810: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00026820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026830: 2020 2020 2020 2031 3120 2020 2020 2020         11       
│ │ │ │ +00026840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026850: 2020 2020 2020 2020 7c0a 7c6f 3420 3a20          |.|o4 : 
│ │ │ │ +00026860: 522d 6d6f 6475 6c65 2c20 7175 6f74 6965  R-module, quotie
│ │ │ │ +00026870: 6e74 206f 6620 5220 2020 2020 2020 2020  nt of R         
│ │ │ │ +00026880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026890: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +000268a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000268b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000268c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000268d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000268e0: 2d2d 2d2b 0a7c 6935 203a 2045 203d 2065  ---+.|i5 : E = e
│ │ │ │ -000268f0: 7874 5673 436f 686f 6d6f 6c6f 6779 2866  xtVsCohomology(f
│ │ │ │ -00026900: 662c 6869 6768 5379 7a79 6779 204e 293b  f,highSyzygy N);
│ │ │ │ -00026910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026920: 2020 207c 0a7c 5461 7465 2052 6573 6f6c     |.|Tate Resol
│ │ │ │ -00026930: 7574 696f 6e20 6f66 2045 7874 5f53 284d  ution of Ext_S(M
│ │ │ │ -00026940: 2c6b 2920 6173 2065 7874 6572 696f 7220  ,k) as exterior 
│ │ │ │ -00026950: 6d6f 6475 6c65 3a20 2020 2020 2020 2020  module:         
│ │ │ │ -00026960: 2020 207c 0a7c 4e6f 7465 2074 6861 7420     |.|Note that 
│ │ │ │ -00026970: 6d61 7073 2067 6f20 6c65 6674 2074 6f20  maps go left to 
│ │ │ │ -00026980: 7269 6768 7420 2020 2020 2020 2020 2020  right           
│ │ │ │ -00026990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000269a0: 2020 207c 0a7c 2020 2020 2020 202d 3131     |.|       -11
│ │ │ │ -000269b0: 202d 3130 2020 2d39 202d 3820 2d37 202d   -10  -9 -8 -7 -
│ │ │ │ -000269c0: 3620 2d35 202d 3420 2d33 202d 3220 202d  6 -5 -4 -3 -2  -
│ │ │ │ -000269d0: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ -000269e0: 2020 207c 0a7c 746f 7461 6c3a 2031 3938     |.|total: 198
│ │ │ │ -000269f0: 2031 3436 2031 3032 2036 3620 3338 2031   146 102 66 38 1
│ │ │ │ -00026a00: 3820 2039 2031 3620 3336 2036 3420 3130  8  9 16 36 64 10
│ │ │ │ -00026a10: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ -00026a20: 2020 207c 0a7c 2020 2020 383a 2031 3036     |.|    8: 106
│ │ │ │ -00026a30: 2020 3739 2020 3536 2033 3720 3232 2031    79  56 37 22 1
│ │ │ │ -00026a40: 3120 2034 2020 3120 2031 2020 3120 2020  1  4  1  1  1   
│ │ │ │ -00026a50: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ -00026a60: 2020 207c 0a7c 2020 2020 393a 2020 3932     |.|    9:  92
│ │ │ │ -00026a70: 2020 3637 2020 3436 2032 3920 3136 2020    67  46 29 16  
│ │ │ │ -00026a80: 3720 2032 2020 2e20 202e 2020 2e20 2020  7  2  .  .  .   
│ │ │ │ -00026a90: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ -00026aa0: 2020 207c 0a7c 2020 2031 303a 2020 202e     |.|   10:   .
│ │ │ │ -00026ab0: 2020 202e 2020 202e 2020 2e20 202e 2020     .   .  .  .  
│ │ │ │ -00026ac0: 2e20 202e 2020 3520 3134 2032 3720 2034  .  .  5 14 27  4
│ │ │ │ -00026ad0: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ -00026ae0: 2020 207c 0a7c 2020 2031 313a 2020 202e     |.|   11:   .
│ │ │ │ -00026af0: 2020 202e 2020 202e 2020 2e20 202e 2020     .   .  .  .  
│ │ │ │ -00026b00: 2e20 2033 2031 3020 3231 2033 3620 2035  .  3 10 21 36  5
│ │ │ │ -00026b10: 3520 2020 2020 2020 2020 2020 2020 2020  5               
│ │ │ │ -00026b20: 2020 207c 0a7c 2d2d 2d20 2020 2020 2020     |.|---       
│ │ │ │ +000268d0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20  --------+.|i5 : 
│ │ │ │ +000268e0: 4520 3d20 6578 7456 7343 6f68 6f6d 6f6c  E = extVsCohomol
│ │ │ │ +000268f0: 6f67 7928 6666 2c68 6967 6853 797a 7967  ogy(ff,highSyzyg
│ │ │ │ +00026900: 7920 4e29 3b20 2020 2020 2020 2020 2020  y N);           
│ │ │ │ +00026910: 2020 2020 2020 2020 7c0a 7c54 6174 6520          |.|Tate 
│ │ │ │ +00026920: 5265 736f 6c75 7469 6f6e 206f 6620 4578  Resolution of Ex
│ │ │ │ +00026930: 745f 5328 4d2c 6b29 2061 7320 6578 7465  t_S(M,k) as exte
│ │ │ │ +00026940: 7269 6f72 206d 6f64 756c 653a 2020 2020  rior module:    
│ │ │ │ +00026950: 2020 2020 2020 2020 7c0a 7c4e 6f74 6520          |.|Note 
│ │ │ │ +00026960: 7468 6174 206d 6170 7320 676f 206c 6566  that maps go lef
│ │ │ │ +00026970: 7420 746f 2072 6967 6874 2020 2020 2020  t to right      
│ │ │ │ +00026980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026990: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000269a0: 2020 2d31 3120 2d31 3020 202d 3920 2d38    -11 -10  -9 -8
│ │ │ │ +000269b0: 202d 3720 2d36 202d 3520 2d34 202d 3320   -7 -6 -5 -4 -3 
│ │ │ │ +000269c0: 2d32 2020 2d31 2020 2020 2020 2020 2020  -2  -1          
│ │ │ │ +000269d0: 2020 2020 2020 2020 7c0a 7c74 6f74 616c          |.|total
│ │ │ │ +000269e0: 3a20 3139 3820 3134 3620 3130 3220 3636  : 198 146 102 66
│ │ │ │ +000269f0: 2033 3820 3138 2020 3920 3136 2033 3620   38 18  9 16 36 
│ │ │ │ +00026a00: 3634 2031 3030 2020 2020 2020 2020 2020  64 100          
│ │ │ │ +00026a10: 2020 2020 2020 2020 7c0a 7c20 2020 2038          |.|    8
│ │ │ │ +00026a20: 3a20 3130 3620 2037 3920 2035 3620 3337  : 106  79  56 37
│ │ │ │ +00026a30: 2032 3220 3131 2020 3420 2031 2020 3120   22 11  4  1  1 
│ │ │ │ +00026a40: 2031 2020 2031 2020 2020 2020 2020 2020   1   1          
│ │ │ │ +00026a50: 2020 2020 2020 2020 7c0a 7c20 2020 2039          |.|    9
│ │ │ │ +00026a60: 3a20 2039 3220 2036 3720 2034 3620 3239  :  92  67  46 29
│ │ │ │ +00026a70: 2031 3620 2037 2020 3220 202e 2020 2e20   16  7  2  .  . 
│ │ │ │ +00026a80: 202e 2020 202e 2020 2020 2020 2020 2020   .   .          
│ │ │ │ +00026a90: 2020 2020 2020 2020 7c0a 7c20 2020 3130          |.|   10
│ │ │ │ +00026aa0: 3a20 2020 2e20 2020 2e20 2020 2e20 202e  :   .   .   .  .
│ │ │ │ +00026ab0: 2020 2e20 202e 2020 2e20 2035 2031 3420    .  .  .  5 14 
│ │ │ │ +00026ac0: 3237 2020 3434 2020 2020 2020 2020 2020  27  44          
│ │ │ │ +00026ad0: 2020 2020 2020 2020 7c0a 7c20 2020 3131          |.|   11
│ │ │ │ +00026ae0: 3a20 2020 2e20 2020 2e20 2020 2e20 202e  :   .   .   .  .
│ │ │ │ +00026af0: 2020 2e20 202e 2020 3320 3130 2032 3120    .  .  3 10 21 
│ │ │ │ +00026b00: 3336 2020 3535 2020 2020 2020 2020 2020  36  55          
│ │ │ │ +00026b10: 2020 2020 2020 2020 7c0a 7c2d 2d2d 2020          |.|---  
│ │ │ │ +00026b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026b60: 2020 207c 0a7c 436f 686f 6d6f 6c6f 6779     |.|Cohomology
│ │ │ │ -00026b70: 2074 6162 6c65 206f 6620 6576 656e 4578   table of evenEx
│ │ │ │ -00026b80: 744d 6f64 756c 6520 4d3a 2020 2020 2020  tModule M:      
│ │ │ │ -00026b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026ba0: 2020 207c 0a7c 2020 202d 3520 2d34 202d     |.|   -5 -4 -
│ │ │ │ -00026bb0: 3320 2d32 202d 3120 2030 2020 3120 2032  3 -2 -1  0  1  2
│ │ │ │ -00026bc0: 2020 3320 2034 2020 2035 2020 2020 2020    3  4   5      
│ │ │ │ -00026bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026be0: 2020 207c 0a7c 323a 2033 3620 3231 2031     |.|2: 36 21 1
│ │ │ │ -00026bf0: 3020 2033 2020 2e20 202e 2020 2e20 202e  0  3  .  .  .  .
│ │ │ │ -00026c00: 2020 2e20 202e 2020 202e 2020 2020 2020    .  .   .      
│ │ │ │ -00026c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026c20: 2020 207c 0a7c 313a 2020 2e20 202e 2020     |.|1:  .  .  
│ │ │ │ -00026c30: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ -00026c40: 2020 2e20 202e 2020 202e 2020 2020 2020    .  .   .      
│ │ │ │ -00026c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026c60: 2020 207c 0a7c 303a 2020 3120 2031 2020     |.|0:  1  1  
│ │ │ │ -00026c70: 3120 2032 2020 3720 3136 2032 3920 3436  1  2  7 16 29 46
│ │ │ │ -00026c80: 2036 3720 3932 2031 3231 2020 2020 2020   67 92 121      
│ │ │ │ -00026c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026ca0: 2020 207c 0a7c 2d2d 2d20 2020 2020 2020     |.|---       
│ │ │ │ +00026b50: 2020 2020 2020 2020 7c0a 7c43 6f68 6f6d          |.|Cohom
│ │ │ │ +00026b60: 6f6c 6f67 7920 7461 626c 6520 6f66 2065  ology table of e
│ │ │ │ +00026b70: 7665 6e45 7874 4d6f 6475 6c65 204d 3a20  venExtModule M: 
│ │ │ │ +00026b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026b90: 2020 2020 2020 2020 7c0a 7c20 2020 2d35          |.|   -5
│ │ │ │ +00026ba0: 202d 3420 2d33 202d 3220 2d31 2020 3020   -4 -3 -2 -1  0 
│ │ │ │ +00026bb0: 2031 2020 3220 2033 2020 3420 2020 3520   1  2  3  4   5 
│ │ │ │ +00026bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026bd0: 2020 2020 2020 2020 7c0a 7c32 3a20 3336          |.|2: 36
│ │ │ │ +00026be0: 2032 3120 3130 2020 3320 202e 2020 2e20   21 10  3  .  . 
│ │ │ │ +00026bf0: 202e 2020 2e20 202e 2020 2e20 2020 2e20   .  .  .  .   . 
│ │ │ │ +00026c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026c10: 2020 2020 2020 2020 7c0a 7c31 3a20 202e          |.|1:  .
│ │ │ │ +00026c20: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ +00026c30: 202e 2020 2e20 202e 2020 2e20 2020 2e20   .  .  .  .   . 
│ │ │ │ +00026c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026c50: 2020 2020 2020 2020 7c0a 7c30 3a20 2031          |.|0:  1
│ │ │ │ +00026c60: 2020 3120 2031 2020 3220 2037 2031 3620    1  1  2  7 16 
│ │ │ │ +00026c70: 3239 2034 3620 3637 2039 3220 3132 3120  29 46 67 92 121 
│ │ │ │ +00026c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026c90: 2020 2020 2020 2020 7c0a 7c2d 2d2d 2020          |.|---  
│ │ │ │ +00026ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026ce0: 2020 207c 0a7c 436f 686f 6d6f 6c6f 6779     |.|Cohomology
│ │ │ │ -00026cf0: 2074 6162 6c65 206f 6620 6f64 6445 7874   table of oddExt
│ │ │ │ -00026d00: 4d6f 6475 6c65 204d 3a20 2020 2020 2020  Module M:       
│ │ │ │ -00026d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026d20: 2020 207c 0a7c 2020 202d 3520 2d34 202d     |.|   -5 -4 -
│ │ │ │ -00026d30: 3320 2d32 202d 3120 2030 2020 3120 2032  3 -2 -1  0  1  2
│ │ │ │ -00026d40: 2020 3320 2020 3420 2020 3520 2020 2020    3   4   5     
│ │ │ │ -00026d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026d60: 2020 207c 0a7c 323a 2032 3820 3135 2020     |.|2: 28 15  
│ │ │ │ -00026d70: 3620 2031 2020 2e20 202e 2020 2e20 202e  6  1  .  .  .  .
│ │ │ │ -00026d80: 2020 2e20 2020 2e20 2020 2e20 2020 2020    .   .   .     
│ │ │ │ -00026d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026da0: 2020 207c 0a7c 313a 2020 2e20 202e 2020     |.|1:  .  .  
│ │ │ │ -00026db0: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ -00026dc0: 2020 2e20 2020 2e20 2020 2e20 2020 2020    .   .   .     
│ │ │ │ -00026dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026de0: 2020 207c 0a7c 303a 2020 3120 2031 2020     |.|0:  1  1  
│ │ │ │ -00026df0: 3120 2034 2031 3120 3232 2033 3720 3536  1  4 11 22 37 56
│ │ │ │ -00026e00: 2037 3920 3130 3620 3133 3720 2020 2020   79 106 137     
│ │ │ │ -00026e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026e20: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00026cd0: 2020 2020 2020 2020 7c0a 7c43 6f68 6f6d          |.|Cohom
│ │ │ │ +00026ce0: 6f6c 6f67 7920 7461 626c 6520 6f66 206f  ology table of o
│ │ │ │ +00026cf0: 6464 4578 744d 6f64 756c 6520 4d3a 2020  ddExtModule M:  
│ │ │ │ +00026d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026d10: 2020 2020 2020 2020 7c0a 7c20 2020 2d35          |.|   -5
│ │ │ │ +00026d20: 202d 3420 2d33 202d 3220 2d31 2020 3020   -4 -3 -2 -1  0 
│ │ │ │ +00026d30: 2031 2020 3220 2033 2020 2034 2020 2035   1  2  3   4   5
│ │ │ │ +00026d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026d50: 2020 2020 2020 2020 7c0a 7c32 3a20 3238          |.|2: 28
│ │ │ │ +00026d60: 2031 3520 2036 2020 3120 202e 2020 2e20   15  6  1  .  . 
│ │ │ │ +00026d70: 202e 2020 2e20 202e 2020 202e 2020 202e   .  .  .   .   .
│ │ │ │ +00026d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026d90: 2020 2020 2020 2020 7c0a 7c31 3a20 202e          |.|1:  .
│ │ │ │ +00026da0: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ +00026db0: 202e 2020 2e20 202e 2020 202e 2020 202e   .  .  .   .   .
│ │ │ │ +00026dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026dd0: 2020 2020 2020 2020 7c0a 7c30 3a20 2031          |.|0:  1
│ │ │ │ +00026de0: 2020 3120 2031 2020 3420 3131 2032 3220    1  1  4 11 22 
│ │ │ │ +00026df0: 3337 2035 3620 3739 2031 3036 2031 3337  37 56 79 106 137
│ │ │ │ +00026e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026e10: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00026e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026e60: 2d2d 2d2b 0a0a 5365 6520 616c 736f 0a3d  ---+..See also.=
│ │ │ │ -00026e70: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f  =======..  * *no
│ │ │ │ -00026e80: 7465 2068 6967 6853 797a 7967 793a 2068  te highSyzygy: h
│ │ │ │ -00026e90: 6967 6853 797a 7967 792c 202d 2d20 5265  ighSyzygy, -- Re
│ │ │ │ -00026ea0: 7475 726e 7320 6120 7379 7a79 6779 206d  turns a syzygy m
│ │ │ │ -00026eb0: 6f64 756c 6520 6f6e 6520 6265 796f 6e64  odule one beyond
│ │ │ │ -00026ec0: 2074 6865 0a20 2020 2072 6567 756c 6172   the.    regular
│ │ │ │ -00026ed0: 6974 7920 6f66 2045 7874 284d 2c6b 290a  ity of Ext(M,k).
│ │ │ │ -00026ee0: 2020 2a20 2a6e 6f74 6520 6578 7465 7269    * *note exteri
│ │ │ │ -00026ef0: 6f72 4578 744d 6f64 756c 653a 2065 7874  orExtModule: ext
│ │ │ │ -00026f00: 6572 696f 7245 7874 4d6f 6475 6c65 2c20  eriorExtModule, 
│ │ │ │ -00026f10: 2d2d 2045 7874 284d 2c6b 2920 6f72 2045  -- Ext(M,k) or E
│ │ │ │ -00026f20: 7874 284d 2c4e 2920 6173 2061 0a20 2020  xt(M,N) as a.   
│ │ │ │ -00026f30: 206d 6f64 756c 6520 6f76 6572 2061 6e20   module over an 
│ │ │ │ -00026f40: 6578 7465 7269 6f72 2061 6c67 6562 7261  exterior algebra
│ │ │ │ -00026f50: 0a0a 5761 7973 2074 6f20 7573 6520 6578  ..Ways to use ex
│ │ │ │ -00026f60: 7456 7343 6f68 6f6d 6f6c 6f67 793a 0a3d  tVsCohomology:.=
│ │ │ │ +00026e50: 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061  --------+..See a
│ │ │ │ +00026e60: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020  lso.========..  
│ │ │ │ +00026e70: 2a20 2a6e 6f74 6520 6869 6768 5379 7a79  * *note highSyzy
│ │ │ │ +00026e80: 6779 3a20 6869 6768 5379 7a79 6779 2c20  gy: highSyzygy, 
│ │ │ │ +00026e90: 2d2d 2052 6574 7572 6e73 2061 2073 797a  -- Returns a syz
│ │ │ │ +00026ea0: 7967 7920 6d6f 6475 6c65 206f 6e65 2062  ygy module one b
│ │ │ │ +00026eb0: 6579 6f6e 6420 7468 650a 2020 2020 7265  eyond the.    re
│ │ │ │ +00026ec0: 6775 6c61 7269 7479 206f 6620 4578 7428  gularity of Ext(
│ │ │ │ +00026ed0: 4d2c 6b29 0a20 202a 202a 6e6f 7465 2065  M,k).  * *note e
│ │ │ │ +00026ee0: 7874 6572 696f 7245 7874 4d6f 6475 6c65  xteriorExtModule
│ │ │ │ +00026ef0: 3a20 6578 7465 7269 6f72 4578 744d 6f64  : exteriorExtMod
│ │ │ │ +00026f00: 756c 652c 202d 2d20 4578 7428 4d2c 6b29  ule, -- Ext(M,k)
│ │ │ │ +00026f10: 206f 7220 4578 7428 4d2c 4e29 2061 7320   or Ext(M,N) as 
│ │ │ │ +00026f20: 610a 2020 2020 6d6f 6475 6c65 206f 7665  a.    module ove
│ │ │ │ +00026f30: 7220 616e 2065 7874 6572 696f 7220 616c  r an exterior al
│ │ │ │ +00026f40: 6765 6272 610a 0a57 6179 7320 746f 2075  gebra..Ways to u
│ │ │ │ +00026f50: 7365 2065 7874 5673 436f 686f 6d6f 6c6f  se extVsCohomolo
│ │ │ │ +00026f60: 6779 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  gy:.============
│ │ │ │  00026f70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00026f80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ -00026f90: 2022 6578 7456 7343 6f68 6f6d 6f6c 6f67   "extVsCohomolog
│ │ │ │ -00026fa0: 7928 4d61 7472 6978 2c4d 6f64 756c 6529  y(Matrix,Module)
│ │ │ │ -00026fb0: 220a 0a46 6f72 2074 6865 2070 726f 6772  "..For the progr
│ │ │ │ -00026fc0: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d  ammer.==========
│ │ │ │ -00026fd0: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62  ========..The ob
│ │ │ │ -00026fe0: 6a65 6374 202a 6e6f 7465 2065 7874 5673  ject *note extVs
│ │ │ │ -00026ff0: 436f 686f 6d6f 6c6f 6779 3a20 6578 7456  Cohomology: extV
│ │ │ │ -00027000: 7343 6f68 6f6d 6f6c 6f67 792c 2069 7320  sCohomology, is 
│ │ │ │ -00027010: 6120 2a6e 6f74 6520 6d65 7468 6f64 2066  a *note method f
│ │ │ │ -00027020: 756e 6374 696f 6e3a 0a28 4d61 6361 756c  unction:.(Macaul
│ │ │ │ -00027030: 6179 3244 6f63 294d 6574 686f 6446 756e  ay2Doc)MethodFun
│ │ │ │ -00027040: 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d  ction,...-------
│ │ │ │ +00026f80: 0a0a 2020 2a20 2265 7874 5673 436f 686f  ..  * "extVsCoho
│ │ │ │ +00026f90: 6d6f 6c6f 6779 284d 6174 7269 782c 4d6f  mology(Matrix,Mo
│ │ │ │ +00026fa0: 6475 6c65 2922 0a0a 466f 7220 7468 6520  dule)"..For the 
│ │ │ │ +00026fb0: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d  programmer.=====
│ │ │ │ +00026fc0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  =============..T
│ │ │ │ +00026fd0: 6865 206f 626a 6563 7420 2a6e 6f74 6520  he object *note 
│ │ │ │ +00026fe0: 6578 7456 7343 6f68 6f6d 6f6c 6f67 793a  extVsCohomology:
│ │ │ │ +00026ff0: 2065 7874 5673 436f 686f 6d6f 6c6f 6779   extVsCohomology
│ │ │ │ +00027000: 2c20 6973 2061 202a 6e6f 7465 206d 6574  , is a *note met
│ │ │ │ +00027010: 686f 6420 6675 6e63 7469 6f6e 3a0a 284d  hod function:.(M
│ │ │ │ +00027020: 6163 6175 6c61 7932 446f 6329 4d65 7468  acaulay2Doc)Meth
│ │ │ │ +00027030: 6f64 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d  odFunction,...--
│ │ │ │ +00027040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027090: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f  --------..The so
│ │ │ │ -000270a0: 7572 6365 206f 6620 7468 6973 2064 6f63  urce of this doc
│ │ │ │ -000270b0: 756d 656e 7420 6973 2069 6e0a 2f62 7569  ument is in./bui
│ │ │ │ -000270c0: 6c64 2f72 6570 726f 6475 6369 626c 652d  ld/reproducible-
│ │ │ │ -000270d0: 7061 7468 2f6d 6163 6175 6c61 7932 2d31  path/macaulay2-1
│ │ │ │ -000270e0: 2e32 352e 3131 2b64 732f 4d32 2f4d 6163  .25.11+ds/M2/Mac
│ │ │ │ -000270f0: 6175 6c61 7932 2f70 6163 6b61 6765 732f  aulay2/packages/
│ │ │ │ -00027100: 0a43 6f6d 706c 6574 6549 6e74 6572 7365  .CompleteInterse
│ │ │ │ -00027110: 6374 696f 6e52 6573 6f6c 7574 696f 6e73  ctionResolutions
│ │ │ │ -00027120: 2e6d 323a 3238 3236 3a30 2e0a 1f0a 4669  .m2:2826:0....Fi
│ │ │ │ -00027130: 6c65 3a20 436f 6d70 6c65 7465 496e 7465  le: CompleteInte
│ │ │ │ -00027140: 7273 6563 7469 6f6e 5265 736f 6c75 7469  rsectionResoluti
│ │ │ │ -00027150: 6f6e 732e 696e 666f 2c20 4e6f 6465 3a20  ons.info, Node: 
│ │ │ │ -00027160: 6669 6e69 7465 4265 7474 694e 756d 6265  finiteBettiNumbe
│ │ │ │ -00027170: 7273 2c20 4e65 7874 3a20 6672 6565 4578  rs, Next: freeEx
│ │ │ │ -00027180: 7465 7269 6f72 5375 6d6d 616e 642c 2050  teriorSummand, P
│ │ │ │ -00027190: 7265 763a 2065 7874 5673 436f 686f 6d6f  rev: extVsCohomo
│ │ │ │ -000271a0: 6c6f 6779 2c20 5570 3a20 546f 700a 0a66  logy, Up: Top..f
│ │ │ │ -000271b0: 696e 6974 6542 6574 7469 4e75 6d62 6572  initeBettiNumber
│ │ │ │ -000271c0: 7320 2d2d 2062 6574 7469 206e 756d 6265  s -- betti numbe
│ │ │ │ -000271d0: 7273 206f 6620 6669 6e69 7465 2072 6573  rs of finite res
│ │ │ │ -000271e0: 6f6c 7574 696f 6e20 636f 6d70 7574 6564  olution computed
│ │ │ │ -000271f0: 2066 726f 6d20 6120 6d61 7472 6978 2066   from a matrix f
│ │ │ │ -00027200: 6163 746f 7269 7a61 7469 6f6e 0a2a 2a2a  actorization.***
│ │ │ │ +00027080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54  -------------..T
│ │ │ │ +00027090: 6865 2073 6f75 7263 6520 6f66 2074 6869  he source of thi
│ │ │ │ +000270a0: 7320 646f 6375 6d65 6e74 2069 7320 696e  s document is in
│ │ │ │ +000270b0: 0a2f 6275 696c 642f 7265 7072 6f64 7563  ./build/reproduc
│ │ │ │ +000270c0: 6962 6c65 2d70 6174 682f 6d61 6361 756c  ible-path/macaul
│ │ │ │ +000270d0: 6179 322d 312e 3235 2e31 312b 6473 2f4d  ay2-1.25.11+ds/M
│ │ │ │ +000270e0: 322f 4d61 6361 756c 6179 322f 7061 636b  2/Macaulay2/pack
│ │ │ │ +000270f0: 6167 6573 2f0a 436f 6d70 6c65 7465 496e  ages/.CompleteIn
│ │ │ │ +00027100: 7465 7273 6563 7469 6f6e 5265 736f 6c75  tersectionResolu
│ │ │ │ +00027110: 7469 6f6e 732e 6d32 3a32 3832 363a 302e  tions.m2:2826:0.
│ │ │ │ +00027120: 0a1f 0a46 696c 653a 2043 6f6d 706c 6574  ...File: Complet
│ │ │ │ +00027130: 6549 6e74 6572 7365 6374 696f 6e52 6573  eIntersectionRes
│ │ │ │ +00027140: 6f6c 7574 696f 6e73 2e69 6e66 6f2c 204e  olutions.info, N
│ │ │ │ +00027150: 6f64 653a 2066 696e 6974 6542 6574 7469  ode: finiteBetti
│ │ │ │ +00027160: 4e75 6d62 6572 732c 204e 6578 743a 2066  Numbers, Next: f
│ │ │ │ +00027170: 7265 6545 7874 6572 696f 7253 756d 6d61  reeExteriorSumma
│ │ │ │ +00027180: 6e64 2c20 5072 6576 3a20 6578 7456 7343  nd, Prev: extVsC
│ │ │ │ +00027190: 6f68 6f6d 6f6c 6f67 792c 2055 703a 2054  ohomology, Up: T
│ │ │ │ +000271a0: 6f70 0a0a 6669 6e69 7465 4265 7474 694e  op..finiteBettiN
│ │ │ │ +000271b0: 756d 6265 7273 202d 2d20 6265 7474 6920  umbers -- betti 
│ │ │ │ +000271c0: 6e75 6d62 6572 7320 6f66 2066 696e 6974  numbers of finit
│ │ │ │ +000271d0: 6520 7265 736f 6c75 7469 6f6e 2063 6f6d  e resolution com
│ │ │ │ +000271e0: 7075 7465 6420 6672 6f6d 2061 206d 6174  puted from a mat
│ │ │ │ +000271f0: 7269 7820 6661 6374 6f72 697a 6174 696f  rix factorizatio
│ │ │ │ +00027200: 6e0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  n.**************
│ │ │ │  00027210: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00027220: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00027230: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00027240: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00027250: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00027260: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ -00027270: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ -00027280: 4c20 3d20 6669 6e69 7465 4265 7474 694e  L = finiteBettiN
│ │ │ │ -00027290: 756d 6265 7273 204d 460a 2020 2a20 496e  umbers MF.  * In
│ │ │ │ -000272a0: 7075 7473 3a0a 2020 2020 2020 2a20 4d46  puts:.      * MF
│ │ │ │ -000272b0: 2c20 6120 2a6e 6f74 6520 6c69 7374 3a20  , a *note list: 
│ │ │ │ -000272c0: 284d 6163 6175 6c61 7932 446f 6329 4c69  (Macaulay2Doc)Li
│ │ │ │ -000272d0: 7374 2c2c 204c 6973 7420 6f66 2048 6173  st,, List of Has
│ │ │ │ -000272e0: 6854 6162 6c65 7320 6173 2063 6f6d 7075  hTables as compu
│ │ │ │ -000272f0: 7465 640a 2020 2020 2020 2020 6279 2022  ted.        by "
│ │ │ │ -00027300: 6d61 7472 6978 4661 6374 6f72 697a 6174  matrixFactorizat
│ │ │ │ -00027310: 696f 6e22 0a20 202a 204f 7574 7075 7473  ion".  * Outputs
│ │ │ │ -00027320: 3a0a 2020 2020 2020 2a20 4c2c 2061 202a  :.      * L, a *
│ │ │ │ -00027330: 6e6f 7465 206c 6973 743a 2028 4d61 6361  note list: (Maca
│ │ │ │ -00027340: 756c 6179 3244 6f63 294c 6973 742c 2c20  ulay2Doc)List,, 
│ │ │ │ -00027350: 4c69 7374 206f 6620 6265 7474 6920 6e75  List of betti nu
│ │ │ │ -00027360: 6d62 6572 730a 0a44 6573 6372 6970 7469  mbers..Descripti
│ │ │ │ -00027370: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  on.===========..
│ │ │ │ -00027380: 5573 6573 2074 6865 2072 616e 6b73 206f  Uses the ranks o
│ │ │ │ -00027390: 6620 7468 6520 4220 6d61 7472 6963 6573  f the B matrices
│ │ │ │ -000273a0: 2069 6e20 6120 6d61 7472 6978 2066 6163   in a matrix fac
│ │ │ │ -000273b0: 746f 7269 7a61 7469 6f6e 2066 6f72 2061  torization for a
│ │ │ │ -000273c0: 206d 6f64 756c 6520 4d20 6f76 6572 0a53   module M over.S
│ │ │ │ -000273d0: 2f28 665f 312c 2e2e 2c66 5f63 2920 746f  /(f_1,..,f_c) to
│ │ │ │ -000273e0: 2063 6f6d 7075 7465 2074 6865 2062 6574   compute the bet
│ │ │ │ -000273f0: 7469 206e 756d 6265 7273 206f 6620 7468  ti numbers of th
│ │ │ │ -00027400: 6520 6d69 6e69 6d61 6c20 7265 736f 6c75  e minimal resolu
│ │ │ │ -00027410: 7469 6f6e 206f 6620 4d20 6f76 6572 0a53  tion of M over.S
│ │ │ │ -00027420: 2c20 7768 6963 6820 6973 2074 6865 2073  , which is the s
│ │ │ │ -00027430: 756d 206f 6620 7468 6520 4b6f 737a 756c  um of the Koszul
│ │ │ │ -00027440: 2063 6f6d 706c 6578 6573 204b 2866 5f31   complexes K(f_1
│ │ │ │ -00027450: 2e2e 665f 7b6a 2d31 7d29 2074 656e 736f  ..f_{j-1}) tenso
│ │ │ │ -00027460: 7265 6420 7769 7468 2042 286a 290a 0a2b  red with B(j)..+
│ │ │ │ +00027250: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a  ***************.
│ │ │ │ +00027260: 0a20 202a 2055 7361 6765 3a20 0a20 2020  .  * Usage: .   
│ │ │ │ +00027270: 2020 2020 204c 203d 2066 696e 6974 6542       L = finiteB
│ │ │ │ +00027280: 6574 7469 4e75 6d62 6572 7320 4d46 0a20  ettiNumbers MF. 
│ │ │ │ +00027290: 202a 2049 6e70 7574 733a 0a20 2020 2020   * Inputs:.     
│ │ │ │ +000272a0: 202a 204d 462c 2061 202a 6e6f 7465 206c   * MF, a *note l
│ │ │ │ +000272b0: 6973 743a 2028 4d61 6361 756c 6179 3244  ist: (Macaulay2D
│ │ │ │ +000272c0: 6f63 294c 6973 742c 2c20 4c69 7374 206f  oc)List,, List o
│ │ │ │ +000272d0: 6620 4861 7368 5461 626c 6573 2061 7320  f HashTables as 
│ │ │ │ +000272e0: 636f 6d70 7574 6564 0a20 2020 2020 2020  computed.       
│ │ │ │ +000272f0: 2062 7920 226d 6174 7269 7846 6163 746f   by "matrixFacto
│ │ │ │ +00027300: 7269 7a61 7469 6f6e 220a 2020 2a20 4f75  rization".  * Ou
│ │ │ │ +00027310: 7470 7574 733a 0a20 2020 2020 202a 204c  tputs:.      * L
│ │ │ │ +00027320: 2c20 6120 2a6e 6f74 6520 6c69 7374 3a20  , a *note list: 
│ │ │ │ +00027330: 284d 6163 6175 6c61 7932 446f 6329 4c69  (Macaulay2Doc)Li
│ │ │ │ +00027340: 7374 2c2c 204c 6973 7420 6f66 2062 6574  st,, List of bet
│ │ │ │ +00027350: 7469 206e 756d 6265 7273 0a0a 4465 7363  ti numbers..Desc
│ │ │ │ +00027360: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
│ │ │ │ +00027370: 3d3d 3d0a 0a55 7365 7320 7468 6520 7261  ===..Uses the ra
│ │ │ │ +00027380: 6e6b 7320 6f66 2074 6865 2042 206d 6174  nks of the B mat
│ │ │ │ +00027390: 7269 6365 7320 696e 2061 206d 6174 7269  rices in a matri
│ │ │ │ +000273a0: 7820 6661 6374 6f72 697a 6174 696f 6e20  x factorization 
│ │ │ │ +000273b0: 666f 7220 6120 6d6f 6475 6c65 204d 206f  for a module M o
│ │ │ │ +000273c0: 7665 720a 532f 2866 5f31 2c2e 2e2c 665f  ver.S/(f_1,..,f_
│ │ │ │ +000273d0: 6329 2074 6f20 636f 6d70 7574 6520 7468  c) to compute th
│ │ │ │ +000273e0: 6520 6265 7474 6920 6e75 6d62 6572 7320  e betti numbers 
│ │ │ │ +000273f0: 6f66 2074 6865 206d 696e 696d 616c 2072  of the minimal r
│ │ │ │ +00027400: 6573 6f6c 7574 696f 6e20 6f66 204d 206f  esolution of M o
│ │ │ │ +00027410: 7665 720a 532c 2077 6869 6368 2069 7320  ver.S, which is 
│ │ │ │ +00027420: 7468 6520 7375 6d20 6f66 2074 6865 204b  the sum of the K
│ │ │ │ +00027430: 6f73 7a75 6c20 636f 6d70 6c65 7865 7320  oszul complexes 
│ │ │ │ +00027440: 4b28 665f 312e 2e66 5f7b 6a2d 317d 2920  K(f_1..f_{j-1}) 
│ │ │ │ +00027450: 7465 6e73 6f72 6564 2077 6974 6820 4228  tensored with B(
│ │ │ │ +00027460: 6a29 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  j)..+-----------
│ │ │ │  00027470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000274a0: 2d2d 2b0a 7c69 3120 3a20 7365 7452 616e  --+.|i1 : setRan
│ │ │ │ -000274b0: 646f 6d53 6565 6420 3020 2020 2020 2020  domSeed 0       
│ │ │ │ -000274c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000274d0: 2020 2020 2020 207c 0a7c 202d 2d20 7365         |.| -- se
│ │ │ │ -000274e0: 7474 696e 6720 7261 6e64 6f6d 2073 6565  tting random see
│ │ │ │ -000274f0: 6420 746f 2030 2020 2020 2020 2020 2020  d to 0          
│ │ │ │ -00027500: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00027490: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2073  -------+.|i1 : s
│ │ │ │ +000274a0: 6574 5261 6e64 6f6d 5365 6564 2030 2020  etRandomSeed 0  
│ │ │ │ +000274b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000274c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000274d0: 2d2d 2073 6574 7469 6e67 2072 616e 646f  -- setting rando
│ │ │ │ +000274e0: 6d20 7365 6564 2074 6f20 3020 2020 2020  m seed to 0     
│ │ │ │ +000274f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027500: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00027510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027540: 207c 0a7c 6f31 203d 2030 2020 2020 2020   |.|o1 = 0      
│ │ │ │ +00027530: 2020 2020 2020 7c0a 7c6f 3120 3d20 3020        |.|o1 = 0 
│ │ │ │ +00027540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027570: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00027560: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +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 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ -000275b0: 203a 206b 6b20 3d20 5a5a 2f31 3031 2020   : kk = ZZ/101  
│ │ │ │ +000275a0: 2b0a 7c69 3220 3a20 6b6b 203d 205a 5a2f  +.|i2 : kk = ZZ/
│ │ │ │ +000275b0: 3130 3120 2020 2020 2020 2020 2020 2020  101             
│ │ │ │  000275c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000275d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000275e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000275d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000275e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000275f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027610: 2020 2020 207c 0a7c 6f32 203d 206b 6b20       |.|o2 = kk 
│ │ │ │ +00027600: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ +00027610: 3d20 6b6b 2020 2020 2020 2020 2020 2020  = kk            
│ │ │ │  00027620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027640: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00027630: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00027640: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00027650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027670: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00027680: 0a7c 6f32 203a 2051 756f 7469 656e 7452  .|o2 : QuotientR
│ │ │ │ -00027690: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │ -000276a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000276b0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00027670: 2020 2020 7c0a 7c6f 3220 3a20 5175 6f74      |.|o2 : Quot
│ │ │ │ +00027680: 6965 6e74 5269 6e67 2020 2020 2020 2020  ientRing        
│ │ │ │ +00027690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000276a0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +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 2d2b 0a7c 6933 203a  ---------+.|i3 :
│ │ │ │ -000276f0: 2053 203d 206b 6b5b 612c 622c 752c 765d   S = kk[a,b,u,v]
│ │ │ │ +000276d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000276e0: 7c69 3320 3a20 5320 3d20 6b6b 5b61 2c62  |i3 : S = kk[a,b
│ │ │ │ +000276f0: 2c75 2c76 5d20 2020 2020 2020 2020 2020  ,u,v]           
│ │ │ │  00027700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027710: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00027720: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00027710: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00027720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027750: 2020 207c 0a7c 6f33 203d 2053 2020 2020     |.|o3 = S    
│ │ │ │ +00027740: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20          |.|o3 = 
│ │ │ │ +00027750: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │  00027760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027780: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00027770: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00027780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000277a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000277b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000277c0: 6f33 203a 2050 6f6c 796e 6f6d 6961 6c52  o3 : PolynomialR
│ │ │ │ -000277d0: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │ -000277e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000277f0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +000277b0: 2020 7c0a 7c6f 3320 3a20 506f 6c79 6e6f    |.|o3 : Polyno
│ │ │ │ +000277c0: 6d69 616c 5269 6e67 2020 2020 2020 2020  mialRing        
│ │ │ │ +000277d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000277e0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +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 2d2b 0a7c 6934 203a 2066  -------+.|i4 : f
│ │ │ │ -00027830: 6620 3d20 6d61 7472 6978 2261 752c 6276  f = matrix"au,bv
│ │ │ │ -00027840: 2220 2020 2020 2020 2020 2020 2020 2020  "               
│ │ │ │ -00027850: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00027810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00027820: 3420 3a20 6666 203d 206d 6174 7269 7822  4 : ff = matrix"
│ │ │ │ +00027830: 6175 2c62 7622 2020 2020 2020 2020 2020  au,bv"          
│ │ │ │ +00027840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027850: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00027860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027890: 207c 0a7c 6f34 203d 207c 2061 7520 6276   |.|o4 = | au bv
│ │ │ │ -000278a0: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ -000278b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000278c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00027880: 2020 2020 2020 7c0a 7c6f 3420 3d20 7c20        |.|o4 = | 
│ │ │ │ +00027890: 6175 2062 7620 7c20 2020 2020 2020 2020  au bv |         
│ │ │ │ +000278a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000278b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000278c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000278d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000278e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000278f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00027900: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ -00027910: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -00027920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027930: 7c0a 7c6f 3420 3a20 4d61 7472 6978 2053  |.|o4 : Matrix S
│ │ │ │ -00027940: 2020 3c2d 2d20 5320 2020 2020 2020 2020    <-- S         
│ │ │ │ -00027950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027960: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +000278f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00027900: 3120 2020 2020 2032 2020 2020 2020 2020  1      2        
│ │ │ │ +00027910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027920: 2020 2020 207c 0a7c 6f34 203a 204d 6174       |.|o4 : Mat
│ │ │ │ +00027930: 7269 7820 5320 203c 2d2d 2053 2020 2020  rix S  <-- S    
│ │ │ │ +00027940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027950: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00027960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027990: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520  ----------+.|i5 
│ │ │ │ -000279a0: 3a20 5220 3d20 532f 6964 6561 6c20 6666  : R = S/ideal ff
│ │ │ │ +00027980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00027990: 0a7c 6935 203a 2052 203d 2053 2f69 6465  .|i5 : R = S/ide
│ │ │ │ +000279a0: 616c 2066 6620 2020 2020 2020 2020 2020  al ff           
│ │ │ │  000279b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000279c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000279d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000279c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000279d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000279e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000279f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027a00: 2020 2020 7c0a 7c6f 3520 3d20 5220 2020      |.|o5 = R   
│ │ │ │ +000279f0: 2020 2020 2020 2020 207c 0a7c 6f35 203d           |.|o5 =
│ │ │ │ +00027a00: 2052 2020 2020 2020 2020 2020 2020 2020   R              
│ │ │ │  00027a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027a30: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00027a20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00027a30: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00027a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027a60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00027a70: 7c6f 3520 3a20 5175 6f74 6965 6e74 5269  |o5 : QuotientRi
│ │ │ │ -00027a80: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │ -00027a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027aa0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00027a60: 2020 207c 0a7c 6f35 203a 2051 756f 7469     |.|o5 : Quoti
│ │ │ │ +00027a70: 656e 7452 696e 6720 2020 2020 2020 2020  entRing         
│ │ │ │ +00027a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027a90: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00027aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027ad0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20  --------+.|i6 : 
│ │ │ │ -00027ae0: 4d30 203d 2052 5e31 2f69 6465 616c 2261  M0 = R^1/ideal"a
│ │ │ │ -00027af0: 2c62 2220 2020 2020 2020 2020 2020 2020  ,b"             
│ │ │ │ -00027b00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00027ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00027ad0: 6936 203a 204d 3020 3d20 525e 312f 6964  i6 : M0 = R^1/id
│ │ │ │ +00027ae0: 6561 6c22 612c 6222 2020 2020 2020 2020  eal"a,b"        
│ │ │ │ +00027af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027b00: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00027b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027b40: 2020 7c0a 7c6f 3620 3d20 636f 6b65 726e    |.|o6 = cokern
│ │ │ │ -00027b50: 656c 207c 2061 2062 207c 2020 2020 2020  el | a b |      
│ │ │ │ -00027b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027b70: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00027b30: 2020 2020 2020 207c 0a7c 6f36 203d 2063         |.|o6 = c
│ │ │ │ +00027b40: 6f6b 6572 6e65 6c20 7c20 6120 6220 7c20  okernel | a b | 
│ │ │ │ +00027b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027b60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00027b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027ba0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00027ba0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00027bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027bc0: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ -00027bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027be0: 207c 0a7c 6f36 203a 2052 2d6d 6f64 756c   |.|o6 : R-modul
│ │ │ │ -00027bf0: 652c 2071 756f 7469 656e 7420 6f66 2052  e, quotient of R
│ │ │ │ -00027c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027c10: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00027bc0: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +00027bd0: 2020 2020 2020 7c0a 7c6f 3620 3a20 522d        |.|o6 : R-
│ │ │ │ +00027be0: 6d6f 6475 6c65 2c20 7175 6f74 6965 6e74  module, quotient
│ │ │ │ +00027bf0: 206f 6620 5220 2020 2020 2020 2020 2020   of R           
│ │ │ │ +00027c00: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00027c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937  -----------+.|i7
│ │ │ │ -00027c50: 203a 2046 203d 2066 7265 6552 6573 6f6c   : F = freeResol
│ │ │ │ -00027c60: 7574 696f 6e28 4d30 2c20 4c65 6e67 7468  ution(M0, Length
│ │ │ │ -00027c70: 4c69 6d69 7420 3d3e 3329 2020 2020 2020  Limit =>3)      
│ │ │ │ -00027c80: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00027c40: 2b0a 7c69 3720 3a20 4620 3d20 6672 6565  +.|i7 : F = free
│ │ │ │ +00027c50: 5265 736f 6c75 7469 6f6e 284d 302c 204c  Resolution(M0, L
│ │ │ │ +00027c60: 656e 6774 684c 696d 6974 203d 3e33 2920  engthLimit =>3) 
│ │ │ │ +00027c70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00027c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027cb0: 2020 2020 207c 0a7c 2020 2020 2020 3120       |.|      1 
│ │ │ │ -00027cc0: 2020 2020 2032 2020 2020 2020 3320 2020       2      3   
│ │ │ │ -00027cd0: 2020 2034 2020 2020 2020 2020 2020 2020     4            
│ │ │ │ -00027ce0: 2020 2020 2020 2020 2020 7c0a 7c6f 3720            |.|o7 
│ │ │ │ -00027cf0: 3d20 5220 203c 2d2d 2052 2020 3c2d 2d20  = R  <-- R  <-- 
│ │ │ │ -00027d00: 5220 203c 2d2d 2052 2020 2020 2020 2020  R  <-- R        
│ │ │ │ -00027d10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00027d20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00027ca0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00027cb0: 2020 2031 2020 2020 2020 3220 2020 2020     1      2     
│ │ │ │ +00027cc0: 2033 2020 2020 2020 3420 2020 2020 2020   3      4       
│ │ │ │ +00027cd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00027ce0: 0a7c 6f37 203d 2052 2020 3c2d 2d20 5220  .|o7 = R  <-- R 
│ │ │ │ +00027cf0: 203c 2d2d 2052 2020 3c2d 2d20 5220 2020   <-- R  <-- R   
│ │ │ │ +00027d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027d10: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00027d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027d50: 2020 2020 7c0a 7c20 2020 2020 3020 2020      |.|     0   
│ │ │ │ -00027d60: 2020 2031 2020 2020 2020 3220 2020 2020     1      2     
│ │ │ │ -00027d70: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ -00027d80: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00027d40: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00027d50: 2030 2020 2020 2020 3120 2020 2020 2032   0      1      2
│ │ │ │ +00027d60: 2020 2020 2020 3320 2020 2020 2020 2020        3         
│ │ │ │ +00027d70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00027d80: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00027d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027db0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00027dc0: 7c6f 3720 3a20 436f 6d70 6c65 7820 2020  |o7 : Complex   
│ │ │ │ +00027db0: 2020 207c 0a7c 6f37 203a 2043 6f6d 706c     |.|o7 : Compl
│ │ │ │ +00027dc0: 6578 2020 2020 2020 2020 2020 2020 2020  ex              
│ │ │ │  00027dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027df0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00027de0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00027df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027e20: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20  --------+.|i8 : 
│ │ │ │ -00027e30: 4d20 3d20 636f 6b65 7220 462e 6464 5f33  M = coker F.dd_3
│ │ │ │ -00027e40: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ -00027e50: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00027e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00027e20: 6938 203a 204d 203d 2063 6f6b 6572 2046  i8 : M = coker F
│ │ │ │ +00027e30: 2e64 645f 333b 2020 2020 2020 2020 2020  .dd_3;          
│ │ │ │ +00027e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027e50: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00027e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027e90: 2d2d 2b0a 7c69 3920 3a20 4d46 203d 206d  --+.|i9 : MF = m
│ │ │ │ -00027ea0: 6174 7269 7846 6163 746f 7269 7a61 7469  atrixFactorizati
│ │ │ │ -00027eb0: 6f6e 2866 662c 4d29 3b20 2020 2020 2020  on(ff,M);       
│ │ │ │ -00027ec0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00027e80: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 204d  -------+.|i9 : M
│ │ │ │ +00027e90: 4620 3d20 6d61 7472 6978 4661 6374 6f72  F = matrixFactor
│ │ │ │ +00027ea0: 697a 6174 696f 6e28 6666 2c4d 293b 2020  ization(ff,M);  
│ │ │ │ +00027eb0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00027ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  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 2b0a 7c69  ------------+.|i
│ │ │ │ -00027f00: 3130 203a 2062 6574 7469 2066 7265 6552  10 : betti freeR
│ │ │ │ -00027f10: 6573 6f6c 7574 696f 6e20 7075 7368 466f  esolution pushFo
│ │ │ │ -00027f20: 7277 6172 6428 6d61 7028 522c 5329 2c4d  rward(map(R,S),M
│ │ │ │ -00027f30: 297c 0a7c 2020 2020 2020 2020 2020 2020  )|.|            
│ │ │ │ +00027ef0: 2d2b 0a7c 6931 3020 3a20 6265 7474 6920  -+.|i10 : betti 
│ │ │ │ +00027f00: 6672 6565 5265 736f 6c75 7469 6f6e 2070  freeResolution p
│ │ │ │ +00027f10: 7573 6846 6f72 7761 7264 286d 6170 2852  ushForward(map(R
│ │ │ │ +00027f20: 2c53 292c 4d29 7c0a 7c20 2020 2020 2020  ,S),M)|.|       
│ │ │ │ +00027f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027f60: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00027f70: 2020 2020 2020 3020 3120 3220 2020 2020        0 1 2     
│ │ │ │ +00027f50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00027f60: 2020 2020 2020 2020 2020 2030 2031 2032             0 1 2
│ │ │ │ +00027f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027f90: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -00027fa0: 3020 3d20 746f 7461 6c3a 2033 2035 2032  0 = total: 3 5 2
│ │ │ │ +00027f90: 7c0a 7c6f 3130 203d 2074 6f74 616c 3a20  |.|o10 = total: 
│ │ │ │ +00027fa0: 3320 3520 3220 2020 2020 2020 2020 2020  3 5 2           
│ │ │ │  00027fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027fd0: 7c0a 7c20 2020 2020 2020 2020 2032 3a20  |.|          2: 
│ │ │ │ -00027fe0: 3320 3420 2e20 2020 2020 2020 2020 2020  3 4 .           
│ │ │ │ -00027ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028000: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00028010: 2020 333a 202e 2031 2032 2020 2020 2020    3: . 1 2      
│ │ │ │ -00028020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028030: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00027fc0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00027fd0: 2020 323a 2033 2034 202e 2020 2020 2020    2: 3 4 .      
│ │ │ │ +00027fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027ff0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00028000: 2020 2020 2020 2033 3a20 2e20 3120 3220         3: . 1 2 
│ │ │ │ +00028010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028020: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00028030: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00028040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028060: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00028070: 0a7c 6f31 3020 3a20 4265 7474 6954 616c  .|o10 : BettiTal
│ │ │ │ -00028080: 6c79 2020 2020 2020 2020 2020 2020 2020  ly              
│ │ │ │ -00028090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000280a0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00028060: 2020 2020 7c0a 7c6f 3130 203a 2042 6574      |.|o10 : Bet
│ │ │ │ +00028070: 7469 5461 6c6c 7920 2020 2020 2020 2020  tiTally         
│ │ │ │ +00028080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028090: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +000280a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000280b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000280c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000280d0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120  ---------+.|i11 
│ │ │ │ -000280e0: 3a20 6669 6e69 7465 4265 7474 694e 756d  : finiteBettiNum
│ │ │ │ -000280f0: 6265 7273 204d 4620 2020 2020 2020 2020  bers MF         
│ │ │ │ -00028100: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00028110: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000280c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000280d0: 7c69 3131 203a 2066 696e 6974 6542 6574  |i11 : finiteBet
│ │ │ │ +000280e0: 7469 4e75 6d62 6572 7320 4d46 2020 2020  tiNumbers MF    
│ │ │ │ +000280f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028100: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00028110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028140: 2020 207c 0a7c 6f31 3120 3d20 7b33 2c20     |.|o11 = {3, 
│ │ │ │ -00028150: 352c 2032 7d20 2020 2020 2020 2020 2020  5, 2}           
│ │ │ │ -00028160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028170: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00028130: 2020 2020 2020 2020 7c0a 7c6f 3131 203d          |.|o11 =
│ │ │ │ +00028140: 207b 332c 2035 2c20 327d 2020 2020 2020   {3, 5, 2}      
│ │ │ │ +00028150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028160: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00028170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000281a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000281b0: 6f31 3120 3a20 4c69 7374 2020 2020 2020  o11 : List      
│ │ │ │ +000281a0: 2020 7c0a 7c6f 3131 203a 204c 6973 7420    |.|o11 : List 
│ │ │ │ +000281b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000281c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000281d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000281e0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +000281d0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +000281e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000281f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028210: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220 3a20  -------+.|i12 : 
│ │ │ │ -00028220: 696e 6669 6e69 7465 4265 7474 694e 756d  infiniteBettiNum
│ │ │ │ -00028230: 6265 7273 284d 462c 3529 2020 2020 2020  bers(MF,5)      
│ │ │ │ -00028240: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00028200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00028210: 3132 203a 2069 6e66 696e 6974 6542 6574  12 : infiniteBet
│ │ │ │ +00028220: 7469 4e75 6d62 6572 7328 4d46 2c35 2920  tiNumbers(MF,5) 
│ │ │ │ +00028230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028240: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00028250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028280: 207c 0a7c 6f31 3220 3d20 7b33 2c20 342c   |.|o12 = {3, 4,
│ │ │ │ -00028290: 2035 2c20 362c 2037 2c20 387d 2020 2020   5, 6, 7, 8}    
│ │ │ │ -000282a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000282b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00028270: 2020 2020 2020 7c0a 7c6f 3132 203d 207b        |.|o12 = {
│ │ │ │ +00028280: 332c 2034 2c20 352c 2036 2c20 372c 2038  3, 4, 5, 6, 7, 8
│ │ │ │ +00028290: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +000282a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000282b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000282c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000282d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000282e0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -000282f0: 3220 3a20 4c69 7374 2020 2020 2020 2020  2 : List        
│ │ │ │ +000282e0: 7c0a 7c6f 3132 203a 204c 6973 7420 2020  |.|o12 : List   
│ │ │ │ +000282f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028320: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00028310: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00028320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028350: 2d2d 2d2d 2d2b 0a7c 6931 3320 3a20 6265  -----+.|i13 : be
│ │ │ │ -00028360: 7474 6920 6672 6565 5265 736f 6c75 7469  tti freeResoluti
│ │ │ │ -00028370: 6f6e 2028 4d2c 204c 656e 6774 684c 696d  on (M, LengthLim
│ │ │ │ -00028380: 6974 203d 3e20 3529 2020 7c0a 7c20 2020  it => 5)  |.|   
│ │ │ │ +00028340: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3133  ----------+.|i13
│ │ │ │ +00028350: 203a 2062 6574 7469 2066 7265 6552 6573   : betti freeRes
│ │ │ │ +00028360: 6f6c 7574 696f 6e20 284d 2c20 4c65 6e67  olution (M, Leng
│ │ │ │ +00028370: 7468 4c69 6d69 7420 3d3e 2035 2920 207c  thLimit => 5)  |
│ │ │ │ +00028380: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00028390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000283a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000283b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000283c0: 0a7c 2020 2020 2020 2020 2020 2020 2030  .|             0
│ │ │ │ -000283d0: 2031 2032 2033 2034 2035 2020 2020 2020   1 2 3 4 5      
│ │ │ │ -000283e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000283f0: 2020 2020 7c0a 7c6f 3133 203d 2074 6f74      |.|o13 = tot
│ │ │ │ -00028400: 616c 3a20 3320 3420 3520 3620 3720 3820  al: 3 4 5 6 7 8 
│ │ │ │ -00028410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028420: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00028430: 2020 2020 2020 323a 2033 2034 2035 2036        2: 3 4 5 6
│ │ │ │ -00028440: 2037 2038 2020 2020 2020 2020 2020 2020   7 8            
│ │ │ │ -00028450: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00028460: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000283b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000283c0: 2020 2020 3020 3120 3220 3320 3420 3520      0 1 2 3 4 5 
│ │ │ │ +000283d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000283e0: 2020 2020 2020 2020 207c 0a7c 6f31 3320           |.|o13 
│ │ │ │ +000283f0: 3d20 746f 7461 6c3a 2033 2034 2035 2036  = total: 3 4 5 6
│ │ │ │ +00028400: 2037 2038 2020 2020 2020 2020 2020 2020   7 8            
│ │ │ │ +00028410: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00028420: 7c20 2020 2020 2020 2020 2032 3a20 3320  |          2: 3 
│ │ │ │ +00028430: 3420 3520 3620 3720 3820 2020 2020 2020  4 5 6 7 8       
│ │ │ │ +00028440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028450: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00028460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028490: 2020 207c 0a7c 6f31 3320 3a20 4265 7474     |.|o13 : Bett
│ │ │ │ -000284a0: 6954 616c 6c79 2020 2020 2020 2020 2020  iTally          
│ │ │ │ -000284b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000284c0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00028480: 2020 2020 2020 2020 7c0a 7c6f 3133 203a          |.|o13 :
│ │ │ │ +00028490: 2042 6574 7469 5461 6c6c 7920 2020 2020   BettiTally     
│ │ │ │ +000284a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000284b0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +000284c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000284d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000284e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000284f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ -00028500: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d  See also.=======
│ │ │ │ -00028510: 3d0a 0a20 202a 202a 6e6f 7465 206d 6174  =..  * *note mat
│ │ │ │ -00028520: 7269 7846 6163 746f 7269 7a61 7469 6f6e  rixFactorization
│ │ │ │ -00028530: 3a20 6d61 7472 6978 4661 6374 6f72 697a  : matrixFactoriz
│ │ │ │ -00028540: 6174 696f 6e2c 202d 2d20 4d61 7073 2069  ation, -- Maps i
│ │ │ │ -00028550: 6e20 6120 6869 6768 6572 0a20 2020 2063  n a higher.    c
│ │ │ │ -00028560: 6f64 696d 656e 7369 6f6e 206d 6174 7269  odimension matri
│ │ │ │ -00028570: 7820 6661 6374 6f72 697a 6174 696f 6e0a  x factorization.
│ │ │ │ -00028580: 2020 2a20 2a6e 6f74 6520 696e 6669 6e69    * *note infini
│ │ │ │ -00028590: 7465 4265 7474 694e 756d 6265 7273 3a20  teBettiNumbers: 
│ │ │ │ -000285a0: 696e 6669 6e69 7465 4265 7474 694e 756d  infiniteBettiNum
│ │ │ │ -000285b0: 6265 7273 2c20 2d2d 2062 6574 7469 206e  bers, -- betti n
│ │ │ │ -000285c0: 756d 6265 7273 206f 660a 2020 2020 6669  umbers of.    fi
│ │ │ │ -000285d0: 6e69 7465 2072 6573 6f6c 7574 696f 6e20  nite resolution 
│ │ │ │ -000285e0: 636f 6d70 7574 6564 2066 726f 6d20 6120  computed from a 
│ │ │ │ -000285f0: 6d61 7472 6978 2066 6163 746f 7269 7a61  matrix factoriza
│ │ │ │ -00028600: 7469 6f6e 0a0a 5761 7973 2074 6f20 7573  tion..Ways to us
│ │ │ │ -00028610: 6520 6669 6e69 7465 4265 7474 694e 756d  e finiteBettiNum
│ │ │ │ -00028620: 6265 7273 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d  bers:.==========
│ │ │ │ -00028630: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00028640: 3d3d 3d3d 3d0a 0a20 202a 2022 6669 6e69  =====..  * "fini
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│ │ │ │  00028710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00028790: 3235 2e31 312b 6473 2f4d 322f 4d61 6361  25.11+ds/M2/Maca
│ │ │ │ +000287a0: 756c 6179 322f 7061 636b 6167 6573 2f0a  ulay2/packages/.
│ │ │ │ +000287b0: 436f 6d70 6c65 7465 496e 7465 7273 6563  CompleteIntersec
│ │ │ │ +000287c0: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e  tionResolutions.
│ │ │ │ +000287d0: 6d32 3a34 3037 323a 302e 0a1f 0a46 696c  m2:4072:0....Fil
│ │ │ │ +000287e0: 653a 2043 6f6d 706c 6574 6549 6e74 6572  e: CompleteInter
│ │ │ │ +000287f0: 7365 6374 696f 6e52 6573 6f6c 7574 696f  sectionResolutio
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│ │ │ │ +00028810: 7265 6545 7874 6572 696f 7253 756d 6d61  reeExteriorSumma
│ │ │ │ +00028820: 6e64 2c20 4e65 7874 3a20 4772 6164 696e  nd, Next: Gradin
│ │ │ │ +00028830: 672c 2050 7265 763a 2066 696e 6974 6542  g, Prev: finiteB
│ │ │ │ +00028840: 6574 7469 4e75 6d62 6572 732c 2055 703a  ettiNumbers, Up:
│ │ │ │ +00028850: 2054 6f70 0a0a 6672 6565 4578 7465 7269   Top..freeExteri
│ │ │ │ +00028860: 6f72 5375 6d6d 616e 6420 2d2d 2066 696e  orSummand -- fin
│ │ │ │ +00028870: 6420 7468 6520 6672 6565 2073 756d 6d61  d the free summa
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│ │ │ │ +00028890: 6f76 6572 2061 6e20 6578 7465 7269 6f72  over an exterior
│ │ │ │ +000288a0: 2061 6c67 6562 7261 0a2a 2a2a 2a2a 2a2a   algebra.*******
│ │ │ │ +000288b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000288c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000288d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000288e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000288f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00028900: 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 6167  ******..  * Usag
│ │ │ │ -00028910: 653a 200a 2020 2020 2020 2020 4620 3d20  e: .        F = 
│ │ │ │ -00028920: 6672 6565 4578 7465 7269 6f72 5375 6d6d  freeExteriorSumm
│ │ │ │ -00028930: 616e 6420 4d0a 2020 2a20 496e 7075 7473  and M.  * Inputs
│ │ │ │ -00028940: 3a0a 2020 2020 2020 2a20 4d2c 2061 202a  :.      * M, a *
│ │ │ │ -00028950: 6e6f 7465 206d 6f64 756c 653a 2028 4d61  note module: (Ma
│ │ │ │ -00028960: 6361 756c 6179 3244 6f63 294d 6f64 756c  caulay2Doc)Modul
│ │ │ │ -00028970: 652c 2c20 6f76 6572 2061 6e20 6578 7465  e,, over an exte
│ │ │ │ -00028980: 7269 6f72 2061 6c67 6562 7261 0a20 202a  rior algebra.  *
│ │ │ │ -00028990: 204f 7574 7075 7473 3a0a 2020 2020 2020   Outputs:.      
│ │ │ │ -000289a0: 2a20 462c 2061 202a 6e6f 7465 206d 6174  * F, a *note mat
│ │ │ │ -000289b0: 7269 783a 2028 4d61 6361 756c 6179 3244  rix: (Macaulay2D
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│ │ │ │ -000289d0: 6672 6f6d 2061 2066 7265 6520 6d6f 6475  from a free modu
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│ │ │ │ -000289f0: 2049 6d61 6765 2069 7320 7468 6520 6c61   Image is the la
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│ │ │ │ +000288f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a  ***********..  *
│ │ │ │ +00028900: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ +00028910: 2046 203d 2066 7265 6545 7874 6572 696f   F = freeExterio
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│ │ │ │ +00028930: 6e70 7574 733a 0a20 2020 2020 202a 204d  nputs:.      * M
│ │ │ │ +00028940: 2c20 6120 2a6e 6f74 6520 6d6f 6475 6c65  , a *note module
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│ │ │ │ +00028990: 2020 2020 202a 2046 2c20 6120 2a6e 6f74       * F, a *not
│ │ │ │ +000289a0: 6520 6d61 7472 6978 3a20 284d 6163 6175  e matrix: (Macau
│ │ │ │ +000289b0: 6c61 7932 446f 6329 4d61 7472 6978 2c2c  lay2Doc)Matrix,,
│ │ │ │ +000289c0: 204d 6170 2066 726f 6d20 6120 6672 6565   Map from a free
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│ │ │ │ +000289e0: 2020 2020 2020 496d 6167 6520 6973 2074        Image is t
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│ │ │ │ +00028a20: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..+-------------
│ │ │ │  00028a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00028a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028a90: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00028a50: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 6b6b  ------+.|i1 : kk
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│ │ │ │ +00028a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028a80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00028a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028ac0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00028ad0: 6f31 203d 206b 6b20 2020 2020 2020 2020  o1 = kk         
│ │ │ │ +00028ac0: 2020 7c0a 7c6f 3120 3d20 6b6b 2020 2020    |.|o1 = kk    
│ │ │ │ +00028ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028b00: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00028af0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00028b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028b30: 2020 2020 2020 2020 207c 0a7c 6f31 203a           |.|o1 :
│ │ │ │ -00028b40: 2051 756f 7469 656e 7452 696e 6720 2020   QuotientRing   
│ │ │ │ +00028b20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00028b30: 7c6f 3120 3a20 5175 6f74 6965 6e74 5269  |o1 : QuotientRi
│ │ │ │ +00028b40: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │  00028b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028b60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00028b70: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00028b60: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00028b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028ba0: 2d2d 2d2d 2d2b 0a7c 6932 203a 2045 203d  -----+.|i2 : E =
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│ │ │ │ +00028ba0: 3a20 4520 3d20 6b6b 5b65 2c66 2c67 2c20  : E = kk[e,f,g, 
│ │ │ │ +00028bb0: 536b 6577 436f 6d6d 7574 6174 6976 6520  SkewCommutative 
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│ │ │ │ +00028bd0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
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│ │ │ │  00028bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00028c00: 2020 2020 2020 7c0a 7c6f 3220 3d20 4520        |.|o2 = E 
│ │ │ │ +00028c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028c40: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00028c30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00028c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028c70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │ +000298c0: 6865 2068 696c 6265 7274 2066 756e 6374  he hilbert funct
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│ │ │ │ +000298f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00029900: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00029910: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00029920: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a  **************..
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│ │ │ │ -00029950: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ -00029960: 2020 2a20 732c 2061 202a 6e6f 7465 2073    * s, a *note s
│ │ │ │ -00029970: 6571 7565 6e63 653a 2028 4d61 6361 756c  equence: (Macaul
│ │ │ │ -00029980: 6179 3244 6f63 2953 6571 7565 6e63 652c  ay2Doc)Sequence,
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│ │ │ │ -000299a0: 2a20 502c 2061 202a 6e6f 7465 206d 6f64  * P, a *note mod
│ │ │ │ -000299b0: 756c 653a 2028 4d61 6361 756c 6179 3244  ule: (Macaulay2D
│ │ │ │ -000299c0: 6f63 294d 6f64 756c 652c 2c20 6772 6164  oc)Module,, grad
│ │ │ │ -000299d0: 6564 206d 6f64 756c 650a 2020 2a20 4f75  ed module.  * Ou
│ │ │ │ -000299e0: 7470 7574 733a 0a20 2020 2020 202a 2048  tputs:.      * H
│ │ │ │ -000299f0: 2c20 6120 2a6e 6f74 6520 6c69 7374 3a20  , a *note list: 
│ │ │ │ -00029a00: 284d 6163 6175 6c61 7932 446f 6329 4c69  (Macaulay2Doc)Li
│ │ │ │ -00029a10: 7374 2c2c 200a 0a57 6179 7320 746f 2075  st,, ..Ways to u
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│ │ │ │ -00029a30: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2268 6628  ======..  * "hf(
│ │ │ │ -00029a40: 4c69 7374 2c4d 6f64 756c 6529 220a 2020  List,Module)".  
│ │ │ │ -00029a50: 2a20 2268 6628 5365 7175 656e 6365 2c4d  * "hf(Sequence,M
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│ │ │ │ -00029a80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
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│ │ │ │ +00029a20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ +00029a30: 2022 6866 284c 6973 742c 4d6f 6475 6c65   "hf(List,Module
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│ │ │ │ +00029ab0: 6675 6e63 7469 6f6e 3a0a 284d 6163 6175  function:.(Macau
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│ │ │ │ -00029b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029b30: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365  ----..The source
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│ │ │ │ -00029b50: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72  t is in./build/r
│ │ │ │ -00029b60: 6570 726f 6475 6369 626c 652d 7061 7468  eproducible-path
│ │ │ │ -00029b70: 2f6d 6163 6175 6c61 7932 2d31 2e32 352e  /macaulay2-1.25.
│ │ │ │ -00029b80: 3131 2b64 732f 4d32 2f4d 6163 6175 6c61  11+ds/M2/Macaula
│ │ │ │ -00029b90: 7932 2f70 6163 6b61 6765 732f 0a43 6f6d  y2/packages/.Com
│ │ │ │ -00029ba0: 706c 6574 6549 6e74 6572 7365 6374 696f  pleteIntersectio
│ │ │ │ -00029bb0: 6e52 6573 6f6c 7574 696f 6e73 2e6d 323a  nResolutions.m2:
│ │ │ │ -00029bc0: 3435 3932 3a30 2e0a 1f0a 4669 6c65 3a20  4592:0....File: 
│ │ │ │ -00029bd0: 436f 6d70 6c65 7465 496e 7465 7273 6563  CompleteIntersec
│ │ │ │ -00029be0: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e  tionResolutions.
│ │ │ │ -00029bf0: 696e 666f 2c20 4e6f 6465 3a20 6866 4d6f  info, Node: hfMo
│ │ │ │ -00029c00: 6475 6c65 4173 4578 742c 204e 6578 743a  duleAsExt, Next:
│ │ │ │ -00029c10: 2068 6967 6853 797a 7967 792c 2050 7265   highSyzygy, Pre
│ │ │ │ -00029c20: 763a 2068 662c 2055 703a 2054 6f70 0a0a  v: hf, Up: Top..
│ │ │ │ -00029c30: 6866 4d6f 6475 6c65 4173 4578 7420 2d2d  hfModuleAsExt --
│ │ │ │ -00029c40: 2070 7265 6469 6374 2062 6574 7469 206e   predict betti n
│ │ │ │ -00029c50: 756d 6265 7273 206f 6620 6d6f 6475 6c65  umbers of module
│ │ │ │ -00029c60: 4173 4578 7428 4d2c 5229 0a2a 2a2a 2a2a  AsExt(M,R).*****
│ │ │ │ +00029b20: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073  ---------..The s
│ │ │ │ +00029b30: 6f75 7263 6520 6f66 2074 6869 7320 646f  ource of this do
│ │ │ │ +00029b40: 6375 6d65 6e74 2069 7320 696e 0a2f 6275  cument is in./bu
│ │ │ │ +00029b50: 696c 642f 7265 7072 6f64 7563 6962 6c65  ild/reproducible
│ │ │ │ +00029b60: 2d70 6174 682f 6d61 6361 756c 6179 322d  -path/macaulay2-
│ │ │ │ +00029b70: 312e 3235 2e31 312b 6473 2f4d 322f 4d61  1.25.11+ds/M2/Ma
│ │ │ │ +00029b80: 6361 756c 6179 322f 7061 636b 6167 6573  caulay2/packages
│ │ │ │ +00029b90: 2f0a 436f 6d70 6c65 7465 496e 7465 7273  /.CompleteInters
│ │ │ │ +00029ba0: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e  ectionResolution
│ │ │ │ +00029bb0: 732e 6d32 3a34 3539 323a 302e 0a1f 0a46  s.m2:4592:0....F
│ │ │ │ +00029bc0: 696c 653a 2043 6f6d 706c 6574 6549 6e74  ile: CompleteInt
│ │ │ │ +00029bd0: 6572 7365 6374 696f 6e52 6573 6f6c 7574  ersectionResolut
│ │ │ │ +00029be0: 696f 6e73 2e69 6e66 6f2c 204e 6f64 653a  ions.info, Node:
│ │ │ │ +00029bf0: 2068 664d 6f64 756c 6541 7345 7874 2c20   hfModuleAsExt, 
│ │ │ │ +00029c00: 4e65 7874 3a20 6869 6768 5379 7a79 6779  Next: highSyzygy
│ │ │ │ +00029c10: 2c20 5072 6576 3a20 6866 2c20 5570 3a20  , Prev: hf, Up: 
│ │ │ │ +00029c20: 546f 700a 0a68 664d 6f64 756c 6541 7345  Top..hfModuleAsE
│ │ │ │ +00029c30: 7874 202d 2d20 7072 6564 6963 7420 6265  xt -- predict be
│ │ │ │ +00029c40: 7474 6920 6e75 6d62 6572 7320 6f66 206d  tti numbers of m
│ │ │ │ +00029c50: 6f64 756c 6541 7345 7874 284d 2c52 290a  oduleAsExt(M,R).
│ │ │ │ +00029c60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00029c70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00029c80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00029c90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00029ca0: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765  *****..  * Usage
│ │ │ │ -00029cb0: 3a20 0a20 2020 2020 2020 2073 6571 203d  : .        seq =
│ │ │ │ -00029cc0: 2068 664d 6f64 756c 6541 7345 7874 286e   hfModuleAsExt(n
│ │ │ │ -00029cd0: 756d 5661 6c75 6573 2c4d 2c6e 756d 6765  umValues,M,numge
│ │ │ │ -00029ce0: 6e73 5229 0a20 202a 2049 6e70 7574 733a  nsR).  * Inputs:
│ │ │ │ -00029cf0: 0a20 2020 2020 202a 206e 756d 5661 6c75  .      * numValu
│ │ │ │ -00029d00: 6573 2c20 616e 202a 6e6f 7465 2069 6e74  es, an *note int
│ │ │ │ -00029d10: 6567 6572 3a20 284d 6163 6175 6c61 7932  eger: (Macaulay2
│ │ │ │ -00029d20: 446f 6329 5a5a 2c2c 206e 756d 6265 7220  Doc)ZZ,, number 
│ │ │ │ -00029d30: 6f66 2076 616c 7565 7320 746f 0a20 2020  of values to.   
│ │ │ │ -00029d40: 2020 2020 2063 6f6d 7075 7465 0a20 2020       compute.   
│ │ │ │ -00029d50: 2020 202a 204d 2c20 6120 2a6e 6f74 6520     * M, a *note 
│ │ │ │ -00029d60: 6d6f 6475 6c65 3a20 284d 6163 6175 6c61  module: (Macaula
│ │ │ │ -00029d70: 7932 446f 6329 4d6f 6475 6c65 2c2c 206d  y2Doc)Module,, m
│ │ │ │ -00029d80: 6f64 756c 6520 6f76 6572 2074 6865 2072  odule over the r
│ │ │ │ -00029d90: 696e 6720 6f66 0a20 2020 2020 2020 206f  ing of.        o
│ │ │ │ -00029da0: 7065 7261 746f 7273 0a20 2020 2020 202a  perators.      *
│ │ │ │ -00029db0: 206e 756d 6765 6e73 522c 2061 6e20 2a6e   numgensR, an *n
│ │ │ │ -00029dc0: 6f74 6520 696e 7465 6765 723a 2028 4d61  ote integer: (Ma
│ │ │ │ -00029dd0: 6361 756c 6179 3244 6f63 295a 5a2c 2c20  caulay2Doc)ZZ,, 
│ │ │ │ -00029de0: 6e75 6d62 6572 206f 6620 6765 6e65 7261  number of genera
│ │ │ │ -00029df0: 746f 7273 206f 660a 2020 2020 2020 2020  tors of.        
│ │ │ │ -00029e00: 7468 6520 7461 7267 6574 2072 696e 670a  the target ring.
│ │ │ │ -00029e10: 2020 2a20 4f75 7470 7574 733a 0a20 2020    * Outputs:.   
│ │ │ │ -00029e20: 2020 202a 2073 6571 2c20 6120 2a6e 6f74     * seq, a *not
│ │ │ │ -00029e30: 6520 7365 7175 656e 6365 3a20 284d 6163  e sequence: (Mac
│ │ │ │ -00029e40: 6175 6c61 7932 446f 6329 5365 7175 656e  aulay2Doc)Sequen
│ │ │ │ -00029e50: 6365 2c2c 2073 6571 7565 6e63 6520 6f66  ce,, sequence of
│ │ │ │ -00029e60: 206e 756d 5661 6c75 6573 0a20 2020 2020   numValues.     
│ │ │ │ -00029e70: 2020 2069 6e74 6567 6572 732c 2074 6865     integers, the
│ │ │ │ -00029e80: 2065 7870 6563 7465 6420 746f 7461 6c20   expected total 
│ │ │ │ -00029e90: 4265 7474 6920 6e75 6d62 6572 730a 0a44  Betti numbers..D
│ │ │ │ -00029ea0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ -00029eb0: 3d3d 3d3d 3d3d 0a0a 4769 7665 6e20 6120  ======..Given a 
│ │ │ │ -00029ec0: 6d6f 6475 6c65 204d 206f 7665 7220 7468  module M over th
│ │ │ │ -00029ed0: 6520 7269 6e67 206f 6620 6f70 6572 6174  e ring of operat
│ │ │ │ -00029ee0: 6f72 7320 246b 5b78 5f31 2e2e 785f 635d  ors $k[x_1..x_c]
│ │ │ │ -00029ef0: 242c 2074 6865 2063 616c 6c20 244e 203d  $, the call $N =
│ │ │ │ -00029f00: 0a6d 6f64 756c 6541 7345 7874 284d 2c52  .moduleAsExt(M,R
│ │ │ │ -00029f10: 2924 2070 726f 6475 6365 7320 6120 6d6f  )$ produces a mo
│ │ │ │ -00029f20: 6475 6c65 204e 206f 7665 7220 7468 6520  dule N over the 
│ │ │ │ -00029f30: 7269 6e67 2052 2077 686f 7365 2065 7874  ring R whose ext
│ │ │ │ -00029f40: 206d 6f64 756c 6520 6973 2074 6865 0a65   module is the.e
│ │ │ │ -00029f50: 7874 6572 696f 7220 616c 6765 6272 6120  xterior algebra 
│ │ │ │ -00029f60: 6f6e 206e 3d6e 756d 6765 6e73 5220 6765  on n=numgensR ge
│ │ │ │ -00029f70: 6e65 7261 746f 7273 2074 656e 736f 7265  nerators tensore
│ │ │ │ -00029f80: 6420 7769 7468 204d 2e20 5468 6973 2073  d with M. This s
│ │ │ │ -00029f90: 6372 6970 7420 636f 6d70 7574 6573 0a6e  cript computes.n
│ │ │ │ -00029fa0: 756d 5661 6c75 6573 2076 616c 7565 7320  umValues values 
│ │ │ │ -00029fb0: 6f66 2074 6865 2048 696c 6265 7274 2066  of the Hilbert f
│ │ │ │ -00029fc0: 756e 6374 696f 6e20 6f66 2024 2420 4d20  unction of $$ M 
│ │ │ │ -00029fd0: 5c6f 7469 6d65 7320 5c77 6564 6765 206b  \otimes \wedge k
│ │ │ │ -00029fe0: 5e6e 2c20 2424 2077 6869 6368 0a73 686f  ^n, $$ which.sho
│ │ │ │ -00029ff0: 756c 6420 6265 2065 7175 616c 2074 6f20  uld be equal to 
│ │ │ │ -0002a000: 7468 6520 746f 7461 6c20 6265 7474 6920  the total betti 
│ │ │ │ -0002a010: 6e75 6d62 6572 7320 6f66 204e 2e0a 0a2b  numbers of N...+
│ │ │ │ +00029c90: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ +00029ca0: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ +00029cb0: 7365 7120 3d20 6866 4d6f 6475 6c65 4173  seq = hfModuleAs
│ │ │ │ +00029cc0: 4578 7428 6e75 6d56 616c 7565 732c 4d2c  Ext(numValues,M,
│ │ │ │ +00029cd0: 6e75 6d67 656e 7352 290a 2020 2a20 496e  numgensR).  * In
│ │ │ │ +00029ce0: 7075 7473 3a0a 2020 2020 2020 2a20 6e75  puts:.      * nu
│ │ │ │ +00029cf0: 6d56 616c 7565 732c 2061 6e20 2a6e 6f74  mValues, an *not
│ │ │ │ +00029d00: 6520 696e 7465 6765 723a 2028 4d61 6361  e integer: (Maca
│ │ │ │ +00029d10: 756c 6179 3244 6f63 295a 5a2c 2c20 6e75  ulay2Doc)ZZ,, nu
│ │ │ │ +00029d20: 6d62 6572 206f 6620 7661 6c75 6573 2074  mber of values t
│ │ │ │ +00029d30: 6f0a 2020 2020 2020 2020 636f 6d70 7574  o.        comput
│ │ │ │ +00029d40: 650a 2020 2020 2020 2a20 4d2c 2061 202a  e.      * M, a *
│ │ │ │ +00029d50: 6e6f 7465 206d 6f64 756c 653a 2028 4d61  note module: (Ma
│ │ │ │ +00029d60: 6361 756c 6179 3244 6f63 294d 6f64 756c  caulay2Doc)Modul
│ │ │ │ +00029d70: 652c 2c20 6d6f 6475 6c65 206f 7665 7220  e,, module over 
│ │ │ │ +00029d80: 7468 6520 7269 6e67 206f 660a 2020 2020  the ring of.    
│ │ │ │ +00029d90: 2020 2020 6f70 6572 6174 6f72 730a 2020      operators.  
│ │ │ │ +00029da0: 2020 2020 2a20 6e75 6d67 656e 7352 2c20      * numgensR, 
│ │ │ │ +00029db0: 616e 202a 6e6f 7465 2069 6e74 6567 6572  an *note integer
│ │ │ │ +00029dc0: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +00029dd0: 5a5a 2c2c 206e 756d 6265 7220 6f66 2067  ZZ,, number of g
│ │ │ │ +00029de0: 656e 6572 6174 6f72 7320 6f66 0a20 2020  enerators of.   
│ │ │ │ +00029df0: 2020 2020 2074 6865 2074 6172 6765 7420       the target 
│ │ │ │ +00029e00: 7269 6e67 0a20 202a 204f 7574 7075 7473  ring.  * Outputs
│ │ │ │ +00029e10: 3a0a 2020 2020 2020 2a20 7365 712c 2061  :.      * seq, a
│ │ │ │ +00029e20: 202a 6e6f 7465 2073 6571 7565 6e63 653a   *note sequence:
│ │ │ │ +00029e30: 2028 4d61 6361 756c 6179 3244 6f63 2953   (Macaulay2Doc)S
│ │ │ │ +00029e40: 6571 7565 6e63 652c 2c20 7365 7175 656e  equence,, sequen
│ │ │ │ +00029e50: 6365 206f 6620 6e75 6d56 616c 7565 730a  ce of numValues.
│ │ │ │ +00029e60: 2020 2020 2020 2020 696e 7465 6765 7273          integers
│ │ │ │ +00029e70: 2c20 7468 6520 6578 7065 6374 6564 2074  , the expected t
│ │ │ │ +00029e80: 6f74 616c 2042 6574 7469 206e 756d 6265  otal Betti numbe
│ │ │ │ +00029e90: 7273 0a0a 4465 7363 7269 7074 696f 6e0a  rs..Description.
│ │ │ │ +00029ea0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a47 6976  ===========..Giv
│ │ │ │ +00029eb0: 656e 2061 206d 6f64 756c 6520 4d20 6f76  en a module M ov
│ │ │ │ +00029ec0: 6572 2074 6865 2072 696e 6720 6f66 206f  er the ring of o
│ │ │ │ +00029ed0: 7065 7261 746f 7273 2024 6b5b 785f 312e  perators $k[x_1.
│ │ │ │ +00029ee0: 2e78 5f63 5d24 2c20 7468 6520 6361 6c6c  .x_c]$, the call
│ │ │ │ +00029ef0: 2024 4e20 3d0a 6d6f 6475 6c65 4173 4578   $N =.moduleAsEx
│ │ │ │ +00029f00: 7428 4d2c 5229 2420 7072 6f64 7563 6573  t(M,R)$ produces
│ │ │ │ +00029f10: 2061 206d 6f64 756c 6520 4e20 6f76 6572   a module N over
│ │ │ │ +00029f20: 2074 6865 2072 696e 6720 5220 7768 6f73   the ring R whos
│ │ │ │ +00029f30: 6520 6578 7420 6d6f 6475 6c65 2069 7320  e ext module is 
│ │ │ │ +00029f40: 7468 650a 6578 7465 7269 6f72 2061 6c67  the.exterior alg
│ │ │ │ +00029f50: 6562 7261 206f 6e20 6e3d 6e75 6d67 656e  ebra on n=numgen
│ │ │ │ +00029f60: 7352 2067 656e 6572 6174 6f72 7320 7465  sR generators te
│ │ │ │ +00029f70: 6e73 6f72 6564 2077 6974 6820 4d2e 2054  nsored with M. T
│ │ │ │ +00029f80: 6869 7320 7363 7269 7074 2063 6f6d 7075  his script compu
│ │ │ │ +00029f90: 7465 730a 6e75 6d56 616c 7565 7320 7661  tes.numValues va
│ │ │ │ +00029fa0: 6c75 6573 206f 6620 7468 6520 4869 6c62  lues of the Hilb
│ │ │ │ +00029fb0: 6572 7420 6675 6e63 7469 6f6e 206f 6620  ert function of 
│ │ │ │ +00029fc0: 2424 204d 205c 6f74 696d 6573 205c 7765  $$ M \otimes \we
│ │ │ │ +00029fd0: 6467 6520 6b5e 6e2c 2024 2420 7768 6963  dge k^n, $$ whic
│ │ │ │ +00029fe0: 680a 7368 6f75 6c64 2062 6520 6571 7561  h.should be equa
│ │ │ │ +00029ff0: 6c20 746f 2074 6865 2074 6f74 616c 2062  l to the total b
│ │ │ │ +0002a000: 6574 7469 206e 756d 6265 7273 206f 6620  etti numbers of 
│ │ │ │ +0002a010: 4e2e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  N...+-----------
│ │ │ │  0002a020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a050: 2b0a 7c69 3120 3a20 6b6b 203d 205a 5a2f  +.|i1 : kk = ZZ/
│ │ │ │ -0002a060: 3130 313b 2020 2020 2020 2020 2020 2020  101;            
│ │ │ │ -0002a070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a080: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0002a040: 2d2d 2d2d 2d2b 0a7c 6931 203a 206b 6b20  -----+.|i1 : kk 
│ │ │ │ +0002a050: 3d20 5a5a 2f31 3031 3b20 2020 2020 2020  = ZZ/101;       
│ │ │ │ +0002a060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a070: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0002a080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a0b0: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 5320  ------+.|i2 : S 
│ │ │ │ -0002a0c0: 3d20 6b6b 5b61 2c62 2c63 5d3b 2020 2020  = kk[a,b,c];    
│ │ │ │ -0002a0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a0e0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0002a0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +0002a0b0: 203a 2053 203d 206b 6b5b 612c 622c 635d   : S = kk[a,b,c]
│ │ │ │ +0002a0c0: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ +0002a0d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002a0e0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  0002a0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0002a120: 3320 3a20 6666 203d 206d 6174 7269 787b  3 : ff = matrix{
│ │ │ │ -0002a130: 7b61 5e34 2c20 625e 342c 635e 347d 7d3b  {a^4, b^4,c^4}};
│ │ │ │ -0002a140: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002a150: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002a110: 2d2b 0a7c 6933 203a 2066 6620 3d20 6d61  -+.|i3 : ff = ma
│ │ │ │ +0002a120: 7472 6978 7b7b 615e 342c 2062 5e34 2c63  trix{{a^4, b^4,c
│ │ │ │ +0002a130: 5e34 7d7d 3b20 2020 2020 2020 2020 2020  ^4}};           
│ │ │ │ +0002a140: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002a150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a180: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0002a190: 2020 3120 2020 2020 2033 2020 2020 2020    1      3      
│ │ │ │ -0002a1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a1b0: 2020 2020 207c 0a7c 6f33 203a 204d 6174       |.|o3 : Mat
│ │ │ │ -0002a1c0: 7269 7820 5320 203c 2d2d 2053 2020 2020  rix S  <-- S    
│ │ │ │ -0002a1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a1e0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0002a170: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002a180: 2020 2020 2020 2031 2020 2020 2020 3320         1      3 
│ │ │ │ +0002a190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a1a0: 2020 2020 2020 2020 2020 7c0a 7c6f 3320            |.|o3 
│ │ │ │ +0002a1b0: 3a20 4d61 7472 6978 2053 2020 3c2d 2d20  : Matrix S  <-- 
│ │ │ │ +0002a1c0: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │ +0002a1d0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0002a1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934  -----------+.|i4
│ │ │ │ -0002a220: 203a 2052 203d 2053 2f69 6465 616c 2066   : R = S/ideal f
│ │ │ │ -0002a230: 663b 2020 2020 2020 2020 2020 2020 2020  f;              
│ │ │ │ -0002a240: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002a250: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0002a210: 2b0a 7c69 3420 3a20 5220 3d20 532f 6964  +.|i4 : R = S/id
│ │ │ │ +0002a220: 6561 6c20 6666 3b20 2020 2020 2020 2020  eal ff;         
│ │ │ │ +0002a230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a240: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0002a250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a280: 2d2b 0a7c 6935 203a 204f 7073 203d 206b  -+.|i5 : Ops = k
│ │ │ │ -0002a290: 6b5b 785f 312c 785f 322c 785f 335d 3b20  k[x_1,x_2,x_3]; 
│ │ │ │ -0002a2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a2b0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0002a270: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 4f70  ------+.|i5 : Op
│ │ │ │ +0002a280: 7320 3d20 6b6b 5b78 5f31 2c78 5f32 2c78  s = kk[x_1,x_2,x
│ │ │ │ +0002a290: 5f33 5d3b 2020 2020 2020 2020 2020 2020  _3];            
│ │ │ │ +0002a2a0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0002a2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a2e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 204d  -------+.|i6 : M
│ │ │ │ -0002a2f0: 4d20 3d20 4f70 735e 312f 2878 5f31 2a69  M = Ops^1/(x_1*i
│ │ │ │ -0002a300: 6465 616c 2878 5f32 5e32 2c78 5f33 2929  deal(x_2^2,x_3))
│ │ │ │ -0002a310: 3b20 2020 2020 2020 2020 7c0a 2b2d 2d2d  ;         |.+---
│ │ │ │ +0002a2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0002a2e0: 3620 3a20 4d4d 203d 204f 7073 5e31 2f28  6 : MM = Ops^1/(
│ │ │ │ +0002a2f0: 785f 312a 6964 6561 6c28 785f 325e 322c  x_1*ideal(x_2^2,
│ │ │ │ +0002a300: 785f 3329 293b 2020 2020 2020 2020 207c  x_3));         |
│ │ │ │ +0002a310: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  0002a320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -0002a350: 6937 203a 204e 203d 206d 6f64 756c 6541  i7 : N = moduleA
│ │ │ │ -0002a360: 7345 7874 284d 4d2c 5229 3b20 2020 2020  sExt(MM,R);     
│ │ │ │ -0002a370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a380: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0002a340: 2d2d 2b0a 7c69 3720 3a20 4e20 3d20 6d6f  --+.|i7 : N = mo
│ │ │ │ +0002a350: 6475 6c65 4173 4578 7428 4d4d 2c52 293b  duleAsExt(MM,R);
│ │ │ │ +0002a360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a370: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0002a380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a3b0: 2d2d 2d2b 0a7c 6938 203a 2062 6574 7469  ---+.|i8 : betti
│ │ │ │ -0002a3c0: 2066 7265 6552 6573 6f6c 7574 696f 6e28   freeResolution(
│ │ │ │ -0002a3d0: 204e 2c20 4c65 6e67 7468 4c69 6d69 7420   N, LengthLimit 
│ │ │ │ -0002a3e0: 3d3e 2031 3029 7c0a 7c20 2020 2020 2020  => 10)|.|       
│ │ │ │ +0002a3a0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20  --------+.|i8 : 
│ │ │ │ +0002a3b0: 6265 7474 6920 6672 6565 5265 736f 6c75  betti freeResolu
│ │ │ │ +0002a3c0: 7469 6f6e 2820 4e2c 204c 656e 6774 684c  tion( N, LengthL
│ │ │ │ +0002a3d0: 696d 6974 203d 3e20 3130 297c 0a7c 2020  imit => 10)|.|  
│ │ │ │ +0002a3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a410: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0002a420: 2020 2020 2020 2020 2030 2020 3120 2032           0  1  2
│ │ │ │ -0002a430: 2020 3320 2034 2020 3520 2036 2020 3720    3  4  5  6  7 
│ │ │ │ -0002a440: 2038 2020 3920 3130 2020 2020 7c0a 7c6f   8  9 10    |.|o
│ │ │ │ -0002a450: 3820 3d20 746f 7461 6c3a 2033 3620 3237  8 = total: 36 27
│ │ │ │ -0002a460: 2032 3920 3331 2033 3320 3335 2033 3720   29 31 33 35 37 
│ │ │ │ -0002a470: 3339 2034 3120 3433 2034 3520 2020 207c  39 41 43 45    |
│ │ │ │ -0002a480: 0a7c 2020 2020 2020 2020 2d36 3a20 3138  .|        -6: 18
│ │ │ │ -0002a490: 2020 3620 202e 2020 2e20 202e 2020 2e20    6  .  .  .  . 
│ │ │ │ -0002a4a0: 202e 2020 2e20 202e 2020 2e20 202e 2020   .  .  .  .  .  
│ │ │ │ -0002a4b0: 2020 7c0a 7c20 2020 2020 2020 202d 353a    |.|        -5:
│ │ │ │ -0002a4c0: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ -0002a4d0: 202e 2020 2e20 202e 2020 2e20 202e 2020   .  .  .  .  .  
│ │ │ │ -0002a4e0: 2e20 2020 207c 0a7c 2020 2020 2020 2020  .    |.|        
│ │ │ │ -0002a4f0: 2d34 3a20 3138 2032 3120 3231 2020 3720  -4: 18 21 21  7 
│ │ │ │ -0002a500: 202e 2020 2e20 202e 2020 2e20 202e 2020   .  .  .  .  .  
│ │ │ │ -0002a510: 2e20 202e 2020 2020 7c0a 7c20 2020 2020  .  .    |.|     
│ │ │ │ -0002a520: 2020 202d 333a 2020 2e20 202e 2020 2e20     -3:  .  .  . 
│ │ │ │ -0002a530: 202e 2020 2e20 202e 2020 2e20 202e 2020   .  .  .  .  .  
│ │ │ │ -0002a540: 2e20 202e 2020 2e20 2020 207c 0a7c 2020  .  .  .    |.|  
│ │ │ │ -0002a550: 2020 2020 2020 2d32 3a20 202e 2020 2e20        -2:  .  . 
│ │ │ │ -0002a560: 2038 2032 3420 3234 2020 3820 202e 2020   8 24 24  8  .  
│ │ │ │ -0002a570: 2e20 202e 2020 2e20 202e 2020 2020 7c0a  .  .  .  .    |.
│ │ │ │ -0002a580: 7c20 2020 2020 2020 202d 313a 2020 2e20  |        -1:  . 
│ │ │ │ -0002a590: 202e 2020 2e20 202e 2020 2e20 202e 2020   .  .  .  .  .  
│ │ │ │ -0002a5a0: 2e20 202e 2020 2e20 202e 2020 2e20 2020  .  .  .  .  .   
│ │ │ │ -0002a5b0: 207c 0a7c 2020 2020 2020 2020 2030 3a20   |.|         0: 
│ │ │ │ -0002a5c0: 202e 2020 2e20 202e 2020 2e20 2039 2032   .  .  .  .  9 2
│ │ │ │ -0002a5d0: 3720 3237 2020 3920 202e 2020 2e20 202e  7 27  9  .  .  .
│ │ │ │ -0002a5e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002a5f0: 313a 2020 2e20 202e 2020 2e20 202e 2020  1:  .  .  .  .  
│ │ │ │ -0002a600: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ -0002a610: 2020 2e20 2020 207c 0a7c 2020 2020 2020    .    |.|      
│ │ │ │ -0002a620: 2020 2032 3a20 202e 2020 2e20 202e 2020     2:  .  .  .  
│ │ │ │ -0002a630: 2e20 202e 2020 2e20 3130 2033 3020 3330  .  .  . 10 30 30
│ │ │ │ -0002a640: 2031 3020 202e 2020 2020 7c0a 7c20 2020   10  .    |.|   
│ │ │ │ -0002a650: 2020 2020 2020 333a 2020 2e20 202e 2020        3:  .  .  
│ │ │ │ -0002a660: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ -0002a670: 2020 2e20 202e 2020 2e20 2020 207c 0a7c    .  .  .    |.|
│ │ │ │ -0002a680: 2020 2020 2020 2020 2034 3a20 202e 2020           4:  .  
│ │ │ │ -0002a690: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ -0002a6a0: 2020 2e20 3131 2033 3320 3333 2020 2020    . 11 33 33    
│ │ │ │ -0002a6b0: 7c0a 7c20 2020 2020 2020 2020 353a 2020  |.|         5:  
│ │ │ │ -0002a6c0: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ -0002a6d0: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ -0002a6e0: 2020 207c 0a7c 2020 2020 2020 2020 2036     |.|         6
│ │ │ │ -0002a6f0: 3a20 202e 2020 2e20 202e 2020 2e20 202e  :  .  .  .  .  .
│ │ │ │ -0002a700: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ -0002a710: 3132 2020 2020 7c0a 7c20 2020 2020 2020  12    |.|       
│ │ │ │ +0002a400: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002a410: 7c20 2020 2020 2020 2020 2020 2020 3020  |             0 
│ │ │ │ +0002a420: 2031 2020 3220 2033 2020 3420 2035 2020   1  2  3  4  5  
│ │ │ │ +0002a430: 3620 2037 2020 3820 2039 2031 3020 2020  6  7  8  9 10   
│ │ │ │ +0002a440: 207c 0a7c 6f38 203d 2074 6f74 616c 3a20   |.|o8 = total: 
│ │ │ │ +0002a450: 3336 2032 3720 3239 2033 3120 3333 2033  36 27 29 31 33 3
│ │ │ │ +0002a460: 3520 3337 2033 3920 3431 2034 3320 3435  5 37 39 41 43 45
│ │ │ │ +0002a470: 2020 2020 7c0a 7c20 2020 2020 2020 202d      |.|        -
│ │ │ │ +0002a480: 363a 2031 3820 2036 2020 2e20 202e 2020  6: 18  6  .  .  
│ │ │ │ +0002a490: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ +0002a4a0: 2020 2e20 2020 207c 0a7c 2020 2020 2020    .    |.|      
│ │ │ │ +0002a4b0: 2020 2d35 3a20 202e 2020 2e20 202e 2020    -5:  .  .  .  
│ │ │ │ +0002a4c0: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ +0002a4d0: 2020 2e20 202e 2020 2020 7c0a 7c20 2020    .  .    |.|   
│ │ │ │ +0002a4e0: 2020 2020 202d 343a 2031 3820 3231 2032       -4: 18 21 2
│ │ │ │ +0002a4f0: 3120 2037 2020 2e20 202e 2020 2e20 202e  1  7  .  .  .  .
│ │ │ │ +0002a500: 2020 2e20 202e 2020 2e20 2020 207c 0a7c    .  .  .    |.|
│ │ │ │ +0002a510: 2020 2020 2020 2020 2d33 3a20 202e 2020          -3:  .  
│ │ │ │ +0002a520: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ +0002a530: 2020 2e20 202e 2020 2e20 202e 2020 2020    .  .  .  .    
│ │ │ │ +0002a540: 7c0a 7c20 2020 2020 2020 202d 323a 2020  |.|        -2:  
│ │ │ │ +0002a550: 2e20 202e 2020 3820 3234 2032 3420 2038  .  .  8 24 24  8
│ │ │ │ +0002a560: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ +0002a570: 2020 207c 0a7c 2020 2020 2020 2020 2d31     |.|        -1
│ │ │ │ +0002a580: 3a20 202e 2020 2e20 202e 2020 2e20 202e  :  .  .  .  .  .
│ │ │ │ +0002a590: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ +0002a5a0: 202e 2020 2020 7c0a 7c20 2020 2020 2020   .    |.|       
│ │ │ │ +0002a5b0: 2020 303a 2020 2e20 202e 2020 2e20 202e    0:  .  .  .  .
│ │ │ │ +0002a5c0: 2020 3920 3237 2032 3720 2039 2020 2e20    9 27 27  9  . 
│ │ │ │ +0002a5d0: 202e 2020 2e20 2020 207c 0a7c 2020 2020   .  .    |.|    
│ │ │ │ +0002a5e0: 2020 2020 2031 3a20 202e 2020 2e20 202e       1:  .  .  .
│ │ │ │ +0002a5f0: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ +0002a600: 202e 2020 2e20 202e 2020 2020 7c0a 7c20   .  .  .    |.| 
│ │ │ │ +0002a610: 2020 2020 2020 2020 323a 2020 2e20 202e          2:  .  .
│ │ │ │ +0002a620: 2020 2e20 202e 2020 2e20 202e 2031 3020    .  .  .  . 10 
│ │ │ │ +0002a630: 3330 2033 3020 3130 2020 2e20 2020 207c  30 30 10  .    |
│ │ │ │ +0002a640: 0a7c 2020 2020 2020 2020 2033 3a20 202e  .|         3:  .
│ │ │ │ +0002a650: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ +0002a660: 202e 2020 2e20 202e 2020 2e20 202e 2020   .  .  .  .  .  
│ │ │ │ +0002a670: 2020 7c0a 7c20 2020 2020 2020 2020 343a    |.|         4:
│ │ │ │ +0002a680: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ +0002a690: 202e 2020 2e20 202e 2031 3120 3333 2033   .  .  . 11 33 3
│ │ │ │ +0002a6a0: 3320 2020 207c 0a7c 2020 2020 2020 2020  3    |.|        
│ │ │ │ +0002a6b0: 2035 3a20 202e 2020 2e20 202e 2020 2e20   5:  .  .  .  . 
│ │ │ │ +0002a6c0: 202e 2020 2e20 202e 2020 2e20 202e 2020   .  .  .  .  .  
│ │ │ │ +0002a6d0: 2e20 202e 2020 2020 7c0a 7c20 2020 2020  .  .    |.|     
│ │ │ │ +0002a6e0: 2020 2020 363a 2020 2e20 202e 2020 2e20      6:  .  .  . 
│ │ │ │ +0002a6f0: 202e 2020 2e20 202e 2020 2e20 202e 2020   .  .  .  .  .  
│ │ │ │ +0002a700: 2e20 202e 2031 3220 2020 207c 0a7c 2020  .  . 12    |.|  
│ │ │ │ +0002a710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a740: 2020 2020 2020 2020 207c 0a7c 6f38 203a           |.|o8 :
│ │ │ │ -0002a750: 2042 6574 7469 5461 6c6c 7920 2020 2020   BettiTally     
│ │ │ │ +0002a730: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002a740: 7c6f 3820 3a20 4265 7474 6954 616c 6c79  |o8 : BettiTally
│ │ │ │ +0002a750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a770: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0002a770: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0002a780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0002a7b0: 0a7c 6939 203a 2068 664d 6f64 756c 6541  .|i9 : hfModuleA
│ │ │ │ -0002a7c0: 7345 7874 2831 322c 4d4d 2c33 2920 2020  sExt(12,MM,3)   
│ │ │ │ -0002a7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a7e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0002a7a0: 2d2d 2d2d 2b0a 7c69 3920 3a20 6866 4d6f  ----+.|i9 : hfMo
│ │ │ │ +0002a7b0: 6475 6c65 4173 4578 7428 3132 2c4d 4d2c  duleAsExt(12,MM,
│ │ │ │ +0002a7c0: 3329 2020 2020 2020 2020 2020 2020 2020  3)              
│ │ │ │ +0002a7d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002a7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a810: 2020 2020 207c 0a7c 6f39 203d 2028 3233       |.|o9 = (23
│ │ │ │ -0002a820: 2c20 3235 2c20 3237 2c20 3239 2c20 3331  , 25, 27, 29, 31
│ │ │ │ -0002a830: 2c20 3333 2c20 3335 2c20 3337 2c20 3339  , 33, 35, 37, 39
│ │ │ │ -0002a840: 2c20 3431 2920 2020 7c0a 7c20 2020 2020  , 41)   |.|     
│ │ │ │ +0002a800: 2020 2020 2020 2020 2020 7c0a 7c6f 3920            |.|o9 
│ │ │ │ +0002a810: 3d20 2832 332c 2032 352c 2032 372c 2032  = (23, 25, 27, 2
│ │ │ │ +0002a820: 392c 2033 312c 2033 332c 2033 352c 2033  9, 31, 33, 35, 3
│ │ │ │ +0002a830: 372c 2033 392c 2034 3129 2020 207c 0a7c  7, 39, 41)   |.|
│ │ │ │ +0002a840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a870: 2020 2020 2020 2020 2020 207c 0a7c 6f39             |.|o9
│ │ │ │ -0002a880: 203a 2053 6571 7565 6e63 6520 2020 2020   : Sequence     
│ │ │ │ +0002a870: 7c0a 7c6f 3920 3a20 5365 7175 656e 6365  |.|o9 : Sequence
│ │ │ │ +0002a880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a8a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002a8b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0002a8a0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0002a8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a8e0: 2d2b 0a0a 5365 6520 616c 736f 0a3d 3d3d  -+..See also.===
│ │ │ │ -0002a8f0: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
│ │ │ │ -0002a900: 206d 6f64 756c 6541 7345 7874 3a20 6d6f   moduleAsExt: mo
│ │ │ │ -0002a910: 6475 6c65 4173 4578 742c 202d 2d20 4669  duleAsExt, -- Fi
│ │ │ │ -0002a920: 6e64 2061 206d 6f64 756c 6520 7769 7468  nd a module with
│ │ │ │ -0002a930: 2067 6976 656e 2061 7379 6d70 746f 7469   given asymptoti
│ │ │ │ -0002a940: 630a 2020 2020 7265 736f 6c75 7469 6f6e  c.    resolution
│ │ │ │ -0002a950: 0a0a 5761 7973 2074 6f20 7573 6520 6866  ..Ways to use hf
│ │ │ │ -0002a960: 4d6f 6475 6c65 4173 4578 743a 0a3d 3d3d  ModuleAsExt:.===
│ │ │ │ -0002a970: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0002a980: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 6866  =======..  * "hf
│ │ │ │ -0002a990: 4d6f 6475 6c65 4173 4578 7428 5a5a 2c4d  ModuleAsExt(ZZ,M
│ │ │ │ -0002a9a0: 6f64 756c 652c 5a5a 2922 0a0a 466f 7220  odule,ZZ)"..For 
│ │ │ │ -0002a9b0: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d  the programmer.=
│ │ │ │ -0002a9c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ -0002ab60: 204e 6578 743a 2068 4d61 7073 2c20 5072   Next: hMaps, Pr
│ │ │ │ -0002ab70: 6576 3a20 6866 4d6f 6475 6c65 4173 4578  ev: hfModuleAsEx
│ │ │ │ -0002ab80: 742c 2055 703a 2054 6f70 0a0a 6869 6768  t, Up: Top..high
│ │ │ │ -0002ab90: 5379 7a79 6779 202d 2d20 5265 7475 726e  Syzygy -- Return
│ │ │ │ -0002aba0: 7320 6120 7379 7a79 6779 206d 6f64 756c  s a syzygy modul
│ │ │ │ -0002abb0: 6520 6f6e 6520 6265 796f 6e64 2074 6865  e one beyond the
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│ │ │ │ +0002aa80: 2d2d 0a0a 5468 6520 736f 7572 6365 206f  --..The source o
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│ │ │ │ +0002aac0: 6163 6175 6c61 7932 2d31 2e32 352e 3131  acaulay2-1.25.11
│ │ │ │ +0002aad0: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932  +ds/M2/Macaulay2
│ │ │ │ +0002aae0: 2f70 6163 6b61 6765 732f 0a43 6f6d 706c  /packages/.Compl
│ │ │ │ +0002aaf0: 6574 6549 6e74 6572 7365 6374 696f 6e52  eteIntersectionR
│ │ │ │ +0002ab00: 6573 6f6c 7574 696f 6e73 2e6d 323a 3331  esolutions.m2:31
│ │ │ │ +0002ab10: 3431 3a30 2e0a 1f0a 4669 6c65 3a20 436f  41:0....File: Co
│ │ │ │ +0002ab20: 6d70 6c65 7465 496e 7465 7273 6563 7469  mpleteIntersecti
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│ │ │ │ +0002ab40: 666f 2c20 4e6f 6465 3a20 6869 6768 5379  fo, Node: highSy
│ │ │ │ +0002ab50: 7a79 6779 2c20 4e65 7874 3a20 684d 6170  zygy, Next: hMap
│ │ │ │ +0002ab60: 732c 2050 7265 763a 2068 664d 6f64 756c  s, Prev: hfModul
│ │ │ │ +0002ab70: 6541 7345 7874 2c20 5570 3a20 546f 700a  eAsExt, Up: Top.
│ │ │ │ +0002ab80: 0a68 6967 6853 797a 7967 7920 2d2d 2052  .highSyzygy -- R
│ │ │ │ +0002ab90: 6574 7572 6e73 2061 2073 797a 7967 7920  eturns a syzygy 
│ │ │ │ +0002aba0: 6d6f 6475 6c65 206f 6e65 2062 6579 6f6e  module one beyon
│ │ │ │ +0002abb0: 6420 7468 6520 7265 6775 6c61 7269 7479  d the regularity
│ │ │ │ +0002abc0: 206f 6620 4578 7428 4d2c 6b29 0a2a 2a2a   of Ext(M,k).***
│ │ │ │ +0002abd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002abe0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002abf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002ac00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002ac10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002ac20: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20  ***..  * Usage: 
│ │ │ │ -0002ac30: 0a20 2020 2020 2020 204d 203d 2068 6967  .        M = hig
│ │ │ │ -0002ac40: 6853 797a 7967 7920 4d30 0a20 202a 2049  hSyzygy M0.  * I
│ │ │ │ -0002ac50: 6e70 7574 733a 0a20 2020 2020 202a 204d  nputs:.      * M
│ │ │ │ -0002ac60: 302c 2061 202a 6e6f 7465 206d 6f64 756c  0, a *note modul
│ │ │ │ -0002ac70: 653a 2028 4d61 6361 756c 6179 3244 6f63  e: (Macaulay2Doc
│ │ │ │ -0002ac80: 294d 6f64 756c 652c 2c20 6f76 6572 2061  )Module,, over a
│ │ │ │ -0002ac90: 2063 6f6d 706c 6574 6520 696e 7465 7273   complete inters
│ │ │ │ -0002aca0: 6563 7469 6f6e 0a20 2020 2020 2020 2072  ection.        r
│ │ │ │ -0002acb0: 696e 670a 2020 2a20 2a6e 6f74 6520 4f70  ing.  * *note Op
│ │ │ │ -0002acc0: 7469 6f6e 616c 2069 6e70 7574 733a 2028  tional inputs: (
│ │ │ │ -0002acd0: 4d61 6361 756c 6179 3244 6f63 2975 7369  Macaulay2Doc)usi
│ │ │ │ -0002ace0: 6e67 2066 756e 6374 696f 6e73 2077 6974  ng functions wit
│ │ │ │ -0002acf0: 6820 6f70 7469 6f6e 616c 2069 6e70 7574  h optional input
│ │ │ │ -0002ad00: 732c 3a0a 2020 2020 2020 2a20 4f70 7469  s,:.      * Opti
│ │ │ │ -0002ad10: 6d69 736d 203d 3e20 2e2e 2e2c 2064 6566  mism => ..., def
│ │ │ │ -0002ad20: 6175 6c74 2076 616c 7565 2030 0a20 202a  ault value 0.  *
│ │ │ │ -0002ad30: 204f 7574 7075 7473 3a0a 2020 2020 2020   Outputs:.      
│ │ │ │ -0002ad40: 2a20 4d2c 2061 202a 6e6f 7465 206d 6f64  * M, a *note mod
│ │ │ │ -0002ad50: 756c 653a 2028 4d61 6361 756c 6179 3244  ule: (Macaulay2D
│ │ │ │ -0002ad60: 6f63 294d 6f64 756c 652c 2c20 6120 7379  oc)Module,, a sy
│ │ │ │ -0002ad70: 7a79 6779 206d 6f64 756c 6520 6f66 204d  zygy module of M
│ │ │ │ -0002ad80: 300a 0a44 6573 6372 6970 7469 6f6e 0a3d  0..Description.=
│ │ │ │ -0002ad90: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 4120 2268  ==========..A "h
│ │ │ │ -0002ada0: 6967 6820 7379 7a79 6779 2220 6f76 6572  igh syzygy" over
│ │ │ │ -0002adb0: 2061 2063 6f6d 706c 6574 6520 696e 7465   a complete inte
│ │ │ │ -0002adc0: 7273 6563 7469 6f6e 2069 7320 6f6e 6520  rsection is one 
│ │ │ │ -0002add0: 7375 6368 2074 6861 7420 6765 6e65 7261  such that genera
│ │ │ │ -0002ade0: 6c0a 6369 2d6f 7065 7261 746f 7273 2068  l.ci-operators h
│ │ │ │ -0002adf0: 6176 6520 7370 6c69 7420 6b65 726e 656c  ave split kernel
│ │ │ │ -0002ae00: 7320 7768 656e 2061 7070 6c69 6564 2072  s when applied r
│ │ │ │ -0002ae10: 6563 7572 7369 7665 6c79 206f 6e20 636f  ecursively on co
│ │ │ │ -0002ae20: 7379 7a79 6779 2063 6861 696e 7320 6f66  syzygy chains of
│ │ │ │ -0002ae30: 0a70 7265 7669 6f75 7320 6b65 726e 656c  .previous kernel
│ │ │ │ -0002ae40: 732e 0a0a 4966 2070 203d 206d 6642 6f75  s...If p = mfBou
│ │ │ │ -0002ae50: 6e64 204d 302c 2074 6865 6e20 6869 6768  nd M0, then high
│ │ │ │ -0002ae60: 5379 7a79 6779 204d 3020 7265 7475 726e  Syzygy M0 return
│ │ │ │ -0002ae70: 7320 7468 6520 702d 7468 2073 797a 7967  s the p-th syzyg
│ │ │ │ -0002ae80: 7920 6f66 204d 302e 2028 6966 2046 2069  y of M0. (if F i
│ │ │ │ -0002ae90: 7320 610a 7265 736f 6c75 7469 6f6e 206f  s a.resolution o
│ │ │ │ -0002aea0: 6620 4d20 7468 6973 2069 7320 7468 6520  f M this is the 
│ │ │ │ -0002aeb0: 636f 6b65 726e 656c 206f 6620 462e 6464  cokernel of F.dd
│ │ │ │ -0002aec0: 5f7b 702b 317d 292e 204f 7074 696d 6973  _{p+1}). Optimis
│ │ │ │ -0002aed0: 6d20 3d3e 2072 2061 7320 6f70 7469 6f6e  m => r as option
│ │ │ │ -0002aee0: 616c 0a61 7267 756d 656e 742c 2068 6967  al.argument, hig
│ │ │ │ -0002aef0: 6853 797a 7967 7928 4d30 2c4f 7074 696d  hSyzygy(M0,Optim
│ │ │ │ -0002af00: 6973 6d3d 3e72 2920 7265 7475 726e 7320  ism=>r) returns 
│ │ │ │ -0002af10: 7468 6520 2870 2d72 292d 7468 2073 797a  the (p-r)-th syz
│ │ │ │ -0002af20: 7967 792e 2054 6865 2073 6372 6970 7420  ygy. The script 
│ │ │ │ -0002af30: 6973 0a75 7365 6675 6c20 7769 7468 206d  is.useful with m
│ │ │ │ -0002af40: 6174 7269 7846 6163 746f 7269 7a61 7469  atrixFactorizati
│ │ │ │ -0002af50: 6f6e 2866 662c 2068 6967 6853 797a 7967  on(ff, highSyzyg
│ │ │ │ -0002af60: 7920 4d30 292e 0a0a 2b2d 2d2d 2d2d 2d2d  y M0)...+-------
│ │ │ │ +0002ac10: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573  ********..  * Us
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│ │ │ │ +0002ac30: 3d20 6869 6768 5379 7a79 6779 204d 300a  = highSyzygy M0.
│ │ │ │ +0002ac40: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
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│ │ │ │ +0002ac60: 6d6f 6475 6c65 3a20 284d 6163 6175 6c61  module: (Macaula
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│ │ │ │ +0002acb0: 7465 204f 7074 696f 6e61 6c20 696e 7075  te Optional inpu
│ │ │ │ +0002acc0: 7473 3a20 284d 6163 6175 6c61 7932 446f  ts: (Macaulay2Do
│ │ │ │ +0002acd0: 6329 7573 696e 6720 6675 6e63 7469 6f6e  c)using function
│ │ │ │ +0002ace0: 7320 7769 7468 206f 7074 696f 6e61 6c20  s with optional 
│ │ │ │ +0002acf0: 696e 7075 7473 2c3a 0a20 2020 2020 202a  inputs,:.      *
│ │ │ │ +0002ad00: 204f 7074 696d 6973 6d20 3d3e 202e 2e2e   Optimism => ...
│ │ │ │ +0002ad10: 2c20 6465 6661 756c 7420 7661 6c75 6520  , default value 
│ │ │ │ +0002ad20: 300a 2020 2a20 4f75 7470 7574 733a 0a20  0.  * Outputs:. 
│ │ │ │ +0002ad30: 2020 2020 202a 204d 2c20 6120 2a6e 6f74       * M, a *not
│ │ │ │ +0002ad40: 6520 6d6f 6475 6c65 3a20 284d 6163 6175  e module: (Macau
│ │ │ │ +0002ad50: 6c61 7932 446f 6329 4d6f 6475 6c65 2c2c  lay2Doc)Module,,
│ │ │ │ +0002ad60: 2061 2073 797a 7967 7920 6d6f 6475 6c65   a syzygy module
│ │ │ │ +0002ad70: 206f 6620 4d30 0a0a 4465 7363 7269 7074   of M0..Descript
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│ │ │ │ +0002ad90: 0a41 2022 6869 6768 2073 797a 7967 7922  .A "high syzygy"
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│ │ │ │ +0002ae10: 6f6e 2063 6f73 797a 7967 7920 6368 6169  on cosyzygy chai
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│ │ │ │ +0002ae50: 2068 6967 6853 797a 7967 7920 4d30 2072   highSyzygy M0 r
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│ │ │ │ +0002ae70: 7379 7a79 6779 206f 6620 4d30 2e20 2869  syzygy of M0. (i
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│ │ │ │ +0002aee0: 2c20 6869 6768 5379 7a79 6779 284d 302c  , highSyzygy(M0,
│ │ │ │ +0002aef0: 4f70 7469 6d69 736d 3d3e 7229 2072 6574  Optimism=>r) ret
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│ │ │ │ +0002af60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002af70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002af80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002af90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002afa0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2073  -------+.|i1 : s
│ │ │ │ -0002afb0: 6574 5261 6e64 6f6d 5365 6564 2031 3030  etRandomSeed 100
│ │ │ │ +0002af90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0002afa0: 3120 3a20 7365 7452 616e 646f 6d53 6565  1 : setRandomSee
│ │ │ │ +0002afb0: 6420 3130 3020 2020 2020 2020 2020 2020  d 100           
│ │ │ │  0002afc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002afd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002afe0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2073          |.| -- s
│ │ │ │ -0002aff0: 6574 7469 6e67 2072 616e 646f 6d20 7365  etting random se
│ │ │ │ -0002b000: 6564 2074 6f20 3130 3020 2020 2020 2020  ed to 100       
│ │ │ │ -0002b010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b020: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0002afd0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002afe0: 202d 2d20 7365 7474 696e 6720 7261 6e64   -- setting rand
│ │ │ │ +0002aff0: 6f6d 2073 6565 6420 746f 2031 3030 2020  om seed to 100  
│ │ │ │ +0002b000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b010: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002b020: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0002b030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b060: 2020 2020 2020 2020 2020 7c0a 7c6f 3120            |.|o1 
│ │ │ │ -0002b070: 3d20 3130 3020 2020 2020 2020 2020 2020  = 100           
│ │ │ │ +0002b050: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002b060: 0a7c 6f31 203d 2031 3030 2020 2020 2020  .|o1 = 100      
│ │ │ │ +0002b070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b0a0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0002b0a0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  0002b0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0002b0f0: 3220 3a20 5320 3d20 5a5a 2f31 3031 5b78  2 : S = ZZ/101[x
│ │ │ │ -0002b100: 2c79 2c7a 5d20 2020 2020 2020 2020 2020  ,y,z]           
│ │ │ │ +0002b0e0: 2d2b 0a7c 6932 203a 2053 203d 205a 5a2f  -+.|i2 : S = ZZ/
│ │ │ │ +0002b0f0: 3130 315b 782c 792c 7a5d 2020 2020 2020  101[x,y,z]      
│ │ │ │ +0002b100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b120: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002b120: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0002b130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b160: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002b170: 7c6f 3220 3d20 5320 2020 2020 2020 2020  |o2 = S         
│ │ │ │ +0002b160: 2020 207c 0a7c 6f32 203d 2053 2020 2020     |.|o2 = S    
│ │ │ │ +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 207c                 |
│ │ │ │ -0002b1b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002b1a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002b1b0: 2020 2020 2020 2020 2020 2020 2020 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: 7c0a 7c6f 3220 3a20 506f 6c79 6e6f 6d69  |.|o2 : Polynomi
│ │ │ │ -0002b200: 616c 5269 6e67 2020 2020 2020 2020 2020  alRing          
│ │ │ │ +0002b1e0: 2020 2020 207c 0a7c 6f32 203a 2050 6f6c       |.|o2 : Pol
│ │ │ │ +0002b1f0: 796e 6f6d 6961 6c52 696e 6720 2020 2020  ynomialRing     
│ │ │ │ +0002b200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b230: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0002b220: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0002b230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b270: 2d2d 2b0a 7c69 3320 3a20 6620 3d20 6d61  --+.|i3 : f = ma
│ │ │ │ -0002b280: 7472 6978 2278 332c 7933 2b78 332c 7a33  trix"x3,y3+x3,z3
│ │ │ │ -0002b290: 2b78 332b 7933 2220 2020 2020 2020 2020  +x3+y3"         
│ │ │ │ -0002b2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b2b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0002b260: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2066  -------+.|i3 : f
│ │ │ │ +0002b270: 203d 206d 6174 7269 7822 7833 2c79 332b   = matrix"x3,y3+
│ │ │ │ +0002b280: 7833 2c7a 332b 7833 2b79 3322 2020 2020  x3,z3+x3+y3"    
│ │ │ │ +0002b290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b2a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +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 7c6f 3320 3d20 7c20 7833      |.|o3 = | x3
│ │ │ │ -0002b300: 2078 332b 7933 2078 332b 7933 2b7a 3320   x3+y3 x3+y3+z3 
│ │ │ │ -0002b310: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0002b320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b330: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002b2e0: 2020 2020 2020 2020 207c 0a7c 6f33 203d           |.|o3 =
│ │ │ │ +0002b2f0: 207c 2078 3320 7833 2b79 3320 7833 2b79   | x3 x3+y3 x3+y
│ │ │ │ +0002b300: 332b 7a33 207c 2020 2020 2020 2020 2020  3+z3 |          
│ │ │ │ +0002b310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b320: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002b330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b370: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0002b380: 2020 2020 2020 3120 2020 2020 2033 2020        1      3  
│ │ │ │ +0002b360: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0002b370: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ +0002b380: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │  0002b390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b3b0: 2020 2020 2020 207c 0a7c 6f33 203a 204d         |.|o3 : M
│ │ │ │ -0002b3c0: 6174 7269 7820 5320 203c 2d2d 2053 2020  atrix S  <-- S  
│ │ │ │ +0002b3a0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0002b3b0: 3320 3a20 4d61 7472 6978 2053 2020 3c2d  3 : Matrix S  <-
│ │ │ │ +0002b3c0: 2d20 5320 2020 2020 2020 2020 2020 2020  - S             
│ │ │ │  0002b3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b3f0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0002b3e0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0002b3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b430: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a  ---------+.|i4 :
│ │ │ │ -0002b440: 2066 6620 3d20 662a 7261 6e64 6f6d 2873   ff = f*random(s
│ │ │ │ -0002b450: 6f75 7263 6520 662c 2073 6f75 7263 6520  ource f, source 
│ │ │ │ -0002b460: 6629 2020 2020 2020 2020 2020 2020 2020  f)              
│ │ │ │ -0002b470: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002b420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0002b430: 7c69 3420 3a20 6666 203d 2066 2a72 616e  |i4 : ff = f*ran
│ │ │ │ +0002b440: 646f 6d28 736f 7572 6365 2066 2c20 736f  dom(source f, so
│ │ │ │ +0002b450: 7572 6365 2066 2920 2020 2020 2020 2020  urce f)         
│ │ │ │ +0002b460: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002b470: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002b480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b4b0: 2020 2020 2020 2020 2020 207c 0a7c 6f34             |.|o4
│ │ │ │ -0002b4c0: 203d 207c 2031 3078 332d 3232 7933 2d34   = | 10x3-22y3-4
│ │ │ │ -0002b4d0: 7a33 202d 3230 7833 2d32 3079 332d 367a  z3 -20x3-20y3-6z
│ │ │ │ -0002b4e0: 3320 2d32 3778 332d 3431 7933 2b7a 3320  3 -27x3-41y3+z3 
│ │ │ │ -0002b4f0: 7c20 2020 2020 2020 2020 2020 7c0a 7c20  |           |.| 
│ │ │ │ +0002b4b0: 7c0a 7c6f 3420 3d20 7c20 3130 7833 2d32  |.|o4 = | 10x3-2
│ │ │ │ +0002b4c0: 3279 332d 347a 3320 2d32 3078 332d 3230  2y3-4z3 -20x3-20
│ │ │ │ +0002b4d0: 7933 2d36 7a33 202d 3237 7833 2d34 3179  y3-6z3 -27x3-41y
│ │ │ │ +0002b4e0: 332b 7a33 207c 2020 2020 2020 2020 2020  3+z3 |          
│ │ │ │ +0002b4f0: 207c 0a7c 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b530: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0002b540: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │ -0002b550: 2020 2020 3320 2020 2020 2020 2020 2020      3           
│ │ │ │ +0002b530: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0002b540: 2020 3120 2020 2020 2033 2020 2020 2020    1      3      
│ │ │ │ +0002b550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b570: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002b580: 7c6f 3420 3a20 4d61 7472 6978 2053 2020  |o4 : Matrix S  
│ │ │ │ -0002b590: 3c2d 2d20 5320 2020 2020 2020 2020 2020  <-- S           
│ │ │ │ +0002b570: 2020 207c 0a7c 6f34 203a 204d 6174 7269     |.|o4 : Matri
│ │ │ │ +0002b580: 7820 5320 203c 2d2d 2053 2020 2020 2020  x S  <-- S      
│ │ │ │ +0002b590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b5b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002b5c0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0002b5b0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0002b5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  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: 2b0a 7c69 3520 3a20 5220 3d20 532f 6964  +.|i5 : R = S/id
│ │ │ │ -0002b610: 6561 6c20 6620 2020 2020 2020 2020 2020  eal f           
│ │ │ │ +0002b5f0: 2d2d 2d2d 2d2b 0a7c 6935 203a 2052 203d  -----+.|i5 : R =
│ │ │ │ +0002b600: 2053 2f69 6465 616c 2066 2020 2020 2020   S/ideal f      
│ │ │ │ +0002b610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b640: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002b630: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0002b640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b680: 2020 7c0a 7c6f 3520 3d20 5220 2020 2020    |.|o5 = R     
│ │ │ │ +0002b670: 2020 2020 2020 207c 0a7c 6f35 203d 2052         |.|o5 = R
│ │ │ │ +0002b680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b6c0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0002b6b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +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 7c6f 3520 3a20 5175 6f74      |.|o5 : Quot
│ │ │ │ -0002b710: 6965 6e74 5269 6e67 2020 2020 2020 2020  ientRing        
│ │ │ │ +0002b6f0: 2020 2020 2020 2020 207c 0a7c 6f35 203a           |.|o5 :
│ │ │ │ +0002b700: 2051 756f 7469 656e 7452 696e 6720 2020   QuotientRing   
│ │ │ │ +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 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0002b730: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0002b740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b780: 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 4d30  ------+.|i6 : M0
│ │ │ │ -0002b790: 203d 2052 5e31 2f69 6465 616c 2278 327a   = R^1/ideal"x2z
│ │ │ │ -0002b7a0: 322c 7879 7a22 2020 2020 2020 2020 2020  2,xyz"          
│ │ │ │ -0002b7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b7c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002b770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936  -----------+.|i6
│ │ │ │ +0002b780: 203a 204d 3020 3d20 525e 312f 6964 6561   : M0 = R^1/idea
│ │ │ │ +0002b790: 6c22 7832 7a32 2c78 797a 2220 2020 2020  l"x2z2,xyz"     
│ │ │ │ +0002b7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b7b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0002b7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b800: 2020 2020 2020 2020 7c0a 7c6f 3620 3d20          |.|o6 = 
│ │ │ │ -0002b810: 636f 6b65 726e 656c 207c 2078 327a 3220  cokernel | x2z2 
│ │ │ │ -0002b820: 7879 7a20 7c20 2020 2020 2020 2020 2020  xyz |           
│ │ │ │ -0002b830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b840: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0002b7f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002b800: 6f36 203d 2063 6f6b 6572 6e65 6c20 7c20  o6 = cokernel | 
│ │ │ │ +0002b810: 7832 7a32 2078 797a 207c 2020 2020 2020  x2z2 xyz |      
│ │ │ │ +0002b820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b830: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002b840: 7c20 2020 2020 2020 2020 2020 2020 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 7c0a 7c20 2020            |.|   
│ │ │ │ -0002b890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b8a0: 2020 2020 2020 2020 2031 2020 2020 2020           1      
│ │ │ │ +0002b870: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002b880: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002b890: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ +0002b8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b8c0: 2020 2020 2020 2020 2020 207c 0a7c 6f36             |.|o6
│ │ │ │ -0002b8d0: 203a 2052 2d6d 6f64 756c 652c 2071 756f   : R-module, quo
│ │ │ │ -0002b8e0: 7469 656e 7420 6f66 2052 2020 2020 2020  tient of R      
│ │ │ │ +0002b8c0: 7c0a 7c6f 3620 3a20 522d 6d6f 6475 6c65  |.|o6 : R-module
│ │ │ │ +0002b8d0: 2c20 7175 6f74 6965 6e74 206f 6620 5220  , quotient of R 
│ │ │ │ +0002b8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b900: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0002b900: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0002b910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -0002b950: 6937 203a 2062 6574 7469 2066 7265 6552  i7 : betti freeR
│ │ │ │ -0002b960: 6573 6f6c 7574 696f 6e20 284d 302c 204c  esolution (M0, L
│ │ │ │ -0002b970: 656e 6774 684c 696d 6974 203d 3e20 3729  engthLimit => 7)
│ │ │ │ -0002b980: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002b990: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0002b940: 2d2d 2b0a 7c69 3720 3a20 6265 7474 6920  --+.|i7 : betti 
│ │ │ │ +0002b950: 6672 6565 5265 736f 6c75 7469 6f6e 2028  freeResolution (
│ │ │ │ +0002b960: 4d30 2c20 4c65 6e67 7468 4c69 6d69 7420  M0, LengthLimit 
│ │ │ │ +0002b970: 3d3e 2037 2920 2020 2020 2020 2020 2020  => 7)           
│ │ │ │ +0002b980: 2020 207c 0a7c 2020 2020 2020 2020 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 207c                 |
│ │ │ │ -0002b9d0: 0a7c 2020 2020 2020 2020 2020 2020 3020  .|            0 
│ │ │ │ -0002b9e0: 3120 3220 2033 2020 3420 2035 2020 3620  1 2  3  4  5  6 
│ │ │ │ -0002b9f0: 2037 2020 2020 2020 2020 2020 2020 2020   7              
│ │ │ │ -0002ba00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ba10: 7c0a 7c6f 3720 3d20 746f 7461 6c3a 2031  |.|o7 = total: 1
│ │ │ │ -0002ba20: 2032 2036 2031 3120 3138 2032 3620 3336   2 6 11 18 26 36
│ │ │ │ -0002ba30: 2034 3720 2020 2020 2020 2020 2020 2020   47             
│ │ │ │ -0002ba40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ba50: 207c 0a7c 2020 2020 2020 2020 2030 3a20   |.|         0: 
│ │ │ │ -0002ba60: 3120 2e20 2e20 202e 2020 2e20 202e 2020  1 . .  .  .  .  
│ │ │ │ -0002ba70: 2e20 202e 2020 2020 2020 2020 2020 2020  .  .            
│ │ │ │ -0002ba80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ba90: 2020 7c0a 7c20 2020 2020 2020 2020 313a    |.|         1:
│ │ │ │ -0002baa0: 202e 202e 202e 2020 2e20 202e 2020 2e20   . . .  .  .  . 
│ │ │ │ -0002bab0: 202e 2020 2e20 2020 2020 2020 2020 2020   .  .           
│ │ │ │ -0002bac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bad0: 2020 207c 0a7c 2020 2020 2020 2020 2032     |.|         2
│ │ │ │ -0002bae0: 3a20 2e20 3120 2e20 202e 2020 2e20 202e  : . 1 .  .  .  .
│ │ │ │ -0002baf0: 2020 2e20 202e 2020 2020 2020 2020 2020    .  .          
│ │ │ │ -0002bb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bb10: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002bb20: 333a 202e 2031 2036 2020 3620 202e 2020  3: . 1 6  6  .  
│ │ │ │ -0002bb30: 2e20 202e 2020 2e20 2020 2020 2020 2020  .  .  .         
│ │ │ │ -0002bb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bb50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0002bb60: 2034 3a20 2e20 2e20 2e20 2035 2031 3820   4: . . .  5 18 
│ │ │ │ -0002bb70: 3134 2020 2e20 202e 2020 2020 2020 2020  14  .  .        
│ │ │ │ -0002bb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bb90: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0002bba0: 2020 353a 202e 202e 202e 2020 2e20 202e    5: . . .  .  .
│ │ │ │ -0002bbb0: 2031 3220 3336 2032 3520 2020 2020 2020   12 36 25       
│ │ │ │ -0002bbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bbd0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0002bbe0: 2020 2036 3a20 2e20 2e20 2e20 202e 2020     6: . . .  .  
│ │ │ │ -0002bbf0: 2e20 202e 2020 2e20 3232 2020 2020 2020  .  .  . 22      
│ │ │ │ -0002bc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bc10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002b9c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002b9d0: 2020 2030 2031 2032 2020 3320 2034 2020     0 1 2  3  4  
│ │ │ │ +0002b9e0: 3520 2036 2020 3720 2020 2020 2020 2020  5  6  7         
│ │ │ │ +0002b9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ba00: 2020 2020 207c 0a7c 6f37 203d 2074 6f74       |.|o7 = tot
│ │ │ │ +0002ba10: 616c 3a20 3120 3220 3620 3131 2031 3820  al: 1 2 6 11 18 
│ │ │ │ +0002ba20: 3236 2033 3620 3437 2020 2020 2020 2020  26 36 47        
│ │ │ │ +0002ba30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ba40: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0002ba50: 2020 303a 2031 202e 202e 2020 2e20 202e    0: 1 . .  .  .
│ │ │ │ +0002ba60: 2020 2e20 202e 2020 2e20 2020 2020 2020    .  .  .       
│ │ │ │ +0002ba70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ba80: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002ba90: 2020 2031 3a20 2e20 2e20 2e20 202e 2020     1: . . .  .  
│ │ │ │ +0002baa0: 2e20 202e 2020 2e20 202e 2020 2020 2020  .  .  .  .      
│ │ │ │ +0002bab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bac0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002bad0: 2020 2020 323a 202e 2031 202e 2020 2e20      2: . 1 .  . 
│ │ │ │ +0002bae0: 202e 2020 2e20 202e 2020 2e20 2020 2020   .  .  .  .     
│ │ │ │ +0002baf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bb00: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0002bb10: 2020 2020 2033 3a20 2e20 3120 3620 2036       3: . 1 6  6
│ │ │ │ +0002bb20: 2020 2e20 202e 2020 2e20 202e 2020 2020    .  .  .  .    
│ │ │ │ +0002bb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bb40: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002bb50: 2020 2020 2020 343a 202e 202e 202e 2020        4: . . .  
│ │ │ │ +0002bb60: 3520 3138 2031 3420 202e 2020 2e20 2020  5 18 14  .  .   
│ │ │ │ +0002bb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bb80: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0002bb90: 2020 2020 2020 2035 3a20 2e20 2e20 2e20         5: . . . 
│ │ │ │ +0002bba0: 202e 2020 2e20 3132 2033 3620 3235 2020   .  . 12 36 25  
│ │ │ │ +0002bbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bbc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0002bbd0: 2020 2020 2020 2020 363a 202e 202e 202e          6: . . .
│ │ │ │ +0002bbe0: 2020 2e20 202e 2020 2e20 202e 2032 3220    .  .  .  . 22 
│ │ │ │ +0002bbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bc00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +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 2020 2020 207c 0a7c 6f37 203a           |.|o7 :
│ │ │ │ -0002bc60: 2042 6574 7469 5461 6c6c 7920 2020 2020   BettiTally     
│ │ │ │ +0002bc40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002bc50: 7c6f 3720 3a20 4265 7474 6954 616c 6c79  |o7 : BettiTally
│ │ │ │ +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 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0002bc80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002bc90: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  0002bca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002bcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002bcc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002bcd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938  -----------+.|i8
│ │ │ │ -0002bce0: 203a 206d 6642 6f75 6e64 204d 3020 2020   : mfBound M0   
│ │ │ │ +0002bcd0: 2b0a 7c69 3820 3a20 6d66 426f 756e 6420  +.|i8 : mfBound 
│ │ │ │ +0002bce0: 4d30 2020 2020 2020 2020 2020 2020 2020  M0              
│ │ │ │  0002bcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bd10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0002bd10: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002bd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bd50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0002bd60: 6f38 203d 2033 2020 2020 2020 2020 2020  o8 = 3          
│ │ │ │ +0002bd50: 2020 7c0a 7c6f 3820 3d20 3320 2020 2020    |.|o8 = 3     
│ │ │ │ +0002bd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bd90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002bda0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0002bd90: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0002bda0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002bdb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002bdc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002bdd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0002bde0: 0a7c 6939 203a 204d 203d 2062 6574 7469  .|i9 : M = betti
│ │ │ │ -0002bdf0: 2066 7265 6552 6573 6f6c 7574 696f 6e28   freeResolution(
│ │ │ │ -0002be00: 6869 6768 5379 7a79 6779 204d 302c 204c  highSyzygy M0, L
│ │ │ │ -0002be10: 656e 6774 684c 696d 6974 203d 3e20 3729  engthLimit => 7)
│ │ │ │ -0002be20: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0002bdd0: 2d2d 2d2d 2b0a 7c69 3920 3a20 4d20 3d20  ----+.|i9 : M = 
│ │ │ │ +0002bde0: 6265 7474 6920 6672 6565 5265 736f 6c75  betti freeResolu
│ │ │ │ +0002bdf0: 7469 6f6e 2868 6967 6853 797a 7967 7920  tion(highSyzygy 
│ │ │ │ +0002be00: 4d30 2c20 4c65 6e67 7468 4c69 6d69 7420  M0, LengthLimit 
│ │ │ │ +0002be10: 3d3e 2037 297c 0a7c 2020 2020 2020 2020  => 7)|.|        
│ │ │ │ +0002be20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002be30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002be40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002be50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002be60: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0002be70: 2030 2020 3120 2032 2020 3320 2034 2020   0  1  2  3  4  
│ │ │ │ -0002be80: 3520 2036 2020 3720 2020 2020 2020 2020  5  6  7         
│ │ │ │ -0002be90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bea0: 2020 7c0a 7c6f 3920 3d20 746f 7461 6c3a    |.|o9 = total:
│ │ │ │ -0002beb0: 2031 3120 3138 2032 3620 3336 2034 3720   11 18 26 36 47 
│ │ │ │ -0002bec0: 3630 2037 3420 3930 2020 2020 2020 2020  60 74 90        
│ │ │ │ -0002bed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bee0: 2020 207c 0a7c 2020 2020 2020 2020 2036     |.|         6
│ │ │ │ -0002bef0: 3a20 2036 2020 2e20 202e 2020 2e20 202e  :  6  .  .  .  .
│ │ │ │ -0002bf00: 2020 2e20 202e 2020 2e20 2020 2020 2020    .  .  .       
│ │ │ │ -0002bf10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bf20: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002bf30: 373a 2020 3520 3138 2031 3420 202e 2020  7:  5 18 14  .  
│ │ │ │ -0002bf40: 2e20 202e 2020 2e20 202e 2020 2020 2020  .  .  .  .      
│ │ │ │ -0002bf50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bf60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0002bf70: 2038 3a20 202e 2020 2e20 3132 2033 3620   8:  .  . 12 36 
│ │ │ │ -0002bf80: 3235 2020 2e20 202e 2020 2e20 2020 2020  25  .  .  .     
│ │ │ │ -0002bf90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bfa0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0002bfb0: 2020 393a 2020 2e20 202e 2020 2e20 202e    9:  .  .  .  .
│ │ │ │ -0002bfc0: 2032 3220 3630 2033 3920 202e 2020 2020   22 60 39  .    
│ │ │ │ -0002bfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bfe0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0002bff0: 2020 3130 3a20 202e 2020 2e20 202e 2020    10:  .  .  .  
│ │ │ │ -0002c000: 2e20 202e 2020 2e20 3335 2039 3020 2020  .  .  . 35 90   
│ │ │ │ -0002c010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c020: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002be50: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0002be60: 2020 2020 2020 3020 2031 2020 3220 2033        0  1  2  3
│ │ │ │ +0002be70: 2020 3420 2035 2020 3620 2037 2020 2020    4  5  6  7    
│ │ │ │ +0002be80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002be90: 2020 2020 2020 207c 0a7c 6f39 203d 2074         |.|o9 = t
│ │ │ │ +0002bea0: 6f74 616c 3a20 3131 2031 3820 3236 2033  otal: 11 18 26 3
│ │ │ │ +0002beb0: 3620 3437 2036 3020 3734 2039 3020 2020  6 47 60 74 90   
│ │ │ │ +0002bec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bed0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002bee0: 2020 2020 363a 2020 3620 202e 2020 2e20      6:  6  .  . 
│ │ │ │ +0002bef0: 202e 2020 2e20 202e 2020 2e20 202e 2020   .  .  .  .  .  
│ │ │ │ +0002bf00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bf10: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0002bf20: 2020 2020 2037 3a20 2035 2031 3820 3134       7:  5 18 14
│ │ │ │ +0002bf30: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ +0002bf40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bf50: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002bf60: 2020 2020 2020 383a 2020 2e20 202e 2031        8:  .  . 1
│ │ │ │ +0002bf70: 3220 3336 2032 3520 202e 2020 2e20 202e  2 36 25  .  .  .
│ │ │ │ +0002bf80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bf90: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0002bfa0: 2020 2020 2020 2039 3a20 202e 2020 2e20         9:  .  . 
│ │ │ │ +0002bfb0: 202e 2020 2e20 3232 2036 3020 3339 2020   .  . 22 60 39  
│ │ │ │ +0002bfc0: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ +0002bfd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0002bfe0: 2020 2020 2020 2031 303a 2020 2e20 202e         10:  .  .
│ │ │ │ +0002bff0: 2020 2e20 202e 2020 2e20 202e 2033 3520    .  .  .  . 35 
│ │ │ │ +0002c000: 3930 2020 2020 2020 2020 2020 2020 2020  90              
│ │ │ │ +0002c010: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002c020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c060: 2020 2020 2020 2020 207c 0a7c 6f39 203a           |.|o9 :
│ │ │ │ -0002c070: 2042 6574 7469 5461 6c6c 7920 2020 2020   BettiTally     
│ │ │ │ +0002c050: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002c060: 7c6f 3920 3a20 4265 7474 6954 616c 6c79  |o9 : BettiTally
│ │ │ │ +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 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0002c090: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002c0a0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  0002c0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -0002c0f0: 3020 3a20 6e65 744c 6973 7420 4252 616e  0 : netList BRan
│ │ │ │ -0002c100: 6b73 206d 6174 7269 7846 6163 746f 7269  ks matrixFactori
│ │ │ │ -0002c110: 7a61 7469 6f6e 2866 662c 2068 6967 6853  zation(ff, highS
│ │ │ │ -0002c120: 797a 7967 7920 4d30 2920 2020 7c0a 7c20  yzygy M0)   |.| 
│ │ │ │ +0002c0e0: 2b0a 7c69 3130 203a 206e 6574 4c69 7374  +.|i10 : netList
│ │ │ │ +0002c0f0: 2042 5261 6e6b 7320 6d61 7472 6978 4661   BRanks matrixFa
│ │ │ │ +0002c100: 6374 6f72 697a 6174 696f 6e28 6666 2c20  ctorization(ff, 
│ │ │ │ +0002c110: 6869 6768 5379 7a79 6779 204d 3029 2020  highSyzygy M0)  
│ │ │ │ +0002c120: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002c130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c160: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0002c170: 2020 2020 2020 2b2d 2b2d 2b20 2020 2020        +-+-+     
│ │ │ │ +0002c160: 2020 7c0a 7c20 2020 2020 202b 2d2b 2d2b    |.|      +-+-+
│ │ │ │ +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 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002c1b0: 7c6f 3130 203d 207c 367c 367c 2020 2020  |o10 = |6|6|    
│ │ │ │ +0002c1a0: 2020 207c 0a7c 6f31 3020 3d20 7c36 7c36     |.|o10 = |6|6
│ │ │ │ +0002c1b0: 7c20 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 207c                 |
│ │ │ │ -0002c1f0: 0a7c 2020 2020 2020 2b2d 2b2d 2b20 2020  .|      +-+-+   
│ │ │ │ +0002c1e0: 2020 2020 7c0a 7c20 2020 2020 202b 2d2b      |.|      +-+
│ │ │ │ +0002c1f0: 2d2b 2020 2020 2020 2020 2020 2020 2020  -+              
│ │ │ │  0002c200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c230: 7c0a 7c20 2020 2020 207c 337c 367c 2020  |.|      |3|6|  
│ │ │ │ +0002c220: 2020 2020 207c 0a7c 2020 2020 2020 7c33       |.|      |3
│ │ │ │ +0002c230: 7c36 7c20 2020 2020 2020 2020 2020 2020  |6|             
│ │ │ │  0002c240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c270: 207c 0a7c 2020 2020 2020 2b2d 2b2d 2b20   |.|      +-+-+ 
│ │ │ │ +0002c260: 2020 2020 2020 7c0a 7c20 2020 2020 202b        |.|      +
│ │ │ │ +0002c270: 2d2b 2d2b 2020 2020 2020 2020 2020 2020  -+-+            
│ │ │ │  0002c280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c2b0: 2020 7c0a 7c20 2020 2020 207c 327c 367c    |.|      |2|6|
│ │ │ │ +0002c2a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002c2b0: 7c32 7c36 7c20 2020 2020 2020 2020 2020  |2|6|           
│ │ │ │  0002c2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c2f0: 2020 207c 0a7c 2020 2020 2020 2b2d 2b2d     |.|      +-+-
│ │ │ │ -0002c300: 2b20 2020 2020 2020 2020 2020 2020 2020  +               
│ │ │ │ +0002c2e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002c2f0: 202b 2d2b 2d2b 2020 2020 2020 2020 2020   +-+-+          
│ │ │ │ +0002c300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c330: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0002c320: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0002c330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c370: 2d2d 2d2d 2d2b 0a0a 496e 2074 6869 7320  -----+..In this 
│ │ │ │ -0002c380: 6361 7365 2061 7320 696e 2061 6c6c 206f  case as in all o
│ │ │ │ -0002c390: 7468 6572 7320 7765 2068 6176 6520 6578  thers we have ex
│ │ │ │ -0002c3a0: 616d 696e 6564 2c20 6772 6561 7465 7220  amined, greater 
│ │ │ │ -0002c3b0: 224f 7074 696d 6973 6d22 2069 7320 6e6f  "Optimism" is no
│ │ │ │ -0002c3c0: 740a 6a75 7374 6966 6965 642c 2061 6e64  t.justified, and
│ │ │ │ -0002c3d0: 2074 6875 7320 6d61 7472 6978 4661 6374   thus matrixFact
│ │ │ │ -0002c3e0: 6f72 697a 6174 696f 6e28 6666 2c20 6869  orization(ff, hi
│ │ │ │ -0002c3f0: 6768 5379 7a79 6779 284d 302c 204f 7074  ghSyzygy(M0, Opt
│ │ │ │ -0002c400: 696d 6973 6d3d 3e31 2929 3b20 776f 756c  imism=>1)); woul
│ │ │ │ -0002c410: 640a 7072 6f64 7563 6520 616e 2065 7272  d.produce an err
│ │ │ │ -0002c420: 6f72 2e0a 0a43 6176 6561 740a 3d3d 3d3d  or...Caveat.====
│ │ │ │ -0002c430: 3d3d 0a0a 4120 6275 6720 696e 2074 6865  ==..A bug in the
│ │ │ │ -0002c440: 2074 6f74 616c 2045 7874 2073 6372 6970   total Ext scrip
│ │ │ │ -0002c450: 7420 6d65 616e 7320 7468 6174 2074 6865  t means that the
│ │ │ │ -0002c460: 206f 6464 4578 744d 6f64 756c 6520 6973   oddExtModule is
│ │ │ │ -0002c470: 2073 6f6d 6574 696d 6573 207a 6572 6f2c   sometimes zero,
│ │ │ │ -0002c480: 0a61 6e64 2074 6869 7320 6361 6e20 6361  .and this can ca
│ │ │ │ -0002c490: 7573 6520 6120 7772 6f6e 6720 7661 6c75  use a wrong valu
│ │ │ │ -0002c4a0: 6520 746f 2062 6520 7265 7475 726e 6564  e to be returned
│ │ │ │ -0002c4b0: 2e0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d  ...See also.====
│ │ │ │ -0002c4c0: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520  ====..  * *note 
│ │ │ │ -0002c4d0: 6576 656e 4578 744d 6f64 756c 653a 2065  evenExtModule: e
│ │ │ │ -0002c4e0: 7665 6e45 7874 4d6f 6475 6c65 2c20 2d2d  venExtModule, --
│ │ │ │ -0002c4f0: 2065 7665 6e20 7061 7274 206f 6620 4578   even part of Ex
│ │ │ │ -0002c500: 745e 2a28 4d2c 6b29 206f 7665 7220 610a  t^*(M,k) over a.
│ │ │ │ -0002c510: 2020 2020 636f 6d70 6c65 7465 2069 6e74      complete int
│ │ │ │ -0002c520: 6572 7365 6374 696f 6e20 6173 206d 6f64  ersection as mod
│ │ │ │ -0002c530: 756c 6520 6f76 6572 2043 4920 6f70 6572  ule over CI oper
│ │ │ │ -0002c540: 6174 6f72 2072 696e 670a 2020 2a20 2a6e  ator ring.  * *n
│ │ │ │ -0002c550: 6f74 6520 6f64 6445 7874 4d6f 6475 6c65  ote oddExtModule
│ │ │ │ -0002c560: 3a20 6f64 6445 7874 4d6f 6475 6c65 2c20  : oddExtModule, 
│ │ │ │ -0002c570: 2d2d 206f 6464 2070 6172 7420 6f66 2045  -- odd part of E
│ │ │ │ -0002c580: 7874 5e2a 284d 2c6b 2920 6f76 6572 2061  xt^*(M,k) over a
│ │ │ │ -0002c590: 2063 6f6d 706c 6574 650a 2020 2020 696e   complete.    in
│ │ │ │ -0002c5a0: 7465 7273 6563 7469 6f6e 2061 7320 6d6f  tersection as mo
│ │ │ │ -0002c5b0: 6475 6c65 206f 7665 7220 4349 206f 7065  dule over CI ope
│ │ │ │ -0002c5c0: 7261 746f 7220 7269 6e67 0a20 202a 202a  rator ring.  * *
│ │ │ │ -0002c5d0: 6e6f 7465 206d 6642 6f75 6e64 3a20 6d66  note mfBound: mf
│ │ │ │ -0002c5e0: 426f 756e 642c 202d 2d20 6465 7465 726d  Bound, -- determ
│ │ │ │ -0002c5f0: 696e 6573 2068 6f77 2068 6967 6820 6120  ines how high a 
│ │ │ │ -0002c600: 7379 7a79 6779 2074 6f20 7461 6b65 2066  syzygy to take f
│ │ │ │ -0002c610: 6f72 0a20 2020 2022 6d61 7472 6978 4661  or.    "matrixFa
│ │ │ │ -0002c620: 6374 6f72 697a 6174 696f 6e22 0a20 202a  ctorization".  *
│ │ │ │ -0002c630: 202a 6e6f 7465 206d 6174 7269 7846 6163   *note matrixFac
│ │ │ │ -0002c640: 746f 7269 7a61 7469 6f6e 3a20 6d61 7472  torization: matr
│ │ │ │ -0002c650: 6978 4661 6374 6f72 697a 6174 696f 6e2c  ixFactorization,
│ │ │ │ -0002c660: 202d 2d20 4d61 7073 2069 6e20 6120 6869   -- Maps in a hi
│ │ │ │ -0002c670: 6768 6572 0a20 2020 2063 6f64 696d 656e  gher.    codimen
│ │ │ │ -0002c680: 7369 6f6e 206d 6174 7269 7820 6661 6374  sion matrix fact
│ │ │ │ -0002c690: 6f72 697a 6174 696f 6e0a 0a57 6179 7320  orization..Ways 
│ │ │ │ -0002c6a0: 746f 2075 7365 2068 6967 6853 797a 7967  to use highSyzyg
│ │ │ │ -0002c6b0: 793a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  y:.=============
│ │ │ │ -0002c6c0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ -0002c6d0: 2268 6967 6853 797a 7967 7928 4d6f 6475  "highSyzygy(Modu
│ │ │ │ -0002c6e0: 6c65 2922 0a0a 466f 7220 7468 6520 7072  le)"..For the pr
│ │ │ │ -0002c6f0: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d  ogrammer.=======
│ │ │ │ -0002c700: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865  ===========..The
│ │ │ │ -0002c710: 206f 626a 6563 7420 2a6e 6f74 6520 6869   object *note hi
│ │ │ │ -0002c720: 6768 5379 7a79 6779 3a20 6869 6768 5379  ghSyzygy: highSy
│ │ │ │ -0002c730: 7a79 6779 2c20 6973 2061 202a 6e6f 7465  zygy, is a *note
│ │ │ │ -0002c740: 206d 6574 686f 6420 6675 6e63 7469 6f6e   method function
│ │ │ │ -0002c750: 2077 6974 680a 6f70 7469 6f6e 733a 2028   with.options: (
│ │ │ │ -0002c760: 4d61 6361 756c 6179 3244 6f63 294d 6574  Macaulay2Doc)Met
│ │ │ │ -0002c770: 686f 6446 756e 6374 696f 6e57 6974 684f  hodFunctionWithO
│ │ │ │ -0002c780: 7074 696f 6e73 2c2e 0a0a 2d2d 2d2d 2d2d  ptions,...------
│ │ │ │ +0002c360: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a49 6e20  ----------+..In 
│ │ │ │ +0002c370: 7468 6973 2063 6173 6520 6173 2069 6e20  this case as in 
│ │ │ │ +0002c380: 616c 6c20 6f74 6865 7273 2077 6520 6861  all others we ha
│ │ │ │ +0002c390: 7665 2065 7861 6d69 6e65 642c 2067 7265  ve examined, gre
│ │ │ │ +0002c3a0: 6174 6572 2022 4f70 7469 6d69 736d 2220  ater "Optimism" 
│ │ │ │ +0002c3b0: 6973 206e 6f74 0a6a 7573 7469 6669 6564  is not.justified
│ │ │ │ +0002c3c0: 2c20 616e 6420 7468 7573 206d 6174 7269  , and thus matri
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│ │ │ │ +0002c3e0: 662c 2068 6967 6853 797a 7967 7928 4d30  f, highSyzygy(M0
│ │ │ │ +0002c3f0: 2c20 4f70 7469 6d69 736d 3d3e 3129 293b  , Optimism=>1));
│ │ │ │ +0002c400: 2077 6f75 6c64 0a70 726f 6475 6365 2061   would.produce a
│ │ │ │ +0002c410: 6e20 6572 726f 722e 0a0a 4361 7665 6174  n error...Caveat
│ │ │ │ +0002c420: 0a3d 3d3d 3d3d 3d0a 0a41 2062 7567 2069  .======..A bug i
│ │ │ │ +0002c430: 6e20 7468 6520 746f 7461 6c20 4578 7420  n the total Ext 
│ │ │ │ +0002c440: 7363 7269 7074 206d 6561 6e73 2074 6861  script means tha
│ │ │ │ +0002c450: 7420 7468 6520 6f64 6445 7874 4d6f 6475  t the oddExtModu
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│ │ │ │ +0002c470: 7a65 726f 2c0a 616e 6420 7468 6973 2063  zero,.and this c
│ │ │ │ +0002c480: 616e 2063 6175 7365 2061 2077 726f 6e67  an cause a wrong
│ │ │ │ +0002c490: 2076 616c 7565 2074 6f20 6265 2072 6574   value to be ret
│ │ │ │ +0002c4a0: 7572 6e65 642e 0a0a 5365 6520 616c 736f  urned...See also
│ │ │ │ +0002c4b0: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a  .========..  * *
│ │ │ │ +0002c4c0: 6e6f 7465 2065 7665 6e45 7874 4d6f 6475  note evenExtModu
│ │ │ │ +0002c4d0: 6c65 3a20 6576 656e 4578 744d 6f64 756c  le: evenExtModul
│ │ │ │ +0002c4e0: 652c 202d 2d20 6576 656e 2070 6172 7420  e, -- even part 
│ │ │ │ +0002c4f0: 6f66 2045 7874 5e2a 284d 2c6b 2920 6f76  of Ext^*(M,k) ov
│ │ │ │ +0002c500: 6572 2061 0a20 2020 2063 6f6d 706c 6574  er a.    complet
│ │ │ │ +0002c510: 6520 696e 7465 7273 6563 7469 6f6e 2061  e intersection a
│ │ │ │ +0002c520: 7320 6d6f 6475 6c65 206f 7665 7220 4349  s module over CI
│ │ │ │ +0002c530: 206f 7065 7261 746f 7220 7269 6e67 0a20   operator ring. 
│ │ │ │ +0002c540: 202a 202a 6e6f 7465 206f 6464 4578 744d   * *note oddExtM
│ │ │ │ +0002c550: 6f64 756c 653a 206f 6464 4578 744d 6f64  odule: oddExtMod
│ │ │ │ +0002c560: 756c 652c 202d 2d20 6f64 6420 7061 7274  ule, -- odd part
│ │ │ │ +0002c570: 206f 6620 4578 745e 2a28 4d2c 6b29 206f   of Ext^*(M,k) o
│ │ │ │ +0002c580: 7665 7220 6120 636f 6d70 6c65 7465 0a20  ver a complete. 
│ │ │ │ +0002c590: 2020 2069 6e74 6572 7365 6374 696f 6e20     intersection 
│ │ │ │ +0002c5a0: 6173 206d 6f64 756c 6520 6f76 6572 2043  as module over C
│ │ │ │ +0002c5b0: 4920 6f70 6572 6174 6f72 2072 696e 670a  I operator ring.
│ │ │ │ +0002c5c0: 2020 2a20 2a6e 6f74 6520 6d66 426f 756e    * *note mfBoun
│ │ │ │ +0002c5d0: 643a 206d 6642 6f75 6e64 2c20 2d2d 2064  d: mfBound, -- d
│ │ │ │ +0002c5e0: 6574 6572 6d69 6e65 7320 686f 7720 6869  etermines how hi
│ │ │ │ +0002c5f0: 6768 2061 2073 797a 7967 7920 746f 2074  gh a syzygy to t
│ │ │ │ +0002c600: 616b 6520 666f 720a 2020 2020 226d 6174  ake for.    "mat
│ │ │ │ +0002c610: 7269 7846 6163 746f 7269 7a61 7469 6f6e  rixFactorization
│ │ │ │ +0002c620: 220a 2020 2a20 2a6e 6f74 6520 6d61 7472  ".  * *note matr
│ │ │ │ +0002c630: 6978 4661 6374 6f72 697a 6174 696f 6e3a  ixFactorization:
│ │ │ │ +0002c640: 206d 6174 7269 7846 6163 746f 7269 7a61   matrixFactoriza
│ │ │ │ +0002c650: 7469 6f6e 2c20 2d2d 204d 6170 7320 696e  tion, -- Maps in
│ │ │ │ +0002c660: 2061 2068 6967 6865 720a 2020 2020 636f   a higher.    co
│ │ │ │ +0002c670: 6469 6d65 6e73 696f 6e20 6d61 7472 6978  dimension matrix
│ │ │ │ +0002c680: 2066 6163 746f 7269 7a61 7469 6f6e 0a0a   factorization..
│ │ │ │ +0002c690: 5761 7973 2074 6f20 7573 6520 6869 6768  Ways to use high
│ │ │ │ +0002c6a0: 5379 7a79 6779 3a0a 3d3d 3d3d 3d3d 3d3d  Syzygy:.========
│ │ │ │ +0002c6b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ +0002c6c0: 0a20 202a 2022 6869 6768 5379 7a79 6779  .  * "highSyzygy
│ │ │ │ +0002c6d0: 284d 6f64 756c 6529 220a 0a46 6f72 2074  (Module)"..For t
│ │ │ │ +0002c6e0: 6865 2070 726f 6772 616d 6d65 720a 3d3d  he programmer.==
│ │ │ │ +0002c6f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0002c700: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f  ..The object *no
│ │ │ │ +0002c710: 7465 2068 6967 6853 797a 7967 793a 2068  te highSyzygy: h
│ │ │ │ +0002c720: 6967 6853 797a 7967 792c 2069 7320 6120  ighSyzygy, is a 
│ │ │ │ +0002c730: 2a6e 6f74 6520 6d65 7468 6f64 2066 756e  *note method fun
│ │ │ │ +0002c740: 6374 696f 6e20 7769 7468 0a6f 7074 696f  ction with.optio
│ │ │ │ +0002c750: 6e73 3a20 284d 6163 6175 6c61 7932 446f  ns: (Macaulay2Do
│ │ │ │ +0002c760: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e  c)MethodFunction
│ │ │ │ +0002c770: 5769 7468 4f70 7469 6f6e 732c 2e0a 0a2d  WithOptions,...-
│ │ │ │ +0002c780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c7d0: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073  ---------..The s
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│ │ │ │ -0002c800: 696c 642f 7265 7072 6f64 7563 6962 6c65  ild/reproducible
│ │ │ │ -0002c810: 2d70 6174 682f 6d61 6361 756c 6179 322d  -path/macaulay2-
│ │ │ │ -0002c820: 312e 3235 2e31 312b 6473 2f4d 322f 4d61  1.25.11+ds/M2/Ma
│ │ │ │ -0002c830: 6361 756c 6179 322f 7061 636b 6167 6573  caulay2/packages
│ │ │ │ -0002c840: 2f0a 436f 6d70 6c65 7465 496e 7465 7273  /.CompleteInters
│ │ │ │ -0002c850: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e  ectionResolution
│ │ │ │ -0002c860: 732e 6d32 3a33 3330 393a 302e 0a1f 0a46  s.m2:3309:0....F
│ │ │ │ -0002c870: 696c 653a 2043 6f6d 706c 6574 6549 6e74  ile: CompleteInt
│ │ │ │ -0002c880: 6572 7365 6374 696f 6e52 6573 6f6c 7574  ersectionResolut
│ │ │ │ -0002c890: 696f 6e73 2e69 6e66 6f2c 204e 6f64 653a  ions.info, Node:
│ │ │ │ -0002c8a0: 2068 4d61 7073 2c20 4e65 7874 3a20 486f   hMaps, Next: Ho
│ │ │ │ -0002c8b0: 6d57 6974 6843 6f6d 706f 6e65 6e74 732c  mWithComponents,
│ │ │ │ -0002c8c0: 2050 7265 763a 2068 6967 6853 797a 7967   Prev: highSyzyg
│ │ │ │ -0002c8d0: 792c 2055 703a 2054 6f70 0a0a 684d 6170  y, Up: Top..hMap
│ │ │ │ -0002c8e0: 7320 2d2d 206c 6973 7420 7468 6520 6d61  s -- list the ma
│ │ │ │ -0002c8f0: 7073 2020 6828 7029 3a20 415f 3028 7029  ps  h(p): A_0(p)
│ │ │ │ -0002c900: 2d2d 3e20 415f 3128 7029 2069 6e20 6120  --> A_1(p) in a 
│ │ │ │ -0002c910: 6d61 7472 6978 4661 6374 6f72 697a 6174  matrixFactorizat
│ │ │ │ -0002c920: 696f 6e0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ion.************
│ │ │ │ +0002c7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a  --------------..
│ │ │ │ +0002c7d0: 5468 6520 736f 7572 6365 206f 6620 7468  The source of th
│ │ │ │ +0002c7e0: 6973 2064 6f63 756d 656e 7420 6973 2069  is document is i
│ │ │ │ +0002c7f0: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475  n./build/reprodu
│ │ │ │ +0002c800: 6369 626c 652d 7061 7468 2f6d 6163 6175  cible-path/macau
│ │ │ │ +0002c810: 6c61 7932 2d31 2e32 352e 3131 2b64 732f  lay2-1.25.11+ds/
│ │ │ │ +0002c820: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163  M2/Macaulay2/pac
│ │ │ │ +0002c830: 6b61 6765 732f 0a43 6f6d 706c 6574 6549  kages/.CompleteI
│ │ │ │ +0002c840: 6e74 6572 7365 6374 696f 6e52 6573 6f6c  ntersectionResol
│ │ │ │ +0002c850: 7574 696f 6e73 2e6d 323a 3333 3039 3a30  utions.m2:3309:0
│ │ │ │ +0002c860: 2e0a 1f0a 4669 6c65 3a20 436f 6d70 6c65  ....File: Comple
│ │ │ │ +0002c870: 7465 496e 7465 7273 6563 7469 6f6e 5265  teIntersectionRe
│ │ │ │ +0002c880: 736f 6c75 7469 6f6e 732e 696e 666f 2c20  solutions.info, 
│ │ │ │ +0002c890: 4e6f 6465 3a20 684d 6170 732c 204e 6578  Node: hMaps, Nex
│ │ │ │ +0002c8a0: 743a 2048 6f6d 5769 7468 436f 6d70 6f6e  t: HomWithCompon
│ │ │ │ +0002c8b0: 656e 7473 2c20 5072 6576 3a20 6869 6768  ents, Prev: high
│ │ │ │ +0002c8c0: 5379 7a79 6779 2c20 5570 3a20 546f 700a  Syzygy, Up: Top.
│ │ │ │ +0002c8d0: 0a68 4d61 7073 202d 2d20 6c69 7374 2074  .hMaps -- list t
│ │ │ │ +0002c8e0: 6865 206d 6170 7320 2068 2870 293a 2041  he maps  h(p): A
│ │ │ │ +0002c8f0: 5f30 2870 292d 2d3e 2041 5f31 2870 2920  _0(p)--> A_1(p) 
│ │ │ │ +0002c900: 696e 2061 206d 6174 7269 7846 6163 746f  in a matrixFacto
│ │ │ │ +0002c910: 7269 7a61 7469 6f6e 0a2a 2a2a 2a2a 2a2a  rization.*******
│ │ │ │ +0002c920: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002c930: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002c940: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002c950: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002c960: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a  ***********..  *
│ │ │ │ -0002c970: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ -0002c980: 2068 4d61 7073 203d 2068 4d61 7073 206d   hMaps = hMaps m
│ │ │ │ -0002c990: 660a 2020 2a20 496e 7075 7473 3a0a 2020  f.  * Inputs:.  
│ │ │ │ -0002c9a0: 2020 2020 2a20 6d66 2c20 6120 2a6e 6f74      * mf, a *not
│ │ │ │ -0002c9b0: 6520 6c69 7374 3a20 284d 6163 6175 6c61  e list: (Macaula
│ │ │ │ -0002c9c0: 7932 446f 6329 4c69 7374 2c2c 206f 7574  y2Doc)List,, out
│ │ │ │ -0002c9d0: 7075 7420 6f66 2061 206d 6174 7269 7846  put of a matrixF
│ │ │ │ -0002c9e0: 6163 746f 7269 7a61 7469 6f6e 0a20 2020  actorization.   
│ │ │ │ -0002c9f0: 2020 2020 2063 6f6d 7075 7461 7469 6f6e       computation
│ │ │ │ -0002ca00: 0a20 202a 204f 7574 7075 7473 3a0a 2020  .  * Outputs:.  
│ │ │ │ -0002ca10: 2020 2020 2a20 684d 6170 732c 2061 202a      * hMaps, a *
│ │ │ │ -0002ca20: 6e6f 7465 206c 6973 743a 2028 4d61 6361  note list: (Maca
│ │ │ │ -0002ca30: 756c 6179 3244 6f63 294c 6973 742c 2c20  ulay2Doc)List,, 
│ │ │ │ -0002ca40: 6c69 7374 206d 6174 7269 6365 7320 2468  list matrices $h
│ │ │ │ -0002ca50: 5f70 3a20 415f 3028 7029 5c74 6f0a 2020  _p: A_0(p)\to.  
│ │ │ │ -0002ca60: 2020 2020 2020 415f 3128 7029 242e 2054        A_1(p)$. T
│ │ │ │ -0002ca70: 6865 2073 6f75 7263 6573 2061 6e64 2074  he sources and t
│ │ │ │ -0002ca80: 6172 6765 7473 206f 6620 7468 6573 6520  argets of these 
│ │ │ │ -0002ca90: 6d61 7073 2068 6176 6520 7468 6520 636f  maps have the co
│ │ │ │ -0002caa0: 6d70 6f6e 656e 7473 0a20 2020 2020 2020  mponents.       
│ │ │ │ -0002cab0: 2042 5f73 2870 292e 0a0a 4465 7363 7269   B_s(p)...Descri
│ │ │ │ -0002cac0: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ -0002cad0: 3d0a 0a53 6565 2074 6865 2064 6f63 756d  =..See the docum
│ │ │ │ -0002cae0: 656e 7461 7469 6f6e 2066 6f72 206d 6174  entation for mat
│ │ │ │ -0002caf0: 7269 7846 6163 746f 7269 7a61 7469 6f6e  rixFactorization
│ │ │ │ -0002cb00: 2066 6f72 2061 6e20 6578 616d 706c 652e   for an example.
│ │ │ │ -0002cb10: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d  ..See also.=====
│ │ │ │ -0002cb20: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 206d  ===..  * *note m
│ │ │ │ -0002cb30: 6174 7269 7846 6163 746f 7269 7a61 7469  atrixFactorizati
│ │ │ │ -0002cb40: 6f6e 3a20 6d61 7472 6978 4661 6374 6f72  on: matrixFactor
│ │ │ │ -0002cb50: 697a 6174 696f 6e2c 202d 2d20 4d61 7073  ization, -- Maps
│ │ │ │ -0002cb60: 2069 6e20 6120 6869 6768 6572 0a20 2020   in a higher.   
│ │ │ │ -0002cb70: 2063 6f64 696d 656e 7369 6f6e 206d 6174   codimension mat
│ │ │ │ -0002cb80: 7269 7820 6661 6374 6f72 697a 6174 696f  rix factorizatio
│ │ │ │ -0002cb90: 6e0a 2020 2a20 2a6e 6f74 6520 644d 6170  n.  * *note dMap
│ │ │ │ -0002cba0: 733a 2064 4d61 7073 2c20 2d2d 206c 6973  s: dMaps, -- lis
│ │ │ │ -0002cbb0: 7420 7468 6520 6d61 7073 2020 6428 7029  t the maps  d(p)
│ │ │ │ -0002cbc0: 3a41 5f31 2870 292d 2d3e 2041 5f30 2870  :A_1(p)--> A_0(p
│ │ │ │ -0002cbd0: 2920 696e 2061 0a20 2020 206d 6174 7269  ) in a.    matri
│ │ │ │ -0002cbe0: 7846 6163 746f 7269 7a61 7469 6f6e 0a20  xFactorization. 
│ │ │ │ -0002cbf0: 202a 202a 6e6f 7465 2042 5261 6e6b 733a   * *note BRanks:
│ │ │ │ -0002cc00: 2042 5261 6e6b 732c 202d 2d20 7261 6e6b   BRanks, -- rank
│ │ │ │ -0002cc10: 7320 6f66 2074 6865 206d 6f64 756c 6573  s of the modules
│ │ │ │ -0002cc20: 2042 5f69 2864 2920 696e 2061 0a20 2020   B_i(d) in a.   
│ │ │ │ -0002cc30: 206d 6174 7269 7846 6163 746f 7269 7a61   matrixFactoriza
│ │ │ │ -0002cc40: 7469 6f6e 0a20 202a 202a 6e6f 7465 2062  tion.  * *note b
│ │ │ │ -0002cc50: 4d61 7073 3a20 624d 6170 732c 202d 2d20  Maps: bMaps, -- 
│ │ │ │ -0002cc60: 6c69 7374 2074 6865 206d 6170 7320 2064  list the maps  d
│ │ │ │ -0002cc70: 5f70 3a42 5f31 2870 292d 2d3e 425f 3028  _p:B_1(p)-->B_0(
│ │ │ │ -0002cc80: 7029 2069 6e20 610a 2020 2020 6d61 7472  p) in a.    matr
│ │ │ │ -0002cc90: 6978 4661 6374 6f72 697a 6174 696f 6e0a  ixFactorization.
│ │ │ │ -0002cca0: 2020 2a20 2a6e 6f74 6520 7073 694d 6170    * *note psiMap
│ │ │ │ -0002ccb0: 733a 2070 7369 4d61 7073 2c20 2d2d 206c  s: psiMaps, -- l
│ │ │ │ -0002ccc0: 6973 7420 7468 6520 6d61 7073 2020 7073  ist the maps  ps
│ │ │ │ -0002ccd0: 6928 7029 3a20 425f 3128 7029 202d 2d3e  i(p): B_1(p) -->
│ │ │ │ -0002cce0: 2041 5f30 2870 2d31 2920 696e 2061 0a20   A_0(p-1) in a. 
│ │ │ │ -0002ccf0: 2020 206d 6174 7269 7846 6163 746f 7269     matrixFactori
│ │ │ │ -0002cd00: 7a61 7469 6f6e 0a0a 5761 7973 2074 6f20  zation..Ways to 
│ │ │ │ -0002cd10: 7573 6520 684d 6170 733a 0a3d 3d3d 3d3d  use hMaps:.=====
│ │ │ │ -0002cd20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20  =============.. 
│ │ │ │ -0002cd30: 202a 2022 684d 6170 7328 4c69 7374 2922   * "hMaps(List)"
│ │ │ │ -0002cd40: 0a0a 466f 7220 7468 6520 7072 6f67 7261  ..For the progra
│ │ │ │ -0002cd50: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  mmer.===========
│ │ │ │ -0002cd60: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a  =======..The obj
│ │ │ │ -0002cd70: 6563 7420 2a6e 6f74 6520 684d 6170 733a  ect *note hMaps:
│ │ │ │ -0002cd80: 2068 4d61 7073 2c20 6973 2061 202a 6e6f   hMaps, is a *no
│ │ │ │ -0002cd90: 7465 206d 6574 686f 6420 6675 6e63 7469  te method functi
│ │ │ │ -0002cda0: 6f6e 3a0a 284d 6163 6175 6c61 7932 446f  on:.(Macaulay2Do
│ │ │ │ -0002cdb0: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e  c)MethodFunction
│ │ │ │ -0002cdc0: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ,...------------
│ │ │ │ +0002c960: 0a0a 2020 2a20 5573 6167 653a 200a 2020  ..  * Usage: .  
│ │ │ │ +0002c970: 2020 2020 2020 684d 6170 7320 3d20 684d        hMaps = hM
│ │ │ │ +0002c980: 6170 7320 6d66 0a20 202a 2049 6e70 7574  aps mf.  * Input
│ │ │ │ +0002c990: 733a 0a20 2020 2020 202a 206d 662c 2061  s:.      * mf, a
│ │ │ │ +0002c9a0: 202a 6e6f 7465 206c 6973 743a 2028 4d61   *note list: (Ma
│ │ │ │ +0002c9b0: 6361 756c 6179 3244 6f63 294c 6973 742c  caulay2Doc)List,
│ │ │ │ +0002c9c0: 2c20 6f75 7470 7574 206f 6620 6120 6d61  , output of a ma
│ │ │ │ +0002c9d0: 7472 6978 4661 6374 6f72 697a 6174 696f  trixFactorizatio
│ │ │ │ +0002c9e0: 6e0a 2020 2020 2020 2020 636f 6d70 7574  n.        comput
│ │ │ │ +0002c9f0: 6174 696f 6e0a 2020 2a20 4f75 7470 7574  ation.  * Output
│ │ │ │ +0002ca00: 733a 0a20 2020 2020 202a 2068 4d61 7073  s:.      * hMaps
│ │ │ │ +0002ca10: 2c20 6120 2a6e 6f74 6520 6c69 7374 3a20  , a *note list: 
│ │ │ │ +0002ca20: 284d 6163 6175 6c61 7932 446f 6329 4c69  (Macaulay2Doc)Li
│ │ │ │ +0002ca30: 7374 2c2c 206c 6973 7420 6d61 7472 6963  st,, list matric
│ │ │ │ +0002ca40: 6573 2024 685f 703a 2041 5f30 2870 295c  es $h_p: A_0(p)\
│ │ │ │ +0002ca50: 746f 0a20 2020 2020 2020 2041 5f31 2870  to.        A_1(p
│ │ │ │ +0002ca60: 2924 2e20 5468 6520 736f 7572 6365 7320  )$. The sources 
│ │ │ │ +0002ca70: 616e 6420 7461 7267 6574 7320 6f66 2074  and targets of t
│ │ │ │ +0002ca80: 6865 7365 206d 6170 7320 6861 7665 2074  hese maps have t
│ │ │ │ +0002ca90: 6865 2063 6f6d 706f 6e65 6e74 730a 2020  he components.  
│ │ │ │ +0002caa0: 2020 2020 2020 425f 7328 7029 2e0a 0a44        B_s(p)...D
│ │ │ │ +0002cab0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ +0002cac0: 3d3d 3d3d 3d3d 0a0a 5365 6520 7468 6520  ======..See the 
│ │ │ │ +0002cad0: 646f 6375 6d65 6e74 6174 696f 6e20 666f  documentation fo
│ │ │ │ +0002cae0: 7220 6d61 7472 6978 4661 6374 6f72 697a  r matrixFactoriz
│ │ │ │ +0002caf0: 6174 696f 6e20 666f 7220 616e 2065 7861  ation for an exa
│ │ │ │ +0002cb00: 6d70 6c65 2e0a 0a53 6565 2061 6c73 6f0a  mple...See also.
│ │ │ │ +0002cb10: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ +0002cb20: 6f74 6520 6d61 7472 6978 4661 6374 6f72  ote matrixFactor
│ │ │ │ +0002cb30: 697a 6174 696f 6e3a 206d 6174 7269 7846  ization: matrixF
│ │ │ │ +0002cb40: 6163 746f 7269 7a61 7469 6f6e 2c20 2d2d  actorization, --
│ │ │ │ +0002cb50: 204d 6170 7320 696e 2061 2068 6967 6865   Maps in a highe
│ │ │ │ +0002cb60: 720a 2020 2020 636f 6469 6d65 6e73 696f  r.    codimensio
│ │ │ │ +0002cb70: 6e20 6d61 7472 6978 2066 6163 746f 7269  n matrix factori
│ │ │ │ +0002cb80: 7a61 7469 6f6e 0a20 202a 202a 6e6f 7465  zation.  * *note
│ │ │ │ +0002cb90: 2064 4d61 7073 3a20 644d 6170 732c 202d   dMaps: dMaps, -
│ │ │ │ +0002cba0: 2d20 6c69 7374 2074 6865 206d 6170 7320  - list the maps 
│ │ │ │ +0002cbb0: 2064 2870 293a 415f 3128 7029 2d2d 3e20   d(p):A_1(p)--> 
│ │ │ │ +0002cbc0: 415f 3028 7029 2069 6e20 610a 2020 2020  A_0(p) in a.    
│ │ │ │ +0002cbd0: 6d61 7472 6978 4661 6374 6f72 697a 6174  matrixFactorizat
│ │ │ │ +0002cbe0: 696f 6e0a 2020 2a20 2a6e 6f74 6520 4252  ion.  * *note BR
│ │ │ │ +0002cbf0: 616e 6b73 3a20 4252 616e 6b73 2c20 2d2d  anks: BRanks, --
│ │ │ │ +0002cc00: 2072 616e 6b73 206f 6620 7468 6520 6d6f   ranks of the mo
│ │ │ │ +0002cc10: 6475 6c65 7320 425f 6928 6429 2069 6e20  dules B_i(d) in 
│ │ │ │ +0002cc20: 610a 2020 2020 6d61 7472 6978 4661 6374  a.    matrixFact
│ │ │ │ +0002cc30: 6f72 697a 6174 696f 6e0a 2020 2a20 2a6e  orization.  * *n
│ │ │ │ +0002cc40: 6f74 6520 624d 6170 733a 2062 4d61 7073  ote bMaps: bMaps
│ │ │ │ +0002cc50: 2c20 2d2d 206c 6973 7420 7468 6520 6d61  , -- list the ma
│ │ │ │ +0002cc60: 7073 2020 645f 703a 425f 3128 7029 2d2d  ps  d_p:B_1(p)--
│ │ │ │ +0002cc70: 3e42 5f30 2870 2920 696e 2061 0a20 2020  >B_0(p) in a.   
│ │ │ │ +0002cc80: 206d 6174 7269 7846 6163 746f 7269 7a61   matrixFactoriza
│ │ │ │ +0002cc90: 7469 6f6e 0a20 202a 202a 6e6f 7465 2070  tion.  * *note p
│ │ │ │ +0002cca0: 7369 4d61 7073 3a20 7073 694d 6170 732c  siMaps: psiMaps,
│ │ │ │ +0002ccb0: 202d 2d20 6c69 7374 2074 6865 206d 6170   -- list the map
│ │ │ │ +0002ccc0: 7320 2070 7369 2870 293a 2042 5f31 2870  s  psi(p): B_1(p
│ │ │ │ +0002ccd0: 2920 2d2d 3e20 415f 3028 702d 3129 2069  ) --> A_0(p-1) i
│ │ │ │ +0002cce0: 6e20 610a 2020 2020 6d61 7472 6978 4661  n a.    matrixFa
│ │ │ │ +0002ccf0: 6374 6f72 697a 6174 696f 6e0a 0a57 6179  ctorization..Way
│ │ │ │ +0002cd00: 7320 746f 2075 7365 2068 4d61 7073 3a0a  s to use hMaps:.
│ │ │ │ +0002cd10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0002cd20: 3d3d 0a0a 2020 2a20 2268 4d61 7073 284c  ==..  * "hMaps(L
│ │ │ │ +0002cd30: 6973 7429 220a 0a46 6f72 2074 6865 2070  ist)"..For the p
│ │ │ │ +0002cd40: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d  rogrammer.======
│ │ │ │ +0002cd50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  ============..Th
│ │ │ │ +0002cd60: 6520 6f62 6a65 6374 202a 6e6f 7465 2068  e object *note h
│ │ │ │ +0002cd70: 4d61 7073 3a20 684d 6170 732c 2069 7320  Maps: hMaps, is 
│ │ │ │ +0002cd80: 6120 2a6e 6f74 6520 6d65 7468 6f64 2066  a *note method f
│ │ │ │ +0002cd90: 756e 6374 696f 6e3a 0a28 4d61 6361 756c  unction:.(Macaul
│ │ │ │ +0002cda0: 6179 3244 6f63 294d 6574 686f 6446 756e  ay2Doc)MethodFun
│ │ │ │ +0002cdb0: 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d  ction,...-------
│ │ │ │ +0002cdc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002cdd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002cde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002cdf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ce00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ce10: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520  ---..The source 
│ │ │ │ -0002ce20: 6f66 2074 6869 7320 646f 6375 6d65 6e74  of this document
│ │ │ │ -0002ce30: 2069 7320 696e 0a2f 6275 696c 642f 7265   is in./build/re
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│ │ │ │ -0002ce50: 6d61 6361 756c 6179 322d 312e 3235 2e31  macaulay2-1.25.1
│ │ │ │ -0002ce60: 312b 6473 2f4d 322f 4d61 6361 756c 6179  1+ds/M2/Macaulay
│ │ │ │ -0002ce70: 322f 7061 636b 6167 6573 2f0a 436f 6d70  2/packages/.Comp
│ │ │ │ -0002ce80: 6c65 7465 496e 7465 7273 6563 7469 6f6e  leteIntersection
│ │ │ │ -0002ce90: 5265 736f 6c75 7469 6f6e 732e 6d32 3a34  Resolutions.m2:4
│ │ │ │ -0002cea0: 3435 373a 302e 0a1f 0a46 696c 653a 2043  457:0....File: C
│ │ │ │ -0002ceb0: 6f6d 706c 6574 6549 6e74 6572 7365 6374  ompleteIntersect
│ │ │ │ -0002cec0: 696f 6e52 6573 6f6c 7574 696f 6e73 2e69  ionResolutions.i
│ │ │ │ -0002ced0: 6e66 6f2c 204e 6f64 653a 2048 6f6d 5769  nfo, Node: HomWi
│ │ │ │ -0002cee0: 7468 436f 6d70 6f6e 656e 7473 2c20 4e65  thComponents, Ne
│ │ │ │ -0002cef0: 7874 3a20 696e 6669 6e69 7465 4265 7474  xt: infiniteBett
│ │ │ │ -0002cf00: 694e 756d 6265 7273 2c20 5072 6576 3a20  iNumbers, Prev: 
│ │ │ │ -0002cf10: 684d 6170 732c 2055 703a 2054 6f70 0a0a  hMaps, Up: Top..
│ │ │ │ -0002cf20: 486f 6d57 6974 6843 6f6d 706f 6e65 6e74  HomWithComponent
│ │ │ │ -0002cf30: 7320 2d2d 2063 6f6d 7075 7465 7320 486f  s -- computes Ho
│ │ │ │ -0002cf40: 6d2c 2070 7265 7365 7276 696e 6720 6469  m, preserving di
│ │ │ │ -0002cf50: 7265 6374 2073 756d 2069 6e66 6f72 6d61  rect sum informa
│ │ │ │ -0002cf60: 7469 6f6e 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a  tion.***********
│ │ │ │ +0002ce00: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f  --------..The so
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│ │ │ │ +0002ce20: 756d 656e 7420 6973 2069 6e0a 2f62 7569  ument is in./bui
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│ │ │ │ +0002ce60: 6175 6c61 7932 2f70 6163 6b61 6765 732f  aulay2/packages/
│ │ │ │ +0002ce70: 0a43 6f6d 706c 6574 6549 6e74 6572 7365  .CompleteInterse
│ │ │ │ +0002ce80: 6374 696f 6e52 6573 6f6c 7574 696f 6e73  ctionResolutions
│ │ │ │ +0002ce90: 2e6d 323a 3434 3537 3a30 2e0a 1f0a 4669  .m2:4457:0....Fi
│ │ │ │ +0002cea0: 6c65 3a20 436f 6d70 6c65 7465 496e 7465  le: CompleteInte
│ │ │ │ +0002ceb0: 7273 6563 7469 6f6e 5265 736f 6c75 7469  rsectionResoluti
│ │ │ │ +0002cec0: 6f6e 732e 696e 666f 2c20 4e6f 6465 3a20  ons.info, Node: 
│ │ │ │ +0002ced0: 486f 6d57 6974 6843 6f6d 706f 6e65 6e74  HomWithComponent
│ │ │ │ +0002cee0: 732c 204e 6578 743a 2069 6e66 696e 6974  s, Next: infinit
│ │ │ │ +0002cef0: 6542 6574 7469 4e75 6d62 6572 732c 2050  eBettiNumbers, P
│ │ │ │ +0002cf00: 7265 763a 2068 4d61 7073 2c20 5570 3a20  rev: hMaps, Up: 
│ │ │ │ +0002cf10: 546f 700a 0a48 6f6d 5769 7468 436f 6d70  Top..HomWithComp
│ │ │ │ +0002cf20: 6f6e 656e 7473 202d 2d20 636f 6d70 7574  onents -- comput
│ │ │ │ +0002cf30: 6573 2048 6f6d 2c20 7072 6573 6572 7669  es Hom, preservi
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│ │ │ │ +0002cf50: 666f 726d 6174 696f 6e0a 2a2a 2a2a 2a2a  formation.******
│ │ │ │ +0002cf60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002cf70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002cf80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002cf90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002cfa0: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055  *********..  * U
│ │ │ │ -0002cfb0: 7361 6765 3a20 0a20 2020 2020 2020 2048  sage: .        H
│ │ │ │ -0002cfc0: 203d 2048 6f6d 284d 2c4e 290a 2020 2a20   = Hom(M,N).  * 
│ │ │ │ -0002cfd0: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ -0002cfe0: 4d2c 2061 202a 6e6f 7465 206d 6f64 756c  M, a *note modul
│ │ │ │ -0002cff0: 653a 2028 4d61 6361 756c 6179 3244 6f63  e: (Macaulay2Doc
│ │ │ │ -0002d000: 294d 6f64 756c 652c 2c20 0a20 2020 2020  )Module,, .     
│ │ │ │ -0002d010: 202a 204e 2c20 6120 2a6e 6f74 6520 6d6f   * N, a *note mo
│ │ │ │ -0002d020: 6475 6c65 3a20 284d 6163 6175 6c61 7932  dule: (Macaulay2
│ │ │ │ -0002d030: 446f 6329 4d6f 6475 6c65 2c2c 200a 2020  Doc)Module,, .  
│ │ │ │ -0002d040: 2a20 4f75 7470 7574 733a 0a20 2020 2020  * Outputs:.     
│ │ │ │ -0002d050: 202a 2048 2c20 6120 2a6e 6f74 6520 6d6f   * H, a *note mo
│ │ │ │ -0002d060: 6475 6c65 3a20 284d 6163 6175 6c61 7932  dule: (Macaulay2
│ │ │ │ -0002d070: 446f 6329 4d6f 6475 6c65 2c2c 200a 0a44  Doc)Module,, ..D
│ │ │ │ -0002d080: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ -0002d090: 3d3d 3d3d 3d3d 0a0a 4966 204d 2061 6e64  ======..If M and
│ │ │ │ -0002d0a0: 2f6f 7220 4e20 6172 6520 6469 7265 6374  /or N are direct
│ │ │ │ -0002d0b0: 2073 756d 206d 6f64 756c 6573 2028 6973   sum modules (is
│ │ │ │ -0002d0c0: 4469 7265 6374 5375 6d20 4d20 3d3d 2074  DirectSum M == t
│ │ │ │ -0002d0d0: 7275 6529 2074 6865 6e20 4820 6973 2074  rue) then H is t
│ │ │ │ -0002d0e0: 6865 0a64 6972 6563 7420 7375 6d20 6f66  he.direct sum of
│ │ │ │ -0002d0f0: 2074 6865 2048 6f6d 7320 6265 7477 6565   the Homs betwee
│ │ │ │ -0002d100: 6e20 7468 6520 636f 6d70 6f6e 656e 7473  n the components
│ │ │ │ -0002d110: 2e20 5468 6973 2053 484f 554c 4420 6265  . This SHOULD be
│ │ │ │ -0002d120: 2062 7569 6c74 2069 6e74 6f0a 486f 6d28   built into.Hom(
│ │ │ │ -0002d130: 4d2c 4e29 2c20 6275 7420 6973 6e27 7420  M,N), but isn't 
│ │ │ │ -0002d140: 6173 206f 6620 4d32 2c20 762e 2031 2e37  as of M2, v. 1.7
│ │ │ │ -0002d150: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d  ..See also.=====
│ │ │ │ -0002d160: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2074  ===..  * *note t
│ │ │ │ -0002d170: 656e 736f 7257 6974 6843 6f6d 706f 6e65  ensorWithCompone
│ │ │ │ -0002d180: 6e74 733a 2074 656e 736f 7257 6974 6843  nts: tensorWithC
│ │ │ │ -0002d190: 6f6d 706f 6e65 6e74 732c 202d 2d20 666f  omponents, -- fo
│ │ │ │ -0002d1a0: 726d 7320 7468 6520 7465 6e73 6f72 0a20  rms the tensor. 
│ │ │ │ -0002d1b0: 2020 2070 726f 6475 6374 2c20 7072 6573     product, pres
│ │ │ │ -0002d1c0: 6572 7669 6e67 2064 6972 6563 7420 7375  erving direct su
│ │ │ │ -0002d1d0: 6d20 696e 666f 726d 6174 696f 6e0a 2020  m information.  
│ │ │ │ -0002d1e0: 2a20 2a6e 6f74 6520 6475 616c 5769 7468  * *note dualWith
│ │ │ │ -0002d1f0: 436f 6d70 6f6e 656e 7473 3a20 6475 616c  Components: dual
│ │ │ │ -0002d200: 5769 7468 436f 6d70 6f6e 656e 7473 2c20  WithComponents, 
│ │ │ │ -0002d210: 2d2d 2064 7561 6c20 6d6f 6475 6c65 2070  -- dual module p
│ │ │ │ -0002d220: 7265 7365 7276 696e 670a 2020 2020 6469  reserving.    di
│ │ │ │ -0002d230: 7265 6374 2073 756d 2069 6e66 6f72 6d61  rect sum informa
│ │ │ │ -0002d240: 7469 6f6e 0a0a 5761 7973 2074 6f20 7573  tion..Ways to us
│ │ │ │ -0002d250: 6520 486f 6d57 6974 6843 6f6d 706f 6e65  e HomWithCompone
│ │ │ │ -0002d260: 6e74 733a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  nts:.===========
│ │ │ │ -0002d270: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0002d280: 3d3d 3d0a 0a20 202a 2022 486f 6d57 6974  ===..  * "HomWit
│ │ │ │ -0002d290: 6843 6f6d 706f 6e65 6e74 7328 4d6f 6475  hComponents(Modu
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│ │ │ │ -0002d2c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0002d2d0: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a  ==..The object *
│ │ │ │ -0002d2e0: 6e6f 7465 2048 6f6d 5769 7468 436f 6d70  note HomWithComp
│ │ │ │ -0002d2f0: 6f6e 656e 7473 3a20 486f 6d57 6974 6843  onents: HomWithC
│ │ │ │ -0002d300: 6f6d 706f 6e65 6e74 732c 2069 7320 6120  omponents, is a 
│ │ │ │ -0002d310: 2a6e 6f74 6520 6d65 7468 6f64 0a66 756e  *note method.fun
│ │ │ │ -0002d320: 6374 696f 6e3a 2028 4d61 6361 756c 6179  ction: (Macaulay
│ │ │ │ -0002d330: 3244 6f63 294d 6574 686f 6446 756e 6374  2Doc)MethodFunct
│ │ │ │ -0002d340: 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d  ion,...---------
│ │ │ │ +0002cf90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a  **************..
│ │ │ │ +0002cfa0: 2020 2a20 5573 6167 653a 200a 2020 2020    * Usage: .    
│ │ │ │ +0002cfb0: 2020 2020 4820 3d20 486f 6d28 4d2c 4e29      H = Hom(M,N)
│ │ │ │ +0002cfc0: 0a20 202a 2049 6e70 7574 733a 0a20 2020  .  * Inputs:.   
│ │ │ │ +0002cfd0: 2020 202a 204d 2c20 6120 2a6e 6f74 6520     * M, a *note 
│ │ │ │ +0002cfe0: 6d6f 6475 6c65 3a20 284d 6163 6175 6c61  module: (Macaula
│ │ │ │ +0002cff0: 7932 446f 6329 4d6f 6475 6c65 2c2c 200a  y2Doc)Module,, .
│ │ │ │ +0002d000: 2020 2020 2020 2a20 4e2c 2061 202a 6e6f        * N, a *no
│ │ │ │ +0002d010: 7465 206d 6f64 756c 653a 2028 4d61 6361  te module: (Maca
│ │ │ │ +0002d020: 756c 6179 3244 6f63 294d 6f64 756c 652c  ulay2Doc)Module,
│ │ │ │ +0002d030: 2c20 0a20 202a 204f 7574 7075 7473 3a0a  , .  * Outputs:.
│ │ │ │ +0002d040: 2020 2020 2020 2a20 482c 2061 202a 6e6f        * H, a *no
│ │ │ │ +0002d050: 7465 206d 6f64 756c 653a 2028 4d61 6361  te module: (Maca
│ │ │ │ +0002d060: 756c 6179 3244 6f63 294d 6f64 756c 652c  ulay2Doc)Module,
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│ │ │ │ +0002d090: 4d20 616e 642f 6f72 204e 2061 7265 2064  M and/or N are d
│ │ │ │ +0002d0a0: 6972 6563 7420 7375 6d20 6d6f 6475 6c65  irect sum module
│ │ │ │ +0002d0b0: 7320 2869 7344 6972 6563 7453 756d 204d  s (isDirectSum M
│ │ │ │ +0002d0c0: 203d 3d20 7472 7565 2920 7468 656e 2048   == true) then H
│ │ │ │ +0002d0d0: 2069 7320 7468 650a 6469 7265 6374 2073   is the.direct s
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│ │ │ │ +0002d0f0: 6574 7765 656e 2074 6865 2063 6f6d 706f  etween the compo
│ │ │ │ +0002d100: 6e65 6e74 732e 2054 6869 7320 5348 4f55  nents. This SHOU
│ │ │ │ +0002d110: 4c44 2062 6520 6275 696c 7420 696e 746f  LD be built into
│ │ │ │ +0002d120: 0a48 6f6d 284d 2c4e 292c 2062 7574 2069  .Hom(M,N), but i
│ │ │ │ +0002d130: 736e 2774 2061 7320 6f66 204d 322c 2076  sn't as of M2, v
│ │ │ │ +0002d140: 2e20 312e 370a 0a53 6565 2061 6c73 6f0a  . 1.7..See also.
│ │ │ │ +0002d150: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ +0002d160: 6f74 6520 7465 6e73 6f72 5769 7468 436f  ote tensorWithCo
│ │ │ │ +0002d170: 6d70 6f6e 656e 7473 3a20 7465 6e73 6f72  mponents: tensor
│ │ │ │ +0002d180: 5769 7468 436f 6d70 6f6e 656e 7473 2c20  WithComponents, 
│ │ │ │ +0002d190: 2d2d 2066 6f72 6d73 2074 6865 2074 656e  -- forms the ten
│ │ │ │ +0002d1a0: 736f 720a 2020 2020 7072 6f64 7563 742c  sor.    product,
│ │ │ │ +0002d1b0: 2070 7265 7365 7276 696e 6720 6469 7265   preserving dire
│ │ │ │ +0002d1c0: 6374 2073 756d 2069 6e66 6f72 6d61 7469  ct sum informati
│ │ │ │ +0002d1d0: 6f6e 0a20 202a 202a 6e6f 7465 2064 7561  on.  * *note dua
│ │ │ │ +0002d1e0: 6c57 6974 6843 6f6d 706f 6e65 6e74 733a  lWithComponents:
│ │ │ │ +0002d1f0: 2064 7561 6c57 6974 6843 6f6d 706f 6e65   dualWithCompone
│ │ │ │ +0002d200: 6e74 732c 202d 2d20 6475 616c 206d 6f64  nts, -- dual mod
│ │ │ │ +0002d210: 756c 6520 7072 6573 6572 7669 6e67 0a20  ule preserving. 
│ │ │ │ +0002d220: 2020 2064 6972 6563 7420 7375 6d20 696e     direct sum in
│ │ │ │ +0002d230: 666f 726d 6174 696f 6e0a 0a57 6179 7320  formation..Ways 
│ │ │ │ +0002d240: 746f 2075 7365 2048 6f6d 5769 7468 436f  to use HomWithCo
│ │ │ │ +0002d250: 6d70 6f6e 656e 7473 3a0a 3d3d 3d3d 3d3d  mponents:.======
│ │ │ │ +0002d260: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0002d270: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2248  ========..  * "H
│ │ │ │ +0002d280: 6f6d 5769 7468 436f 6d70 6f6e 656e 7473  omWithComponents
│ │ │ │ +0002d290: 284d 6f64 756c 652c 4d6f 6475 6c65 2922  (Module,Module)"
│ │ │ │ +0002d2a0: 0a0a 466f 7220 7468 6520 7072 6f67 7261  ..For the progra
│ │ │ │ +0002d2b0: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  mmer.===========
│ │ │ │ +0002d2c0: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a  =======..The obj
│ │ │ │ +0002d2d0: 6563 7420 2a6e 6f74 6520 486f 6d57 6974  ect *note HomWit
│ │ │ │ +0002d2e0: 6843 6f6d 706f 6e65 6e74 733a 2048 6f6d  hComponents: Hom
│ │ │ │ +0002d2f0: 5769 7468 436f 6d70 6f6e 656e 7473 2c20  WithComponents, 
│ │ │ │ +0002d300: 6973 2061 202a 6e6f 7465 206d 6574 686f  is a *note metho
│ │ │ │ +0002d310: 640a 6675 6e63 7469 6f6e 3a20 284d 6163  d.function: (Mac
│ │ │ │ +0002d320: 6175 6c61 7932 446f 6329 4d65 7468 6f64  aulay2Doc)Method
│ │ │ │ +0002d330: 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d  Function,...----
│ │ │ │ +0002d340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002d380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002d390: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572  ------..The sour
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│ │ │ │ -0002d3c0: 2f72 6570 726f 6475 6369 626c 652d 7061  /reproducible-pa
│ │ │ │ -0002d3d0: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32  th/macaulay2-1.2
│ │ │ │ -0002d3e0: 352e 3131 2b64 732f 4d32 2f4d 6163 6175  5.11+ds/M2/Macau
│ │ │ │ -0002d3f0: 6c61 7932 2f70 6163 6b61 6765 732f 0a43  lay2/packages/.C
│ │ │ │ -0002d400: 6f6d 706c 6574 6549 6e74 6572 7365 6374  ompleteIntersect
│ │ │ │ -0002d410: 696f 6e52 6573 6f6c 7574 696f 6e73 2e6d  ionResolutions.m
│ │ │ │ -0002d420: 323a 3236 3435 3a30 2e0a 1f0a 4669 6c65  2:2645:0....File
│ │ │ │ -0002d430: 3a20 436f 6d70 6c65 7465 496e 7465 7273  : CompleteInters
│ │ │ │ -0002d440: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e  ectionResolution
│ │ │ │ -0002d450: 732e 696e 666f 2c20 4e6f 6465 3a20 696e  s.info, Node: in
│ │ │ │ -0002d460: 6669 6e69 7465 4265 7474 694e 756d 6265  finiteBettiNumbe
│ │ │ │ -0002d470: 7273 2c20 4e65 7874 3a20 6973 4c69 6e65  rs, Next: isLine
│ │ │ │ -0002d480: 6172 2c20 5072 6576 3a20 486f 6d57 6974  ar, Prev: HomWit
│ │ │ │ -0002d490: 6843 6f6d 706f 6e65 6e74 732c 2055 703a  hComponents, Up:
│ │ │ │ -0002d4a0: 2054 6f70 0a0a 696e 6669 6e69 7465 4265   Top..infiniteBe
│ │ │ │ -0002d4b0: 7474 694e 756d 6265 7273 202d 2d20 6265  ttiNumbers -- be
│ │ │ │ -0002d4c0: 7474 6920 6e75 6d62 6572 7320 6f66 2066  tti numbers of f
│ │ │ │ -0002d4d0: 696e 6974 6520 7265 736f 6c75 7469 6f6e  inite resolution
│ │ │ │ -0002d4e0: 2063 6f6d 7075 7465 6420 6672 6f6d 2061   computed from a
│ │ │ │ -0002d4f0: 206d 6174 7269 7820 6661 6374 6f72 697a   matrix factoriz
│ │ │ │ -0002d500: 6174 696f 6e0a 2a2a 2a2a 2a2a 2a2a 2a2a  ation.**********
│ │ │ │ +0002d380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865  -----------..The
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│ │ │ │ +0002d3b0: 6275 696c 642f 7265 7072 6f64 7563 6962  build/reproducib
│ │ │ │ +0002d3c0: 6c65 2d70 6174 682f 6d61 6361 756c 6179  le-path/macaulay
│ │ │ │ +0002d3d0: 322d 312e 3235 2e31 312b 6473 2f4d 322f  2-1.25.11+ds/M2/
│ │ │ │ +0002d3e0: 4d61 6361 756c 6179 322f 7061 636b 6167  Macaulay2/packag
│ │ │ │ +0002d3f0: 6573 2f0a 436f 6d70 6c65 7465 496e 7465  es/.CompleteInte
│ │ │ │ +0002d400: 7273 6563 7469 6f6e 5265 736f 6c75 7469  rsectionResoluti
│ │ │ │ +0002d410: 6f6e 732e 6d32 3a32 3634 353a 302e 0a1f  ons.m2:2645:0...
│ │ │ │ +0002d420: 0a46 696c 653a 2043 6f6d 706c 6574 6549  .File: CompleteI
│ │ │ │ +0002d430: 6e74 6572 7365 6374 696f 6e52 6573 6f6c  ntersectionResol
│ │ │ │ +0002d440: 7574 696f 6e73 2e69 6e66 6f2c 204e 6f64  utions.info, Nod
│ │ │ │ +0002d450: 653a 2069 6e66 696e 6974 6542 6574 7469  e: infiniteBetti
│ │ │ │ +0002d460: 4e75 6d62 6572 732c 204e 6578 743a 2069  Numbers, Next: i
│ │ │ │ +0002d470: 734c 696e 6561 722c 2050 7265 763a 2048  sLinear, Prev: H
│ │ │ │ +0002d480: 6f6d 5769 7468 436f 6d70 6f6e 656e 7473  omWithComponents
│ │ │ │ +0002d490: 2c20 5570 3a20 546f 700a 0a69 6e66 696e  , Up: Top..infin
│ │ │ │ +0002d4a0: 6974 6542 6574 7469 4e75 6d62 6572 7320  iteBettiNumbers 
│ │ │ │ +0002d4b0: 2d2d 2062 6574 7469 206e 756d 6265 7273  -- betti numbers
│ │ │ │ +0002d4c0: 206f 6620 6669 6e69 7465 2072 6573 6f6c   of finite resol
│ │ │ │ +0002d4d0: 7574 696f 6e20 636f 6d70 7574 6564 2066  ution computed f
│ │ │ │ +0002d4e0: 726f 6d20 6120 6d61 7472 6978 2066 6163  rom a matrix fac
│ │ │ │ +0002d4f0: 746f 7269 7a61 7469 6f6e 0a2a 2a2a 2a2a  torization.*****
│ │ │ │ +0002d500: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002d510: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002d520: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002d530: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002d540: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002d550: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002d560: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765  *****..  * Usage
│ │ │ │ -0002d570: 3a20 0a20 2020 2020 2020 204c 203d 2066  : .        L = f
│ │ │ │ -0002d580: 696e 6974 6542 6574 7469 4e75 6d62 6572  initeBettiNumber
│ │ │ │ -0002d590: 7320 284d 462c 6c65 6e29 0a20 202a 2049  s (MF,len).  * I
│ │ │ │ -0002d5a0: 6e70 7574 733a 0a20 2020 2020 202a 204d  nputs:.      * M
│ │ │ │ -0002d5b0: 462c 2061 202a 6e6f 7465 206c 6973 743a  F, a *note list:
│ │ │ │ -0002d5c0: 2028 4d61 6361 756c 6179 3244 6f63 294c   (Macaulay2Doc)L
│ │ │ │ -0002d5d0: 6973 742c 2c20 4c69 7374 206f 6620 4861  ist,, List of Ha
│ │ │ │ -0002d5e0: 7368 5461 626c 6573 2061 7320 636f 6d70  shTables as comp
│ │ │ │ -0002d5f0: 7574 6564 0a20 2020 2020 2020 2062 7920  uted.        by 
│ │ │ │ -0002d600: 226d 6174 7269 7846 6163 746f 7269 7a61  "matrixFactoriza
│ │ │ │ -0002d610: 7469 6f6e 220a 2020 2020 2020 2a20 6c65  tion".      * le
│ │ │ │ -0002d620: 6e2c 2061 6e20 2a6e 6f74 6520 696e 7465  n, an *note inte
│ │ │ │ -0002d630: 6765 723a 2028 4d61 6361 756c 6179 3244  ger: (Macaulay2D
│ │ │ │ -0002d640: 6f63 295a 5a2c 2c20 6c65 6e67 7468 206f  oc)ZZ,, length o
│ │ │ │ -0002d650: 6620 6265 7474 6920 6e75 6d62 6572 0a20  f betti number. 
│ │ │ │ -0002d660: 2020 2020 2020 2073 6571 7565 6e63 6520         sequence 
│ │ │ │ -0002d670: 746f 2070 726f 6475 6365 0a20 202a 204f  to produce.  * O
│ │ │ │ -0002d680: 7574 7075 7473 3a0a 2020 2020 2020 2a20  utputs:.      * 
│ │ │ │ -0002d690: 4c2c 2061 202a 6e6f 7465 206c 6973 743a  L, a *note list:
│ │ │ │ -0002d6a0: 2028 4d61 6361 756c 6179 3244 6f63 294c   (Macaulay2Doc)L
│ │ │ │ -0002d6b0: 6973 742c 2c20 4c69 7374 206f 6620 6265  ist,, List of be
│ │ │ │ -0002d6c0: 7474 6920 6e75 6d62 6572 730a 0a44 6573  tti numbers..Des
│ │ │ │ -0002d6d0: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ -0002d6e0: 3d3d 3d3d 0a0a 5573 6573 2074 6865 2072  ====..Uses the r
│ │ │ │ -0002d6f0: 616e 6b73 206f 6620 7468 6520 4220 6d61  anks of the B ma
│ │ │ │ -0002d700: 7472 6963 6573 2069 6e20 6120 6d61 7472  trices in a matr
│ │ │ │ -0002d710: 6978 2066 6163 746f 7269 7a61 7469 6f6e  ix factorization
│ │ │ │ -0002d720: 2066 6f72 2061 206d 6f64 756c 6520 4d20   for a module M 
│ │ │ │ -0002d730: 6f76 6572 0a53 2f28 665f 312c 2e2e 2c66  over.S/(f_1,..,f
│ │ │ │ -0002d740: 5f63 2920 746f 2063 6f6d 7075 7465 2074  _c) to compute t
│ │ │ │ -0002d750: 6865 2062 6574 7469 206e 756d 6265 7273  he betti numbers
│ │ │ │ -0002d760: 206f 6620 7468 6520 6d69 6e69 6d61 6c20   of the minimal 
│ │ │ │ -0002d770: 7265 736f 6c75 7469 6f6e 206f 6620 4d20  resolution of M 
│ │ │ │ -0002d780: 6f76 6572 0a52 2c20 7768 6963 6820 6973  over.R, which is
│ │ │ │ -0002d790: 2074 6865 2073 756d 206f 6620 7468 6520   the sum of the 
│ │ │ │ -0002d7a0: 6469 7669 6465 6420 706f 7765 7220 616c  divided power al
│ │ │ │ -0002d7b0: 6765 6272 6173 206f 6e20 632d 6a2b 3120  gebras on c-j+1 
│ │ │ │ -0002d7c0: 7661 7269 6162 6c65 7320 7465 6e73 6f72  variables tensor
│ │ │ │ -0002d7d0: 6564 0a77 6974 6820 4228 6a29 2e0a 0a2b  ed.with B(j)...+
│ │ │ │ +0002d550: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ +0002d560: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ +0002d570: 4c20 3d20 6669 6e69 7465 4265 7474 694e  L = finiteBettiN
│ │ │ │ +0002d580: 756d 6265 7273 2028 4d46 2c6c 656e 290a  umbers (MF,len).
│ │ │ │ +0002d590: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ +0002d5a0: 2020 2a20 4d46 2c20 6120 2a6e 6f74 6520    * MF, a *note 
│ │ │ │ +0002d5b0: 6c69 7374 3a20 284d 6163 6175 6c61 7932  list: (Macaulay2
│ │ │ │ +0002d5c0: 446f 6329 4c69 7374 2c2c 204c 6973 7420  Doc)List,, List 
│ │ │ │ +0002d5d0: 6f66 2048 6173 6854 6162 6c65 7320 6173  of HashTables as
│ │ │ │ +0002d5e0: 2063 6f6d 7075 7465 640a 2020 2020 2020   computed.      
│ │ │ │ +0002d5f0: 2020 6279 2022 6d61 7472 6978 4661 6374    by "matrixFact
│ │ │ │ +0002d600: 6f72 697a 6174 696f 6e22 0a20 2020 2020  orization".     
│ │ │ │ +0002d610: 202a 206c 656e 2c20 616e 202a 6e6f 7465   * len, an *note
│ │ │ │ +0002d620: 2069 6e74 6567 6572 3a20 284d 6163 6175   integer: (Macau
│ │ │ │ +0002d630: 6c61 7932 446f 6329 5a5a 2c2c 206c 656e  lay2Doc)ZZ,, len
│ │ │ │ +0002d640: 6774 6820 6f66 2062 6574 7469 206e 756d  gth of betti num
│ │ │ │ +0002d650: 6265 720a 2020 2020 2020 2020 7365 7175  ber.        sequ
│ │ │ │ +0002d660: 656e 6365 2074 6f20 7072 6f64 7563 650a  ence to produce.
│ │ │ │ +0002d670: 2020 2a20 4f75 7470 7574 733a 0a20 2020    * Outputs:.   
│ │ │ │ +0002d680: 2020 202a 204c 2c20 6120 2a6e 6f74 6520     * L, a *note 
│ │ │ │ +0002d690: 6c69 7374 3a20 284d 6163 6175 6c61 7932  list: (Macaulay2
│ │ │ │ +0002d6a0: 446f 6329 4c69 7374 2c2c 204c 6973 7420  Doc)List,, List 
│ │ │ │ +0002d6b0: 6f66 2062 6574 7469 206e 756d 6265 7273  of betti numbers
│ │ │ │ +0002d6c0: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d  ..Description.==
│ │ │ │ +0002d6d0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a55 7365 7320  =========..Uses 
│ │ │ │ +0002d6e0: 7468 6520 7261 6e6b 7320 6f66 2074 6865  the ranks of the
│ │ │ │ +0002d6f0: 2042 206d 6174 7269 6365 7320 696e 2061   B matrices in a
│ │ │ │ +0002d700: 206d 6174 7269 7820 6661 6374 6f72 697a   matrix factoriz
│ │ │ │ +0002d710: 6174 696f 6e20 666f 7220 6120 6d6f 6475  ation for a modu
│ │ │ │ +0002d720: 6c65 204d 206f 7665 720a 532f 2866 5f31  le M over.S/(f_1
│ │ │ │ +0002d730: 2c2e 2e2c 665f 6329 2074 6f20 636f 6d70  ,..,f_c) to comp
│ │ │ │ +0002d740: 7574 6520 7468 6520 6265 7474 6920 6e75  ute the betti nu
│ │ │ │ +0002d750: 6d62 6572 7320 6f66 2074 6865 206d 696e  mbers of the min
│ │ │ │ +0002d760: 696d 616c 2072 6573 6f6c 7574 696f 6e20  imal resolution 
│ │ │ │ +0002d770: 6f66 204d 206f 7665 720a 522c 2077 6869  of M over.R, whi
│ │ │ │ +0002d780: 6368 2069 7320 7468 6520 7375 6d20 6f66  ch is the sum of
│ │ │ │ +0002d790: 2074 6865 2064 6976 6964 6564 2070 6f77   the divided pow
│ │ │ │ +0002d7a0: 6572 2061 6c67 6562 7261 7320 6f6e 2063  er algebras on c
│ │ │ │ +0002d7b0: 2d6a 2b31 2076 6172 6961 626c 6573 2074  -j+1 variables t
│ │ │ │ +0002d7c0: 656e 736f 7265 640a 7769 7468 2042 286a  ensored.with B(j
│ │ │ │ +0002d7d0: 292e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  )...+-----------
│ │ │ │  0002d7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002d800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002d810: 2d2d 2b0a 7c69 3120 3a20 7365 7452 616e  --+.|i1 : setRan
│ │ │ │ -0002d820: 646f 6d53 6565 6420 3020 2020 2020 2020  domSeed 0       
│ │ │ │ -0002d830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d840: 2020 2020 2020 207c 0a7c 202d 2d20 7365         |.| -- se
│ │ │ │ -0002d850: 7474 696e 6720 7261 6e64 6f6d 2073 6565  tting random see
│ │ │ │ -0002d860: 6420 746f 2030 2020 2020 2020 2020 2020  d to 0          
│ │ │ │ -0002d870: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0002d800: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2073  -------+.|i1 : s
│ │ │ │ +0002d810: 6574 5261 6e64 6f6d 5365 6564 2030 2020  etRandomSeed 0  
│ │ │ │ +0002d820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d830: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0002d840: 2d2d 2073 6574 7469 6e67 2072 616e 646f  -- setting rando
│ │ │ │ +0002d850: 6d20 7365 6564 2074 6f20 3020 2020 2020  m seed to 0     
│ │ │ │ +0002d860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d870: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002d880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d8b0: 207c 0a7c 6f31 203d 2030 2020 2020 2020   |.|o1 = 0      
│ │ │ │ +0002d8a0: 2020 2020 2020 7c0a 7c6f 3120 3d20 3020        |.|o1 = 0 
│ │ │ │ +0002d8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d8e0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0002d8d0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0002d8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002d910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ -0002d920: 203a 206b 6b20 3d20 5a5a 2f31 3031 2020   : kk = ZZ/101  
│ │ │ │ +0002d910: 2b0a 7c69 3220 3a20 6b6b 203d 205a 5a2f  +.|i2 : kk = ZZ/
│ │ │ │ +0002d920: 3130 3120 2020 2020 2020 2020 2020 2020  101             
│ │ │ │  0002d930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d950: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0002d940: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002d950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d980: 2020 2020 207c 0a7c 6f32 203d 206b 6b20       |.|o2 = kk 
│ │ │ │ +0002d970: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ +0002d980: 3d20 6b6b 2020 2020 2020 2020 2020 2020  = kk            
│ │ │ │  0002d990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d9b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002d9a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002d9b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002d9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d9e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002d9f0: 0a7c 6f32 203a 2051 756f 7469 656e 7452  .|o2 : QuotientR
│ │ │ │ -0002da00: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │ -0002da10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002da20: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0002d9e0: 2020 2020 7c0a 7c6f 3220 3a20 5175 6f74      |.|o2 : Quot
│ │ │ │ +0002d9f0: 6965 6e74 5269 6e67 2020 2020 2020 2020  ientRing        
│ │ │ │ +0002da00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002da10: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0002da20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002da30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002da40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002da50: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a  ---------+.|i3 :
│ │ │ │ -0002da60: 2053 203d 206b 6b5b 612c 622c 752c 765d   S = kk[a,b,u,v]
│ │ │ │ +0002da40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0002da50: 7c69 3320 3a20 5320 3d20 6b6b 5b61 2c62  |i3 : S = kk[a,b
│ │ │ │ +0002da60: 2c75 2c76 5d20 2020 2020 2020 2020 2020  ,u,v]           
│ │ │ │  0002da70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002da80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002da90: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0002da80: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0002da90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002daa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dac0: 2020 207c 0a7c 6f33 203d 2053 2020 2020     |.|o3 = S    
│ │ │ │ +0002dab0: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20          |.|o3 = 
│ │ │ │ +0002dac0: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │  0002dad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002daf0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002dae0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002daf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002db00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002db10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002db20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0002db30: 6f33 203a 2050 6f6c 796e 6f6d 6961 6c52  o3 : PolynomialR
│ │ │ │ -0002db40: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │ -0002db50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002db60: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0002db20: 2020 7c0a 7c6f 3320 3a20 506f 6c79 6e6f    |.|o3 : Polyno
│ │ │ │ +0002db30: 6d69 616c 5269 6e67 2020 2020 2020 2020  mialRing        
│ │ │ │ +0002db40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002db50: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002db60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002db70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002db80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002db90: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2066  -------+.|i4 : f
│ │ │ │ -0002dba0: 6620 3d20 6d61 7472 6978 2261 752c 6276  f = matrix"au,bv
│ │ │ │ -0002dbb0: 2220 2020 2020 2020 2020 2020 2020 2020  "               
│ │ │ │ -0002dbc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0002db80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0002db90: 3420 3a20 6666 203d 206d 6174 7269 7822  4 : ff = matrix"
│ │ │ │ +0002dba0: 6175 2c62 7622 2020 2020 2020 2020 2020  au,bv"          
│ │ │ │ +0002dbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dbc0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002dbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dc00: 207c 0a7c 6f34 203d 207c 2061 7520 6276   |.|o4 = | au bv
│ │ │ │ -0002dc10: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ -0002dc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dc30: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0002dbf0: 2020 2020 2020 7c0a 7c6f 3420 3d20 7c20        |.|o4 = | 
│ │ │ │ +0002dc00: 6175 2062 7620 7c20 2020 2020 2020 2020  au bv |         
│ │ │ │ +0002dc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dc20: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0002dc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dc60: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002dc70: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ -0002dc80: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -0002dc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dca0: 7c0a 7c6f 3420 3a20 4d61 7472 6978 2053  |.|o4 : Matrix S
│ │ │ │ -0002dcb0: 2020 3c2d 2d20 5320 2020 2020 2020 2020    <-- S         
│ │ │ │ -0002dcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dcd0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0002dc60: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0002dc70: 3120 2020 2020 2032 2020 2020 2020 2020  1      2        
│ │ │ │ +0002dc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dc90: 2020 2020 207c 0a7c 6f34 203a 204d 6174       |.|o4 : Mat
│ │ │ │ +0002dca0: 7269 7820 5320 203c 2d2d 2053 2020 2020  rix S  <-- S    
│ │ │ │ +0002dcb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dcc0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0002dcd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002dce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002dcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002dd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520  ----------+.|i5 
│ │ │ │ -0002dd10: 3a20 5220 3d20 532f 6964 6561 6c20 6666  : R = S/ideal ff
│ │ │ │ +0002dcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0002dd00: 0a7c 6935 203a 2052 203d 2053 2f69 6465  .|i5 : R = S/ide
│ │ │ │ +0002dd10: 616c 2066 6620 2020 2020 2020 2020 2020  al ff           
│ │ │ │  0002dd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dd30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002dd40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002dd30: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002dd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dd70: 2020 2020 7c0a 7c6f 3520 3d20 5220 2020      |.|o5 = R   
│ │ │ │ +0002dd60: 2020 2020 2020 2020 207c 0a7c 6f35 203d           |.|o5 =
│ │ │ │ +0002dd70: 2052 2020 2020 2020 2020 2020 2020 2020   R              
│ │ │ │  0002dd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dda0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0002dd90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002dda0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0002ddb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ddc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ddd0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002dde0: 7c6f 3520 3a20 5175 6f74 6965 6e74 5269  |o5 : QuotientRi
│ │ │ │ -0002ddf0: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │ -0002de00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002de10: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0002ddd0: 2020 207c 0a7c 6f35 203a 2051 756f 7469     |.|o5 : Quoti
│ │ │ │ +0002dde0: 656e 7452 696e 6720 2020 2020 2020 2020  entRing         
│ │ │ │ +0002ddf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002de00: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0002de10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002de20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002de30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002de40: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20  --------+.|i6 : 
│ │ │ │ -0002de50: 4d30 203d 2052 5e31 2f69 6465 616c 2261  M0 = R^1/ideal"a
│ │ │ │ -0002de60: 2c62 2220 2020 2020 2020 2020 2020 2020  ,b"             
│ │ │ │ -0002de70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002de30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +0002de40: 6936 203a 204d 3020 3d20 525e 312f 6964  i6 : M0 = R^1/id
│ │ │ │ +0002de50: 6561 6c22 612c 6222 2020 2020 2020 2020  eal"a,b"        
│ │ │ │ +0002de60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002de70: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0002de80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002de90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002deb0: 2020 7c0a 7c6f 3620 3d20 636f 6b65 726e    |.|o6 = cokern
│ │ │ │ -0002dec0: 656c 207c 2061 2062 207c 2020 2020 2020  el | a b |      
│ │ │ │ -0002ded0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dee0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002dea0: 2020 2020 2020 207c 0a7c 6f36 203d 2063         |.|o6 = c
│ │ │ │ +0002deb0: 6f6b 6572 6e65 6c20 7c20 6120 6220 7c20  okernel | a b | 
│ │ │ │ +0002dec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ded0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0002dee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002def0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002df00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002df10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0002df10: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002df20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002df30: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ -0002df40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002df50: 207c 0a7c 6f36 203a 2052 2d6d 6f64 756c   |.|o6 : R-modul
│ │ │ │ -0002df60: 652c 2071 756f 7469 656e 7420 6f66 2052  e, quotient of R
│ │ │ │ -0002df70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002df80: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0002df30: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +0002df40: 2020 2020 2020 7c0a 7c6f 3620 3a20 522d        |.|o6 : R-
│ │ │ │ +0002df50: 6d6f 6475 6c65 2c20 7175 6f74 6965 6e74  module, quotient
│ │ │ │ +0002df60: 206f 6620 5220 2020 2020 2020 2020 2020   of R           
│ │ │ │ +0002df70: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0002df80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002df90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002dfa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002dfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937  -----------+.|i7
│ │ │ │ -0002dfc0: 203a 2046 203d 2066 7265 6552 6573 6f6c   : F = freeResol
│ │ │ │ -0002dfd0: 7574 696f 6e28 4d30 2c20 4c65 6e67 7468  ution(M0, Length
│ │ │ │ -0002dfe0: 4c69 6d69 7420 3d3e 3329 2020 2020 2020  Limit =>3)      
│ │ │ │ -0002dff0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0002dfb0: 2b0a 7c69 3720 3a20 4620 3d20 6672 6565  +.|i7 : F = free
│ │ │ │ +0002dfc0: 5265 736f 6c75 7469 6f6e 284d 302c 204c  Resolution(M0, L
│ │ │ │ +0002dfd0: 656e 6774 684c 696d 6974 203d 3e33 2920  engthLimit =>3) 
│ │ │ │ +0002dfe0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002dff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e020: 2020 2020 207c 0a7c 2020 2020 2020 3120       |.|      1 
│ │ │ │ -0002e030: 2020 2020 2032 2020 2020 2020 3320 2020       2      3   
│ │ │ │ -0002e040: 2020 2034 2020 2020 2020 2020 2020 2020     4            
│ │ │ │ -0002e050: 2020 2020 2020 2020 2020 7c0a 7c6f 3720            |.|o7 
│ │ │ │ -0002e060: 3d20 5220 203c 2d2d 2052 2020 3c2d 2d20  = R  <-- R  <-- 
│ │ │ │ -0002e070: 5220 203c 2d2d 2052 2020 2020 2020 2020  R  <-- R        
│ │ │ │ -0002e080: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002e090: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002e010: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002e020: 2020 2031 2020 2020 2020 3220 2020 2020     1      2     
│ │ │ │ +0002e030: 2033 2020 2020 2020 3420 2020 2020 2020   3      4       
│ │ │ │ +0002e040: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002e050: 0a7c 6f37 203d 2052 2020 3c2d 2d20 5220  .|o7 = R  <-- R 
│ │ │ │ +0002e060: 203c 2d2d 2052 2020 3c2d 2d20 5220 2020   <-- R  <-- R   
│ │ │ │ +0002e070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e080: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002e090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e0c0: 2020 2020 7c0a 7c20 2020 2020 3020 2020      |.|     0   
│ │ │ │ -0002e0d0: 2020 2031 2020 2020 2020 3220 2020 2020     1      2     
│ │ │ │ -0002e0e0: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ -0002e0f0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0002e0b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0002e0c0: 2030 2020 2020 2020 3120 2020 2020 2032   0      1      2
│ │ │ │ +0002e0d0: 2020 2020 2020 3320 2020 2020 2020 2020        3         
│ │ │ │ +0002e0e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002e0f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0002e100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e120: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002e130: 7c6f 3720 3a20 436f 6d70 6c65 7820 2020  |o7 : Complex   
│ │ │ │ +0002e120: 2020 207c 0a7c 6f37 203a 2043 6f6d 706c     |.|o7 : Compl
│ │ │ │ +0002e130: 6578 2020 2020 2020 2020 2020 2020 2020  ex              
│ │ │ │  0002e140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e160: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0002e150: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0002e160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e190: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20  --------+.|i8 : 
│ │ │ │ -0002e1a0: 4d20 3d20 636f 6b65 7220 462e 6464 5f33  M = coker F.dd_3
│ │ │ │ -0002e1b0: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ -0002e1c0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0002e180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +0002e190: 6938 203a 204d 203d 2063 6f6b 6572 2046  i8 : M = coker F
│ │ │ │ +0002e1a0: 2e64 645f 333b 2020 2020 2020 2020 2020  .dd_3;          
│ │ │ │ +0002e1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e1c0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  0002e1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e200: 2d2d 2b0a 7c69 3920 3a20 4d46 203d 206d  --+.|i9 : MF = m
│ │ │ │ -0002e210: 6174 7269 7846 6163 746f 7269 7a61 7469  atrixFactorizati
│ │ │ │ -0002e220: 6f6e 2866 662c 4d29 3b20 2020 2020 2020  on(ff,M);       
│ │ │ │ -0002e230: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002e1f0: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 204d  -------+.|i9 : M
│ │ │ │ +0002e200: 4620 3d20 6d61 7472 6978 4661 6374 6f72  F = matrixFactor
│ │ │ │ +0002e210: 697a 6174 696f 6e28 6666 2c4d 293b 2020  ization(ff,M);  
│ │ │ │ +0002e220: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0002e230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0002e270: 3130 203a 2062 6574 7469 2066 7265 6552  10 : betti freeR
│ │ │ │ -0002e280: 6573 6f6c 7574 696f 6e20 7075 7368 466f  esolution pushFo
│ │ │ │ -0002e290: 7277 6172 6428 6d61 7028 522c 5329 2c4d  rward(map(R,S),M
│ │ │ │ -0002e2a0: 297c 0a7c 2020 2020 2020 2020 2020 2020  )|.|            
│ │ │ │ +0002e260: 2d2b 0a7c 6931 3020 3a20 6265 7474 6920  -+.|i10 : betti 
│ │ │ │ +0002e270: 6672 6565 5265 736f 6c75 7469 6f6e 2070  freeResolution p
│ │ │ │ +0002e280: 7573 6846 6f72 7761 7264 286d 6170 2852  ushForward(map(R
│ │ │ │ +0002e290: 2c53 292c 4d29 7c0a 7c20 2020 2020 2020  ,S),M)|.|       
│ │ │ │ +0002e2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e2d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0002e2e0: 2020 2020 2020 3020 3120 3220 2020 2020        0 1 2     
│ │ │ │ +0002e2c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0002e2d0: 2020 2020 2020 2020 2020 2030 2031 2032             0 1 2
│ │ │ │ +0002e2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e300: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -0002e310: 3020 3d20 746f 7461 6c3a 2033 2035 2032  0 = total: 3 5 2
│ │ │ │ +0002e300: 7c0a 7c6f 3130 203d 2074 6f74 616c 3a20  |.|o10 = total: 
│ │ │ │ +0002e310: 3320 3520 3220 2020 2020 2020 2020 2020  3 5 2           
│ │ │ │  0002e320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e340: 7c0a 7c20 2020 2020 2020 2020 2032 3a20  |.|          2: 
│ │ │ │ -0002e350: 3320 3420 2e20 2020 2020 2020 2020 2020  3 4 .           
│ │ │ │ -0002e360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e370: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0002e380: 2020 333a 202e 2031 2032 2020 2020 2020    3: . 1 2      
│ │ │ │ -0002e390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e3a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002e330: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002e340: 2020 323a 2033 2034 202e 2020 2020 2020    2: 3 4 .      
│ │ │ │ +0002e350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e360: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002e370: 2020 2020 2020 2033 3a20 2e20 3120 3220         3: . 1 2 
│ │ │ │ +0002e380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e390: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002e3a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002e3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e3d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002e3e0: 0a7c 6f31 3020 3a20 4265 7474 6954 616c  .|o10 : BettiTal
│ │ │ │ -0002e3f0: 6c79 2020 2020 2020 2020 2020 2020 2020  ly              
│ │ │ │ -0002e400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e410: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0002e3d0: 2020 2020 7c0a 7c6f 3130 203a 2042 6574      |.|o10 : Bet
│ │ │ │ +0002e3e0: 7469 5461 6c6c 7920 2020 2020 2020 2020  tiTally         
│ │ │ │ +0002e3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e400: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0002e410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e440: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120  ---------+.|i11 
│ │ │ │ -0002e450: 3a20 6669 6e69 7465 4265 7474 694e 756d  : finiteBettiNum
│ │ │ │ -0002e460: 6265 7273 204d 4620 2020 2020 2020 2020  bers MF         
│ │ │ │ -0002e470: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002e480: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0002e430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0002e440: 7c69 3131 203a 2066 696e 6974 6542 6574  |i11 : finiteBet
│ │ │ │ +0002e450: 7469 4e75 6d62 6572 7320 4d46 2020 2020  tiNumbers MF    
│ │ │ │ +0002e460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e470: 2020 207c 0a7c 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e4b0: 2020 207c 0a7c 6f31 3120 3d20 7b33 2c20     |.|o11 = {3, 
│ │ │ │ -0002e4c0: 352c 2032 7d20 2020 2020 2020 2020 2020  5, 2}           
│ │ │ │ -0002e4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e4e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002e4a0: 2020 2020 2020 2020 7c0a 7c6f 3131 203d          |.|o11 =
│ │ │ │ +0002e4b0: 207b 332c 2035 2c20 327d 2020 2020 2020   {3, 5, 2}      
│ │ │ │ +0002e4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e4d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002e4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e510: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0002e520: 6f31 3120 3a20 4c69 7374 2020 2020 2020  o11 : List      
│ │ │ │ +0002e510: 2020 7c0a 7c6f 3131 203a 204c 6973 7420    |.|o11 : List 
│ │ │ │ +0002e520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e550: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0002e540: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002e550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e580: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220 3a20  -------+.|i12 : 
│ │ │ │ -0002e590: 696e 6669 6e69 7465 4265 7474 694e 756d  infiniteBettiNum
│ │ │ │ -0002e5a0: 6265 7273 284d 462c 3529 2020 2020 2020  bers(MF,5)      
│ │ │ │ -0002e5b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0002e570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0002e580: 3132 203a 2069 6e66 696e 6974 6542 6574  12 : infiniteBet
│ │ │ │ +0002e590: 7469 4e75 6d62 6572 7328 4d46 2c35 2920  tiNumbers(MF,5) 
│ │ │ │ +0002e5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e5b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002e5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e5f0: 207c 0a7c 6f31 3220 3d20 7b33 2c20 342c   |.|o12 = {3, 4,
│ │ │ │ -0002e600: 2035 2c20 362c 2037 2c20 387d 2020 2020   5, 6, 7, 8}    
│ │ │ │ -0002e610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e620: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0002e5e0: 2020 2020 2020 7c0a 7c6f 3132 203d 207b        |.|o12 = {
│ │ │ │ +0002e5f0: 332c 2034 2c20 352c 2036 2c20 372c 2038  3, 4, 5, 6, 7, 8
│ │ │ │ +0002e600: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +0002e610: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0002e620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e650: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -0002e660: 3220 3a20 4c69 7374 2020 2020 2020 2020  2 : List        
│ │ │ │ +0002e650: 7c0a 7c6f 3132 203a 204c 6973 7420 2020  |.|o12 : List   
│ │ │ │ +0002e660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e690: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0002e680: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0002e690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e6c0: 2d2d 2d2d 2d2b 0a7c 6931 3320 3a20 6265  -----+.|i13 : be
│ │ │ │ -0002e6d0: 7474 6920 6672 6565 5265 736f 6c75 7469  tti freeResoluti
│ │ │ │ -0002e6e0: 6f6e 2028 4d2c 204c 656e 6774 684c 696d  on (M, LengthLim
│ │ │ │ -0002e6f0: 6974 203d 3e20 3529 2020 7c0a 7c20 2020  it => 5)  |.|   
│ │ │ │ +0002e6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3133  ----------+.|i13
│ │ │ │ +0002e6c0: 203a 2062 6574 7469 2066 7265 6552 6573   : betti freeRes
│ │ │ │ +0002e6d0: 6f6c 7574 696f 6e20 284d 2c20 4c65 6e67  olution (M, Leng
│ │ │ │ +0002e6e0: 7468 4c69 6d69 7420 3d3e 2035 2920 207c  thLimit => 5)  |
│ │ │ │ +0002e6f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002e700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e720: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002e730: 0a7c 2020 2020 2020 2020 2020 2020 2030  .|             0
│ │ │ │ -0002e740: 2031 2032 2033 2034 2035 2020 2020 2020   1 2 3 4 5      
│ │ │ │ -0002e750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e760: 2020 2020 7c0a 7c6f 3133 203d 2074 6f74      |.|o13 = tot
│ │ │ │ -0002e770: 616c 3a20 3320 3420 3520 3620 3720 3820  al: 3 4 5 6 7 8 
│ │ │ │ -0002e780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e790: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0002e7a0: 2020 2020 2020 323a 2033 2034 2035 2036        2: 3 4 5 6
│ │ │ │ -0002e7b0: 2037 2038 2020 2020 2020 2020 2020 2020   7 8            
│ │ │ │ -0002e7c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002e7d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0002e720: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002e730: 2020 2020 3020 3120 3220 3320 3420 3520      0 1 2 3 4 5 
│ │ │ │ +0002e740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e750: 2020 2020 2020 2020 207c 0a7c 6f31 3320           |.|o13 
│ │ │ │ +0002e760: 3d20 746f 7461 6c3a 2033 2034 2035 2036  = total: 3 4 5 6
│ │ │ │ +0002e770: 2037 2038 2020 2020 2020 2020 2020 2020   7 8            
│ │ │ │ +0002e780: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002e790: 7c20 2020 2020 2020 2020 2032 3a20 3320  |          2: 3 
│ │ │ │ +0002e7a0: 3420 3520 3620 3720 3820 2020 2020 2020  4 5 6 7 8       
│ │ │ │ +0002e7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e7c0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0002e7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e800: 2020 207c 0a7c 6f31 3320 3a20 4265 7474     |.|o13 : Bett
│ │ │ │ -0002e810: 6954 616c 6c79 2020 2020 2020 2020 2020  iTally          
│ │ │ │ -0002e820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e830: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0002e7f0: 2020 2020 2020 2020 7c0a 7c6f 3133 203a          |.|o13 :
│ │ │ │ +0002e800: 2042 6574 7469 5461 6c6c 7920 2020 2020   BettiTally     
│ │ │ │ +0002e810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e820: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0002e830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ -0002e870: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d  See also.=======
│ │ │ │ -0002e880: 3d0a 0a20 202a 202a 6e6f 7465 206d 6174  =..  * *note mat
│ │ │ │ -0002e890: 7269 7846 6163 746f 7269 7a61 7469 6f6e  rixFactorization
│ │ │ │ -0002e8a0: 3a20 6d61 7472 6978 4661 6374 6f72 697a  : matrixFactoriz
│ │ │ │ -0002e8b0: 6174 696f 6e2c 202d 2d20 4d61 7073 2069  ation, -- Maps i
│ │ │ │ -0002e8c0: 6e20 6120 6869 6768 6572 0a20 2020 2063  n a higher.    c
│ │ │ │ -0002e8d0: 6f64 696d 656e 7369 6f6e 206d 6174 7269  odimension matri
│ │ │ │ -0002e8e0: 7820 6661 6374 6f72 697a 6174 696f 6e0a  x factorization.
│ │ │ │ -0002e8f0: 2020 2a20 2a6e 6f74 6520 6669 6e69 7465    * *note finite
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│ │ │ │ -0002e9a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ +0002e920: 6e75 6d62 6572 7320 6f66 2066 696e 6974  numbers of finit
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│ │ │ │ +0002e990: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0002e9a0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ +0002e9b0: 2269 6e66 696e 6974 6542 6574 7469 4e75  "infiniteBettiNu
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│ │ │ │ +0002e9d0: 0a46 6f72 2074 6865 2070 726f 6772 616d  .For the program
│ │ │ │ +0002e9e0: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  mer.============
│ │ │ │ +0002e9f0: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65  ======..The obje
│ │ │ │ +0002ea00: 6374 202a 6e6f 7465 2069 6e66 696e 6974  ct *note infinit
│ │ │ │ +0002ea10: 6542 6574 7469 4e75 6d62 6572 733a 2069  eBettiNumbers: i
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│ │ │ │ +0002ea40: 6d65 7468 6f64 0a66 756e 6374 696f 6e3a  method.function:
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│ │ │ │  0002ea80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ea90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002eaa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002eab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002eac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865  -----------..The
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│ │ │ │ -0002eb90: 653a 2069 734c 696e 6561 722c 204e 6578  e: isLinear, Nex
│ │ │ │ -0002eba0: 743a 2069 7351 7561 7369 5265 6775 6c61  t: isQuasiRegula
│ │ │ │ -0002ebb0: 722c 2050 7265 763a 2069 6e66 696e 6974  r, Prev: infinit
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│ │ │ │ -0002ec00: 2068 6176 6520 6465 6772 6565 2031 0a2a   have degree 1.*
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│ │ │ │ +0002eb50: 3a30 2e0a 1f0a 4669 6c65 3a20 436f 6d70  :0....File: Comp
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│ │ │ │ +0002eb70: 5265 736f 6c75 7469 6f6e 732e 696e 666f  Resolutions.info
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│ │ │ │ +0002eb90: 2c20 4e65 7874 3a20 6973 5175 6173 6952  , Next: isQuasiR
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│ │ │ │ +0002ebb0: 6669 6e69 7465 4265 7474 694e 756d 6265  finiteBettiNumbe
│ │ │ │ +0002ebc0: 7273 2c20 5570 3a20 546f 700a 0a69 734c  rs, Up: Top..isL
│ │ │ │ +0002ebd0: 696e 6561 7220 2d2d 2063 6865 636b 2077  inear -- check w
│ │ │ │ +0002ebe0: 6865 7468 6572 206d 6174 7269 7820 656e  hether matrix en
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│ │ │ │ +0002ec00: 6520 310a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  e 1.************
│ │ │ │  0002ec10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002ec20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002ec30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002ec40: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765  *****..  * Usage
│ │ │ │ -0002ec50: 3a20 0a20 2020 2020 2020 2062 203d 2069  : .        b = i
│ │ │ │ -0002ec60: 734c 696e 6561 7220 4d0a 2020 2a20 496e  sLinear M.  * In
│ │ │ │ -0002ec70: 7075 7473 3a0a 2020 2020 2020 2a20 4d2c  puts:.      * M,
│ │ │ │ -0002ec80: 2061 202a 6e6f 7465 206d 6174 7269 783a   a *note matrix:
│ │ │ │ -0002ec90: 2028 4d61 6361 756c 6179 3244 6f63 294d   (Macaulay2Doc)M
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│ │ │ │ -0002ecb0: 7075 7473 3a0a 2020 2020 2020 2a20 622c  puts:.      * b,
│ │ │ │ -0002ecc0: 2061 202a 6e6f 7465 2042 6f6f 6c65 616e   a *note Boolean
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│ │ │ │ -0002ecf0: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d  ..Description.==
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│ │ │ │ -0002ed10: 7468 6174 2061 206c 696e 6561 7220 6d61  that a linear ma
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│ │ │ │ -0002ed70: 636f 6b65 726e 656c 2064 6563 6f6d 706f  cokernel decompo
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│ │ │ │ +0002ec40: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
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│ │ │ │ +0002ee70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ee80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ee90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002eea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002eeb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0002efd0: 0a69 7351 7561 7369 5265 6775 6c61 7220  .isQuasiRegular 
│ │ │ │ -0002efe0: 2d2d 2074 6573 7473 2061 206d 6174 7269  -- tests a matri
│ │ │ │ -0002eff0: 7820 6f72 2073 6571 7565 6e63 6520 6f72  x or sequence or
│ │ │ │ -0002f000: 206c 6973 7420 666f 7220 7175 6173 692d   list for quasi-
│ │ │ │ -0002f010: 7265 6775 6c61 7269 7479 206f 6e20 6120  regularity on a 
│ │ │ │ -0002f020: 6d6f 6475 6c65 0a2a 2a2a 2a2a 2a2a 2a2a  module.*********
│ │ │ │ +0002eeb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a  --------------..
│ │ │ │ +0002eec0: 5468 6520 736f 7572 6365 206f 6620 7468  The source of th
│ │ │ │ +0002eed0: 6973 2064 6f63 756d 656e 7420 6973 2069  is document is i
│ │ │ │ +0002eee0: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475  n./build/reprodu
│ │ │ │ +0002eef0: 6369 626c 652d 7061 7468 2f6d 6163 6175  cible-path/macau
│ │ │ │ +0002ef00: 6c61 7932 2d31 2e32 352e 3131 2b64 732f  lay2-1.25.11+ds/
│ │ │ │ +0002ef10: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163  M2/Macaulay2/pac
│ │ │ │ +0002ef20: 6b61 6765 732f 0a43 6f6d 706c 6574 6549  kages/.CompleteI
│ │ │ │ +0002ef30: 6e74 6572 7365 6374 696f 6e52 6573 6f6c  ntersectionResol
│ │ │ │ +0002ef40: 7574 696f 6e73 2e6d 323a 3334 3633 3a30  utions.m2:3463:0
│ │ │ │ +0002ef50: 2e0a 1f0a 4669 6c65 3a20 436f 6d70 6c65  ....File: Comple
│ │ │ │ +0002ef60: 7465 496e 7465 7273 6563 7469 6f6e 5265  teIntersectionRe
│ │ │ │ +0002ef70: 736f 6c75 7469 6f6e 732e 696e 666f 2c20  solutions.info, 
│ │ │ │ +0002ef80: 4e6f 6465 3a20 6973 5175 6173 6952 6567  Node: isQuasiReg
│ │ │ │ +0002ef90: 756c 6172 2c20 4e65 7874 3a20 6973 5374  ular, Next: isSt
│ │ │ │ +0002efa0: 6162 6c79 5472 6976 6961 6c2c 2050 7265  ablyTrivial, Pre
│ │ │ │ +0002efb0: 763a 2069 734c 696e 6561 722c 2055 703a  v: isLinear, Up:
│ │ │ │ +0002efc0: 2054 6f70 0a0a 6973 5175 6173 6952 6567   Top..isQuasiReg
│ │ │ │ +0002efd0: 756c 6172 202d 2d20 7465 7374 7320 6120  ular -- tests a 
│ │ │ │ +0002efe0: 6d61 7472 6978 206f 7220 7365 7175 656e  matrix or sequen
│ │ │ │ +0002eff0: 6365 206f 7220 6c69 7374 2066 6f72 2071  ce or list for q
│ │ │ │ +0002f000: 7561 7369 2d72 6567 756c 6172 6974 7920  uasi-regularity 
│ │ │ │ +0002f010: 6f6e 2061 206d 6f64 756c 650a 2a2a 2a2a  on a module.****
│ │ │ │ +0002f020: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002f030: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002f040: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002f050: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002f060: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002f070: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020  ************..  
│ │ │ │ -0002f080: 2a20 5573 6167 653a 200a 2020 2020 2020  * Usage: .      
│ │ │ │ -0002f090: 2020 7420 3d20 6973 5175 6173 6952 6567    t = isQuasiReg
│ │ │ │ -0002f0a0: 756c 6172 2866 662c 4d29 0a20 202a 2049  ular(ff,M).  * I
│ │ │ │ -0002f0b0: 6e70 7574 733a 0a20 2020 2020 202a 2066  nputs:.      * f
│ │ │ │ -0002f0c0: 662c 2061 202a 6e6f 7465 206d 6174 7269  f, a *note matri
│ │ │ │ -0002f0d0: 783a 2028 4d61 6361 756c 6179 3244 6f63  x: (Macaulay2Doc
│ │ │ │ -0002f0e0: 294d 6174 7269 782c 2c20 0a20 2020 2020  )Matrix,, .     
│ │ │ │ -0002f0f0: 202a 2066 662c 2061 202a 6e6f 7465 206c   * ff, a *note l
│ │ │ │ -0002f100: 6973 743a 2028 4d61 6361 756c 6179 3244  ist: (Macaulay2D
│ │ │ │ -0002f110: 6f63 294c 6973 742c 2c20 0a20 2020 2020  oc)List,, .     
│ │ │ │ -0002f120: 202a 2066 662c 2061 202a 6e6f 7465 2073   * ff, a *note s
│ │ │ │ -0002f130: 6571 7565 6e63 653a 2028 4d61 6361 756c  equence: (Macaul
│ │ │ │ -0002f140: 6179 3244 6f63 2953 6571 7565 6e63 652c  ay2Doc)Sequence,
│ │ │ │ -0002f150: 2c20 0a20 2020 2020 202a 204d 2c20 6120  , .      * M, a 
│ │ │ │ -0002f160: 2a6e 6f74 6520 6d6f 6475 6c65 3a20 284d  *note module: (M
│ │ │ │ -0002f170: 6163 6175 6c61 7932 446f 6329 4d6f 6475  acaulay2Doc)Modu
│ │ │ │ -0002f180: 6c65 2c2c 200a 2020 2a20 4f75 7470 7574  le,, .  * Output
│ │ │ │ -0002f190: 733a 0a20 2020 2020 202a 2074 2c20 6120  s:.      * t, a 
│ │ │ │ -0002f1a0: 2a6e 6f74 6520 426f 6f6c 6561 6e20 7661  *note Boolean va
│ │ │ │ -0002f1b0: 6c75 653a 2028 4d61 6361 756c 6179 3244  lue: (Macaulay2D
│ │ │ │ -0002f1c0: 6f63 2942 6f6f 6c65 616e 2c2c 200a 0a44  oc)Boolean,, ..D
│ │ │ │ -0002f1d0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ -0002f1e0: 3d3d 3d3d 3d3d 0a0a 6666 2069 7320 7175  ======..ff is qu
│ │ │ │ -0002f1f0: 6173 692d 7265 6775 6c61 7220 6966 2074  asi-regular if t
│ │ │ │ -0002f200: 6865 206c 656e 6774 6820 6f66 2066 6620  he length of ff 
│ │ │ │ -0002f210: 6973 203c 3d20 6469 6d20 4d20 616e 6420  is <= dim M and 
│ │ │ │ -0002f220: 7468 6520 616e 6e69 6869 6c61 746f 7220  the annihilator 
│ │ │ │ -0002f230: 6f66 2066 665f 690a 6f6e 204d 2f28 6666  of ff_i.on M/(ff
│ │ │ │ -0002f240: 5f30 2e2e 6666 5f7b 2869 2d31 2929 7d4d  _0..ff_{(i-1))}M
│ │ │ │ -0002f250: 2068 6173 2066 696e 6974 6520 6c65 6e67   has finite leng
│ │ │ │ -0002f260: 7468 2066 6f72 2061 6c6c 2069 3d30 2e2e  th for all i=0..
│ │ │ │ -0002f270: 286c 656e 6774 6820 6666 292d 312e 0a0a  (length ff)-1...
│ │ │ │ -0002f280: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -0002f290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120  ----------+.|i1 
│ │ │ │ -0002f2b0: 3a20 6b6b 3d5a 5a2f 3130 313b 2020 2020  : kk=ZZ/101;    
│ │ │ │ -0002f2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f2d0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0002f070: 2a0a 0a20 202a 2055 7361 6765 3a20 0a20  *..  * Usage: . 
│ │ │ │ +0002f080: 2020 2020 2020 2074 203d 2069 7351 7561         t = isQua
│ │ │ │ +0002f090: 7369 5265 6775 6c61 7228 6666 2c4d 290a  siRegular(ff,M).
│ │ │ │ +0002f0a0: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ +0002f0b0: 2020 2a20 6666 2c20 6120 2a6e 6f74 6520    * ff, a *note 
│ │ │ │ +0002f0c0: 6d61 7472 6978 3a20 284d 6163 6175 6c61  matrix: (Macaula
│ │ │ │ +0002f0d0: 7932 446f 6329 4d61 7472 6978 2c2c 200a  y2Doc)Matrix,, .
│ │ │ │ +0002f0e0: 2020 2020 2020 2a20 6666 2c20 6120 2a6e        * ff, a *n
│ │ │ │ +0002f0f0: 6f74 6520 6c69 7374 3a20 284d 6163 6175  ote list: (Macau
│ │ │ │ +0002f100: 6c61 7932 446f 6329 4c69 7374 2c2c 200a  lay2Doc)List,, .
│ │ │ │ +0002f110: 2020 2020 2020 2a20 6666 2c20 6120 2a6e        * ff, a *n
│ │ │ │ +0002f120: 6f74 6520 7365 7175 656e 6365 3a20 284d  ote sequence: (M
│ │ │ │ +0002f130: 6163 6175 6c61 7932 446f 6329 5365 7175  acaulay2Doc)Sequ
│ │ │ │ +0002f140: 656e 6365 2c2c 200a 2020 2020 2020 2a20  ence,, .      * 
│ │ │ │ +0002f150: 4d2c 2061 202a 6e6f 7465 206d 6f64 756c  M, a *note modul
│ │ │ │ +0002f160: 653a 2028 4d61 6361 756c 6179 3244 6f63  e: (Macaulay2Doc
│ │ │ │ +0002f170: 294d 6f64 756c 652c 2c20 0a20 202a 204f  )Module,, .  * O
│ │ │ │ +0002f180: 7574 7075 7473 3a0a 2020 2020 2020 2a20  utputs:.      * 
│ │ │ │ +0002f190: 742c 2061 202a 6e6f 7465 2042 6f6f 6c65  t, a *note Boole
│ │ │ │ +0002f1a0: 616e 2076 616c 7565 3a20 284d 6163 6175  an value: (Macau
│ │ │ │ +0002f1b0: 6c61 7932 446f 6329 426f 6f6c 6561 6e2c  lay2Doc)Boolean,
│ │ │ │ +0002f1c0: 2c20 0a0a 4465 7363 7269 7074 696f 6e0a  , ..Description.
│ │ │ │ +0002f1d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a66 6620  ===========..ff 
│ │ │ │ +0002f1e0: 6973 2071 7561 7369 2d72 6567 756c 6172  is quasi-regular
│ │ │ │ +0002f1f0: 2069 6620 7468 6520 6c65 6e67 7468 206f   if the length o
│ │ │ │ +0002f200: 6620 6666 2069 7320 3c3d 2064 696d 204d  f ff is <= dim M
│ │ │ │ +0002f210: 2061 6e64 2074 6865 2061 6e6e 6968 696c   and the annihil
│ │ │ │ +0002f220: 6174 6f72 206f 6620 6666 5f69 0a6f 6e20  ator of ff_i.on 
│ │ │ │ +0002f230: 4d2f 2866 665f 302e 2e66 665f 7b28 692d  M/(ff_0..ff_{(i-
│ │ │ │ +0002f240: 3129 297d 4d20 6861 7320 6669 6e69 7465  1))}M has finite
│ │ │ │ +0002f250: 206c 656e 6774 6820 666f 7220 616c 6c20   length for all 
│ │ │ │ +0002f260: 693d 302e 2e28 6c65 6e67 7468 2066 6629  i=0..(length ff)
│ │ │ │ +0002f270: 2d31 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  -1...+----------
│ │ │ │ +0002f280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002f290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0002f2a0: 0a7c 6931 203a 206b 6b3d 5a5a 2f31 3031  .|i1 : kk=ZZ/101
│ │ │ │ +0002f2b0: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ +0002f2c0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0002f2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002f2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f300: 2d2d 2b0a 7c69 3220 3a20 5320 3d20 6b6b  --+.|i2 : S = kk
│ │ │ │ -0002f310: 5b61 2c62 2c63 5d3b 2020 2020 2020 2020  [a,b,c];        
│ │ │ │ -0002f320: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002f330: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -0002f340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f350: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320  ----------+.|i3 
│ │ │ │ -0002f360: 3a20 4520 3d20 535e 312f 6964 6561 6c22  : E = S^1/ideal"
│ │ │ │ -0002f370: 6162 222b 2b53 5e31 2f69 6465 616c 2076  ab"++S^1/ideal v
│ │ │ │ -0002f380: 6172 7320 533b 7c0a 2b2d 2d2d 2d2d 2d2d  ars S;|.+-------
│ │ │ │ +0002f2f0: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2053  -------+.|i2 : S
│ │ │ │ +0002f300: 203d 206b 6b5b 612c 622c 635d 3b20 2020   = kk[a,b,c];   
│ │ │ │ +0002f310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f320: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0002f330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002f340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0002f350: 0a7c 6933 203a 2045 203d 2053 5e31 2f69  .|i3 : E = S^1/i
│ │ │ │ +0002f360: 6465 616c 2261 6222 2b2b 535e 312f 6964  deal"ab"++S^1/id
│ │ │ │ +0002f370: 6561 6c20 7661 7273 2053 3b7c 0a2b 2d2d  eal vars S;|.+--
│ │ │ │ +0002f380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002f390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f3b0: 2d2d 2b0a 7c69 3420 3a20 6631 203d 6d61  --+.|i4 : f1 =ma
│ │ │ │ -0002f3c0: 7472 6978 2261 223b 2020 2020 2020 2020  trix"a";        
│ │ │ │ -0002f3d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002f3e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0002f3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f400: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0002f410: 2020 2020 2020 2020 2020 3120 2020 2020            1     
│ │ │ │ -0002f420: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │ -0002f430: 2020 2020 2020 7c0a 7c6f 3420 3a20 4d61        |.|o4 : Ma
│ │ │ │ -0002f440: 7472 6978 2053 2020 3c2d 2d20 5320 2020  trix S  <-- S   
│ │ │ │ -0002f450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f460: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0002f3a0: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2066  -------+.|i4 : f
│ │ │ │ +0002f3b0: 3120 3d6d 6174 7269 7822 6122 3b20 2020  1 =matrix"a";   
│ │ │ │ +0002f3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f3d0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0002f3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f3f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002f400: 0a7c 2020 2020 2020 2020 2020 2020 2031  .|             1
│ │ │ │ +0002f410: 2020 2020 2020 3120 2020 2020 2020 2020        1         
│ │ │ │ +0002f420: 2020 2020 2020 2020 2020 207c 0a7c 6f34             |.|o4
│ │ │ │ +0002f430: 203a 204d 6174 7269 7820 5320 203c 2d2d   : Matrix S  <--
│ │ │ │ +0002f440: 2053 2020 2020 2020 2020 2020 2020 2020   S              
│ │ │ │ +0002f450: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002f460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002f470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0002f490: 7c69 3520 3a20 6632 203d 6d61 7472 6978  |i5 : f2 =matrix
│ │ │ │ -0002f4a0: 2261 2b62 2c63 223b 2020 2020 2020 2020  "a+b,c";        
│ │ │ │ -0002f4b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002f480: 2d2d 2d2b 0a7c 6935 203a 2066 3220 3d6d  ---+.|i5 : f2 =m
│ │ │ │ +0002f490: 6174 7269 7822 612b 622c 6322 3b20 2020  atrix"a+b,c";   
│ │ │ │ +0002f4a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002f4b0: 0a7c 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 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0002f4f0: 2020 2020 2020 3120 2020 2020 2032 2020        1      2  
│ │ │ │ -0002f500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f510: 2020 7c0a 7c6f 3520 3a20 4d61 7472 6978    |.|o5 : Matrix
│ │ │ │ -0002f520: 2053 2020 3c2d 2d20 5320 2020 2020 2020   S  <-- S       
│ │ │ │ -0002f530: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002f540: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -0002f550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f560: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620  ----------+.|i6 
│ │ │ │ -0002f570: 3a20 6633 203d 206d 6174 7269 7822 612b  : f3 = matrix"a+
│ │ │ │ -0002f580: 6222 3b20 2020 2020 2020 2020 2020 2020  b";             
│ │ │ │ -0002f590: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0002f4d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0002f4e0: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ +0002f4f0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +0002f500: 2020 2020 2020 207c 0a7c 6f35 203a 204d         |.|o5 : M
│ │ │ │ +0002f510: 6174 7269 7820 5320 203c 2d2d 2053 2020  atrix S  <-- S  
│ │ │ │ +0002f520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f530: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0002f540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002f550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0002f560: 0a7c 6936 203a 2066 3320 3d20 6d61 7472  .|i6 : f3 = matr
│ │ │ │ +0002f570: 6978 2261 2b62 223b 2020 2020 2020 2020  ix"a+b";        
│ │ │ │ +0002f580: 2020 2020 2020 2020 2020 207c 0a7c 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 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0002f5d0: 2020 3120 2020 2020 2031 2020 2020 2020    1      1      
│ │ │ │ -0002f5e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002f5f0: 7c6f 3620 3a20 4d61 7472 6978 2053 2020  |o6 : Matrix S  
│ │ │ │ -0002f600: 3c2d 2d20 5320 2020 2020 2020 2020 2020  <-- S           
│ │ │ │ -0002f610: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0002f5b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002f5c0: 2020 2020 2020 2031 2020 2020 2020 3120         1      1 
│ │ │ │ +0002f5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f5e0: 2020 207c 0a7c 6f36 203a 204d 6174 7269     |.|o6 : Matri
│ │ │ │ +0002f5f0: 7820 5320 203c 2d2d 2053 2020 2020 2020  x S  <-- S      
│ │ │ │ +0002f600: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002f610: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  0002f620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f640: 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20 6634  ------+.|i7 : f4
│ │ │ │ -0002f650: 203d 206d 6174 7269 7822 612b 622c 2061   = matrix"a+b, a
│ │ │ │ -0002f660: 322b 6222 3b20 2020 2020 2020 2020 2020  2+b";           
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│ │ │ │  0002f680: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0002f6b0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
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│ │ │ │ -0002f6d0: 3a20 4d61 7472 6978 2053 2020 3c2d 2d20  : Matrix S  <-- 
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│ │ │ │ -0002f6f0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0002f690: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
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│ │ │ │ +0002f6b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002f6c0: 0a7c 6f37 203a 204d 6174 7269 7820 5320  .|o7 : Matrix S 
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│ │ │ │ -0002f760: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0002f870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0002f8a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
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│ │ │ │ -0002fe10: 7373 6962 6c65 206f 6273 7472 7563 7469  ssible obstructi
│ │ │ │ -0002fe20: 6f6e 2074 6f20 7468 6520 636f 6d6d 7574  on to the commut
│ │ │ │ -0002fe30: 6174 6976 6974 7920 6f66 2074 6865 2043  ativity of the C
│ │ │ │ -0002fe40: 4920 6f70 6572 6174 6f72 7320 696e 2063  I operators in c
│ │ │ │ -0002fe50: 6f64 696d 2063 2c0a 6576 656e 2061 7379  odim c,.even asy
│ │ │ │ -0002fe60: 6d70 746f 7469 6361 6c6c 792c 2077 6f75  mptotically, wou
│ │ │ │ -0002fe70: 6c64 2062 6520 7468 6520 6e6f 6e2d 7472  ld be the non-tr
│ │ │ │ -0002fe80: 6976 6961 6c69 7479 206f 6620 7468 6520  iviality of the 
│ │ │ │ -0002fe90: 6d61 7020 4d5f 7b28 6b2b 3429 7d20 2d2d  map M_{(k+4)} --
│ │ │ │ -0002fea0: 3e20 4d5f 6b0a 5c6f 7469 6d65 7320 5c77  > M_k.\otimes \w
│ │ │ │ -0002feb0: 6564 6765 5e32 2853 5e63 2920 696e 2074  edge^2(S^c) in t
│ │ │ │ -0002fec0: 6865 2073 7461 626c 6520 6361 7465 676f  he stable catego
│ │ │ │ -0002fed0: 7279 206f 6620 6d61 7869 6d61 6c20 436f  ry of maximal Co
│ │ │ │ -0002fee0: 6865 6e2d 4d61 6361 756c 6179 206d 6f64  hen-Macaulay mod
│ │ │ │ -0002fef0: 756c 6573 2e0a 0a49 6e20 7468 6520 666f  ules...In the fo
│ │ │ │ -0002ff00: 6c6c 6f77 696e 6720 6578 616d 706c 652c  llowing example,
│ │ │ │ -0002ff10: 2073 7475 6469 6564 2069 6e20 7468 6520   studied in the 
│ │ │ │ -0002ff20: 7061 7065 7220 2254 6f72 2061 7320 6120  paper "Tor as a 
│ │ │ │ -0002ff30: 6d6f 6475 6c65 206f 7665 7220 616e 0a65  module over an.e
│ │ │ │ -0002ff40: 7874 6572 696f 7220 616c 6765 6272 6122  xterior algebra"
│ │ │ │ -0002ff50: 206f 6620 4569 7365 6e62 7564 2c20 5065   of Eisenbud, Pe
│ │ │ │ -0002ff60: 6576 6120 616e 6420 5363 6872 6579 6572  eva and Schreyer
│ │ │ │ -0002ff70: 2c20 7468 6520 6d61 7020 6973 206e 6f6e  , the map is non
│ │ │ │ -0002ff80: 2d74 7269 7669 616c 2e2e 2e62 7574 0a69  -trivial...but.i
│ │ │ │ -0002ff90: 7420 6973 2073 7461 626c 7920 7472 6976  t is stably triv
│ │ │ │ -0002ffa0: 6961 6c2e 2054 6865 2073 616d 6520 676f  ial. The same go
│ │ │ │ -0002ffb0: 6573 2066 6f72 2068 6967 6865 7220 7661  es for higher va
│ │ │ │ -0002ffc0: 6c75 6573 206f 6620 6b20 2877 6869 6368  lues of k (which
│ │ │ │ -0002ffd0: 2074 616b 6520 6c6f 6e67 6572 0a74 6f20   take longer.to 
│ │ │ │ -0002ffe0: 636f 6d70 7574 6529 2e20 286e 6f74 6520  compute). (note 
│ │ │ │ -0002fff0: 7468 6174 2069 6e20 7468 6973 2063 6173  that in this cas
│ │ │ │ -00030000: 652c 2077 6974 6820 6320 3d20 332c 2074  e, with c = 3, t
│ │ │ │ -00030010: 776f 206f 6620 7468 6520 7468 7265 6520  wo of the three 
│ │ │ │ -00030020: 616c 7465 726e 6174 696e 670a 7072 6f64  alternating.prod
│ │ │ │ -00030030: 7563 7473 2061 7265 2061 6374 7561 6c6c  ucts are actuall
│ │ │ │ -00030040: 7920 6571 7561 6c20 746f 2030 2c20 736f  y equal to 0, so
│ │ │ │ -00030050: 2077 6520 7465 7374 206f 6e6c 7920 7468   we test only th
│ │ │ │ -00030060: 6520 7468 6972 642e 290a 0a4e 6f74 6520  e third.)..Note 
│ │ │ │ -00030070: 7468 6174 2054 2069 7320 7765 6c6c 2d64  that T is well-d
│ │ │ │ -00030080: 6566 696e 6564 2075 7020 746f 2068 6f6d  efined up to hom
│ │ │ │ -00030090: 6f74 6f70 793b 2073 6f20 545e 3220 6973  otopy; so T^2 is
│ │ │ │ -000300a0: 2077 656c 6c2d 6465 6669 6e65 6420 6d6f   well-defined mo
│ │ │ │ -000300b0: 6420 6d6d 5e32 2e0a 0a2b 2d2d 2d2d 2d2d  d mm^2...+------
│ │ │ │ +0002fcf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a  ***********..  *
│ │ │ │ +0002fd00: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ +0002fd10: 2062 203d 2069 7353 7461 626c 7954 7269   b = isStablyTri
│ │ │ │ +0002fd20: 7669 616c 2066 0a20 202a 2049 6e70 7574  vial f.  * Input
│ │ │ │ +0002fd30: 733a 0a20 2020 2020 202a 2066 2c20 6120  s:.      * f, a 
│ │ │ │ +0002fd40: 2a6e 6f74 6520 6d61 7472 6978 3a20 284d  *note matrix: (M
│ │ │ │ +0002fd50: 6163 6175 6c61 7932 446f 6329 4d61 7472  acaulay2Doc)Matr
│ │ │ │ +0002fd60: 6978 2c2c 206d 6170 204d 2074 6f20 4e0a  ix,, map M to N.
│ │ │ │ +0002fd70: 2020 2a20 4f75 7470 7574 733a 0a20 2020    * Outputs:.   
│ │ │ │ +0002fd80: 2020 202a 2062 2c20 6120 2a6e 6f74 6520     * b, a *note 
│ │ │ │ +0002fd90: 426f 6f6c 6561 6e20 7661 6c75 653a 2028  Boolean value: (
│ │ │ │ +0002fda0: 4d61 6361 756c 6179 3244 6f63 2942 6f6f  Macaulay2Doc)Boo
│ │ │ │ +0002fdb0: 6c65 616e 2c2c 2074 7275 6520 6966 6620  lean,, true iff 
│ │ │ │ +0002fdc0: 6620 6661 6374 6f72 730a 2020 2020 2020  f factors.      
│ │ │ │ +0002fdd0: 2020 7468 726f 7567 6820 6120 7072 6f6a    through a proj
│ │ │ │ +0002fde0: 6563 7469 7665 0a0a 4465 7363 7269 7074  ective..Descript
│ │ │ │ +0002fdf0: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ion.===========.
│ │ │ │ +0002fe00: 0a41 2070 6f73 7369 626c 6520 6f62 7374  .A possible obst
│ │ │ │ +0002fe10: 7275 6374 696f 6e20 746f 2074 6865 2063  ruction to the c
│ │ │ │ +0002fe20: 6f6d 6d75 7461 7469 7669 7479 206f 6620  ommutativity of 
│ │ │ │ +0002fe30: 7468 6520 4349 206f 7065 7261 746f 7273  the CI operators
│ │ │ │ +0002fe40: 2069 6e20 636f 6469 6d20 632c 0a65 7665   in codim c,.eve
│ │ │ │ +0002fe50: 6e20 6173 796d 7074 6f74 6963 616c 6c79  n asymptotically
│ │ │ │ +0002fe60: 2c20 776f 756c 6420 6265 2074 6865 206e  , would be the n
│ │ │ │ +0002fe70: 6f6e 2d74 7269 7669 616c 6974 7920 6f66  on-triviality of
│ │ │ │ +0002fe80: 2074 6865 206d 6170 204d 5f7b 286b 2b34   the map M_{(k+4
│ │ │ │ +0002fe90: 297d 202d 2d3e 204d 5f6b 0a5c 6f74 696d  )} --> M_k.\otim
│ │ │ │ +0002fea0: 6573 205c 7765 6467 655e 3228 535e 6329  es \wedge^2(S^c)
│ │ │ │ +0002feb0: 2069 6e20 7468 6520 7374 6162 6c65 2063   in the stable c
│ │ │ │ +0002fec0: 6174 6567 6f72 7920 6f66 206d 6178 696d  ategory of maxim
│ │ │ │ +0002fed0: 616c 2043 6f68 656e 2d4d 6163 6175 6c61  al Cohen-Macaula
│ │ │ │ +0002fee0: 7920 6d6f 6475 6c65 732e 0a0a 496e 2074  y modules...In t
│ │ │ │ +0002fef0: 6865 2066 6f6c 6c6f 7769 6e67 2065 7861  he following exa
│ │ │ │ +0002ff00: 6d70 6c65 2c20 7374 7564 6965 6420 696e  mple, studied in
│ │ │ │ +0002ff10: 2074 6865 2070 6170 6572 2022 546f 7220   the paper "Tor 
│ │ │ │ +0002ff20: 6173 2061 206d 6f64 756c 6520 6f76 6572  as a module over
│ │ │ │ +0002ff30: 2061 6e0a 6578 7465 7269 6f72 2061 6c67   an.exterior alg
│ │ │ │ +0002ff40: 6562 7261 2220 6f66 2045 6973 656e 6275  ebra" of Eisenbu
│ │ │ │ +0002ff50: 642c 2050 6565 7661 2061 6e64 2053 6368  d, Peeva and Sch
│ │ │ │ +0002ff60: 7265 7965 722c 2074 6865 206d 6170 2069  reyer, the map i
│ │ │ │ +0002ff70: 7320 6e6f 6e2d 7472 6976 6961 6c2e 2e2e  s non-trivial...
│ │ │ │ +0002ff80: 6275 740a 6974 2069 7320 7374 6162 6c79  but.it is stably
│ │ │ │ +0002ff90: 2074 7269 7669 616c 2e20 5468 6520 7361   trivial. The sa
│ │ │ │ +0002ffa0: 6d65 2067 6f65 7320 666f 7220 6869 6768  me goes for high
│ │ │ │ +0002ffb0: 6572 2076 616c 7565 7320 6f66 206b 2028  er values of k (
│ │ │ │ +0002ffc0: 7768 6963 6820 7461 6b65 206c 6f6e 6765  which take longe
│ │ │ │ +0002ffd0: 720a 746f 2063 6f6d 7075 7465 292e 2028  r.to compute). (
│ │ │ │ +0002ffe0: 6e6f 7465 2074 6861 7420 696e 2074 6869  note that in thi
│ │ │ │ +0002fff0: 7320 6361 7365 2c20 7769 7468 2063 203d  s case, with c =
│ │ │ │ +00030000: 2033 2c20 7477 6f20 6f66 2074 6865 2074   3, two of the t
│ │ │ │ +00030010: 6872 6565 2061 6c74 6572 6e61 7469 6e67  hree alternating
│ │ │ │ +00030020: 0a70 726f 6475 6374 7320 6172 6520 6163  .products are ac
│ │ │ │ +00030030: 7475 616c 6c79 2065 7175 616c 2074 6f20  tually equal to 
│ │ │ │ +00030040: 302c 2073 6f20 7765 2074 6573 7420 6f6e  0, so we test on
│ │ │ │ +00030050: 6c79 2074 6865 2074 6869 7264 2e29 0a0a  ly the third.)..
│ │ │ │ +00030060: 4e6f 7465 2074 6861 7420 5420 6973 2077  Note that T is w
│ │ │ │ +00030070: 656c 6c2d 6465 6669 6e65 6420 7570 2074  ell-defined up t
│ │ │ │ +00030080: 6f20 686f 6d6f 746f 7079 3b20 736f 2054  o homotopy; so T
│ │ │ │ +00030090: 5e32 2069 7320 7765 6c6c 2d64 6566 696e  ^2 is well-defin
│ │ │ │ +000300a0: 6564 206d 6f64 206d 6d5e 322e 0a0a 2b2d  ed mod mm^2...+-
│ │ │ │ +000300b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000300c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000300d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000300e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000300f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00030100: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 206b  -------+.|i1 : k
│ │ │ │ -00030110: 6b20 3d20 5a5a 2f31 3031 2020 2020 2020  k = ZZ/101      
│ │ │ │ +000300f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00030100: 3120 3a20 6b6b 203d 205a 5a2f 3130 3120  1 : kk = ZZ/101 
│ │ │ │ +00030110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030150: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00030140: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00030150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000301a0: 2020 2020 2020 207c 0a7c 6f31 203d 206b         |.|o1 = k
│ │ │ │ -000301b0: 6b20 2020 2020 2020 2020 2020 2020 2020  k               
│ │ │ │ +00030190: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +000301a0: 3120 3d20 6b6b 2020 2020 2020 2020 2020  1 = kk          
│ │ │ │ +000301b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000301e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000301f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030240: 2020 2020 2020 207c 0a7c 6f31 203a 2051         |.|o1 : Q
│ │ │ │ -00030250: 756f 7469 656e 7452 696e 6720 2020 2020  uotientRing     
│ │ │ │ +00030230: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00030240: 3120 3a20 5175 6f74 6965 6e74 5269 6e67  1 : QuotientRing
│ │ │ │ +00030250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030290: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00030280: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00030290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000302a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000302b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000302c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000302d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000302e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2053  -------+.|i2 : S
│ │ │ │ -000302f0: 203d 206b 6b5b 612c 622c 635d 2020 2020   = kk[a,b,c]    
│ │ │ │ +000302d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +000302e0: 3220 3a20 5320 3d20 6b6b 5b61 2c62 2c63  2 : S = kk[a,b,c
│ │ │ │ +000302f0: 5d20 2020 2020 2020 2020 2020 2020 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 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00030320: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00030330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030340: 2020 2020 2020 2020 2020 2020 2020 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 207c 0a7c 6f32 203d 2053         |.|o2 = S
│ │ │ │ +00030370: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00030380: 3220 3d20 5320 2020 2020 2020 2020 2020  2 = S           
│ │ │ │  00030390: 2020 2020 2020 2020 2020 2020 2020 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 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000303c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000303d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000303e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000303f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030420: 2020 2020 2020 207c 0a7c 6f32 203a 2050         |.|o2 : P
│ │ │ │ -00030430: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020  olynomialRing   
│ │ │ │ +00030410: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00030420: 3220 3a20 506f 6c79 6e6f 6d69 616c 5269  2 : PolynomialRi
│ │ │ │ +00030430: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │  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 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00030460: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00030470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00030480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00030490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000304a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000304b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000304c0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2066  -------+.|i3 : f
│ │ │ │ -000304d0: 6620 3d20 6d61 7472 6978 2261 322c 6232  f = matrix"a2,b2
│ │ │ │ -000304e0: 2c63 3222 2020 2020 2020 2020 2020 2020  ,c2"            
│ │ │ │ +000304b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +000304c0: 3320 3a20 6666 203d 206d 6174 7269 7822  3 : ff = matrix"
│ │ │ │ +000304d0: 6132 2c62 322c 6332 2220 2020 2020 2020  a2,b2,c2"       
│ │ │ │ +000304e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000304f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030510: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00030500: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00030510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030560: 2020 2020 2020 207c 0a7c 6f33 203d 207c         |.|o3 = |
│ │ │ │ -00030570: 2061 3220 6232 2063 3220 7c20 2020 2020   a2 b2 c2 |     
│ │ │ │ +00030550: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00030560: 3320 3d20 7c20 6132 2062 3220 6332 207c  3 = | a2 b2 c2 |
│ │ │ │ +00030570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000305a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000305b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000305a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000305b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000305c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000305d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000305e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000305f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030600: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00030610: 2020 2020 2020 2031 2020 2020 2020 3320         1      3 
│ │ │ │ +000305f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00030600: 2020 2020 2020 2020 2020 2020 3120 2020              1   
│ │ │ │ +00030610: 2020 2033 2020 2020 2020 2020 2020 2020     3            
│ │ │ │  00030620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030650: 2020 2020 2020 207c 0a7c 6f33 203a 204d         |.|o3 : M
│ │ │ │ -00030660: 6174 7269 7820 5320 203c 2d2d 2053 2020  atrix S  <-- S  
│ │ │ │ +00030640: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00030650: 3320 3a20 4d61 7472 6978 2053 2020 3c2d  3 : Matrix S  <-
│ │ │ │ +00030660: 2d20 5320 2020 2020 2020 2020 2020 2020  - S             
│ │ │ │  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 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00030690: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +000306a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000306b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000306c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000306d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000306e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000306f0: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2052  -------+.|i4 : R
│ │ │ │ -00030700: 203d 2053 2f69 6465 616c 2066 6620 2020   = S/ideal ff   
│ │ │ │ +000306e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +000306f0: 3420 3a20 5220 3d20 532f 6964 6561 6c20  4 : R = S/ideal 
│ │ │ │ +00030700: 6666 2020 2020 2020 2020 2020 2020 2020  ff              
│ │ │ │  00030710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030740: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00030730: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00030740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030790: 2020 2020 2020 207c 0a7c 6f34 203d 2052         |.|o4 = R
│ │ │ │ +00030780: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00030790: 3420 3d20 5220 2020 2020 2020 2020 2020  4 = R           
│ │ │ │  000307a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000307d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000307e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000307f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030830: 2020 2020 2020 207c 0a7c 6f34 203a 2051         |.|o4 : Q
│ │ │ │ -00030840: 756f 7469 656e 7452 696e 6720 2020 2020  uotientRing     
│ │ │ │ +00030820: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00030830: 3420 3a20 5175 6f74 6965 6e74 5269 6e67  4 : QuotientRing
│ │ │ │ +00030840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030880: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00030870: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00030880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00030890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000308a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000308b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000308c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000308d0: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 204d  -------+.|i5 : M
│ │ │ │ -000308e0: 203d 2052 5e31 2f69 6465 616c 2261 2c62   = R^1/ideal"a,b
│ │ │ │ -000308f0: 6322 2020 2020 2020 2020 2020 2020 2020  c"              
│ │ │ │ +000308c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +000308d0: 3520 3a20 4d20 3d20 525e 312f 6964 6561  5 : M = R^1/idea
│ │ │ │ +000308e0: 6c22 612c 6263 2220 2020 2020 2020 2020  l"a,bc"         
│ │ │ │ +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 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00030910: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00030920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030970: 2020 2020 2020 207c 0a7c 6f35 203d 2063         |.|o5 = c
│ │ │ │ -00030980: 6f6b 6572 6e65 6c20 7c20 6120 6263 207c  okernel | a bc |
│ │ │ │ +00030960: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00030970: 3520 3d20 636f 6b65 726e 656c 207c 2061  5 = cokernel | a
│ │ │ │ +00030980: 2062 6320 7c20 2020 2020 2020 2020 2020   bc |           
│ │ │ │  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 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000309b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000309c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000309d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000309e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000309f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030a10: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00030a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030a30: 2020 2020 2020 3120 2020 2020 2020 2020        1         
│ │ │ │ +00030a00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00030a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030a20: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ +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 207c 0a7c 6f35 203a 2052         |.|o5 : R
│ │ │ │ -00030a70: 2d6d 6f64 756c 652c 2071 756f 7469 656e  -module, quotien
│ │ │ │ -00030a80: 7420 6f66 2052 2020 2020 2020 2020 2020  t of R          
│ │ │ │ +00030a50: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00030a60: 3520 3a20 522d 6d6f 6475 6c65 2c20 7175  5 : R-module, qu
│ │ │ │ +00030a70: 6f74 6965 6e74 206f 6620 5220 2020 2020  otient of R     
│ │ │ │ +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 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00030aa0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00030ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00030ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00030ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00030ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00030af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00030b00: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 206b  -------+.|i6 : k
│ │ │ │ -00030b10: 203d 2031 2020 2020 2020 2020 2020 2020   = 1            
│ │ │ │ +00030af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00030b00: 3620 3a20 6b20 3d20 3120 2020 2020 2020  6 : k = 1       
│ │ │ │ +00030b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030b50: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00030b40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00030b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 207c 0a7c 6f36 203d 2031         |.|o6 = 1
│ │ │ │ +00030b90: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00030ba0: 3620 3d20 3120 2020 2020 2020 2020 2020  6 = 1           
│ │ │ │  00030bb0: 2020 2020 2020 2020 2020 2020 2020 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 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00030be0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00030bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00030c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00030c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00030c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00030c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00030c40: 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a 206d  -------+.|i7 : m
│ │ │ │ -00030c50: 203d 206b 2b35 2020 2020 2020 2020 2020   = k+5          
│ │ │ │ +00030c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00030c40: 3720 3a20 6d20 3d20 6b2b 3520 2020 2020  7 : m = k+5     
│ │ │ │ +00030c50: 2020 2020 2020 2020 2020 2020 2020 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 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00030c80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00030c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030ce0: 2020 2020 2020 207c 0a7c 6f37 203d 2036         |.|o7 = 6
│ │ │ │ +00030cd0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00030ce0: 3720 3d20 3620 2020 2020 2020 2020 2020  7 = 6           
│ │ │ │  00030cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030d30: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00030d20: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00030d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00030d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00030d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00030d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00030d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00030d80: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 2046  -------+.|i8 : F
│ │ │ │ -00030d90: 203d 2066 7265 6552 6573 6f6c 7574 696f   = freeResolutio
│ │ │ │ -00030da0: 6e28 4d2c 204c 656e 6774 684c 696d 6974  n(M, LengthLimit
│ │ │ │ -00030db0: 203d 3e20 6d29 2020 2020 2020 2020 2020   => m)          
│ │ │ │ -00030dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030dd0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00030d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00030d80: 3820 3a20 4620 3d20 6672 6565 5265 736f  8 : F = freeReso
│ │ │ │ +00030d90: 6c75 7469 6f6e 284d 2c20 4c65 6e67 7468  lution(M, Length
│ │ │ │ +00030da0: 4c69 6d69 7420 3d3e 206d 2920 2020 2020  Limit => m)     
│ │ │ │ +00030db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030dc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00030dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030e20: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00030e30: 3120 2020 2020 2032 2020 2020 2020 3420  1      2      4 
│ │ │ │ -00030e40: 2020 2020 2037 2020 2020 2020 3131 2020       7      11  
│ │ │ │ -00030e50: 2020 2020 3136 2020 2020 2020 3232 2020      16      22  
│ │ │ │ -00030e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030e70: 2020 2020 2020 207c 0a7c 6f38 203d 2052         |.|o8 = R
│ │ │ │ -00030e80: 2020 3c2d 2d20 5220 203c 2d2d 2052 2020    <-- R  <-- R  
│ │ │ │ -00030e90: 3c2d 2d20 5220 203c 2d2d 2052 2020 203c  <-- R  <-- R   <
│ │ │ │ -00030ea0: 2d2d 2052 2020 203c 2d2d 2052 2020 2020  -- R   <-- R    
│ │ │ │ -00030eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030ec0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00030e10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00030e20: 2020 2020 2031 2020 2020 2020 3220 2020       1      2   
│ │ │ │ +00030e30: 2020 2034 2020 2020 2020 3720 2020 2020     4      7     
│ │ │ │ +00030e40: 2031 3120 2020 2020 2031 3620 2020 2020   11      16     
│ │ │ │ +00030e50: 2032 3220 2020 2020 2020 2020 2020 2020   22             
│ │ │ │ +00030e60: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00030e70: 3820 3d20 5220 203c 2d2d 2052 2020 3c2d  8 = R  <-- R  <-
│ │ │ │ +00030e80: 2d20 5220 203c 2d2d 2052 2020 3c2d 2d20  - R  <-- R  <-- 
│ │ │ │ +00030e90: 5220 2020 3c2d 2d20 5220 2020 3c2d 2d20  R   <-- R   <-- 
│ │ │ │ +00030ea0: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │ +00030eb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00030ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030f10: 2020 2020 2020 207c 0a7c 2020 2020 2030         |.|     0
│ │ │ │ -00030f20: 2020 2020 2020 3120 2020 2020 2032 2020        1      2  
│ │ │ │ -00030f30: 2020 2020 3320 2020 2020 2034 2020 2020      3      4    
│ │ │ │ -00030f40: 2020 2035 2020 2020 2020 2036 2020 2020     5       6    
│ │ │ │ -00030f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030f60: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00030f00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00030f10: 2020 2020 3020 2020 2020 2031 2020 2020      0      1    
│ │ │ │ +00030f20: 2020 3220 2020 2020 2033 2020 2020 2020    2      3      
│ │ │ │ +00030f30: 3420 2020 2020 2020 3520 2020 2020 2020  4       5       
│ │ │ │ +00030f40: 3620 2020 2020 2020 2020 2020 2020 2020  6               
│ │ │ │ +00030f50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00030f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030fb0: 2020 2020 2020 207c 0a7c 6f38 203a 2043         |.|o8 : C
│ │ │ │ -00030fc0: 6f6d 706c 6578 2020 2020 2020 2020 2020  omplex          
│ │ │ │ +00030fa0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00030fb0: 3820 3a20 436f 6d70 6c65 7820 2020 2020  8 : Complex     
│ │ │ │ +00030fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00030ff0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00031000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031050: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2073  -------+.|i9 : s
│ │ │ │ -00031060: 797a 7967 6965 7320 3d20 6170 706c 7928  yzygies = apply(
│ │ │ │ -00031070: 312e 2e6d 2c20 692d 3e63 6f6b 6572 2046  1..m, i->coker F
│ │ │ │ -00031080: 2e64 645f 6929 3b20 2020 2020 2020 2020  .dd_i);         
│ │ │ │ -00031090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000310a0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00031040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00031050: 3920 3a20 7379 7a79 6769 6573 203d 2061  9 : syzygies = a
│ │ │ │ +00031060: 7070 6c79 2831 2e2e 6d2c 2069 2d3e 636f  pply(1..m, i->co
│ │ │ │ +00031070: 6b65 7220 462e 6464 5f69 293b 2020 2020  ker F.dd_i);    
│ │ │ │ +00031080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031090: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +000310a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000310b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000310c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000310d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000310e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000310f0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20  -------+.|i10 : 
│ │ │ │ -00031100: 7431 203d 206d 616b 6554 2866 662c 462c  t1 = makeT(ff,F,
│ │ │ │ -00031110: 6b2b 3429 3b20 2020 2020 2020 2020 2020  k+4);           
│ │ │ │ +000310e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +000310f0: 3130 203a 2074 3120 3d20 6d61 6b65 5428  10 : t1 = makeT(
│ │ │ │ +00031100: 6666 2c46 2c6b 2b34 293b 2020 2020 2020  ff,F,k+4);      
│ │ │ │ +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                  
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│ │ │ │ +00031130: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00031140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -000311a0: 7432 203d 206d 616b 6554 2866 662c 462c  t2 = makeT(ff,F,
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│ │ │ │ +00031180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00031190: 3131 203a 2074 3220 3d20 6d61 6b65 5428  11 : t2 = makeT(
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│ │ │ │ +000311e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000311f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00031240: 5432 436f 6d70 6f6e 656e 7473 203d 2066  T2Components = f
│ │ │ │ -00031250: 6c61 7474 656e 2066 6f72 2069 2066 726f  latten for i fro
│ │ │ │ -00031260: 6d20 3020 746f 2031 206c 6973 7428 666f  m 0 to 1 list(fo
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│ │ │ │ +00031230: 3132 203a 2054 3243 6f6d 706f 6e65 6e74  12 : T2Component
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│ │ │ │ +00031260: 7374 2866 6f72 206a 2066 726f 6d20 692b  st(for j from i+
│ │ │ │ +00031270: 3120 746f 2032 206c 6973 7420 7c0a 7c2d  1 to 2 list |.|-
│ │ │ │ +00031280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  000312a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000312b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000312c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -000312e0: 6b2c 2046 5f28 6b2b 3429 2c20 7432 5f69  k, F_(k+4), t2_i
│ │ │ │ -000312f0: 2a74 315f 6a2d 7432 5f6a 2a74 315f 6929  *t1_j-t2_j*t1_i)
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│ │ │ │ +000312c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c6d  ------------|.|m
│ │ │ │ +000312d0: 6170 2846 5f6b 2c20 465f 286b 2b34 292c  ap(F_k, F_(k+4),
│ │ │ │ +000312e0: 2074 325f 692a 7431 5f6a 2d74 325f 6a2a   t2_i*t1_j-t2_j*
│ │ │ │ +000312f0: 7431 5f69 2929 3b20 2020 2020 2020 2020  t1_i));         
│ │ │ │ +00031300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031310: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00031320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031370: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3320 3a20  -------+.|i13 : 
│ │ │ │ -00031380: 6720 3d20 6d61 7028 7379 7a79 6769 6573  g = map(syzygies
│ │ │ │ -00031390: 5f6b 2c20 7379 7a79 6769 6573 5f28 6b2b  _k, syzygies_(k+
│ │ │ │ -000313a0: 3429 2c20 5432 436f 6d70 6f6e 656e 7473  4), T2Components
│ │ │ │ -000313b0: 5f32 2920 2020 2020 2020 2020 2020 2020  _2)             
│ │ │ │ -000313c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00031360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00031370: 3133 203a 2067 203d 206d 6170 2873 797a  13 : g = map(syz
│ │ │ │ +00031380: 7967 6965 735f 6b2c 2073 797a 7967 6965  ygies_k, syzygie
│ │ │ │ +00031390: 735f 286b 2b34 292c 2054 3243 6f6d 706f  s_(k+4), T2Compo
│ │ │ │ +000313a0: 6e65 6e74 735f 3229 2020 2020 2020 2020  nents_2)        
│ │ │ │ +000313b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
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│ │ │ │  000313d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00031400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031410: 2020 2020 2020 207c 0a7c 6f31 3320 3d20         |.|o13 = 
│ │ │ │ -00031420: 7b31 7d20 7c20 3020 3020 3020 3020 3020  {1} | 0 0 0 0 0 
│ │ │ │ -00031430: 2d63 2030 2030 2062 2030 2030 2030 2030  -c 0 0 b 0 0 0 0
│ │ │ │ -00031440: 2030 2030 2030 207c 2020 2020 2020 2020   0 0 0 |        
│ │ │ │ -00031450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031460: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00031470: 7b32 7d20 7c20 3020 3020 3020 3020 3020  {2} | 0 0 0 0 0 
│ │ │ │ -00031480: 3020 2030 2030 2030 2030 2030 2030 2030  0  0 0 0 0 0 0 0
│ │ │ │ -00031490: 2030 2030 2030 207c 2020 2020 2020 2020   0 0 0 |        
│ │ │ │ -000314a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000314b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00031400: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00031410: 3133 203d 207b 317d 207c 2030 2030 2030  13 = {1} | 0 0 0
│ │ │ │ +00031420: 2030 2030 202d 6320 3020 3020 6220 3020   0 0 -c 0 0 b 0 
│ │ │ │ +00031430: 3020 3020 3020 3020 3020 3020 7c20 2020  0 0 0 0 0 0 |   
│ │ │ │ +00031440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031450: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00031460: 2020 2020 207b 327d 207c 2030 2030 2030       {2} | 0 0 0
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│ │ │ │ +00031490: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00031500: 2020 2020 2020 207c 0a7c 6f31 3320 3a20         |.|o13 : 
│ │ │ │ -00031510: 4d61 7472 6978 2020 2020 2020 2020 2020  Matrix          
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│ │ │ │ +00031500: 3133 203a 204d 6174 7269 7820 2020 2020  13 : Matrix     
│ │ │ │ +00031510: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00031530: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00031550: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00031540: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00031550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000315a0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3420 3a20  -------+.|i14 : 
│ │ │ │ -000315b0: 6973 5374 6162 6c79 5472 6976 6961 6c20  isStablyTrivial 
│ │ │ │ -000315c0: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ +00031590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +000315a0: 3134 203a 2069 7353 7461 626c 7954 7269  14 : isStablyTri
│ │ │ │ +000315b0: 7669 616c 2067 2020 2020 2020 2020 2020  vial g          
│ │ │ │ +000315c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000315d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000315f0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000315e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000315f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00031640: 2020 2020 2020 207c 0a7c 6f31 3420 3d20         |.|o14 = 
│ │ │ │ -00031650: 7472 7565 2020 2020 2020 2020 2020 2020  true            
│ │ │ │ +00031630: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00031640: 3134 203d 2074 7275 6520 2020 2020 2020  14 = true       
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│ │ │ │  00031670: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00031690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000316a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000316b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000316c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000316d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000316e0: 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 616c  -------+..See al
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│ │ │ │ -00031720: 6d61 7020 6672 6f6d 2048 6f6d 284d 2c4e  map from Hom(M,N
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│ │ │ │ -00031740: 486f 6d20 6d6f 6475 6c65 0a0a 5761 7973  Hom module..Ways
│ │ │ │ -00031750: 2074 6f20 7573 6520 6973 5374 6162 6c79   to use isStably
│ │ │ │ -00031760: 5472 6976 6961 6c3a 0a3d 3d3d 3d3d 3d3d  Trivial:.=======
│ │ │ │ -00031770: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00031780: 3d3d 3d3d 3d0a 0a20 202a 2022 6973 5374  =====..  * "isSt
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│ │ │ │ -000317a0: 6978 2922 0a0a 466f 7220 7468 6520 7072  ix)"..For the pr
│ │ │ │ -000317b0: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d  ogrammer.=======
│ │ │ │ -000317c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865  ===========..The
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│ │ │ │ -00031830: 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d  Function,...----
│ │ │ │ +000316d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53  ------------+..S
│ │ │ │ +000316e0: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d  ee also.========
│ │ │ │ +000316f0: 0a0a 2020 2a20 2a6e 6f74 6520 7374 6162  ..  * *note stab
│ │ │ │ +00031700: 6c65 486f 6d3a 2073 7461 626c 6548 6f6d  leHom: stableHom
│ │ │ │ +00031710: 2c20 2d2d 206d 6170 2066 726f 6d20 486f  , -- map from Ho
│ │ │ │ +00031720: 6d28 4d2c 4e29 2074 6f20 7468 6520 7374  m(M,N) to the st
│ │ │ │ +00031730: 6162 6c65 2048 6f6d 206d 6f64 756c 650a  able Hom module.
│ │ │ │ +00031740: 0a57 6179 7320 746f 2075 7365 2069 7353  .Ways to use isS
│ │ │ │ +00031750: 7461 626c 7954 7269 7669 616c 3a0a 3d3d  tablyTrivial:.==
│ │ │ │ +00031760: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00031770: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ +00031780: 2269 7353 7461 626c 7954 7269 7669 616c  "isStablyTrivial
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│ │ │ │ +000317a0: 6865 2070 726f 6772 616d 6d65 720a 3d3d  he programmer.==
│ │ │ │ +000317b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +000317c0: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f  ..The object *no
│ │ │ │ +000317d0: 7465 2069 7353 7461 626c 7954 7269 7669  te isStablyTrivi
│ │ │ │ +000317e0: 616c 3a20 6973 5374 6162 6c79 5472 6976  al: isStablyTriv
│ │ │ │ +000317f0: 6961 6c2c 2069 7320 6120 2a6e 6f74 6520  ial, is a *note 
│ │ │ │ +00031800: 6d65 7468 6f64 2066 756e 6374 696f 6e3a  method function:
│ │ │ │ +00031810: 0a28 4d61 6361 756c 6179 3244 6f63 294d  .(Macaulay2Doc)M
│ │ │ │ +00031820: 6574 686f 6446 756e 6374 696f 6e2c 2e0a  ethodFunction,..
│ │ │ │ +00031830: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .---------------
│ │ │ │  00031840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865  -----------..The
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│ │ │ │ -00031940: 7574 696f 6e73 2e69 6e66 6f2c 204e 6f64  utions.info, Nod
│ │ │ │ -00031950: 653a 206b 6f73 7a75 6c45 7874 656e 7369  e: koszulExtensi
│ │ │ │ -00031960: 6f6e 2c20 4e65 7874 3a20 4c61 7965 7265  on, Next: Layere
│ │ │ │ -00031970: 642c 2050 7265 763a 2069 7353 7461 626c  d, Prev: isStabl
│ │ │ │ -00031980: 7954 7269 7669 616c 2c20 5570 3a20 546f  yTrivial, Up: To
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│ │ │ │ -000319a0: 6f6e 202d 2d20 6372 6561 7465 7320 7468  on -- creates th
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│ │ │ │ +000318e0: 6163 6b61 6765 732f 0a43 6f6d 706c 6574  ackages/.Complet
│ │ │ │ +000318f0: 6549 6e74 6572 7365 6374 696f 6e52 6573  eIntersectionRes
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│ │ │ │ +00031930: 5265 736f 6c75 7469 6f6e 732e 696e 666f  Resolutions.info
│ │ │ │ +00031940: 2c20 4e6f 6465 3a20 6b6f 737a 756c 4578  , Node: koszulEx
│ │ │ │ +00031950: 7465 6e73 696f 6e2c 204e 6578 743a 204c  tension, Next: L
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│ │ │ │ +00031970: 5374 6162 6c79 5472 6976 6961 6c2c 2055  StablyTrivial, U
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│ │ │ │ +00031990: 7465 6e73 696f 6e20 2d2d 2063 7265 6174  tension -- creat
│ │ │ │ +000319a0: 6573 2074 6865 204b 6f73 7a75 6c20 6578  es the Koszul ex
│ │ │ │ +000319b0: 7465 6e73 696f 6e20 636f 6d70 6c65 7820  tension complex 
│ │ │ │ +000319c0: 6f66 2061 206d 6170 0a2a 2a2a 2a2a 2a2a  of a map.*******
│ │ │ │ +000319d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000319e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000319f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00031a00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00031a10: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a  ****..  * Usage:
│ │ │ │ -00031a20: 200a 2020 2020 2020 2020 4d4d 203d 206b   .        MM = k
│ │ │ │ -00031a30: 6f73 7a75 6c45 7874 656e 7369 6f6e 2846  oszulExtension(F
│ │ │ │ -00031a40: 462c 4242 2c70 7369 312c 6666 290a 2020  F,BB,psi1,ff).  
│ │ │ │ -00031a50: 2a20 496e 7075 7473 3a0a 2020 2020 2020  * Inputs:.      
│ │ │ │ -00031a60: 2a20 4646 2c20 6120 2a6e 6f74 6520 636f  * FF, a *note co
│ │ │ │ -00031a70: 6d70 6c65 783a 2028 436f 6d70 6c65 7865  mplex: (Complexe
│ │ │ │ -00031a80: 7329 436f 6d70 6c65 782c 2c20 7265 736f  s)Complex,, reso
│ │ │ │ -00031a90: 6c75 7469 6f6e 206f 7665 7220 530a 2020  lution over S.  
│ │ │ │ -00031aa0: 2020 2020 2a20 4242 2c20 6120 2a6e 6f74      * BB, a *not
│ │ │ │ -00031ab0: 6520 636f 6d70 6c65 783a 2028 436f 6d70  e complex: (Comp
│ │ │ │ -00031ac0: 6c65 7865 7329 436f 6d70 6c65 782c 2c20  lexes)Complex,, 
│ │ │ │ -00031ad0: 7477 6f2d 7465 726d 2063 6f6d 706c 6578  two-term complex
│ │ │ │ -00031ae0: 2042 425f 312d 2d3e 4242 5f30 0a20 2020   BB_1-->BB_0.   
│ │ │ │ -00031af0: 2020 202a 2070 7369 312c 2061 202a 6e6f     * psi1, a *no
│ │ │ │ -00031b00: 7465 206d 6174 7269 783a 2028 4d61 6361  te matrix: (Maca
│ │ │ │ -00031b10: 756c 6179 3244 6f63 294d 6174 7269 782c  ulay2Doc)Matrix,
│ │ │ │ -00031b20: 2c20 6672 6f6d 2042 425f 3120 746f 2046  , from BB_1 to F
│ │ │ │ -00031b30: 465f 300a 2020 2020 2020 2a20 6666 2c20  F_0.      * ff, 
│ │ │ │ -00031b40: 6120 2a6e 6f74 6520 6d61 7472 6978 3a20  a *note matrix: 
│ │ │ │ -00031b50: 284d 6163 6175 6c61 7932 446f 6329 4d61  (Macaulay2Doc)Ma
│ │ │ │ -00031b60: 7472 6978 2c2c 2072 6567 756c 6172 2073  trix,, regular s
│ │ │ │ -00031b70: 6571 7565 6e63 650a 2020 2020 2020 2020  equence.        
│ │ │ │ -00031b80: 616e 6e69 6869 6c61 7469 6e67 2074 6865  annihilating the
│ │ │ │ -00031b90: 206d 6f64 756c 6520 7265 736f 6c76 6564   module resolved
│ │ │ │ -00031ba0: 2062 7920 4646 0a20 202a 204f 7574 7075   by FF.  * Outpu
│ │ │ │ -00031bb0: 7473 3a0a 2020 2020 2020 2a20 4d4d 2c20  ts:.      * MM, 
│ │ │ │ -00031bc0: 6120 2a6e 6f74 6520 636f 6d70 6c65 783a  a *note complex:
│ │ │ │ -00031bd0: 2028 436f 6d70 6c65 7865 7329 436f 6d70   (Complexes)Comp
│ │ │ │ -00031be0: 6c65 782c 2c20 7468 6520 6d61 7070 696e  lex,, the mappin
│ │ │ │ -00031bf0: 6720 636f 6e65 206f 6620 7468 650a 2020  g cone of the.  
│ │ │ │ -00031c00: 2020 2020 2020 696e 6475 6365 6420 6d61        induced ma
│ │ │ │ -00031c10: 7020 425b 2d31 5d5c 6f74 696d 6573 204b  p B[-1]\otimes K
│ │ │ │ -00031c20: 4b28 6666 2920 746f 2057 2065 7874 656e  K(ff) to W exten
│ │ │ │ -00031c30: 6469 6e67 2070 7369 0a0a 4465 7363 7269  ding psi..Descri
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│ │ │ │ -00031c50: 3d0a 0a49 6d70 6c65 6d65 6e74 7320 7468  =..Implements th
│ │ │ │ -00031c60: 6520 636f 6e73 7472 7563 7469 6f6e 2069  e construction i
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│ │ │ │ -00031cb0: 656e 6275 6420 616e 6420 5065 6576 612e  enbud and Peeva.
│ │ │ │ -00031cc0: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d  ..See also.=====
│ │ │ │ -00031cd0: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 206d  ===..  * *note m
│ │ │ │ -00031ce0: 616b 6546 696e 6974 6552 6573 6f6c 7574  akeFiniteResolut
│ │ │ │ -00031cf0: 696f 6e3a 206d 616b 6546 696e 6974 6552  ion: makeFiniteR
│ │ │ │ -00031d00: 6573 6f6c 7574 696f 6e2c 202d 2d20 6669  esolution, -- fi
│ │ │ │ -00031d10: 6e69 7465 2072 6573 6f6c 7574 696f 6e20  nite resolution 
│ │ │ │ -00031d20: 6f66 2061 0a20 2020 206d 6174 7269 7820  of a.    matrix 
│ │ │ │ -00031d30: 6661 6374 6f72 697a 6174 696f 6e20 6d6f  factorization mo
│ │ │ │ -00031d40: 6475 6c65 204d 0a0a 5761 7973 2074 6f20  dule M..Ways to 
│ │ │ │ -00031d50: 7573 6520 6b6f 737a 756c 4578 7465 6e73  use koszulExtens
│ │ │ │ -00031d60: 696f 6e3a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  ion:.===========
│ │ │ │ -00031d70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00031d80: 3d0a 0a20 202a 2022 6b6f 737a 756c 4578  =..  * "koszulEx
│ │ │ │ -00031d90: 7465 6e73 696f 6e28 436f 6d70 6c65 782c  tension(Complex,
│ │ │ │ -00031da0: 436f 6d70 6c65 782c 4d61 7472 6978 2c4d  Complex,Matrix,M
│ │ │ │ -00031db0: 6174 7269 7829 220a 0a46 6f72 2074 6865  atrix)"..For the
│ │ │ │ -00031dc0: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d   programmer.====
│ │ │ │ -00031dd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ -00031de0: 5468 6520 6f62 6a65 6374 202a 6e6f 7465  The object *note
│ │ │ │ -00031df0: 206b 6f73 7a75 6c45 7874 656e 7369 6f6e   koszulExtension
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│ │ │ │ -00031e20: 7468 6f64 2066 756e 6374 696f 6e3a 0a28  thod function:.(
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│ │ │ │ +00031a00: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055  *********..  * U
│ │ │ │ +00031a10: 7361 6765 3a20 0a20 2020 2020 2020 204d  sage: .        M
│ │ │ │ +00031a20: 4d20 3d20 6b6f 737a 756c 4578 7465 6e73  M = koszulExtens
│ │ │ │ +00031a30: 696f 6e28 4646 2c42 422c 7073 6931 2c66  ion(FF,BB,psi1,f
│ │ │ │ +00031a40: 6629 0a20 202a 2049 6e70 7574 733a 0a20  f).  * Inputs:. 
│ │ │ │ +00031a50: 2020 2020 202a 2046 462c 2061 202a 6e6f       * FF, a *no
│ │ │ │ +00031a60: 7465 2063 6f6d 706c 6578 3a20 2843 6f6d  te complex: (Com
│ │ │ │ +00031a70: 706c 6578 6573 2943 6f6d 706c 6578 2c2c  plexes)Complex,,
│ │ │ │ +00031a80: 2072 6573 6f6c 7574 696f 6e20 6f76 6572   resolution over
│ │ │ │ +00031a90: 2053 0a20 2020 2020 202a 2042 422c 2061   S.      * BB, a
│ │ │ │ +00031aa0: 202a 6e6f 7465 2063 6f6d 706c 6578 3a20   *note complex: 
│ │ │ │ +00031ab0: 2843 6f6d 706c 6578 6573 2943 6f6d 706c  (Complexes)Compl
│ │ │ │ +00031ac0: 6578 2c2c 2074 776f 2d74 6572 6d20 636f  ex,, two-term co
│ │ │ │ +00031ad0: 6d70 6c65 7820 4242 5f31 2d2d 3e42 425f  mplex BB_1-->BB_
│ │ │ │ +00031ae0: 300a 2020 2020 2020 2a20 7073 6931 2c20  0.      * psi1, 
│ │ │ │ +00031af0: 6120 2a6e 6f74 6520 6d61 7472 6978 3a20  a *note matrix: 
│ │ │ │ +00031b00: 284d 6163 6175 6c61 7932 446f 6329 4d61  (Macaulay2Doc)Ma
│ │ │ │ +00031b10: 7472 6978 2c2c 2066 726f 6d20 4242 5f31  trix,, from BB_1
│ │ │ │ +00031b20: 2074 6f20 4646 5f30 0a20 2020 2020 202a   to FF_0.      *
│ │ │ │ +00031b30: 2066 662c 2061 202a 6e6f 7465 206d 6174   ff, a *note mat
│ │ │ │ +00031b40: 7269 783a 2028 4d61 6361 756c 6179 3244  rix: (Macaulay2D
│ │ │ │ +00031b50: 6f63 294d 6174 7269 782c 2c20 7265 6775  oc)Matrix,, regu
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│ │ │ │ +00031b70: 2020 2020 2061 6e6e 6968 696c 6174 696e       annihilatin
│ │ │ │ +00031b80: 6720 7468 6520 6d6f 6475 6c65 2072 6573  g the module res
│ │ │ │ +00031b90: 6f6c 7665 6420 6279 2046 460a 2020 2a20  olved by FF.  * 
│ │ │ │ +00031ba0: 4f75 7470 7574 733a 0a20 2020 2020 202a  Outputs:.      *
│ │ │ │ +00031bb0: 204d 4d2c 2061 202a 6e6f 7465 2063 6f6d   MM, a *note com
│ │ │ │ +00031bc0: 706c 6578 3a20 2843 6f6d 706c 6578 6573  plex: (Complexes
│ │ │ │ +00031bd0: 2943 6f6d 706c 6578 2c2c 2074 6865 206d  )Complex,, the m
│ │ │ │ +00031be0: 6170 7069 6e67 2063 6f6e 6520 6f66 2074  apping cone of t
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│ │ │ │ +00031c20: 6578 7465 6e64 696e 6720 7073 690a 0a44  extending psi..D
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│ │ │ │ +00031c40: 3d3d 3d3d 3d3d 0a0a 496d 706c 656d 656e  ======..Implemen
│ │ │ │ +00031c50: 7473 2074 6865 2063 6f6e 7374 7275 6374  ts the construct
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│ │ │ │ +00031c70: 2022 4d61 7472 6978 2046 6163 746f 7269   "Matrix Factori
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│ │ │ │ +00031c90: 720a 436f 6469 6d65 6e73 696f 6e22 2062  r.Codimension" b
│ │ │ │ +00031ca0: 7920 4569 7365 6e62 7564 2061 6e64 2050  y Eisenbud and P
│ │ │ │ +00031cb0: 6565 7661 2e0a 0a53 6565 2061 6c73 6f0a  eeva...See also.
│ │ │ │ +00031cc0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ +00031cd0: 6f74 6520 6d61 6b65 4669 6e69 7465 5265  ote makeFiniteRe
│ │ │ │ +00031ce0: 736f 6c75 7469 6f6e 3a20 6d61 6b65 4669  solution: makeFi
│ │ │ │ +00031cf0: 6e69 7465 5265 736f 6c75 7469 6f6e 2c20  niteResolution, 
│ │ │ │ +00031d00: 2d2d 2066 696e 6974 6520 7265 736f 6c75  -- finite resolu
│ │ │ │ +00031d10: 7469 6f6e 206f 6620 610a 2020 2020 6d61  tion of a.    ma
│ │ │ │ +00031d20: 7472 6978 2066 6163 746f 7269 7a61 7469  trix factorizati
│ │ │ │ +00031d30: 6f6e 206d 6f64 756c 6520 4d0a 0a57 6179  on module M..Way
│ │ │ │ +00031d40: 7320 746f 2075 7365 206b 6f73 7a75 6c45  s to use koszulE
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│ │ │ │ +00031d60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00031d70: 3d3d 3d3d 3d3d 0a0a 2020 2a20 226b 6f73  ======..  * "kos
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│ │ │ │ +00031db0: 7220 7468 6520 7072 6f67 7261 6d6d 6572  r the programmer
│ │ │ │ +00031dc0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ +00031dd0: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420  ===..The object 
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│ │ │ │ +00031e30: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e  c)MethodFunction
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│ │ │ │  00031e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a  --------------..
│ │ │ │ -00031ea0: 5468 6520 736f 7572 6365 206f 6620 7468  The source of th
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│ │ │ │ -00031ee0: 6c61 7932 2d31 2e32 352e 3131 2b64 732f  lay2-1.25.11+ds/
│ │ │ │ -00031ef0: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163  M2/Macaulay2/pac
│ │ │ │ -00031f00: 6b61 6765 732f 0a43 6f6d 706c 6574 6549  kages/.CompleteI
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│ │ │ │ -00031f30: 2e0a 1f0a 4669 6c65 3a20 436f 6d70 6c65  ....File: Comple
│ │ │ │ -00031f40: 7465 496e 7465 7273 6563 7469 6f6e 5265  teIntersectionRe
│ │ │ │ -00031f50: 736f 6c75 7469 6f6e 732e 696e 666f 2c20  solutions.info, 
│ │ │ │ -00031f60: 4e6f 6465 3a20 4c61 7965 7265 642c 204e  Node: Layered, N
│ │ │ │ -00031f70: 6578 743a 206c 6179 6572 6564 5265 736f  ext: layeredReso
│ │ │ │ -00031f80: 6c75 7469 6f6e 2c20 5072 6576 3a20 6b6f  lution, Prev: ko
│ │ │ │ -00031f90: 737a 756c 4578 7465 6e73 696f 6e2c 2055  szulExtension, U
│ │ │ │ -00031fa0: 703a 2054 6f70 0a0a 4c61 7965 7265 6420  p: Top..Layered 
│ │ │ │ -00031fb0: 2d2d 204f 7074 696f 6e20 666f 7220 6d61  -- Option for ma
│ │ │ │ -00031fc0: 7472 6978 4661 6374 6f72 697a 6174 696f  trixFactorizatio
│ │ │ │ -00031fd0: 6e0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  n.**************
│ │ │ │ +00031e90: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520  ---..The source 
│ │ │ │ +00031ea0: 6f66 2074 6869 7320 646f 6375 6d65 6e74  of this document
│ │ │ │ +00031eb0: 2069 7320 696e 0a2f 6275 696c 642f 7265   is in./build/re
│ │ │ │ +00031ec0: 7072 6f64 7563 6962 6c65 2d70 6174 682f  producible-path/
│ │ │ │ +00031ed0: 6d61 6361 756c 6179 322d 312e 3235 2e31  macaulay2-1.25.1
│ │ │ │ +00031ee0: 312b 6473 2f4d 322f 4d61 6361 756c 6179  1+ds/M2/Macaulay
│ │ │ │ +00031ef0: 322f 7061 636b 6167 6573 2f0a 436f 6d70  2/packages/.Comp
│ │ │ │ +00031f00: 6c65 7465 496e 7465 7273 6563 7469 6f6e  leteIntersection
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│ │ │ │ -00032a60: 6475 6c65 7320 425f 730a 696e 2074 6865  dules B_s.in the
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│ │ │ │ -00032a80: 6f75 7470 7574 2e0a 0a48 6572 6520 6973  output...Here is
│ │ │ │ -00032a90: 2061 6e20 6578 616d 706c 6520 636f 6d70   an example comp
│ │ │ │ -00032aa0: 7574 696e 6720 3520 7465 726d 7320 6f66  uting 5 terms of
│ │ │ │ -00032ab0: 2061 6e20 696e 6669 6e69 7465 2072 6573   an infinite res
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│ │ │ │ +000325a0: 2020 2020 2028 4646 2c20 6175 6729 203d       (FF, aug) =
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│ │ │ │ +000325f0: 662c 4d2c 6c65 6e29 0a20 202a 2049 6e70  f,M,len).  * Inp
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│ │ │ │ +00032630: 6174 7269 782c 2c20 3120 7820 6320 6d61  atrix,, 1 x c ma
│ │ │ │ +00032640: 7472 6978 2077 686f 7365 2065 6e74 7269  trix whose entri
│ │ │ │ +00032650: 6573 0a20 2020 2020 2020 2061 7265 2061  es.        are a
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│ │ │ │ +00032680: 6569 6e20 7269 6e67 2053 0a20 2020 2020  ein ring S.     
│ │ │ │ +00032690: 202a 204d 2c20 6120 2a6e 6f74 6520 6d6f   * M, a *note mo
│ │ │ │ +000326a0: 6475 6c65 3a20 284d 6163 6175 6c61 7932  dule: (Macaulay2
│ │ │ │ +000326b0: 446f 6329 4d6f 6475 6c65 2c2c 204d 434d  Doc)Module,, MCM
│ │ │ │ +000326c0: 206d 6f64 756c 6520 6f76 6572 2052 2c0a   module over R,.
│ │ │ │ +000326d0: 2020 2020 2020 2020 7265 7072 6573 656e          represen
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│ │ │ │ +00032700: 6361 7365 2061 6e64 2061 7320 616e 2052  case and as an R
│ │ │ │ +00032710: 2d6d 6f64 756c 6520 696e 2074 6865 0a20  -module in the. 
│ │ │ │ +00032720: 2020 2020 2020 2073 6563 6f6e 640a 2020         second.  
│ │ │ │ +00032730: 2020 2020 2a20 6c65 6e2c 2061 6e20 2a6e      * len, an *n
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│ │ │ │ +00032760: 6c65 6e67 7468 206f 6620 7468 6520 7365  length of the se
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│ │ │ │ +00032790: 746f 2062 6520 636f 6d70 7574 6564 206f  to be computed o
│ │ │ │ +000327a0: 7665 7220 522c 2069 6e20 7468 6520 7365  ver R, in the se
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│ │ │ │ +000327c0: 6e6f 7465 204f 7074 696f 6e61 6c20 696e  note Optional in
│ │ │ │ +000327d0: 7075 7473 3a20 284d 6163 6175 6c61 7932  puts: (Macaulay2
│ │ │ │ +000327e0: 446f 6329 7573 696e 6720 6675 6e63 7469  Doc)using functi
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│ │ │ │ +00032800: 6c20 696e 7075 7473 2c3a 0a20 2020 2020  l inputs,:.     
│ │ │ │ +00032810: 202a 2043 6865 636b 203d 3e20 2e2e 2e2c   * Check => ...,
│ │ │ │ +00032820: 2064 6566 6175 6c74 2076 616c 7565 2066   default value f
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│ │ │ │ +00032840: 626f 7365 203d 3e20 2e2e 2e2c 2064 6566  bose => ..., def
│ │ │ │ +00032850: 6175 6c74 2076 616c 7565 2066 616c 7365  ault value false
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│ │ │ │ +00032870: 2020 2020 2a20 4646 2c20 6120 2a6e 6f74      * FF, a *not
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│ │ │ │ +000328a0: 7265 736f 6c75 7469 6f6e 206f 6620 4d20  resolution of M 
│ │ │ │ +000328b0: 6f76 6572 2053 2069 6e20 7468 650a 2020  over S in the.  
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│ │ │ │ +00032980: 6163 6175 6c61 7920 6d6f 6475 6c65 7322  acaulay modules"
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│ │ │ │ +000329a0: 2050 6565 7661 2e20 5468 6579 2061 7265   Peeva. They are
│ │ │ │ +000329b0: 2062 6f74 6820 6d69 6e69 6d61 6c20 7768   both minimal wh
│ │ │ │ +000329c0: 656e 204d 0a69 7320 6120 7375 6666 6963  en M.is a suffic
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│ │ │ │ +00032a00: 5665 7262 6f73 653d 3e74 7275 6520 6973  Verbose=>true is
│ │ │ │ +00032a10: 0a73 6574 2c20 7468 656e 2028 696e 2074  .set, then (in t
│ │ │ │ +00032a20: 6865 2063 6173 6520 6f66 2074 6865 2072  he case of the r
│ │ │ │ +00032a30: 6573 6f6c 7574 696f 6e20 6f76 6572 2053  esolution over S
│ │ │ │ +00032a40: 2920 7468 6520 7261 6e6b 7320 6f66 2074  ) the ranks of t
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│ │ │ │ +00032a70: 2061 7265 206f 7574 7075 742e 0a0a 4865   are output...He
│ │ │ │ +00032a80: 7265 2069 7320 616e 2065 7861 6d70 6c65  re is an example
│ │ │ │ +00032a90: 2063 6f6d 7075 7469 6e67 2035 2074 6572   computing 5 ter
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│ │ │ │ +00032ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00032b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00032b10: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20  --------+.|i1 : 
│ │ │ │ -00032b20: 5320 3d20 5a5a 2f31 3031 5b61 2c62 2c63  S = ZZ/101[a,b,c
│ │ │ │ -00032b30: 5d20 2020 2020 2020 2020 2020 2020 2020  ]               
│ │ │ │ +00032b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00032b10: 6931 203a 2053 203d 205a 5a2f 3130 315b  i1 : S = ZZ/101[
│ │ │ │ +00032b20: 612c 622c 635d 2020 2020 2020 2020 2020  a,b,c]          
│ │ │ │ +00032b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032b60: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00032b50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00032b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032bb0: 2020 2020 2020 2020 7c0a 7c6f 3120 3d20          |.|o1 = 
│ │ │ │ -00032bc0: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │ +00032ba0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00032bb0: 6f31 203d 2053 2020 2020 2020 2020 2020  o1 = S          
│ │ │ │ +00032bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032c00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00032bf0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00032c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032c50: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20          |.|o1 : 
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│ │ │ │ +00032c50: 6f31 203a 2050 6f6c 796e 6f6d 6961 6c52  o1 : PolynomialR
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│ │ │ │  00032c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032ca0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00032c90: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00032ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00032ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00032cf0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20  --------+.|i2 : 
│ │ │ │ -00032d00: 6666 203d 206d 6174 7269 7822 6133 2c20  ff = matrix"a3, 
│ │ │ │ -00032d10: 6233 2c20 6333 2220 2020 2020 2020 2020  b3, c3"         
│ │ │ │ +00032ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00032cf0: 6932 203a 2066 6620 3d20 6d61 7472 6978  i2 : ff = matrix
│ │ │ │ +00032d00: 2261 332c 2062 332c 2063 3322 2020 2020  "a3, b3, c3"    
│ │ │ │ +00032d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032d40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00032d30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00032d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032d90: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20          |.|o2 = 
│ │ │ │ -00032da0: 7c20 6133 2062 3320 6333 207c 2020 2020  | a3 b3 c3 |    
│ │ │ │ +00032d80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00032d90: 6f32 203d 207c 2061 3320 6233 2063 3320  o2 = | a3 b3 c3 
│ │ │ │ +00032da0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00032db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032de0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00032dd0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00032de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032e30: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00032e40: 2020 2020 2020 2020 3120 2020 2020 2033          1      3
│ │ │ │ +00032e20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00032e30: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │ +00032e40: 2020 2020 3320 2020 2020 2020 2020 2020      3           
│ │ │ │  00032e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032e80: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20          |.|o2 : 
│ │ │ │ -00032e90: 4d61 7472 6978 2053 2020 3c2d 2d20 5320  Matrix S  <-- S 
│ │ │ │ +00032e70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00032e80: 6f32 203a 204d 6174 7269 7820 5320 203c  o2 : Matrix S  <
│ │ │ │ +00032e90: 2d2d 2053 2020 2020 2020 2020 2020 2020  -- S            
│ │ │ │  00032ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032ed0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00032ec0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00032ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00032f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00032f20: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │ -00032f30: 5220 3d20 532f 6964 6561 6c20 6666 2020  R = S/ideal ff  
│ │ │ │ +00032f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00032f20: 6933 203a 2052 203d 2053 2f69 6465 616c  i3 : R = S/ideal
│ │ │ │ +00032f30: 2066 6620 2020 2020 2020 2020 2020 2020   ff             
│ │ │ │  00032f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032f70: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00032f60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00032f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032fc0: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20          |.|o3 = 
│ │ │ │ -00032fd0: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │ +00032fb0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00032fc0: 6f33 203d 2052 2020 2020 2020 2020 2020  o3 = R          
│ │ │ │ +00032fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033010: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00033000: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033060: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20          |.|o3 : 
│ │ │ │ -00033070: 5175 6f74 6965 6e74 5269 6e67 2020 2020  QuotientRing    
│ │ │ │ +00033050: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033060: 6f33 203a 2051 756f 7469 656e 7452 696e  o3 : QuotientRin
│ │ │ │ +00033070: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │  00033080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000330a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000330b0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +000330a0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +000330b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000330c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000330d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000330e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000330f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033100: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20  --------+.|i4 : 
│ │ │ │ -00033110: 4d20 3d20 7379 7a79 6779 4d6f 6475 6c65  M = syzygyModule
│ │ │ │ -00033120: 2832 2c63 6f6b 6572 2076 6172 7320 5229  (2,coker vars R)
│ │ │ │ +000330f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00033100: 6934 203a 204d 203d 2073 797a 7967 794d  i4 : M = syzygyM
│ │ │ │ +00033110: 6f64 756c 6528 322c 636f 6b65 7220 7661  odule(2,coker va
│ │ │ │ +00033120: 7273 2052 2920 2020 2020 2020 2020 2020  rs R)           
│ │ │ │  00033130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033150: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00033140: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000331a0: 2020 2020 2020 2020 7c0a 7c6f 3420 3d20          |.|o4 = 
│ │ │ │ -000331b0: 636f 6b65 726e 656c 207b 327d 207c 2061  cokernel {2} | a
│ │ │ │ -000331c0: 2020 3020 2d63 3220 3020 2020 6232 2030    0 -c2 0   b2 0
│ │ │ │ -000331d0: 2030 2020 2030 2020 3020 2030 207c 2020   0   0  0  0 |  
│ │ │ │ -000331e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000331f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00033200: 2020 2020 2020 2020 207b 327d 207c 202d           {2} | -
│ │ │ │ -00033210: 6220 3020 3020 2020 2d63 3220 3020 2030  b 0 0   -c2 0  0
│ │ │ │ -00033220: 2030 2020 2061 3220 3020 2030 207c 2020   0   a2 0  0 |  
│ │ │ │ -00033230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033240: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00033250: 2020 2020 2020 2020 207b 327d 207c 2063           {2} | c
│ │ │ │ -00033260: 2020 3020 3020 2020 3020 2020 3020 2030    0 0   0   0  0
│ │ │ │ -00033270: 202d 6232 2030 2020 6132 2030 207c 2020   -b2 0  a2 0 |  
│ │ │ │ -00033280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033290: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000332a0: 2020 2020 2020 2020 207b 337d 207c 2030           {3} | 0
│ │ │ │ -000332b0: 2020 6320 6220 2020 6120 2020 3020 2030    c b   a   0  0
│ │ │ │ -000332c0: 2030 2020 2030 2020 3020 2030 207c 2020   0   0  0  0 |  
│ │ │ │ -000332d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000332e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000332f0: 2020 2020 2020 2020 207b 337d 207c 2030           {3} | 0
│ │ │ │ -00033300: 2020 3020 3020 2020 3020 2020 6320 2062    0 0   0   c  b
│ │ │ │ -00033310: 2061 2020 2030 2020 3020 2030 207c 2020   a   0  0  0 |  
│ │ │ │ -00033320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033330: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00033340: 2020 2020 2020 2020 207b 337d 207c 2030           {3} | 0
│ │ │ │ -00033350: 2020 3020 3020 2020 3020 2020 3020 2030    0 0   0   0  0
│ │ │ │ -00033360: 2030 2020 2063 2020 6220 2061 207c 2020   0   c  b  a |  
│ │ │ │ -00033370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033380: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00033190: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000331a0: 6f34 203d 2063 6f6b 6572 6e65 6c20 7b32  o4 = cokernel {2
│ │ │ │ +000331b0: 7d20 7c20 6120 2030 202d 6332 2030 2020  } | a  0 -c2 0  
│ │ │ │ +000331c0: 2062 3220 3020 3020 2020 3020 2030 2020   b2 0 0   0  0  
│ │ │ │ +000331d0: 3020 7c20 2020 2020 2020 2020 2020 2020  0 |             
│ │ │ │ +000331e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000331f0: 2020 2020 2020 2020 2020 2020 2020 7b32                {2
│ │ │ │ +00033200: 7d20 7c20 2d62 2030 2030 2020 202d 6332  } | -b 0 0   -c2
│ │ │ │ +00033210: 2030 2020 3020 3020 2020 6132 2030 2020   0  0 0   a2 0  
│ │ │ │ +00033220: 3020 7c20 2020 2020 2020 2020 2020 2020  0 |             
│ │ │ │ +00033230: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033240: 2020 2020 2020 2020 2020 2020 2020 7b32                {2
│ │ │ │ +00033250: 7d20 7c20 6320 2030 2030 2020 2030 2020  } | c  0 0   0  
│ │ │ │ +00033260: 2030 2020 3020 2d62 3220 3020 2061 3220   0  0 -b2 0  a2 
│ │ │ │ +00033270: 3020 7c20 2020 2020 2020 2020 2020 2020  0 |             
│ │ │ │ +00033280: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033290: 2020 2020 2020 2020 2020 2020 2020 7b33                {3
│ │ │ │ +000332a0: 7d20 7c20 3020 2063 2062 2020 2061 2020  } | 0  c b   a  
│ │ │ │ +000332b0: 2030 2020 3020 3020 2020 3020 2030 2020   0  0 0   0  0  
│ │ │ │ +000332c0: 3020 7c20 2020 2020 2020 2020 2020 2020  0 |             
│ │ │ │ +000332d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000332e0: 2020 2020 2020 2020 2020 2020 2020 7b33                {3
│ │ │ │ +000332f0: 7d20 7c20 3020 2030 2030 2020 2030 2020  } | 0  0 0   0  
│ │ │ │ +00033300: 2063 2020 6220 6120 2020 3020 2030 2020   c  b a   0  0  
│ │ │ │ +00033310: 3020 7c20 2020 2020 2020 2020 2020 2020  0 |             
│ │ │ │ +00033320: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033330: 2020 2020 2020 2020 2020 2020 2020 7b33                {3
│ │ │ │ +00033340: 7d20 7c20 3020 2030 2030 2020 2030 2020  } | 0  0 0   0  
│ │ │ │ +00033350: 2030 2020 3020 3020 2020 6320 2062 2020   0  0 0   c  b  
│ │ │ │ +00033360: 6120 7c20 2020 2020 2020 2020 2020 2020  a |             
│ │ │ │ +00033370: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000333a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000333b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000333c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000333d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000333e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000333f0: 2020 2020 2020 2036 2020 2020 2020 2020         6        
│ │ │ │ +000333c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000333d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000333e0: 2020 2020 2020 2020 2020 2020 3620 2020              6   
│ │ │ │ +000333f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033420: 2020 2020 2020 2020 7c0a 7c6f 3420 3a20          |.|o4 : 
│ │ │ │ -00033430: 522d 6d6f 6475 6c65 2c20 7175 6f74 6965  R-module, quotie
│ │ │ │ -00033440: 6e74 206f 6620 5220 2020 2020 2020 2020  nt of R         
│ │ │ │ +00033410: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033420: 6f34 203a 2052 2d6d 6f64 756c 652c 2071  o4 : R-module, q
│ │ │ │ +00033430: 756f 7469 656e 7420 6f66 2052 2020 2020  uotient of R    
│ │ │ │ +00033440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033470: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00033460: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00033470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00033480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00033490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000334a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000334b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000334c0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20  --------+.|i5 : 
│ │ │ │ -000334d0: 2846 462c 2061 7567 2920 3d20 6c61 7965  (FF, aug) = laye
│ │ │ │ -000334e0: 7265 6452 6573 6f6c 7574 696f 6e28 6666  redResolution(ff
│ │ │ │ -000334f0: 2c4d 2c35 2920 2020 2020 2020 2020 2020  ,M,5)           
│ │ │ │ -00033500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033510: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00033520: 2020 3020 3120 3220 3320 3420 3520 2020    0 1 2 3 4 5   
│ │ │ │ +000334b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +000334c0: 6935 203a 2028 4646 2c20 6175 6729 203d  i5 : (FF, aug) =
│ │ │ │ +000334d0: 206c 6179 6572 6564 5265 736f 6c75 7469   layeredResoluti
│ │ │ │ +000334e0: 6f6e 2866 662c 4d2c 3529 2020 2020 2020  on(ff,M,5)      
│ │ │ │ +000334f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033500: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033510: 2020 2020 2020 2030 2031 2032 2033 2034         0 1 2 3 4
│ │ │ │ +00033520: 2035 2020 2020 2020 2020 2020 2020 2020   5              
│ │ │ │  00033530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033560: 2020 2020 2020 2020 7c0a 7c74 6f74 616c          |.|total
│ │ │ │ -00033570: 3a20 3420 3420 3420 3420 3420 3420 2020  : 4 4 4 4 4 4   
│ │ │ │ +00033550: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033560: 746f 7461 6c3a 2034 2034 2034 2034 2034  total: 4 4 4 4 4
│ │ │ │ +00033570: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │  00033580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000335a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000335b0: 2020 2020 2020 2020 7c0a 7c20 2020 2032          |.|    2
│ │ │ │ -000335c0: 3a20 3320 3120 2e20 2e20 2e20 2e20 2020  : 3 1 . . . .   
│ │ │ │ +000335a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000335b0: 2020 2020 323a 2033 2031 202e 202e 202e      2: 3 1 . . .
│ │ │ │ +000335c0: 202e 2020 2020 2020 2020 2020 2020 2020   .              
│ │ │ │  000335d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000335e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000335f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033600: 2020 2020 2020 2020 7c0a 7c20 2020 2033          |.|    3
│ │ │ │ -00033610: 3a20 3120 3320 3320 3120 2e20 2e20 2020  : 1 3 3 1 . .   
│ │ │ │ +000335f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033600: 2020 2020 333a 2031 2033 2033 2031 202e      3: 1 3 3 1 .
│ │ │ │ +00033610: 202e 2020 2020 2020 2020 2020 2020 2020   .              
│ │ │ │  00033620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033650: 2020 2020 2020 2020 7c0a 7c20 2020 2034          |.|    4
│ │ │ │ -00033660: 3a20 2e20 2e20 3120 3320 3320 3120 2020  : . . 1 3 3 1   
│ │ │ │ +00033640: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033650: 2020 2020 343a 202e 202e 2031 2033 2033      4: . . 1 3 3
│ │ │ │ +00033660: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │  00033670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000336a0: 2020 2020 2020 2020 7c0a 7c20 2020 2035          |.|    5
│ │ │ │ -000336b0: 3a20 2e20 2e20 2e20 2e20 3120 3320 2020  : . . . . 1 3   
│ │ │ │ +00033690: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000336a0: 2020 2020 353a 202e 202e 202e 202e 2031      5: . . . . 1
│ │ │ │ +000336b0: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │  000336c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000336d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000336e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000336f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00033700: 2020 3020 3120 3220 2033 2020 3420 2035    0 1 2  3  4  5
│ │ │ │ +000336e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000336f0: 2020 2020 2020 2030 2031 2032 2020 3320         0 1 2  3 
│ │ │ │ +00033700: 2034 2020 3520 2020 2020 2020 2020 2020   4  5           
│ │ │ │  00033710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033740: 2020 2020 2020 2020 7c0a 7c74 6f74 616c          |.|total
│ │ │ │ -00033750: 3a20 3520 3720 3920 3131 2031 3320 3135  : 5 7 9 11 13 15
│ │ │ │ +00033730: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033740: 746f 7461 6c3a 2035 2037 2039 2031 3120  total: 5 7 9 11 
│ │ │ │ +00033750: 3133 2031 3520 2020 2020 2020 2020 2020  13 15           
│ │ │ │  00033760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033790: 2020 2020 2020 2020 7c0a 7c20 2020 2032          |.|    2
│ │ │ │ -000337a0: 3a20 3320 3120 2e20 202e 2020 2e20 202e  : 3 1 .  .  .  .
│ │ │ │ +00033780: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033790: 2020 2020 323a 2033 2031 202e 2020 2e20      2: 3 1 .  . 
│ │ │ │ +000337a0: 202e 2020 2e20 2020 2020 2020 2020 2020   .  .           
│ │ │ │  000337b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000337c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000337d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000337e0: 2020 2020 2020 2020 7c0a 7c20 2020 2033          |.|    3
│ │ │ │ -000337f0: 3a20 3220 3620 3620 2032 2020 2e20 202e  : 2 6 6  2  .  .
│ │ │ │ +000337d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000337e0: 2020 2020 333a 2032 2036 2036 2020 3220      3: 2 6 6  2 
│ │ │ │ +000337f0: 202e 2020 2e20 2020 2020 2020 2020 2020   .  .           
│ │ │ │  00033800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033830: 2020 2020 2020 2020 7c0a 7c20 2020 2034          |.|    4
│ │ │ │ -00033840: 3a20 2e20 2e20 3320 2039 2020 3920 2033  : . . 3  9  9  3
│ │ │ │ +00033820: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033830: 2020 2020 343a 202e 202e 2033 2020 3920      4: . . 3  9 
│ │ │ │ +00033840: 2039 2020 3320 2020 2020 2020 2020 2020   9  3           
│ │ │ │  00033850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033880: 2020 2020 2020 2020 7c0a 7c20 2020 2035          |.|    5
│ │ │ │ -00033890: 3a20 2e20 2e20 2e20 202e 2020 3420 3132  : . . .  .  4 12
│ │ │ │ +00033870: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033880: 2020 2020 353a 202e 202e 202e 2020 2e20      5: . . .  . 
│ │ │ │ +00033890: 2034 2031 3220 2020 2020 2020 2020 2020   4 12           
│ │ │ │  000338a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000338b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000338c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000338d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000338e0: 2020 3020 2031 2020 3220 2033 2020 3420    0  1  2  3  4 
│ │ │ │ -000338f0: 2035 2020 2020 2020 2020 2020 2020 2020   5              
│ │ │ │ +000338c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │ +000338e0: 3320 2034 2020 3520 2020 2020 2020 2020  3  4  5         
│ │ │ │ +000338f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033920: 2020 2020 2020 2020 7c0a 7c74 6f74 616c          |.|total
│ │ │ │ -00033930: 3a20 3620 3130 2031 3520 3231 2032 3820  : 6 10 15 21 28 
│ │ │ │ -00033940: 3336 2020 2020 2020 2020 2020 2020 2020  36              
│ │ │ │ +00033910: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033920: 746f 7461 6c3a 2036 2031 3020 3135 2032  total: 6 10 15 2
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│ │ │ │ +00033940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033970: 2020 2020 2020 2020 7c0a 7c20 2020 2032          |.|    2
│ │ │ │ -00033980: 3a20 3320 2031 2020 2e20 202e 2020 2e20  : 3  1  .  .  . 
│ │ │ │ -00033990: 202e 2020 2020 2020 2020 2020 2020 2020   .              
│ │ │ │ +00033960: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033970: 2020 2020 323a 2033 2020 3120 202e 2020      2: 3  1  .  
│ │ │ │ +00033980: 2e20 202e 2020 2e20 2020 2020 2020 2020  .  .  .         
│ │ │ │ +00033990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000339a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000339b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000339c0: 2020 2020 2020 2020 7c0a 7c20 2020 2033          |.|    3
│ │ │ │ -000339d0: 3a20 3320 2039 2020 3920 2033 2020 2e20  : 3  9  9  3  . 
│ │ │ │ -000339e0: 202e 2020 2020 2020 2020 2020 2020 2020   .              
│ │ │ │ +000339b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │ +000339e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000339f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033a10: 2020 2020 2020 2020 7c0a 7c20 2020 2034          |.|    4
│ │ │ │ -00033a20: 3a20 2e20 202e 2020 3620 3138 2031 3820  : .  .  6 18 18 
│ │ │ │ -00033a30: 2036 2020 2020 2020 2020 2020 2020 2020   6              
│ │ │ │ +00033a00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033a10: 2020 2020 343a 202e 2020 2e20 2036 2031      4: .  .  6 1
│ │ │ │ +00033a20: 3820 3138 2020 3620 2020 2020 2020 2020  8 18  6         
│ │ │ │ +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                  
│ │ │ │ -00033a60: 2020 2020 2020 2020 7c0a 7c20 2020 2035          |.|    5
│ │ │ │ -00033a70: 3a20 2e20 202e 2020 2e20 202e 2031 3020  : .  .  .  . 10 
│ │ │ │ -00033a80: 3330 2020 2020 2020 2020 2020 2020 2020  30              
│ │ │ │ +00033a50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │ +00033a70: 2e20 3130 2033 3020 2020 2020 2020 2020  . 10 30         
│ │ │ │ +00033a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033ab0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00033aa0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033b00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00033b10: 2020 3620 2020 2020 2031 3020 2020 2020    6      10     
│ │ │ │ -00033b20: 2031 3520 2020 2020 2032 3120 2020 2020   15      21     
│ │ │ │ -00033b30: 2032 3820 2020 2020 2033 3620 2020 2020   28      36     
│ │ │ │ -00033b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033b50: 2020 2020 2020 2020 7c0a 7c6f 3520 3d20          |.|o5 = 
│ │ │ │ -00033b60: 2852 2020 3c2d 2d20 5220 2020 3c2d 2d20  (R  <-- R   <-- 
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│ │ │ │ -00033b90: 207c 2030 2030 2020 3020 2031 2030 2030   | 0 0  0  1 0 0
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│ │ │ │ +00033af0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033b00: 2020 2020 2020 2036 2020 2020 2020 3130         6      10
│ │ │ │ +00033b10: 2020 2020 2020 3135 2020 2020 2020 3231        15      21
│ │ │ │ +00033b20: 2020 2020 2020 3238 2020 2020 2020 3336        28      36
│ │ │ │ +00033b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033b40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033b50: 6f35 203d 2028 5220 203c 2d2d 2052 2020  o5 = (R  <-- R  
│ │ │ │ +00033b60: 203c 2d2d 2052 2020 203c 2d2d 2052 2020   <-- R   <-- R  
│ │ │ │ +00033b70: 203c 2d2d 2052 2020 203c 2d2d 2052 2020   <-- R   <-- R  
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│ │ │ │ +00033b90: 3120 3020 3020 7c29 2020 2020 207c 0a7c  1 0 0 |)     |.|
│ │ │ │ +00033ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033bd0: 2020 2020 2020 2020 2020 2020 207b 327d               {2}
│ │ │ │ -00033be0: 207c 2030 202d 3120 3020 2030 2030 2030   | 0 -1 0  0 0 0
│ │ │ │ -00033bf0: 207c 2020 2020 2020 7c0a 7c20 2020 2020   |      |.|     
│ │ │ │ -00033c00: 2030 2020 2020 2020 3120 2020 2020 2020   0      1       
│ │ │ │ -00033c10: 3220 2020 2020 2020 3320 2020 2020 2020  2       3       
│ │ │ │ -00033c20: 3420 2020 2020 2020 3520 2020 207b 327d  4       5    {2}
│ │ │ │ -00033c30: 207c 2030 2030 2020 2d31 2030 2030 2030   | 0 0  -1 0 0 0
│ │ │ │ -00033c40: 207c 2020 2020 2020 7c0a 7c20 2020 2020   |      |.|     
│ │ │ │ +00033bd0: 2020 7b32 7d20 7c20 3020 2d31 2030 2020    {2} | 0 -1 0  
│ │ │ │ +00033be0: 3020 3020 3020 7c20 2020 2020 207c 0a7c  0 0 0 |      |.|
│ │ │ │ +00033bf0: 2020 2020 2020 3020 2020 2020 2031 2020        0      1  
│ │ │ │ +00033c00: 2020 2020 2032 2020 2020 2020 2033 2020       2       3  
│ │ │ │ +00033c10: 2020 2020 2034 2020 2020 2020 2035 2020       4       5  
│ │ │ │ +00033c20: 2020 7b32 7d20 7c20 3020 3020 202d 3120    {2} | 0 0  -1 
│ │ │ │ +00033c30: 3020 3020 3020 7c20 2020 2020 207c 0a7c  0 0 0 |      |.|
│ │ │ │ +00033c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033c70: 2020 2020 2020 2020 2020 2020 207b 337d               {3}
│ │ │ │ -00033c80: 207c 2030 2030 2020 3020 2030 2030 2031   | 0 0  0  0 0 1
│ │ │ │ -00033c90: 207c 2020 2020 2020 7c0a 7c20 2020 2020   |      |.|     
│ │ │ │ +00033c70: 2020 7b33 7d20 7c20 3020 3020 2030 2020    {3} | 0 0  0  
│ │ │ │ +00033c80: 3020 3020 3120 7c20 2020 2020 207c 0a7c  0 0 1 |      |.|
│ │ │ │ +00033c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033cc0: 2020 2020 2020 2020 2020 2020 207b 337d               {3}
│ │ │ │ -00033cd0: 207c 2030 2030 2020 3020 2030 2031 2030   | 0 0  0  0 1 0
│ │ │ │ -00033ce0: 207c 2020 2020 2020 7c0a 7c20 2020 2020   |      |.|     
│ │ │ │ +00033cc0: 2020 7b33 7d20 7c20 3020 3020 2030 2020    {3} | 0 0  0  
│ │ │ │ +00033cd0: 3020 3120 3020 7c20 2020 2020 207c 0a7c  0 1 0 |      |.|
│ │ │ │ +00033ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033d10: 2020 2020 2020 2020 2020 2020 207b 337d               {3}
│ │ │ │ -00033d20: 207c 2031 2030 2020 3020 2030 2030 2030   | 1 0  0  0 0 0
│ │ │ │ -00033d30: 207c 2020 2020 2020 7c0a 7c20 2020 2020   |      |.|     
│ │ │ │ +00033d10: 2020 7b33 7d20 7c20 3120 3020 2030 2020    {3} | 1 0  0  
│ │ │ │ +00033d20: 3020 3020 3020 7c20 2020 2020 207c 0a7c  0 0 0 |      |.|
│ │ │ │ +00033d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033d80: 2020 2020 2020 2020 7c0a 7c6f 3520 3a20          |.|o5 : 
│ │ │ │ -00033d90: 5365 7175 656e 6365 2020 2020 2020 2020  Sequence        
│ │ │ │ +00033d70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033d80: 6f35 203a 2053 6571 7565 6e63 6520 2020  o5 : Sequence   
│ │ │ │ +00033d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033dd0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00033dc0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00033dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00033de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00033df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00033e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033e20: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20  --------+.|i6 : 
│ │ │ │ -00033e30: 6265 7474 6920 4646 2020 2020 2020 2020  betti FF        
│ │ │ │ +00033e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00033e20: 6936 203a 2062 6574 7469 2046 4620 2020  i6 : betti FF   
│ │ │ │ +00033e30: 2020 2020 2020 2020 2020 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 2020 2020 2020                  
│ │ │ │ -00033e70: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00033e60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033ec0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00033ed0: 2020 2020 2020 2030 2020 3120 2032 2020         0  1  2  
│ │ │ │ -00033ee0: 3320 2034 2020 3520 2020 2020 2020 2020  3  4  5         
│ │ │ │ +00033eb0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033ec0: 2020 2020 2020 2020 2020 2020 3020 2031              0  1
│ │ │ │ +00033ed0: 2020 3220 2033 2020 3420 2035 2020 2020    2  3  4  5    
│ │ │ │ +00033ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033f10: 2020 2020 2020 2020 7c0a 7c6f 3620 3d20          |.|o6 = 
│ │ │ │ -00033f20: 746f 7461 6c3a 2036 2031 3020 3135 2032  total: 6 10 15 2
│ │ │ │ -00033f30: 3120 3238 2033 3620 2020 2020 2020 2020  1 28 36         
│ │ │ │ +00033f00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033f10: 6f36 203d 2074 6f74 616c 3a20 3620 3130  o6 = total: 6 10
│ │ │ │ +00033f20: 2031 3520 3231 2032 3820 3336 2020 2020   15 21 28 36    
│ │ │ │ +00033f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033f60: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00033f70: 2020 2020 323a 2033 2020 3120 202e 2020      2: 3  1  .  
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│ │ │ │ +00033f50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033f60: 2020 2020 2020 2020 2032 3a20 3320 2031           2: 3  1
│ │ │ │ +00033f70: 2020 2e20 202e 2020 2e20 202e 2020 2020    .  .  .  .    
│ │ │ │ +00033f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033fb0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00033fc0: 2020 2020 333a 2033 2020 3920 2039 2020      3: 3  9  9  
│ │ │ │ -00033fd0: 3320 202e 2020 2e20 2020 2020 2020 2020  3  .  .         
│ │ │ │ +00033fa0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033fb0: 2020 2020 2020 2020 2033 3a20 3320 2039           3: 3  9
│ │ │ │ +00033fc0: 2020 3920 2033 2020 2e20 202e 2020 2020    9  3  .  .    
│ │ │ │ +00033fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034000: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00034010: 2020 2020 343a 202e 2020 2e20 2036 2031      4: .  .  6 1
│ │ │ │ -00034020: 3820 3138 2020 3620 2020 2020 2020 2020  8 18  6         
│ │ │ │ +00033ff0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00034000: 2020 2020 2020 2020 2034 3a20 2e20 202e           4: .  .
│ │ │ │ +00034010: 2020 3620 3138 2031 3820 2036 2020 2020    6 18 18  6    
│ │ │ │ +00034020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034050: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00034060: 2020 2020 353a 202e 2020 2e20 202e 2020      5: .  .  .  
│ │ │ │ -00034070: 2e20 3130 2033 3020 2020 2020 2020 2020  . 10 30         
│ │ │ │ +00034040: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00034050: 2020 2020 2020 2020 2035 3a20 2e20 202e           5: .  .
│ │ │ │ +00034060: 2020 2e20 202e 2031 3020 3330 2020 2020    .  . 10 30    
│ │ │ │ +00034070: 2020 2020 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 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00034090: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000340a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000340b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000340c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000340d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000340e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000340f0: 2020 2020 2020 2020 7c0a 7c6f 3620 3a20          |.|o6 : 
│ │ │ │ -00034100: 4265 7474 6954 616c 6c79 2020 2020 2020  BettiTally      
│ │ │ │ +000340e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000340f0: 6f36 203a 2042 6574 7469 5461 6c6c 7920  o6 : BettiTally 
│ │ │ │ +00034100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034140: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00034130: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00034140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034190: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20  --------+.|i7 : 
│ │ │ │ -000341a0: 6265 7474 6920 6672 6565 5265 736f 6c75  betti freeResolu
│ │ │ │ -000341b0: 7469 6f6e 284d 2c20 4c65 6e67 7468 4c69  tion(M, LengthLi
│ │ │ │ -000341c0: 6d69 743d 3e35 2920 2020 2020 2020 2020  mit=>5)         
│ │ │ │ -000341d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000341e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00034180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00034190: 6937 203a 2062 6574 7469 2066 7265 6552  i7 : betti freeR
│ │ │ │ +000341a0: 6573 6f6c 7574 696f 6e28 4d2c 204c 656e  esolution(M, Len
│ │ │ │ +000341b0: 6774 684c 696d 6974 3d3e 3529 2020 2020  gthLimit=>5)    
│ │ │ │ +000341c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000341d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000341e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000341f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034230: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00034240: 2020 2020 2020 2030 2020 3120 2032 2020         0  1  2  
│ │ │ │ -00034250: 3320 2034 2020 3520 2020 2020 2020 2020  3  4  5         
│ │ │ │ +00034220: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00034230: 2020 2020 2020 2020 2020 2020 3020 2031              0  1
│ │ │ │ +00034240: 2020 3220 2033 2020 3420 2035 2020 2020    2  3  4  5    
│ │ │ │ +00034250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034280: 2020 2020 2020 2020 7c0a 7c6f 3720 3d20          |.|o7 = 
│ │ │ │ -00034290: 746f 7461 6c3a 2036 2031 3020 3135 2032  total: 6 10 15 2
│ │ │ │ -000342a0: 3120 3238 2033 3620 2020 2020 2020 2020  1 28 36         
│ │ │ │ +00034270: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00034280: 6f37 203d 2074 6f74 616c 3a20 3620 3130  o7 = total: 6 10
│ │ │ │ +00034290: 2031 3520 3231 2032 3820 3336 2020 2020   15 21 28 36    
│ │ │ │ +000342a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000342b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000342c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000342d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000342e0: 2020 2020 323a 2033 2020 3120 202e 2020      2: 3  1  .  
│ │ │ │ -000342f0: 2e20 202e 2020 2e20 2020 2020 2020 2020  .  .  .         
│ │ │ │ +000342c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000342d0: 2020 2020 2020 2020 2032 3a20 3320 2031           2: 3  1
│ │ │ │ +000342e0: 2020 2e20 202e 2020 2e20 202e 2020 2020    .  .  .  .    
│ │ │ │ +000342f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034320: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00034330: 2020 2020 333a 2033 2020 3920 2039 2020      3: 3  9  9  
│ │ │ │ -00034340: 3320 202e 2020 2e20 2020 2020 2020 2020  3  .  .         
│ │ │ │ +00034310: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00034320: 2020 2020 2020 2020 2033 3a20 3320 2039           3: 3  9
│ │ │ │ +00034330: 2020 3920 2033 2020 2e20 202e 2020 2020    9  3  .  .    
│ │ │ │ +00034340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034370: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00034380: 2020 2020 343a 202e 2020 2e20 2036 2031      4: .  .  6 1
│ │ │ │ -00034390: 3820 3138 2020 3620 2020 2020 2020 2020  8 18  6         
│ │ │ │ +00034360: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00034370: 2020 2020 2020 2020 2034 3a20 2e20 202e           4: .  .
│ │ │ │ +00034380: 2020 3620 3138 2031 3820 2036 2020 2020    6 18 18  6    
│ │ │ │ +00034390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000343a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000343b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000343c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000343d0: 2020 2020 353a 202e 2020 2e20 202e 2020      5: .  .  .  
│ │ │ │ -000343e0: 2e20 3130 2033 3020 2020 2020 2020 2020  . 10 30         
│ │ │ │ +000343b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000343c0: 2020 2020 2020 2020 2035 3a20 2e20 202e           5: .  .
│ │ │ │ +000343d0: 2020 2e20 202e 2031 3020 3330 2020 2020    .  . 10 30    
│ │ │ │ +000343e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000343f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034410: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00034400: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00034410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034460: 2020 2020 2020 2020 7c0a 7c6f 3720 3a20          |.|o7 : 
│ │ │ │ -00034470: 4265 7474 6954 616c 6c79 2020 2020 2020  BettiTally      
│ │ │ │ +00034450: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00034460: 6f37 203a 2042 6574 7469 5461 6c6c 7920  o7 : BettiTally 
│ │ │ │ +00034470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000344a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000344b0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +000344a0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +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 2d2d  ----------------
│ │ │ │  000344e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000344f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034500: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20  --------+.|i8 : 
│ │ │ │ -00034510: 4320 3d20 636f 6d70 6c65 7820 666c 6174  C = complex flat
│ │ │ │ -00034520: 7465 6e20 7b7b 6175 677d 207c 6170 706c  ten {{aug} |appl
│ │ │ │ -00034530: 7928 342c 2069 2d3e 2046 462e 6464 5f28  y(4, i-> FF.dd_(
│ │ │ │ -00034540: 692b 3129 297d 2020 2020 2020 2020 2020  i+1))}          
│ │ │ │ -00034550: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000344f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00034500: 6938 203a 2043 203d 2063 6f6d 706c 6578  i8 : C = complex
│ │ │ │ +00034510: 2066 6c61 7474 656e 207b 7b61 7567 7d20   flatten {{aug} 
│ │ │ │ +00034520: 7c61 7070 6c79 2834 2c20 692d 3e20 4646  |apply(4, i-> FF
│ │ │ │ +00034530: 2e64 645f 2869 2b31 2929 7d20 2020 2020  .dd_(i+1))}     
│ │ │ │ +00034540: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00034550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000345a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000345b0: 2020 2020 2020 2036 2020 2020 2020 3130         6      10
│ │ │ │ -000345c0: 2020 2020 2020 3135 2020 2020 2020 3231        15      21
│ │ │ │ -000345d0: 2020 2020 2020 3238 2020 2020 2020 2020        28        
│ │ │ │ -000345e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000345f0: 2020 2020 2020 2020 7c0a 7c6f 3820 3d20          |.|o8 = 
│ │ │ │ -00034600: 4d20 3c2d 2d20 5220 203c 2d2d 2052 2020  M <-- R  <-- R  
│ │ │ │ -00034610: 203c 2d2d 2052 2020 203c 2d2d 2052 2020   <-- R   <-- R  
│ │ │ │ -00034620: 203c 2d2d 2052 2020 2020 2020 2020 2020   <-- R          
│ │ │ │ -00034630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034640: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00034590: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000345a0: 2020 2020 2020 2020 2020 2020 3620 2020              6   
│ │ │ │ +000345b0: 2020 2031 3020 2020 2020 2031 3520 2020     10      15   
│ │ │ │ +000345c0: 2020 2032 3120 2020 2020 2032 3820 2020     21      28   
│ │ │ │ +000345d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000345e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000345f0: 6f38 203d 204d 203c 2d2d 2052 2020 3c2d  o8 = M <-- R  <-
│ │ │ │ +00034600: 2d20 5220 2020 3c2d 2d20 5220 2020 3c2d  - R   <-- R   <-
│ │ │ │ +00034610: 2d20 5220 2020 3c2d 2d20 5220 2020 2020  - R   <-- R     
│ │ │ │ +00034620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034630: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00034640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034690: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000346a0: 3020 2020 2020 3120 2020 2020 2032 2020  0     1      2  
│ │ │ │ -000346b0: 2020 2020 2033 2020 2020 2020 2034 2020       3       4  
│ │ │ │ -000346c0: 2020 2020 2035 2020 2020 2020 2020 2020       5          
│ │ │ │ -000346d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000346e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00034680: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00034690: 2020 2020 2030 2020 2020 2031 2020 2020       0     1    
│ │ │ │ +000346a0: 2020 3220 2020 2020 2020 3320 2020 2020    2       3     
│ │ │ │ +000346b0: 2020 3420 2020 2020 2020 3520 2020 2020    4       5     
│ │ │ │ +000346c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000346d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000346e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000346f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034730: 2020 2020 2020 2020 7c0a 7c6f 3820 3a20          |.|o8 : 
│ │ │ │ -00034740: 436f 6d70 6c65 7820 2020 2020 2020 2020  Complex         
│ │ │ │ +00034720: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00034730: 6f38 203a 2043 6f6d 706c 6578 2020 2020  o8 : Complex    
│ │ │ │ +00034740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034780: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00034770: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +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 2d2d  ----------------
│ │ │ │  000347b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000347c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000347d0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3920 3a20  --------+.|i9 : 
│ │ │ │ -000347e0: 6170 706c 7928 342c 2069 202d 3e46 462e  apply(4, i ->FF.
│ │ │ │ -000347f0: 6464 5f28 692b 3129 2920 2020 2020 2020  dd_(i+1))       
│ │ │ │ +000347c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +000347d0: 6939 203a 2061 7070 6c79 2834 2c20 6920  i9 : apply(4, i 
│ │ │ │ +000347e0: 2d3e 4646 2e64 645f 2869 2b31 2929 2020  ->FF.dd_(i+1))  
│ │ │ │ +000347f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034820: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00034810: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00034820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034870: 2020 2020 2020 2020 7c0a 7c6f 3920 3d20          |.|o9 = 
│ │ │ │ -00034880: 7b7b 337d 207c 2063 202d 6220 6120 2030  {{3} | c -b a  0
│ │ │ │ -00034890: 2020 3020 3020 2030 2020 3020 3020 2030    0 0  0  0 0  0
│ │ │ │ -000348a0: 2020 7c2c 207b 347d 207c 2063 3220 3020    |, {4} | c2 0 
│ │ │ │ -000348b0: 2d61 2030 2062 2030 2030 2020 3020 2020  -a 0 b 0 0  0   
│ │ │ │ -000348c0: 3020 2030 2020 3020 7c0a 7c20 2020 2020  0  0  0 |.|     
│ │ │ │ -000348d0: 207b 337d 207c 2030 2030 2020 3020 2063   {3} | 0 0  0  c
│ │ │ │ -000348e0: 2020 6220 6120 2030 2020 3020 3020 2030    b a  0  0 0  0
│ │ │ │ -000348f0: 2020 7c20 207b 347d 207c 2030 2020 3020    |  {4} | 0  0 
│ │ │ │ -00034900: 3020 2030 2063 2030 2030 2020 3020 2020  0  0 c 0 0  0   
│ │ │ │ -00034910: 3020 2030 2020 3020 7c0a 7c20 2020 2020  0  0  0 |.|     
│ │ │ │ -00034920: 207b 337d 207c 2030 2030 2020 6332 2030   {3} | 0 0  c2 0
│ │ │ │ -00034930: 2020 3020 3020 2030 2020 6120 2d63 202d    0 0  0  a -c -
│ │ │ │ -00034940: 6220 7c20 207b 347d 207c 2030 2020 3020  b |  {4} | 0  0 
│ │ │ │ -00034950: 6320 2030 2030 2030 2030 2020 3020 2020  c  0 0 0 0  0   
│ │ │ │ -00034960: 3020 2030 2020 3020 7c0a 7c20 2020 2020  0  0  0 |.|     
│ │ │ │ -00034970: 207b 327d 207c 2030 2030 2020 3020 2030   {2} | 0 0  0  0
│ │ │ │ -00034980: 2020 3020 3020 2062 2020 3020 6132 2030    0 0  b  0 a2 0
│ │ │ │ -00034990: 2020 7c20 207b 347d 207c 2030 2020 3020    |  {4} | 0  0 
│ │ │ │ -000349a0: 3020 2030 2030 2030 2030 2020 3020 2020  0  0 0 0 0  0   
│ │ │ │ -000349b0: 2d61 2030 2020 6220 7c0a 7c20 2020 2020  -a 0  b |.|     
│ │ │ │ -000349c0: 207b 327d 207c 2030 2063 3220 3020 2030   {2} | 0 c2 0  0
│ │ │ │ -000349d0: 2020 3020 6232 202d 6320 3020 3020 2061    0 b2 -c 0 0  a
│ │ │ │ -000349e0: 3220 7c20 207b 347d 207c 2030 2020 3020  2 |  {4} | 0  0 
│ │ │ │ -000349f0: 3020 2030 2030 2030 2062 3220 3020 2020  0  0 0 0 b2 0   
│ │ │ │ -00034a00: 3020 202d 6120 2d63 7c0a 7c20 2020 2020  0  -a -c|.|     
│ │ │ │ -00034a10: 207b 327d 207c 2030 2030 2020 3020 2062   {2} | 0 0  0  b
│ │ │ │ -00034a20: 3220 3020 3020 2061 2020 3020 3020 2030  2 0 0  a  0 0  0
│ │ │ │ -00034a30: 2020 7c20 207b 347d 207c 2030 2020 3020    |  {4} | 0  0 
│ │ │ │ -00034a40: 3020 2030 2030 2030 2030 2020 6132 2020  0  0 0 0 0  a2  
│ │ │ │ -00034a50: 6320 2062 2020 3020 7c0a 7c20 2020 2020  c  b  0 |.|     
│ │ │ │ +00034860: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00034870: 6f39 203d 207b 7b33 7d20 7c20 6320 2d62  o9 = {{3} | c -b
│ │ │ │ +00034880: 2061 2020 3020 2030 2030 2020 3020 2030   a  0  0 0  0  0
│ │ │ │ +00034890: 2030 2020 3020 207c 2c20 7b34 7d20 7c20   0  0  |, {4} | 
│ │ │ │ +000348a0: 6332 2030 202d 6120 3020 6220 3020 3020  c2 0 -a 0 b 0 0 
│ │ │ │ +000348b0: 2030 2020 2030 2020 3020 2030 207c 0a7c   0   0  0  0 |.|
│ │ │ │ +000348c0: 2020 2020 2020 7b33 7d20 7c20 3020 3020        {3} | 0 0 
│ │ │ │ +000348d0: 2030 2020 6320 2062 2061 2020 3020 2030   0  c  b a  0  0
│ │ │ │ +000348e0: 2030 2020 3020 207c 2020 7b34 7d20 7c20   0  0  |  {4} | 
│ │ │ │ +000348f0: 3020 2030 2030 2020 3020 6320 3020 3020  0  0 0  0 c 0 0 
│ │ │ │ +00034900: 2030 2020 2030 2020 3020 2030 207c 0a7c   0   0  0  0 |.|
│ │ │ │ +00034910: 2020 2020 2020 7b33 7d20 7c20 3020 3020        {3} | 0 0 
│ │ │ │ +00034920: 2063 3220 3020 2030 2030 2020 3020 2061   c2 0  0 0  0  a
│ │ │ │ +00034930: 202d 6320 2d62 207c 2020 7b34 7d20 7c20   -c -b |  {4} | 
│ │ │ │ +00034940: 3020 2030 2063 2020 3020 3020 3020 3020  0  0 c  0 0 0 0 
│ │ │ │ +00034950: 2030 2020 2030 2020 3020 2030 207c 0a7c   0   0  0  0 |.|
│ │ │ │ +00034960: 2020 2020 2020 7b32 7d20 7c20 3020 3020        {2} | 0 0 
│ │ │ │ +00034970: 2030 2020 3020 2030 2030 2020 6220 2030   0  0  0 0  b  0
│ │ │ │ +00034980: 2061 3220 3020 207c 2020 7b34 7d20 7c20   a2 0  |  {4} | 
│ │ │ │ +00034990: 3020 2030 2030 2020 3020 3020 3020 3020  0  0 0  0 0 0 0 
│ │ │ │ +000349a0: 2030 2020 202d 6120 3020 2062 207c 0a7c   0   -a 0  b |.|
│ │ │ │ +000349b0: 2020 2020 2020 7b32 7d20 7c20 3020 6332        {2} | 0 c2
│ │ │ │ +000349c0: 2030 2020 3020 2030 2062 3220 2d63 2030   0  0  0 b2 -c 0
│ │ │ │ +000349d0: 2030 2020 6132 207c 2020 7b34 7d20 7c20   0  a2 |  {4} | 
│ │ │ │ +000349e0: 3020 2030 2030 2020 3020 3020 3020 6232  0  0 0  0 0 0 b2
│ │ │ │ +000349f0: 2030 2020 2030 2020 2d61 202d 637c 0a7c   0   0  -a -c|.|
│ │ │ │ +00034a00: 2020 2020 2020 7b32 7d20 7c20 3020 3020        {2} | 0 0 
│ │ │ │ +00034a10: 2030 2020 6232 2030 2030 2020 6120 2030   0  b2 0 0  a  0
│ │ │ │ +00034a20: 2030 2020 3020 207c 2020 7b34 7d20 7c20   0  0  |  {4} | 
│ │ │ │ +00034a30: 3020 2030 2030 2020 3020 3020 3020 3020  0  0 0  0 0 0 0 
│ │ │ │ +00034a40: 2061 3220 2063 2020 6220 2030 207c 0a7c   a2  c  b  0 |.|
│ │ │ │ +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 2020                  
│ │ │ │ -00034a80: 2020 2020 207b 337d 207c 2030 2020 3020       {3} | 0  0 
│ │ │ │ -00034a90: 3020 2030 2030 2030 2030 2020 3020 2020  0  0 0 0 0  0   
│ │ │ │ -00034aa0: 6232 2030 2020 3020 7c0a 7c20 2020 2020  b2 0  0 |.|     
│ │ │ │ +00034a70: 2020 2020 2020 2020 2020 7b33 7d20 7c20            {3} | 
│ │ │ │ +00034a80: 3020 2030 2030 2020 3020 3020 3020 3020  0  0 0  0 0 0 0 
│ │ │ │ +00034a90: 2030 2020 2062 3220 3020 2030 207c 0a7c   0   b2 0  0 |.|
│ │ │ │ +00034aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034ad0: 2020 2020 207b 347d 207c 2030 2020 3020       {4} | 0  0 
│ │ │ │ -00034ae0: 3020 2030 2030 2030 2030 2020 3020 2020  0  0 0 0 0  0   
│ │ │ │ -00034af0: 3020 2030 2020 3020 7c0a 7c20 2020 2020  0  0  0 |.|     
│ │ │ │ +00034ac0: 2020 2020 2020 2020 2020 7b34 7d20 7c20            {4} | 
│ │ │ │ +00034ad0: 3020 2030 2030 2020 3020 3020 3020 3020  0  0 0  0 0 0 0 
│ │ │ │ +00034ae0: 2030 2020 2030 2020 3020 2030 207c 0a7c   0   0  0  0 |.|
│ │ │ │ +00034af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034b20: 2020 2020 207b 347d 207c 2030 2020 3020       {4} | 0  0 
│ │ │ │ -00034b30: 3020 2030 2030 2030 2030 2020 3020 2020  0  0 0 0 0  0   
│ │ │ │ -00034b40: 3020 2030 2020 3020 7c0a 7c20 2020 2020  0  0  0 |.|     
│ │ │ │ +00034b10: 2020 2020 2020 2020 2020 7b34 7d20 7c20            {4} | 
│ │ │ │ +00034b20: 3020 2030 2030 2020 3020 3020 3020 3020  0  0 0  0 0 0 0 
│ │ │ │ +00034b30: 2030 2020 2030 2020 3020 2030 207c 0a7c   0   0  0  0 |.|
│ │ │ │ +00034b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034b70: 2020 2020 207b 347d 207c 2030 2020 3020       {4} | 0  0 
│ │ │ │ -00034b80: 3020 2030 2030 2030 2030 2020 2d62 3220  0  0 0 0 0  -b2 
│ │ │ │ -00034b90: 3020 2030 2020 3020 7c0a 7c20 2020 2020  0  0  0 |.|     
│ │ │ │ +00034b60: 2020 2020 2020 2020 2020 7b34 7d20 7c20            {4} | 
│ │ │ │ +00034b70: 3020 2030 2030 2020 3020 3020 3020 3020  0  0 0  0 0 0 0 
│ │ │ │ +00034b80: 202d 6232 2030 2020 3020 2030 207c 0a7c   -b2 0  0  0 |.|
│ │ │ │ +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 2020                  
│ │ │ │  00034bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034be0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00034bd0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00034be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034c30: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00034c20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00034c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034c80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00034c70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +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 2020                  
│ │ │ │  00034cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034cd0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00034cc0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +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 2020                  
│ │ │ │  00034d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034d20: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00034d10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +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 2020                  
│ │ │ │  00034d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034d70: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00034d60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +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 2020                  
│ │ │ │  00034da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034dc0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00034db0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +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 2020                  
│ │ │ │  00034df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034e10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00034e00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00034e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034e60: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00034e50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00034e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034eb0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00034ea0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00034eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034f00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00034ef0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00034f00: 2020 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d       -----------
│ │ │ │  00034f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034f50: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ -00034f60: 3020 2030 2030 2030 2020 7c2c 207b 367d  0  0 0 0  |, {6}
│ │ │ │ -00034f70: 207c 2063 202d 6220 6120 2030 2020 3020   | c -b a  0  0 
│ │ │ │ +00034f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c  -------------|.|
│ │ │ │ +00034f50: 2020 2020 2030 2020 3020 3020 3020 207c       0  0 0 0  |
│ │ │ │ +00034f60: 2c20 7b36 7d20 7c20 6320 2d62 2061 2020  , {6} | c -b a  
│ │ │ │ +00034f70: 3020 2030 2030 2020 3020 2030 2030 2020  0  0 0  0  0 0  
│ │ │ │  00034f80: 3020 2030 2020 3020 3020 2030 2020 3020  0  0  0 0  0  0 
│ │ │ │ -00034f90: 2030 2030 2020 3020 2030 2020 3020 2020   0 0  0  0  0   
│ │ │ │ -00034fa0: 3020 2030 2020 3020 7c0a 7c20 2020 2020  0  0  0 |.|     
│ │ │ │ -00034fb0: 3020 2030 2030 2030 2020 7c20 207b 367d  0  0 0 0  |  {6}
│ │ │ │ -00034fc0: 207c 2030 2030 2020 3020 2063 2020 6220   | 0 0  0  c  b 
│ │ │ │ -00034fd0: 6120 2030 2020 3020 3020 2030 2020 3020  a  0  0 0  0  0 
│ │ │ │ -00034fe0: 2030 2030 2020 3020 2030 2020 3020 2020   0 0  0  0  0   
│ │ │ │ -00034ff0: 3020 2030 2020 3020 7c0a 7c20 2020 2020  0  0  0 |.|     
│ │ │ │ -00035000: 3020 2030 2030 2030 2020 7c20 207b 367d  0  0 0 0  |  {6}
│ │ │ │ -00035010: 207c 2030 2030 2020 6332 2030 2020 3020   | 0 0  c2 0  0 
│ │ │ │ -00035020: 3020 2030 2020 6120 2d63 202d 6220 3020  0  0  a -c -b 0 
│ │ │ │ -00035030: 2030 2030 2020 3020 2030 2020 3020 2020   0 0  0  0  0   
│ │ │ │ -00035040: 3020 2030 2020 3020 7c0a 7c20 2020 2020  0  0  0 |.|     
│ │ │ │ -00035050: 3020 2030 2030 2030 2020 7c20 207b 357d  0  0 0 0  |  {5}
│ │ │ │ -00035060: 207c 2030 2030 2020 3020 2030 2020 3020   | 0 0  0  0  0 
│ │ │ │ -00035070: 3020 2062 2020 3020 6132 2030 2020 3020  0  b  0 a2 0  0 
│ │ │ │ -00035080: 2030 2030 2020 3020 2030 2020 3020 2020   0 0  0  0  0   
│ │ │ │ -00035090: 3020 2030 2020 3020 7c0a 7c20 2020 2020  0  0  0 |.|     
│ │ │ │ -000350a0: 3020 2030 2030 2030 2020 7c20 207b 357d  0  0 0 0  |  {5}
│ │ │ │ -000350b0: 207c 2030 2063 3220 3020 2030 2020 3020   | 0 c2 0  0  0 
│ │ │ │ -000350c0: 6232 202d 6320 3020 3020 2061 3220 3020  b2 -c 0 0  a2 0 
│ │ │ │ -000350d0: 2030 2030 2020 3020 2030 2020 3020 2020   0 0  0  0  0   
│ │ │ │ -000350e0: 3020 2030 2020 3020 7c0a 7c20 2020 2020  0  0  0 |.|     
│ │ │ │ -000350f0: 3020 2030 2030 2030 2020 7c20 207b 357d  0  0 0 0  |  {5}
│ │ │ │ -00035100: 207c 2030 2030 2020 3020 2062 3220 3020   | 0 0  0  b2 0 
│ │ │ │ -00035110: 3020 2061 2020 3020 3020 2030 2020 3020  0  a  0 0  0  0 
│ │ │ │ -00035120: 2030 2030 2020 3020 2030 2020 3020 2020   0 0  0  0  0   
│ │ │ │ -00035130: 3020 2030 2020 3020 7c0a 7c20 2020 2020  0  0  0 |.|     
│ │ │ │ -00035140: 3020 2030 2030 2061 3220 7c20 207b 367d  0  0 0 a2 |  {6}
│ │ │ │ -00035150: 207c 2030 2030 2020 3020 2030 2020 3020   | 0 0  0  0  0 
│ │ │ │ -00035160: 3020 2030 2020 3020 3020 2030 2020 6320  0  0  0 0  0  c 
│ │ │ │ -00035170: 2062 2061 2020 3020 2030 2020 3020 2020   b a  0  0  0   
│ │ │ │ -00035180: 3020 2030 2020 3020 7c0a 7c20 2020 2020  0  0  0 |.|     
│ │ │ │ -00035190: 6132 2063 2062 2030 2020 7c20 207b 367d  a2 c b 0  |  {6}
│ │ │ │ -000351a0: 207c 2030 2030 2020 3020 2030 2020 3020   | 0 0  0  0  0 
│ │ │ │ -000351b0: 3020 2030 2020 3020 3020 2030 2020 3020  0  0  0 0  0  0 
│ │ │ │ -000351c0: 2030 2030 2020 3020 2061 2020 2d63 2020   0 0  0  a  -c  
│ │ │ │ -000351d0: 2d62 2030 2020 3020 7c0a 7c20 2020 2020  -b 0  0 |.|     
│ │ │ │ -000351e0: 3020 2061 2030 202d 6220 7c20 207b 357d  0  a 0 -b |  {5}
│ │ │ │ -000351f0: 207c 2030 2030 2020 3020 2030 2020 3020   | 0 0  0  0  0 
│ │ │ │ -00035200: 3020 2030 2020 3020 3020 2030 2020 3020  0  0  0 0  0  0 
│ │ │ │ -00035210: 2030 2030 2020 6220 2030 2020 6132 2020   0 0  b  0  a2  
│ │ │ │ -00035220: 3020 2030 2020 3020 7c0a 7c20 2020 2020  0  0  0 |.|     
│ │ │ │ -00035230: 3020 2030 2061 2063 2020 7c20 207b 357d  0  0 a c  |  {5}
│ │ │ │ -00035240: 207c 2030 2030 2020 3020 2030 2020 3020   | 0 0  0  0  0 
│ │ │ │ -00035250: 3020 2030 2020 3020 3020 2030 2020 3020  0  0  0 0  0  0 
│ │ │ │ -00035260: 2030 2062 3220 2d63 2030 2020 3020 2020   0 b2 -c 0  0   
│ │ │ │ -00035270: 6132 2030 2020 3020 7c0a 7c20 2020 2020  a2 0  0 |.|     
│ │ │ │ -00035280: 2020 2020 2020 2020 2020 2020 207b 357d               {5}
│ │ │ │ -00035290: 207c 2030 2030 2020 3020 2030 2020 3020   | 0 0  0  0  0 
│ │ │ │ -000352a0: 3020 2030 2020 3020 3020 2030 2020 6232  0  0  0 0  0  b2
│ │ │ │ -000352b0: 2030 2030 2020 6120 2030 2020 3020 2020   0 0  a  0  0   
│ │ │ │ -000352c0: 3020 2030 2020 3020 7c0a 7c20 2020 2020  0  0  0 |.|     
│ │ │ │ -000352d0: 2020 2020 2020 2020 2020 2020 207b 367d               {6}
│ │ │ │ -000352e0: 207c 2030 2030 2020 3020 2030 2020 3020   | 0 0  0  0  0 
│ │ │ │ +00034f90: 2030 2020 2030 2020 3020 2030 207c 0a7c   0   0  0  0 |.|
│ │ │ │ +00034fa0: 2020 2020 2030 2020 3020 3020 3020 207c       0  0 0 0  |
│ │ │ │ +00034fb0: 2020 7b36 7d20 7c20 3020 3020 2030 2020    {6} | 0 0  0  
│ │ │ │ +00034fc0: 6320 2062 2061 2020 3020 2030 2030 2020  c  b a  0  0 0  
│ │ │ │ +00034fd0: 3020 2030 2020 3020 3020 2030 2020 3020  0  0  0 0  0  0 
│ │ │ │ +00034fe0: 2030 2020 2030 2020 3020 2030 207c 0a7c   0   0  0  0 |.|
│ │ │ │ +00034ff0: 2020 2020 2030 2020 3020 3020 3020 207c       0  0 0 0  |
│ │ │ │ +00035000: 2020 7b36 7d20 7c20 3020 3020 2063 3220    {6} | 0 0  c2 
│ │ │ │ +00035010: 3020 2030 2030 2020 3020 2061 202d 6320  0  0 0  0  a -c 
│ │ │ │ +00035020: 2d62 2030 2020 3020 3020 2030 2020 3020  -b 0  0 0  0  0 
│ │ │ │ +00035030: 2030 2020 2030 2020 3020 2030 207c 0a7c   0   0  0  0 |.|
│ │ │ │ +00035040: 2020 2020 2030 2020 3020 3020 3020 207c       0  0 0 0  |
│ │ │ │ +00035050: 2020 7b35 7d20 7c20 3020 3020 2030 2020    {5} | 0 0  0  
│ │ │ │ +00035060: 3020 2030 2030 2020 6220 2030 2061 3220  0  0 0  b  0 a2 
│ │ │ │ +00035070: 3020 2030 2020 3020 3020 2030 2020 3020  0  0  0 0  0  0 
│ │ │ │ +00035080: 2030 2020 2030 2020 3020 2030 207c 0a7c   0   0  0  0 |.|
│ │ │ │ +00035090: 2020 2020 2030 2020 3020 3020 3020 207c       0  0 0 0  |
│ │ │ │ +000350a0: 2020 7b35 7d20 7c20 3020 6332 2030 2020    {5} | 0 c2 0  
│ │ │ │ +000350b0: 3020 2030 2062 3220 2d63 2030 2030 2020  0  0 b2 -c 0 0  
│ │ │ │ +000350c0: 6132 2030 2020 3020 3020 2030 2020 3020  a2 0  0 0  0  0 
│ │ │ │ +000350d0: 2030 2020 2030 2020 3020 2030 207c 0a7c   0   0  0  0 |.|
│ │ │ │ +000350e0: 2020 2020 2030 2020 3020 3020 3020 207c       0  0 0 0  |
│ │ │ │ +000350f0: 2020 7b35 7d20 7c20 3020 3020 2030 2020    {5} | 0 0  0  
│ │ │ │ +00035100: 6232 2030 2030 2020 6120 2030 2030 2020  b2 0 0  a  0 0  
│ │ │ │ +00035110: 3020 2030 2020 3020 3020 2030 2020 3020  0  0  0 0  0  0 
│ │ │ │ +00035120: 2030 2020 2030 2020 3020 2030 207c 0a7c   0   0  0  0 |.|
│ │ │ │ +00035130: 2020 2020 2030 2020 3020 3020 6132 207c       0  0 0 a2 |
│ │ │ │ +00035140: 2020 7b36 7d20 7c20 3020 3020 2030 2020    {6} | 0 0  0  
│ │ │ │ +00035150: 3020 2030 2030 2020 3020 2030 2030 2020  0  0 0  0  0 0  
│ │ │ │ +00035160: 3020 2063 2020 6220 6120 2030 2020 3020  0  c  b a  0  0 
│ │ │ │ +00035170: 2030 2020 2030 2020 3020 2030 207c 0a7c   0   0  0  0 |.|
│ │ │ │ +00035180: 2020 2020 2061 3220 6320 6220 3020 207c       a2 c b 0  |
│ │ │ │ +00035190: 2020 7b36 7d20 7c20 3020 3020 2030 2020    {6} | 0 0  0  
│ │ │ │ +000351a0: 3020 2030 2030 2020 3020 2030 2030 2020  0  0 0  0  0 0  
│ │ │ │ +000351b0: 3020 2030 2020 3020 3020 2030 2020 6120  0  0  0 0  0  a 
│ │ │ │ +000351c0: 202d 6320 202d 6220 3020 2030 207c 0a7c   -c  -b 0  0 |.|
│ │ │ │ +000351d0: 2020 2020 2030 2020 6120 3020 2d62 207c       0  a 0 -b |
│ │ │ │ +000351e0: 2020 7b35 7d20 7c20 3020 3020 2030 2020    {5} | 0 0  0  
│ │ │ │ +000351f0: 3020 2030 2030 2020 3020 2030 2030 2020  0  0 0  0  0 0  
│ │ │ │ +00035200: 3020 2030 2020 3020 3020 2062 2020 3020  0  0  0 0  b  0 
│ │ │ │ +00035210: 2061 3220 2030 2020 3020 2030 207c 0a7c   a2  0  0  0 |.|
│ │ │ │ +00035220: 2020 2020 2030 2020 3020 6120 6320 207c       0  0 a c  |
│ │ │ │ +00035230: 2020 7b35 7d20 7c20 3020 3020 2030 2020    {5} | 0 0  0  
│ │ │ │ +00035240: 3020 2030 2030 2020 3020 2030 2030 2020  0  0 0  0  0 0  
│ │ │ │ +00035250: 3020 2030 2020 3020 6232 202d 6320 3020  0  0  0 b2 -c 0 
│ │ │ │ +00035260: 2030 2020 2061 3220 3020 2030 207c 0a7c   0   a2 0  0 |.|
│ │ │ │ +00035270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035280: 2020 7b35 7d20 7c20 3020 3020 2030 2020    {5} | 0 0  0  
│ │ │ │ +00035290: 3020 2030 2030 2020 3020 2030 2030 2020  0  0 0  0  0 0  
│ │ │ │ +000352a0: 3020 2062 3220 3020 3020 2061 2020 3020  0  b2 0 0  a  0 
│ │ │ │ +000352b0: 2030 2020 2030 2020 3020 2030 207c 0a7c   0   0  0  0 |.|
│ │ │ │ +000352c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000352d0: 2020 7b36 7d20 7c20 3020 3020 2030 2020    {6} | 0 0  0  
│ │ │ │ +000352e0: 3020 2030 2030 2020 3020 2030 2030 2020  0  0 0  0  0 0  
│ │ │ │  000352f0: 3020 2030 2020 3020 3020 2030 2020 3020  0  0  0 0  0  0 
│ │ │ │ -00035300: 2030 2030 2020 3020 2030 2020 3020 2020   0 0  0  0  0   
│ │ │ │ -00035310: 3020 2030 2020 6120 7c0a 7c20 2020 2020  0  0  a |.|     
│ │ │ │ -00035320: 2020 2020 2020 2020 2020 2020 207b 357d               {5}
│ │ │ │ -00035330: 207c 2030 2030 2020 3020 2030 2020 3020   | 0 0  0  0  0 
│ │ │ │ +00035300: 2030 2020 2030 2020 3020 2061 207c 0a7c   0   0  0  a |.|
│ │ │ │ +00035310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035320: 2020 7b35 7d20 7c20 3020 3020 2030 2020    {5} | 0 0  0  
│ │ │ │ +00035330: 3020 2030 2030 2020 3020 2030 2030 2020  0  0 0  0  0 0  
│ │ │ │  00035340: 3020 2030 2020 3020 3020 2030 2020 3020  0  0  0 0  0  0 
│ │ │ │ -00035350: 2030 2030 2020 3020 2030 2020 3020 2020   0 0  0  0  0   
│ │ │ │ -00035360: 3020 2062 2020 3020 7c0a 7c20 2020 2020  0  b  0 |.|     
│ │ │ │ -00035370: 2020 2020 2020 2020 2020 2020 207b 357d               {5}
│ │ │ │ -00035380: 207c 2030 2030 2020 3020 2030 2020 3020   | 0 0  0  0  0 
│ │ │ │ -00035390: 3020 2030 2020 3020 3020 2030 2020 3020  0  0  0 0  0  0 
│ │ │ │ -000353a0: 2030 2030 2020 3020 2062 3220 3020 2020   0 0  0  b2 0   
│ │ │ │ -000353b0: 3020 202d 6320 3020 7c0a 7c20 2020 2020  0  -c 0 |.|     
│ │ │ │ -000353c0: 2020 2020 2020 2020 2020 2020 207b 357d               {5}
│ │ │ │ -000353d0: 207c 2030 2030 2020 3020 2030 2020 3020   | 0 0  0  0  0 
│ │ │ │ +00035350: 2030 2020 2030 2020 6220 2030 207c 0a7c   0   0  b  0 |.|
│ │ │ │ +00035360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035370: 2020 7b35 7d20 7c20 3020 3020 2030 2020    {5} | 0 0  0  
│ │ │ │ +00035380: 3020 2030 2030 2020 3020 2030 2030 2020  0  0 0  0  0 0  
│ │ │ │ +00035390: 3020 2030 2020 3020 3020 2030 2020 6232  0  0  0 0  0  b2
│ │ │ │ +000353a0: 2030 2020 2030 2020 2d63 2030 207c 0a7c   0   0  -c 0 |.|
│ │ │ │ +000353b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000353c0: 2020 7b35 7d20 7c20 3020 3020 2030 2020    {5} | 0 0  0  
│ │ │ │ +000353d0: 3020 2030 2030 2020 3020 2030 2030 2020  0  0 0  0  0 0  
│ │ │ │  000353e0: 3020 2030 2020 3020 3020 2030 2020 3020  0  0  0 0  0  0 
│ │ │ │ -000353f0: 2030 2030 2020 3020 2030 2020 2d62 3220   0 0  0  0  -b2 
│ │ │ │ -00035400: 3020 2061 2020 3020 7c0a 7c20 2020 2020  0  a  0 |.|     
│ │ │ │ +000353f0: 202d 6232 2030 2020 6120 2030 207c 0a7c   -b2 0  a  0 |.|
│ │ │ │ +00035400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035450: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00035440: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00035450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000354a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00035490: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000354a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000354b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000354c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000354d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000354e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000354f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000354e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000354f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035540: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00035530: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00035540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035590: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00035580: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00035590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000355a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000355b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000355c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000355d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000355e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000355d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000355e0: 2020 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d       -----------
│ │ │ │  000355f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00035d60: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00035d70: 3020 2020 3020 2030 2030 2020 3020 2030  0   0  0 0  0  0
│ │ │ │ -00035d80: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ +00035d50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00035d60: 2020 2020 2030 2020 2030 2020 3020 3020       0   0  0 0 
│ │ │ │ +00035d70: 2030 2020 3020 3020 3020 207c 2020 2020   0  0 0 0  |    
│ │ │ │ +00035d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035db0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00035dc0: 3020 2020 3020 2030 2030 2020 3020 2030  0   0  0 0  0  0
│ │ │ │ -00035dd0: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ +00035da0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00035db0: 2020 2020 2030 2020 2030 2020 3020 3020       0   0  0 0 
│ │ │ │ +00035dc0: 2030 2020 3020 3020 3020 207c 2020 2020   0  0 0 0  |    
│ │ │ │ +00035dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035e00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00035e10: 3020 2020 3020 2030 2030 2020 3020 2030  0   0  0 0  0  0
│ │ │ │ -00035e20: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ +00035df0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00035e00: 2020 2020 2030 2020 2030 2020 3020 3020       0   0  0 0 
│ │ │ │ +00035e10: 2030 2020 3020 3020 3020 207c 2020 2020   0  0 0 0  |    
│ │ │ │ +00035e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035e50: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00035e60: 3020 2020 3020 2030 2030 2020 3020 2030  0   0  0 0  0  0
│ │ │ │ -00035e70: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ +00035e40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00035e50: 2020 2020 2030 2020 2030 2020 3020 3020       0   0  0 0 
│ │ │ │ +00035e60: 2030 2020 3020 3020 3020 207c 2020 2020   0  0 0 0  |    
│ │ │ │ +00035e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035ea0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00035eb0: 3020 2020 3020 2030 2030 2020 3020 2030  0   0  0 0  0  0
│ │ │ │ -00035ec0: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ +00035e90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00035ea0: 2020 2020 2030 2020 2030 2020 3020 3020       0   0  0 0 
│ │ │ │ +00035eb0: 2030 2020 3020 3020 3020 207c 2020 2020   0  0 0 0  |    
│ │ │ │ +00035ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035ef0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00035f00: 3020 2020 3020 2030 2030 2020 3020 2030  0   0  0 0  0  0
│ │ │ │ -00035f10: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ +00035ee0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00035ef0: 2020 2020 2030 2020 2030 2020 3020 3020       0   0  0 0 
│ │ │ │ +00035f00: 2030 2020 3020 3020 3020 207c 2020 2020   0  0 0 0  |    
│ │ │ │ +00035f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035f40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00035f50: 3020 2020 3020 2030 2030 2020 3020 2030  0   0  0 0  0  0
│ │ │ │ -00035f60: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ +00035f30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00035f40: 2020 2020 2030 2020 2030 2020 3020 3020       0   0  0 0 
│ │ │ │ +00035f50: 2030 2020 3020 3020 3020 207c 2020 2020   0  0 0 0  |    
│ │ │ │ +00035f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035f90: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00035fa0: 3020 2020 3020 2030 2030 2020 3020 2030  0   0  0 0  0  0
│ │ │ │ -00035fb0: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ +00035f80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00035f90: 2020 2020 2030 2020 2030 2020 3020 3020       0   0  0 0 
│ │ │ │ +00035fa0: 2030 2020 3020 3020 3020 207c 2020 2020   0  0 0 0  |    
│ │ │ │ +00035fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035fe0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00035ff0: 3020 2020 3020 2030 2030 2020 3020 2030  0   0  0 0  0  0
│ │ │ │ -00036000: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ +00035fd0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00035fe0: 2020 2020 2030 2020 2030 2020 3020 3020       0   0  0 0 
│ │ │ │ +00035ff0: 2030 2020 3020 3020 3020 207c 2020 2020   0  0 0 0  |    
│ │ │ │ +00036000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036030: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00036040: 3020 2020 3020 2030 2030 2020 3020 2030  0   0  0 0  0  0
│ │ │ │ -00036050: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ +00036020: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00036030: 2020 2020 2030 2020 2030 2020 3020 3020       0   0  0 0 
│ │ │ │ +00036040: 2030 2020 3020 3020 3020 207c 2020 2020   0  0 0 0  |    
│ │ │ │ +00036050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036080: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00036090: 3020 2020 3020 2030 2030 2020 3020 2030  0   0  0 0  0  0
│ │ │ │ -000360a0: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ +00036070: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00036080: 2020 2020 2030 2020 2030 2020 3020 3020       0   0  0 0 
│ │ │ │ +00036090: 2030 2020 3020 3020 3020 207c 2020 2020   0  0 0 0  |    
│ │ │ │ +000360a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000360b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000360c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000360d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000360e0: 3020 2020 3020 2030 2030 2020 3020 2030  0   0  0 0  0  0
│ │ │ │ -000360f0: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ +000360c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000360d0: 2020 2020 2030 2020 2030 2020 3020 3020       0   0  0 0 
│ │ │ │ +000360e0: 2030 2020 3020 3020 3020 207c 2020 2020   0  0 0 0  |    
│ │ │ │ +000360f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036120: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00036130: 3020 2020 3020 2030 2061 3220 3020 2030  0   0  0 a2 0  0
│ │ │ │ -00036140: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ +00036110: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00036120: 2020 2020 2030 2020 2030 2020 3020 6132       0   0  0 a2
│ │ │ │ +00036130: 2030 2020 3020 3020 3020 207c 2020 2020   0  0 0 0  |    
│ │ │ │ +00036140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036170: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00036180: 6132 2020 6320 2062 2030 2020 3020 2030  a2  c  b 0  0  0
│ │ │ │ -00036190: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ +00036160: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00036170: 2020 2020 2061 3220 2063 2020 6220 3020       a2  c  b 0 
│ │ │ │ +00036180: 2030 2020 3020 3020 3020 207c 2020 2020   0  0 0 0  |    
│ │ │ │ +00036190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000361a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000361b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000361c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000361d0: 3020 2020 6120 2030 202d 6220 3020 2030  0   a  0 -b 0  0
│ │ │ │ -000361e0: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ +000361b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000361c0: 2020 2020 2030 2020 2061 2020 3020 2d62       0   a  0 -b
│ │ │ │ +000361d0: 2030 2020 3020 3020 3020 207c 2020 2020   0  0 0 0  |    
│ │ │ │ +000361e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000361f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036210: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00036220: 3020 2020 3020 2061 2063 2020 3020 2030  0   0  a c  0  0
│ │ │ │ -00036230: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ +00036200: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00036210: 2020 2020 2030 2020 2030 2020 6120 6320       0   0  a c 
│ │ │ │ +00036220: 2030 2020 3020 3020 3020 207c 2020 2020   0  0 0 0  |    
│ │ │ │ +00036230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036260: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00036270: 3020 2020 6232 2030 2030 2020 3020 2030  0   b2 0 0  0  0
│ │ │ │ -00036280: 2030 2061 3220 7c20 2020 2020 2020 2020   0 a2 |         
│ │ │ │ +00036250: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00036260: 2020 2020 2030 2020 2062 3220 3020 3020       0   b2 0 0 
│ │ │ │ +00036270: 2030 2020 3020 3020 6132 207c 2020 2020   0  0 0 a2 |    
│ │ │ │ +00036280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000362a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000362b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000362c0: 3020 2020 3020 2030 2030 2020 6132 2063  0   0  0 0  a2 c
│ │ │ │ -000362d0: 2062 2030 2020 7c20 2020 2020 2020 2020   b 0  |         
│ │ │ │ +000362a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000362b0: 2020 2020 2030 2020 2030 2020 3020 3020       0   0  0 0 
│ │ │ │ +000362c0: 2061 3220 6320 6220 3020 207c 2020 2020   a2 c b 0  |    
│ │ │ │ +000362d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000362e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000362f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036300: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00036310: 3020 2020 3020 2030 2030 2020 3020 2061  0   0  0 0  0  a
│ │ │ │ -00036320: 2030 202d 6220 7c20 2020 2020 2020 2020   0 -b |         
│ │ │ │ +000362f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00036300: 2020 2020 2030 2020 2030 2020 3020 3020       0   0  0 0 
│ │ │ │ +00036310: 2030 2020 6120 3020 2d62 207c 2020 2020   0  a 0 -b |    
│ │ │ │ +00036320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036350: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00036360: 2d62 3220 3020 2030 2030 2020 3020 2030  -b2 0  0 0  0  0
│ │ │ │ -00036370: 2061 2063 2020 7c20 2020 2020 2020 2020   a c  |         
│ │ │ │ +00036340: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00036350: 2020 2020 202d 6232 2030 2020 3020 3020       -b2 0  0 0 
│ │ │ │ +00036360: 2030 2020 3020 6120 6320 207c 2020 2020   0  0 a c  |    
│ │ │ │ +00036370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000363a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00036390: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000363a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000363b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000363c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000363d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000363e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000363f0: 2020 2020 2020 2020 7c0a 7c6f 3920 3a20          |.|o9 : 
│ │ │ │ -00036400: 4c69 7374 2020 2020 2020 2020 2020 2020  List            
│ │ │ │ +000363e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000363f0: 6f39 203a 204c 6973 7420 2020 2020 2020  o9 : List       
│ │ │ │ +00036400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036440: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00036430: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00036440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036490: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3130 203a  --------+.|i10 :
│ │ │ │ -000364a0: 2061 7070 6c79 2835 2c20 6a2d 3e20 7072   apply(5, j-> pr
│ │ │ │ -000364b0: 756e 6520 4848 5f6a 2043 203d 3d20 3029  une HH_j C == 0)
│ │ │ │ +00036480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00036490: 6931 3020 3a20 6170 706c 7928 352c 206a  i10 : apply(5, j
│ │ │ │ +000364a0: 2d3e 2070 7275 6e65 2048 485f 6a20 4320  -> prune HH_j C 
│ │ │ │ +000364b0: 3d3d 2030 2920 2020 2020 2020 2020 2020  == 0)           
│ │ │ │  000364c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000364d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000364e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000364d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000364e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000364f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036530: 2020 2020 2020 2020 7c0a 7c6f 3130 203d          |.|o10 =
│ │ │ │ -00036540: 207b 7472 7565 2c20 6661 6c73 652c 2066   {true, false, f
│ │ │ │ -00036550: 616c 7365 2c20 6661 6c73 652c 2066 616c  alse, false, fal
│ │ │ │ -00036560: 7365 7d20 2020 2020 2020 2020 2020 2020  se}             
│ │ │ │ -00036570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036580: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00036520: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00036530: 6f31 3020 3d20 7b74 7275 652c 2066 616c  o10 = {true, fal
│ │ │ │ +00036540: 7365 2c20 6661 6c73 652c 2066 616c 7365  se, false, false
│ │ │ │ +00036550: 2c20 6661 6c73 657d 2020 2020 2020 2020  , false}        
│ │ │ │ +00036560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036570: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00036580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000365a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000365b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000365c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000365d0: 2020 2020 2020 2020 7c0a 7c6f 3130 203a          |.|o10 :
│ │ │ │ -000365e0: 204c 6973 7420 2020 2020 2020 2020 2020   List           
│ │ │ │ +000365c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000365d0: 6f31 3020 3a20 4c69 7374 2020 2020 2020  o10 : List      
│ │ │ │ +000365e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000365f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036620: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00036610: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00036620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036630: 2d2d 2d2d 2d2d 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 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036670: 2d2d 2d2d 2d2d 2d2d 2b0a 0a41 6e64 206f  --------+..And o
│ │ │ │ -00036680: 6e65 2063 6f6d 7075 7469 6e67 2074 6865  ne computing the
│ │ │ │ -00036690: 2077 686f 6c65 2066 696e 6974 6520 7265   whole finite re
│ │ │ │ -000366a0: 736f 6c75 7469 6f6e 3a0a 0a2b 2d2d 2d2d  solution:..+----
│ │ │ │ +00036660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ +00036670: 416e 6420 6f6e 6520 636f 6d70 7574 696e  And one computin
│ │ │ │ +00036680: 6720 7468 6520 7768 6f6c 6520 6669 6e69  g the whole fini
│ │ │ │ +00036690: 7465 2072 6573 6f6c 7574 696f 6e3a 0a0a  te resolution:..
│ │ │ │ +000366a0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 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 2d2d 2d2d  ----------------
│ │ │ │ -000366e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000366f0: 2d2d 2d2b 0a7c 6931 3120 3a20 4d53 203d  ---+.|i11 : MS =
│ │ │ │ -00036700: 2070 7573 6846 6f72 7761 7264 286d 6170   pushForward(map
│ │ │ │ -00036710: 2852 2c53 292c 204d 293b 2020 2020 2020  (R,S), M);      
│ │ │ │ +000366e0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131 203a  --------+.|i11 :
│ │ │ │ +000366f0: 204d 5320 3d20 7075 7368 466f 7277 6172   MS = pushForwar
│ │ │ │ +00036700: 6428 6d61 7028 522c 5329 2c20 4d29 3b20  d(map(R,S), M); 
│ │ │ │ +00036710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036730: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00036730: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00036740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036780: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220 3a20  -------+.|i12 : 
│ │ │ │ -00036790: 2847 472c 2061 7567 2920 3d20 6c61 7965  (GG, aug) = laye
│ │ │ │ -000367a0: 7265 6452 6573 6f6c 7574 696f 6e28 6666  redResolution(ff
│ │ │ │ -000367b0: 2c4d 5329 2020 2020 2020 2020 2020 2020  ,MS)            
│ │ │ │ -000367c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000367d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00036770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00036780: 3132 203a 2028 4747 2c20 6175 6729 203d  12 : (GG, aug) =
│ │ │ │ +00036790: 206c 6179 6572 6564 5265 736f 6c75 7469   layeredResoluti
│ │ │ │ +000367a0: 6f6e 2866 662c 4d53 2920 2020 2020 2020  on(ff,MS)       
│ │ │ │ +000367b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000367c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000367d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000367e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000367f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036810: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00036820: 2020 2020 2020 3620 2020 2020 2031 3320        6      13 
│ │ │ │ -00036830: 2020 2020 2031 3020 2020 2020 2033 2020       10      3  
│ │ │ │ +00036810: 7c0a 7c20 2020 2020 2020 2036 2020 2020  |.|        6    
│ │ │ │ +00036820: 2020 3133 2020 2020 2020 3130 2020 2020    13      10    
│ │ │ │ +00036830: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │  00036840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036860: 2020 2020 207c 0a7c 6f31 3220 3d20 2853       |.|o12 = (S
│ │ │ │ -00036870: 2020 3c2d 2d20 5320 2020 3c2d 2d20 5320    <-- S   <-- S 
│ │ │ │ -00036880: 2020 3c2d 2d20 5320 2c20 7b32 7d20 7c20    <-- S , {2} | 
│ │ │ │ -00036890: 3020 3020 3020 3020 2030 2020 3120 7c29  0 0 0 0  0  1 |)
│ │ │ │ -000368a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000368b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000368c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000368d0: 2020 2020 7b32 7d20 7c20 3020 3020 3020      {2} | 0 0 0 
│ │ │ │ -000368e0: 2d31 2030 2020 3020 7c20 2020 2020 2020  -1 0  0 |       
│ │ │ │ -000368f0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00036900: 2020 2030 2020 2020 2020 3120 2020 2020     0      1     
│ │ │ │ -00036910: 2020 3220 2020 2020 2020 3320 2020 7b32    2       3   {2
│ │ │ │ -00036920: 7d20 7c20 3020 3020 3020 3020 202d 3120  } | 0 0 0 0  -1 
│ │ │ │ -00036930: 3020 7c20 2020 2020 2020 2020 2020 2020  0 |             
│ │ │ │ -00036940: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00036950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036960: 2020 2020 2020 2020 7b33 7d20 7c20 3120          {3} | 1 
│ │ │ │ -00036970: 3020 3020 3020 2030 2020 3020 7c20 2020  0 0 0  0  0 |   
│ │ │ │ -00036980: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00036850: 2020 2020 2020 2020 2020 7c0a 7c6f 3132            |.|o12
│ │ │ │ +00036860: 203d 2028 5320 203c 2d2d 2053 2020 203c   = (S  <-- S   <
│ │ │ │ +00036870: 2d2d 2053 2020 203c 2d2d 2053 202c 207b  -- S   <-- S , {
│ │ │ │ +00036880: 327d 207c 2030 2030 2030 2030 2020 3020  2} | 0 0 0 0  0 
│ │ │ │ +00036890: 2031 207c 2920 2020 2020 2020 2020 2020   1 |)           
│ │ │ │ +000368a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000368b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000368c0: 2020 2020 2020 2020 207b 327d 207c 2030           {2} | 0
│ │ │ │ +000368d0: 2030 2030 202d 3120 3020 2030 207c 2020   0 0 -1 0  0 |  
│ │ │ │ +000368e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000368f0: 7c20 2020 2020 2020 3020 2020 2020 2031  |       0      1
│ │ │ │ +00036900: 2020 2020 2020 2032 2020 2020 2020 2033         2       3
│ │ │ │ +00036910: 2020 207b 327d 207c 2030 2030 2030 2030     {2} | 0 0 0 0
│ │ │ │ +00036920: 2020 2d31 2030 207c 2020 2020 2020 2020    -1 0 |        
│ │ │ │ +00036930: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00036940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036950: 2020 2020 2020 2020 2020 2020 207b 337d               {3}
│ │ │ │ +00036960: 207c 2031 2030 2030 2030 2020 3020 2030   | 1 0 0 0  0  0
│ │ │ │ +00036970: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ +00036980: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00036990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000369a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000369b0: 2020 7b33 7d20 7c20 3020 3120 3020 3020    {3} | 0 1 0 0 
│ │ │ │ -000369c0: 2030 2020 3020 7c20 2020 2020 2020 2020   0  0 |         
│ │ │ │ -000369d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000369a0: 2020 2020 2020 207b 337d 207c 2030 2031         {3} | 0 1
│ │ │ │ +000369b0: 2030 2030 2020 3020 2030 207c 2020 2020   0 0  0  0 |    
│ │ │ │ +000369c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000369d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000369e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000369f0: 2020 2020 2020 2020 2020 2020 7b33 7d20              {3} 
│ │ │ │ -00036a00: 7c20 3020 3020 3120 3020 2030 2020 3020  | 0 0 1 0  0  0 
│ │ │ │ -00036a10: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00036a20: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000369f0: 207b 337d 207c 2030 2030 2031 2030 2020   {3} | 0 0 1 0  
│ │ │ │ +00036a00: 3020 2030 207c 2020 2020 2020 2020 2020  0  0 |          
│ │ │ │ +00036a10: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00036a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036a60: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -00036a70: 3220 3a20 5365 7175 656e 6365 2020 2020  2 : Sequence    
│ │ │ │ +00036a60: 7c0a 7c6f 3132 203a 2053 6571 7565 6e63  |.|o12 : Sequenc
│ │ │ │ +00036a70: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │  00036a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036ab0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00036aa0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00036ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00036b00: 0a7c 6931 3320 3a20 2847 472c 2061 7567  .|i13 : (GG, aug
│ │ │ │ -00036b10: 2920 3d20 6c61 7965 7265 6452 6573 6f6c  ) = layeredResol
│ │ │ │ -00036b20: 7574 696f 6e28 6666 2c4d 532c 2056 6572  ution(ff,MS, Ver
│ │ │ │ -00036b30: 626f 7365 203d 3e74 7275 6529 2020 2020  bose =>true)    
│ │ │ │ -00036b40: 2020 2020 2020 2020 207c 0a7c 7b33 2c20           |.|{3, 
│ │ │ │ -00036b50: 317d 2069 6e20 636f 6469 6d65 6e73 696f  1} in codimensio
│ │ │ │ -00036b60: 6e20 3320 2020 2020 2020 2020 2020 2020  n 3             
│ │ │ │ +00036af0: 2d2d 2d2d 2b0a 7c69 3133 203a 2028 4747  ----+.|i13 : (GG
│ │ │ │ +00036b00: 2c20 6175 6729 203d 206c 6179 6572 6564  , aug) = layered
│ │ │ │ +00036b10: 5265 736f 6c75 7469 6f6e 2866 662c 4d53  Resolution(ff,MS
│ │ │ │ +00036b20: 2c20 5665 7262 6f73 6520 3d3e 7472 7565  , Verbose =>true
│ │ │ │ +00036b30: 2920 2020 2020 2020 2020 2020 2020 7c0a  )             |.
│ │ │ │ +00036b40: 7c7b 332c 2031 7d20 696e 2063 6f64 696d  |{3, 1} in codim
│ │ │ │ +00036b50: 656e 7369 6f6e 2033 2020 2020 2020 2020  ension 3        
│ │ │ │ +00036b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036b90: 2020 207c 0a7c 7b33 2c20 317d 2069 6e20     |.|{3, 1} in 
│ │ │ │ -00036ba0: 636f 6469 6d65 6e73 696f 6e20 3220 2020  codimension 2   
│ │ │ │ +00036b80: 2020 2020 2020 2020 7c0a 7c7b 332c 2031          |.|{3, 1
│ │ │ │ +00036b90: 7d20 696e 2063 6f64 696d 656e 7369 6f6e  } in codimension
│ │ │ │ +00036ba0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │  00036bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036bd0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00036bd0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00036be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036c20: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00036c30: 2020 3620 2020 2020 2031 3320 2020 2020    6      13     
│ │ │ │ -00036c40: 2031 3020 2020 2020 2033 2020 2020 2020   10      3      
│ │ │ │ +00036c10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00036c20: 2020 2020 2020 2036 2020 2020 2020 3133         6      13
│ │ │ │ +00036c30: 2020 2020 2020 3130 2020 2020 2020 3320        10      3 
│ │ │ │ +00036c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036c70: 207c 0a7c 6f31 3320 3d20 2853 2020 3c2d   |.|o13 = (S  <-
│ │ │ │ -00036c80: 2d20 5320 2020 3c2d 2d20 5320 2020 3c2d  - S   <-- S   <-
│ │ │ │ -00036c90: 2d20 5320 2c20 7b32 7d20 7c20 3020 3020  - S , {2} | 0 0 
│ │ │ │ -00036ca0: 3020 3020 2030 2020 3120 7c29 2020 2020  0 0  0  1 |)    
│ │ │ │ -00036cb0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00036c60: 2020 2020 2020 7c0a 7c6f 3133 203d 2028        |.|o13 = (
│ │ │ │ +00036c70: 5320 203c 2d2d 2053 2020 203c 2d2d 2053  S  <-- S   <-- S
│ │ │ │ +00036c80: 2020 203c 2d2d 2053 202c 207b 327d 207c     <-- S , {2} |
│ │ │ │ +00036c90: 2030 2030 2030 2030 2020 3020 2031 207c   0 0 0 0  0  1 |
│ │ │ │ +00036ca0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +00036cb0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00036cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036ce0: 7b32 7d20 7c20 3020 3020 3020 2d31 2030  {2} | 0 0 0 -1 0
│ │ │ │ -00036cf0: 2020 3020 7c20 2020 2020 2020 2020 2020    0 |           
│ │ │ │ -00036d00: 2020 2020 207c 0a7c 2020 2020 2020 2030       |.|       0
│ │ │ │ -00036d10: 2020 2020 2020 3120 2020 2020 2020 3220        1       2 
│ │ │ │ -00036d20: 2020 2020 2020 3320 2020 7b32 7d20 7c20        3   {2} | 
│ │ │ │ -00036d30: 3020 3020 3020 3020 202d 3120 3020 7c20  0 0 0 0  -1 0 | 
│ │ │ │ -00036d40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00036d50: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00036d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036d70: 2020 2020 7b33 7d20 7c20 3120 3020 3020      {3} | 1 0 0 
│ │ │ │ -00036d80: 3020 2030 2020 3020 7c20 2020 2020 2020  0  0  0 |       
│ │ │ │ -00036d90: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00036cd0: 2020 2020 207b 327d 207c 2030 2030 2030       {2} | 0 0 0
│ │ │ │ +00036ce0: 202d 3120 3020 2030 207c 2020 2020 2020   -1 0  0 |      
│ │ │ │ +00036cf0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00036d00: 2020 2020 3020 2020 2020 2031 2020 2020      0      1    
│ │ │ │ +00036d10: 2020 2032 2020 2020 2020 2033 2020 207b     2       3   {
│ │ │ │ +00036d20: 327d 207c 2030 2030 2030 2030 2020 2d31  2} | 0 0 0 0  -1
│ │ │ │ +00036d30: 2030 207c 2020 2020 2020 2020 2020 2020   0 |            
│ │ │ │ +00036d40: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00036d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036d60: 2020 2020 2020 2020 207b 337d 207c 2031           {3} | 1
│ │ │ │ +00036d70: 2030 2030 2030 2020 3020 2030 207c 2020   0 0 0  0  0 |  
│ │ │ │ +00036d80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00036d90: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00036da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036db0: 2020 2020 2020 2020 2020 2020 2020 7b33                {3
│ │ │ │ -00036dc0: 7d20 7c20 3020 3120 3020 3020 2030 2020  } | 0 1 0 0  0  
│ │ │ │ -00036dd0: 3020 7c20 2020 2020 2020 2020 2020 2020  0 |             
│ │ │ │ -00036de0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00036df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036e00: 2020 2020 2020 2020 7b33 7d20 7c20 3020          {3} | 0 
│ │ │ │ -00036e10: 3020 3120 3020 2030 2020 3020 7c20 2020  0 1 0  0  0 |   
│ │ │ │ -00036e20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00036db0: 2020 207b 337d 207c 2030 2031 2030 2030     {3} | 0 1 0 0
│ │ │ │ +00036dc0: 2020 3020 2030 207c 2020 2020 2020 2020    0  0 |        
│ │ │ │ +00036dd0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00036de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036df0: 2020 2020 2020 2020 2020 2020 207b 337d               {3}
│ │ │ │ +00036e00: 207c 2030 2030 2031 2030 2020 3020 2030   | 0 0 1 0  0  0
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│ │ │ │ +00036e20: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00036e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036e40: 2020 2020 2020 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 207c 0a7c 6f31 3320 3a20         |.|o13 : 
│ │ │ │ -00036e80: 5365 7175 656e 6365 2020 2020 2020 2020  Sequence        
│ │ │ │ +00036e60: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00036e70: 3133 203a 2053 6571 7565 6e63 6520 2020  13 : Sequence   
│ │ │ │ +00036e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036ec0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00036eb0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00036ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -00036f10: 3420 3a20 6265 7474 6920 4747 2020 2020  4 : betti GG    
│ │ │ │ +00036f00: 2b0a 7c69 3134 203a 2062 6574 7469 2047  +.|i14 : betti G
│ │ │ │ +00036f10: 4720 2020 2020 2020 2020 2020 2020 2020  G               
│ │ │ │  00036f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036f50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00036f40: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00036f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036f60: 2020 2020 2020 2020 2020 2020 2020 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 2020 2020 2020 2020 2020 2020 2030  .|             0
│ │ │ │ -00036fb0: 2020 3120 2032 2033 2020 2020 2020 2020    1  2 3        
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│ │ │ │ +00036fa0: 2020 2020 3020 2031 2020 3220 3320 2020      0  1  2 3   
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│ │ │ │  00036fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036fe0: 2020 2020 2020 2020 207c 0a7c 6f31 3420           |.|o14 
│ │ │ │ -00036ff0: 3d20 746f 7461 6c3a 2036 2031 3320 3130  = total: 6 13 10
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│ │ │ │ +00036fe0: 7c6f 3134 203d 2074 6f74 616c 3a20 3620  |o14 = total: 6 
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│ │ │ │  00037010: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00037040: 323a 2033 2020 3120 202e 202e 2020 2020  2: 3  1  . .    
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│ │ │ │ +00037030: 2020 2020 2032 3a20 3320 2031 2020 2e20       2: 3  1  . 
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│ │ │ │  00037050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037070: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │ -00037090: 3920 202e 202e 2020 2020 2020 2020 2020  9  . .          
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│ │ │ │ -000370d0: 2020 2020 343a 202e 2020 2e20 202e 202e      4: .  .  . .
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│ │ │ │ -00037100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037110: 207c 0a7c 2020 2020 2020 2020 2020 353a   |.|          5:
│ │ │ │ -00037120: 202e 2020 3320 2039 202e 2020 2020 2020   .  3  9 .      
│ │ │ │ +00037100: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00037110: 2020 2035 3a20 2e20 2033 2020 3920 2e20     5: .  3  9 . 
│ │ │ │ +00037120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037150: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00037160: 2020 2020 2020 2020 363a 202e 2020 2e20          6: .  . 
│ │ │ │ -00037170: 202e 202e 2020 2020 2020 2020 2020 2020   . .            
│ │ │ │ +00037150: 7c0a 7c20 2020 2020 2020 2020 2036 3a20  |.|          6: 
│ │ │ │ +00037160: 2e20 202e 2020 2e20 2e20 2020 2020 2020  .  .  . .       
│ │ │ │ +00037170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000371a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000371b0: 2020 373a 202e 2020 2e20 2031 2033 2020    7: .  .  1 3  
│ │ │ │ +00037190: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000371a0: 2020 2020 2020 2037 3a20 2e20 202e 2020         7: .  .  
│ │ │ │ +000371b0: 3120 3320 2020 2020 2020 2020 2020 2020  1 3             
│ │ │ │  000371c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000371d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000371e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000371f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000371e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000371f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037230: 2020 2020 2020 2020 207c 0a7c 6f31 3420           |.|o14 
│ │ │ │ -00037240: 3a20 4265 7474 6954 616c 6c79 2020 2020  : BettiTally    
│ │ │ │ +00037220: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00037230: 7c6f 3134 203a 2042 6574 7469 5461 6c6c  |o14 : BettiTall
│ │ │ │ +00037240: 7920 2020 2020 2020 2020 2020 2020 2020  y               
│ │ │ │  00037250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037280: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00037270: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00037280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000372a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000372b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000372c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -000372d0: 6931 3520 3a20 6265 7474 6920 6672 6565  i15 : betti free
│ │ │ │ -000372e0: 5265 736f 6c75 7469 6f6e 204d 5320 2020  Resolution MS   
│ │ │ │ +000372c0: 2d2d 2b0a 7c69 3135 203a 2062 6574 7469  --+.|i15 : betti
│ │ │ │ +000372d0: 2066 7265 6552 6573 6f6c 7574 696f 6e20   freeResolution 
│ │ │ │ +000372e0: 4d53 2020 2020 2020 2020 2020 2020 2020  MS              
│ │ │ │  000372f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037310: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00037300: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00037310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037360: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00037370: 2030 2020 3120 2032 2033 2020 2020 2020   0  1  2 3      
│ │ │ │ +00037350: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00037360: 2020 2020 2020 3020 2031 2020 3220 3320        0  1  2 3 
│ │ │ │ +00037370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000373a0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -000373b0: 3520 3d20 746f 7461 6c3a 2036 2031 3320  5 = total: 6 13 
│ │ │ │ -000373c0: 3130 2033 2020 2020 2020 2020 2020 2020  10 3            
│ │ │ │ +000373a0: 7c0a 7c6f 3135 203d 2074 6f74 616c 3a20  |.|o15 = total: 
│ │ │ │ +000373b0: 3620 3133 2031 3020 3320 2020 2020 2020  6 13 10 3       
│ │ │ │ +000373c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000373d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000373e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000373f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00037400: 2020 323a 2033 2020 3120 202e 202e 2020    2: 3  1  . .  
│ │ │ │ +000373e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000373f0: 2020 2020 2020 2032 3a20 3320 2031 2020         2: 3  1  
│ │ │ │ +00037400: 2e20 2e20 2020 2020 2020 2020 2020 2020  . .             
│ │ │ │  00037410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037430: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00037440: 0a7c 2020 2020 2020 2020 2020 333a 2033  .|          3: 3
│ │ │ │ -00037450: 2020 3920 202e 202e 2020 2020 2020 2020    9  . .        
│ │ │ │ +00037430: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00037440: 2033 3a20 3320 2039 2020 2e20 2e20 2020   3: 3  9  . .   
│ │ │ │ +00037450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037480: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00037490: 2020 2020 2020 343a 202e 2020 2e20 202e        4: .  .  .
│ │ │ │ -000374a0: 202e 2020 2020 2020 2020 2020 2020 2020   .              
│ │ │ │ +00037470: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00037480: 7c20 2020 2020 2020 2020 2034 3a20 2e20  |          4: . 
│ │ │ │ +00037490: 202e 2020 2e20 2e20 2020 2020 2020 2020   .  . .         
│ │ │ │ +000374a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000374b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000374c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000374d0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -000374e0: 353a 202e 2020 3320 2039 202e 2020 2020  5: .  3  9 .    
│ │ │ │ +000374c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000374d0: 2020 2020 2035 3a20 2e20 2033 2020 3920       5: .  3  9 
│ │ │ │ +000374e0: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │  000374f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037510: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00037520: 2020 2020 2020 2020 2020 363a 202e 2020            6: .  
│ │ │ │ -00037530: 2e20 202e 202e 2020 2020 2020 2020 2020  .  . .          
│ │ │ │ +00037510: 2020 7c0a 7c20 2020 2020 2020 2020 2036    |.|          6
│ │ │ │ +00037520: 3a20 2e20 202e 2020 2e20 2e20 2020 2020  : .  .  . .     
│ │ │ │ +00037530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037560: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00037570: 2020 2020 373a 202e 2020 2e20 2031 2033      7: .  .  1 3
│ │ │ │ +00037550: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00037560: 2020 2020 2020 2020 2037 3a20 2e20 202e           7: .  .
│ │ │ │ +00037570: 2020 3120 3320 2020 2020 2020 2020 2020    1 3           
│ │ │ │  00037580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000375a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000375b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000375a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000375b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000375c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000375d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000375e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000375f0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -00037600: 3520 3a20 4265 7474 6954 616c 6c79 2020  5 : BettiTally  
│ │ │ │ +000375f0: 7c0a 7c6f 3135 203a 2042 6574 7469 5461  |.|o15 : BettiTa
│ │ │ │ +00037600: 6c6c 7920 2020 2020 2020 2020 2020 2020  lly             
│ │ │ │  00037610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037640: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00037630: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00037640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00037680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00037690: 0a7c 6931 3620 3a20 4320 3d20 636f 6d70  .|i16 : C = comp
│ │ │ │ -000376a0: 6c65 7820 666c 6174 7465 6e20 7b7b 6175  lex flatten {{au
│ │ │ │ -000376b0: 677d 207c 6170 706c 7928 6c65 6e67 7468  g} |apply(length
│ │ │ │ -000376c0: 2047 4720 2d31 2c20 692d 3e20 4747 2e64   GG -1, i-> GG.d
│ │ │ │ -000376d0: 645f 2869 2b31 2929 7d7c 0a7c 2020 2020  d_(i+1))}|.|    
│ │ │ │ +00037680: 2d2d 2d2d 2b0a 7c69 3136 203a 2043 203d  ----+.|i16 : C =
│ │ │ │ +00037690: 2063 6f6d 706c 6578 2066 6c61 7474 656e   complex flatten
│ │ │ │ +000376a0: 207b 7b61 7567 7d20 7c61 7070 6c79 286c   {{aug} |apply(l
│ │ │ │ +000376b0: 656e 6774 6820 4747 202d 312c 2069 2d3e  ength GG -1, i->
│ │ │ │ +000376c0: 2047 472e 6464 5f28 692b 3129 297d 7c0a   GG.dd_(i+1))}|.
│ │ │ │ +000376d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000376e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000376f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037720: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00037730: 2020 2020 3620 2020 2020 2031 3320 2020      6      13   
│ │ │ │ -00037740: 2020 2031 3020 2020 2020 2020 2020 2020     10           
│ │ │ │ +00037710: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037720: 2020 2020 2020 2020 2036 2020 2020 2020           6      
│ │ │ │ +00037730: 3133 2020 2020 2020 3130 2020 2020 2020  13      10      
│ │ │ │ +00037740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037760: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00037770: 6f31 3620 3d20 4d53 203c 2d2d 2053 2020  o16 = MS <-- S  
│ │ │ │ -00037780: 3c2d 2d20 5320 2020 3c2d 2d20 5320 2020  <-- S   <-- S   
│ │ │ │ +00037760: 2020 7c0a 7c6f 3136 203d 204d 5320 3c2d    |.|o16 = MS <-
│ │ │ │ +00037770: 2d20 5320 203c 2d2d 2053 2020 203c 2d2d  - S  <-- S   <--
│ │ │ │ +00037780: 2053 2020 2020 2020 2020 2020 2020 2020   S              
│ │ │ │  00037790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000377a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000377b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000377a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000377b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000377c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000377d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000377e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000377f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037800: 207c 0a7c 2020 2020 2020 3020 2020 2020   |.|      0     
│ │ │ │ -00037810: 2031 2020 2020 2020 3220 2020 2020 2020   1      2       
│ │ │ │ -00037820: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ +000377f0: 2020 2020 2020 7c0a 7c20 2020 2020 2030        |.|      0
│ │ │ │ +00037800: 2020 2020 2020 3120 2020 2020 2032 2020        1      2  
│ │ │ │ +00037810: 2020 2020 2033 2020 2020 2020 2020 2020       3          
│ │ │ │ +00037820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037840: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00037840: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00037850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037890: 2020 2020 207c 0a7c 6f31 3620 3a20 436f       |.|o16 : Co
│ │ │ │ -000378a0: 6d70 6c65 7820 2020 2020 2020 2020 2020  mplex           
│ │ │ │ +00037880: 2020 2020 2020 2020 2020 7c0a 7c6f 3136            |.|o16
│ │ │ │ +00037890: 203a 2043 6f6d 706c 6578 2020 2020 2020   : Complex      
│ │ │ │ +000378a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000378b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000378c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000378d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000378e0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000378d0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +000378e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000378f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00037910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00037920: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3720  ---------+.|i17 
│ │ │ │ -00037930: 3a20 6170 706c 7928 6c65 6e67 7468 2047  : apply(length G
│ │ │ │ -00037940: 4720 2b31 202c 206a 2d3e 2070 7275 6e65  G +1 , j-> prune
│ │ │ │ -00037950: 2048 485f 6a20 4320 3d3d 2030 2920 2020   HH_j C == 0)   
│ │ │ │ -00037960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037970: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00037910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00037920: 7c69 3137 203a 2061 7070 6c79 286c 656e  |i17 : apply(len
│ │ │ │ +00037930: 6774 6820 4747 202b 3120 2c20 6a2d 3e20  gth GG +1 , j-> 
│ │ │ │ +00037940: 7072 756e 6520 4848 5f6a 2043 203d 3d20  prune HH_j C == 
│ │ │ │ +00037950: 3029 2020 2020 2020 2020 2020 2020 2020  0)              
│ │ │ │ +00037960: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000379a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000379b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000379c0: 6f31 3720 3d20 7b74 7275 652c 2074 7275  o17 = {true, tru
│ │ │ │ -000379d0: 652c 2074 7275 652c 2066 616c 7365 7d20  e, true, false} 
│ │ │ │ +000379b0: 2020 7c0a 7c6f 3137 203d 207b 7472 7565    |.|o17 = {true
│ │ │ │ +000379c0: 2c20 7472 7565 2c20 7472 7565 2c20 6661  , true, true, fa
│ │ │ │ +000379d0: 6c73 657d 2020 2020 2020 2020 2020 2020  lse}            
│ │ │ │  000379e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000379f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037a00: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000379f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00037a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037a50: 207c 0a7c 6f31 3720 3a20 4c69 7374 2020   |.|o17 : List  
│ │ │ │ +00037a40: 2020 2020 2020 7c0a 7c6f 3137 203a 204c        |.|o17 : L
│ │ │ │ +00037a50: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │  00037a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037a90: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00037a90: 7c0a 2b2d 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 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00037ae0: 2d2d 2d2d 2d2b 0a0a 5761 7973 2074 6f20  -----+..Ways to 
│ │ │ │ -00037af0: 7573 6520 6c61 7965 7265 6452 6573 6f6c  use layeredResol
│ │ │ │ -00037b00: 7574 696f 6e3a 0a3d 3d3d 3d3d 3d3d 3d3d  ution:.=========
│ │ │ │ -00037b10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00037b20: 3d3d 3d3d 3d0a 0a20 202a 2022 6c61 7965  =====..  * "laye
│ │ │ │ -00037b30: 7265 6452 6573 6f6c 7574 696f 6e28 4d61  redResolution(Ma
│ │ │ │ -00037b40: 7472 6978 2c4d 6f64 756c 6529 220a 2020  trix,Module)".  
│ │ │ │ -00037b50: 2a20 226c 6179 6572 6564 5265 736f 6c75  * "layeredResolu
│ │ │ │ -00037b60: 7469 6f6e 284d 6174 7269 782c 4d6f 6475  tion(Matrix,Modu
│ │ │ │ -00037b70: 6c65 2c5a 5a29 220a 0a46 6f72 2074 6865  le,ZZ)"..For the
│ │ │ │ -00037b80: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d   programmer.====
│ │ │ │ -00037b90: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ -00037ba0: 5468 6520 6f62 6a65 6374 202a 6e6f 7465  The object *note
│ │ │ │ -00037bb0: 206c 6179 6572 6564 5265 736f 6c75 7469   layeredResoluti
│ │ │ │ -00037bc0: 6f6e 3a20 6c61 7965 7265 6452 6573 6f6c  on: layeredResol
│ │ │ │ -00037bd0: 7574 696f 6e2c 2069 7320 6120 2a6e 6f74  ution, is a *not
│ │ │ │ -00037be0: 6520 6d65 7468 6f64 0a66 756e 6374 696f  e method.functio
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│ │ │ │  00037f20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00037f30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ +00038180: 700a 0a6d 616b 6546 696e 6974 6552 6573  p..makeFiniteRes
│ │ │ │ +00038190: 6f6c 7574 696f 6e20 2d2d 2066 696e 6974  olution -- finit
│ │ │ │ +000381a0: 6520 7265 736f 6c75 7469 6f6e 206f 6620  e resolution of 
│ │ │ │ +000381b0: 6120 6d61 7472 6978 2066 6163 746f 7269  a matrix factori
│ │ │ │ +000381c0: 7a61 7469 6f6e 206d 6f64 756c 6520 4d0a  zation module M.
│ │ │ │ +000381d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000381e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000381f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00038200: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00038210: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00038220: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 7361  *******..  * Usa
│ │ │ │ -00038230: 6765 3a20 0a20 2020 2020 2020 2041 203d  ge: .        A =
│ │ │ │ -00038240: 206d 616b 6546 696e 6974 6552 6573 6f6c   makeFiniteResol
│ │ │ │ -00038250: 7574 696f 6e28 6666 2c6d 6629 0a20 202a  ution(ff,mf).  *
│ │ │ │ -00038260: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ -00038270: 206d 662c 2061 202a 6e6f 7465 206c 6973   mf, a *note lis
│ │ │ │ -00038280: 743a 2028 4d61 6361 756c 6179 3244 6f63  t: (Macaulay2Doc
│ │ │ │ -00038290: 294c 6973 742c 2c20 6f75 7470 7574 206f  )List,, output o
│ │ │ │ -000382a0: 6620 6d61 7472 6978 4661 6374 6f72 697a  f matrixFactoriz
│ │ │ │ -000382b0: 6174 696f 6e0a 2020 2020 2020 2a20 6666  ation.      * ff
│ │ │ │ -000382c0: 2c20 6120 2a6e 6f74 6520 6d61 7472 6978  , a *note matrix
│ │ │ │ -000382d0: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ -000382e0: 4d61 7472 6978 2c2c 2074 6865 2072 6567  Matrix,, the reg
│ │ │ │ -000382f0: 756c 6172 2073 6571 7565 6e63 6520 7573  ular sequence us
│ │ │ │ -00038300: 6564 0a20 2020 2020 2020 2066 6f72 2074  ed.        for t
│ │ │ │ -00038310: 6865 206d 6174 7269 7846 6163 746f 7269  he matrixFactori
│ │ │ │ -00038320: 7a61 7469 6f6e 2063 6f6d 7075 7461 7469  zation computati
│ │ │ │ -00038330: 6f6e 0a20 202a 204f 7574 7075 7473 3a0a  on.  * Outputs:.
│ │ │ │ -00038340: 2020 2020 2020 2a20 412c 2061 202a 6e6f        * A, a *no
│ │ │ │ -00038350: 7465 2063 6f6d 706c 6578 3a20 2843 6f6d  te complex: (Com
│ │ │ │ -00038360: 706c 6578 6573 2943 6f6d 706c 6578 2c2c  plexes)Complex,,
│ │ │ │ -00038370: 2041 2069 7320 7468 6520 6d69 6e69 6d61   A is the minima
│ │ │ │ -00038380: 6c20 6669 6e69 7465 0a20 2020 2020 2020  l finite.       
│ │ │ │ -00038390: 2072 6573 6f6c 7574 696f 6e20 6f66 204d   resolution of M
│ │ │ │ -000383a0: 206f 7665 7220 522e 0a0a 4465 7363 7269   over R...Descri
│ │ │ │ -000383b0: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ -000383c0: 3d0a 0a53 7570 706f 7365 2074 6861 7420  =..Suppose that 
│ │ │ │ -000383d0: 665f 312e 2e66 5f63 2069 7320 6120 686f  f_1..f_c is a ho
│ │ │ │ -000383e0: 6d6f 6765 6e65 6f75 7320 7265 6775 6c61  mogeneous regula
│ │ │ │ -000383f0: 7220 7365 7175 656e 6365 206f 6620 666f  r sequence of fo
│ │ │ │ -00038400: 726d 7320 6f66 2074 6865 2073 616d 650a  rms of the same.
│ │ │ │ -00038410: 6465 6772 6565 2069 6e20 6120 706f 6c79  degree in a poly
│ │ │ │ -00038420: 6e6f 6d69 616c 2072 696e 6720 5320 616e  nomial ring S an
│ │ │ │ -00038430: 6420 4d20 6973 2061 2068 6967 6820 7379  d M is a high sy
│ │ │ │ -00038440: 7a79 6779 206d 6f64 756c 6520 6f76 6572  zygy module over
│ │ │ │ -00038450: 2053 2f28 665f 312c 2e2e 2c66 5f63 290a   S/(f_1,..,f_c).
│ │ │ │ -00038460: 3d20 5228 6329 2c20 616e 6420 6d66 203d  = R(c), and mf =
│ │ │ │ -00038470: 2028 642c 6829 2069 7320 7468 6520 6f75   (d,h) is the ou
│ │ │ │ -00038480: 7470 7574 206f 6620 6d61 7472 6978 4661  tput of matrixFa
│ │ │ │ -00038490: 6374 6f72 697a 6174 696f 6e28 4d2c 6666  ctorization(M,ff
│ │ │ │ -000384a0: 292e 2049 6620 7468 650a 636f 6d70 6c65  ). If the.comple
│ │ │ │ -000384b0: 7869 7479 206f 6620 4d20 6973 2063 272c  xity of M is c',
│ │ │ │ -000384c0: 2074 6865 6e20 4d20 6861 7320 6120 6669   then M has a fi
│ │ │ │ -000384d0: 6e69 7465 2066 7265 6520 7265 736f 6c75  nite free resolu
│ │ │ │ -000384e0: 7469 6f6e 206f 7665 7220 5220 3d0a 532f  tion over R =.S/
│ │ │ │ -000384f0: 2866 5f31 2c2e 2e2c 665f 7b28 632d 6327  (f_1,..,f_{(c-c'
│ │ │ │ -00038500: 297d 2920 2861 6e64 2c20 6d6f 7265 2067  )}) (and, more g
│ │ │ │ -00038510: 656e 6572 616c 6c79 2c20 6861 7320 636f  enerally, has co
│ │ │ │ -00038520: 6d70 6c65 7869 7479 2063 2d64 206f 7665  mplexity c-d ove
│ │ │ │ -00038530: 720a 532f 2866 5f31 2c2e 2e2c 665f 7b28  r.S/(f_1,..,f_{(
│ │ │ │ -00038540: 632d 6429 7d29 2066 6f72 2064 3e3d 6327  c-d)}) for d>=c'
│ │ │ │ -00038550: 292e 0a0a 5468 6520 636f 6d70 6c65 7820  )...The complex 
│ │ │ │ -00038560: 4120 6973 2074 6865 206d 696e 696d 616c  A is the minimal
│ │ │ │ -00038570: 2066 696e 6974 6520 6672 6565 2072 6573   finite free res
│ │ │ │ -00038580: 6f6c 7574 696f 6e20 6f66 204d 206f 7665  olution of M ove
│ │ │ │ -00038590: 7220 412c 2063 6f6e 7374 7275 6374 6564  r A, constructed
│ │ │ │ -000385a0: 2061 730a 616e 2069 7465 7261 7465 6420   as.an iterated 
│ │ │ │ -000385b0: 4b6f 737a 756c 2065 7874 656e 7369 6f6e  Koszul extension
│ │ │ │ -000385c0: 2c20 6d61 6465 2066 726f 6d20 7468 6520  , made from the 
│ │ │ │ -000385d0: 6d61 7073 2069 6e20 624d 6170 7320 6d66  maps in bMaps mf
│ │ │ │ -000385e0: 2061 6e64 2070 7369 4d61 7073 206d 662c   and psiMaps mf,
│ │ │ │ -000385f0: 2061 730a 6465 7363 7269 6265 6420 696e   as.described in
│ │ │ │ -00038600: 2045 6973 656e 6275 642d 5065 6576 612e   Eisenbud-Peeva.
│ │ │ │ -00038610: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..+-------------
│ │ │ │ +00038210: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020  ************..  
│ │ │ │ +00038220: 2a20 5573 6167 653a 200a 2020 2020 2020  * Usage: .      
│ │ │ │ +00038230: 2020 4120 3d20 6d61 6b65 4669 6e69 7465    A = makeFinite
│ │ │ │ +00038240: 5265 736f 6c75 7469 6f6e 2866 662c 6d66  Resolution(ff,mf
│ │ │ │ +00038250: 290a 2020 2a20 496e 7075 7473 3a0a 2020  ).  * Inputs:.  
│ │ │ │ +00038260: 2020 2020 2a20 6d66 2c20 6120 2a6e 6f74      * mf, a *not
│ │ │ │ +00038270: 6520 6c69 7374 3a20 284d 6163 6175 6c61  e list: (Macaula
│ │ │ │ +00038280: 7932 446f 6329 4c69 7374 2c2c 206f 7574  y2Doc)List,, out
│ │ │ │ +00038290: 7075 7420 6f66 206d 6174 7269 7846 6163  put of matrixFac
│ │ │ │ +000382a0: 746f 7269 7a61 7469 6f6e 0a20 2020 2020  torization.     
│ │ │ │ +000382b0: 202a 2066 662c 2061 202a 6e6f 7465 206d   * ff, a *note m
│ │ │ │ +000382c0: 6174 7269 783a 2028 4d61 6361 756c 6179  atrix: (Macaulay
│ │ │ │ +000382d0: 3244 6f63 294d 6174 7269 782c 2c20 7468  2Doc)Matrix,, th
│ │ │ │ +000382e0: 6520 7265 6775 6c61 7220 7365 7175 656e  e regular sequen
│ │ │ │ +000382f0: 6365 2075 7365 640a 2020 2020 2020 2020  ce used.        
│ │ │ │ +00038300: 666f 7220 7468 6520 6d61 7472 6978 4661  for the matrixFa
│ │ │ │ +00038310: 6374 6f72 697a 6174 696f 6e20 636f 6d70  ctorization comp
│ │ │ │ +00038320: 7574 6174 696f 6e0a 2020 2a20 4f75 7470  utation.  * Outp
│ │ │ │ +00038330: 7574 733a 0a20 2020 2020 202a 2041 2c20  uts:.      * A, 
│ │ │ │ +00038340: 6120 2a6e 6f74 6520 636f 6d70 6c65 783a  a *note complex:
│ │ │ │ +00038350: 2028 436f 6d70 6c65 7865 7329 436f 6d70   (Complexes)Comp
│ │ │ │ +00038360: 6c65 782c 2c20 4120 6973 2074 6865 206d  lex,, A is the m
│ │ │ │ +00038370: 696e 696d 616c 2066 696e 6974 650a 2020  inimal finite.  
│ │ │ │ +00038380: 2020 2020 2020 7265 736f 6c75 7469 6f6e        resolution
│ │ │ │ +00038390: 206f 6620 4d20 6f76 6572 2052 2e0a 0a44   of M over R...D
│ │ │ │ +000383a0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ +000383b0: 3d3d 3d3d 3d3d 0a0a 5375 7070 6f73 6520  ======..Suppose 
│ │ │ │ +000383c0: 7468 6174 2066 5f31 2e2e 665f 6320 6973  that f_1..f_c is
│ │ │ │ +000383d0: 2061 2068 6f6d 6f67 656e 656f 7573 2072   a homogeneous r
│ │ │ │ +000383e0: 6567 756c 6172 2073 6571 7565 6e63 6520  egular sequence 
│ │ │ │ +000383f0: 6f66 2066 6f72 6d73 206f 6620 7468 6520  of forms of the 
│ │ │ │ +00038400: 7361 6d65 0a64 6567 7265 6520 696e 2061  same.degree in a
│ │ │ │ +00038410: 2070 6f6c 796e 6f6d 6961 6c20 7269 6e67   polynomial ring
│ │ │ │ +00038420: 2053 2061 6e64 204d 2069 7320 6120 6869   S and M is a hi
│ │ │ │ +00038430: 6768 2073 797a 7967 7920 6d6f 6475 6c65  gh syzygy module
│ │ │ │ +00038440: 206f 7665 7220 532f 2866 5f31 2c2e 2e2c   over S/(f_1,..,
│ │ │ │ +00038450: 665f 6329 0a3d 2052 2863 292c 2061 6e64  f_c).= R(c), and
│ │ │ │ +00038460: 206d 6620 3d20 2864 2c68 2920 6973 2074   mf = (d,h) is t
│ │ │ │ +00038470: 6865 206f 7574 7075 7420 6f66 206d 6174  he output of mat
│ │ │ │ +00038480: 7269 7846 6163 746f 7269 7a61 7469 6f6e  rixFactorization
│ │ │ │ +00038490: 284d 2c66 6629 2e20 4966 2074 6865 0a63  (M,ff). If the.c
│ │ │ │ +000384a0: 6f6d 706c 6578 6974 7920 6f66 204d 2069  omplexity of M i
│ │ │ │ +000384b0: 7320 6327 2c20 7468 656e 204d 2068 6173  s c', then M has
│ │ │ │ +000384c0: 2061 2066 696e 6974 6520 6672 6565 2072   a finite free r
│ │ │ │ +000384d0: 6573 6f6c 7574 696f 6e20 6f76 6572 2052  esolution over R
│ │ │ │ +000384e0: 203d 0a53 2f28 665f 312c 2e2e 2c66 5f7b   =.S/(f_1,..,f_{
│ │ │ │ +000384f0: 2863 2d63 2729 7d29 2028 616e 642c 206d  (c-c')}) (and, m
│ │ │ │ +00038500: 6f72 6520 6765 6e65 7261 6c6c 792c 2068  ore generally, h
│ │ │ │ +00038510: 6173 2063 6f6d 706c 6578 6974 7920 632d  as complexity c-
│ │ │ │ +00038520: 6420 6f76 6572 0a53 2f28 665f 312c 2e2e  d over.S/(f_1,..
│ │ │ │ +00038530: 2c66 5f7b 2863 2d64 297d 2920 666f 7220  ,f_{(c-d)}) for 
│ │ │ │ +00038540: 643e 3d63 2729 2e0a 0a54 6865 2063 6f6d  d>=c')...The com
│ │ │ │ +00038550: 706c 6578 2041 2069 7320 7468 6520 6d69  plex A is the mi
│ │ │ │ +00038560: 6e69 6d61 6c20 6669 6e69 7465 2066 7265  nimal finite fre
│ │ │ │ +00038570: 6520 7265 736f 6c75 7469 6f6e 206f 6620  e resolution of 
│ │ │ │ +00038580: 4d20 6f76 6572 2041 2c20 636f 6e73 7472  M over A, constr
│ │ │ │ +00038590: 7563 7465 6420 6173 0a61 6e20 6974 6572  ucted as.an iter
│ │ │ │ +000385a0: 6174 6564 204b 6f73 7a75 6c20 6578 7465  ated Koszul exte
│ │ │ │ +000385b0: 6e73 696f 6e2c 206d 6164 6520 6672 6f6d  nsion, made from
│ │ │ │ +000385c0: 2074 6865 206d 6170 7320 696e 2062 4d61   the maps in bMa
│ │ │ │ +000385d0: 7073 206d 6620 616e 6420 7073 694d 6170  ps mf and psiMap
│ │ │ │ +000385e0: 7320 6d66 2c20 6173 0a64 6573 6372 6962  s mf, as.describ
│ │ │ │ +000385f0: 6564 2069 6e20 4569 7365 6e62 7564 2d50  ed in Eisenbud-P
│ │ │ │ +00038600: 6565 7661 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  eeva...+--------
│ │ │ │ +00038610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038660: 2b0a 7c69 3120 3a20 7365 7452 616e 646f  +.|i1 : setRando
│ │ │ │ -00038670: 6d53 6565 6420 3020 2020 2020 2020 2020  mSeed 0         
│ │ │ │ +00038650: 2d2d 2d2d 2d2b 0a7c 6931 203a 2073 6574  -----+.|i1 : set
│ │ │ │ +00038660: 5261 6e64 6f6d 5365 6564 2030 2020 2020  RandomSeed 0    
│ │ │ │ +00038670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000386a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000386b0: 7c0a 7c20 2d2d 2073 6574 7469 6e67 2072  |.| -- setting r
│ │ │ │ -000386c0: 616e 646f 6d20 7365 6564 2074 6f20 3020  andom seed to 0 
│ │ │ │ +000386a0: 2020 2020 207c 0a7c 202d 2d20 7365 7474       |.| -- sett
│ │ │ │ +000386b0: 696e 6720 7261 6e64 6f6d 2073 6565 6420  ing random seed 
│ │ │ │ +000386c0: 746f 2030 2020 2020 2020 2020 2020 2020  to 0            
│ │ │ │  000386d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000386e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000386f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038700: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000386f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00038700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038750: 7c0a 7c6f 3120 3d20 3020 2020 2020 2020  |.|o1 = 0       
│ │ │ │ +00038740: 2020 2020 207c 0a7c 6f31 203d 2030 2020       |.|o1 = 0  
│ │ │ │ +00038750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000387a0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00038790: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +000387a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000387b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000387c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000387d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000387e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000387f0: 2b0a 7c69 3220 3a20 5320 3d20 5a5a 2f31  +.|i2 : S = ZZ/1
│ │ │ │ -00038800: 3031 5b61 2c62 2c63 5d3b 2020 2020 2020  01[a,b,c];      
│ │ │ │ +000387e0: 2d2d 2d2d 2d2b 0a7c 6932 203a 2053 203d  -----+.|i2 : S =
│ │ │ │ +000387f0: 205a 5a2f 3130 315b 612c 622c 635d 3b20   ZZ/101[a,b,c]; 
│ │ │ │ +00038800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038840: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00038830: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00038840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038890: 2b0a 7c69 3320 3a20 6666 203d 206d 6174  +.|i3 : ff = mat
│ │ │ │ -000388a0: 7269 7822 6133 2c62 3322 3b20 2020 2020  rix"a3,b3";     
│ │ │ │ +00038880: 2d2d 2d2d 2d2b 0a7c 6933 203a 2066 6620  -----+.|i3 : ff 
│ │ │ │ +00038890: 3d20 6d61 7472 6978 2261 332c 6233 223b  = matrix"a3,b3";
│ │ │ │ +000388a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000388b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000388c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000388d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000388e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000388d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000388e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000388f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038930: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00038940: 3120 2020 2020 2032 2020 2020 2020 2020  1      2        
│ │ │ │ +00038920: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00038930: 2020 2020 2031 2020 2020 2020 3220 2020       1      2   
│ │ │ │ +00038940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038980: 7c0a 7c6f 3320 3a20 4d61 7472 6978 2053  |.|o3 : Matrix S
│ │ │ │ -00038990: 2020 3c2d 2d20 5320 2020 2020 2020 2020    <-- S         
│ │ │ │ +00038970: 2020 2020 207c 0a7c 6f33 203a 204d 6174       |.|o3 : Mat
│ │ │ │ +00038980: 7269 7820 5320 203c 2d2d 2053 2020 2020  rix S  <-- S    
│ │ │ │ +00038990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000389a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000389b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000389c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000389d0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +000389c0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +000389d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000389e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000389f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00038a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038a20: 2b0a 7c69 3420 3a20 5220 3d20 532f 6964  +.|i4 : R = S/id
│ │ │ │ -00038a30: 6561 6c20 6666 3b20 2020 2020 2020 2020  eal ff;         
│ │ │ │ +00038a10: 2d2d 2d2d 2d2b 0a7c 6934 203a 2052 203d  -----+.|i4 : R =
│ │ │ │ +00038a20: 2053 2f69 6465 616c 2066 663b 2020 2020   S/ideal ff;    
│ │ │ │ +00038a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00038a70: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00038a60: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00038a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038ac0: 2b0a 7c69 3520 3a20 4d20 3d20 6869 6768  +.|i5 : M = high
│ │ │ │ -00038ad0: 5379 7a79 6779 2028 525e 312f 6964 6561  Syzygy (R^1/idea
│ │ │ │ -00038ae0: 6c20 7661 7273 2052 293b 2020 2020 2020  l vars R);      
│ │ │ │ +00038ab0: 2d2d 2d2d 2d2b 0a7c 6935 203a 204d 203d  -----+.|i5 : M =
│ │ │ │ +00038ac0: 2068 6967 6853 797a 7967 7920 2852 5e31   highSyzygy (R^1
│ │ │ │ +00038ad0: 2f69 6465 616c 2076 6172 7320 5229 3b20  /ideal vars R); 
│ │ │ │ +00038ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00038b00: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00038b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038b60: 2b0a 7c69 3620 3a20 6d66 203d 206d 6174  +.|i6 : mf = mat
│ │ │ │ -00038b70: 7269 7846 6163 746f 7269 7a61 7469 6f6e  rixFactorization
│ │ │ │ -00038b80: 2028 6666 2c20 4d29 2020 2020 2020 2020   (ff, M)        
│ │ │ │ +00038b50: 2d2d 2d2d 2d2b 0a7c 6936 203a 206d 6620  -----+.|i6 : mf 
│ │ │ │ +00038b60: 3d20 6d61 7472 6978 4661 6374 6f72 697a  = matrixFactoriz
│ │ │ │ +00038b70: 6174 696f 6e20 2866 662c 204d 2920 2020  ation (ff, M)   
│ │ │ │ +00038b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00038ba0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
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│ │ │ │ -00038c90: 2030 2020 2030 2020 2030 2030 2020 3020   0   0   0 0  0 
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│ │ │ │ -00038cb0: 2020 6120 6320 2030 2020 3020 2030 2020    a c  0  0  0  
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│ │ │ │ -00038d90: 7c0a 7c20 2020 2020 207b 347d 207c 2030  |.|      {4} | 0
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│ │ │ │ -00038de0: 7c0a 7c20 2020 2020 207b 347d 207c 2030  |.|      {4} | 0
│ │ │ │ -00038df0: 2020 3020 3020 2030 2020 3020 2063 2020    0 0  0  0  c  
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│ │ │ │ +00038bf0: 2020 2020 207c 0a7c 6f36 203d 207b 7b34       |.|o6 = {{4
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│ │ │ │ +00038d10: 2020 7b36 7d20 7c20 6120 6320 202d 6220    {6} | a c  -b 
│ │ │ │ +00038d20: 3020 2030 2020 3020 2020 3020 2020 3020  0  0  0   0   0 
│ │ │ │ +00038d30: 3020 2030 207c 0a7c 2020 2020 2020 7b34  0  0 |.|      {4
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│ │ │ │  00038ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00039140: 2020 2020 207c 0a7c 2020 2020 2061 2020       |.|     a  
│ │ │ │ +00039150: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │  00039160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039170: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00039190: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000391a0: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │  000391b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000391c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000391d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000391e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000391e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000391f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039200: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00039220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039240: 7c0a 7c6f 3620 3a20 4c69 7374 2020 2020  |.|o6 : List    
│ │ │ │ +00039230: 2020 2020 207c 0a7c 6f36 203a 204c 6973       |.|o6 : Lis
│ │ │ │ +00039240: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │  00039250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039290: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00039280: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00039290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000392a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000392b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000392c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000392d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000392e0: 2b0a 7c69 3720 3a20 4720 3d20 6d61 6b65  +.|i7 : G = make
│ │ │ │ -000392f0: 4669 6e69 7465 5265 736f 6c75 7469 6f6e  FiniteResolution
│ │ │ │ -00039300: 2866 662c 6d66 2920 2020 2020 2020 2020  (ff,mf)         
│ │ │ │ +000392d0: 2d2d 2d2d 2d2b 0a7c 6937 203a 2047 203d  -----+.|i7 : G =
│ │ │ │ +000392e0: 206d 616b 6546 696e 6974 6552 6573 6f6c   makeFiniteResol
│ │ │ │ +000392f0: 7574 696f 6e28 6666 2c6d 6629 2020 2020  ution(ff,mf)    
│ │ │ │ +00039300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039330: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00039320: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00039330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039380: 7c0a 7c20 2020 2020 2037 2020 2020 2020  |.|      7      
│ │ │ │ -00039390: 3132 2020 2020 2020 3520 2020 2020 2020  12      5       
│ │ │ │ +00039370: 2020 2020 207c 0a7c 2020 2020 2020 3720       |.|      7 
│ │ │ │ +00039380: 2020 2020 2031 3220 2020 2020 2035 2020       12      5  
│ │ │ │ +00039390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000393a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000393b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000393c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000393d0: 7c0a 7c6f 3720 3d20 5320 203c 2d2d 2053  |.|o7 = S  <-- S
│ │ │ │ -000393e0: 2020 203c 2d2d 2053 2020 2020 2020 2020     <-- S        
│ │ │ │ +000393c0: 2020 2020 207c 0a7c 6f37 203d 2053 2020       |.|o7 = S  
│ │ │ │ +000393d0: 3c2d 2d20 5320 2020 3c2d 2d20 5320 2020  <-- S   <-- S   
│ │ │ │ +000393e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000393f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039420: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00039410: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00039420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039470: 7c0a 7c20 2020 2020 3020 2020 2020 2031  |.|     0      1
│ │ │ │ -00039480: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +00039460: 2020 2020 207c 0a7c 2020 2020 2030 2020       |.|     0  
│ │ │ │ +00039470: 2020 2020 3120 2020 2020 2020 3220 2020      1       2   
│ │ │ │ +00039480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000394a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000394b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000394c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000394b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000394c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000394d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000394e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000394f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039510: 7c0a 7c6f 3720 3a20 436f 6d70 6c65 7820  |.|o7 : Complex 
│ │ │ │ +00039500: 2020 2020 207c 0a7c 6f37 203a 2043 6f6d       |.|o7 : Com
│ │ │ │ +00039510: 706c 6578 2020 2020 2020 2020 2020 2020  plex            
│ │ │ │  00039520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039560: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
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│ │ │ │ +00039560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00039570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00039580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00039590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000395a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000395b0: 2b0a 7c69 3820 3a20 4620 3d20 6672 6565  +.|i8 : F = free
│ │ │ │ -000395c0: 5265 736f 6c75 7469 6f6e 2070 7573 6846  Resolution pushF
│ │ │ │ -000395d0: 6f72 7761 7264 286d 6170 2852 2c53 292c  orward(map(R,S),
│ │ │ │ -000395e0: 4d29 2020 2020 2020 2020 2020 2020 2020  M)              
│ │ │ │ -000395f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039600: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000395a0: 2d2d 2d2d 2d2b 0a7c 6938 203a 2046 203d  -----+.|i8 : F =
│ │ │ │ +000395b0: 2066 7265 6552 6573 6f6c 7574 696f 6e20   freeResolution 
│ │ │ │ +000395c0: 7075 7368 466f 7277 6172 6428 6d61 7028  pushForward(map(
│ │ │ │ +000395d0: 522c 5329 2c4d 2920 2020 2020 2020 2020  R,S),M)         
│ │ │ │ +000395e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000395f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00039600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039650: 7c0a 7c20 2020 2020 2037 2020 2020 2020  |.|      7      
│ │ │ │ -00039660: 3132 2020 2020 2020 3520 2020 2020 2020  12      5       
│ │ │ │ +00039640: 2020 2020 207c 0a7c 2020 2020 2020 3720       |.|      7 
│ │ │ │ +00039650: 2020 2020 2031 3220 2020 2020 2035 2020       12      5  
│ │ │ │ +00039660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000396a0: 7c0a 7c6f 3820 3d20 5320 203c 2d2d 2053  |.|o8 = S  <-- S
│ │ │ │ -000396b0: 2020 203c 2d2d 2053 2020 2020 2020 2020     <-- S        
│ │ │ │ +00039690: 2020 2020 207c 0a7c 6f38 203d 2053 2020       |.|o8 = S  
│ │ │ │ +000396a0: 3c2d 2d20 5320 2020 3c2d 2d20 5320 2020  <-- S   <-- S   
│ │ │ │ +000396b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000396c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000396d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000396e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000396f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000396e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000396f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039740: 7c0a 7c20 2020 2020 3020 2020 2020 2031  |.|     0      1
│ │ │ │ -00039750: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +00039730: 2020 2020 207c 0a7c 2020 2020 2030 2020       |.|     0  
│ │ │ │ +00039740: 2020 2020 3120 2020 2020 2020 3220 2020      1       2   
│ │ │ │ +00039750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039790: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00039780: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00039790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000397a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000397b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000397c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000397d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000397e0: 7c0a 7c6f 3820 3a20 436f 6d70 6c65 7820  |.|o8 : Complex 
│ │ │ │ +000397d0: 2020 2020 207c 0a7c 6f38 203a 2043 6f6d       |.|o8 : Com
│ │ │ │ +000397e0: 706c 6578 2020 2020 2020 2020 2020 2020  plex            
│ │ │ │  000397f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039830: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00039820: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00039830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00039840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00039850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00039860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00039870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00039880: 2b0a 7c69 3920 3a20 472e 6464 5f31 2020  +.|i9 : G.dd_1  
│ │ │ │ +00039870: 2d2d 2d2d 2d2b 0a7c 6939 203a 2047 2e64  -----+.|i9 : G.d
│ │ │ │ +00039880: 645f 3120 2020 2020 2020 2020 2020 2020  d_1             
│ │ │ │  00039890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000398a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000398b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000398c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000398d0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000398c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000398d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000398e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000398f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039920: 7c0a 7c6f 3920 3d20 7b34 7d20 7c20 6220  |.|o9 = {4} | b 
│ │ │ │ -00039930: 202d 6120 3020 2030 2020 3020 2020 2d61   -a 0  0  0   -a
│ │ │ │ -00039940: 3320 3020 2020 3020 2020 3020 2030 2030  3 0   0   0  0 0
│ │ │ │ -00039950: 2020 3020 207c 2020 2020 2020 2020 2020    0  |          
│ │ │ │ -00039960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039970: 7c0a 7c20 2020 2020 7b34 7d20 7c20 2d63  |.|     {4} | -c
│ │ │ │ -00039980: 2030 2020 6120 2062 3220 3020 2020 3020   0  a  b2 0   0 
│ │ │ │ -00039990: 2020 2d61 3320 3020 2020 3020 2030 2030    -a3 0   0  0 0
│ │ │ │ -000399a0: 2020 3020 207c 2020 2020 2020 2020 2020    0  |          
│ │ │ │ -000399b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000399c0: 7c0a 7c20 2020 2020 7b34 7d20 7c20 3020  |.|     {4} | 0 
│ │ │ │ -000399d0: 2063 2020 2d62 2030 2020 6132 2020 3020   c  -b 0  a2  0 
│ │ │ │ -000399e0: 2020 3020 2020 2d61 3320 3020 2030 2030    0   -a3 0  0 0
│ │ │ │ -000399f0: 2020 3020 207c 2020 2020 2020 2020 2020    0  |          
│ │ │ │ -00039a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039a10: 7c0a 7c20 2020 2020 7b34 7d20 7c20 3020  |.|     {4} | 0 
│ │ │ │ -00039a20: 2030 2020 3020 2030 2020 3020 2020 3020   0  0  0  0   0 
│ │ │ │ -00039a30: 2020 3020 2020 3020 2020 2d63 2062 2030    0   0   -c b 0
│ │ │ │ -00039a40: 2020 6132 207c 2020 2020 2020 2020 2020    a2 |          
│ │ │ │ -00039a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039a60: 7c0a 7c20 2020 2020 7b34 7d20 7c20 3020  |.|     {4} | 0 
│ │ │ │ -00039a70: 2030 2020 3020 2030 2020 3020 2020 3020   0  0  0  0   0 
│ │ │ │ -00039a80: 2020 3020 2020 3020 2020 6120 2030 2062    0   0   a  0 b
│ │ │ │ -00039a90: 2020 3020 207c 2020 2020 2020 2020 2020    0  |          
│ │ │ │ -00039aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039ab0: 7c0a 7c20 2020 2020 7b34 7d20 7c20 3020  |.|     {4} | 0 
│ │ │ │ -00039ac0: 2030 2020 3020 2030 2020 2d62 3220 3020   0  0  0  -b2 0 
│ │ │ │ -00039ad0: 2020 3020 2020 3020 2020 3020 2061 2063    0   0   0  a c
│ │ │ │ -00039ae0: 2020 3020 207c 2020 2020 2020 2020 2020    0  |          
│ │ │ │ -00039af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039b00: 7c0a 7c20 2020 2020 7b33 7d20 7c20 3020  |.|     {3} | 0 
│ │ │ │ -00039b10: 2062 3220 3020 2030 2020 3020 2020 3020   b2 0  0  0   0 
│ │ │ │ -00039b20: 2020 3020 2020 3020 2020 3020 2030 2061    0   0   0  0 a
│ │ │ │ -00039b30: 3220 3020 207c 2020 2020 2020 2020 2020  2 0  |          
│ │ │ │ -00039b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039b50: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00039910: 2020 2020 207c 0a7c 6f39 203d 207b 347d       |.|o9 = {4}
│ │ │ │ +00039920: 207c 2062 2020 2d61 2030 2020 3020 2030   | b  -a 0  0  0
│ │ │ │ +00039930: 2020 202d 6133 2030 2020 2030 2020 2030     -a3 0   0   0
│ │ │ │ +00039940: 2020 3020 3020 2030 2020 7c20 2020 2020    0 0  0  |     
│ │ │ │ +00039950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039960: 2020 2020 207c 0a7c 2020 2020 207b 347d       |.|     {4}
│ │ │ │ +00039970: 207c 202d 6320 3020 2061 2020 6232 2030   | -c 0  a  b2 0
│ │ │ │ +00039980: 2020 2030 2020 202d 6133 2030 2020 2030     0   -a3 0   0
│ │ │ │ +00039990: 2020 3020 3020 2030 2020 7c20 2020 2020    0 0  0  |     
│ │ │ │ +000399a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000399b0: 2020 2020 207c 0a7c 2020 2020 207b 347d       |.|     {4}
│ │ │ │ +000399c0: 207c 2030 2020 6320 202d 6220 3020 2061   | 0  c  -b 0  a
│ │ │ │ +000399d0: 3220 2030 2020 2030 2020 202d 6133 2030  2  0   0   -a3 0
│ │ │ │ +000399e0: 2020 3020 3020 2030 2020 7c20 2020 2020    0 0  0  |     
│ │ │ │ +000399f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039a00: 2020 2020 207c 0a7c 2020 2020 207b 347d       |.|     {4}
│ │ │ │ +00039a10: 207c 2030 2020 3020 2030 2020 3020 2030   | 0  0  0  0  0
│ │ │ │ +00039a20: 2020 2030 2020 2030 2020 2030 2020 202d     0   0   0   -
│ │ │ │ +00039a30: 6320 6220 3020 2061 3220 7c20 2020 2020  c b 0  a2 |     
│ │ │ │ +00039a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039a50: 2020 2020 207c 0a7c 2020 2020 207b 347d       |.|     {4}
│ │ │ │ +00039a60: 207c 2030 2020 3020 2030 2020 3020 2030   | 0  0  0  0  0
│ │ │ │ +00039a70: 2020 2030 2020 2030 2020 2030 2020 2061     0   0   0   a
│ │ │ │ +00039a80: 2020 3020 6220 2030 2020 7c20 2020 2020    0 b  0  |     
│ │ │ │ +00039a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039aa0: 2020 2020 207c 0a7c 2020 2020 207b 347d       |.|     {4}
│ │ │ │ +00039ab0: 207c 2030 2020 3020 2030 2020 3020 202d   | 0  0  0  0  -
│ │ │ │ +00039ac0: 6232 2030 2020 2030 2020 2030 2020 2030  b2 0   0   0   0
│ │ │ │ +00039ad0: 2020 6120 6320 2030 2020 7c20 2020 2020    a c  0  |     
│ │ │ │ +00039ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039af0: 2020 2020 207c 0a7c 2020 2020 207b 337d       |.|     {3}
│ │ │ │ +00039b00: 207c 2030 2020 6232 2030 2020 3020 2030   | 0  b2 0  0  0
│ │ │ │ +00039b10: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ +00039b20: 2020 3020 6132 2030 2020 7c20 2020 2020    0 a2 0  |     
│ │ │ │ +00039b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039b40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00039b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039ba0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00039bb0: 3720 2020 2020 2031 3220 2020 2020 2020  7      12       
│ │ │ │ +00039b90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00039ba0: 2020 2020 2037 2020 2020 2020 3132 2020       7      12  
│ │ │ │ +00039bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039bf0: 7c0a 7c6f 3920 3a20 4d61 7472 6978 2053  |.|o9 : Matrix S
│ │ │ │ -00039c00: 2020 3c2d 2d20 5320 2020 2020 2020 2020    <-- S         
│ │ │ │ +00039be0: 2020 2020 207c 0a7c 6f39 203a 204d 6174       |.|o9 : Mat
│ │ │ │ +00039bf0: 7269 7820 5320 203c 2d2d 2053 2020 2020  rix S  <-- S    
│ │ │ │ +00039c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039c40: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00039c30: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00039c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00039c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00039c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00039c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00039c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00039c90: 2b0a 7c69 3130 203a 2046 2e64 645f 3120  +.|i10 : F.dd_1 
│ │ │ │ +00039c80: 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20 462e  -----+.|i10 : F.
│ │ │ │ +00039c90: 6464 5f31 2020 2020 2020 2020 2020 2020  dd_1            
│ │ │ │  00039ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039ce0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00039cd0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00039ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00039f00: 2020 2020 207c 0a7c 2020 2020 2020 7b34       |.|      {4
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│ │ │ │ -00039fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039fb0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00039fc0: 2037 2020 2020 2020 3132 2020 2020 2020   7      12      
│ │ │ │ +00039fa0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00039fb0: 2020 2020 2020 3720 2020 2020 2031 3220        7      12 
│ │ │ │ +00039fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00039fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003a010: 5320 203c 2d2d 2053 2020 2020 2020 2020  S  <-- S        
│ │ │ │ +00039ff0: 2020 2020 207c 0a7c 6f31 3020 3a20 4d61       |.|o10 : Ma
│ │ │ │ +0003a000: 7472 6978 2053 2020 3c2d 2d20 5320 2020  trix S  <-- S   
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│ │ │ │ -0003a0a0: 2b0a 7c69 3131 203a 2047 2e64 645f 3220  +.|i11 : G.dd_2 
│ │ │ │ +0003a090: 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20 472e  -----+.|i11 : G.
│ │ │ │ +0003a0a0: 6464 5f32 2020 2020 2020 2020 2020 2020  dd_2            
│ │ │ │  0003a0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003a140: 7c0a 7c6f 3131 203d 207b 357d 207c 202d  |.|o11 = {5} | -
│ │ │ │ -0003a150: 6133 2030 2020 2020 3020 2020 3020 2020  a3 0    0   0   
│ │ │ │ -0003a160: 3020 2020 2020 7c20 2020 2020 2020 2020  0     |         
│ │ │ │ +0003a130: 2020 2020 207c 0a7c 6f31 3120 3d20 7b35       |.|o11 = {5
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│ │ │ │ +0003a150: 2030 2020 2030 2020 2020 207c 2020 2020   0   0     |    
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│ │ │ │ -0003a180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a190: 7c0a 7c20 2020 2020 207b 357d 207c 2030  |.|      {5} | 0
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│ │ │ │ -0003a1b0: 3020 2020 2020 7c20 2020 2020 2020 2020  0     |         
│ │ │ │ +0003a180: 2020 2020 207c 0a7c 2020 2020 2020 7b35       |.|      {5
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│ │ │ │ +0003a1a0: 2030 2020 2030 2020 2020 207c 2020 2020   0   0     |    
│ │ │ │ +0003a1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003a1e0: 7c0a 7c20 2020 2020 207b 357d 207c 2030  |.|      {5} | 0
│ │ │ │ -0003a1f0: 2020 2030 2020 2020 2d61 3320 3020 2020     0    -a3 0   
│ │ │ │ -0003a200: 3020 2020 2020 7c20 2020 2020 2020 2020  0     |         
│ │ │ │ +0003a1d0: 2020 2020 207c 0a7c 2020 2020 2020 7b35       |.|      {5
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│ │ │ │ +0003a1f0: 2030 2020 2030 2020 2020 207c 2020 2020   0   0     |    
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│ │ │ │  0003a210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a230: 7c0a 7c20 2020 2020 207b 367d 207c 2030  |.|      {6} | 0
│ │ │ │ -0003a240: 2020 2030 2020 2020 3020 2020 2d61 3320     0    0   -a3 
│ │ │ │ -0003a250: 3020 2020 2020 7c20 2020 2020 2020 2020  0     |         
│ │ │ │ +0003a220: 2020 2020 207c 0a7c 2020 2020 2020 7b36       |.|      {6
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│ │ │ │ +0003a240: 202d 6133 2030 2020 2020 207c 2020 2020   -a3 0     |    
│ │ │ │ +0003a250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a280: 7c0a 7c20 2020 2020 207b 367d 207c 2030  |.|      {6} | 0
│ │ │ │ -0003a290: 2020 2030 2020 2020 3020 2020 3020 2020     0    0   0   
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│ │ │ │ +0003a270: 2020 2020 207c 0a7c 2020 2020 2020 7b36       |.|      {6
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│ │ │ │ +0003a290: 2030 2020 202d 6133 2020 207c 2020 2020   0   -a3   |    
│ │ │ │ +0003a2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a2d0: 7c0a 7c20 2020 2020 207b 377d 207c 202d  |.|      {7} | -
│ │ │ │ -0003a2e0: 6220 2061 2020 2020 3020 2020 3020 2020  b  a    0   0   
│ │ │ │ -0003a2f0: 3020 2020 2020 7c20 2020 2020 2020 2020  0     |         
│ │ │ │ +0003a2c0: 2020 2020 207c 0a7c 2020 2020 2020 7b37       |.|      {7
│ │ │ │ +0003a2d0: 7d20 7c20 2d62 2020 6120 2020 2030 2020  } | -b  a    0  
│ │ │ │ +0003a2e0: 2030 2020 2030 2020 2020 207c 2020 2020   0   0     |    
│ │ │ │ +0003a2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a320: 7c0a 7c20 2020 2020 207b 377d 207c 2063  |.|      {7} | c
│ │ │ │ -0003a330: 2020 2030 2020 2020 2d61 2020 2d62 3220     0    -a  -b2 
│ │ │ │ -0003a340: 3020 2020 2020 7c20 2020 2020 2020 2020  0     |         
│ │ │ │ +0003a310: 2020 2020 207c 0a7c 2020 2020 2020 7b37       |.|      {7
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│ │ │ │ +0003a330: 202d 6232 2030 2020 2020 207c 2020 2020   -b2 0     |    
│ │ │ │ +0003a340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a370: 7c0a 7c20 2020 2020 207b 377d 207c 2030  |.|      {7} | 0
│ │ │ │ -0003a380: 2020 202d 6320 2020 6220 2020 3020 2020     -c   b   0   
│ │ │ │ -0003a390: 2d61 3220 2020 7c20 2020 2020 2020 2020  -a2   |         
│ │ │ │ +0003a360: 2020 2020 207c 0a7c 2020 2020 2020 7b37       |.|      {7
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│ │ │ │ +0003a380: 2030 2020 202d 6132 2020 207c 2020 2020   0   -a2   |    
│ │ │ │ +0003a390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a3c0: 7c0a 7c20 2020 2020 207b 357d 207c 2030  |.|      {5} | 0
│ │ │ │ -0003a3d0: 2020 202d 6233 2020 3020 2020 3020 2020     -b3  0   0   
│ │ │ │ -0003a3e0: 3020 2020 2020 7c20 2020 2020 2020 2020  0     |         
│ │ │ │ +0003a3b0: 2020 2020 207c 0a7c 2020 2020 2020 7b35       |.|      {5
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│ │ │ │ +0003a3d0: 2030 2020 2030 2020 2020 207c 2020 2020   0   0     |    
│ │ │ │ +0003a3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a410: 7c0a 7c20 2020 2020 207b 357d 207c 2030  |.|      {5} | 0
│ │ │ │ -0003a420: 2020 202d 6232 6320 3020 2020 3020 2020     -b2c 0   0   
│ │ │ │ -0003a430: 2d61 3262 3220 7c20 2020 2020 2020 2020  -a2b2 |         
│ │ │ │ +0003a400: 2020 2020 207c 0a7c 2020 2020 2020 7b35       |.|      {5
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│ │ │ │ +0003a420: 2030 2020 202d 6132 6232 207c 2020 2020   0   -a2b2 |    
│ │ │ │ +0003a430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a460: 7c0a 7c20 2020 2020 207b 357d 207c 2030  |.|      {5} | 0
│ │ │ │ -0003a470: 2020 2061 6232 2020 3020 2020 3020 2020     ab2  0   0   
│ │ │ │ -0003a480: 3020 2020 2020 7c20 2020 2020 2020 2020  0     |         
│ │ │ │ +0003a450: 2020 2020 207c 0a7c 2020 2020 2020 7b35       |.|      {5
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│ │ │ │ +0003a470: 2030 2020 2030 2020 2020 207c 2020 2020   0   0     |    
│ │ │ │ +0003a480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a4b0: 7c0a 7c20 2020 2020 207b 367d 207c 2030  |.|      {6} | 0
│ │ │ │ -0003a4c0: 2020 2030 2020 2020 3020 2020 3020 2020     0    0   0   
│ │ │ │ -0003a4d0: 6233 2020 2020 7c20 2020 2020 2020 2020  b3    |         
│ │ │ │ +0003a4a0: 2020 2020 207c 0a7c 2020 2020 2020 7b36       |.|      {6
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│ │ │ │ +0003a4c0: 2030 2020 2062 3320 2020 207c 2020 2020   0   b3    |    
│ │ │ │ +0003a4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003a550: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0003a560: 2031 3220 2020 2020 2035 2020 2020 2020   12      5      
│ │ │ │ +0003a540: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0003a550: 2020 2020 2020 3132 2020 2020 2020 3520        12      5 
│ │ │ │ +0003a560: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003a590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a5a0: 7c0a 7c6f 3131 203a 204d 6174 7269 7820  |.|o11 : Matrix 
│ │ │ │ -0003a5b0: 5320 2020 3c2d 2d20 5320 2020 2020 2020  S   <-- S       
│ │ │ │ +0003a590: 2020 2020 207c 0a7c 6f31 3120 3a20 4d61       |.|o11 : Ma
│ │ │ │ +0003a5a0: 7472 6978 2053 2020 203c 2d2d 2053 2020  trix S   <-- S  
│ │ │ │ +0003a5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003a5f0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0003a5e0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0003a5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003a600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003a610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003a620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003a630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003a640: 2b0a 7c69 3132 203a 2046 2e64 645f 3220  +.|i12 : F.dd_2 
│ │ │ │ +0003a630: 2d2d 2d2d 2d2b 0a7c 6931 3220 3a20 462e  -----+.|i12 : F.
│ │ │ │ +0003a640: 6464 5f32 2020 2020 2020 2020 2020 2020  dd_2            
│ │ │ │  0003a650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a660: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003a680: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0003a680: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0003a690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a6e0: 7c0a 7c6f 3132 203d 207b 357d 207c 2062  |.|o12 = {5} | b
│ │ │ │ -0003a6f0: 3320 2020 3020 2020 3020 2020 3020 2020  3   0   0   0   
│ │ │ │ -0003a700: 2020 3020 2020 7c20 2020 2020 2020 2020    0   |         
│ │ │ │ +0003a6d0: 2020 2020 207c 0a7c 6f31 3220 3d20 7b35       |.|o12 = {5
│ │ │ │ +0003a6e0: 7d20 7c20 6233 2020 2030 2020 2030 2020  } | b3   0   0  
│ │ │ │ +0003a6f0: 2030 2020 2020 2030 2020 207c 2020 2020   0     0   |    
│ │ │ │ +0003a700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a730: 7c0a 7c20 2020 2020 207b 357d 207c 202d  |.|      {5} | -
│ │ │ │ -0003a740: 6132 6220 3020 2020 3020 2020 3020 2020  a2b 0   0   0   
│ │ │ │ -0003a750: 2020 3020 2020 7c20 2020 2020 2020 2020    0   |         
│ │ │ │ +0003a720: 2020 2020 207c 0a7c 2020 2020 2020 7b35       |.|      {5
│ │ │ │ +0003a730: 7d20 7c20 2d61 3262 2030 2020 2030 2020  } | -a2b 0   0  
│ │ │ │ +0003a740: 2030 2020 2020 2030 2020 207c 2020 2020   0     0   |    
│ │ │ │ +0003a750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a780: 7c0a 7c20 2020 2020 207b 357d 207c 202d  |.|      {5} | -
│ │ │ │ -0003a790: 6132 6320 3020 2020 3020 2020 2d61 3262  a2c 0   0   -a2b
│ │ │ │ -0003a7a0: 3220 3020 2020 7c20 2020 2020 2020 2020  2 0   |         
│ │ │ │ +0003a770: 2020 2020 207c 0a7c 2020 2020 2020 7b35       |.|      {5
│ │ │ │ +0003a780: 7d20 7c20 2d61 3263 2030 2020 2030 2020  } | -a2c 0   0  
│ │ │ │ +0003a790: 202d 6132 6232 2030 2020 207c 2020 2020   -a2b2 0   |    
│ │ │ │ +0003a7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a7d0: 7c0a 7c20 2020 2020 207b 357d 207c 2030  |.|      {5} | 0
│ │ │ │ -0003a7e0: 2020 2020 2d62 3320 3020 2020 3020 2020      -b3 0   0   
│ │ │ │ -0003a7f0: 2020 3020 2020 7c20 2020 2020 2020 2020    0   |         
│ │ │ │ +0003a7c0: 2020 2020 207c 0a7c 2020 2020 2020 7b35       |.|      {5
│ │ │ │ +0003a7d0: 7d20 7c20 3020 2020 202d 6233 2030 2020  } | 0    -b3 0  
│ │ │ │ +0003a7e0: 2030 2020 2020 2030 2020 207c 2020 2020   0     0   |    
│ │ │ │ +0003a7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a820: 7c0a 7c20 2020 2020 207b 357d 207c 2030  |.|      {5} | 0
│ │ │ │ -0003a830: 2020 2020 3020 2020 2d62 3320 3020 2020      0   -b3 0   
│ │ │ │ -0003a840: 2020 3020 2020 7c20 2020 2020 2020 2020    0   |         
│ │ │ │ +0003a810: 2020 2020 207c 0a7c 2020 2020 2020 7b35       |.|      {5
│ │ │ │ +0003a820: 7d20 7c20 3020 2020 2030 2020 202d 6233  } | 0    0   -b3
│ │ │ │ +0003a830: 2030 2020 2020 2030 2020 207c 2020 2020   0     0   |    
│ │ │ │ +0003a840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a870: 7c0a 7c20 2020 2020 207b 357d 207c 202d  |.|      {5} | -
│ │ │ │ -0003a880: 6133 2020 3020 2020 3020 2020 3020 2020  a3  0   0   0   
│ │ │ │ -0003a890: 2020 3020 2020 7c20 2020 2020 2020 2020    0   |         
│ │ │ │ +0003a860: 2020 2020 207c 0a7c 2020 2020 2020 7b35       |.|      {5
│ │ │ │ +0003a870: 7d20 7c20 2d61 3320 2030 2020 2030 2020  } | -a3  0   0  
│ │ │ │ +0003a880: 2030 2020 2020 2030 2020 207c 2020 2020   0     0   |    
│ │ │ │ +0003a890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a8c0: 7c0a 7c20 2020 2020 207b 367d 207c 2030  |.|      {6} | 0
│ │ │ │ -0003a8d0: 2020 2020 3020 2020 3020 2020 2d62 3320      0   0   -b3 
│ │ │ │ -0003a8e0: 2020 3020 2020 7c20 2020 2020 2020 2020    0   |         
│ │ │ │ +0003a8b0: 2020 2020 207c 0a7c 2020 2020 2020 7b36       |.|      {6
│ │ │ │ +0003a8c0: 7d20 7c20 3020 2020 2030 2020 2030 2020  } | 0    0   0  
│ │ │ │ +0003a8d0: 202d 6233 2020 2030 2020 207c 2020 2020   -b3   0   |    
│ │ │ │ +0003a8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a910: 7c0a 7c20 2020 2020 207b 367d 207c 2030  |.|      {6} | 0
│ │ │ │ -0003a920: 2020 2020 3020 2020 3020 2020 3020 2020      0   0   0   
│ │ │ │ -0003a930: 2020 2d62 3320 7c20 2020 2020 2020 2020    -b3 |         
│ │ │ │ +0003a900: 2020 2020 207c 0a7c 2020 2020 2020 7b36       |.|      {6
│ │ │ │ +0003a910: 7d20 7c20 3020 2020 2030 2020 2030 2020  } | 0    0   0  
│ │ │ │ +0003a920: 2030 2020 2020 202d 6233 207c 2020 2020   0     -b3 |    
│ │ │ │ +0003a930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a960: 7c0a 7c20 2020 2020 207b 367d 207c 2030  |.|      {6} | 0
│ │ │ │ -0003a970: 2020 2020 3020 2020 3020 2020 6133 2020      0   0   a3  
│ │ │ │ -0003a980: 2020 3020 2020 7c20 2020 2020 2020 2020    0   |         
│ │ │ │ +0003a950: 2020 2020 207c 0a7c 2020 2020 2020 7b36       |.|      {6
│ │ │ │ +0003a960: 7d20 7c20 3020 2020 2030 2020 2030 2020  } | 0    0   0  
│ │ │ │ +0003a970: 2061 3320 2020 2030 2020 207c 2020 2020   a3    0   |    
│ │ │ │ +0003a980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a9b0: 7c0a 7c20 2020 2020 207b 377d 207c 202d  |.|      {7} | -
│ │ │ │ -0003a9c0: 6220 2020 6120 2020 3020 2020 3020 2020  b   a   0   0   
│ │ │ │ -0003a9d0: 2020 3020 2020 7c20 2020 2020 2020 2020    0   |         
│ │ │ │ +0003a9a0: 2020 2020 207c 0a7c 2020 2020 2020 7b37       |.|      {7
│ │ │ │ +0003a9b0: 7d20 7c20 2d62 2020 2061 2020 2030 2020  } | -b   a   0  
│ │ │ │ +0003a9c0: 2030 2020 2020 2030 2020 207c 2020 2020   0     0   |    
│ │ │ │ +0003a9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003aa00: 7c0a 7c20 2020 2020 207b 377d 207c 2063  |.|      {7} | c
│ │ │ │ -0003aa10: 2020 2020 3020 2020 6120 2020 6232 2020      0   a   b2  
│ │ │ │ -0003aa20: 2020 3020 2020 7c20 2020 2020 2020 2020    0   |         
│ │ │ │ +0003a9f0: 2020 2020 207c 0a7c 2020 2020 2020 7b37       |.|      {7
│ │ │ │ +0003aa00: 7d20 7c20 6320 2020 2030 2020 2061 2020  } | c    0   a  
│ │ │ │ +0003aa10: 2062 3220 2020 2030 2020 207c 2020 2020   b2    0   |    
│ │ │ │ +0003aa20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003aa30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003aa40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003aa50: 7c0a 7c20 2020 2020 207b 377d 207c 2030  |.|      {7} | 0
│ │ │ │ -0003aa60: 2020 2020 2d63 2020 2d62 2020 3020 2020      -c  -b  0   
│ │ │ │ -0003aa70: 2020 6132 2020 7c20 2020 2020 2020 2020    a2  |         
│ │ │ │ +0003aa40: 2020 2020 207c 0a7c 2020 2020 2020 7b37       |.|      {7
│ │ │ │ +0003aa50: 7d20 7c20 3020 2020 202d 6320 202d 6220  } | 0    -c  -b 
│ │ │ │ +0003aa60: 2030 2020 2020 2061 3220 207c 2020 2020   0     a2  |    
│ │ │ │ +0003aa70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003aa80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003aa90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003aaa0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0003aa90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0003aaa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003aab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003aac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003aad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003aae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003aaf0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0003ab00: 2031 3220 2020 2020 2035 2020 2020 2020   12      5      
│ │ │ │ +0003aae0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0003aaf0: 2020 2020 2020 3132 2020 2020 2020 3520        12      5 
│ │ │ │ +0003ab00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ab10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ab20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ab30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ab40: 7c0a 7c6f 3132 203a 204d 6174 7269 7820  |.|o12 : Matrix 
│ │ │ │ -0003ab50: 5320 2020 3c2d 2d20 5320 2020 2020 2020  S   <-- S       
│ │ │ │ +0003ab30: 2020 2020 207c 0a7c 6f31 3220 3a20 4d61       |.|o12 : Ma
│ │ │ │ +0003ab40: 7472 6978 2053 2020 203c 2d2d 2053 2020  trix S   <-- S  
│ │ │ │ +0003ab50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ab60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ab70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ab80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ab90: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0003ab80: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0003ab90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003aba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003abb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003abc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003abd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003abe0: 2b0a 0a49 6620 7468 6520 636f 6d70 6c65  +..If the comple
│ │ │ │ -0003abf0: 7869 7479 206f 6620 4d20 6973 206e 6f74  xity of M is not
│ │ │ │ -0003ac00: 206d 6178 696d 616c 2c20 7468 656e 2074   maximal, then t
│ │ │ │ -0003ac10: 6865 2066 696e 6974 6520 7265 736f 6c75  he finite resolu
│ │ │ │ -0003ac20: 7469 6f6e 2074 616b 6573 2070 6c61 6365  tion takes place
│ │ │ │ -0003ac30: 0a6f 7665 7220 616e 2069 6e74 6572 6d65  .over an interme
│ │ │ │ -0003ac40: 6469 6174 6520 636f 6d70 6c65 7465 2069  diate complete i
│ │ │ │ -0003ac50: 6e74 6572 7365 6374 696f 6e3a 0a0a 2b2d  ntersection:..+-
│ │ │ │ +0003abd0: 2d2d 2d2d 2d2b 0a0a 4966 2074 6865 2063  -----+..If the c
│ │ │ │ +0003abe0: 6f6d 706c 6578 6974 7920 6f66 204d 2069  omplexity of M i
│ │ │ │ +0003abf0: 7320 6e6f 7420 6d61 7869 6d61 6c2c 2074  s not maximal, t
│ │ │ │ +0003ac00: 6865 6e20 7468 6520 6669 6e69 7465 2072  hen the finite r
│ │ │ │ +0003ac10: 6573 6f6c 7574 696f 6e20 7461 6b65 7320  esolution takes 
│ │ │ │ +0003ac20: 706c 6163 650a 6f76 6572 2061 6e20 696e  place.over an in
│ │ │ │ +0003ac30: 7465 726d 6564 6961 7465 2063 6f6d 706c  termediate compl
│ │ │ │ +0003ac40: 6574 6520 696e 7465 7273 6563 7469 6f6e  ete intersection
│ │ │ │ +0003ac50: 3a0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  :..+------------
│ │ │ │  0003ac60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ac70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ac80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ac90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003aca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0003acb0: 3133 203a 2053 203d 205a 5a2f 3130 315b  13 : S = ZZ/101[
│ │ │ │ -0003acc0: 612c 622c 632c 645d 2020 2020 2020 2020  a,b,c,d]        
│ │ │ │ +0003aca0: 2d2b 0a7c 6931 3320 3a20 5320 3d20 5a5a  -+.|i13 : S = ZZ
│ │ │ │ +0003acb0: 2f31 3031 5b61 2c62 2c63 2c64 5d20 2020  /101[a,b,c,d]   
│ │ │ │ +0003acc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003acd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ace0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003acf0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003acf0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003ad00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ad10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ad20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ad30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ad40: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003ad50: 3133 203d 2053 2020 2020 2020 2020 2020  13 = S          
│ │ │ │ +0003ad40: 207c 0a7c 6f31 3320 3d20 5320 2020 2020   |.|o13 = S     
│ │ │ │ +0003ad50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ad60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ad70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ad80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ad90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003ad90: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003ada0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003adb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003adc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003add0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ade0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003adf0: 3133 203a 2050 6f6c 796e 6f6d 6961 6c52  13 : PolynomialR
│ │ │ │ -0003ae00: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │ +0003ade0: 207c 0a7c 6f31 3320 3a20 506f 6c79 6e6f   |.|o13 : Polyno
│ │ │ │ +0003adf0: 6d69 616c 5269 6e67 2020 2020 2020 2020  mialRing        
│ │ │ │ +0003ae00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ae10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ae20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ae30: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0003ae30: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0003ae40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ae50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ae60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ae70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003ae80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0003ae90: 3134 203a 2066 6631 203d 206d 6174 7269  14 : ff1 = matri
│ │ │ │ -0003aea0: 7822 6133 2c62 332c 6333 2c64 3322 2020  x"a3,b3,c3,d3"  
│ │ │ │ +0003ae80: 2d2b 0a7c 6931 3420 3a20 6666 3120 3d20  -+.|i14 : ff1 = 
│ │ │ │ +0003ae90: 6d61 7472 6978 2261 332c 6233 2c63 332c  matrix"a3,b3,c3,
│ │ │ │ +0003aea0: 6433 2220 2020 2020 2020 2020 2020 2020  d3"             
│ │ │ │  0003aeb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003aec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003aed0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003aed0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003aee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003aef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003af00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003af10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003af20: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003af30: 3134 203d 207c 2061 3320 6233 2063 3320  14 = | a3 b3 c3 
│ │ │ │ -0003af40: 6433 207c 2020 2020 2020 2020 2020 2020  d3 |            
│ │ │ │ +0003af20: 207c 0a7c 6f31 3420 3d20 7c20 6133 2062   |.|o14 = | a3 b
│ │ │ │ +0003af30: 3320 6333 2064 3320 7c20 2020 2020 2020  3 c3 d3 |       
│ │ │ │ +0003af40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003af50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003af60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003af70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003af70: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003af80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003af90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003afa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003afb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003afc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003afd0: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │ -0003afe0: 2020 2020 3420 2020 2020 2020 2020 2020      4           
│ │ │ │ +0003afc0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0003afd0: 2020 3120 2020 2020 2034 2020 2020 2020    1      4      
│ │ │ │ +0003afe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003aff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b010: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003b020: 3134 203a 204d 6174 7269 7820 5320 203c  14 : Matrix S  <
│ │ │ │ -0003b030: 2d2d 2053 2020 2020 2020 2020 2020 2020  -- S            
│ │ │ │ +0003b010: 207c 0a7c 6f31 3420 3a20 4d61 7472 6978   |.|o14 : Matrix
│ │ │ │ +0003b020: 2053 2020 3c2d 2d20 5320 2020 2020 2020   S  <-- S       
│ │ │ │ +0003b030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b060: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0003b060: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0003b070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003b0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0003b0c0: 3135 203a 2066 6620 3d66 6631 2a72 616e  15 : ff =ff1*ran
│ │ │ │ -0003b0d0: 646f 6d28 736f 7572 6365 2066 6631 2c20  dom(source ff1, 
│ │ │ │ -0003b0e0: 736f 7572 6365 2066 6631 2920 2020 2020  source ff1)     
│ │ │ │ +0003b0b0: 2d2b 0a7c 6931 3520 3a20 6666 203d 6666  -+.|i15 : ff =ff
│ │ │ │ +0003b0c0: 312a 7261 6e64 6f6d 2873 6f75 7263 6520  1*random(source 
│ │ │ │ +0003b0d0: 6666 312c 2073 6f75 7263 6520 6666 3129  ff1, source ff1)
│ │ │ │ +0003b0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b100: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003b100: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003b110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b150: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003b160: 3135 203d 207c 2032 3461 332d 3336 6233  15 = | 24a3-36b3
│ │ │ │ -0003b170: 2d33 3063 332d 3239 6433 2031 3961 332b  -30c3-29d3 19a3+
│ │ │ │ -0003b180: 3139 6233 2d31 3063 332d 3239 6433 202d  19b3-10c3-29d3 -
│ │ │ │ -0003b190: 3861 332d 3232 6233 2d32 3963 332d 3234  8a3-22b3-29c3-24
│ │ │ │ -0003b1a0: 6433 2020 2020 2020 2020 2020 7c0a 7c20  d3          |.| 
│ │ │ │ -0003b1b0: 2020 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d       -----------
│ │ │ │ +0003b150: 207c 0a7c 6f31 3520 3d20 7c20 3234 6133   |.|o15 = | 24a3
│ │ │ │ +0003b160: 2d33 3662 332d 3330 6333 2d32 3964 3320  -36b3-30c3-29d3 
│ │ │ │ +0003b170: 3139 6133 2b31 3962 332d 3130 6333 2d32  19a3+19b3-10c3-2
│ │ │ │ +0003b180: 3964 3320 2d38 6133 2d32 3262 332d 3239  9d3 -8a3-22b3-29
│ │ │ │ +0003b190: 6333 2d32 3464 3320 2020 2020 2020 2020  c3-24d3         
│ │ │ │ +0003b1a0: 207c 0a7c 2020 2020 2020 2d2d 2d2d 2d2d   |.|      ------
│ │ │ │ +0003b1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003b1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ -0003b200: 2020 2020 202d 3338 6133 2d31 3662 332b       -38a3-16b3+
│ │ │ │ -0003b210: 3339 6333 2b32 3164 3320 7c20 2020 2020  39c3+21d3 |     
│ │ │ │ +0003b1f0: 2d7c 0a7c 2020 2020 2020 2d33 3861 332d  -|.|      -38a3-
│ │ │ │ +0003b200: 3136 6233 2b33 3963 332b 3231 6433 207c  16b3+39c3+21d3 |
│ │ │ │ +0003b210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b240: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003b240: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003b250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b290: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003b2a0: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │ -0003b2b0: 2020 2020 3420 2020 2020 2020 2020 2020      4           
│ │ │ │ +0003b290: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0003b2a0: 2020 3120 2020 2020 2034 2020 2020 2020    1      4      
│ │ │ │ +0003b2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b2e0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003b2f0: 3135 203a 204d 6174 7269 7820 5320 203c  15 : Matrix S  <
│ │ │ │ -0003b300: 2d2d 2053 2020 2020 2020 2020 2020 2020  -- S            
│ │ │ │ +0003b2e0: 207c 0a7c 6f31 3520 3a20 4d61 7472 6978   |.|o15 : Matrix
│ │ │ │ +0003b2f0: 2053 2020 3c2d 2d20 5320 2020 2020 2020   S  <-- S       
│ │ │ │ +0003b300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b330: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0003b330: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0003b340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003b380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0003b390: 3136 203a 2052 203d 2053 2f69 6465 616c  16 : R = S/ideal
│ │ │ │ -0003b3a0: 2066 6620 2020 2020 2020 2020 2020 2020   ff             
│ │ │ │ +0003b380: 2d2b 0a7c 6931 3620 3a20 5220 3d20 532f  -+.|i16 : R = S/
│ │ │ │ +0003b390: 6964 6561 6c20 6666 2020 2020 2020 2020  ideal ff        
│ │ │ │ +0003b3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b3d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003b3d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003b3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b420: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003b430: 3136 203d 2052 2020 2020 2020 2020 2020  16 = R          
│ │ │ │ +0003b420: 207c 0a7c 6f31 3620 3d20 5220 2020 2020   |.|o16 = R     
│ │ │ │ +0003b430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b470: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003b470: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003b480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b4c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003b4d0: 3136 203a 2051 756f 7469 656e 7452 696e  16 : QuotientRin
│ │ │ │ -0003b4e0: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ +0003b4c0: 207c 0a7c 6f31 3620 3a20 5175 6f74 6965   |.|o16 : Quotie
│ │ │ │ +0003b4d0: 6e74 5269 6e67 2020 2020 2020 2020 2020  ntRing          
│ │ │ │ +0003b4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b510: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0003b510: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0003b520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003b560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0003b570: 3137 203a 204d 203d 2068 6967 6853 797a  17 : M = highSyz
│ │ │ │ -0003b580: 7967 7920 2852 5e31 2f69 6465 616c 2261  ygy (R^1/ideal"a
│ │ │ │ -0003b590: 3262 3222 2920 2020 2020 2020 2020 2020  2b2")           
│ │ │ │ +0003b560: 2d2b 0a7c 6931 3720 3a20 4d20 3d20 6869  -+.|i17 : M = hi
│ │ │ │ +0003b570: 6768 5379 7a79 6779 2028 525e 312f 6964  ghSyzygy (R^1/id
│ │ │ │ +0003b580: 6561 6c22 6132 6232 2229 2020 2020 2020  eal"a2b2")      
│ │ │ │ +0003b590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b5b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003b5b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003b5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b600: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003b610: 3137 203d 2063 6f6b 6572 6e65 6c20 7b36  17 = cokernel {6
│ │ │ │ -0003b620: 7d20 7c20 6232 2030 202d 6132 2030 207c  } | b2 0 -a2 0 |
│ │ │ │ +0003b600: 207c 0a7c 6f31 3720 3d20 636f 6b65 726e   |.|o17 = cokern
│ │ │ │ +0003b610: 656c 207b 367d 207c 2062 3220 3020 2d61  el {6} | b2 0 -a
│ │ │ │ +0003b620: 3220 3020 7c20 2020 2020 2020 2020 2020  2 0 |           
│ │ │ │  0003b630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b650: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003b660: 2020 2020 2020 2020 2020 2020 2020 7b37                {7
│ │ │ │ -0003b670: 7d20 7c20 6120 2062 2030 2020 2030 207c  } | a  b 0   0 |
│ │ │ │ +0003b650: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0003b660: 2020 207b 377d 207c 2061 2020 6220 3020     {7} | a  b 0 
│ │ │ │ +0003b670: 2020 3020 7c20 2020 2020 2020 2020 2020    0 |           
│ │ │ │  0003b680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b6a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003b6b0: 2020 2020 2020 2020 2020 2020 2020 7b37                {7
│ │ │ │ -0003b6c0: 7d20 7c20 3020 2030 2062 2020 2061 207c  } | 0  0 b   a |
│ │ │ │ +0003b6a0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0003b6b0: 2020 207b 377d 207c 2030 2020 3020 6220     {7} | 0  0 b 
│ │ │ │ +0003b6c0: 2020 6120 7c20 2020 2020 2020 2020 2020    a |           
│ │ │ │  0003b6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b6f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003b6f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003b700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b740: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003b740: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003b750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b760: 2020 2020 2020 2020 2020 2020 3320 2020              3   
│ │ │ │ +0003b760: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │  0003b770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b790: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003b7a0: 3137 203a 2052 2d6d 6f64 756c 652c 2071  17 : R-module, q
│ │ │ │ -0003b7b0: 756f 7469 656e 7420 6f66 2052 2020 2020  uotient of R    
│ │ │ │ +0003b790: 207c 0a7c 6f31 3720 3a20 522d 6d6f 6475   |.|o17 : R-modu
│ │ │ │ +0003b7a0: 6c65 2c20 7175 6f74 6965 6e74 206f 6620  le, quotient of 
│ │ │ │ +0003b7b0: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │  0003b7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b7e0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0003b7e0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0003b7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003b830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0003b840: 3138 203a 2063 6f6d 706c 6578 6974 7920  18 : complexity 
│ │ │ │ -0003b850: 4d20 2020 2020 2020 2020 2020 2020 2020  M               
│ │ │ │ +0003b830: 2d2b 0a7c 6931 3820 3a20 636f 6d70 6c65  -+.|i18 : comple
│ │ │ │ +0003b840: 7869 7479 204d 2020 2020 2020 2020 2020  xity M          
│ │ │ │ +0003b850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b880: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003b880: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003b890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b8d0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003b8e0: 3138 203d 2032 2020 2020 2020 2020 2020  18 = 2          
│ │ │ │ +0003b8d0: 207c 0a7c 6f31 3820 3d20 3220 2020 2020   |.|o18 = 2     
│ │ │ │ +0003b8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b920: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0003b920: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0003b930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003b970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0003b980: 3139 203a 206d 6620 3d20 6d61 7472 6978  19 : mf = matrix
│ │ │ │ -0003b990: 4661 6374 6f72 697a 6174 696f 6e20 2866  Factorization (f
│ │ │ │ -0003b9a0: 662c 204d 2920 2020 2020 2020 2020 2020  f, M)           
│ │ │ │ +0003b970: 2d2b 0a7c 6931 3920 3a20 6d66 203d 206d  -+.|i19 : mf = m
│ │ │ │ +0003b980: 6174 7269 7846 6163 746f 7269 7a61 7469  atrixFactorizati
│ │ │ │ +0003b990: 6f6e 2028 6666 2c20 4d29 2020 2020 2020  on (ff, M)      
│ │ │ │ +0003b9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b9c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003b9c0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003b9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ba00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ba10: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003ba20: 3139 203d 207b 7b37 7d20 7c20 2d61 202d  19 = {{7} | -a -
│ │ │ │ -0003ba30: 3336 6220 3020 6120 7c2c 207b 387d 207c  36b 0 a |, {8} |
│ │ │ │ -0003ba40: 2033 3561 3220 2034 3862 2020 3020 2020   35a2  48b  0   
│ │ │ │ -0003ba50: 2020 2d33 3362 2030 2020 2020 207c 2c20    -33b 0     |, 
│ │ │ │ -0003ba60: 7b36 7d20 7c20 3020 2020 3336 7c0a 7c20  {6} | 0   36|.| 
│ │ │ │ -0003ba70: 2020 2020 2020 7b36 7d20 7c20 6232 2061        {6} | b2 a
│ │ │ │ -0003ba80: 3220 2020 3020 3020 7c20 207b 387d 207c  2   0 0 |  {8} |
│ │ │ │ -0003ba90: 202d 3335 6232 202d 3335 6120 3020 2020   -35b2 -35a 0   
│ │ │ │ -0003baa0: 2020 3020 2020 2030 2020 2020 207c 2020    0    0     |  
│ │ │ │ -0003bab0: 7b37 7d20 7c20 2d33 3620 3020 7c0a 7c20  {7} | -36 0 |.| 
│ │ │ │ -0003bac0: 2020 2020 2020 7b37 7d20 7c20 3020 2030        {7} | 0  0
│ │ │ │ -0003bad0: 2020 2020 6220 6120 7c20 207b 387d 207c      b a |  {8} |
│ │ │ │ -0003bae0: 2030 2020 2020 2030 2020 2020 3333 6232   0     0    33b2
│ │ │ │ -0003baf0: 2020 3333 6120 202d 3333 6232 207c 2020    33a  -33b2 |  
│ │ │ │ -0003bb00: 7b37 7d20 7c20 3120 2020 3020 7c0a 7c20  {7} | 1   0 |.| 
│ │ │ │ +0003ba10: 207c 0a7c 6f31 3920 3d20 7b7b 377d 207c   |.|o19 = {{7} |
│ │ │ │ +0003ba20: 202d 6120 2d33 3662 2030 2061 207c 2c20   -a -36b 0 a |, 
│ │ │ │ +0003ba30: 7b38 7d20 7c20 3335 6132 2020 3438 6220  {8} | 35a2  48b 
│ │ │ │ +0003ba40: 2030 2020 2020 202d 3333 6220 3020 2020   0     -33b 0   
│ │ │ │ +0003ba50: 2020 7c2c 207b 367d 207c 2030 2020 2033    |, {6} | 0   3
│ │ │ │ +0003ba60: 367c 0a7c 2020 2020 2020 207b 367d 207c  6|.|       {6} |
│ │ │ │ +0003ba70: 2062 3220 6132 2020 2030 2030 207c 2020   b2 a2   0 0 |  
│ │ │ │ +0003ba80: 7b38 7d20 7c20 2d33 3562 3220 2d33 3561  {8} | -35b2 -35a
│ │ │ │ +0003ba90: 2030 2020 2020 2030 2020 2020 3020 2020   0     0    0   
│ │ │ │ +0003baa0: 2020 7c20 207b 377d 207c 202d 3336 2030    |  {7} | -36 0
│ │ │ │ +0003bab0: 207c 0a7c 2020 2020 2020 207b 377d 207c   |.|       {7} |
│ │ │ │ +0003bac0: 2030 2020 3020 2020 2062 2061 207c 2020   0  0    b a |  
│ │ │ │ +0003bad0: 7b38 7d20 7c20 3020 2020 2020 3020 2020  {8} | 0     0   
│ │ │ │ +0003bae0: 2033 3362 3220 2033 3361 2020 2d33 3362   33b2  33a  -33b
│ │ │ │ +0003baf0: 3220 7c20 207b 377d 207c 2031 2020 2030  2 |  {7} | 1   0
│ │ │ │ +0003bb00: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003bb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bb20: 2020 2020 2020 2020 2020 207b 387d 207c             {8} |
│ │ │ │ -0003bb30: 2030 2020 2020 2030 2020 2020 2d34 3361   0     0    -43a
│ │ │ │ -0003bb40: 3220 2d33 3362 2030 2020 2020 207c 2020  2 -33b 0     |  
│ │ │ │ -0003bb50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003bb60: 2020 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d       -----------
│ │ │ │ +0003bb20: 7b38 7d20 7c20 3020 2020 2020 3020 2020  {8} | 0     0   
│ │ │ │ +0003bb30: 202d 3433 6132 202d 3333 6220 3020 2020   -43a2 -33b 0   
│ │ │ │ +0003bb40: 2020 7c20 2020 2020 2020 2020 2020 2020    |             
│ │ │ │ +0003bb50: 207c 0a7c 2020 2020 2020 2d2d 2d2d 2d2d   |.|      ------
│ │ │ │ +0003bb60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003bb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003bb80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003bb90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003bba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ -0003bbb0: 2020 2020 2030 2020 7c7d 2020 2020 2020       0  |}      
│ │ │ │ +0003bba0: 2d7c 0a7c 2020 2020 2020 3020 207c 7d20  -|.|      0  |} 
│ │ │ │ +0003bbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bbf0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003bc00: 2020 2020 2033 3620 7c20 2020 2020 2020       36 |       
│ │ │ │ +0003bbf0: 207c 0a7c 2020 2020 2020 3336 207c 2020   |.|      36 |  
│ │ │ │ +0003bc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bc40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003bc50: 2020 2020 2030 2020 7c20 2020 2020 2020       0  |       
│ │ │ │ +0003bc40: 207c 0a7c 2020 2020 2020 3020 207c 2020   |.|      0  |  
│ │ │ │ +0003bc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bc90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003bc90: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003bca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bcb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bcd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bce0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003bcf0: 3139 203a 204c 6973 7420 2020 2020 2020  19 : List       
│ │ │ │ +0003bce0: 207c 0a7c 6f31 3920 3a20 4c69 7374 2020   |.|o19 : List  
│ │ │ │ +0003bcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bd30: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0003bd30: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0003bd40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003bd50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003bd60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003bd70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003bd80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0003bd90: 3230 203a 2063 6f6d 706c 6578 6974 7920  20 : complexity 
│ │ │ │ -0003bda0: 6d66 2020 2020 2020 2020 2020 2020 2020  mf              
│ │ │ │ +0003bd80: 2d2b 0a7c 6932 3020 3a20 636f 6d70 6c65  -+.|i20 : comple
│ │ │ │ +0003bd90: 7869 7479 206d 6620 2020 2020 2020 2020  xity mf         
│ │ │ │ +0003bda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bdd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003bdd0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003bde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003be00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003be10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003be20: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003be30: 3230 203d 2032 2020 2020 2020 2020 2020  20 = 2          
│ │ │ │ +0003be20: 207c 0a7c 6f32 3020 3d20 3220 2020 2020   |.|o20 = 2     
│ │ │ │ +0003be30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003be40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003be50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003be60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003be70: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0003be70: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0003be80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003be90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003bea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003beb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003bec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0003bed0: 3231 203a 2042 5261 6e6b 7320 6d66 2020  21 : BRanks mf  
│ │ │ │ +0003bec0: 2d2b 0a7c 6932 3120 3a20 4252 616e 6b73  -+.|i21 : BRanks
│ │ │ │ +0003bed0: 206d 6620 2020 2020 2020 2020 2020 2020   mf             
│ │ │ │  0003bee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bf00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bf10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003bf10: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003bf20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bf30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bf40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bf50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bf60: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003bf70: 3231 203d 207b 7b32 2c20 327d 2c20 7b31  21 = {{2, 2}, {1
│ │ │ │ -0003bf80: 2c20 327d 7d20 2020 2020 2020 2020 2020  , 2}}           
│ │ │ │ +0003bf60: 207c 0a7c 6f32 3120 3d20 7b7b 322c 2032   |.|o21 = {{2, 2
│ │ │ │ +0003bf70: 7d2c 207b 312c 2032 7d7d 2020 2020 2020  }, {1, 2}}      
│ │ │ │ +0003bf80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bf90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bfb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003bfb0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003bfc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c000: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003c010: 3231 203a 204c 6973 7420 2020 2020 2020  21 : List       
│ │ │ │ +0003c000: 207c 0a7c 6f32 3120 3a20 4c69 7374 2020   |.|o21 : List  
│ │ │ │ +0003c010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c050: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0003c050: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0003c060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003c0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0003c0b0: 3232 203a 2047 203d 206d 616b 6546 696e  22 : G = makeFin
│ │ │ │ -0003c0c0: 6974 6552 6573 6f6c 7574 696f 6e28 6666  iteResolution(ff
│ │ │ │ -0003c0d0: 2c6d 6629 3b20 2020 2020 2020 2020 2020  ,mf);           
│ │ │ │ +0003c0a0: 2d2b 0a7c 6932 3220 3a20 4720 3d20 6d61  -+.|i22 : G = ma
│ │ │ │ +0003c0b0: 6b65 4669 6e69 7465 5265 736f 6c75 7469  keFiniteResoluti
│ │ │ │ +0003c0c0: 6f6e 2866 662c 6d66 293b 2020 2020 2020  on(ff,mf);      
│ │ │ │ +0003c0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c0f0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0003c0f0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0003c100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003c140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0003c150: 3233 203a 2063 6f64 696d 2072 696e 6720  23 : codim ring 
│ │ │ │ -0003c160: 4720 2020 2020 2020 2020 2020 2020 2020  G               
│ │ │ │ +0003c140: 2d2b 0a7c 6932 3320 3a20 636f 6469 6d20  -+.|i23 : codim 
│ │ │ │ +0003c150: 7269 6e67 2047 2020 2020 2020 2020 2020  ring G          
│ │ │ │ +0003c160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c190: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003c190: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003c1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c1e0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003c1f0: 3233 203d 2032 2020 2020 2020 2020 2020  23 = 2          
│ │ │ │ +0003c1e0: 207c 0a7c 6f32 3320 3d20 3220 2020 2020   |.|o23 = 2     
│ │ │ │ +0003c1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c230: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0003c230: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0003c240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003c280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0003c290: 3234 203a 2052 3120 3d20 7269 6e67 2047  24 : R1 = ring G
│ │ │ │ +0003c280: 2d2b 0a7c 6932 3420 3a20 5231 203d 2072  -+.|i24 : R1 = r
│ │ │ │ +0003c290: 696e 6720 4720 2020 2020 2020 2020 2020  ing G           
│ │ │ │  0003c2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c2d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003c2d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003c2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c320: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003c330: 3234 203d 2052 3120 2020 2020 2020 2020  24 = R1         
│ │ │ │ +0003c320: 207c 0a7c 6f32 3420 3d20 5231 2020 2020   |.|o24 = R1    
│ │ │ │ +0003c330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c370: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003c370: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003c380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c3c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003c3d0: 3234 203a 2051 756f 7469 656e 7452 696e  24 : QuotientRin
│ │ │ │ -0003c3e0: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ +0003c3c0: 207c 0a7c 6f32 3420 3a20 5175 6f74 6965   |.|o24 : Quotie
│ │ │ │ +0003c3d0: 6e74 5269 6e67 2020 2020 2020 2020 2020  ntRing          
│ │ │ │ +0003c3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c410: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0003c410: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0003c420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003c460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0003c470: 3235 203a 2046 203d 2066 7265 6552 6573  25 : F = freeRes
│ │ │ │ -0003c480: 6f6c 7574 696f 6e28 7072 756e 6520 7075  olution(prune pu
│ │ │ │ -0003c490: 7368 466f 7277 6172 6428 6d61 7028 522c  shForward(map(R,
│ │ │ │ -0003c4a0: 5231 292c 4d29 2c20 4c65 6e67 7468 4c69  R1),M), LengthLi
│ │ │ │ -0003c4b0: 6d69 7420 3d3e 2034 2920 2020 7c0a 7c20  mit => 4)   |.| 
│ │ │ │ +0003c460: 2d2b 0a7c 6932 3520 3a20 4620 3d20 6672  -+.|i25 : F = fr
│ │ │ │ +0003c470: 6565 5265 736f 6c75 7469 6f6e 2870 7275  eeResolution(pru
│ │ │ │ +0003c480: 6e65 2070 7573 6846 6f72 7761 7264 286d  ne pushForward(m
│ │ │ │ +0003c490: 6170 2852 2c52 3129 2c4d 292c 204c 656e  ap(R,R1),M), Len
│ │ │ │ +0003c4a0: 6774 684c 696d 6974 203d 3e20 3429 2020  gthLimit => 4)  
│ │ │ │ +0003c4b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003c4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c500: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003c510: 2020 2020 2020 2033 2020 2020 2020 2035         3       5
│ │ │ │ -0003c520: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +0003c500: 207c 0a7c 2020 2020 2020 2020 3320 2020   |.|        3   
│ │ │ │ +0003c510: 2020 2020 3520 2020 2020 2020 3220 2020      5       2   
│ │ │ │ +0003c520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c550: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003c560: 3235 203d 2052 3120 203c 2d2d 2052 3120  25 = R1  <-- R1 
│ │ │ │ -0003c570: 203c 2d2d 2052 3120 2020 2020 2020 2020   <-- R1         
│ │ │ │ +0003c550: 207c 0a7c 6f32 3520 3d20 5231 2020 3c2d   |.|o25 = R1  <-
│ │ │ │ +0003c560: 2d20 5231 2020 3c2d 2d20 5231 2020 2020  - R1  <-- R1    
│ │ │ │ +0003c570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c5a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003c5a0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003c5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c5f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003c600: 2020 2020 2030 2020 2020 2020 2031 2020       0       1  
│ │ │ │ -0003c610: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +0003c5f0: 207c 0a7c 2020 2020 2020 3020 2020 2020   |.|      0     
│ │ │ │ +0003c600: 2020 3120 2020 2020 2020 3220 2020 2020    1       2     
│ │ │ │ +0003c610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c640: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003c640: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003c650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c690: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003c6a0: 3235 203a 2043 6f6d 706c 6578 2020 2020  25 : Complex    
│ │ │ │ +0003c690: 207c 0a7c 6f32 3520 3a20 436f 6d70 6c65   |.|o25 : Comple
│ │ │ │ +0003c6a0: 7820 2020 2020 2020 2020 2020 2020 2020  x               
│ │ │ │  0003c6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c6e0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0003c6e0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0003c6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003c730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0003c740: 3236 203a 2062 6574 7469 2046 2020 2020  26 : betti F    
│ │ │ │ +0003c730: 2d2b 0a7c 6932 3620 3a20 6265 7474 6920  -+.|i26 : betti 
│ │ │ │ +0003c740: 4620 2020 2020 2020 2020 2020 2020 2020  F               
│ │ │ │  0003c750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c780: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003c780: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003c790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c7d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003c7e0: 2020 2020 2020 2020 2020 2020 3020 3120              0 1 
│ │ │ │ -0003c7f0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +0003c7d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0003c7e0: 2030 2031 2032 2020 2020 2020 2020 2020   0 1 2          
│ │ │ │ +0003c7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c820: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003c830: 3236 203d 2074 6f74 616c 3a20 3320 3520  26 = total: 3 5 
│ │ │ │ -0003c840: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +0003c820: 207c 0a7c 6f32 3620 3d20 746f 7461 6c3a   |.|o26 = total:
│ │ │ │ +0003c830: 2033 2035 2032 2020 2020 2020 2020 2020   3 5 2          
│ │ │ │ +0003c840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c870: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003c880: 2020 2020 2020 2020 2036 3a20 3120 2e20           6: 1 . 
│ │ │ │ -0003c890: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ +0003c870: 207c 0a7c 2020 2020 2020 2020 2020 363a   |.|          6:
│ │ │ │ +0003c880: 2031 202e 202e 2020 2020 2020 2020 2020   1 . .          
│ │ │ │ +0003c890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c8c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003c8d0: 2020 2020 2020 2020 2037 3a20 3220 3420           7: 2 4 
│ │ │ │ -0003c8e0: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ +0003c8c0: 207c 0a7c 2020 2020 2020 2020 2020 373a   |.|          7:
│ │ │ │ +0003c8d0: 2032 2034 202e 2020 2020 2020 2020 2020   2 4 .          
│ │ │ │ +0003c8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c910: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003c920: 2020 2020 2020 2020 2038 3a20 2e20 2e20           8: . . 
│ │ │ │ -0003c930: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ +0003c910: 207c 0a7c 2020 2020 2020 2020 2020 383a   |.|          8:
│ │ │ │ +0003c920: 202e 202e 202e 2020 2020 2020 2020 2020   . . .          
│ │ │ │ +0003c930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c960: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003c970: 2020 2020 2020 2020 2039 3a20 2e20 3120           9: . 1 
│ │ │ │ -0003c980: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +0003c960: 207c 0a7c 2020 2020 2020 2020 2020 393a   |.|          9:
│ │ │ │ +0003c970: 202e 2031 2032 2020 2020 2020 2020 2020   . 1 2          
│ │ │ │ +0003c980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c9b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003c9b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003c9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ca00: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003ca10: 3236 203a 2042 6574 7469 5461 6c6c 7920  26 : BettiTally 
│ │ │ │ +0003ca00: 207c 0a7c 6f32 3620 3a20 4265 7474 6954   |.|o26 : BettiT
│ │ │ │ +0003ca10: 616c 6c79 2020 2020 2020 2020 2020 2020  ally            
│ │ │ │  0003ca20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ca30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ca40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ca50: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0003ca50: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0003ca60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ca70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ca80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ca90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003caa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0003cab0: 3237 203a 2062 6574 7469 2047 2020 2020  27 : betti G    
│ │ │ │ +0003caa0: 2d2b 0a7c 6932 3720 3a20 6265 7474 6920  -+.|i27 : betti 
│ │ │ │ +0003cab0: 4720 2020 2020 2020 2020 2020 2020 2020  G               
│ │ │ │  0003cac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003caf0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003caf0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003cb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cb40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003cb50: 2020 2020 2020 2020 2020 2020 3020 3120              0 1 
│ │ │ │ -0003cb60: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +0003cb40: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0003cb50: 2030 2031 2032 2020 2020 2020 2020 2020   0 1 2          
│ │ │ │ +0003cb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cb90: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003cba0: 3237 203d 2074 6f74 616c 3a20 3320 3520  27 = total: 3 5 
│ │ │ │ -0003cbb0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +0003cb90: 207c 0a7c 6f32 3720 3d20 746f 7461 6c3a   |.|o27 = total:
│ │ │ │ +0003cba0: 2033 2035 2032 2020 2020 2020 2020 2020   3 5 2          
│ │ │ │ +0003cbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cbe0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003cbf0: 2020 2020 2020 2020 2036 3a20 3120 2e20           6: 1 . 
│ │ │ │ -0003cc00: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ +0003cbe0: 207c 0a7c 2020 2020 2020 2020 2020 363a   |.|          6:
│ │ │ │ +0003cbf0: 2031 202e 202e 2020 2020 2020 2020 2020   1 . .          
│ │ │ │ +0003cc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cc30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003cc40: 2020 2020 2020 2020 2037 3a20 3220 3420           7: 2 4 
│ │ │ │ -0003cc50: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ +0003cc30: 207c 0a7c 2020 2020 2020 2020 2020 373a   |.|          7:
│ │ │ │ +0003cc40: 2032 2034 202e 2020 2020 2020 2020 2020   2 4 .          
│ │ │ │ +0003cc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cc80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003cc90: 2020 2020 2020 2020 2038 3a20 2e20 2e20           8: . . 
│ │ │ │ -0003cca0: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ +0003cc80: 207c 0a7c 2020 2020 2020 2020 2020 383a   |.|          8:
│ │ │ │ +0003cc90: 202e 202e 202e 2020 2020 2020 2020 2020   . . .          
│ │ │ │ +0003cca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ccb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ccc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ccd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003cce0: 2020 2020 2020 2020 2039 3a20 2e20 3120           9: . 1 
│ │ │ │ -0003ccf0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +0003ccd0: 207c 0a7c 2020 2020 2020 2020 2020 393a   |.|          9:
│ │ │ │ +0003cce0: 202e 2031 2032 2020 2020 2020 2020 2020   . 1 2          
│ │ │ │ +0003ccf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cd20: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003cd20: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003cd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cd70: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003cd80: 3237 203a 2042 6574 7469 5461 6c6c 7920  27 : BettiTally 
│ │ │ │ +0003cd70: 207c 0a7c 6f32 3720 3a20 4265 7474 6954   |.|o27 : BettiT
│ │ │ │ +0003cd80: 616c 6c79 2020 2020 2020 2020 2020 2020  ally            
│ │ │ │  0003cd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cdc0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0003cdc0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0003cdd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003cde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003cdf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ce00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003ce10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53  ------------+..S
│ │ │ │ -0003ce20: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d  ee also.========
│ │ │ │ -0003ce30: 0a0a 2020 2a20 2a6e 6f74 6520 6d61 7472  ..  * *note matr
│ │ │ │ -0003ce40: 6978 4661 6374 6f72 697a 6174 696f 6e3a  ixFactorization:
│ │ │ │ -0003ce50: 206d 6174 7269 7846 6163 746f 7269 7a61   matrixFactoriza
│ │ │ │ -0003ce60: 7469 6f6e 2c20 2d2d 204d 6170 7320 696e  tion, -- Maps in
│ │ │ │ -0003ce70: 2061 2068 6967 6865 720a 2020 2020 636f   a higher.    co
│ │ │ │ -0003ce80: 6469 6d65 6e73 696f 6e20 6d61 7472 6978  dimension matrix
│ │ │ │ -0003ce90: 2066 6163 746f 7269 7a61 7469 6f6e 0a20   factorization. 
│ │ │ │ -0003cea0: 202a 202a 6e6f 7465 2062 4d61 7073 3a20   * *note bMaps: 
│ │ │ │ -0003ceb0: 624d 6170 732c 202d 2d20 6c69 7374 2074  bMaps, -- list t
│ │ │ │ -0003cec0: 6865 206d 6170 7320 2064 5f70 3a42 5f31  he maps  d_p:B_1
│ │ │ │ -0003ced0: 2870 292d 2d3e 425f 3028 7029 2069 6e20  (p)-->B_0(p) in 
│ │ │ │ -0003cee0: 610a 2020 2020 6d61 7472 6978 4661 6374  a.    matrixFact
│ │ │ │ -0003cef0: 6f72 697a 6174 696f 6e0a 2020 2a20 2a6e  orization.  * *n
│ │ │ │ -0003cf00: 6f74 6520 7073 694d 6170 733a 2070 7369  ote psiMaps: psi
│ │ │ │ -0003cf10: 4d61 7073 2c20 2d2d 206c 6973 7420 7468  Maps, -- list th
│ │ │ │ -0003cf20: 6520 6d61 7073 2020 7073 6928 7029 3a20  e maps  psi(p): 
│ │ │ │ -0003cf30: 425f 3128 7029 202d 2d3e 2041 5f30 2870  B_1(p) --> A_0(p
│ │ │ │ -0003cf40: 2d31 2920 696e 2061 0a20 2020 206d 6174  -1) in a.    mat
│ │ │ │ -0003cf50: 7269 7846 6163 746f 7269 7a61 7469 6f6e  rixFactorization
│ │ │ │ -0003cf60: 0a20 202a 202a 6e6f 7465 2068 4d61 7073  .  * *note hMaps
│ │ │ │ -0003cf70: 3a20 684d 6170 732c 202d 2d20 6c69 7374  : hMaps, -- list
│ │ │ │ -0003cf80: 2074 6865 206d 6170 7320 2068 2870 293a   the maps  h(p):
│ │ │ │ -0003cf90: 2041 5f30 2870 292d 2d3e 2041 5f31 2870   A_0(p)--> A_1(p
│ │ │ │ -0003cfa0: 2920 696e 2061 0a20 2020 206d 6174 7269  ) in a.    matri
│ │ │ │ -0003cfb0: 7846 6163 746f 7269 7a61 7469 6f6e 0a20  xFactorization. 
│ │ │ │ -0003cfc0: 202a 202a 6e6f 7465 2063 6f6d 706c 6578   * *note complex
│ │ │ │ -0003cfd0: 6974 793a 2063 6f6d 706c 6578 6974 792c  ity: complexity,
│ │ │ │ -0003cfe0: 202d 2d20 636f 6d70 6c65 7869 7479 206f   -- complexity o
│ │ │ │ -0003cff0: 6620 6120 6d6f 6475 6c65 206f 7665 7220  f a module over 
│ │ │ │ -0003d000: 6120 636f 6d70 6c65 7465 0a20 2020 2069  a complete.    i
│ │ │ │ -0003d010: 6e74 6572 7365 6374 696f 6e0a 0a57 6179  ntersection..Way
│ │ │ │ -0003d020: 7320 746f 2075 7365 206d 616b 6546 696e  s to use makeFin
│ │ │ │ -0003d030: 6974 6552 6573 6f6c 7574 696f 6e3a 0a3d  iteResolution:.=
│ │ │ │ +0003ce10: 2d2b 0a0a 5365 6520 616c 736f 0a3d 3d3d  -+..See also.===
│ │ │ │ +0003ce20: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
│ │ │ │ +0003ce30: 206d 6174 7269 7846 6163 746f 7269 7a61   matrixFactoriza
│ │ │ │ +0003ce40: 7469 6f6e 3a20 6d61 7472 6978 4661 6374  tion: matrixFact
│ │ │ │ +0003ce50: 6f72 697a 6174 696f 6e2c 202d 2d20 4d61  orization, -- Ma
│ │ │ │ +0003ce60: 7073 2069 6e20 6120 6869 6768 6572 0a20  ps in a higher. 
│ │ │ │ +0003ce70: 2020 2063 6f64 696d 656e 7369 6f6e 206d     codimension m
│ │ │ │ +0003ce80: 6174 7269 7820 6661 6374 6f72 697a 6174  atrix factorizat
│ │ │ │ +0003ce90: 696f 6e0a 2020 2a20 2a6e 6f74 6520 624d  ion.  * *note bM
│ │ │ │ +0003cea0: 6170 733a 2062 4d61 7073 2c20 2d2d 206c  aps: bMaps, -- l
│ │ │ │ +0003ceb0: 6973 7420 7468 6520 6d61 7073 2020 645f  ist the maps  d_
│ │ │ │ +0003cec0: 703a 425f 3128 7029 2d2d 3e42 5f30 2870  p:B_1(p)-->B_0(p
│ │ │ │ +0003ced0: 2920 696e 2061 0a20 2020 206d 6174 7269  ) in a.    matri
│ │ │ │ +0003cee0: 7846 6163 746f 7269 7a61 7469 6f6e 0a20  xFactorization. 
│ │ │ │ +0003cef0: 202a 202a 6e6f 7465 2070 7369 4d61 7073   * *note psiMaps
│ │ │ │ +0003cf00: 3a20 7073 694d 6170 732c 202d 2d20 6c69  : psiMaps, -- li
│ │ │ │ +0003cf10: 7374 2074 6865 206d 6170 7320 2070 7369  st the maps  psi
│ │ │ │ +0003cf20: 2870 293a 2042 5f31 2870 2920 2d2d 3e20  (p): B_1(p) --> 
│ │ │ │ +0003cf30: 415f 3028 702d 3129 2069 6e20 610a 2020  A_0(p-1) in a.  
│ │ │ │ +0003cf40: 2020 6d61 7472 6978 4661 6374 6f72 697a    matrixFactoriz
│ │ │ │ +0003cf50: 6174 696f 6e0a 2020 2a20 2a6e 6f74 6520  ation.  * *note 
│ │ │ │ +0003cf60: 684d 6170 733a 2068 4d61 7073 2c20 2d2d  hMaps: hMaps, --
│ │ │ │ +0003cf70: 206c 6973 7420 7468 6520 6d61 7073 2020   list the maps  
│ │ │ │ +0003cf80: 6828 7029 3a20 415f 3028 7029 2d2d 3e20  h(p): A_0(p)--> 
│ │ │ │ +0003cf90: 415f 3128 7029 2069 6e20 610a 2020 2020  A_1(p) in a.    
│ │ │ │ +0003cfa0: 6d61 7472 6978 4661 6374 6f72 697a 6174  matrixFactorizat
│ │ │ │ +0003cfb0: 696f 6e0a 2020 2a20 2a6e 6f74 6520 636f  ion.  * *note co
│ │ │ │ +0003cfc0: 6d70 6c65 7869 7479 3a20 636f 6d70 6c65  mplexity: comple
│ │ │ │ +0003cfd0: 7869 7479 2c20 2d2d 2063 6f6d 706c 6578  xity, -- complex
│ │ │ │ +0003cfe0: 6974 7920 6f66 2061 206d 6f64 756c 6520  ity of a module 
│ │ │ │ +0003cff0: 6f76 6572 2061 2063 6f6d 706c 6574 650a  over a complete.
│ │ │ │ +0003d000: 2020 2020 696e 7465 7273 6563 7469 6f6e      intersection
│ │ │ │ +0003d010: 0a0a 5761 7973 2074 6f20 7573 6520 6d61  ..Ways to use ma
│ │ │ │ +0003d020: 6b65 4669 6e69 7465 5265 736f 6c75 7469  keFiniteResoluti
│ │ │ │ +0003d030: 6f6e 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  on:.============
│ │ │ │  0003d040: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0003d050: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0003d060: 0a0a 2020 2a20 226d 616b 6546 696e 6974  ..  * "makeFinit
│ │ │ │ -0003d070: 6552 6573 6f6c 7574 696f 6e28 4d61 7472  eResolution(Matr
│ │ │ │ -0003d080: 6978 2c4c 6973 7429 220a 0a46 6f72 2074  ix,List)"..For t
│ │ │ │ -0003d090: 6865 2070 726f 6772 616d 6d65 720a 3d3d  he programmer.==
│ │ │ │ -0003d0a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0003d0b0: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f  ..The object *no
│ │ │ │ -0003d0c0: 7465 206d 616b 6546 696e 6974 6552 6573  te makeFiniteRes
│ │ │ │ -0003d0d0: 6f6c 7574 696f 6e3a 206d 616b 6546 696e  olution: makeFin
│ │ │ │ -0003d0e0: 6974 6552 6573 6f6c 7574 696f 6e2c 2069  iteResolution, i
│ │ │ │ -0003d0f0: 7320 6120 2a6e 6f74 6520 6d65 7468 6f64  s a *note method
│ │ │ │ -0003d100: 0a66 756e 6374 696f 6e3a 2028 4d61 6361  .function: (Maca
│ │ │ │ -0003d110: 756c 6179 3244 6f63 294d 6574 686f 6446  ulay2Doc)MethodF
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│ │ │ │ +0003d050: 3d3d 3d3d 3d0a 0a20 202a 2022 6d61 6b65  =====..  * "make
│ │ │ │ +0003d060: 4669 6e69 7465 5265 736f 6c75 7469 6f6e  FiniteResolution
│ │ │ │ +0003d070: 284d 6174 7269 782c 4c69 7374 2922 0a0a  (Matrix,List)"..
│ │ │ │ +0003d080: 466f 7220 7468 6520 7072 6f67 7261 6d6d  For the programm
│ │ │ │ +0003d090: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  er.=============
│ │ │ │ +0003d0a0: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563  =====..The objec
│ │ │ │ +0003d0b0: 7420 2a6e 6f74 6520 6d61 6b65 4669 6e69  t *note makeFini
│ │ │ │ +0003d0c0: 7465 5265 736f 6c75 7469 6f6e 3a20 6d61  teResolution: ma
│ │ │ │ +0003d0d0: 6b65 4669 6e69 7465 5265 736f 6c75 7469  keFiniteResoluti
│ │ │ │ +0003d0e0: 6f6e 2c20 6973 2061 202a 6e6f 7465 206d  on, is a *note m
│ │ │ │ +0003d0f0: 6574 686f 640a 6675 6e63 7469 6f6e 3a20  ethod.function: 
│ │ │ │ +0003d100: 284d 6163 6175 6c61 7932 446f 6329 4d65  (Macaulay2Doc)Me
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│ │ │ │  0003d310: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0003d320: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0003d330: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0003d340: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0003d350: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0003d360: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020  ************..  
│ │ │ │ -0003d370: 2a20 5573 6167 653a 200a 2020 2020 2020  * Usage: .      
│ │ │ │ -0003d380: 2020 6d61 7073 203d 206d 616b 6546 696e    maps = makeFin
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│ │ │ │ -0003d3a0: 696d 3228 6666 2c6d 6629 0a20 202a 2049  im2(ff,mf).  * I
│ │ │ │ -0003d3b0: 6e70 7574 733a 0a20 2020 2020 202a 206d  nputs:.      * m
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│ │ │ │ -0003d400: 202a 2066 662c 2061 202a 6e6f 7465 206d   * ff, a *note m
│ │ │ │ -0003d410: 6174 7269 783a 2028 4d61 6361 756c 6179  atrix: (Macaulay
│ │ │ │ -0003d420: 3244 6f63 294d 6174 7269 782c 2c20 7265  2Doc)Matrix,, re
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│ │ │ │ -0003d440: 202a 202a 6e6f 7465 204f 7074 696f 6e61   * *note Optiona
│ │ │ │ -0003d450: 6c20 696e 7075 7473 3a20 284d 6163 6175  l inputs: (Macau
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│ │ │ │ -0003d470: 6e63 7469 6f6e 7320 7769 7468 206f 7074  nctions with opt
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│ │ │ │ -0003d490: 2020 2020 202a 2043 6865 636b 203d 3e20       * Check => 
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│ │ │ │ -0003d4b0: 7565 2066 616c 7365 0a20 202a 204f 7574  ue false.  * Out
│ │ │ │ -0003d4c0: 7075 7473 3a0a 2020 2020 2020 2a20 6d61  puts:.      * ma
│ │ │ │ -0003d4d0: 7073 2c20 6120 2a6e 6f74 6520 6861 7368  ps, a *note hash
│ │ │ │ -0003d4e0: 2074 6162 6c65 3a20 284d 6163 6175 6c61   table: (Macaula
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│ │ │ │ -0003d500: 2c20 6d61 6e79 206d 6170 730a 0a44 6573  , many maps..Des
│ │ │ │ -0003d510: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ -0003d520: 3d3d 3d3d 0a0a 4769 7665 6e20 6120 636f  ====..Given a co
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│ │ │ │ -0003d540: 746f 7269 7a61 7469 6f6e 2c20 6d61 6b65  torization, make
│ │ │ │ -0003d550: 7320 616c 6c20 7468 6520 636f 6d70 6f6e  s all the compon
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│ │ │ │ -0003d580: 7468 6520 686f 6d6f 746f 7069 6573 2074  the homotopies t
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│ │ │ │ -0003d5c0: 2034 2e32 2e33 206f 6620 4569 7365 6e62   4.2.3 of Eisenb
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│ │ │ │ -0003d5e0: 6c20 4672 6565 2052 6573 6f6c 7574 696f  l Free Resolutio
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│ │ │ │ +0003d360: 2a0a 0a20 202a 2055 7361 6765 3a20 0a20  *..  * Usage: . 
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│ │ │ │ +0003d380: 6b65 4669 6e69 7465 5265 736f 6c75 7469  keFiniteResoluti
│ │ │ │ +0003d390: 6f6e 436f 6469 6d32 2866 662c 6d66 290a  onCodim2(ff,mf).
│ │ │ │ +0003d3a0: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ +0003d3b0: 2020 2a20 6d66 2c20 6120 2a6e 6f74 6520    * mf, a *note 
│ │ │ │ +0003d3c0: 6c69 7374 3a20 284d 6163 6175 6c61 7932  list: (Macaulay2
│ │ │ │ +0003d3d0: 446f 6329 4c69 7374 2c2c 206d 6174 7269  Doc)List,, matri
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│ │ │ │ +0003d3f0: 2020 2020 2020 2a20 6666 2c20 6120 2a6e        * ff, a *n
│ │ │ │ +0003d400: 6f74 6520 6d61 7472 6978 3a20 284d 6163  ote matrix: (Mac
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│ │ │ │ +0003d4c0: 202a 206d 6170 732c 2061 202a 6e6f 7465   * maps, a *note
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│ │ │ │ +0003d5b0: 6173 2069 6e20 342e 322e 3320 6f66 2045  as in 4.2.3 of E
│ │ │ │ +0003d5c0: 6973 656e 6275 642d 5065 6576 6120 224d  isenbud-Peeva "M
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│ │ │ │  0003d630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003d640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0003d6d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0003d6e0: 6f31 203d 206b 6b20 2020 2020 2020 2020  o1 = kk         
│ │ │ │ +0003d6d0: 2020 7c0a 7c6f 3120 3d20 6b6b 2020 2020    |.|o1 = kk    
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│ │ │ │ +0003d710: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
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│ │ │ │  0003d730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d740: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0003d7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003d7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0003d7f0: 2053 203d 206b 6b5b 612c 625d 2020 2020   S = kk[a,b]    
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│ │ │ │ +0003d7e0: 7c69 3220 3a20 5320 3d20 6b6b 5b61 2c62  |i2 : S = kk[a,b
│ │ │ │ +0003d7f0: 5d20 2020 2020 2020 2020 2020 2020 2020  ]               
│ │ │ │  0003d800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d820: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
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│ │ │ │ -0003d870: 0a7c 6f32 203d 2053 2020 2020 2020 2020  .|o2 = S        
│ │ │ │ +0003d860: 2020 2020 7c0a 7c6f 3220 3d20 5320 2020      |.|o2 = S   
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│ │ │ │  0003d880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d890: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003d8b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0003d8a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0003d8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d8f0: 2020 2020 207c 0a7c 6f32 203a 2050 6f6c       |.|o2 : Pol
│ │ │ │ -0003d900: 796e 6f6d 6961 6c52 696e 6720 2020 2020  ynomialRing     
│ │ │ │ +0003d8e0: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ +0003d8f0: 3a20 506f 6c79 6e6f 6d69 616c 5269 6e67  : PolynomialRing
│ │ │ │ +0003d900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d930: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0003d920: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0003d930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003d940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003d950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003d960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003d970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933  -----------+.|i3
│ │ │ │ -0003d980: 203a 2066 6620 3d20 6d61 7472 6978 2261   : ff = matrix"a
│ │ │ │ -0003d990: 342c 6234 2220 2020 2020 2020 2020 2020  4,b4"           
│ │ │ │ +0003d970: 2b0a 7c69 3320 3a20 6666 203d 206d 6174  +.|i3 : ff = mat
│ │ │ │ +0003d980: 7269 7822 6134 2c62 3422 2020 2020 2020  rix"a4,b4"      
│ │ │ │ +0003d990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d9b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0003d9c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0003d9b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003d9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003da00: 207c 0a7c 6f33 203d 207c 2061 3420 6234   |.|o3 = | a4 b4
│ │ │ │ -0003da10: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ +0003d9f0: 2020 2020 2020 7c0a 7c6f 3320 3d20 7c20        |.|o3 = | 
│ │ │ │ +0003da00: 6134 2062 3420 7c20 2020 2020 2020 2020  a4 b4 |         
│ │ │ │ +0003da10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003da20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003da30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003da40: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0003da30: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003da40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003da50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003da60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003da70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003da80: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0003da90: 2020 2020 2020 2031 2020 2020 2020 3220         1      2 
│ │ │ │ +0003da70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003da80: 2020 2020 2020 2020 2020 2020 3120 2020              1   
│ │ │ │ +0003da90: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │  0003daa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dac0: 2020 2020 2020 2020 2020 7c0a 7c6f 3320            |.|o3 
│ │ │ │ -0003dad0: 3a20 4d61 7472 6978 2053 2020 3c2d 2d20  : Matrix S  <-- 
│ │ │ │ -0003dae0: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │ +0003dab0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003dac0: 0a7c 6f33 203a 204d 6174 7269 7820 5320  .|o3 : Matrix S 
│ │ │ │ +0003dad0: 203c 2d2d 2053 2020 2020 2020 2020 2020   <-- S          
│ │ │ │ +0003dae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003daf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003db00: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0003db00: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  0003db10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003db20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003db30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003db40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003db50: 2b0a 7c69 3420 3a20 5220 3d20 532f 6964  +.|i4 : R = S/id
│ │ │ │ -0003db60: 6561 6c20 6666 2020 2020 2020 2020 2020  eal ff          
│ │ │ │ +0003db40: 2d2d 2d2d 2d2b 0a7c 6934 203a 2052 203d  -----+.|i4 : R =
│ │ │ │ +0003db50: 2053 2f69 6465 616c 2066 6620 2020 2020   S/ideal ff     
│ │ │ │ +0003db60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003db70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003db80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003db90: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003db80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003db90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003dba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003dbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dbd0: 2020 2020 2020 7c0a 7c6f 3420 3d20 5220        |.|o4 = R 
│ │ │ │ +0003dbc0: 2020 2020 2020 2020 2020 207c 0a7c 6f34             |.|o4
│ │ │ │ +0003dbd0: 203d 2052 2020 2020 2020 2020 2020 2020   = R            
│ │ │ │  0003dbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003dbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dc10: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003dc00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0003dc10: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0003dc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003dc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003dc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dc50: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003dc60: 3420 3a20 5175 6f74 6965 6e74 5269 6e67  4 : QuotientRing
│ │ │ │ +0003dc50: 207c 0a7c 6f34 203a 2051 756f 7469 656e   |.|o4 : Quotien
│ │ │ │ +0003dc60: 7452 696e 6720 2020 2020 2020 2020 2020  tRing           
│ │ │ │  0003dc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003dc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dc90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0003dca0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0003dc90: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0003dca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003dcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003dcc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003dcd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003dce0: 2d2d 2b0a 7c69 3520 3a20 4e20 3d20 525e  --+.|i5 : N = R^
│ │ │ │ -0003dcf0: 312f 6964 6561 6c22 6132 2c20 6162 2c20  1/ideal"a2, ab, 
│ │ │ │ -0003dd00: 6233 2220 2020 2020 2020 2020 2020 2020  b3"             
│ │ │ │ -0003dd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dd20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0003dcd0: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 204e  -------+.|i5 : N
│ │ │ │ +0003dce0: 203d 2052 5e31 2f69 6465 616c 2261 322c   = R^1/ideal"a2,
│ │ │ │ +0003dcf0: 2061 622c 2062 3322 2020 2020 2020 2020   ab, b3"        
│ │ │ │ +0003dd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003dd10: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0003dd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003dd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003dd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dd60: 2020 2020 2020 2020 7c0a 7c6f 3520 3d20          |.|o5 = 
│ │ │ │ -0003dd70: 636f 6b65 726e 656c 207c 2061 3220 6162  cokernel | a2 ab
│ │ │ │ -0003dd80: 2062 3320 7c20 2020 2020 2020 2020 2020   b3 |           
│ │ │ │ +0003dd50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0003dd60: 6f35 203d 2063 6f6b 6572 6e65 6c20 7c20  o5 = cokernel | 
│ │ │ │ +0003dd70: 6132 2061 6220 6233 207c 2020 2020 2020  a2 ab b3 |      
│ │ │ │ +0003dd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003dd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dda0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0003dda0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0003ddb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ddc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ddd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dde0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0003ddf0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0003de00: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │ +0003dde0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003ddf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003de00: 2020 3120 2020 2020 2020 2020 2020 2020    1             
│ │ │ │  0003de10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003de20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003de30: 207c 0a7c 6f35 203a 2052 2d6d 6f64 756c   |.|o5 : R-modul
│ │ │ │ -0003de40: 652c 2071 756f 7469 656e 7420 6f66 2052  e, quotient of R
│ │ │ │ +0003de20: 2020 2020 2020 7c0a 7c6f 3520 3a20 522d        |.|o5 : R-
│ │ │ │ +0003de30: 6d6f 6475 6c65 2c20 7175 6f74 6965 6e74  module, quotient
│ │ │ │ +0003de40: 206f 6620 5220 2020 2020 2020 2020 2020   of R           
│ │ │ │  0003de50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003de60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003de70: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0003de60: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0003de70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003de80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003de90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003dea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003deb0: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 204e  -------+.|i6 : N
│ │ │ │ -0003dec0: 203d 2063 6f6b 6572 2076 6172 7320 5220   = coker vars R 
│ │ │ │ +0003dea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0003deb0: 3620 3a20 4e20 3d20 636f 6b65 7220 7661  6 : N = coker va
│ │ │ │ +0003dec0: 7273 2052 2020 2020 2020 2020 2020 2020  rs R            
│ │ │ │  0003ded0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003def0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0003dee0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003def0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0003df00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003df10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003df20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003df30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0003df40: 6f36 203d 2063 6f6b 6572 6e65 6c20 7c20  o6 = cokernel | 
│ │ │ │ -0003df50: 6120 6220 7c20 2020 2020 2020 2020 2020  a b |           
│ │ │ │ +0003df30: 2020 7c0a 7c6f 3620 3d20 636f 6b65 726e    |.|o6 = cokern
│ │ │ │ +0003df40: 656c 207c 2061 2062 207c 2020 2020 2020  el | a b |      
│ │ │ │ +0003df50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003df60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003df70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003df80: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0003df70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0003df80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003df90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003dfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dfc0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003dfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dfe0: 2020 3120 2020 2020 2020 2020 2020 2020    1             
│ │ │ │ -0003dff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e000: 2020 2020 2020 7c0a 7c6f 3620 3a20 522d        |.|o6 : R-
│ │ │ │ -0003e010: 6d6f 6475 6c65 2c20 7175 6f74 6965 6e74  module, quotient
│ │ │ │ -0003e020: 206f 6620 5220 2020 2020 2020 2020 2020   of R           
│ │ │ │ -0003e030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e040: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0003dfb0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003dfc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003dfd0: 2020 2020 2020 2031 2020 2020 2020 2020         1        
│ │ │ │ +0003dfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003dff0: 2020 2020 2020 2020 2020 207c 0a7c 6f36             |.|o6
│ │ │ │ +0003e000: 203a 2052 2d6d 6f64 756c 652c 2071 756f   : R-module, quo
│ │ │ │ +0003e010: 7469 656e 7420 6f66 2052 2020 2020 2020  tient of R      
│ │ │ │ +0003e020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e030: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0003e040: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  0003e050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0003e090: 3720 3a20 4d20 3d20 6869 6768 5379 7a79  7 : M = highSyzy
│ │ │ │ -0003e0a0: 6779 204e 2020 2020 2020 2020 2020 2020  gy N            
│ │ │ │ +0003e080: 2d2b 0a7c 6937 203a 204d 203d 2068 6967  -+.|i7 : M = hig
│ │ │ │ +0003e090: 6853 797a 7967 7920 4e20 2020 2020 2020  hSyzygy N       
│ │ │ │ +0003e0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e0c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0003e0d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0003e0c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0003e0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e110: 2020 7c0a 7c6f 3720 3d20 636f 6b65 726e    |.|o7 = cokern
│ │ │ │ -0003e120: 656c 207b 327d 207c 2030 202d 6233 2061  el {2} | 0 -b3 a
│ │ │ │ -0003e130: 3320 3020 7c20 2020 2020 2020 2020 2020  3 0 |           
│ │ │ │ -0003e140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e150: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0003e160: 2020 2020 2020 7b34 7d20 7c20 6220 6120        {4} | b a 
│ │ │ │ -0003e170: 2020 3020 2030 207c 2020 2020 2020 2020    0  0 |        
│ │ │ │ -0003e180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e190: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0003e1a0: 2020 2020 2020 2020 207b 347d 207c 2030           {4} | 0
│ │ │ │ -0003e1b0: 2030 2020 2062 2020 6120 7c20 2020 2020   0   b  a |     
│ │ │ │ +0003e100: 2020 2020 2020 207c 0a7c 6f37 203d 2063         |.|o7 = c
│ │ │ │ +0003e110: 6f6b 6572 6e65 6c20 7b32 7d20 7c20 3020  okernel {2} | 0 
│ │ │ │ +0003e120: 2d62 3320 6133 2030 207c 2020 2020 2020  -b3 a3 0 |      
│ │ │ │ +0003e130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e140: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0003e150: 2020 2020 2020 2020 2020 207b 347d 207c             {4} |
│ │ │ │ +0003e160: 2062 2061 2020 2030 2020 3020 7c20 2020   b a   0  0 |   
│ │ │ │ +0003e170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e180: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0003e190: 2020 2020 2020 2020 2020 2020 2020 7b34                {4
│ │ │ │ +0003e1a0: 7d20 7c20 3020 3020 2020 6220 2061 207c  } | 0 0   b  a |
│ │ │ │ +0003e1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e1d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0003e1d0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0003e1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e210: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0003e220: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0003e230: 2020 2020 2020 2020 2020 2020 2033 2020               3  
│ │ │ │ +0003e210: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003e220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e230: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │  0003e240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e260: 207c 0a7c 6f37 203a 2052 2d6d 6f64 756c   |.|o7 : R-modul
│ │ │ │ -0003e270: 652c 2071 756f 7469 656e 7420 6f66 2052  e, quotient of R
│ │ │ │ +0003e250: 2020 2020 2020 7c0a 7c6f 3720 3a20 522d        |.|o7 : R-
│ │ │ │ +0003e260: 6d6f 6475 6c65 2c20 7175 6f74 6965 6e74  module, quotient
│ │ │ │ +0003e270: 206f 6620 5220 2020 2020 2020 2020 2020   of R           
│ │ │ │  0003e280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e2a0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0003e290: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0003e2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e2e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 204d  -------+.|i8 : M
│ │ │ │ -0003e2f0: 5320 3d20 7075 7368 466f 7277 6172 6428  S = pushForward(
│ │ │ │ -0003e300: 6d61 7028 522c 5329 2c4d 2920 2020 2020  map(R,S),M)     
│ │ │ │ -0003e310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e320: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0003e2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0003e2e0: 3820 3a20 4d53 203d 2070 7573 6846 6f72  8 : MS = pushFor
│ │ │ │ +0003e2f0: 7761 7264 286d 6170 2852 2c53 292c 4d29  ward(map(R,S),M)
│ │ │ │ +0003e300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e310: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003e320: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0003e330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e360: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0003e370: 6f38 203d 2063 6f6b 6572 6e65 6c20 7b32  o8 = cokernel {2
│ │ │ │ -0003e380: 7d20 7c20 3020 6233 2061 3320 3020 3020  } | 0 b3 a3 0 0 
│ │ │ │ -0003e390: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ -0003e3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e3b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0003e3c0: 207b 347d 207c 2062 202d 6120 3020 2030   {4} | b -a 0  0
│ │ │ │ -0003e3d0: 2030 2020 7c20 2020 2020 2020 2020 2020   0  |           
│ │ │ │ -0003e3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e3f0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003e400: 2020 2020 7b34 7d20 7c20 3020 3020 2062      {4} | 0 0  b
│ │ │ │ -0003e410: 2020 6120 6234 207c 2020 2020 2020 2020    a b4 |        
│ │ │ │ -0003e420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e430: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0003e360: 2020 7c0a 7c6f 3820 3d20 636f 6b65 726e    |.|o8 = cokern
│ │ │ │ +0003e370: 656c 207b 327d 207c 2030 2062 3320 6133  el {2} | 0 b3 a3
│ │ │ │ +0003e380: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ +0003e390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e3a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0003e3b0: 2020 2020 2020 7b34 7d20 7c20 6220 2d61        {4} | b -a
│ │ │ │ +0003e3c0: 2030 2020 3020 3020 207c 2020 2020 2020   0  0 0  |      
│ │ │ │ +0003e3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e3e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003e3f0: 2020 2020 2020 2020 207b 347d 207c 2030           {4} | 0
│ │ │ │ +0003e400: 2030 2020 6220 2061 2062 3420 7c20 2020   0  b  a b4 |   
│ │ │ │ +0003e410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e420: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0003e430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e470: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0003e480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e490: 2020 2020 2020 2020 3320 2020 2020 2020          3       
│ │ │ │ +0003e460: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0003e470: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0003e480: 2020 2020 2020 2020 2020 2020 2033 2020               3  
│ │ │ │ +0003e490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e4b0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003e4c0: 3820 3a20 532d 6d6f 6475 6c65 2c20 7175  8 : S-module, qu
│ │ │ │ -0003e4d0: 6f74 6965 6e74 206f 6620 5320 2020 2020  otient of S     
│ │ │ │ +0003e4b0: 207c 0a7c 6f38 203a 2053 2d6d 6f64 756c   |.|o8 : S-modul
│ │ │ │ +0003e4c0: 652c 2071 756f 7469 656e 7420 6f66 2053  e, quotient of S
│ │ │ │ +0003e4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e4f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0003e500: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0003e4f0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0003e500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e540: 2d2d 2b0a 7c69 3920 3a20 6d66 203d 206d  --+.|i9 : mf = m
│ │ │ │ -0003e550: 6174 7269 7846 6163 746f 7269 7a61 7469  atrixFactorizati
│ │ │ │ -0003e560: 6f6e 2866 662c 204d 2920 2020 2020 2020  on(ff, M)       
│ │ │ │ -0003e570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e580: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0003e530: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 206d  -------+.|i9 : m
│ │ │ │ +0003e540: 6620 3d20 6d61 7472 6978 4661 6374 6f72  f = matrixFactor
│ │ │ │ +0003e550: 697a 6174 696f 6e28 6666 2c20 4d29 2020  ization(ff, M)  
│ │ │ │ +0003e560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e570: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0003e580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e5c0: 2020 2020 2020 2020 7c0a 7c6f 3920 3d20          |.|o9 = 
│ │ │ │ -0003e5d0: 7b7b 347d 207c 2061 202d 6220 3020 3020  {{4} | a -b 0 0 
│ │ │ │ -0003e5e0: 207c 2c20 7b35 7d20 7c20 6133 2062 2030   |, {5} | a3 b 0
│ │ │ │ -0003e5f0: 2020 2030 2020 3020 207c 2c20 7b32 7d20     0  0  |, {2} 
│ │ │ │ -0003e600: 7c20 3020 2d31 2030 207c 7d7c 0a7c 2020  | 0 -1 0 |}|.|  
│ │ │ │ -0003e610: 2020 2020 7b32 7d20 7c20 3020 6133 2030      {2} | 0 a3 0
│ │ │ │ -0003e620: 2062 3320 7c20 207b 357d 207c 2030 2020   b3 |  {5} | 0  
│ │ │ │ -0003e630: 6120 2d62 3320 3020 2030 2020 7c20 207b  a -b3 0  0  |  {
│ │ │ │ -0003e640: 347d 207c 2030 2030 2020 3120 7c20 7c0a  4} | 0 0  1 | |.
│ │ │ │ -0003e650: 7c20 2020 2020 207b 347d 207c 2030 2030  |      {4} | 0 0
│ │ │ │ -0003e660: 2020 6220 6120 207c 2020 7b35 7d20 7c20    b a  |  {5} | 
│ │ │ │ -0003e670: 3020 2030 2030 2020 202d 6120 6233 207c  0  0 0   -a b3 |
│ │ │ │ -0003e680: 2020 7b34 7d20 7c20 3120 3020 2030 207c    {4} | 1 0  0 |
│ │ │ │ -0003e690: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0003e6a0: 2020 2020 2020 2020 2020 2020 207b 357d               {5}
│ │ │ │ -0003e6b0: 207c 2030 2020 3020 6133 2020 6220 2030   | 0  0 a3  b  0
│ │ │ │ -0003e6c0: 2020 7c20 2020 2020 2020 2020 2020 2020    |             
│ │ │ │ -0003e6d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0003e5b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0003e5c0: 6f39 203d 207b 7b34 7d20 7c20 6120 2d62  o9 = {{4} | a -b
│ │ │ │ +0003e5d0: 2030 2030 2020 7c2c 207b 357d 207c 2061   0 0  |, {5} | a
│ │ │ │ +0003e5e0: 3320 6220 3020 2020 3020 2030 2020 7c2c  3 b 0   0  0  |,
│ │ │ │ +0003e5f0: 207b 327d 207c 2030 202d 3120 3020 7c7d   {2} | 0 -1 0 |}
│ │ │ │ +0003e600: 7c0a 7c20 2020 2020 207b 327d 207c 2030  |.|      {2} | 0
│ │ │ │ +0003e610: 2061 3320 3020 6233 207c 2020 7b35 7d20   a3 0 b3 |  {5} 
│ │ │ │ +0003e620: 7c20 3020 2061 202d 6233 2030 2020 3020  | 0  a -b3 0  0 
│ │ │ │ +0003e630: 207c 2020 7b34 7d20 7c20 3020 3020 2031   |  {4} | 0 0  1
│ │ │ │ +0003e640: 207c 207c 0a7c 2020 2020 2020 7b34 7d20   | |.|      {4} 
│ │ │ │ +0003e650: 7c20 3020 3020 2062 2061 2020 7c20 207b  | 0 0  b a  |  {
│ │ │ │ +0003e660: 357d 207c 2030 2020 3020 3020 2020 2d61  5} | 0  0 0   -a
│ │ │ │ +0003e670: 2062 3320 7c20 207b 347d 207c 2031 2030   b3 |  {4} | 1 0
│ │ │ │ +0003e680: 2020 3020 7c20 7c0a 7c20 2020 2020 2020    0 | |.|       
│ │ │ │ +0003e690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e6a0: 2020 7b35 7d20 7c20 3020 2030 2061 3320    {5} | 0  0 a3 
│ │ │ │ +0003e6b0: 2062 2020 3020 207c 2020 2020 2020 2020   b  0  |        
│ │ │ │ +0003e6c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003e6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e710: 2020 2020 2020 207c 0a7c 6f39 203a 204c         |.|o9 : L
│ │ │ │ -0003e720: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ +0003e700: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0003e710: 3920 3a20 4c69 7374 2020 2020 2020 2020  9 : List        
│ │ │ │ +0003e720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e750: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0003e740: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003e750: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  0003e760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -0003e7a0: 6931 3020 3a20 4720 3d20 6d61 6b65 4669  i10 : G = makeFi
│ │ │ │ -0003e7b0: 6e69 7465 5265 736f 6c75 7469 6f6e 436f  niteResolutionCo
│ │ │ │ -0003e7c0: 6469 6d32 2866 662c 6d66 2920 2020 2020  dim2(ff,mf)     
│ │ │ │ -0003e7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e7e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0003e790: 2d2d 2b0a 7c69 3130 203a 2047 203d 206d  --+.|i10 : G = m
│ │ │ │ +0003e7a0: 616b 6546 696e 6974 6552 6573 6f6c 7574  akeFiniteResolut
│ │ │ │ +0003e7b0: 696f 6e43 6f64 696d 3228 6666 2c6d 6629  ionCodim2(ff,mf)
│ │ │ │ +0003e7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e7d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0003e7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e820: 2020 207c 0a7c 6f31 3020 3d20 4861 7368     |.|o10 = Hash
│ │ │ │ -0003e830: 5461 626c 657b 2261 6c70 6861 2220 3d3e  Table{"alpha" =>
│ │ │ │ -0003e840: 207b 357d 207c 2030 2020 2030 207c 2020   {5} | 0   0 |  
│ │ │ │ -0003e850: 2020 2020 2020 2020 2020 2020 7d20 2020              }   
│ │ │ │ -0003e860: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0003e870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e880: 2020 2020 7b35 7d20 7c20 2d62 3320 3020      {5} | -b3 0 
│ │ │ │ -0003e890: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0003e8a0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0003e8b0: 2020 2020 2020 2020 2020 2020 2262 2220              "b" 
│ │ │ │ -0003e8c0: 3d3e 207b 347d 207c 2062 2061 207c 2020  => {4} | b a |  
│ │ │ │ +0003e810: 2020 2020 2020 2020 7c0a 7c6f 3130 203d          |.|o10 =
│ │ │ │ +0003e820: 2048 6173 6854 6162 6c65 7b22 616c 7068   HashTable{"alph
│ │ │ │ +0003e830: 6122 203d 3e20 7b35 7d20 7c20 3020 2020  a" => {5} | 0   
│ │ │ │ +0003e840: 3020 7c20 2020 2020 2020 2020 2020 2020  0 |             
│ │ │ │ +0003e850: 207d 2020 2020 2020 2020 207c 0a7c 2020   }         |.|  
│ │ │ │ +0003e860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e870: 2020 2020 2020 2020 207b 357d 207c 202d           {5} | -
│ │ │ │ +0003e880: 6233 2030 207c 2020 2020 2020 2020 2020  b3 0 |          
│ │ │ │ +0003e890: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0003e8a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0003e8b0: 2022 6222 203d 3e20 7b34 7d20 7c20 6220   "b" => {4} | b 
│ │ │ │ +0003e8c0: 6120 7c20 2020 2020 2020 2020 2020 2020  a |             
│ │ │ │  0003e8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e8e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003e8f0: 2020 2020 2020 2020 2020 2020 2020 2022                 "
│ │ │ │ -0003e900: 6831 2722 203d 3e20 7b35 7d20 7c20 3020  h1'" => {5} | 0 
│ │ │ │ -0003e910: 2020 3020 2030 2020 7c20 2020 2020 2020    0  0  |       
│ │ │ │ -0003e920: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0003e930: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0003e940: 2020 2020 2020 2020 2020 207b 357d 207c             {5} |
│ │ │ │ -0003e950: 202d 6233 2030 2020 3020 207c 2020 2020   -b3 0  0  |    
│ │ │ │ -0003e960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e970: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0003e980: 2020 2020 2020 2020 2020 2020 2020 7b35                {5
│ │ │ │ -0003e990: 7d20 7c20 3020 2020 2d61 2062 3320 7c20  } | 0   -a b3 | 
│ │ │ │ -0003e9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e9b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0003e9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e9d0: 207b 357d 207c 2061 3320 2062 2020 3020   {5} | a3  b  0 
│ │ │ │ -0003e9e0: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ -0003e9f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0003ea00: 2020 2020 2020 2020 2020 2022 6831 2220             "h1" 
│ │ │ │ -0003ea10: 3d3e 207b 357d 207c 2030 2020 2030 2020  => {5} | 0   0  
│ │ │ │ -0003ea20: 3020 207c 2020 2020 2020 2020 2020 2020  0  |            
│ │ │ │ -0003ea30: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0003ea40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ea50: 2020 2020 2020 7b35 7d20 7c20 2d62 3320        {5} | -b3 
│ │ │ │ -0003ea60: 3020 2030 2020 7c20 2020 2020 2020 2020  0  0  |         
│ │ │ │ -0003ea70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0003ea80: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0003ea90: 2020 2020 2020 2020 207b 357d 207c 2030           {5} | 0
│ │ │ │ -0003eaa0: 2020 202d 6120 6233 207c 2020 2020 2020     -a b3 |      
│ │ │ │ -0003eab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003eac0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0003ead0: 2020 2020 2020 2020 2020 2020 7b35 7d20              {5} 
│ │ │ │ -0003eae0: 7c20 6133 2020 6220 2030 2020 7c20 2020  | a3  b  0  |   
│ │ │ │ -0003eaf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003eb00: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0003eb10: 2020 2020 2020 2022 6d75 2220 3d3e 207b         "mu" => {
│ │ │ │ -0003eb20: 357d 207c 2061 3320 6220 7c20 2020 2020  5} | a3 b |     
│ │ │ │ -0003eb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003eb40: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0003eb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003eb60: 2020 7b35 7d20 7c20 3020 2061 207c 2020    {5} | 0  a |  
│ │ │ │ -0003eb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003eb80: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0003eb90: 2020 2020 2020 2020 2020 2020 2022 7061               "pa
│ │ │ │ -0003eba0: 7274 6961 6c22 203d 3e20 7b34 7d20 7c20  rtial" => {4} | 
│ │ │ │ -0003ebb0: 6120 2d62 207c 2020 2020 2020 2020 2020  a -b |          
│ │ │ │ -0003ebc0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0003e8e0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0003e8f0: 2020 2020 2268 3127 2220 3d3e 207b 357d      "h1'" => {5}
│ │ │ │ +0003e900: 207c 2030 2020 2030 2020 3020 207c 2020   | 0   0  0  |  
│ │ │ │ +0003e910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e920: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0003e930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e940: 7b35 7d20 7c20 2d62 3320 3020 2030 2020  {5} | -b3 0  0  
│ │ │ │ +0003e950: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0003e960: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0003e970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e980: 2020 207b 357d 207c 2030 2020 202d 6120     {5} | 0   -a 
│ │ │ │ +0003e990: 6233 207c 2020 2020 2020 2020 2020 2020  b3 |            
│ │ │ │ +0003e9a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0003e9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e9c0: 2020 2020 2020 7b35 7d20 7c20 6133 2020        {5} | a3  
│ │ │ │ +0003e9d0: 6220 2030 2020 7c20 2020 2020 2020 2020  b  0  |         
│ │ │ │ +0003e9e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0003e9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ea00: 2268 3122 203d 3e20 7b35 7d20 7c20 3020  "h1" => {5} | 0 
│ │ │ │ +0003ea10: 2020 3020 2030 2020 7c20 2020 2020 2020    0  0  |       
│ │ │ │ +0003ea20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ea30: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0003ea40: 2020 2020 2020 2020 2020 207b 357d 207c             {5} |
│ │ │ │ +0003ea50: 202d 6233 2030 2020 3020 207c 2020 2020   -b3 0  0  |    
│ │ │ │ +0003ea60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ea70: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003ea80: 2020 2020 2020 2020 2020 2020 2020 7b35                {5
│ │ │ │ +0003ea90: 7d20 7c20 3020 2020 2d61 2062 3320 7c20  } | 0   -a b3 | 
│ │ │ │ +0003eaa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003eab0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0003eac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ead0: 207b 357d 207c 2061 3320 2062 2020 3020   {5} | a3  b  0 
│ │ │ │ +0003eae0: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ +0003eaf0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003eb00: 2020 2020 2020 2020 2020 2020 226d 7522              "mu"
│ │ │ │ +0003eb10: 203d 3e20 7b35 7d20 7c20 6133 2062 207c   => {5} | a3 b |
│ │ │ │ +0003eb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003eb30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003eb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003eb50: 2020 2020 2020 207b 357d 207c 2030 2020         {5} | 0  
│ │ │ │ +0003eb60: 6120 7c20 2020 2020 2020 2020 2020 2020  a |             
│ │ │ │ +0003eb70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003eb80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0003eb90: 2020 2270 6172 7469 616c 2220 3d3e 207b    "partial" => {
│ │ │ │ +0003eba0: 347d 207c 2061 202d 6220 7c20 2020 2020  4} | a -b |     
│ │ │ │ +0003ebb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ebc0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0003ebd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ebe0: 2020 2020 2020 2020 2020 2020 207b 327d               {2}
│ │ │ │ -0003ebf0: 207c 2030 2061 3320 7c20 2020 2020 2020   | 0 a3 |       
│ │ │ │ -0003ec00: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003ec40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ec50: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003ec60: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -0003ec70: 327d 207c 2030 2062 3320 7c20 2020 2020  2} | 0 b3 |     
│ │ │ │ -0003ec80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ec90: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0003eca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ecb0: 2020 2020 2020 2020 2020 3320 2020 2020            3     
│ │ │ │ -0003ecc0: 2035 2020 2020 2020 3220 2020 2020 2020   5      2       
│ │ │ │ -0003ecd0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0003ece0: 2020 2020 2020 2020 2020 2020 2272 6573              "res
│ │ │ │ -0003ecf0: 6f6c 7574 696f 6e22 203d 3e20 5320 203c  olution" => S  <
│ │ │ │ -0003ed00: 2d2d 2053 2020 3c2d 2d20 5320 203c 2d2d  -- S  <-- S  <--
│ │ │ │ -0003ed10: 2030 2020 2020 2020 2020 2020 7c0a 7c20   0          |.| 
│ │ │ │ +0003ebe0: 2020 7b32 7d20 7c20 3020 6133 207c 2020    {2} | 0 a3 |  
│ │ │ │ +0003ebf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ec00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0003ec10: 2020 2020 2020 2020 2270 7369 2220 3d3e          "psi" =>
│ │ │ │ +0003ec20: 207b 347d 207c 2030 2030 2020 7c20 2020   {4} | 0 0  |   
│ │ │ │ +0003ec30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ec40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003ec50: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0003ec70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ec80: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0003ec90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003eca0: 2020 2020 2020 2020 2020 2020 2020 2033                 3
│ │ │ │ +0003ecb0: 2020 2020 2020 3520 2020 2020 2032 2020        5      2  
│ │ │ │ +0003ecc0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0003ecd0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0003ece0: 2022 7265 736f 6c75 7469 6f6e 2220 3d3e   "resolution" =>
│ │ │ │ +0003ecf0: 2053 2020 3c2d 2d20 5320 203c 2d2d 2053   S  <-- S  <-- S
│ │ │ │ +0003ed00: 2020 3c2d 2d20 3020 2020 2020 2020 2020    <-- 0         
│ │ │ │ +0003ed10: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003ed20: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0003ed40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003ed60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0003ed70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ed80: 2020 3020 2020 2020 2031 2020 2020 2020    0      1      
│ │ │ │ -0003ed90: 3220 2020 2020 2033 2020 2020 2020 2020  2      3        
│ │ │ │ -0003eda0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0003edb0: 2020 2020 2022 7369 676d 6122 203d 3e20       "sigma" => 
│ │ │ │ -0003edc0: 7b35 7d20 7c20 6233 207c 2020 2020 2020  {5} | b3 |      
│ │ │ │ -0003edd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003edf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003ee20: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0003ee30: 2020 2020 2020 2020 2020 2022 7461 7522             "tau"
│ │ │ │ -0003ee40: 203d 3e20 3020 2020 2020 2020 2020 2020   => 0           
│ │ │ │ +0003ed50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0003ed60: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0003ed80: 2020 2020 2032 2020 2020 2020 3320 2020       2      3   
│ │ │ │ +0003ed90: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0003eda0: 2020 2020 2020 2020 2020 2273 6967 6d61            "sigma
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│ │ │ │ +0003edc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003edd0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0003ede0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003edf0: 2020 2020 2020 2020 7b35 7d20 7c20 3020          {5} | 0 
│ │ │ │ +0003ee00: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ +0003ee10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0003ee20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ee30: 2274 6175 2220 3d3e 2030 2020 2020 2020  "tau" => 0      
│ │ │ │ +0003ee40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ee50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ee60: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0003ee70: 2020 2020 2020 2020 2020 2020 2020 2275                "u
│ │ │ │ -0003ee80: 2220 3d3e 207b 387d 207c 2031 2030 207c  " => {8} | 1 0 |
│ │ │ │ +0003ee60: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0003ee70: 2020 2022 7522 203d 3e20 7b38 7d20 7c20     "u" => {8} | 
│ │ │ │ +0003ee80: 3120 3020 7c20 2020 2020 2020 2020 2020  1 0 |           
│ │ │ │  0003ee90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003eea0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0003eeb0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0003eec0: 2022 7622 203d 3e20 7b39 7d20 7c20 3020   "v" => {9} | 0 
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│ │ │ │ -0003eee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003eef0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0003ef00: 2020 2020 2020 2020 2020 207b 397d 207c             {9} |
│ │ │ │ -0003ef10: 2031 2030 2020 7c20 2020 2020 2020 2020   1 0  |         
│ │ │ │ -0003ef20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ef30: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0003ef40: 2020 2020 2020 2022 5822 203d 3e20 3020         "X" => 0 
│ │ │ │ +0003eea0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003eeb0: 2020 2020 2020 2276 2220 3d3e 207b 397d        "v" => {9}
│ │ │ │ +0003eec0: 207c 2030 202d 3120 7c20 2020 2020 2020   | 0 -1 |       
│ │ │ │ +0003eed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003eee0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0003eef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ef00: 7b39 7d20 7c20 3120 3020 207c 2020 2020  {9} | 1 0  |    
│ │ │ │ +0003ef10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ef20: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003ef30: 2020 2020 2020 2020 2020 2020 2258 2220              "X" 
│ │ │ │ +0003ef40: 3d3e 2030 2020 2020 2020 2020 2020 2020  => 0            
│ │ │ │  0003ef50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ef60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ef70: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0003ef80: 2020 2020 2020 2020 2020 2259 2220 3d3e            "Y" =>
│ │ │ │ -0003ef90: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ -0003efa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003efb0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0003ef60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003ef70: 2020 2020 2020 2020 2020 2020 2020 2022                 "
│ │ │ │ +0003ef80: 5922 203d 3e20 3020 2020 2020 2020 2020  Y" => 0         
│ │ │ │ +0003ef90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003efa0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003efb0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0003efc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 207c 0a7c               |.|
│ │ │ │ -0003f000: 6f31 3020 3a20 4861 7368 5461 626c 6520  o10 : HashTable 
│ │ │ │ +0003eff0: 2020 7c0a 7c6f 3130 203a 2048 6173 6854    |.|o10 : HashT
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│ │ │ │  0003f010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f040: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0003f030: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0003f040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003f050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003f060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003f070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003f080: 2d2d 2d2b 0a7c 6931 3120 3a20 4620 3d20  ---+.|i11 : F = 
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│ │ │ │ -0003f0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f0c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0003f0b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0003f0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f100: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0003f110: 2020 2033 2020 2020 2020 3520 2020 2020     3      5     
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│ │ │ │ +0003f120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f140: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003f150: 3131 203d 2053 2020 3c2d 2d20 5320 203c  11 = S  <-- S  <
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│ │ │ │  0003f170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f180: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0003f190: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
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│ │ │ │  0003f1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0003f220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f230: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003f250: 2020 2020 2020 2020 7c0a 7c6f 3131 203a          |.|o11 :
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│ │ │ │  0003f270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f290: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
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│ │ │ │ -0003f3b0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 226d 616b  ======..  * "mak
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│ │ │ │ -0003f3f0: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d  rogrammer.======
│ │ │ │ -0003f400: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  ============..Th
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│ │ │ │ -000405a0: 2020 7c20 2020 2020 2020 7c0a 7c20 2020    |       |.|   
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│ │ │ │ +00040590: 2030 2020 6420 207c 2020 2020 2020 207c   0  d  |       |
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│ │ │ │ -00040b80: 2020 2020 2020 2020 2020 2020 7b7b 302c              {{0,
│ │ │ │ -00040b90: 2031 2c20 307d 2c20 317d 203d 3e20 7b32   1, 0}, 1} => {2
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│ │ │ │ +00040b80: 207b 7b30 2c20 312c 2030 7d2c 2031 7d20   {{0, 1, 0}, 1} 
│ │ │ │ +00040b90: 3d3e 207b 327d 207c 202d 3120 3020 3020  => {2} | -1 0 0 
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│ │ │ │ -00040d00: 7b7b 302c 2031 2c20 307d 2c20 327d 203d  {{0, 1, 0}, 2} =
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│ │ │ │ -00040df0: 2020 2020 2020 2020 7b7b 302c 2031 2c20          {{0, 1, 
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│ │ │ │ -00040e40: 2c20 2d31 7d20 3d3e 2030 2020 2020 2020  , -1} => 0      
│ │ │ │ -00040e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00040e70: 2020 2020 7b7b 312c 2030 2c20 307d 2c20      {{1, 0, 0}, 
│ │ │ │ -00040e80: 2d31 7d20 3d3e 2030 2020 2020 2020 2020  -1} => 0        
│ │ │ │ -00040e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000412e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000412f0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00041300: 3520 3a20 4861 7368 5461 626c 6520 2020  5 : HashTable   
│ │ │ │ +000412f0: 207c 0a7c 6f35 203a 2048 6173 6854 6162   |.|o5 : HashTab
│ │ │ │ +00041300: 6c65 2020 2020 2020 2020 2020 2020 2020  le              
│ │ │ │  00041310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041330: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00041320: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00041330: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00041340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041370: 2d2d 2d2d 2d2d 2d2d 2b0a 0a49 6e20 7468  --------+..In th
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│ │ │ │ -00041390: 6572 2068 6f6d 6f74 6f70 6965 7320 6172  er homotopies ar
│ │ │ │ -000413a0: 6520 303a 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d  e 0:..+---------
│ │ │ │ +00041360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ +00041370: 496e 2074 6869 7320 6361 7365 2074 6865  In this case the
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│ │ │ │ +000413a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000413b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000413c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000413d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000413e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 204c  -------+.|i6 : L
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│ │ │ │ -00041410: 6f6d 6f74 236b 213d 3020 616e 6420 7375  omot#k!=0 and su
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│ │ │ │ +000413d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +000413e0: 3620 3a20 4c20 3d20 736f 7274 2073 656c  6 : L = sort sel
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│ │ │ │ +00041400: 6b2d 3e28 686f 6d6f 7423 6b21 3d30 2061  k->(homot#k!=0 a
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│ │ │ │ +00041420: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00041430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041460: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00041470: 6f36 203d 207b 7d20 2020 2020 2020 2020  o6 = {}         
│ │ │ │ +00041460: 2020 7c0a 7c6f 3620 3d20 7b7d 2020 2020    |.|o6 = {}    
│ │ │ │ +00041470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000414a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000414b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000414a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000414b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000414c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000414d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000414e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000414f0: 2020 207c 0a7c 6f36 203a 204c 6973 7420     |.|o6 : List 
│ │ │ │ +000414e0: 2020 2020 2020 2020 7c0a 7c6f 3620 3a20          |.|o6 : 
│ │ │ │ +000414f0: 4c69 7374 2020 2020 2020 2020 2020 2020  List            
│ │ │ │  00041500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041530: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00041520: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00041530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041570: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4f6e 2074  ---------+..On t
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│ │ │ │ -000415b0: 6e20 616e 6420 736f 6d65 7468 696e 6720  n and something 
│ │ │ │ -000415c0: 636f 6e74 6169 6e65 640a 696e 2069 7420  contained.in it 
│ │ │ │ -000415d0: 696e 2061 206d 6f72 6520 636f 6d70 6c69  in a more compli
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│ │ │ │ -000415f0: 2074 6865 2070 726f 6772 616d 2067 6976   the program giv
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│ │ │ │ -00041610: 720a 686f 6d6f 746f 7069 6573 3a0a 0a2b  r.homotopies:..+
│ │ │ │ +00041560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00041570: 0a4f 6e20 7468 6520 6f74 6865 7220 6861  .On the other ha
│ │ │ │ +00041580: 6e64 2c20 6966 2077 6520 7461 6b65 2061  nd, if we take a
│ │ │ │ +00041590: 2063 6f6d 706c 6574 6520 696e 7465 7273   complete inters
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│ │ │ │ +000415e0: 7469 6f6e 2c20 7468 6520 7072 6f67 7261  tion, the progra
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│ │ │ │ +00041610: 733a 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  s:..+-----------
│ │ │ │  00041620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00041670: 6937 203a 206b 6b3d 205a 5a2f 3332 3030  i7 : kk= ZZ/3200
│ │ │ │ -00041680: 333b 2020 2020 2020 2020 2020 2020 2020  3;              
│ │ │ │ +00041660: 2d2d 2b0a 7c69 3720 3a20 6b6b 3d20 5a5a  --+.|i7 : kk= ZZ
│ │ │ │ +00041670: 2f33 3230 3033 3b20 2020 2020 2020 2020  /32003;         
│ │ │ │ +00041680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000416a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000416b0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +000416b0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  000416c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000416d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000416e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000416f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00041710: 6938 203a 2053 203d 206b 6b5b 612c 622c  i8 : S = kk[a,b,
│ │ │ │ -00041720: 632c 645d 3b20 2020 2020 2020 2020 2020  c,d];           
│ │ │ │ +00041700: 2d2d 2b0a 7c69 3820 3a20 5320 3d20 6b6b  --+.|i8 : S = kk
│ │ │ │ +00041710: 5b61 2c62 2c63 2c64 5d3b 2020 2020 2020  [a,b,c,d];      
│ │ │ │ +00041720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041750: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00041750: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00041760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000417a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -000417b0: 6939 203a 204d 203d 2053 5e31 2f28 6964  i9 : M = S^1/(id
│ │ │ │ -000417c0: 6561 6c22 6132 2c62 322c 6332 2c64 3222  eal"a2,b2,c2,d2"
│ │ │ │ -000417d0: 293b 2020 2020 2020 2020 2020 2020 2020  );              
│ │ │ │ +000417a0: 2d2d 2b0a 7c69 3920 3a20 4d20 3d20 535e  --+.|i9 : M = S^
│ │ │ │ +000417b0: 312f 2869 6465 616c 2261 322c 6232 2c63  1/(ideal"a2,b2,c
│ │ │ │ +000417c0: 322c 6432 2229 3b20 2020 2020 2020 2020  2,d2");         
│ │ │ │ +000417d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000417e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000417f0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +000417f0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00041800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00041850: 6931 3020 3a20 4620 3d20 6672 6565 5265  i10 : F = freeRe
│ │ │ │ -00041860: 736f 6c75 7469 6f6e 204d 2020 2020 2020  solution M      
│ │ │ │ +00041840: 2d2d 2b0a 7c69 3130 203a 2046 203d 2066  --+.|i10 : F = f
│ │ │ │ +00041850: 7265 6552 6573 6f6c 7574 696f 6e20 4d20  reeResolution M 
│ │ │ │ +00041860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041890: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00041890: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000418a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000418b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000418c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000418d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000418e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000418f0: 2020 2020 2020 2031 2020 2020 2020 3420         1      4 
│ │ │ │ -00041900: 2020 2020 2036 2020 2020 2020 3420 2020       6      4   
│ │ │ │ -00041910: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │ +000418e0: 2020 7c0a 7c20 2020 2020 2020 3120 2020    |.|       1   
│ │ │ │ +000418f0: 2020 2034 2020 2020 2020 3620 2020 2020     4      6     
│ │ │ │ +00041900: 2034 2020 2020 2020 3120 2020 2020 2020   4      1       
│ │ │ │ +00041910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041930: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00041940: 6f31 3020 3d20 5320 203c 2d2d 2053 2020  o10 = S  <-- S  
│ │ │ │ -00041950: 3c2d 2d20 5320 203c 2d2d 2053 2020 3c2d  <-- S  <-- S  <-
│ │ │ │ -00041960: 2d20 5320 2020 2020 2020 2020 2020 2020  - S             
│ │ │ │ +00041930: 2020 7c0a 7c6f 3130 203d 2053 2020 3c2d    |.|o10 = S  <-
│ │ │ │ +00041940: 2d20 5320 203c 2d2d 2053 2020 3c2d 2d20  - S  <-- S  <-- 
│ │ │ │ +00041950: 5320 203c 2d2d 2053 2020 2020 2020 2020  S  <-- S        
│ │ │ │ +00041960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041980: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00041980: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00041990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000419a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000419b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000419c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000419d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000419e0: 2020 2020 2020 3020 2020 2020 2031 2020        0      1  
│ │ │ │ -000419f0: 2020 2020 3220 2020 2020 2033 2020 2020      2      3    
│ │ │ │ -00041a00: 2020 3420 2020 2020 2020 2020 2020 2020    4             
│ │ │ │ +000419d0: 2020 7c0a 7c20 2020 2020 2030 2020 2020    |.|      0    
│ │ │ │ +000419e0: 2020 3120 2020 2020 2032 2020 2020 2020    1      2      
│ │ │ │ +000419f0: 3320 2020 2020 2034 2020 2020 2020 2020  3      4        
│ │ │ │ +00041a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041a20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00041a20: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00041a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041a70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00041a80: 6f31 3020 3a20 436f 6d70 6c65 7820 2020  o10 : Complex   
│ │ │ │ +00041a70: 2020 7c0a 7c6f 3130 203a 2043 6f6d 706c    |.|o10 : Compl
│ │ │ │ +00041a80: 6578 2020 2020 2020 2020 2020 2020 2020  ex              
│ │ │ │  00041a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041ac0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00041ac0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00041ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00041b20: 6931 3120 3a20 7365 7452 616e 646f 6d53  i11 : setRandomS
│ │ │ │ -00041b30: 6565 6420 3020 2020 2020 2020 2020 2020  eed 0           
│ │ │ │ +00041b10: 2d2d 2b0a 7c69 3131 203a 2073 6574 5261  --+.|i11 : setRa
│ │ │ │ +00041b20: 6e64 6f6d 5365 6564 2030 2020 2020 2020  ndomSeed 0      
│ │ │ │ +00041b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041b60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00041b70: 202d 2d20 7365 7474 696e 6720 7261 6e64   -- setting rand
│ │ │ │ -00041b80: 6f6d 2073 6565 6420 746f 2030 2020 2020  om seed to 0    
│ │ │ │ +00041b60: 2020 7c0a 7c20 2d2d 2073 6574 7469 6e67    |.| -- setting
│ │ │ │ +00041b70: 2072 616e 646f 6d20 7365 6564 2074 6f20   random seed to 
│ │ │ │ +00041b80: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │  00041b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041bb0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00041bb0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00041bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041c00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00041c10: 6f31 3120 3d20 3020 2020 2020 2020 2020  o11 = 0         
│ │ │ │ +00041c00: 2020 7c0a 7c6f 3131 203d 2030 2020 2020    |.|o11 = 0    
│ │ │ │ +00041c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041c50: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00041c50: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00041c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00041cb0: 6931 3220 3a20 6620 3d20 7261 6e64 6f6d  i12 : f = random
│ │ │ │ -00041cc0: 2853 5e31 2c53 5e7b 323a 2d35 7d29 3b20  (S^1,S^{2:-5}); 
│ │ │ │ +00041ca0: 2d2d 2b0a 7c69 3132 203a 2066 203d 2072  --+.|i12 : f = r
│ │ │ │ +00041cb0: 616e 646f 6d28 535e 312c 535e 7b32 3a2d  andom(S^1,S^{2:-
│ │ │ │ +00041cc0: 357d 293b 2020 2020 2020 2020 2020 2020  5});            
│ │ │ │  00041cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041cf0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00041cf0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00041d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041d40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00041d50: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ -00041d60: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +00041d40: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00041d50: 2020 2031 2020 2020 2020 3220 2020 2020     1      2     
│ │ │ │ +00041d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041d90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00041da0: 6f31 3220 3a20 4d61 7472 6978 2053 2020  o12 : Matrix S  
│ │ │ │ -00041db0: 3c2d 2d20 5320 2020 2020 2020 2020 2020  <-- S           
│ │ │ │ +00041d90: 2020 7c0a 7c6f 3132 203a 204d 6174 7269    |.|o12 : Matri
│ │ │ │ +00041da0: 7820 5320 203c 2d2d 2053 2020 2020 2020  x S  <-- S      
│ │ │ │ +00041db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041de0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00041de0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00041df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00041e40: 6931 3320 3a20 686f 6d6f 7420 3d20 6d61  i13 : homot = ma
│ │ │ │ -00041e50: 6b65 486f 6d6f 746f 7069 6573 2866 2c46  keHomotopies(f,F
│ │ │ │ -00041e60: 2c35 2920 2020 2020 2020 2020 2020 2020  ,5)             
│ │ │ │ +00041e30: 2d2d 2b0a 7c69 3133 203a 2068 6f6d 6f74  --+.|i13 : homot
│ │ │ │ +00041e40: 203d 206d 616b 6548 6f6d 6f74 6f70 6965   = makeHomotopie
│ │ │ │ +00041e50: 7328 662c 462c 3529 2020 2020 2020 2020  s(f,F,5)        
│ │ │ │ +00041e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041e80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00041e80: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00041e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041ed0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00041ee0: 6f31 3320 3d20 4861 7368 5461 626c 657b  o13 = HashTable{
│ │ │ │ -00041ef0: 7b7b 302c 2030 7d2c 2030 7d20 3d3e 2030  {{0, 0}, 0} => 0
│ │ │ │ +00041ed0: 2020 7c0a 7c6f 3133 203d 2048 6173 6854    |.|o13 = HashT
│ │ │ │ +00041ee0: 6162 6c65 7b7b 7b30 2c20 307d 2c20 307d  able{{{0, 0}, 0}
│ │ │ │ +00041ef0: 203d 3e20 3020 2020 2020 2020 2020 2020   => 0           
│ │ │ │  00041f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041f20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00041f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041f40: 7b7b 302c 2030 7d2c 2031 7d20 3d3e 207c  {{0, 0}, 1} => |
│ │ │ │ -00041f50: 2061 3220 2020 2020 2020 2020 2020 2020   a2             
│ │ │ │ +00041f20: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00041f30: 2020 2020 207b 7b30 2c20 307d 2c20 317d       {{0, 0}, 1}
│ │ │ │ +00041f40: 203d 3e20 7c20 6132 2020 2020 2020 2020   => | a2        
│ │ │ │ +00041f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041f70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00041f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041f90: 7b7b 302c 2030 7d2c 2032 7d20 3d3e 207b  {{0, 0}, 2} => {
│ │ │ │ -00041fa0: 327d 207c 2020 2020 2020 2020 2020 2020  2} |            
│ │ │ │ +00041f70: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00041f80: 2020 2020 207b 7b30 2c20 307d 2c20 327d       {{0, 0}, 2}
│ │ │ │ +00041f90: 203d 3e20 7b32 7d20 7c20 2020 2020 2020   => {2} |       
│ │ │ │ +00041fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041fc0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00041fc0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00041fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041fe0: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00041ff0: 327d 207c 2020 2020 2020 2020 2020 2020  2} |            
│ │ │ │ +00041fe0: 2020 2020 7b32 7d20 7c20 2020 2020 2020      {2} |       
│ │ │ │ +00041ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00042000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042010: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00042010: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00042020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042030: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00042040: 327d 207c 2020 2020 2020 2020 2020 2020  2} |            
│ │ │ │ +00042030: 2020 2020 7b32 7d20 7c20 2020 2020 2020      {2} |       
│ │ │ │ +00042040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00042050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042060: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00042060: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00042070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042080: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00042090: 327d 207c 2020 2020 2020 2020 2020 2020  2} |            
│ │ │ │ +00042080: 2020 2020 7b32 7d20 7c20 2020 2020 2020      {2} |       
│ │ │ │ +00042090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000420a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000420b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000420c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000420d0: 7b7b 302c 2030 7d2c 2033 7d20 3d3e 207b  {{0, 0}, 3} => {
│ │ │ │ -000420e0: 347d 207c 2020 2020 2020 2020 2020 2020  4} |            
│ │ │ │ +000420b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000420c0: 2020 2020 207b 7b30 2c20 307d 2c20 337d       {{0, 0}, 3}
│ │ │ │ +000420d0: 203d 3e20 7b34 7d20 7c20 2020 2020 2020   => {4} |       
│ │ │ │ +000420e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000420f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042100: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00042100: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00042110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042120: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00042130: 347d 207c 2020 2020 2020 2020 2020 2020  4} |            
│ │ │ │ +00042120: 2020 2020 7b34 7d20 7c20 2020 2020 2020      {4} |       
│ │ │ │ +00042130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00042140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042150: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00042150: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00042160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042170: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00042180: 347d 207c 2020 2020 2020 2020 2020 2020  4} |            
│ │ │ │ +00042170: 2020 2020 7b34 7d20 7c20 2020 2020 2020      {4} |       
│ │ │ │ +00042180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00042190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000421a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000421a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000421b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000421c0: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -000421d0: 347d 207c 2020 2020 2020 2020 2020 2020  4} |            
│ │ │ │ +000421c0: 2020 2020 7b34 7d20 7c20 2020 2020 2020      {4} |       
│ │ │ │ +000421d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000421e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000421f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │  00043860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043870: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │ -000438c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000438d0: 202d 6232 202d 6332 2030 2020 202d 6432   -b2 -c2 0   -d2
│ │ │ │ -000438e0: 2030 2020 2030 2020 207c 2020 2020 2020   0   0   |      
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│ │ │ │ -000439d0: 2062 3220 2063 3220 207c 2020 2020 2020   b2  c2  |      
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│ │ │ │ -00043a20: 207c 2020 2020 2020 2020 2020 2020 2020   |              
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│ │ │ │ -00043ab0: 2061 3220 2030 2020 2030 2020 2064 3220   a2  0   0   d2 
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│ │ │ │ +00043aa0: 2020 7c0a 7c20 6132 2020 3020 2020 3020    |.| a2  0   0 
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│ │ │ │ -00043b00: 2030 2020 202d 6232 202d 6332 2030 2020   0   -b2 -c2 0  
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│ │ │ │  00043b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043b40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00043b50: 2030 2020 2061 3220 2030 2020 202d 6332   0   a2  0   -c2
│ │ │ │ -00043b60: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ +00043b40: 2020 7c0a 7c20 3020 2020 6132 2020 3020    |.| 0   a2  0 
│ │ │ │ +00043b50: 2020 2d63 3220 7c20 2020 2020 2020 2020    -c2 |         
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│ │ │ │  00043b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00043b90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00043ba0: 2030 2020 2030 2020 2061 3220 2062 3220   0   0   a2  b2 
│ │ │ │ -00043bb0: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ +00043b90: 2020 7c0a 7c20 3020 2020 3020 2020 6132    |.| 0   0   a2
│ │ │ │ +00043ba0: 2020 6232 2020 7c20 2020 2020 2020 2020    b2  |         
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│ │ │ │ -00043c30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00043c40: 2063 3220 207c 2020 2020 2020 2020 2020   c2  |          
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│ │ │ │  00043c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00043c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00043c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043c80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │ -00043cd0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │  00043d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00043d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043d70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00043d70: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00043d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00043d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00043da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00043db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043dc0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00043dd0: 2031 3233 3661 332b 3839 3232 6132 622d   1236a3+8922a2b-
│ │ │ │ -00043de0: 3335 3839 6162 322b 3839 3731 6233 2d35  3589ab2+8971b3-5
│ │ │ │ -00043df0: 3030 3661 3263 2d35 3539 3961 6263 2d31  006a2c-5599abc-1
│ │ │ │ -00043e00: 3431 3635 6232 632b 3838 3830 6132 642b  4165b2c+8880a2d+
│ │ │ │ -00043e10: 3432 3539 6162 642d 3330 3032 627c 0a7c  4259abd-3002b|.|
│ │ │ │ -00043e20: 2031 3033 3730 6162 322d 3730 3932 6233   10370ab2-7092b3
│ │ │ │ -00043e30: 2d39 3730 3261 6263 2d36 3632 3762 3263  -9702abc-6627b2c
│ │ │ │ -00043e40: 2b38 3838 3661 6264 2b34 3730 3062 3264  +8886abd+4700b2d
│ │ │ │ -00043e50: 2d31 3561 6364 2b35 3936 3962 6364 2020  -15acd+5969bcd  
│ │ │ │ -00043e60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00043e70: 2031 3337 3037 6133 2b32 3137 3761 3262   13707a3+2177a2b
│ │ │ │ -00043e80: 2d37 3032 3861 6232 2b39 3739 3762 332d  -7028ab2+9797b3-
│ │ │ │ -00043e90: 3730 3231 6132 632b 3633 3737 6162 632b  7021a2c+6377abc+
│ │ │ │ -00043ea0: 3538 3734 6232 632d 3736 3030 6163 322d  5874b2c-7600ac2-
│ │ │ │ -00043eb0: 3131 3732 3662 6332 2b31 3134 307c 0a7c  11726bc2+1140|.|
│ │ │ │ -00043ec0: 202d 3939 3461 332d 3139 3436 6132 622d   -994a3-1946a2b-
│ │ │ │ -00043ed0: 3637 3233 6162 322d 3534 3833 6233 2d36  6723ab2-5483b3-6
│ │ │ │ -00043ee0: 3435 3361 3263 2d31 3139 3261 6263 2d31  453a2c-1192abc-1
│ │ │ │ -00043ef0: 3532 3530 6232 632d 3331 3634 6163 322b  5250b2c-3164ac2+
│ │ │ │ -00043f00: 3239 3562 6332 2d37 3635 3063 337c 0a7c  295bc2-7650c3|.|
│ │ │ │ -00043f10: 202d 3130 3337 3061 6232 2b37 3039 3262   -10370ab2+7092b
│ │ │ │ -00043f20: 332b 3937 3032 6162 632b 3636 3237 6232  3+9702abc+6627b2
│ │ │ │ -00043f30: 632b 3730 3238 6163 322d 3937 3937 6263  c+7028ac2-9797bc
│ │ │ │ -00043f40: 322d 3538 3734 6333 2d38 3838 3661 6264  2-5874c3-8886abd
│ │ │ │ -00043f50: 2d34 3730 3062 3264 2b31 3561 637c 0a7c  -4700b2d+15ac|.|
│ │ │ │ -00043f60: 202d 3133 3730 3761 332d 3231 3737 6132   -13707a3-2177a2
│ │ │ │ -00043f70: 622b 3730 3231 6132 632d 3633 3737 6162  b+7021a2c-6377ab
│ │ │ │ -00043f80: 632b 3736 3030 6163 322b 3131 3732 3662  c+7600ac2+11726b
│ │ │ │ -00043f90: 6332 2d31 3134 3063 332d 3132 3036 6132  c2-1140c3-1206a2
│ │ │ │ -00043fa0: 642b 3131 3433 3561 6264 2b36 307c 0a7c  d+11435abd+60|.|
│ │ │ │ -00043fb0: 2037 3032 3861 332d 3937 3937 6132 622d   7028a3-9797a2b-
│ │ │ │ -00043fc0: 3538 3734 6132 632d 3930 3734 6132 6420  5874a2c-9074a2d 
│ │ │ │ +00043dc0: 2020 7c0a 7c20 3132 3336 6133 2b38 3932    |.| 1236a3+892
│ │ │ │ +00043dd0: 3261 3262 2d33 3538 3961 6232 2b38 3937  2a2b-3589ab2+897
│ │ │ │ +00043de0: 3162 332d 3530 3036 6132 632d 3535 3939  1b3-5006a2c-5599
│ │ │ │ +00043df0: 6162 632d 3134 3136 3562 3263 2b38 3838  abc-14165b2c+888
│ │ │ │ +00043e00: 3061 3264 2b34 3235 3961 6264 2d33 3030  0a2d+4259abd-300
│ │ │ │ +00043e10: 3262 7c0a 7c20 3130 3337 3061 6232 2d37  2b|.| 10370ab2-7
│ │ │ │ +00043e20: 3039 3262 332d 3937 3032 6162 632d 3636  092b3-9702abc-66
│ │ │ │ +00043e30: 3237 6232 632b 3838 3836 6162 642b 3437  27b2c+8886abd+47
│ │ │ │ +00043e40: 3030 6232 642d 3135 6163 642b 3539 3639  00b2d-15acd+5969
│ │ │ │ +00043e50: 6263 6420 2020 2020 2020 2020 2020 2020  bcd             
│ │ │ │ +00043e60: 2020 7c0a 7c20 3133 3730 3761 332b 3231    |.| 13707a3+21
│ │ │ │ +00043e70: 3737 6132 622d 3730 3238 6162 322b 3937  77a2b-7028ab2+97
│ │ │ │ +00043e80: 3937 6233 2d37 3032 3161 3263 2b36 3337  97b3-7021a2c+637
│ │ │ │ +00043e90: 3761 6263 2b35 3837 3462 3263 2d37 3630  7abc+5874b2c-760
│ │ │ │ +00043ea0: 3061 6332 2d31 3137 3236 6263 322b 3131  0ac2-11726bc2+11
│ │ │ │ +00043eb0: 3430 7c0a 7c20 2d39 3934 6133 2d31 3934  40|.| -994a3-194
│ │ │ │ +00043ec0: 3661 3262 2d36 3732 3361 6232 2d35 3438  6a2b-6723ab2-548
│ │ │ │ +00043ed0: 3362 332d 3634 3533 6132 632d 3131 3932  3b3-6453a2c-1192
│ │ │ │ +00043ee0: 6162 632d 3135 3235 3062 3263 2d33 3136  abc-15250b2c-316
│ │ │ │ +00043ef0: 3461 6332 2b32 3935 6263 322d 3736 3530  4ac2+295bc2-7650
│ │ │ │ +00043f00: 6333 7c0a 7c20 2d31 3033 3730 6162 322b  c3|.| -10370ab2+
│ │ │ │ +00043f10: 3730 3932 6233 2b39 3730 3261 6263 2b36  7092b3+9702abc+6
│ │ │ │ +00043f20: 3632 3762 3263 2b37 3032 3861 6332 2d39  627b2c+7028ac2-9
│ │ │ │ +00043f30: 3739 3762 6332 2d35 3837 3463 332d 3838  797bc2-5874c3-88
│ │ │ │ +00043f40: 3836 6162 642d 3437 3030 6232 642b 3135  86abd-4700b2d+15
│ │ │ │ +00043f50: 6163 7c0a 7c20 2d31 3337 3037 6133 2d32  ac|.| -13707a3-2
│ │ │ │ +00043f60: 3137 3761 3262 2b37 3032 3161 3263 2d36  177a2b+7021a2c-6
│ │ │ │ +00043f70: 3337 3761 6263 2b37 3630 3061 6332 2b31  377abc+7600ac2+1
│ │ │ │ +00043f80: 3137 3236 6263 322d 3131 3430 6333 2d31  1726bc2-1140c3-1
│ │ │ │ +00043f90: 3230 3661 3264 2b31 3134 3335 6162 642b  206a2d+11435abd+
│ │ │ │ +00043fa0: 3630 7c0a 7c20 3730 3238 6133 2d39 3739  60|.| 7028a3-979
│ │ │ │ +00043fb0: 3761 3262 2d35 3837 3461 3263 2d39 3037  7a2b-5874a2c-907
│ │ │ │ +00043fc0: 3461 3264 2020 2020 2020 2020 2020 2020  4a2d            
│ │ │ │  00043fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00043fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043ff0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00044000: 2039 3934 6133 2b31 3934 3661 3262 2b36   994a3+1946a2b+6
│ │ │ │ -00044010: 3435 3361 3263 2b31 3139 3261 6263 2b31  453a2c+1192abc+1
│ │ │ │ -00044020: 3130 3435 6132 642b 3135 3333 3361 6264  1045a2d+15333abd
│ │ │ │ -00044030: 2b33 3536 3061 6364 2d32 3239 3262 6364  +3560acd-2292bcd
│ │ │ │ -00044040: 2b37 3139 3461 6432 2d36 3234 357c 0a7c  +7194ad2-6245|.|
│ │ │ │ -00044050: 2036 3732 3361 332b 3534 3833 6132 622b   6723a3+5483a2b+
│ │ │ │ -00044060: 3135 3235 3061 3263 2d31 3035 3637 6132  15250a2c-10567a2
│ │ │ │ -00044070: 6420 2020 2020 2020 2020 2020 2020 2020  d               
│ │ │ │ +00043ff0: 2020 7c0a 7c20 3939 3461 332b 3139 3436    |.| 994a3+1946
│ │ │ │ +00044000: 6132 622b 3634 3533 6132 632b 3131 3932  a2b+6453a2c+1192
│ │ │ │ +00044010: 6162 632b 3131 3034 3561 3264 2b31 3533  abc+11045a2d+153
│ │ │ │ +00044020: 3333 6162 642b 3335 3630 6163 642d 3232  33abd+3560acd-22
│ │ │ │ +00044030: 3932 6263 642b 3731 3934 6164 322d 3632  92bcd+7194ad2-62
│ │ │ │ +00044040: 3435 7c0a 7c20 3637 3233 6133 2b35 3438  45|.| 6723a3+548
│ │ │ │ +00044050: 3361 3262 2b31 3532 3530 6132 632d 3130  3a2b+15250a2c-10
│ │ │ │ +00044060: 3536 3761 3264 2020 2020 2020 2020 2020  567a2d          
│ │ │ │ +00044070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044090: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000440a0: 2033 3136 3461 332d 3239 3561 3262 2b37   3164a3-295a2b+7
│ │ │ │ -000440b0: 3635 3061 3263 2d31 3433 3838 6132 6420  650a2c-14388a2d 
│ │ │ │ +00044090: 2020 7c0a 7c20 3331 3634 6133 2d32 3935    |.| 3164a3-295
│ │ │ │ +000440a0: 6132 622b 3736 3530 6132 632d 3134 3338  a2b+7650a2c-1438
│ │ │ │ +000440b0: 3861 3264 2020 2020 2020 2020 2020 2020  8a2d            
│ │ │ │  000440c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000440d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000440e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000440f0: 2031 3337 3037 6133 2b32 3137 3761 3262   13707a3+2177a2b
│ │ │ │ -00044100: 2d37 3032 3161 3263 2b36 3337 3761 6263  -7021a2c+6377abc
│ │ │ │ -00044110: 2d37 3630 3061 6332 2d31 3137 3236 6263  -7600ac2-11726bc
│ │ │ │ -00044120: 322b 3131 3430 6333 2b31 3230 3661 3264  2+1140c3+1206a2d
│ │ │ │ -00044130: 2d31 3134 3335 6162 642d 3630 347c 0a7c  -11435abd-604|.|
│ │ │ │ -00044140: 202d 3939 3461 332d 3139 3436 6132 622d   -994a3-1946a2b-
│ │ │ │ -00044150: 3634 3533 6132 632d 3131 3932 6162 632d  6453a2c-1192abc-
│ │ │ │ -00044160: 3131 3034 3561 3264 2d31 3533 3333 6162  11045a2d-15333ab
│ │ │ │ -00044170: 642d 3335 3630 6163 642b 3232 3932 6263  d-3560acd+2292bc
│ │ │ │ -00044180: 642d 3731 3934 6164 322b 3632 347c 0a7c  d-7194ad2+624|.|
│ │ │ │ -00044190: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ +000440e0: 2020 7c0a 7c20 3133 3730 3761 332b 3231    |.| 13707a3+21
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│ │ │ │  00044f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00045090: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000450d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │ -000452b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │  00047ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00047bb0: 3930 3861 6263 642d 3938 3934 6232 6364  908abcd-9894b2cd
│ │ │ │ -00047bc0: 2b38 3434 3061 6332 642b 3130 3639 3362  +8440ac2d+10693b
│ │ │ │ -00047bd0: 6332 642b 3132 3036 3261 3264 322d 3133  c2d+12062a2d2-13
│ │ │ │ -00047be0: 3632 3661 6264 322b 3432 3437 6232 6432  626abd2+4247b2d2
│ │ │ │ -00047bf0: 2d31 3237 3033 6163 6432 2b32 397c 0a7c  -12703acd2+29|.|
│ │ │ │ -00047c00: 3432 3062 3364 2b35 3233 3461 3263 642b  420b3d+5234a2cd+
│ │ │ │ -00047c10: 3730 3036 6162 6364 2d31 3439 3231 6232  7006abcd-14921b2
│ │ │ │ -00047c20: 6364 2b38 3831 3061 6332 642d 3332 3731  cd+8810ac2d-3271
│ │ │ │ -00047c30: 6263 3264 2d31 3036 3633 6333 642d 3132  bc2d-10663c3d-12
│ │ │ │ -00047c40: 3533 3761 3264 322b 3135 3238 317c 0a7c  537a2d2+15281|.|
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│ │ │ │ +00047bb0: 3462 3263 642b 3834 3430 6163 3264 2b31  4b2cd+8440ac2d+1
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│ │ │ │ +00047bf0: 3239 7c0a 7c34 3230 6233 642b 3532 3334  29|.|420b3d+5234
│ │ │ │ +00047c00: 6132 6364 2b37 3030 3661 6263 642d 3134  a2cd+7006abcd-14
│ │ │ │ +00047c10: 3932 3162 3263 642b 3838 3130 6163 3264  921b2cd+8810ac2d
│ │ │ │ +00047c20: 2d33 3237 3162 6332 642d 3130 3636 3363  -3271bc2d-10663c
│ │ │ │ +00047c30: 3364 2d31 3235 3337 6132 6432 2b31 3532  3d-12537a2d2+152
│ │ │ │ +00047c40: 3831 7c0a 7c20 2020 2020 2020 2020 2020  81|.|           
│ │ │ │  00047c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00047ca0: 3561 6263 642b 3632 3937 6232 6364 2b31  5abcd+6297b2cd+1
│ │ │ │ -00047cb0: 3533 3739 6163 3264 2d31 3238 3831 6263  5379ac2d-12881bc
│ │ │ │ -00047cc0: 3264 2d36 3334 3161 3264 322b 3832 3932  2d-6341a2d2+8292
│ │ │ │ -00047cd0: 6162 6432 2b31 3138 3632 6232 6432 2d31  abd2+11862b2d2-1
│ │ │ │ -00047ce0: 3035 3136 6163 6432 2d31 3234 397c 0a7c  0516acd2-1249|.|
│ │ │ │ -00047cf0: 3639 3435 6132 6232 2d31 3230 3938 6162  6945a2b2-12098ab
│ │ │ │ -00047d00: 332d 3732 3437 6234 2b31 3539 3336 6162  3-7247b4+15936ab
│ │ │ │ -00047d10: 3263 2d35 3338 3562 3363 2d35 3337 3562  2c-5385b3c-5375b
│ │ │ │ -00047d20: 3263 322d 3130 3330 3561 6232 642b 3832  2c2-10305ab2d+82
│ │ │ │ -00047d30: 3937 6233 642b 3130 3636 3362 327c 0a7c  97b3d+10663b2|.|
│ │ │ │ +00047c90: 2020 7c0a 7c35 6162 6364 2b36 3239 3762    |.|5abcd+6297b
│ │ │ │ +00047ca0: 3263 642b 3135 3337 3961 6332 642d 3132  2cd+15379ac2d-12
│ │ │ │ +00047cb0: 3838 3162 6332 642d 3633 3431 6132 6432  881bc2d-6341a2d2
│ │ │ │ +00047cc0: 2b38 3239 3261 6264 322b 3131 3836 3262  +8292abd2+11862b
│ │ │ │ +00047cd0: 3264 322d 3130 3531 3661 6364 322d 3132  2d2-10516acd2-12
│ │ │ │ +00047ce0: 3439 7c0a 7c36 3934 3561 3262 322d 3132  49|.|6945a2b2-12
│ │ │ │ +00047cf0: 3039 3861 6233 2d37 3234 3762 342b 3135  098ab3-7247b4+15
│ │ │ │ +00047d00: 3933 3661 6232 632d 3533 3835 6233 632d  936ab2c-5385b3c-
│ │ │ │ +00047d10: 3533 3735 6232 6332 2d31 3033 3035 6162  5375b2c2-10305ab
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│ │ │ │ +00047d30: 6232 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d  b2|.|-----------
│ │ │ │  00047d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00047d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00047d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00047d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00047d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c  -------------|.|
│ │ │ │ +00047d80: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020  --|.|           
│ │ │ │  00047d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000484a0: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ +000484b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000484c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000484d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000484f0: 2020 2020 2020 2020 2020 2031 3233 3661             1236a
│ │ │ │ -00048500: 332b 3839 3232 6132 622d 3335 387c 0a7c  3+8922a2b-358|.|
│ │ │ │ -00048510: 6232 642b 3630 3430 6163 642d 3830 3232  b2d+6040acd-8022
│ │ │ │ -00048520: 6263 642d 3339 3638 6332 642b 3331 3634  bcd-3968c2d+3164
│ │ │ │ -00048530: 6164 322d 3239 3562 6432 2b37 3635 3063  ad2-295bd2+7650c
│ │ │ │ -00048540: 6432 2d31 3433 3838 6433 2031 3033 3730  d2-14388d3 10370
│ │ │ │ -00048550: 6162 322d 3730 3932 6233 2d39 377c 0a7c  ab2-7092b3-97|.|
│ │ │ │ +000484f0: 3132 3336 6133 2b38 3932 3261 3262 2d33  1236a3+8922a2b-3
│ │ │ │ +00048500: 3538 7c0a 7c62 3264 2b36 3034 3061 6364  58|.|b2d+6040acd
│ │ │ │ +00048510: 2d38 3032 3262 6364 2d33 3936 3863 3264  -8022bcd-3968c2d
│ │ │ │ +00048520: 2b33 3136 3461 6432 2d32 3935 6264 322b  +3164ad2-295bd2+
│ │ │ │ +00048530: 3736 3530 6364 322d 3134 3338 3864 3320  7650cd2-14388d3 
│ │ │ │ +00048540: 3130 3337 3061 6232 2d37 3039 3262 332d  10370ab2-7092b3-
│ │ │ │ +00048550: 3937 7c0a 7c20 2020 2020 2020 2020 2020  97|.|           
│ │ │ │  00048560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048570: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000485a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000485b0: 6164 322d 3632 3435 6264 322b 3836 3339  ad2-6245bd2+8639
│ │ │ │ -000485c0: 6364 322d 3934 3236 6433 2020 2020 2020  cd2-9426d3      
│ │ │ │ +00048590: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ +000485a0: 2020 7c0a 7c61 6432 2d36 3234 3562 6432    |.|ad2-6245bd2
│ │ │ │ +000485b0: 2b38 3633 3963 6432 2d39 3432 3664 3320  +8639cd2-9426d3 
│ │ │ │ +000485c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000485d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000485e0: 2020 2020 2020 2020 2020 2030 2020 2020             0    
│ │ │ │ -000485f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000485e0: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ +000485f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00048600: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00048630: 2020 2020 2020 2020 2020 2039 3934 6133             994a3
│ │ │ │ -00048640: 2b31 3934 3661 3262 2b36 3732 337c 0a7c  +1946a2b+6723|.|
│ │ │ │ -00048650: 3135 3235 3063 6432 2d31 3035 3637 6433  15250cd2-10567d3
│ │ │ │ -00048660: 2031 3233 3661 332b 3839 3232 6132 622d   1236a3+8922a2b-
│ │ │ │ -00048670: 3335 3839 6162 322b 3839 3731 6233 2d35  3589ab2+8971b3-5
│ │ │ │ -00048680: 3030 3661 3263 2d35 3539 3961 6263 2d31  006a2c-5599abc-1
│ │ │ │ -00048690: 3431 3635 6232 632b 3838 3830 617c 0a7c  4165b2c+8880a|.|
│ │ │ │ -000486a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000486b0: 2030 2020 2020 2020 2020 2020 2020 2020   0              
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│ │ │ │ +00048640: 3233 7c0a 7c31 3532 3530 6364 322d 3130  23|.|15250cd2-10
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│ │ │ │ -00048780: 3532 3530 6232 632d 3131 3034 357c 0a7c  5250b2c-11045|.|
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│ │ │ │ -000487b0: 632b 3636 3237 6232 632d 3937 3937 6263  c+6627b2c-9797bc
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│ │ │ │ -000487d0: 2d34 3730 3062 3264 2b31 3561 637c 0a7c  -4700b2d+15ac|.|
│ │ │ │ +00048730: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
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│ │ │ │ +00048870: 2020 7c0a 7c63 6432 2d39 3263 3264 322b    |.|cd2-92c2d2+
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│ │ │ │ -000488f0: 2020 2020 2020 2020 2020 2020 207c 2020               |  
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│ │ │ │ -00048c80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │ -00048d10: 3332 3563 6432 2b31 3137 3430 6433 2038  325cd2+11740d3 8
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│ │ │ │ +00048cd0: 622b 7c0a 7c35 3439 6232 642d 3930 3333  b+|.|549b2d-9033
│ │ │ │ +00048ce0: 6163 642d 3239 3938 6263 642b 3230 3336  acd-2998bcd+2036
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│ │ │ │ +00048d20: 3762 7c0a 7c20 2020 2020 2020 2020 2020  7b|.|           
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│ │ │ │  00048d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048d60: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ -00048d70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │ +00048d80: 6264 322b 3639 3232 6364 322b 3530 3830  bd2+6922cd2+5080
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│ │ │ │ -00048dc0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │ -00048e00: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ -00048e10: 3034 3830 6133 2b36 3230 3361 327c 0a7c  0480a3+6203a2|.|
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│ │ │ │ -00048e30: 2d32 3632 3764 3320 3130 3761 332b 3433  -2627d3 107a3+43
│ │ │ │ -00048e40: 3736 6132 622b 3337 3833 6162 322b 3130  76a2b+3783ab2+10
│ │ │ │ -00048e50: 3335 3962 332d 3535 3730 6132 632d 3533  359b3-5570a2c-53
│ │ │ │ -00048e60: 3037 6162 632d 3734 3634 6232 637c 0a7c  07abc-7464b2c|.|
│ │ │ │ -00048e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048e80: 2020 2020 2020 2020 3020 2020 2020 2020          0       
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│ │ │ │ +00048e10: 6132 7c0a 7c31 3038 3636 6264 322b 3730  a2|.|10866bd2+70
│ │ │ │ +00048e20: 3631 6364 322d 3236 3237 6433 2031 3037  61cd2-2627d3 107
│ │ │ │ +00048e30: 6133 2b34 3337 3661 3262 2b33 3738 3361  a3+4376a2b+3783a
│ │ │ │ +00048e40: 6232 2b31 3033 3539 6233 2d35 3537 3061  b2+10359b3-5570a
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│ │ │ │ +00048e60: 3263 7c0a 7c20 2020 2020 2020 2020 2020  2c|.|           
│ │ │ │ +00048e70: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ +00048e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048eb0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00048ec0: 3530 3830 6433 2020 2020 2020 2020 2020  5080d3          
│ │ │ │ -00048ed0: 2020 2020 2020 2020 3020 2020 2020 2020          0       
│ │ │ │ +00048eb0: 2020 7c0a 7c35 3038 3064 3320 2020 2020    |.|5080d3     
│ │ │ │ +00048ec0: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ +00048ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048f00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00048f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048f20: 2020 2020 2020 2020 2d31 3034 3830 6133          -10480a3
│ │ │ │ -00048f30: 2d36 3230 3361 3262 2b39 3533 3461 6232  -6203a2b+9534ab2
│ │ │ │ -00048f40: 2b31 3038 3636 6233 2d39 3438 3061 3263  +10866b3-9480a2c
│ │ │ │ -00048f50: 2b37 3235 3661 6263 2d37 3036 317c 0a7c  +7256abc-7061|.|
│ │ │ │ -00048f60: 3332 3563 6432 2d31 3137 3430 6433 202d  325cd2-11740d3 -
│ │ │ │ -00048f70: 3832 3331 6162 322d 3133 3137 3762 332d  8231ab2-13177b3-
│ │ │ │ -00048f80: 3538 3634 6162 632d 3133 3939 3062 3263  5864abc-13990b2c
│ │ │ │ -00048f90: 2b31 3330 3962 6332 2d35 3339 3863 332d  +1309bc2-5398c3-
│ │ │ │ -00048fa0: 3530 3236 6162 642b 3131 3532 317c 0a7c  5026abd+11521|.|
│ │ │ │ +00048f00: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00048f10: 2020 2020 2020 2020 2020 2020 202d 3130               -10
│ │ │ │ +00048f20: 3438 3061 332d 3632 3033 6132 622b 3935  480a3-6203a2b+95
│ │ │ │ +00048f30: 3334 6162 322b 3130 3836 3662 332d 3934  34ab2+10866b3-94
│ │ │ │ +00048f40: 3830 6132 632b 3732 3536 6162 632d 3730  80a2c+7256abc-70
│ │ │ │ +00048f50: 3631 7c0a 7c33 3235 6364 322d 3131 3734  61|.|325cd2-1174
│ │ │ │ +00048f60: 3064 3320 2d38 3233 3161 6232 2d31 3331  0d3 -8231ab2-131
│ │ │ │ +00048f70: 3737 6233 2d35 3836 3461 6263 2d31 3339  77b3-5864abc-139
│ │ │ │ +00048f80: 3930 6232 632b 3133 3039 6263 322d 3533  90b2c+1309bc2-53
│ │ │ │ +00048f90: 3938 6333 2d35 3032 3661 6264 2b31 3135  98c3-5026abd+115
│ │ │ │ +00048fa0: 3231 7c0a 7c20 2020 2020 2020 2020 2020  21|.|           
│ │ │ │  00048fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048ff0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00048ff0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00049000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049040: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00049050: 3330 6263 6432 2b38 3532 3763 3264 322b  30bcd2+8527c2d2+
│ │ │ │ -00049060: 3539 3432 6164 332b 3134 3932 3562 6433  5942ad3+14925bd3
│ │ │ │ -00049070: 2d32 3936 6364 332b 3830 3834 6434 207c  -296cd3+8084d4 |
│ │ │ │ +00049040: 2020 7c0a 7c33 3062 6364 322b 3835 3237    |.|30bcd2+8527
│ │ │ │ +00049050: 6332 6432 2b35 3934 3261 6433 2b31 3439  c2d2+5942ad3+149
│ │ │ │ +00049060: 3235 6264 332d 3239 3663 6433 2b38 3038  25bd3-296cd3+808
│ │ │ │ +00049070: 3464 3420 7c20 2020 2020 2020 2020 2020  4d4 |           
│ │ │ │  00049080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049090: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000490a0: 3034 3661 6264 322b 3633 3432 6232 6432  046abd2+6342b2d2
│ │ │ │ -000490b0: 2d37 3438 3061 6364 322d 3738 3433 6263  -7480acd2-7843bc
│ │ │ │ -000490c0: 6432 2d38 3038 3463 3264 3220 2020 207c  d2-8084c2d2    |
│ │ │ │ +00049090: 2020 7c0a 7c30 3436 6162 6432 2b36 3334    |.|046abd2+634
│ │ │ │ +000490a0: 3262 3264 322d 3734 3830 6163 6432 2d37  2b2d2-7480acd2-7
│ │ │ │ +000490b0: 3834 3362 6364 322d 3830 3834 6332 6432  843bcd2-8084c2d2
│ │ │ │ +000490c0: 2020 2020 7c20 2020 2020 2020 2020 2020      |           
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│ │ │ │ -000491a0: 3261 3362 2b31 3339 3936 6132 6232 2d39  2a3b+13996a2b2-9
│ │ │ │ -000491b0: 3333 3861 6233 2d34 3637 3662 342d 3938  338ab3-4676b4-98
│ │ │ │ -000491c0: 3236 6133 632b 3437 3331 6132 6263 2d31  26a3c+4731a2bc-1
│ │ │ │ -000491d0: 3237 3335 6162 3263 2b37 3432 317c 0a7c  2735ab2c+7421|.|
│ │ │ │ +00049180: 2020 7c0a 7c64 2b38 3038 3462 3264 3220    |.|d+8084b2d2 
│ │ │ │ +00049190: 2d31 3034 3932 6133 622b 3133 3939 3661  -10492a3b+13996a
│ │ │ │ +000491a0: 3262 322d 3933 3338 6162 332d 3436 3736  2b2-9338ab3-4676
│ │ │ │ +000491b0: 6234 2d39 3832 3661 3363 2b34 3733 3161  b4-9826a3c+4731a
│ │ │ │ +000491c0: 3262 632d 3132 3733 3561 6232 632b 3734  2bc-12735ab2c+74
│ │ │ │ +000491d0: 3231 7c0a 7c20 2020 2020 2020 2020 2020  21|.|           
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│ │ │ │ -000492d0: 3537 6263 6432 2d35 3337 3563 3264 322d  57bcd2-5375c2d2-
│ │ │ │ -000492e0: 3130 3330 3561 6433 2b38 3239 3762 6433  10305ad3+8297bd3
│ │ │ │ -000492f0: 2b31 3036 3633 6364 332b 3633 3431 6434  +10663cd3+6341d4
│ │ │ │ -00049300: 207c 2020 2020 2020 2020 2020 2020 2020   |              
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│ │ │ │ -00049320: 6162 6432 2d39 3334 3662 3264 322b 3433  abd2-9346b2d2+43
│ │ │ │ -00049330: 3039 6163 6432 2b38 3236 3962 6364 322d  09acd2+8269bcd2-
│ │ │ │ -00049340: 3633 3431 6332 6432 2020 2020 2020 2020  6341c2d2        
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│ │ │ │ +000492c0: 2020 7c0a 7c35 3762 6364 322d 3533 3735    |.|57bcd2-5375
│ │ │ │ +000492d0: 6332 6432 2d31 3033 3035 6164 332b 3832  c2d2-10305ad3+82
│ │ │ │ +000492e0: 3937 6264 332b 3130 3636 3363 6433 2b36  97bd3+10663cd3+6
│ │ │ │ +000492f0: 3334 3164 3420 7c20 2020 2020 2020 2020  341d4 |         
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│ │ │ │ +00049310: 2020 7c0a 7c61 6264 322d 3933 3436 6232    |.|abd2-9346b2
│ │ │ │ +00049320: 6432 2b34 3330 3961 6364 322b 3832 3639  d2+4309acd2+8269
│ │ │ │ +00049330: 6263 6432 2d36 3334 3163 3264 3220 2020  bcd2-6341c2d2   
│ │ │ │ +00049340: 2020 2020 2020 7c20 2020 2020 2020 2020        |         
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│ │ │ │ -000493c0: 3962 6364 3220 2020 2020 2020 2020 2020  9bcd2           
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│ │ │ │ +000493b0: 2020 7c0a 7c39 6263 6432 2020 2020 2020    |.|9bcd2      
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│ │ │ │ -00049420: 3534 6133 622b 3638 3432 6132 6232 2d31  54a3b+6842a2b2-1
│ │ │ │ -00049430: 3432 3236 6162 332d 3633 3433 6234 2d31  4226ab3-6343b4-1
│ │ │ │ -00049440: 3235 3435 6133 632d 3130 3031 3561 3262  2545a3c-10015a2b
│ │ │ │ -00049450: 632d 3134 3238 3661 6232 632d 357c 0a7c  c-14286ab2c-5|.|
│ │ │ │ +000493e0: 2020 2020 2020 7c20 2020 2020 2020 2020        |         
│ │ │ │ +000493f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00049400: 2020 7c0a 7c63 642b 3633 3431 6232 6432    |.|cd+6341b2d2
│ │ │ │ +00049410: 202d 3130 3535 3461 3362 2b36 3834 3261   -10554a3b+6842a
│ │ │ │ +00049420: 3262 322d 3134 3232 3661 6233 2d36 3334  2b2-14226ab3-634
│ │ │ │ +00049430: 3362 342d 3132 3534 3561 3363 2d31 3030  3b4-12545a3c-100
│ │ │ │ +00049440: 3135 6132 6263 2d31 3432 3836 6162 3263  15a2bc-14286ab2c
│ │ │ │ +00049450: 2d35 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d  -5|.|-----------
│ │ │ │  00049460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00049470: 2d2d 2d2d 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 2d7c 0a7c  -------------|.|
│ │ │ │ +000494a0: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020  --|.|           
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│ │ │ │ -00049be0: 3961 6232 2b38 3937 3162 332d 3530 3036  9ab2+8971b3-5006
│ │ │ │ -00049bf0: 6132 632d 3535 3939 6162 632d 3134 3136  a2c-5599abc-1416
│ │ │ │ -00049c00: 3562 3263 2b38 3838 3061 3264 2b34 3235  5b2c+8880a2d+425
│ │ │ │ -00049c10: 3961 6264 2d33 3030 3262 3264 2d31 3338  9abd-3002b2d-138
│ │ │ │ -00049c20: 3932 6163 642d 3130 3532 3162 637c 0a7c  92acd-10521bc|.|
│ │ │ │ -00049c30: 3032 6162 632d 3636 3237 6232 632b 3838  02abc-6627b2c+88
│ │ │ │ -00049c40: 3836 6162 642b 3437 3030 6232 642d 3135  86abd+4700b2d-15
│ │ │ │ -00049c50: 6163 642b 3539 3639 6263 6420 2020 2020  acd+5969bcd     
│ │ │ │ +00049bd0: 2020 7c0a 7c39 6162 322b 3839 3731 6233    |.|9ab2+8971b3
│ │ │ │ +00049be0: 2d35 3030 3661 3263 2d35 3539 3961 6263  -5006a2c-5599abc
│ │ │ │ +00049bf0: 2d31 3431 3635 6232 632b 3838 3830 6132  -14165b2c+8880a2
│ │ │ │ +00049c00: 642b 3432 3539 6162 642d 3330 3032 6232  d+4259abd-3002b2
│ │ │ │ +00049c10: 642d 3133 3839 3261 6364 2d31 3035 3231  d-13892acd-10521
│ │ │ │ +00049c20: 6263 7c0a 7c30 3261 6263 2d36 3632 3762  bc|.|02abc-6627b
│ │ │ │ +00049c30: 3263 2b38 3838 3661 6264 2b34 3730 3062  2c+8886abd+4700b
│ │ │ │ +00049c40: 3264 2d31 3561 6364 2b35 3936 3962 6364  2d-15acd+5969bcd
│ │ │ │ +00049c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00049d20: 6162 322b 3534 3833 6233 2b36 3435 3361  ab2+5483b3+6453a
│ │ │ │ -00049d30: 3263 2b31 3139 3261 6263 2b31 3532 3530  2c+1192abc+15250
│ │ │ │ -00049d40: 6232 632b 3331 3634 6163 322d 3239 3562  b2c+3164ac2-295b
│ │ │ │ -00049d50: 6332 2b37 3635 3063 332b 3131 3034 3561  c2+7650c3+11045a
│ │ │ │ -00049d60: 3264 2b31 3533 3333 6162 642d 317c 0a7c  2d+15333abd-1|.|
│ │ │ │ -00049d70: 3264 2b34 3235 3961 6264 2d33 3030 3262  2d+4259abd-3002b
│ │ │ │ -00049d80: 3264 2d31 3338 3932 6163 642d 3130 3532  2d-13892acd-1052
│ │ │ │ -00049d90: 3162 6364 2020 2020 2020 2020 2020 2020  1bcd            
│ │ │ │ -00049da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049db0: 2020 2020 2037 3032 3861 6432 207c 0a7c       7028ad2 |.|
│ │ │ │ +00049d10: 2020 7c0a 7c61 6232 2b35 3438 3362 332b    |.|ab2+5483b3+
│ │ │ │ +00049d20: 3634 3533 6132 632b 3131 3932 6162 632b  6453a2c+1192abc+
│ │ │ │ +00049d30: 3135 3235 3062 3263 2b33 3136 3461 6332  15250b2c+3164ac2
│ │ │ │ +00049d40: 2d32 3935 6263 322b 3736 3530 6333 2b31  -295bc2+7650c3+1
│ │ │ │ +00049d50: 3130 3435 6132 642b 3135 3333 3361 6264  1045a2d+15333abd
│ │ │ │ +00049d60: 2d31 7c0a 7c32 642b 3432 3539 6162 642d  -1|.|2d+4259abd-
│ │ │ │ +00049d70: 3330 3032 6232 642d 3133 3839 3261 6364  3002b2d-13892acd
│ │ │ │ +00049d80: 2d31 3035 3231 6263 6420 2020 2020 2020  -10521bcd       
│ │ │ │ +00049d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00049da0: 2020 2020 2020 2020 2020 3730 3238 6164            7028ad
│ │ │ │ +00049db0: 3220 7c0a 7c20 2020 2020 2020 2020 2020  2 |.|           
│ │ │ │  00049dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00049ea0: 3236 6433 202d 3937 3937 6132 627c 0a7c  26d3 -9797a2b|.|
│ │ │ │ -00049eb0: 642d 3539 3639 6263 642d 3930 3734 6332  d-5969bcd-9074c2
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│ │ │ │ -00049ed0: 6432 2d31 3035 3637 6433 2031 3233 3661  d2-10567d3 1236a
│ │ │ │ -00049ee0: 332b 3839 3232 6132 622d 3335 3839 6162  3+8922a2b-3589ab
│ │ │ │ -00049ef0: 322b 3839 3731 6233 2d35 3030 367c 0a7c  2+8971b3-5006|.|
│ │ │ │ +00049e40: 2020 2020 2020 2020 2020 2d31 3337 3037            -13707
│ │ │ │ +00049e50: 6133 7c0a 7c61 3264 2d31 3533 3333 6162  a3|.|a2d-15333ab
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│ │ │ │ +00049eb0: 3037 3463 3264 2b35 3438 3362 6432 2b31  074c2d+5483bd2+1
│ │ │ │ +00049ec0: 3532 3530 6364 322d 3130 3536 3764 3320  5250cd2-10567d3 
│ │ │ │ +00049ed0: 3132 3336 6133 2b38 3932 3261 3262 2d33  1236a3+8922a2b-3
│ │ │ │ +00049ee0: 3538 3961 6232 2b38 3937 3162 332d 3530  589ab2+8971b3-50
│ │ │ │ +00049ef0: 3036 7c0a 7c20 2020 2020 2020 2020 2020  06|.|           
│ │ │ │  00049f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0004a080: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0004a090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0004a0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a0d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004a0e0: 6132 6332 2d35 3037 6162 6332 2b31 3338  a2c2-507abc2+138
│ │ │ │ -0004a0f0: 3034 6232 6332 2b38 3431 3661 6333 2d39  04b2c2+8416ac3-9
│ │ │ │ -0004a100: 3263 342d 3335 3838 6133 642b 3134 3534  2c4-3588a3d+1454
│ │ │ │ -0004a110: 3161 3262 642d 3836 3835 6162 3264 2d31  1a2bd-8685ab2d-1
│ │ │ │ -0004a120: 3430 3935 6233 642b 3132 3735 307c 0a7c  4095b3d+12750|.|
│ │ │ │ +0004a0d0: 2020 7c0a 7c61 3263 322d 3530 3761 6263    |.|a2c2-507abc
│ │ │ │ +0004a0e0: 322b 3133 3830 3462 3263 322b 3834 3136  2+13804b2c2+8416
│ │ │ │ +0004a0f0: 6163 332d 3932 6334 2d33 3538 3861 3364  ac3-92c4-3588a3d
│ │ │ │ +0004a100: 2b31 3435 3431 6132 6264 2d38 3638 3561  +14541a2bd-8685a
│ │ │ │ +0004a110: 6232 642d 3134 3039 3562 3364 2b31 3237  b2d-14095b3d+127
│ │ │ │ +0004a120: 3530 7c0a 7c20 2020 2020 2020 2020 2020  50|.|           
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│ │ │ │  0004a140: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0004a170: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
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│ │ │ │  0004a1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0004a1c0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
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│ │ │ │ +0004a210: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0004a220: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0004a2b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004a2b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0004a2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0004a2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a300: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004a300: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0004a310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a350: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004a350: 2020 7c0a 7c20 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 2020 2020 2020 2020                  
│ │ │ │  0004a390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a3a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004a3b0: 3337 3833 6162 322b 3130 3335 3962 332d  3783ab2+10359b3-
│ │ │ │ -0004a3c0: 3535 3730 6132 632d 3533 3037 6162 632d  5570a2c-5307abc-
│ │ │ │ -0004a3d0: 3734 3634 6232 632b 3331 3837 6132 642b  7464b2c+3187a2d+
│ │ │ │ -0004a3e0: 3835 3730 6162 642d 3832 3531 6232 642b  8570abd-8251b2d+
│ │ │ │ -0004a3f0: 3834 3434 6163 642b 3530 3731 627c 0a7c  8444acd+5071b|.|
│ │ │ │ -0004a400: 332b 3538 3634 6162 632b 3133 3939 3062  3+5864abc+13990b
│ │ │ │ -0004a410: 3263 2b35 3032 3661 6264 2d31 3135 3231  2c+5026abd-11521
│ │ │ │ -0004a420: 6232 642d 3735 3031 6163 642d 3137 3739  b2d-7501acd-1779
│ │ │ │ -0004a430: 6263 6420 2020 2020 2020 2020 2020 2020  bcd             
│ │ │ │ -0004a440: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004a3a0: 2020 7c0a 7c33 3738 3361 6232 2b31 3033    |.|3783ab2+103
│ │ │ │ +0004a3b0: 3539 6233 2d35 3537 3061 3263 2d35 3330  59b3-5570a2c-530
│ │ │ │ +0004a3c0: 3761 6263 2d37 3436 3462 3263 2b33 3138  7abc-7464b2c+318
│ │ │ │ +0004a3d0: 3761 3264 2b38 3537 3061 6264 2d38 3235  7a2d+8570abd-825
│ │ │ │ +0004a3e0: 3162 3264 2b38 3434 3461 6364 2b35 3037  1b2d+8444acd+507
│ │ │ │ +0004a3f0: 3162 7c0a 7c33 2b35 3836 3461 6263 2b31  1b|.|3+5864abc+1
│ │ │ │ +0004a400: 3339 3930 6232 632b 3530 3236 6162 642d  3990b2c+5026abd-
│ │ │ │ +0004a410: 3131 3532 3162 3264 2d37 3530 3161 6364  11521b2d-7501acd
│ │ │ │ +0004a420: 2d31 3737 3962 6364 2020 2020 2020 2020  -1779bcd        
│ │ │ │ +0004a430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a440: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0004a450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a490: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004a490: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0004a4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0004a4e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004a4f0: 622d 3935 3334 6162 322d 3130 3836 3662  b-9534ab2-10866b
│ │ │ │ -0004a500: 332b 3934 3830 6132 632d 3732 3536 6162  3+9480a2c-7256ab
│ │ │ │ -0004a510: 632b 3730 3631 6232 632d 3531 3037 6163  c+7061b2c-5107ac
│ │ │ │ -0004a520: 322d 3536 3739 6263 322b 3633 3235 6333  2-5679bc2+6325c3
│ │ │ │ -0004a530: 2b31 3139 3530 6132 642b 3533 327c 0a7c  +11950a2d+532|.|
│ │ │ │ -0004a540: 2b33 3138 3761 3264 2b38 3537 3061 6264  +3187a2d+8570abd
│ │ │ │ -0004a550: 2d38 3235 3162 3264 2b38 3434 3461 6364  -8251b2d+8444acd
│ │ │ │ -0004a560: 2b35 3037 3162 6364 2020 2020 2020 2020  +5071bcd        
│ │ │ │ +0004a4e0: 2020 7c0a 7c62 2d39 3533 3461 6232 2d31    |.|b-9534ab2-1
│ │ │ │ +0004a4f0: 3038 3636 6233 2b39 3438 3061 3263 2d37  0866b3+9480a2c-7
│ │ │ │ +0004a500: 3235 3661 6263 2b37 3036 3162 3263 2d35  256abc+7061b2c-5
│ │ │ │ +0004a510: 3130 3761 6332 2d35 3637 3962 6332 2b36  107ac2-5679bc2+6
│ │ │ │ +0004a520: 3332 3563 332b 3131 3935 3061 3264 2b35  325c3+11950a2d+5
│ │ │ │ +0004a530: 3332 7c0a 7c2b 3331 3837 6132 642b 3835  32|.|+3187a2d+85
│ │ │ │ +0004a540: 3730 6162 642d 3832 3531 6232 642b 3834  70abd-8251b2d+84
│ │ │ │ +0004a550: 3434 6163 642b 3530 3731 6263 6420 2020  44acd+5071bcd   
│ │ │ │ +0004a560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a580: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004a580: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0004a590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a5d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004a5d0: 2020 7c0a 7c20 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 2020 2020 2020 2020                  
│ │ │ │  0004a610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a620: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004a630: 6232 632d 3131 3935 3061 3264 2d35 3332  b2c-11950a2d-532
│ │ │ │ -0004a640: 3161 6264 2b32 3632 3762 3264 2d33 3939  1abd+2627b2d-399
│ │ │ │ -0004a650: 3661 6364 2d37 3135 3262 6364 2b39 3339  6acd-7152bcd+939
│ │ │ │ -0004a660: 3861 6432 2b31 3533 3137 6264 322d 3639  8ad2+15317bd2-69
│ │ │ │ -0004a670: 3232 6364 322d 3530 3830 6433 207c 0a7c  22cd2-5080d3 |.|
│ │ │ │ -0004a680: 6232 642b 3735 3031 6163 642b 3137 3739  b2d+7501acd+1779
│ │ │ │ -0004a690: 6263 642d 3535 3439 6332 642d 3130 3836  bcd-5549c2d-1086
│ │ │ │ -0004a6a0: 3662 6432 2b37 3036 3163 6432 2d32 3632  6bd2+7061cd2-262
│ │ │ │ -0004a6b0: 3764 3320 3130 3761 332b 3433 3736 6132  7d3 107a3+4376a2
│ │ │ │ -0004a6c0: 622b 3337 3833 6162 322b 3130 337c 0a7c  b+3783ab2+103|.|
│ │ │ │ +0004a620: 2020 7c0a 7c62 3263 2d31 3139 3530 6132    |.|b2c-11950a2
│ │ │ │ +0004a630: 642d 3533 3231 6162 642b 3236 3237 6232  d-5321abd+2627b2
│ │ │ │ +0004a640: 642d 3339 3936 6163 642d 3731 3532 6263  d-3996acd-7152bc
│ │ │ │ +0004a650: 642b 3933 3938 6164 322b 3135 3331 3762  d+9398ad2+15317b
│ │ │ │ +0004a660: 6432 2d36 3932 3263 6432 2d35 3038 3064  d2-6922cd2-5080d
│ │ │ │ +0004a670: 3320 7c0a 7c62 3264 2b37 3530 3161 6364  3 |.|b2d+7501acd
│ │ │ │ +0004a680: 2b31 3737 3962 6364 2d35 3534 3963 3264  +1779bcd-5549c2d
│ │ │ │ +0004a690: 2d31 3038 3636 6264 322b 3730 3631 6364  -10866bd2+7061cd
│ │ │ │ +0004a6a0: 322d 3236 3237 6433 2031 3037 6133 2b34  2-2627d3 107a3+4
│ │ │ │ +0004a6b0: 3337 3661 3262 2b33 3738 3361 6232 2b31  376a2b+3783ab2+1
│ │ │ │ +0004a6c0: 3033 7c0a 7c20 2020 2020 2020 2020 2020  03|.|           
│ │ │ │  0004a6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a710: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004a710: 2020 7c0a 7c20 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 2020 2020 2020 2020                  
│ │ │ │  0004a750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a760: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004a760: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0004a770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a7b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004a7b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0004a7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a800: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004a800: 2020 7c0a 7c20 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 2020 2020 2020 2020                  
│ │ │ │  0004a840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a850: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004a850: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0004a860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a8a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004a8b0: 6233 632b 3134 3931 3961 3263 322b 3131  b3c+14919a2c2+11
│ │ │ │ -0004a8c0: 3334 3661 6263 322d 3134 3137 3262 3263  346abc2-14172b2c
│ │ │ │ -0004a8d0: 322d 3139 3130 6163 332d 3131 3635 3662  2-1910ac3-11656b
│ │ │ │ -0004a8e0: 6333 2b38 3532 3763 342b 3637 3730 6133  c3+8527c4+6770a3
│ │ │ │ -0004a8f0: 642d 3931 3037 6132 6264 2d38 387c 0a7c  d-9107a2bd-88|.|
│ │ │ │ +0004a8a0: 2020 7c0a 7c62 3363 2b31 3439 3139 6132    |.|b3c+14919a2
│ │ │ │ +0004a8b0: 6332 2b31 3133 3436 6162 6332 2d31 3431  c2+11346abc2-141
│ │ │ │ +0004a8c0: 3732 6232 6332 2d31 3931 3061 6333 2d31  72b2c2-1910ac3-1
│ │ │ │ +0004a8d0: 3136 3536 6263 332b 3835 3237 6334 2b36  1656bc3+8527c4+6
│ │ │ │ +0004a8e0: 3737 3061 3364 2d39 3130 3761 3262 642d  770a3d-9107a2bd-
│ │ │ │ +0004a8f0: 3838 7c0a 7c20 2020 2020 2020 2020 2020  88|.|           
│ │ │ │  0004a900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a920: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0004b470: 3236 6433 2031 3337 3037 6133 2b32 3137  26d3 13707a3+217
│ │ │ │ +0004b480: 3761 7c0a 7c20 2020 2020 2020 2020 2020  7a|.|           
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│ │ │ │ -0004b540: 2b37 3032 3161 3263 2d36 3337 3761 6263  +7021a2c-6377abc
│ │ │ │ -0004b550: 2b37 3630 3061 6332 2b31 3137 3236 6263  +7600ac2+11726bc
│ │ │ │ -0004b560: 322d 3131 3430 6333 2d31 3230 3661 3264  2-1140c3-1206a2d
│ │ │ │ -0004b570: 2b31 3134 3335 6162 642b 3630 347c 0a7c  +11435abd+604|.|
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│ │ │ │ +0004b4d0: 2020 7c0a 7c32 2b37 3039 3262 332b 3937    |.|2+7092b3+97
│ │ │ │ +0004b4e0: 3032 6162 632b 3636 3237 6232 632d 3937  02abc+6627b2c-97
│ │ │ │ +0004b4f0: 3937 6263 322d 3538 3734 6333 2d38 3838  97bc2-5874c3-888
│ │ │ │ +0004b500: 3661 6264 2d34 3730 3062 3264 2b31 3561  6abd-4700b2d+15a
│ │ │ │ +0004b510: 6364 2d35 3936 3962 6364 2d39 3037 3463  cd-5969bcd-9074c
│ │ │ │ +0004b520: 3264 7c0a 7c2d 3231 3737 6132 622b 3730  2d|.|-2177a2b+70
│ │ │ │ +0004b530: 3238 6162 322b 3730 3231 6132 632d 3633  28ab2+7021a2c-63
│ │ │ │ +0004b540: 3737 6162 632b 3736 3030 6163 322b 3131  77abc+7600ac2+11
│ │ │ │ +0004b550: 3732 3662 6332 2d31 3134 3063 332d 3132  726bc2-1140c3-12
│ │ │ │ +0004b560: 3036 6132 642b 3131 3433 3561 6264 2b36  06a2d+11435abd+6
│ │ │ │ +0004b570: 3034 7c0a 7c2d 3538 3734 6132 632d 3930  04|.|-5874a2c-90
│ │ │ │ +0004b580: 3734 6132 6420 2020 2020 2020 2020 2020  74a2d           
│ │ │ │  0004b590: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0004b5c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004b5d0: 6132 632d 3535 3939 6162 632d 3134 3136  a2c-5599abc-1416
│ │ │ │ -0004b5e0: 3562 3263 2b38 3838 3061 3264 2b34 3235  5b2c+8880a2d+425
│ │ │ │ -0004b5f0: 3961 6264 2d33 3030 3262 3264 2d31 3338  9abd-3002b2d-138
│ │ │ │ -0004b600: 3932 6163 642d 3130 3532 3162 6364 207c  92acd-10521bcd |
│ │ │ │ -0004b610: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │ +0004b5e0: 642b 3432 3539 6162 642d 3330 3032 6232  d+4259abd-3002b2
│ │ │ │ +0004b5f0: 642d 3133 3839 3261 6364 2d31 3035 3231  d-13892acd-10521
│ │ │ │ +0004b600: 6263 6420 7c20 2020 2020 2020 2020 2020  bcd |           
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│ │ │ │ -0004b700: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004b700: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0004b710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b750: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004b750: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0004b760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b7a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004b7a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0004b7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b7f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004b800: 6132 6364 2b34 3035 3361 6263 642b 3230  a2cd+4053abcd+20
│ │ │ │ -0004b810: 3830 6232 6364 2d39 3137 3561 6332 642b  80b2cd-9175ac2d+
│ │ │ │ -0004b820: 3634 3962 6332 642d 3838 3239 6333 642d  649bc2d-8829c3d-
│ │ │ │ -0004b830: 3231 3634 6132 6432 2d38 3633 3561 6264  2164a2d2-8635abd
│ │ │ │ -0004b840: 322b 3731 3631 6232 6432 2d39 397c 0a7c  2+7161b2d2-99|.|
│ │ │ │ +0004b7f0: 2020 7c0a 7c61 3263 642b 3430 3533 6162    |.|a2cd+4053ab
│ │ │ │ +0004b800: 6364 2b32 3038 3062 3263 642d 3931 3735  cd+2080b2cd-9175
│ │ │ │ +0004b810: 6163 3264 2b36 3439 6263 3264 2d38 3832  ac2d+649bc2d-882
│ │ │ │ +0004b820: 3963 3364 2d32 3136 3461 3264 322d 3836  9c3d-2164a2d2-86
│ │ │ │ +0004b830: 3335 6162 6432 2b37 3136 3162 3264 322d  35abd2+7161b2d2-
│ │ │ │ +0004b840: 3939 7c0a 7c20 2020 2020 2020 2020 2020  99|.|           
│ │ │ │  0004b850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b890: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004b890: 2020 7c0a 7c20 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 2020 2020 2020 2020                  
│ │ │ │  0004b8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b8e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004b8e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0004b8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b930: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004b930: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0004b940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b980: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004b980: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0004b990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b9d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004b9d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0004b9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ba00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ba10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ba20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004ba20: 2020 7c0a 7c20 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 2020 2020 2020 2020                  
│ │ │ │  0004ba60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ba70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004ba70: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0004ba80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ba90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004baa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bac0: 2020 2020 2020 3020 2020 2020 207c 0a7c        0      |.|
│ │ │ │ -0004bad0: 6364 2020 2020 2020 2020 2020 2020 2020  cd              
│ │ │ │ +0004bab0: 2020 2020 2020 2020 2020 2030 2020 2020             0    
│ │ │ │ +0004bac0: 2020 7c0a 7c63 6420 2020 2020 2020 2020    |.|cd         
│ │ │ │ +0004bad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004baf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bb10: 2020 2020 2020 3020 2020 2020 207c 0a7c        0      |.|
│ │ │ │ +0004bb00: 2020 2020 2020 2020 2020 2030 2020 2020             0    
│ │ │ │ +0004bb10: 2020 7c0a 7c20 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 2020 2020 2020 2020                  
│ │ │ │ -0004bb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bb60: 2020 2020 2020 3020 2020 2020 207c 0a7c        0      |.|
│ │ │ │ +0004bb50: 2020 2020 2020 2020 2020 2030 2020 2020             0    
│ │ │ │ +0004bb60: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0004bb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bbb0: 2020 2020 2020 3130 3761 332b 347c 0a7c        107a3+4|.|
│ │ │ │ +0004bba0: 2020 2020 2020 2020 2020 2031 3037 6133             107a3
│ │ │ │ +0004bbb0: 2b34 7c0a 7c20 2020 2020 2020 2020 2020  +4|.|           
│ │ │ │  0004bbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bc00: 2020 2020 2020 3832 3331 6162 327c 0a7c        8231ab2|.|
│ │ │ │ -0004bc10: 3161 6264 2d32 3632 3762 3264 2b33 3939  1abd-2627b2d+399
│ │ │ │ -0004bc20: 3661 6364 2b37 3135 3262 6364 2b31 3137  6acd+7152bcd+117
│ │ │ │ -0004bc30: 3430 6332 642d 3933 3938 6164 322d 3135  40c2d-9398ad2-15
│ │ │ │ -0004bc40: 3331 3762 6432 2b36 3932 3263 6432 2b35  317bd2+6922cd2+5
│ │ │ │ -0004bc50: 3038 3064 3320 2d31 3533 3434 617c 0a7c  080d3 -15344a|.|
│ │ │ │ -0004bc60: 2d31 3032 3539 6164 3220 2020 2020 2020  -10259ad2       
│ │ │ │ +0004bbf0: 2020 2020 2020 2020 2020 2038 3233 3161             8231a
│ │ │ │ +0004bc00: 6232 7c0a 7c31 6162 642d 3236 3237 6232  b2|.|1abd-2627b2
│ │ │ │ +0004bc10: 642b 3339 3936 6163 642b 3731 3532 6263  d+3996acd+7152bc
│ │ │ │ +0004bc20: 642b 3131 3734 3063 3264 2d39 3339 3861  d+11740c2d-9398a
│ │ │ │ +0004bc30: 6432 2d31 3533 3137 6264 322b 3639 3232  d2-15317bd2+6922
│ │ │ │ +0004bc40: 6364 322b 3530 3830 6433 202d 3135 3334  cd2+5080d3 -1534
│ │ │ │ +0004bc50: 3461 7c0a 7c2d 3130 3235 3961 6432 2020  4a|.|-10259ad2  
│ │ │ │ +0004bc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bca0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004bcb0: 2d38 3233 3161 6232 2d31 3331 3737 6233  -8231ab2-13177b3
│ │ │ │ -0004bcc0: 2d35 3836 3461 6263 2d31 3339 3930 6232  -5864abc-13990b2
│ │ │ │ -0004bcd0: 632b 3133 3039 6263 322d 3533 3938 6333  c+1309bc2-5398c3
│ │ │ │ -0004bce0: 2d35 3032 3661 6264 2b31 3135 3231 6232  -5026abd+11521b2
│ │ │ │ -0004bcf0: 642b 3735 3031 6163 642b 3137 377c 0a7c  d+7501acd+177|.|
│ │ │ │ -0004bd00: 3135 3334 3461 332d 3236 3533 6132 622d  15344a3-2653a2b-
│ │ │ │ -0004bd10: 3130 3235 3961 6232 2d31 3233 3635 6132  10259ab2-12365a2
│ │ │ │ -0004bd20: 632b 3732 3136 6162 632d 3632 3330 6163  c+7216abc-6230ac
│ │ │ │ -0004bd30: 322b 3533 3236 6263 322d 3130 3331 6333  2+5326bc2-1031c3
│ │ │ │ -0004bd40: 2b31 3335 3038 6132 642b 3130 317c 0a7c  +13508a2d+101|.|
│ │ │ │ -0004bd50: 3133 3039 6132 622d 3533 3938 6132 632d  1309a2b-5398a2c-
│ │ │ │ -0004bd60: 3535 3439 6132 6420 2020 2020 2020 2020  5549a2d         
│ │ │ │ +0004bca0: 2020 7c0a 7c2d 3832 3331 6162 322d 3133    |.|-8231ab2-13
│ │ │ │ +0004bcb0: 3137 3762 332d 3538 3634 6162 632d 3133  177b3-5864abc-13
│ │ │ │ +0004bcc0: 3939 3062 3263 2b31 3330 3962 6332 2d35  990b2c+1309bc2-5
│ │ │ │ +0004bcd0: 3339 3863 332d 3530 3236 6162 642b 3131  398c3-5026abd+11
│ │ │ │ +0004bce0: 3532 3162 3264 2b37 3530 3161 6364 2b31  521b2d+7501acd+1
│ │ │ │ +0004bcf0: 3737 7c0a 7c31 3533 3434 6133 2d32 3635  77|.|15344a3-265
│ │ │ │ +0004bd00: 3361 3262 2d31 3032 3539 6162 322d 3132  3a2b-10259ab2-12
│ │ │ │ +0004bd10: 3336 3561 3263 2b37 3231 3661 6263 2d36  365a2c+7216abc-6
│ │ │ │ +0004bd20: 3233 3061 6332 2b35 3332 3662 6332 2d31  230ac2+5326bc2-1
│ │ │ │ +0004bd30: 3033 3163 332b 3133 3530 3861 3264 2b31  031c3+13508a2d+1
│ │ │ │ +0004bd40: 3031 7c0a 7c31 3330 3961 3262 2d35 3339  01|.|1309a2b-539
│ │ │ │ +0004bd50: 3861 3263 2d35 3534 3961 3264 2020 2020  8a2c-5549a2d    
│ │ │ │ +0004bd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bd90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004bda0: 3539 6233 2d35 3537 3061 3263 2d35 3330  59b3-5570a2c-530
│ │ │ │ -0004bdb0: 3761 6263 2d37 3436 3462 3263 2b33 3138  7abc-7464b2c+318
│ │ │ │ -0004bdc0: 3761 3264 2b38 3537 3061 6264 2d38 3235  7a2d+8570abd-825
│ │ │ │ -0004bdd0: 3162 3264 2b38 3434 3461 6364 2b35 3037  1b2d+8444acd+507
│ │ │ │ -0004bde0: 3162 6364 207c 2020 2020 2020 207c 0a7c  1bcd |       |.|
│ │ │ │ +0004bd90: 2020 7c0a 7c35 3962 332d 3535 3730 6132    |.|59b3-5570a2
│ │ │ │ +0004bda0: 632d 3533 3037 6162 632d 3734 3634 6232  c-5307abc-7464b2
│ │ │ │ +0004bdb0: 632b 3331 3837 6132 642b 3835 3730 6162  c+3187a2d+8570ab
│ │ │ │ +0004bdc0: 642d 3832 3531 6232 642b 3834 3434 6163  d-8251b2d+8444ac
│ │ │ │ +0004bdd0: 642b 3530 3731 6263 6420 7c20 2020 2020  d+5071bcd |     
│ │ │ │ +0004bde0: 2020 7c0a 7c20 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 2020 2020 2020 2020                  
│ │ │ │  0004be20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004be30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004be30: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0004be40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004be50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004be60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004be70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004be80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004be80: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0004be90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004beb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bed0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004bed0: 2020 7c0a 7c20 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 2020 2020 2020 2020                  
│ │ │ │  0004bf10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bf20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004bf20: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0004bf30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bf40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bf50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bf60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bf70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004bf70: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0004bf80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bf90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bfc0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004bfd0: 3436 6162 3264 2d35 3231 3862 3364 2d38  46ab2d-5218b3d-8
│ │ │ │ -0004bfe0: 3032 6132 6364 2d36 3333 3261 6263 642b  02a2cd-6332abcd+
│ │ │ │ -0004bff0: 3537 3238 6232 6364 2d35 3334 3761 6332  5728b2cd-5347ac2
│ │ │ │ -0004c000: 642d 3133 3339 3462 6332 642d 3239 3663  d-13394bc2d-296c
│ │ │ │ -0004c010: 3364 2d37 3434 3661 3264 322d 317c 0a7c  3d-7446a2d2-1|.|
│ │ │ │ +0004bfc0: 2020 7c0a 7c34 3661 6232 642d 3532 3138    |.|46ab2d-5218
│ │ │ │ +0004bfd0: 6233 642d 3830 3261 3263 642d 3633 3332  b3d-802a2cd-6332
│ │ │ │ +0004bfe0: 6162 6364 2b35 3732 3862 3263 642d 3533  abcd+5728b2cd-53
│ │ │ │ +0004bff0: 3437 6163 3264 2d31 3333 3934 6263 3264  47ac2d-13394bc2d
│ │ │ │ +0004c000: 2d32 3936 6333 642d 3734 3436 6132 6432  -296c3d-7446a2d2
│ │ │ │ +0004c010: 2d31 7c0a 7c20 2020 2020 2020 2020 2020  -1|.|           
│ │ │ │  0004c020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c060: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004c060: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0004c070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c0b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004c0b0: 2020 7c0a 7c20 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 2020 2020 2020 2020                  
│ │ │ │  0004c0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c100: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004c100: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0004c110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c150: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004c150: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0004c160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c1a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004c1a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0004c1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c1f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004c1f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0004c200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c240: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004c250: 3031 6162 3264 2b34 3432 3062 3364 2b31  01ab2d+4420b3d+1
│ │ │ │ -0004c260: 3233 3137 6132 6364 2b31 3533 3738 6162  2317a2cd+15378ab
│ │ │ │ -0004c270: 6364 2b31 3439 3231 6232 6364 2d38 3831  cd+14921b2cd-881
│ │ │ │ -0004c280: 3061 6332 642b 3332 3731 6263 3264 2b31  0ac2d+3271bc2d+1
│ │ │ │ -0004c290: 3036 3633 6333 642b 3132 3533 377c 0a7c  0663c3d+12537|.|
│ │ │ │ +0004c240: 2020 7c0a 7c30 3161 6232 642b 3434 3230    |.|01ab2d+4420
│ │ │ │ +0004c250: 6233 642b 3132 3331 3761 3263 642b 3135  b3d+12317a2cd+15
│ │ │ │ +0004c260: 3337 3861 6263 642b 3134 3932 3162 3263  378abcd+14921b2c
│ │ │ │ +0004c270: 642d 3838 3130 6163 3264 2b33 3237 3162  d-8810ac2d+3271b
│ │ │ │ +0004c280: 6332 642b 3130 3636 3363 3364 2b31 3235  c2d+10663c3d+125
│ │ │ │ +0004c290: 3337 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d  37|.|-----------
│ │ │ │  0004c2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004c2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004c2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004c2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0004c2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c  -------------|.|
│ │ │ │ +0004c2e0: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020  --|.|           
│ │ │ │  0004c2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c320: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0004c330: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0004c340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c370: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0004c380: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
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│ │ │ │ -0004d1e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004d1e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
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│ │ │ │ -0004d290: 3337 3661 3262 2b33 3738 3361 6232 2b31  376a2b+3783ab2+1
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│ │ │ │ -0004d2b0: 3330 3761 6263 2d37 3436 3462 3263 2b33  307abc-7464b2c+3
│ │ │ │ -0004d2c0: 3138 3761 3264 2b38 3537 3061 6264 2d38  187a2d+8570abd-8
│ │ │ │ -0004d2d0: 3235 3162 3264 2b38 3434 3461 637c 0a7c  251b2d+8444ac|.|
│ │ │ │ -0004d2e0: 2b31 3331 3737 6233 2b35 3836 3461 6263  +13177b3+5864abc
│ │ │ │ -0004d2f0: 2b31 3339 3930 6232 632b 3530 3236 6162  +13990b2c+5026ab
│ │ │ │ -0004d300: 642d 3131 3532 3162 3264 2d37 3530 3161  d-11521b2d-7501a
│ │ │ │ -0004d310: 6364 2d31 3737 3962 6364 2020 2020 2020  cd-1779bcd      
│ │ │ │ -0004d320: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │ -0004d360: 3263 2b36 3233 3061 6332 2d35 3332 3662  2c+6230ac2-5326b
│ │ │ │ -0004d370: 6332 2b31 3033 3163 332d 3133 357c 0a7c  c2+1031c3-135|.|
│ │ │ │ +0004d280: 2020 7c0a 7c33 3736 6132 622b 3337 3833    |.|376a2b+3783
│ │ │ │ +0004d290: 6162 322b 3130 3335 3962 332d 3535 3730  ab2+10359b3-5570
│ │ │ │ +0004d2a0: 6132 632d 3533 3037 6162 632d 3734 3634  a2c-5307abc-7464
│ │ │ │ +0004d2b0: 6232 632b 3331 3837 6132 642b 3835 3730  b2c+3187a2d+8570
│ │ │ │ +0004d2c0: 6162 642d 3832 3531 6232 642b 3834 3434  abd-8251b2d+8444
│ │ │ │ +0004d2d0: 6163 7c0a 7c2b 3133 3137 3762 332b 3538  ac|.|+13177b3+58
│ │ │ │ +0004d2e0: 3634 6162 632b 3133 3939 3062 3263 2b35  64abc+13990b2c+5
│ │ │ │ +0004d2f0: 3032 3661 6264 2d31 3135 3231 6232 642d  026abd-11521b2d-
│ │ │ │ +0004d300: 3735 3031 6163 642d 3137 3739 6263 6420  7501acd-1779bcd 
│ │ │ │ +0004d310: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0004d330: 3032 3539 6162 322d 3133 3039 6233 2b31  0259ab2-1309b3+1
│ │ │ │ +0004d340: 3233 3635 6132 632d 3732 3136 6162 632b  2365a2c-7216abc+
│ │ │ │ +0004d350: 3533 3938 6232 632b 3632 3330 6163 322d  5398b2c+6230ac2-
│ │ │ │ +0004d360: 3533 3236 6263 322b 3130 3331 6333 2d31  5326bc2+1031c3-1
│ │ │ │ +0004d370: 3335 7c0a 7c20 2020 2020 2020 2020 2020  35|.|           
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│ │ │ │ -0004d3e0: 3461 6432 2d31 3038 3636 6264 322b 3730  4ad2-10866bd2+70
│ │ │ │ -0004d3f0: 3631 6364 322d 3236 3237 6433 2020 2020  61cd2-2627d3    
│ │ │ │ -0004d400: 2020 2020 2020 2020 2020 2020 2020 3130                10
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│ │ │ │ -0004d430: 3938 6263 642b 3230 3336 6332 642d 3531  98bcd+2036c2d-51
│ │ │ │ -0004d440: 3037 6164 322d 3536 3739 6264 322b 3633  07ad2-5679bd2+63
│ │ │ │ -0004d450: 3235 6364 322b 3131 3734 3064 3320 3020  25cd2+11740d3 0 
│ │ │ │ -0004d460: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004d3b0: 2020 2030 2020 2020 2020 2020 2020 2020     0            
│ │ │ │ +0004d3c0: 2020 7c0a 7c39 6263 642d 3535 3439 6332    |.|9bcd-5549c2
│ │ │ │ +0004d3d0: 642d 3935 3334 6164 322d 3130 3836 3662  d-9534ad2-10866b
│ │ │ │ +0004d3e0: 6432 2b37 3036 3163 6432 2d32 3632 3764  d2+7061cd2-2627d
│ │ │ │ +0004d3f0: 3320 2020 2020 2020 2020 2020 2020 2020  3               
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│ │ │ │ +0004d410: 2b33 7c0a 7c32 3561 6264 2d39 3033 3361  +3|.|25abd-9033a
│ │ │ │ +0004d420: 6364 2d32 3939 3862 6364 2b32 3033 3663  cd-2998bcd+2036c
│ │ │ │ +0004d430: 3264 2d35 3130 3761 6432 2d35 3637 3962  2d-5107ad2-5679b
│ │ │ │ +0004d440: 6432 2b36 3332 3563 6432 2b31 3137 3430  d2+6325cd2+11740
│ │ │ │ +0004d450: 6433 2030 2020 2020 2020 2020 2020 2020  d3 0            
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│ │ │ │ +0004d4a0: 2020 2031 3533 3434 6133 2d32 3635 3361     15344a3-2653a
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│ │ │ │ -0004d6f0: 3030 3436 6162 6432 2d36 3334 3262 3264  0046abd2-6342b2d
│ │ │ │ -0004d700: 322b 3734 3830 6163 6432 2b37 3834 3362  2+7480acd2+7843b
│ │ │ │ -0004d710: 6364 322b 3830 3834 6332 6432 2036 3430  cd2+8084c2d2 640
│ │ │ │ -0004d720: 3961 3362 2d31 3031 3531 6132 6232 2d33  9a3b-10151a2b2-3
│ │ │ │ -0004d730: 3836 3361 6233 2b37 3637 3262 347c 0a7c  863ab3+7672b4|.|
│ │ │ │ +0004d6e0: 2020 7c0a 7c30 3034 3661 6264 322d 3633    |.|0046abd2-63
│ │ │ │ +0004d6f0: 3432 6232 6432 2b37 3438 3061 6364 322b  42b2d2+7480acd2+
│ │ │ │ +0004d700: 3738 3433 6263 6432 2b38 3038 3463 3264  7843bcd2+8084c2d
│ │ │ │ +0004d710: 3220 3634 3039 6133 622d 3130 3135 3161  2 6409a3b-10151a
│ │ │ │ +0004d720: 3262 322d 3338 3633 6162 332b 3736 3732  2b2-3863ab3+7672
│ │ │ │ +0004d730: 6234 7c0a 7c20 2020 2020 2020 2020 2020  b4|.|           
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│ │ │ │ -0004d970: 6132 6432 2d31 3532 3831 6162 6432 2b39  a2d2-15281abd2+9
│ │ │ │ -0004d980: 3334 3662 3264 322d 3433 3039 6163 6432  346b2d2-4309acd2
│ │ │ │ -0004d990: 2d38 3236 3962 6364 322b 3633 3431 6332  -8269bcd2+6341c2
│ │ │ │ -0004d9a0: 6432 202d 3132 3035 3161 3362 2b33 3533  d2 -12051a3b+353
│ │ │ │ -0004d9b0: 6132 6232 2b38 3531 3861 6233 2b7c 0a7c  a2b2+8518ab3+|.|
│ │ │ │ +0004d960: 2020 7c0a 7c61 3264 322d 3135 3238 3161    |.|a2d2-15281a
│ │ │ │ +0004d970: 6264 322b 3933 3436 6232 6432 2d34 3330  bd2+9346b2d2-430
│ │ │ │ +0004d980: 3961 6364 322d 3832 3639 6263 6432 2b36  9acd2-8269bcd2+6
│ │ │ │ +0004d990: 3334 3163 3264 3220 2d31 3230 3531 6133  341c2d2 -12051a3
│ │ │ │ +0004d9a0: 622b 3335 3361 3262 322b 3835 3138 6162  b+353a2b2+8518ab
│ │ │ │ +0004d9b0: 332b 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d  3+|.|-----------
│ │ │ │  0004d9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004d9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004d9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004d9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0004da00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c  -------------|.|
│ │ │ │ +0004da00: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020  --|.|           
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│ │ │ │ -0004e3e0: 3630 3061 6332 2b31 3137 3236 6263 322d  600ac2+11726bc2-
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│ │ │ │ +0004e3d0: 6232 632b 3736 3030 6163 322b 3131 3732  b2c+7600ac2+1172
│ │ │ │ +0004e3e0: 3662 6332 2d31 3134 3063 332d 3132 3036  6bc2-1140c3-1206
│ │ │ │ +0004e3f0: 6132 642b 3131 3433 3561 6264 2d39 3037  a2d+11435abd-907
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│ │ │ │ -0004e650: 3531 3361 3263 322b 3130 3535 3861 6263  513a2c2+10558abc
│ │ │ │ -0004e660: 322b 3235 3930 6232 6332 2b31 3434 3134  2+2590b2c2+14414
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│ │ │ │ -0004e680: 3961 6232 642d 3132 3631 3562 337c 0a7c  9ab2d-12615b3|.|
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│ │ │ │ +0004e640: 6233 632b 3535 3133 6132 6332 2b31 3035  b3c+5513a2c2+105
│ │ │ │ +0004e650: 3538 6162 6332 2b32 3539 3062 3263 322b  58abc2+2590b2c2+
│ │ │ │ +0004e660: 3134 3431 3461 3364 2b33 3834 3461 3262  14414a3d+3844a2b
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│ │ │ │ -0004ea50: 3038 6132 642d 3130 3132 3561 6264 2b35  08a2d-10125abd+5
│ │ │ │ -0004ea60: 3534 3962 3264 2b39 3033 3361 6364 2b32  549b2d+9033acd+2
│ │ │ │ -0004ea70: 3939 3862 6364 2d32 3033 3663 3264 207c  998bcd-2036c2d |
│ │ │ │ +0004ea40: 2020 7c0a 7c30 3861 3264 2d31 3031 3235    |.|08a2d-10125
│ │ │ │ +0004ea50: 6162 642b 3535 3439 6232 642b 3930 3333  abd+5549b2d+9033
│ │ │ │ +0004ea60: 6163 642b 3239 3938 6263 642d 3230 3336  acd+2998bcd-2036
│ │ │ │ +0004ea70: 6332 6420 7c20 2020 2020 2020 2020 2020  c2d |           
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│ │ │ │ -0004eaf0: 3738 3361 6232 2b31 3033 3539 6233 2d35  783ab2+10359b3-5
│ │ │ │ -0004eb00: 3537 3061 3263 2d35 3330 3761 6263 2d37  570a2c-5307abc-7
│ │ │ │ -0004eb10: 3436 3462 3263 2b33 3138 3761 3264 2b38  464b2c+3187a2d+8
│ │ │ │ -0004eb20: 3537 3061 6264 2d38 3235 3162 3264 2b38  570abd-8251b2d+8
│ │ │ │ -0004eb30: 3434 3461 6364 2b35 3037 3162 637c 0a7c  444acd+5071bc|.|
│ │ │ │ +0004eae0: 2020 7c0a 7c37 3833 6162 322b 3130 3335    |.|783ab2+1035
│ │ │ │ +0004eaf0: 3962 332d 3535 3730 6132 632d 3533 3037  9b3-5570a2c-5307
│ │ │ │ +0004eb00: 6162 632d 3734 3634 6232 632b 3331 3837  abc-7464b2c+3187
│ │ │ │ +0004eb10: 6132 642b 3835 3730 6162 642d 3832 3531  a2d+8570abd-8251
│ │ │ │ +0004eb20: 6232 642b 3834 3434 6163 642b 3530 3731  b2d+8444acd+5071
│ │ │ │ +0004eb30: 6263 7c0a 7c20 2020 2020 2020 2020 2020  bc|.|           
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│ │ │ │ -0004eb90: 2d31 3032 3539 6162 322b 3133 3039 6233  -10259ab2+1309b3
│ │ │ │ -0004eba0: 2d31 3233 3635 6132 632b 3732 3136 6162  -12365a2c+7216ab
│ │ │ │ -0004ebb0: 632d 3533 3938 6232 632d 3632 3330 6163  c-5398b2c-6230ac
│ │ │ │ -0004ebc0: 322b 3533 3236 6263 322d 3130 3331 6333  2+5326bc2-1031c3
│ │ │ │ -0004ebd0: 2b31 3335 3038 6132 642b 3130 317c 0a7c  +13508a2d+101|.|
│ │ │ │ +0004eb80: 2020 7c0a 7c2d 3130 3235 3961 6232 2b31    |.|-10259ab2+1
│ │ │ │ +0004eb90: 3330 3962 332d 3132 3336 3561 3263 2b37  309b3-12365a2c+7
│ │ │ │ +0004eba0: 3231 3661 6263 2d35 3339 3862 3263 2d36  216abc-5398b2c-6
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│ │ │ │ -000528d0: 207c 2020 2020 2020 2020 2020 2020 2020   |              
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│ │ │ │ +00052dc0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00052dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00052e10: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00052e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00052e60: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00052e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052eb0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00052eb0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00052ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00052f00: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
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│ │ │ │ +00052fa0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
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│ │ │ │  00052fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00053000: 2020 2020 2020 2020 2020 7c20 2020 2020            |     
│ │ │ │ +00052ff0: 2020 7c0a 7c20 2020 2020 2020 2020 207c    |.|          |
│ │ │ │ +00053000: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00053090: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000530a0: 642b 3530 3731 6263 6420 7c20 2020 2020  d+5071bcd |     
│ │ │ │ +00053090: 2020 7c0a 7c64 2b35 3037 3162 6364 207c    |.|d+5071bcd |
│ │ │ │ +000530a0: 2020 2020 2020 2020 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 207c 0a7c               |.|
│ │ │ │ -000530f0: 2020 2020 2020 2020 2020 7c20 2020 2020            |     
│ │ │ │ +000530e0: 2020 7c0a 7c20 2020 2020 2020 2020 207c    |.|          |
│ │ │ │ +000530f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053100: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00053150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053160: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00053180: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
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│ │ │ │ -000531d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000531d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000531e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000531f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00053290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000532a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000532c0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000532d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000532e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000532f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053300: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00053320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053360: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00053370: 3834 6434 207c 2020 2020 2020 2020 2020  84d4 |          
│ │ │ │ +00053360: 2020 7c0a 7c38 3464 3420 7c20 2020 2020    |.|84d4 |     
│ │ │ │ +00053370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000533a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000533b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │  000533c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000533d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 207c 0a7c               |.|
│ │ │ │ +00053400: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00053410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053450: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00053450: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00053460: 2020 2020 2020 2020 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 207c 0a7c               |.|
│ │ │ │ +000534a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000534b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000534c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000534d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000534e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000534f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │  00053500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053540: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00053540: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00053550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053590: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │ -000535e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │ +000535e0: 2020 7c0a 7c36 3363 6433 2b36 3334 3164    |.|63cd3+6341d
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│ │ │ │ -000537e0: 2031 7d2c 207b 7b31 2c20 317d 2c20 307d   1}, {{1, 1}, 0}
│ │ │ │ -000537f0: 2c20 7b7b 312c 2031 7d2c 2031 7d2c 207b  , {{1, 1}, 1}, {
│ │ │ │ -00053800: 7b32 2c20 307d 2c20 307d 2c20 7b7b 322c  {2, 0}, 0}, {{2,
│ │ │ │ -00053810: 207c 0a7c 2020 2020 2020 2d2d 2d2d 2d2d   |.|      ------
│ │ │ │ +000537b0: 2020 2020 2020 7c0a 7c6f 3134 203d 207b        |.|o14 = {
│ │ │ │ +000537c0: 7b7b 302c 2032 7d2c 2030 7d2c 207b 7b30  {{0, 2}, 0}, {{0
│ │ │ │ +000537d0: 2c20 327d 2c20 317d 2c20 7b7b 312c 2031  , 2}, 1}, {{1, 1
│ │ │ │ +000537e0: 7d2c 2030 7d2c 207b 7b31 2c20 317d 2c20  }, 0}, {{1, 1}, 
│ │ │ │ +000537f0: 317d 2c20 7b7b 322c 2030 7d2c 2030 7d2c  1}, {{2, 0}, 0},
│ │ │ │ +00053800: 207b 7b32 2c20 7c0a 7c20 2020 2020 202d   {{2, |.|      -
│ │ │ │ +00053810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053860: 2d7c 0a7c 2020 2020 2020 307d 2c20 317d  -|.|      0}, 1}
│ │ │ │ -00053870: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +00053850: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2030  ------|.|      0
│ │ │ │ +00053860: 7d2c 2031 7d7d 2020 2020 2020 2020 2020  }, 1}}          
│ │ │ │ +00053870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000538a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000538b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000538a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000538b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000538c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000538d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000538e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000538f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053900: 207c 0a7c 6f31 3420 3a20 4c69 7374 2020   |.|o14 : List  
│ │ │ │ +000538f0: 2020 2020 2020 7c0a 7c6f 3134 203a 204c        |.|o14 : L
│ │ │ │ +00053900: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │  00053910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053950: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00053940: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00053950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000539a0: 2d2b 0a7c 6931 3520 3a20 234c 2020 2020  -+.|i15 : #L    
│ │ │ │ +00053990: 2d2d 2d2d 2d2d 2b0a 7c69 3135 203a 2023  ------+.|i15 : #
│ │ │ │ +000539a0: 4c20 2020 2020 2020 2020 2020 2020 2020  L               
│ │ │ │  000539b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000539c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000539d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000539e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000539f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000539e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000539f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053a40: 207c 0a7c 6f31 3520 3d20 3620 2020 2020   |.|o15 = 6     
│ │ │ │ +00053a30: 2020 2020 2020 7c0a 7c6f 3135 203d 2036        |.|o15 = 6
│ │ │ │ +00053a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053a50: 2020 2020 2020 2020 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: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00053a80: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00053a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053ae0: 2d2b 0a7c 6931 3620 3a20 6e65 744c 6973  -+.|i16 : netLis
│ │ │ │ -00053af0: 7420 4c20 2020 2020 2020 2020 2020 2020  t L             
│ │ │ │ +00053ad0: 2d2d 2d2d 2d2d 2b0a 7c69 3136 203a 206e  ------+.|i16 : n
│ │ │ │ +00053ae0: 6574 4c69 7374 204c 2020 2020 2020 2020  etList L        
│ │ │ │ +00053af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053b30: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00053b20: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00053b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053b80: 207c 0a7c 2020 2020 2020 2b2d 2d2d 2d2d   |.|      +-----
│ │ │ │ -00053b90: 2d2b 2d2b 2020 2020 2020 2020 2020 2020  -+-+            
│ │ │ │ +00053b70: 2020 2020 2020 7c0a 7c20 2020 2020 202b        |.|      +
│ │ │ │ +00053b80: 2d2d 2d2d 2d2d 2b2d 2b20 2020 2020 2020  ------+-+       
│ │ │ │ +00053b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053bd0: 207c 0a7c 6f31 3620 3d20 7c7b 302c 2032   |.|o16 = |{0, 2
│ │ │ │ -00053be0: 7d7c 307c 2020 2020 2020 2020 2020 2020  }|0|            
│ │ │ │ +00053bc0: 2020 2020 2020 7c0a 7c6f 3136 203d 207c        |.|o16 = |
│ │ │ │ +00053bd0: 7b30 2c20 327d 7c30 7c20 2020 2020 2020  {0, 2}|0|       
│ │ │ │ +00053be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053c20: 207c 0a7c 2020 2020 2020 2b2d 2d2d 2d2d   |.|      +-----
│ │ │ │ -00053c30: 2d2b 2d2b 2020 2020 2020 2020 2020 2020  -+-+            
│ │ │ │ +00053c10: 2020 2020 2020 7c0a 7c20 2020 2020 202b        |.|      +
│ │ │ │ +00053c20: 2d2d 2d2d 2d2d 2b2d 2b20 2020 2020 2020  ------+-+       
│ │ │ │ +00053c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053c70: 207c 0a7c 2020 2020 2020 7c7b 302c 2032   |.|      |{0, 2
│ │ │ │ -00053c80: 7d7c 317c 2020 2020 2020 2020 2020 2020  }|1|            
│ │ │ │ +00053c60: 2020 2020 2020 7c0a 7c20 2020 2020 207c        |.|      |
│ │ │ │ +00053c70: 7b30 2c20 327d 7c31 7c20 2020 2020 2020  {0, 2}|1|       
│ │ │ │ +00053c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053cc0: 207c 0a7c 2020 2020 2020 2b2d 2d2d 2d2d   |.|      +-----
│ │ │ │ -00053cd0: 2d2b 2d2b 2020 2020 2020 2020 2020 2020  -+-+            
│ │ │ │ +00053cb0: 2020 2020 2020 7c0a 7c20 2020 2020 202b        |.|      +
│ │ │ │ +00053cc0: 2d2d 2d2d 2d2d 2b2d 2b20 2020 2020 2020  ------+-+       
│ │ │ │ +00053cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053d10: 207c 0a7c 2020 2020 2020 7c7b 312c 2031   |.|      |{1, 1
│ │ │ │ -00053d20: 7d7c 307c 2020 2020 2020 2020 2020 2020  }|0|            
│ │ │ │ +00053d00: 2020 2020 2020 7c0a 7c20 2020 2020 207c        |.|      |
│ │ │ │ +00053d10: 7b31 2c20 317d 7c30 7c20 2020 2020 2020  {1, 1}|0|       
│ │ │ │ +00053d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053d60: 207c 0a7c 2020 2020 2020 2b2d 2d2d 2d2d   |.|      +-----
│ │ │ │ -00053d70: 2d2b 2d2b 2020 2020 2020 2020 2020 2020  -+-+            
│ │ │ │ +00053d50: 2020 2020 2020 7c0a 7c20 2020 2020 202b        |.|      +
│ │ │ │ +00053d60: 2d2d 2d2d 2d2d 2b2d 2b20 2020 2020 2020  ------+-+       
│ │ │ │ +00053d70: 2020 2020 2020 2020 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: 207c 0a7c 2020 2020 2020 7c7b 312c 2031   |.|      |{1, 1
│ │ │ │ -00053dc0: 7d7c 317c 2020 2020 2020 2020 2020 2020  }|1|            
│ │ │ │ +00053da0: 2020 2020 2020 7c0a 7c20 2020 2020 207c        |.|      |
│ │ │ │ +00053db0: 7b31 2c20 317d 7c31 7c20 2020 2020 2020  {1, 1}|1|       
│ │ │ │ +00053dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053e00: 207c 0a7c 2020 2020 2020 2b2d 2d2d 2d2d   |.|      +-----
│ │ │ │ -00053e10: 2d2b 2d2b 2020 2020 2020 2020 2020 2020  -+-+            
│ │ │ │ +00053df0: 2020 2020 2020 7c0a 7c20 2020 2020 202b        |.|      +
│ │ │ │ +00053e00: 2d2d 2d2d 2d2d 2b2d 2b20 2020 2020 2020  ------+-+       
│ │ │ │ +00053e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053e50: 207c 0a7c 2020 2020 2020 7c7b 322c 2030   |.|      |{2, 0
│ │ │ │ -00053e60: 7d7c 307c 2020 2020 2020 2020 2020 2020  }|0|            
│ │ │ │ +00053e40: 2020 2020 2020 7c0a 7c20 2020 2020 207c        |.|      |
│ │ │ │ +00053e50: 7b32 2c20 307d 7c30 7c20 2020 2020 2020  {2, 0}|0|       
│ │ │ │ +00053e60: 2020 2020 2020 2020 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: 207c 0a7c 2020 2020 2020 2b2d 2d2d 2d2d   |.|      +-----
│ │ │ │ -00053eb0: 2d2b 2d2b 2020 2020 2020 2020 2020 2020  -+-+            
│ │ │ │ +00053e90: 2020 2020 2020 7c0a 7c20 2020 2020 202b        |.|      +
│ │ │ │ +00053ea0: 2d2d 2d2d 2d2d 2b2d 2b20 2020 2020 2020  ------+-+       
│ │ │ │ +00053eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053ef0: 207c 0a7c 2020 2020 2020 7c7b 322c 2030   |.|      |{2, 0
│ │ │ │ -00053f00: 7d7c 317c 2020 2020 2020 2020 2020 2020  }|1|            
│ │ │ │ +00053ee0: 2020 2020 2020 7c0a 7c20 2020 2020 207c        |.|      |
│ │ │ │ +00053ef0: 7b32 2c20 307d 7c31 7c20 2020 2020 2020  {2, 0}|1|       
│ │ │ │ +00053f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053f40: 207c 0a7c 2020 2020 2020 2b2d 2d2d 2d2d   |.|      +-----
│ │ │ │ -00053f50: 2d2b 2d2b 2020 2020 2020 2020 2020 2020  -+-+            
│ │ │ │ +00053f30: 2020 2020 2020 7c0a 7c20 2020 2020 202b        |.|      +
│ │ │ │ +00053f40: 2d2d 2d2d 2d2d 2b2d 2b20 2020 2020 2020  ------+-+       
│ │ │ │ +00053f50: 2020 2020 2020 2020 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: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00053f80: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00053f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053fe0: 2d2b 0a0a 466f 7220 6578 616d 706c 6520  -+..For example 
│ │ │ │ -00053ff0: 7765 2068 6176 653a 0a0a 2b2d 2d2d 2d2d  we have:..+-----
│ │ │ │ +00053fd0: 2d2d 2d2d 2d2d 2b0a 0a46 6f72 2065 7861  ------+..For exa
│ │ │ │ +00053fe0: 6d70 6c65 2077 6520 6861 7665 3a0a 0a2b  mple we have:..+
│ │ │ │ +00053ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00054000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00054010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00054020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00054030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00054040: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3137 203a  --------+.|i17 :
│ │ │ │ -00054050: 2068 6f6d 6f74 2328 4c5f 3029 2020 2020   homot#(L_0)    
│ │ │ │ +00054030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00054040: 6931 3720 3a20 686f 6d6f 7423 284c 5f30  i17 : homot#(L_0
│ │ │ │ +00054050: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  00054060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054090: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00054080: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00054090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000540a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000540b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000540c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000540d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000540e0: 2020 2020 2020 2020 7c0a 7c6f 3137 203d          |.|o17 =
│ │ │ │ -000540f0: 207b 367d 207c 202d 3133 3739 3561 342b   {6} | -13795a4+
│ │ │ │ -00054100: 3230 3139 6133 622b 3133 3736 3961 3262  2019a3b+13769a2b
│ │ │ │ -00054110: 322b 3735 3836 6162 332b 3836 3439 6234  2+7586ab3+8649b4
│ │ │ │ -00054120: 2b36 3435 3461 3363 2d31 3031 3837 6132  +6454a3c-10187a2
│ │ │ │ -00054130: 6263 2d31 3738 3361 7c0a 7c20 2020 2020  bc-1783a|.|     
│ │ │ │ -00054140: 207b 367d 207c 2031 3131 3532 6134 2d31   {6} | 11152a4-1
│ │ │ │ -00054150: 3333 3661 3362 2b31 3138 3436 6132 6232  336a3b+11846a2b2
│ │ │ │ -00054160: 2b31 3032 3634 6162 332b 3631 3862 342d  +10264ab3+618b4-
│ │ │ │ -00054170: 3131 3035 3161 3363 2b31 3231 3239 6132  11051a3c+12129a2
│ │ │ │ -00054180: 6263 2b35 3932 3761 7c0a 7c20 2020 2020  bc+5927a|.|     
│ │ │ │ -00054190: 207b 367d 207c 202d 3633 3338 6134 2b31   {6} | -6338a4+1
│ │ │ │ -000541a0: 3030 3235 6133 622b 3134 3938 3761 3363  0025a3b+14987a3c
│ │ │ │ -000541b0: 2d39 3935 3961 3262 632d 3131 3639 3161  -9959a2bc-11691a
│ │ │ │ -000541c0: 3263 322b 3132 3333 3661 6263 322d 3737  2c2+12336abc2-77
│ │ │ │ -000541d0: 3836 6133 642d 3131 7c0a 7c20 2020 2020  86a3d-11|.|     
│ │ │ │ -000541e0: 207b 367d 207c 2032 3237 3561 342d 3233   {6} | 2275a4-23
│ │ │ │ -000541f0: 3961 3362 2b31 3435 3934 6132 6232 2d38  9a3b+14594a2b2-8
│ │ │ │ -00054200: 3135 3361 6233 2d31 3139 3435 6234 2d38  153ab3-11945b4-8
│ │ │ │ -00054210: 3431 3661 3363 2b36 3235 3161 3262 632d  416a3c+6251a2bc-
│ │ │ │ -00054220: 3330 3233 6162 3263 7c0a 7c20 2020 2020  3023ab2c|.|     
│ │ │ │ -00054230: 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   ---------------
│ │ │ │ +000540d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000540e0: 6f31 3720 3d20 7b36 7d20 7c20 2d31 3337  o17 = {6} | -137
│ │ │ │ +000540f0: 3935 6134 2b32 3031 3961 3362 2b31 3337  95a4+2019a3b+137
│ │ │ │ +00054100: 3639 6132 6232 2b37 3538 3661 6233 2b38  69a2b2+7586ab3+8
│ │ │ │ +00054110: 3634 3962 342b 3634 3534 6133 632d 3130  649b4+6454a3c-10
│ │ │ │ +00054120: 3138 3761 3262 632d 3137 3833 617c 0a7c  187a2bc-1783a|.|
│ │ │ │ +00054130: 2020 2020 2020 7b36 7d20 7c20 3131 3135        {6} | 1115
│ │ │ │ +00054140: 3261 342d 3133 3336 6133 622b 3131 3834  2a4-1336a3b+1184
│ │ │ │ +00054150: 3661 3262 322b 3130 3236 3461 6233 2b36  6a2b2+10264ab3+6
│ │ │ │ +00054160: 3138 6234 2d31 3130 3531 6133 632b 3132  18b4-11051a3c+12
│ │ │ │ +00054170: 3132 3961 3262 632b 3539 3237 617c 0a7c  129a2bc+5927a|.|
│ │ │ │ +00054180: 2020 2020 2020 7b36 7d20 7c20 2d36 3333        {6} | -633
│ │ │ │ +00054190: 3861 342b 3130 3032 3561 3362 2b31 3439  8a4+10025a3b+149
│ │ │ │ +000541a0: 3837 6133 632d 3939 3539 6132 6263 2d31  87a3c-9959a2bc-1
│ │ │ │ +000541b0: 3136 3931 6132 6332 2b31 3233 3336 6162  1691a2c2+12336ab
│ │ │ │ +000541c0: 6332 2d37 3738 3661 3364 2d31 317c 0a7c  c2-7786a3d-11|.|
│ │ │ │ +000541d0: 2020 2020 2020 7b36 7d20 7c20 3232 3735        {6} | 2275
│ │ │ │ +000541e0: 6134 2d32 3339 6133 622b 3134 3539 3461  a4-239a3b+14594a
│ │ │ │ +000541f0: 3262 322d 3831 3533 6162 332d 3131 3934  2b2-8153ab3-1194
│ │ │ │ +00054200: 3562 342d 3834 3136 6133 632b 3632 3531  5b4-8416a3c+6251
│ │ │ │ +00054210: 6132 6263 2d33 3032 3361 6232 637c 0a7c  a2bc-3023ab2c|.|
│ │ │ │ +00054220: 2020 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d        ----------
│ │ │ │ +00054230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00054240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00054250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00054260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00054270: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ -00054280: 2062 3263 2b39 3231 3962 3363 2b35 3531   b2c+9219b3c+551
│ │ │ │ -00054290: 3361 3263 322b 3130 3535 3861 6263 322b  3a2c2+10558abc2+
│ │ │ │ -000542a0: 3235 3930 6232 6332 2b31 3136 3234 6133  2590b2c2+11624a3
│ │ │ │ -000542b0: 642d 3536 3033 6132 6264 2b31 3430 3538  d-5603a2bd+14058
│ │ │ │ -000542c0: 6162 3264 2d31 3236 7c0a 7c20 2020 2020  ab2d-126|.|     
│ │ │ │ -000542d0: 2062 3263 2b34 3839 6233 632d 3135 3338   b2c+489b3c-1538
│ │ │ │ -000542e0: 3361 3263 322b 3530 3761 6263 322d 3133  3a2c2+507abc2-13
│ │ │ │ -000542f0: 3830 3462 3263 322d 3834 3136 6163 332b  804b2c2-8416ac3+
│ │ │ │ -00054300: 3932 6334 2d31 3130 3537 6133 642d 3531  92c4-11057a3d-51
│ │ │ │ -00054310: 3133 6132 6264 2d32 7c0a 7c20 2020 2020  13a2bd-2|.|     
│ │ │ │ -00054320: 2035 3661 3262 642b 3439 3630 6132 6364   56a2bd+4960a2cd
│ │ │ │ -00054330: 2d35 3538 3961 6263 642d 3831 3633 6163  -5589abcd-8163ac
│ │ │ │ -00054340: 3264 2d31 3839 3562 6332 642b 3934 3634  2d-1895bc2d+9464
│ │ │ │ -00054350: 6132 6432 2d37 3235 3361 6264 322b 3132  a2d2-7253abd2+12
│ │ │ │ -00054360: 3634 3261 6364 322d 7c0a 7c20 2020 2020  642acd2-|.|     
│ │ │ │ -00054370: 202b 3539 3333 6233 632b 3932 6132 6332   +5933b3c+92a2c2
│ │ │ │ -00054380: 2b35 3334 3361 6263 322b 3337 3938 6232  +5343abc2+3798b2
│ │ │ │ -00054390: 6332 2d31 3539 3638 6133 642b 3437 3361  c2-15968a3d+473a
│ │ │ │ -000543a0: 3262 642b 3133 3239 3361 6232 642d 3337  2bd+13293ab2d-37
│ │ │ │ -000543b0: 3631 6233 642d 3737 7c0a 7c20 2020 2020  61b3d-77|.|     
│ │ │ │ -000543c0: 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   ---------------
│ │ │ │ +00054260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c  -------------|.|
│ │ │ │ +00054270: 2020 2020 2020 6232 632b 3932 3139 6233        b2c+9219b3
│ │ │ │ +00054280: 632b 3535 3133 6132 6332 2b31 3035 3538  c+5513a2c2+10558
│ │ │ │ +00054290: 6162 6332 2b32 3539 3062 3263 322b 3131  abc2+2590b2c2+11
│ │ │ │ +000542a0: 3632 3461 3364 2d35 3630 3361 3262 642b  624a3d-5603a2bd+
│ │ │ │ +000542b0: 3134 3035 3861 6232 642d 3132 367c 0a7c  14058ab2d-126|.|
│ │ │ │ +000542c0: 2020 2020 2020 6232 632b 3438 3962 3363        b2c+489b3c
│ │ │ │ +000542d0: 2d31 3533 3833 6132 6332 2b35 3037 6162  -15383a2c2+507ab
│ │ │ │ +000542e0: 6332 2d31 3338 3034 6232 6332 2d38 3431  c2-13804b2c2-841
│ │ │ │ +000542f0: 3661 6333 2b39 3263 342d 3131 3035 3761  6ac3+92c4-11057a
│ │ │ │ +00054300: 3364 2d35 3131 3361 3262 642d 327c 0a7c  3d-5113a2bd-2|.|
│ │ │ │ +00054310: 2020 2020 2020 3536 6132 6264 2b34 3936        56a2bd+496
│ │ │ │ +00054320: 3061 3263 642d 3535 3839 6162 6364 2d38  0a2cd-5589abcd-8
│ │ │ │ +00054330: 3136 3361 6332 642d 3138 3935 6263 3264  163ac2d-1895bc2d
│ │ │ │ +00054340: 2b39 3436 3461 3264 322d 3732 3533 6162  +9464a2d2-7253ab
│ │ │ │ +00054350: 6432 2b31 3236 3432 6163 6432 2d7c 0a7c  d2+12642acd2-|.|
│ │ │ │ +00054360: 2020 2020 2020 2b35 3933 3362 3363 2b39        +5933b3c+9
│ │ │ │ +00054370: 3261 3263 322b 3533 3433 6162 6332 2b33  2a2c2+5343abc2+3
│ │ │ │ +00054380: 3739 3862 3263 322d 3135 3936 3861 3364  798b2c2-15968a3d
│ │ │ │ +00054390: 2b34 3733 6132 6264 2b31 3332 3933 6162  +473a2bd+13293ab
│ │ │ │ +000543a0: 3264 2d33 3736 3162 3364 2d37 377c 0a7c  2d-3761b3d-77|.|
│ │ │ │ +000543b0: 2020 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d        ----------
│ │ │ │ +000543c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000543d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000543e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000543f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00054400: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ -00054410: 2031 3562 3364 2b37 3836 3961 3263 642d   15b3d+7869a2cd-
│ │ │ │ -00054420: 3230 3532 6162 6364 2d31 3833 3162 3263  2052abcd-1831b2c
│ │ │ │ -00054430: 642b 3630 3432 6163 3264 2d32 3536 3162  d+6042ac2d-2561b
│ │ │ │ -00054440: 6332 642d 3837 3039 6132 6432 2d31 3332  c2d-8709a2d2-132
│ │ │ │ -00054450: 3139 6162 6432 2b34 7c0a 7c20 2020 2020  19abd2+4|.|     
│ │ │ │ -00054460: 2037 3632 6162 3264 2b31 3430 3935 6233   762ab2d+14095b3
│ │ │ │ -00054470: 642d 3135 3838 6132 6364 2b32 3030 3061  d-1588a2cd+2000a
│ │ │ │ -00054480: 6263 642d 3230 3830 6232 6364 2b39 3137  bcd-2080b2cd+917
│ │ │ │ -00054490: 3561 6332 642d 3634 3962 6332 642b 3838  5ac2d-649bc2d+88
│ │ │ │ -000544a0: 3239 6333 642b 3231 7c0a 7c20 2020 2020  29c3d+21|.|     
│ │ │ │ -000544b0: 2031 3935 3862 6364 3220 2020 2020 2020   1958bcd2       
│ │ │ │ +000543f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c  -------------|.|
│ │ │ │ +00054400: 2020 2020 2020 3135 6233 642b 3738 3639        15b3d+7869
│ │ │ │ +00054410: 6132 6364 2d32 3035 3261 6263 642d 3138  a2cd-2052abcd-18
│ │ │ │ +00054420: 3331 6232 6364 2b36 3034 3261 6332 642d  31b2cd+6042ac2d-
│ │ │ │ +00054430: 3235 3631 6263 3264 2d38 3730 3961 3264  2561bc2d-8709a2d
│ │ │ │ +00054440: 322d 3133 3231 3961 6264 322b 347c 0a7c  2-13219abd2+4|.|
│ │ │ │ +00054450: 2020 2020 2020 3736 3261 6232 642b 3134        762ab2d+14
│ │ │ │ +00054460: 3039 3562 3364 2d31 3538 3861 3263 642b  095b3d-1588a2cd+
│ │ │ │ +00054470: 3230 3030 6162 6364 2d32 3038 3062 3263  2000abcd-2080b2c
│ │ │ │ +00054480: 642b 3931 3735 6163 3264 2d36 3439 6263  d+9175ac2d-649bc
│ │ │ │ +00054490: 3264 2b38 3832 3963 3364 2b32 317c 0a7c  2d+8829c3d+21|.|
│ │ │ │ +000544a0: 2020 2020 2020 3139 3538 6263 6432 2020        1958bcd2  
│ │ │ │ +000544b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000544c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000544d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000544e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000544f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00054500: 2031 3761 3263 642d 3733 3839 6162 6364   17a2cd-7389abcd
│ │ │ │ -00054510: 2b34 3732 3362 3263 642d 3133 3236 3261  +4723b2cd-13262a
│ │ │ │ -00054520: 6332 642b 3534 3331 6263 3264 2b31 3132  c2d+5431bc2d+112
│ │ │ │ -00054530: 3734 6132 6432 2d32 3137 6162 6432 2b31  74a2d2-217abd2+1
│ │ │ │ -00054540: 3236 3162 3264 322b 7c0a 7c20 2020 2020  261b2d2+|.|     
│ │ │ │ -00054550: 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   ---------------
│ │ │ │ +000544e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000544f0: 2020 2020 2020 3137 6132 6364 2d37 3338        17a2cd-738
│ │ │ │ +00054500: 3961 6263 642b 3437 3233 6232 6364 2d31  9abcd+4723b2cd-1
│ │ │ │ +00054510: 3332 3632 6163 3264 2b35 3433 3162 6332  3262ac2d+5431bc2
│ │ │ │ +00054520: 642b 3131 3237 3461 3264 322d 3231 3761  d+11274a2d2-217a
│ │ │ │ +00054530: 6264 322b 3132 3631 6232 6432 2b7c 0a7c  bd2+1261b2d2+|.|
│ │ │ │ +00054540: 2020 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d        ----------
│ │ │ │ +00054550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00054560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00054570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00054580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00054590: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ -000545a0: 2032 3039 6232 6432 2b31 3232 3235 6163   209b2d2+12225ac
│ │ │ │ -000545b0: 6432 2d32 3630 3562 6364 322d 3932 6332  d2-2605bcd2-92c2
│ │ │ │ -000545c0: 6432 2b31 3539 3638 6164 332b 3134 3836  d2+15968ad3+1486
│ │ │ │ -000545d0: 3062 6433 2d38 3832 3963 6433 2d31 3132  0bd3-8829cd3-112
│ │ │ │ -000545e0: 3734 6434 207c 2020 7c0a 7c20 2020 2020  74d4 |  |.|     
│ │ │ │ -000545f0: 2036 3461 3264 322b 3836 3335 6162 6432   64a2d2+8635abd2
│ │ │ │ -00054600: 2d37 3136 3162 3264 322b 3939 3761 6364  -7161b2d2+997acd
│ │ │ │ -00054610: 322b 3330 3135 6263 6432 2b31 3132 3734  2+3015bcd2+11274
│ │ │ │ -00054620: 6332 6432 2020 2020 2020 2020 2020 2020  c2d2            
│ │ │ │ -00054630: 2020 2020 207c 2020 7c0a 7c20 2020 2020       |  |.|     
│ │ │ │ +00054580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c  -------------|.|
│ │ │ │ +00054590: 2020 2020 2020 3230 3962 3264 322b 3132        209b2d2+12
│ │ │ │ +000545a0: 3232 3561 6364 322d 3236 3035 6263 6432  225acd2-2605bcd2
│ │ │ │ +000545b0: 2d39 3263 3264 322b 3135 3936 3861 6433  -92c2d2+15968ad3
│ │ │ │ +000545c0: 2b31 3438 3630 6264 332d 3838 3239 6364  +14860bd3-8829cd
│ │ │ │ +000545d0: 332d 3131 3237 3464 3420 7c20 207c 0a7c  3-11274d4 |  |.|
│ │ │ │ +000545e0: 2020 2020 2020 3634 6132 6432 2b38 3633        64a2d2+863
│ │ │ │ +000545f0: 3561 6264 322d 3731 3631 6232 6432 2b39  5abd2-7161b2d2+9
│ │ │ │ +00054600: 3937 6163 6432 2b33 3031 3562 6364 322b  97acd2+3015bcd2+
│ │ │ │ +00054610: 3131 3237 3463 3264 3220 2020 2020 2020  11274c2d2       
│ │ │ │ +00054620: 2020 2020 2020 2020 2020 7c20 207c 0a7c            |  |.|
│ │ │ │ +00054630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054680: 2020 2020 207c 2020 7c0a 7c20 2020 2020       |  |.|     
│ │ │ │ -00054690: 2038 3230 3161 6364 322d 3134 3038 3062   8201acd2-14080b
│ │ │ │ -000546a0: 6364 3220 2020 2020 2020 2020 2020 2020  cd2             
│ │ │ │ +00054670: 2020 2020 2020 2020 2020 7c20 207c 0a7c            |  |.|
│ │ │ │ +00054680: 2020 2020 2020 3832 3031 6163 6432 2d31        8201acd2-1
│ │ │ │ +00054690: 3430 3830 6263 6432 2020 2020 2020 2020  4080bcd2        
│ │ │ │ +000546a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000546b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000546c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000546d0: 2020 2020 207c 2020 7c0a 7c20 2020 2020       |  |.|     
│ │ │ │ +000546c0: 2020 2020 2020 2020 2020 7c20 207c 0a7c            |  |.|
│ │ │ │ +000546d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000546e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000546f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054720: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00054730: 2020 2020 2020 2020 2034 2020 2020 2020           4      
│ │ │ │ -00054740: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +00054710: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00054720: 2020 2020 2020 2020 2020 2020 2020 3420                4 
│ │ │ │ +00054730: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ +00054740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054770: 2020 2020 2020 2020 7c0a 7c6f 3137 203a          |.|o17 :
│ │ │ │ -00054780: 204d 6174 7269 7820 5320 203c 2d2d 2053   Matrix S  <-- S
│ │ │ │ +00054760: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00054770: 6f31 3720 3a20 4d61 7472 6978 2053 2020  o17 : Matrix S  
│ │ │ │ +00054780: 3c2d 2d20 5320 2020 2020 2020 2020 2020  <-- S           
│ │ │ │  00054790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000547a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000547b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000547c0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +000547b0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +000547c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000547d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000547e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000547f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00054800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00054810: 2d2d 2d2d 2d2d 2d2d 2b0a 0a42 7574 2061  --------+..But a
│ │ │ │ -00054820: 6c6c 2074 6865 2068 6f6d 6f74 6f70 6965  ll the homotopie
│ │ │ │ -00054830: 7320 6172 6520 6d69 6e69 6d61 6c20 696e  s are minimal in
│ │ │ │ -00054840: 2074 6869 7320 6361 7365 3a0a 0a2b 2d2d   this case:..+--
│ │ │ │ +00054800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ +00054810: 4275 7420 616c 6c20 7468 6520 686f 6d6f  But all the homo
│ │ │ │ +00054820: 746f 7069 6573 2061 7265 206d 696e 696d  topies are minim
│ │ │ │ +00054830: 616c 2069 6e20 7468 6973 2063 6173 653a  al in this case:
│ │ │ │ +00054840: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..+-------------
│ │ │ │  00054850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00054860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00054870: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3138  ----------+.|i18
│ │ │ │ -00054880: 203a 206b 3120 3d20 535e 312f 2869 6465   : k1 = S^1/(ide
│ │ │ │ -00054890: 616c 2076 6172 7320 5329 3b20 2020 2020  al vars S);     
│ │ │ │ -000548a0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00054860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00054870: 0a7c 6931 3820 3a20 6b31 203d 2053 5e31  .|i18 : k1 = S^1
│ │ │ │ +00054880: 2f28 6964 6561 6c20 7661 7273 2053 293b  /(ideal vars S);
│ │ │ │ +00054890: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000548a0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  000548b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000548c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000548d0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3139 203a  --------+.|i19 :
│ │ │ │ -000548e0: 2073 656c 6563 7428 6b65 7973 2068 6f6d   select(keys hom
│ │ │ │ -000548f0: 6f74 2c6b 2d3e 286b 312a 2a68 6f6d 6f74  ot,k->(k1**homot
│ │ │ │ -00054900: 236b 2921 3d30 297c 0a7c 2020 2020 2020  #k)!=0)|.|      
│ │ │ │ +000548c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +000548d0: 6931 3920 3a20 7365 6c65 6374 286b 6579  i19 : select(key
│ │ │ │ +000548e0: 7320 686f 6d6f 742c 6b2d 3e28 6b31 2a2a  s homot,k->(k1**
│ │ │ │ +000548f0: 686f 6d6f 7423 6b29 213d 3029 7c0a 7c20  homot#k)!=0)|.| 
│ │ │ │ +00054900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054930: 2020 2020 2020 7c0a 7c6f 3139 203d 207b        |.|o19 = {
│ │ │ │ -00054940: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
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│ │ │ │ -00054990: 2020 2020 7c0a 7c6f 3139 203a 204c 6973      |.|o19 : Lis
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│ │ │ │ -00054a30: 7331 2c20 2d2d 2072 6574 7572 6e73 2061  s1, -- returns a
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│ │ │ │ -00054a50: 0a20 2020 2068 6f6d 6f74 6f70 6965 730a  .    homotopies.
│ │ │ │ -00054a60: 0a57 6179 7320 746f 2075 7365 206d 616b  .Ways to use mak
│ │ │ │ -00054a70: 6548 6f6d 6f74 6f70 6965 733a 0a3d 3d3d  eHomotopies:.===
│ │ │ │ -00054a80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00054a90: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 226d  ========..  * "m
│ │ │ │ -00054aa0: 616b 6548 6f6d 6f74 6f70 6965 7328 4d61  akeHomotopies(Ma
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│ │ │ │ -00054ac0: 202a 2022 6d61 6b65 486f 6d6f 746f 7069   * "makeHomotopi
│ │ │ │ -00054ad0: 6573 284d 6174 7269 782c 436f 6d70 6c65  es(Matrix,Comple
│ │ │ │ -00054ae0: 782c 5a5a 2922 0a0a 466f 7220 7468 6520  x,ZZ)"..For the 
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│ │ │ │ -00054b00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  =============..T
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│ │ │ │ -00054b30: 6d61 6b65 486f 6d6f 746f 7069 6573 2c20  makeHomotopies, 
│ │ │ │ -00054b40: 6973 2061 202a 6e6f 7465 206d 6574 686f  is a *note metho
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│ │ │ │ -00054b70: 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d  Function,...----
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│ │ │ │ +000549f0: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  so.========..  *
│ │ │ │ +00054a00: 202a 6e6f 7465 206d 616b 6548 6f6d 6f74   *note makeHomot
│ │ │ │ +00054a10: 6f70 6965 7331 3a20 6d61 6b65 486f 6d6f  opies1: makeHomo
│ │ │ │ +00054a20: 746f 7069 6573 312c 202d 2d20 7265 7475  topies1, -- retu
│ │ │ │ +00054a30: 726e 7320 6120 7379 7374 656d 206f 6620  rns a system of 
│ │ │ │ +00054a40: 6669 7273 740a 2020 2020 686f 6d6f 746f  first.    homoto
│ │ │ │ +00054a50: 7069 6573 0a0a 5761 7973 2074 6f20 7573  pies..Ways to us
│ │ │ │ +00054a60: 6520 6d61 6b65 486f 6d6f 746f 7069 6573  e makeHomotopies
│ │ │ │ +00054a70: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  :.==============
│ │ │ │ +00054a80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20  =============.. 
│ │ │ │ +00054a90: 202a 2022 6d61 6b65 486f 6d6f 746f 7069   * "makeHomotopi
│ │ │ │ +00054aa0: 6573 284d 6174 7269 782c 436f 6d70 6c65  es(Matrix,Comple
│ │ │ │ +00054ab0: 7829 220a 2020 2a20 226d 616b 6548 6f6d  x)".  * "makeHom
│ │ │ │ +00054ac0: 6f74 6f70 6965 7328 4d61 7472 6978 2c43  otopies(Matrix,C
│ │ │ │ +00054ad0: 6f6d 706c 6578 2c5a 5a29 220a 0a46 6f72  omplex,ZZ)"..For
│ │ │ │ +00054ae0: 2074 6865 2070 726f 6772 616d 6d65 720a   the programmer.
│ │ │ │ +00054af0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00054b00: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a  ==..The object *
│ │ │ │ +00054b10: 6e6f 7465 206d 616b 6548 6f6d 6f74 6f70  note makeHomotop
│ │ │ │ +00054b20: 6965 733a 206d 616b 6548 6f6d 6f74 6f70  ies: makeHomotop
│ │ │ │ +00054b30: 6965 732c 2069 7320 6120 2a6e 6f74 6520  ies, is a *note 
│ │ │ │ +00054b40: 6d65 7468 6f64 2066 756e 6374 696f 6e3a  method function:
│ │ │ │ +00054b50: 0a28 4d61 6361 756c 6179 3244 6f63 294d  .(Macaulay2Doc)M
│ │ │ │ +00054b60: 6574 686f 6446 756e 6374 696f 6e2c 2e0a  ethodFunction,..
│ │ │ │ +00054b70: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .---------------
│ │ │ │  00054b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00054b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00054ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00054bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00054bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865  -----------..The
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│ │ │ │ -00054c00: 6c65 2d70 6174 682f 6d61 6361 756c 6179  le-path/macaulay
│ │ │ │ -00054c10: 322d 312e 3235 2e31 312b 6473 2f4d 322f  2-1.25.11+ds/M2/
│ │ │ │ -00054c20: 4d61 6361 756c 6179 322f 7061 636b 6167  Macaulay2/packag
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│ │ │ │ -00054c70: 6e74 6572 7365 6374 696f 6e52 6573 6f6c  ntersectionResol
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│ │ │ │ -00054c90: 653a 206d 616b 6548 6f6d 6f74 6f70 6965  e: makeHomotopie
│ │ │ │ -00054ca0: 7331 2c20 4e65 7874 3a20 6d61 6b65 486f  s1, Next: makeHo
│ │ │ │ -00054cb0: 6d6f 746f 7069 6573 4f6e 486f 6d6f 6c6f  motopiesOnHomolo
│ │ │ │ -00054cc0: 6779 2c20 5072 6576 3a20 6d61 6b65 486f  gy, Prev: makeHo
│ │ │ │ -00054cd0: 6d6f 746f 7069 6573 2c20 5570 3a20 546f  motopies, Up: To
│ │ │ │ -00054ce0: 700a 0a6d 616b 6548 6f6d 6f74 6f70 6965  p..makeHomotopie
│ │ │ │ -00054cf0: 7331 202d 2d20 7265 7475 726e 7320 6120  s1 -- returns a 
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│ │ │ │ -00054d10: 686f 6d6f 746f 7069 6573 0a2a 2a2a 2a2a  homotopies.*****
│ │ │ │ +00054bc0: 0a0a 5468 6520 736f 7572 6365 206f 6620  ..The source of 
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│ │ │ │ +00054bf0: 6475 6369 626c 652d 7061 7468 2f6d 6163  ducible-path/mac
│ │ │ │ +00054c00: 6175 6c61 7932 2d31 2e32 352e 3131 2b64  aulay2-1.25.11+d
│ │ │ │ +00054c10: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70  s/M2/Macaulay2/p
│ │ │ │ +00054c20: 6163 6b61 6765 732f 0a43 6f6d 706c 6574  ackages/.Complet
│ │ │ │ +00054c30: 6549 6e74 6572 7365 6374 696f 6e52 6573  eIntersectionRes
│ │ │ │ +00054c40: 6f6c 7574 696f 6e73 2e6d 323a 3337 3731  olutions.m2:3771
│ │ │ │ +00054c50: 3a30 2e0a 1f0a 4669 6c65 3a20 436f 6d70  :0....File: Comp
│ │ │ │ +00054c60: 6c65 7465 496e 7465 7273 6563 7469 6f6e  leteIntersection
│ │ │ │ +00054c70: 5265 736f 6c75 7469 6f6e 732e 696e 666f  Resolutions.info
│ │ │ │ +00054c80: 2c20 4e6f 6465 3a20 6d61 6b65 486f 6d6f  , Node: makeHomo
│ │ │ │ +00054c90: 746f 7069 6573 312c 204e 6578 743a 206d  topies1, Next: m
│ │ │ │ +00054ca0: 616b 6548 6f6d 6f74 6f70 6965 734f 6e48  akeHomotopiesOnH
│ │ │ │ +00054cb0: 6f6d 6f6c 6f67 792c 2050 7265 763a 206d  omology, Prev: m
│ │ │ │ +00054cc0: 616b 6548 6f6d 6f74 6f70 6965 732c 2055  akeHomotopies, U
│ │ │ │ +00054cd0: 703a 2054 6f70 0a0a 6d61 6b65 486f 6d6f  p: Top..makeHomo
│ │ │ │ +00054ce0: 746f 7069 6573 3120 2d2d 2072 6574 7572  topies1 -- retur
│ │ │ │ +00054cf0: 6e73 2061 2073 7973 7465 6d20 6f66 2066  ns a system of f
│ │ │ │ +00054d00: 6972 7374 2068 6f6d 6f74 6f70 6965 730a  irst homotopies.
│ │ │ │ +00054d10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00054d20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00054d30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00054d40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00054d50: 2a2a 0a0a 2020 2a20 5573 6167 653a 200a  **..  * Usage: .
│ │ │ │ -00054d60: 2020 2020 2020 2020 4820 3d20 6d61 6b65          H = make
│ │ │ │ -00054d70: 486f 6d6f 746f 7069 6573 3128 662c 462c  Homotopies1(f,F,
│ │ │ │ -00054d80: 6429 0a20 202a 2049 6e70 7574 733a 0a20  d).  * Inputs:. 
│ │ │ │ -00054d90: 2020 2020 202a 2066 2c20 6120 2a6e 6f74       * f, a *not
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│ │ │ │ -00054db0: 6c61 7932 446f 6329 4d61 7472 6978 2c2c  lay2Doc)Matrix,,
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│ │ │ │ -00054df0: 636f 6d70 6c65 783a 2028 436f 6d70 6c65  complex: (Comple
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│ │ │ │ -00054e90: 2020 2020 686f 6d6f 746f 7069 6573 2028      homotopies (
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│ │ │ │ -00055010: 6f6d 6f74 6f70 6965 732c 202d 2d20 7265  omotopies, -- re
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│ │ │ │ -00055040: 6f74 6f70 6965 730a 2020 2a20 2a6e 6f74  otopies.  * *not
│ │ │ │ -00055050: 6520 6578 7465 7269 6f72 546f 724d 6f64  e exteriorTorMod
│ │ │ │ -00055060: 756c 653a 2065 7874 6572 696f 7254 6f72  ule: exteriorTor
│ │ │ │ -00055070: 4d6f 6475 6c65 2c20 2d2d 2054 6f72 2061  Module, -- Tor a
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│ │ │ │ -000550e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ -00055190: 206d 616b 6548 6f6d 6f74 6f70 6965 7331   makeHomotopies1
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│ │ │ │ +00054db0: 7269 782c 2c20 3178 6e20 6d61 7472 6978  rix,, 1xn matrix
│ │ │ │ +00054dc0: 206f 6620 656c 656d 656e 7473 206f 6620   of elements of 
│ │ │ │ +00054dd0: 530a 2020 2020 2020 2a20 462c 2061 202a  S.      * F, a *
│ │ │ │ +00054de0: 6e6f 7465 2063 6f6d 706c 6578 3a20 2843  note complex: (C
│ │ │ │ +00054df0: 6f6d 706c 6578 6573 2943 6f6d 706c 6578  omplexes)Complex
│ │ │ │ +00054e00: 2c2c 2061 646d 6974 7469 6e67 2068 6f6d  ,, admitting hom
│ │ │ │ +00054e10: 6f74 6f70 6965 7320 666f 7220 7468 650a  otopies for the.
│ │ │ │ +00054e20: 2020 2020 2020 2020 656e 7472 6965 7320          entries 
│ │ │ │ +00054e30: 6f66 2066 0a20 2020 2020 202a 2064 2c20  of f.      * d, 
│ │ │ │ +00054e40: 616e 202a 6e6f 7465 2069 6e74 6567 6572  an *note integer
│ │ │ │ +00054e50: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +00054e60: 5a5a 2c2c 2068 6f77 2066 6172 2062 6163  ZZ,, how far bac
│ │ │ │ +00054e70: 6b20 746f 2063 6f6d 7075 7465 2074 6865  k to compute the
│ │ │ │ +00054e80: 0a20 2020 2020 2020 2068 6f6d 6f74 6f70  .        homotop
│ │ │ │ +00054e90: 6965 7320 2864 6566 6175 6c74 7320 746f  ies (defaults to
│ │ │ │ +00054ea0: 206c 656e 6774 6820 6f66 2046 290a 2020   length of F).  
│ │ │ │ +00054eb0: 2a20 4f75 7470 7574 733a 0a20 2020 2020  * Outputs:.     
│ │ │ │ +00054ec0: 202a 2048 2c20 6120 2a6e 6f74 6520 6861   * H, a *note ha
│ │ │ │ +00054ed0: 7368 2074 6162 6c65 3a20 284d 6163 6175  sh table: (Macau
│ │ │ │ +00054ee0: 6c61 7932 446f 6329 4861 7368 5461 626c  lay2Doc)HashTabl
│ │ │ │ +00054ef0: 652c 2c20 6769 7665 7320 7468 6520 686f  e,, gives the ho
│ │ │ │ +00054f00: 6d6f 746f 7079 0a20 2020 2020 2020 2066  motopy.        f
│ │ │ │ +00054f10: 726f 6d20 465f 6920 636f 7272 6573 706f  rom F_i correspo
│ │ │ │ +00054f20: 6e64 696e 6720 746f 2066 5f6a 2061 7320  nding to f_j as 
│ │ │ │ +00054f30: 7468 6520 7661 6c75 6520 2448 235c 7b6a  the value $H#\{j
│ │ │ │ +00054f40: 2c69 5c7d 240a 0a44 6573 6372 6970 7469  ,i\}$..Descripti
│ │ │ │ +00054f50: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  on.===========..
│ │ │ │ +00054f60: 5361 6d65 2061 7320 6d61 6b65 486f 6d6f  Same as makeHomo
│ │ │ │ +00054f70: 746f 7069 6573 2c20 6275 7420 6f6e 6c79  topies, but only
│ │ │ │ +00054f80: 2063 6f6d 7075 7465 7320 7468 6520 6f72   computes the or
│ │ │ │ +00054f90: 6469 6e61 7279 2068 6f6d 6f74 6f70 6965  dinary homotopie
│ │ │ │ +00054fa0: 732c 206e 6f74 2074 6865 0a68 6967 6865  s, not the.highe
│ │ │ │ +00054fb0: 7220 6f6e 6573 2e20 5573 6564 2069 6e20  r ones. Used in 
│ │ │ │ +00054fc0: 6578 7465 7269 6f72 546f 724d 6f64 756c  exteriorTorModul
│ │ │ │ +00054fd0: 650a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d  e..See also.====
│ │ │ │ +00054fe0: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520  ====..  * *note 
│ │ │ │ +00054ff0: 6d61 6b65 486f 6d6f 746f 7069 6573 3a20  makeHomotopies: 
│ │ │ │ +00055000: 6d61 6b65 486f 6d6f 746f 7069 6573 2c20  makeHomotopies, 
│ │ │ │ +00055010: 2d2d 2072 6574 7572 6e73 2061 2073 7973  -- returns a sys
│ │ │ │ +00055020: 7465 6d20 6f66 2068 6967 6865 720a 2020  tem of higher.  
│ │ │ │ +00055030: 2020 686f 6d6f 746f 7069 6573 0a20 202a    homotopies.  *
│ │ │ │ +00055040: 202a 6e6f 7465 2065 7874 6572 696f 7254   *note exteriorT
│ │ │ │ +00055050: 6f72 4d6f 6475 6c65 3a20 6578 7465 7269  orModule: exteri
│ │ │ │ +00055060: 6f72 546f 724d 6f64 756c 652c 202d 2d20  orTorModule, -- 
│ │ │ │ +00055070: 546f 7220 6173 2061 206d 6f64 756c 6520  Tor as a module 
│ │ │ │ +00055080: 6f76 6572 2061 6e0a 2020 2020 6578 7465  over an.    exte
│ │ │ │ +00055090: 7269 6f72 2061 6c67 6562 7261 206f 7220  rior algebra or 
│ │ │ │ +000550a0: 6269 6772 6164 6564 2061 6c67 6562 7261  bigraded algebra
│ │ │ │ +000550b0: 0a0a 5761 7973 2074 6f20 7573 6520 6d61  ..Ways to use ma
│ │ │ │ +000550c0: 6b65 486f 6d6f 746f 7069 6573 313a 0a3d  keHomotopies1:.=
│ │ │ │ +000550d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +000550e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ +000550f0: 2022 6d61 6b65 486f 6d6f 746f 7069 6573   "makeHomotopies
│ │ │ │ +00055100: 3128 4d61 7472 6978 2c43 6f6d 706c 6578  1(Matrix,Complex
│ │ │ │ +00055110: 2922 0a20 202a 2022 6d61 6b65 486f 6d6f  )".  * "makeHomo
│ │ │ │ +00055120: 746f 7069 6573 3128 4d61 7472 6978 2c43  topies1(Matrix,C
│ │ │ │ +00055130: 6f6d 706c 6578 2c5a 5a29 220a 0a46 6f72  omplex,ZZ)"..For
│ │ │ │ +00055140: 2074 6865 2070 726f 6772 616d 6d65 720a   the programmer.
│ │ │ │ +00055150: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00055160: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a  ==..The object *
│ │ │ │ +00055170: 6e6f 7465 206d 616b 6548 6f6d 6f74 6f70  note makeHomotop
│ │ │ │ +00055180: 6965 7331 3a20 6d61 6b65 486f 6d6f 746f  ies1: makeHomoto
│ │ │ │ +00055190: 7069 6573 312c 2069 7320 6120 2a6e 6f74  pies1, is a *not
│ │ │ │ +000551a0: 6520 6d65 7468 6f64 2066 756e 6374 696f  e method functio
│ │ │ │ +000551b0: 6e3a 0a28 4d61 6361 756c 6179 3244 6f63  n:.(Macaulay2Doc
│ │ │ │ +000551c0: 294d 6574 686f 6446 756e 6374 696f 6e2c  )MethodFunction,
│ │ │ │ +000551d0: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...-------------
│ │ │ │  000551e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000551f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00055200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00055210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00055220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54  -------------..T
│ │ │ │ -00055230: 6865 2073 6f75 7263 6520 6f66 2074 6869  he source of thi
│ │ │ │ -00055240: 7320 646f 6375 6d65 6e74 2069 7320 696e  s document is in
│ │ │ │ -00055250: 0a2f 6275 696c 642f 7265 7072 6f64 7563  ./build/reproduc
│ │ │ │ -00055260: 6962 6c65 2d70 6174 682f 6d61 6361 756c  ible-path/macaul
│ │ │ │ -00055270: 6179 322d 312e 3235 2e31 312b 6473 2f4d  ay2-1.25.11+ds/M
│ │ │ │ -00055280: 322f 4d61 6361 756c 6179 322f 7061 636b  2/Macaulay2/pack
│ │ │ │ -00055290: 6167 6573 2f0a 436f 6d70 6c65 7465 496e  ages/.CompleteIn
│ │ │ │ -000552a0: 7465 7273 6563 7469 6f6e 5265 736f 6c75  tersectionResolu
│ │ │ │ -000552b0: 7469 6f6e 732e 6d32 3a33 3830 313a 302e  tions.m2:3801:0.
│ │ │ │ -000552c0: 0a1f 0a46 696c 653a 2043 6f6d 706c 6574  ...File: Complet
│ │ │ │ -000552d0: 6549 6e74 6572 7365 6374 696f 6e52 6573  eIntersectionRes
│ │ │ │ -000552e0: 6f6c 7574 696f 6e73 2e69 6e66 6f2c 204e  olutions.info, N
│ │ │ │ -000552f0: 6f64 653a 206d 616b 6548 6f6d 6f74 6f70  ode: makeHomotop
│ │ │ │ -00055300: 6965 734f 6e48 6f6d 6f6c 6f67 792c 204e  iesOnHomology, N
│ │ │ │ -00055310: 6578 743a 206d 616b 654d 6f64 756c 652c  ext: makeModule,
│ │ │ │ -00055320: 2050 7265 763a 206d 616b 6548 6f6d 6f74   Prev: makeHomot
│ │ │ │ -00055330: 6f70 6965 7331 2c20 5570 3a20 546f 700a  opies1, Up: Top.
│ │ │ │ -00055340: 0a6d 616b 6548 6f6d 6f74 6f70 6965 734f  .makeHomotopiesO
│ │ │ │ -00055350: 6e48 6f6d 6f6c 6f67 7920 2d2d 2048 6f6d  nHomology -- Hom
│ │ │ │ -00055360: 6f6c 6f67 7920 6f66 2061 2063 6f6d 706c  ology of a compl
│ │ │ │ -00055370: 6578 2061 7320 6578 7465 7269 6f72 206d  ex as exterior m
│ │ │ │ -00055380: 6f64 756c 650a 2a2a 2a2a 2a2a 2a2a 2a2a  odule.**********
│ │ │ │ +00055220: 2d2d 0a0a 5468 6520 736f 7572 6365 206f  --..The source o
│ │ │ │ +00055230: 6620 7468 6973 2064 6f63 756d 656e 7420  f this document 
│ │ │ │ +00055240: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570  is in./build/rep
│ │ │ │ +00055250: 726f 6475 6369 626c 652d 7061 7468 2f6d  roducible-path/m
│ │ │ │ +00055260: 6163 6175 6c61 7932 2d31 2e32 352e 3131  acaulay2-1.25.11
│ │ │ │ +00055270: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932  +ds/M2/Macaulay2
│ │ │ │ +00055280: 2f70 6163 6b61 6765 732f 0a43 6f6d 706c  /packages/.Compl
│ │ │ │ +00055290: 6574 6549 6e74 6572 7365 6374 696f 6e52  eteIntersectionR
│ │ │ │ +000552a0: 6573 6f6c 7574 696f 6e73 2e6d 323a 3338  esolutions.m2:38
│ │ │ │ +000552b0: 3031 3a30 2e0a 1f0a 4669 6c65 3a20 436f  01:0....File: Co
│ │ │ │ +000552c0: 6d70 6c65 7465 496e 7465 7273 6563 7469  mpleteIntersecti
│ │ │ │ +000552d0: 6f6e 5265 736f 6c75 7469 6f6e 732e 696e  onResolutions.in
│ │ │ │ +000552e0: 666f 2c20 4e6f 6465 3a20 6d61 6b65 486f  fo, Node: makeHo
│ │ │ │ +000552f0: 6d6f 746f 7069 6573 4f6e 486f 6d6f 6c6f  motopiesOnHomolo
│ │ │ │ +00055300: 6779 2c20 4e65 7874 3a20 6d61 6b65 4d6f  gy, Next: makeMo
│ │ │ │ +00055310: 6475 6c65 2c20 5072 6576 3a20 6d61 6b65  dule, Prev: make
│ │ │ │ +00055320: 486f 6d6f 746f 7069 6573 312c 2055 703a  Homotopies1, Up:
│ │ │ │ +00055330: 2054 6f70 0a0a 6d61 6b65 486f 6d6f 746f   Top..makeHomoto
│ │ │ │ +00055340: 7069 6573 4f6e 486f 6d6f 6c6f 6779 202d  piesOnHomology -
│ │ │ │ +00055350: 2d20 486f 6d6f 6c6f 6779 206f 6620 6120  - Homology of a 
│ │ │ │ +00055360: 636f 6d70 6c65 7820 6173 2065 7874 6572  complex as exter
│ │ │ │ +00055370: 696f 7220 6d6f 6475 6c65 0a2a 2a2a 2a2a  ior module.*****
│ │ │ │ +00055380: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00055390: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000553a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000553b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000553c0: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ -000553d0: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ -000553e0: 2848 2c68 2920 3d20 6d61 6b65 486f 6d6f  (H,h) = makeHomo
│ │ │ │ -000553f0: 746f 7069 6573 4f6e 486f 6d6f 6c6f 6779  topiesOnHomology
│ │ │ │ -00055400: 2866 662c 2043 290a 2020 2a20 496e 7075  (ff, C).  * Inpu
│ │ │ │ -00055410: 7473 3a0a 2020 2020 2020 2a20 6666 2c20  ts:.      * ff, 
│ │ │ │ -00055420: 6120 2a6e 6f74 6520 6d61 7472 6978 3a20  a *note matrix: 
│ │ │ │ -00055430: 284d 6163 6175 6c61 7932 446f 6329 4d61  (Macaulay2Doc)Ma
│ │ │ │ -00055440: 7472 6978 2c2c 206d 6174 7269 7820 6f66  trix,, matrix of
│ │ │ │ -00055450: 2065 6c65 6d65 6e74 7320 686f 6d6f 746f   elements homoto
│ │ │ │ -00055460: 7069 630a 2020 2020 2020 2020 746f 2030  pic.        to 0
│ │ │ │ -00055470: 206f 6e20 430a 2020 2020 2020 2a20 432c   on C.      * C,
│ │ │ │ -00055480: 2061 202a 6e6f 7465 2063 6f6d 706c 6578   a *note complex
│ │ │ │ -00055490: 3a20 2843 6f6d 706c 6578 6573 2943 6f6d  : (Complexes)Com
│ │ │ │ -000554a0: 706c 6578 2c2c 200a 2020 2a20 4f75 7470  plex,, .  * Outp
│ │ │ │ -000554b0: 7574 733a 0a20 2020 2020 202a 2048 2c20  uts:.      * H, 
│ │ │ │ -000554c0: 6120 2a6e 6f74 6520 6861 7368 2074 6162  a *note hash tab
│ │ │ │ -000554d0: 6c65 3a20 284d 6163 6175 6c61 7932 446f  le: (Macaulay2Do
│ │ │ │ -000554e0: 6329 4861 7368 5461 626c 652c 2c20 486f  c)HashTable,, Ho
│ │ │ │ -000554f0: 6d6f 6c6f 6779 206f 6620 432c 2069 6e64  mology of C, ind
│ │ │ │ -00055500: 6578 6564 0a20 2020 2020 2020 2062 7920  exed.        by 
│ │ │ │ -00055510: 706c 6163 6573 2069 6e20 7468 6520 430a  places in the C.
│ │ │ │ -00055520: 2020 2020 2020 2a20 682c 2061 202a 6e6f        * h, a *no
│ │ │ │ -00055530: 7465 2068 6173 6820 7461 626c 653a 2028  te hash table: (
│ │ │ │ -00055540: 4d61 6361 756c 6179 3244 6f63 2948 6173  Macaulay2Doc)Has
│ │ │ │ -00055550: 6854 6162 6c65 2c2c 2068 6f6d 6f74 6f70  hTable,, homotop
│ │ │ │ -00055560: 6965 7320 666f 720a 2020 2020 2020 2020  ies for.        
│ │ │ │ -00055570: 656c 656d 656e 7473 206f 6620 6620 6f6e  elements of f on
│ │ │ │ -00055580: 2074 6865 2068 6f6d 6f6c 6f67 7920 6f66   the homology of
│ │ │ │ -00055590: 2043 0a0a 4465 7363 7269 7074 696f 6e0a   C..Description.
│ │ │ │ -000555a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865  ===========..The
│ │ │ │ -000555b0: 2073 6372 6970 7420 6361 6c6c 7320 6d61   script calls ma
│ │ │ │ -000555c0: 6b65 486f 6d6f 746f 7069 6573 3120 746f  keHomotopies1 to
│ │ │ │ -000555d0: 2070 726f 6475 6365 2068 6f6d 6f74 6f70   produce homotop
│ │ │ │ -000555e0: 6965 7320 666f 7220 7468 6520 6666 5f69  ies for the ff_i
│ │ │ │ -000555f0: 206f 6e20 432c 2061 6e64 0a74 6865 6e20   on C, and.then 
│ │ │ │ -00055600: 636f 6d70 7574 6573 2074 6865 6972 2061  computes their a
│ │ │ │ -00055610: 6374 696f 6e20 6f6e 2074 6865 2048 6f6d  ction on the Hom
│ │ │ │ -00055620: 6f6c 6f67 7920 6f66 2043 2e0a 0a53 6565  ology of C...See
│ │ │ │ -00055630: 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a   also.========..
│ │ │ │ -00055640: 2020 2a20 2a6e 6f74 6520 6578 7465 7269    * *note exteri
│ │ │ │ -00055650: 6f72 546f 724d 6f64 756c 653a 2065 7874  orTorModule: ext
│ │ │ │ -00055660: 6572 696f 7254 6f72 4d6f 6475 6c65 2c20  eriorTorModule, 
│ │ │ │ -00055670: 2d2d 2054 6f72 2061 7320 6120 6d6f 6475  -- Tor as a modu
│ │ │ │ -00055680: 6c65 206f 7665 7220 616e 0a20 2020 2065  le over an.    e
│ │ │ │ -00055690: 7874 6572 696f 7220 616c 6765 6272 6120  xterior algebra 
│ │ │ │ -000556a0: 6f72 2062 6967 7261 6465 6420 616c 6765  or bigraded alge
│ │ │ │ -000556b0: 6272 610a 2020 2a20 2a6e 6f74 6520 6578  bra.  * *note ex
│ │ │ │ -000556c0: 7465 7269 6f72 4578 744d 6f64 756c 653a  teriorExtModule:
│ │ │ │ -000556d0: 2065 7874 6572 696f 7245 7874 4d6f 6475   exteriorExtModu
│ │ │ │ -000556e0: 6c65 2c20 2d2d 2045 7874 284d 2c6b 2920  le, -- Ext(M,k) 
│ │ │ │ -000556f0: 6f72 2045 7874 284d 2c4e 2920 6173 2061  or Ext(M,N) as a
│ │ │ │ -00055700: 0a20 2020 206d 6f64 756c 6520 6f76 6572  .    module over
│ │ │ │ -00055710: 2061 6e20 6578 7465 7269 6f72 2061 6c67   an exterior alg
│ │ │ │ -00055720: 6562 7261 0a0a 5761 7973 2074 6f20 7573  ebra..Ways to us
│ │ │ │ -00055730: 6520 6d61 6b65 486f 6d6f 746f 7069 6573  e makeHomotopies
│ │ │ │ -00055740: 4f6e 486f 6d6f 6c6f 6779 3a0a 3d3d 3d3d  OnHomology:.====
│ │ │ │ +000553b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a  ***************.
│ │ │ │ +000553c0: 0a20 202a 2055 7361 6765 3a20 0a20 2020  .  * Usage: .   
│ │ │ │ +000553d0: 2020 2020 2028 482c 6829 203d 206d 616b       (H,h) = mak
│ │ │ │ +000553e0: 6548 6f6d 6f74 6f70 6965 734f 6e48 6f6d  eHomotopiesOnHom
│ │ │ │ +000553f0: 6f6c 6f67 7928 6666 2c20 4329 0a20 202a  ology(ff, C).  *
│ │ │ │ +00055400: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ +00055410: 2066 662c 2061 202a 6e6f 7465 206d 6174   ff, a *note mat
│ │ │ │ +00055420: 7269 783a 2028 4d61 6361 756c 6179 3244  rix: (Macaulay2D
│ │ │ │ +00055430: 6f63 294d 6174 7269 782c 2c20 6d61 7472  oc)Matrix,, matr
│ │ │ │ +00055440: 6978 206f 6620 656c 656d 656e 7473 2068  ix of elements h
│ │ │ │ +00055450: 6f6d 6f74 6f70 6963 0a20 2020 2020 2020  omotopic.       
│ │ │ │ +00055460: 2074 6f20 3020 6f6e 2043 0a20 2020 2020   to 0 on C.     
│ │ │ │ +00055470: 202a 2043 2c20 6120 2a6e 6f74 6520 636f   * C, a *note co
│ │ │ │ +00055480: 6d70 6c65 783a 2028 436f 6d70 6c65 7865  mplex: (Complexe
│ │ │ │ +00055490: 7329 436f 6d70 6c65 782c 2c20 0a20 202a  s)Complex,, .  *
│ │ │ │ +000554a0: 204f 7574 7075 7473 3a0a 2020 2020 2020   Outputs:.      
│ │ │ │ +000554b0: 2a20 482c 2061 202a 6e6f 7465 2068 6173  * H, a *note has
│ │ │ │ +000554c0: 6820 7461 626c 653a 2028 4d61 6361 756c  h table: (Macaul
│ │ │ │ +000554d0: 6179 3244 6f63 2948 6173 6854 6162 6c65  ay2Doc)HashTable
│ │ │ │ +000554e0: 2c2c 2048 6f6d 6f6c 6f67 7920 6f66 2043  ,, Homology of C
│ │ │ │ +000554f0: 2c20 696e 6465 7865 640a 2020 2020 2020  , indexed.      
│ │ │ │ +00055500: 2020 6279 2070 6c61 6365 7320 696e 2074    by places in t
│ │ │ │ +00055510: 6865 2043 0a20 2020 2020 202a 2068 2c20  he C.      * h, 
│ │ │ │ +00055520: 6120 2a6e 6f74 6520 6861 7368 2074 6162  a *note hash tab
│ │ │ │ +00055530: 6c65 3a20 284d 6163 6175 6c61 7932 446f  le: (Macaulay2Do
│ │ │ │ +00055540: 6329 4861 7368 5461 626c 652c 2c20 686f  c)HashTable,, ho
│ │ │ │ +00055550: 6d6f 746f 7069 6573 2066 6f72 0a20 2020  motopies for.   
│ │ │ │ +00055560: 2020 2020 2065 6c65 6d65 6e74 7320 6f66       elements of
│ │ │ │ +00055570: 2066 206f 6e20 7468 6520 686f 6d6f 6c6f   f on the homolo
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│ │ │ │ +000555c0: 7331 2074 6f20 7072 6f64 7563 6520 686f  s1 to produce ho
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│ │ │ │ +00055640: 7874 6572 696f 7254 6f72 4d6f 6475 6c65  xteriorTorModule
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│ │ │ │ +00055710: 7220 616c 6765 6272 610a 0a57 6179 7320  r algebra..Ways 
│ │ │ │ +00055720: 746f 2075 7365 206d 616b 6548 6f6d 6f74  to use makeHomot
│ │ │ │ +00055730: 6f70 6965 734f 6e48 6f6d 6f6c 6f67 793a  opiesOnHomology:
│ │ │ │ +00055740: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
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│ │ │ │ -00055760: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00055770: 3d0a 0a20 202a 2022 6d61 6b65 486f 6d6f  =..  * "makeHomo
│ │ │ │ -00055780: 746f 7069 6573 4f6e 486f 6d6f 6c6f 6779  topiesOnHomology
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│ │ │ │ -000557a0: 220a 0a46 6f72 2074 6865 2070 726f 6772  "..For the progr
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│ │ │ │ -00055800: 6965 734f 6e48 6f6d 6f6c 6f67 792c 2069  iesOnHomology, i
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│ │ │ │ -00055820: 2066 756e 6374 696f 6e3a 2028 4d61 6361   function: (Maca
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│ │ │ │ +00055760: 3d3d 3d3d 3d3d 0a0a 2020 2a20 226d 616b  ======..  * "mak
│ │ │ │ +00055770: 6548 6f6d 6f74 6f70 6965 734f 6e48 6f6d  eHomotopiesOnHom
│ │ │ │ +00055780: 6f6c 6f67 7928 4d61 7472 6978 2c43 6f6d  ology(Matrix,Com
│ │ │ │ +00055790: 706c 6578 2922 0a0a 466f 7220 7468 6520  plex)"..For the 
│ │ │ │ +000557a0: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d  programmer.=====
│ │ │ │ +000557b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  =============..T
│ │ │ │ +000557c0: 6865 206f 626a 6563 7420 2a6e 6f74 6520  he object *note 
│ │ │ │ +000557d0: 6d61 6b65 486f 6d6f 746f 7069 6573 4f6e  makeHomotopiesOn
│ │ │ │ +000557e0: 486f 6d6f 6c6f 6779 3a20 6d61 6b65 486f  Homology: makeHo
│ │ │ │ +000557f0: 6d6f 746f 7069 6573 4f6e 486f 6d6f 6c6f  motopiesOnHomolo
│ │ │ │ +00055800: 6779 2c20 6973 2061 202a 6e6f 7465 0a6d  gy, is a *note.m
│ │ │ │ +00055810: 6574 686f 6420 6675 6e63 7469 6f6e 3a20  ethod function: 
│ │ │ │ +00055820: 284d 6163 6175 6c61 7932 446f 6329 4d65  (Macaulay2Doc)Me
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│ │ │ │  00055850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00055860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00055870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00055880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00055890: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520  ----------..The 
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│ │ │ │ -000558e0: 2d31 2e32 352e 3131 2b64 732f 4d32 2f4d  -1.25.11+ds/M2/M
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│ │ │ │ -00055980: 206d 616b 6548 6f6d 6f74 6f70 6965 734f   makeHomotopiesO
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│ │ │ │ -000559a0: 6f70 0a0a 6d61 6b65 4d6f 6475 6c65 202d  op..makeModule -
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│ │ │ │ -000559d0: 7469 6f6e 206f 6620 6d6f 6475 6c65 7320  tion of modules 
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│ │ │ │ +00055880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a  ---------------.
│ │ │ │ +00055890: 0a54 6865 2073 6f75 7263 6520 6f66 2074  .The source of t
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│ │ │ │ +000558c0: 7563 6962 6c65 2d70 6174 682f 6d61 6361  ucible-path/maca
│ │ │ │ +000558d0: 756c 6179 322d 312e 3235 2e31 312b 6473  ulay2-1.25.11+ds
│ │ │ │ +000558e0: 2f4d 322f 4d61 6361 756c 6179 322f 7061  /M2/Macaulay2/pa
│ │ │ │ +000558f0: 636b 6167 6573 2f0a 436f 6d70 6c65 7465  ckages/.Complete
│ │ │ │ +00055900: 496e 7465 7273 6563 7469 6f6e 5265 736f  IntersectionReso
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│ │ │ │ +00055960: 652c 204e 6578 743a 206d 616b 6554 2c20  e, Next: makeT, 
│ │ │ │ +00055970: 5072 6576 3a20 6d61 6b65 486f 6d6f 746f  Prev: makeHomoto
│ │ │ │ +00055980: 7069 6573 4f6e 486f 6d6f 6c6f 6779 2c20  piesOnHomology, 
│ │ │ │ +00055990: 5570 3a20 546f 700a 0a6d 616b 654d 6f64  Up: Top..makeMod
│ │ │ │ +000559a0: 756c 6520 2d2d 206d 616b 6573 2061 204d  ule -- makes a M
│ │ │ │ +000559b0: 6f64 756c 6520 6f75 7420 6f66 2061 2063  odule out of a c
│ │ │ │ +000559c0: 6f6c 6c65 6374 696f 6e20 6f66 206d 6f64  ollection of mod
│ │ │ │ +000559d0: 756c 6573 2061 6e64 206d 6170 730a 2a2a  ules and maps.**
│ │ │ │ +000559e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000559f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00055a00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00055a10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00055a20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20  *************.. 
│ │ │ │ -00055a30: 202a 2055 7361 6765 3a20 0a20 2020 2020   * Usage: .     
│ │ │ │ -00055a40: 2020 204d 203d 206d 616b 654d 6f64 756c     M = makeModul
│ │ │ │ -00055a50: 6528 482c 452c 7068 6929 0a20 202a 2049  e(H,E,phi).  * I
│ │ │ │ -00055a60: 6e70 7574 733a 0a20 2020 2020 202a 2048  nputs:.      * H
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│ │ │ │ -00055aa0: 6772 6164 6564 2063 6f6d 706f 6e65 6e74  graded component
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│ │ │ │ -00055bd0: 6f66 2074 6865 2076 6172 6961 626c 6573  of the variables
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│ │ │ │ +00055a30: 2020 2020 2020 2020 4d20 3d20 6d61 6b65          M = make
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│ │ │ │ +00055a50: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ +00055a60: 2020 2a20 482c 2061 202a 6e6f 7465 2068    * H, a *note h
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│ │ │ │ +00055a80: 756c 6179 3244 6f63 2948 6173 6854 6162  ulay2Doc)HashTab
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│ │ │ │ +00055ae0: 2020 2020 202a 2045 2c20 6120 2a6e 6f74       * E, a *not
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│ │ │ │ +00055b50: 6869 2c20 6120 2a6e 6f74 6520 6861 7368  hi, a *note hash
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│ │ │ │ +00055bb0: 7420 7769 6c6c 2062 6520 7468 6520 6163  t will be the ac
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│ │ │ │ +00055bf0: 4d2c 2061 202a 6e6f 7465 206d 6f64 756c  M, a *note modul
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│ │ │ │ +00055c80: 6420 6861 7665 2063 6f6e 7365 6375 7469  d have consecuti
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│ │ │ │ +00055ca0: 695f 302e 2e69 5f30 2c20 7361 792c 2077  i_0..i_0, say, w
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│ │ │ │  00056160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00056190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -000561f0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00056190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +000561a0: 3120 3a20 5345 203d 205a 5a2f 3130 315b  1 : SE = ZZ/101[
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│ │ │ │ +000561f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056240: 2020 2020 2020 207c 0a7c 6f31 203d 2053         |.|o1 = S
│ │ │ │ -00056250: 4520 2020 2020 2020 2020 2020 2020 2020  E               
│ │ │ │ +00056230: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00056240: 3120 3d20 5345 2020 2020 2020 2020 2020  1 = SE          
│ │ │ │ +00056250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056290: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00056280: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00056290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000562a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000562b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000562c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000562d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000562e0: 2020 2020 2020 207c 0a7c 6f31 203a 2050         |.|o1 : P
│ │ │ │ -000562f0: 6f6c 796e 6f6d 6961 6c52 696e 672c 2032  olynomialRing, 2
│ │ │ │ -00056300: 2073 6b65 7720 636f 6d6d 7574 6174 6976   skew commutativ
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│ │ │ │ -00056330: 2020 2020 2020 207c 0a7c 2d2d 2d2d 2d2d         |.|------
│ │ │ │ +000562d0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +000562e0: 3120 3a20 506f 6c79 6e6f 6d69 616c 5269  1 : PolynomialRi
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│ │ │ │ +00056310: 7329 2020 2020 2020 2020 2020 2020 2020  s)              
│ │ │ │ +00056320: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ +00056330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  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  ----------------
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│ │ │ │ +00056370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c53  ------------|.|S
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│ │ │ │ +000563a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000563b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000563c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000563d0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +000563c0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
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│ │ │ │  000563f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00056440: 6232 2c63 3222 2020 2020 2020 2020 2020  b2,c2"          
│ │ │ │ +00056410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00056420: 3220 3a20 5245 203d 2053 452f 6964 6561  2 : RE = SE/idea
│ │ │ │ +00056430: 6c22 6132 2c62 322c 6332 2220 2020 2020  l"a2,b2,c2"     
│ │ │ │ +00056440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056470: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00056460: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00056470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000564a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000564b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000564c0: 2020 2020 2020 207c 0a7c 6f32 203d 2052         |.|o2 = R
│ │ │ │ -000564d0: 4520 2020 2020 2020 2020 2020 2020 2020  E               
│ │ │ │ +000564b0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +000564c0: 3220 3d20 5245 2020 2020 2020 2020 2020  2 = RE          
│ │ │ │ +000564d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000564e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000564f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056510: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00056500: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00056510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056560: 2020 2020 2020 207c 0a7c 6f32 203a 2051         |.|o2 : Q
│ │ │ │ -00056570: 756f 7469 656e 7452 696e 6720 2020 2020  uotientRing     
│ │ │ │ +00056550: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00056560: 3220 3a20 5175 6f74 6965 6e74 5269 6e67  2 : QuotientRing
│ │ │ │ +00056570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000565a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000565b0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +000565a0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +000565b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000565c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000565d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000565e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000565f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056600: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2054  -------+.|i3 : T
│ │ │ │ -00056610: 203d 2068 6173 6854 6162 6c65 207b 7b30   = hashTable {{0
│ │ │ │ -00056620: 2c52 455e 317d 2c7b 312c 5245 5e7b 323a  ,RE^1},{1,RE^{2:
│ │ │ │ -00056630: 7b20 2d31 2c2d 317d 7d7d 2c20 7b32 2c52  { -1,-1}}}, {2,R
│ │ │ │ -00056640: 455e 7b7b 202d 322c 2d32 7d7d 7d7d 2020  E^{{ -2,-2}}}}  
│ │ │ │ -00056650: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000565f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00056600: 3320 3a20 5420 3d20 6861 7368 5461 626c  3 : T = hashTabl
│ │ │ │ +00056610: 6520 7b7b 302c 5245 5e31 7d2c 7b31 2c52  e {{0,RE^1},{1,R
│ │ │ │ +00056620: 455e 7b32 3a7b 202d 312c 2d31 7d7d 7d2c  E^{2:{ -1,-1}}},
│ │ │ │ +00056630: 207b 322c 5245 5e7b 7b20 2d32 2c2d 327d   {2,RE^{{ -2,-2}
│ │ │ │ +00056640: 7d7d 7d20 2020 2020 2020 2020 7c0a 7c20  }}}         |.| 
│ │ │ │ +00056650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000566a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000566b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000566c0: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +00056690: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000566a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000566b0: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ +000566c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000566d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000566e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000566f0: 2020 2020 2020 207c 0a7c 6f33 203d 2048         |.|o3 = H
│ │ │ │ -00056700: 6173 6854 6162 6c65 7b30 203d 3e20 5245  ashTable{0 => RE
│ │ │ │ -00056710: 207d 2020 2020 2020 2020 2020 2020 2020   }              
│ │ │ │ +000566e0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +000566f0: 3320 3d20 4861 7368 5461 626c 657b 3020  3 = HashTable{0 
│ │ │ │ +00056700: 3d3e 2052 4520 7d20 2020 2020 2020 2020  => RE }         
│ │ │ │ +00056710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056740: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00056750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056760: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00056730: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00056740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056750: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +00056760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056790: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000567a0: 2020 2020 2020 2020 2031 203d 3e20 5245           1 => RE
│ │ │ │ +00056780: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00056790: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ +000567a0: 3d3e 2052 4520 2020 2020 2020 2020 2020  => RE           
│ │ │ │  000567b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000567c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000567d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000567e0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000567f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056800: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +000567d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000567e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000567f0: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ +00056800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056830: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00056840: 2020 2020 2020 2020 2032 203d 3e20 5245           2 => RE
│ │ │ │ +00056820: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00056830: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +00056840: 3d3e 2052 4520 2020 2020 2020 2020 2020  => RE           
│ │ │ │  00056850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056880: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00056870: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00056880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000568a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000568b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000568c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000568d0: 2020 2020 2020 207c 0a7c 6f33 203a 2048         |.|o3 : H
│ │ │ │ -000568e0: 6173 6854 6162 6c65 2020 2020 2020 2020  ashTable        
│ │ │ │ +000568c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +000568d0: 3320 3a20 4861 7368 5461 626c 6520 2020  3 : HashTable   
│ │ │ │ +000568e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000568f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056920: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00056910: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00056920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056970: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2045  -------+.|i4 : E
│ │ │ │ -00056980: 203d 206d 6174 7269 787b 7b78 2c79 7d7d   = matrix{{x,y}}
│ │ │ │ +00056960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00056970: 3420 3a20 4520 3d20 6d61 7472 6978 7b7b  4 : E = matrix{{
│ │ │ │ +00056980: 782c 797d 7d20 2020 2020 2020 2020 2020  x,y}}           
│ │ │ │  00056990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000569a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000569b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000569c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000569b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000569c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000569d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000569e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000569f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056a10: 2020 2020 2020 207c 0a7c 6f34 203d 207c         |.|o4 = |
│ │ │ │ -00056a20: 2078 2079 207c 2020 2020 2020 2020 2020   x y |          
│ │ │ │ +00056a00: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00056a10: 3420 3d20 7c20 7820 7920 7c20 2020 2020  4 = | x y |     
│ │ │ │ +00056a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056a60: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00056a50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00056a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056ab0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00056ac0: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ -00056ad0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00056aa0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00056ab0: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │ +00056ac0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +00056ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056b00: 2020 2020 2020 207c 0a7c 6f34 203a 204d         |.|o4 : M
│ │ │ │ -00056b10: 6174 7269 7820 5245 2020 3c2d 2d20 5245  atrix RE  <-- RE
│ │ │ │ +00056af0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00056b00: 3420 3a20 4d61 7472 6978 2052 4520 203c  4 : Matrix RE  <
│ │ │ │ +00056b10: 2d2d 2052 4520 2020 2020 2020 2020 2020  -- RE           
│ │ │ │  00056b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056b50: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00056b40: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00056b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056ba0: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2046  -------+.|i5 : F
│ │ │ │ -00056bb0: 3d61 7070 6c79 2832 2c20 6a2d 3e20 736f  =apply(2, j-> so
│ │ │ │ -00056bc0: 7572 6365 2045 5f7b 6a7d 2920 2020 2020  urce E_{j})     
│ │ │ │ +00056b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00056ba0: 3520 3a20 463d 6170 706c 7928 322c 206a  5 : F=apply(2, j
│ │ │ │ +00056bb0: 2d3e 2073 6f75 7263 6520 455f 7b6a 7d29  -> source E_{j})
│ │ │ │ +00056bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056bf0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00056be0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00056bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056c40: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00056c50: 2020 3120 2020 2031 2020 2020 2020 2020    1    1        
│ │ │ │ +00056c30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00056c40: 2020 2020 2020 2031 2020 2020 3120 2020         1    1   
│ │ │ │ +00056c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056c90: 2020 2020 2020 207c 0a7c 6f35 203d 207b         |.|o5 = {
│ │ │ │ -00056ca0: 5245 202c 2052 4520 7d20 2020 2020 2020  RE , RE }       
│ │ │ │ +00056c80: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00056c90: 3520 3d20 7b52 4520 2c20 5245 207d 2020  5 = {RE , RE }  
│ │ │ │ +00056ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056ce0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00056cd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00056ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056d30: 2020 2020 2020 207c 0a7c 6f35 203a 204c         |.|o5 : L
│ │ │ │ -00056d40: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ +00056d20: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00056d30: 3520 3a20 4c69 7374 2020 2020 2020 2020  5 : List        
│ │ │ │ +00056d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056d80: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00056d70: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00056d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056dd0: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 2070  -------+.|i6 : p
│ │ │ │ -00056de0: 6869 203d 2068 6173 6854 6162 6c65 7b20  hi = hashTable{ 
│ │ │ │ -00056df0: 7b7b 302c 307d 2c20 6d61 7028 5423 312c  {{0,0}, map(T#1,
│ │ │ │ -00056e00: 2046 5f30 2a2a 5423 302c 2054 2331 5f7b   F_0**T#0, T#1_{
│ │ │ │ -00056e10: 307d 297d 2c7b 7b31 2c30 7d2c 206d 6170  0})},{{1,0}, map
│ │ │ │ -00056e20: 2854 2331 2c20 207c 0a7c 2020 2020 2020  (T#1,  |.|      
│ │ │ │ +00056dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00056dd0: 3620 3a20 7068 6920 3d20 6861 7368 5461  6 : phi = hashTa
│ │ │ │ +00056de0: 626c 657b 207b 7b30 2c30 7d2c 206d 6170  ble{ {{0,0}, map
│ │ │ │ +00056df0: 2854 2331 2c20 465f 302a 2a54 2330 2c20  (T#1, F_0**T#0, 
│ │ │ │ +00056e00: 5423 315f 7b30 7d29 7d2c 7b7b 312c 307d  T#1_{0})},{{1,0}
│ │ │ │ +00056e10: 2c20 6d61 7028 5423 312c 2020 7c0a 7c20  , map(T#1,  |.| 
│ │ │ │ +00056e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056e70: 2020 2020 2020 207c 0a7c 6f36 203d 2048         |.|o6 = H
│ │ │ │ -00056e80: 6173 6854 6162 6c65 7b7b 302c 2030 7d20  ashTable{{0, 0} 
│ │ │ │ -00056e90: 3d3e 207b 312c 2031 7d20 7c20 3120 7c20  => {1, 1} | 1 | 
│ │ │ │ -00056ea0: 2020 7d20 2020 2020 2020 2020 2020 2020    }             
│ │ │ │ -00056eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056ec0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00056ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056ee0: 2020 207b 312c 2031 7d20 7c20 3020 7c20     {1, 1} | 0 | 
│ │ │ │ +00056e60: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00056e70: 3620 3d20 4861 7368 5461 626c 657b 7b30  6 = HashTable{{0
│ │ │ │ +00056e80: 2c20 307d 203d 3e20 7b31 2c20 317d 207c  , 0} => {1, 1} |
│ │ │ │ +00056e90: 2031 207c 2020 207d 2020 2020 2020 2020   1 |   }        
│ │ │ │ +00056ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056eb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00056ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056ed0: 2020 2020 2020 2020 7b31 2c20 317d 207c          {1, 1} |
│ │ │ │ +00056ee0: 2030 207c 2020 2020 2020 2020 2020 2020   0 |            
│ │ │ │  00056ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056f10: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00056f20: 2020 2020 2020 2020 207b 302c 2031 7d20           {0, 1} 
│ │ │ │ -00056f30: 3d3e 207b 322c 2032 7d20 7c20 3020 3120  => {2, 2} | 0 1 
│ │ │ │ -00056f40: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00056f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056f60: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00056f70: 2020 2020 2020 2020 207b 312c 2030 7d20           {1, 0} 
│ │ │ │ -00056f80: 3d3e 207b 312c 2031 7d20 7c20 3020 7c20  => {1, 1} | 0 | 
│ │ │ │ +00056f00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00056f10: 2020 2020 2020 2020 2020 2020 2020 7b30                {0
│ │ │ │ +00056f20: 2c20 317d 203d 3e20 7b32 2c20 327d 207c  , 1} => {2, 2} |
│ │ │ │ +00056f30: 2030 2031 207c 2020 2020 2020 2020 2020   0 1 |          
│ │ │ │ +00056f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056f50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00056f60: 2020 2020 2020 2020 2020 2020 2020 7b31                {1
│ │ │ │ +00056f70: 2c20 307d 203d 3e20 7b31 2c20 317d 207c  , 0} => {1, 1} |
│ │ │ │ +00056f80: 2030 207c 2020 2020 2020 2020 2020 2020   0 |            
│ │ │ │  00056f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056fb0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00056fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056fd0: 2020 207b 312c 2031 7d20 7c20 3120 7c20     {1, 1} | 1 | 
│ │ │ │ +00056fa0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00056fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056fc0: 2020 2020 2020 2020 7b31 2c20 317d 207c          {1, 1} |
│ │ │ │ +00056fd0: 2031 207c 2020 2020 2020 2020 2020 2020   1 |            
│ │ │ │  00056fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057000: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00057010: 2020 2020 2020 2020 207b 312c 2031 7d20           {1, 1} 
│ │ │ │ -00057020: 3d3e 207b 322c 2032 7d20 7c20 2d31 2030  => {2, 2} | -1 0
│ │ │ │ -00057030: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ -00057040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057050: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00056ff0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00057000: 2020 2020 2020 2020 2020 2020 2020 7b31                {1
│ │ │ │ +00057010: 2c20 317d 203d 3e20 7b32 2c20 327d 207c  , 1} => {2, 2} |
│ │ │ │ +00057020: 202d 3120 3020 7c20 2020 2020 2020 2020   -1 0 |         
│ │ │ │ +00057030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057040: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00057050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000570a0: 2020 2020 2020 207c 0a7c 6f36 203a 2048         |.|o6 : H
│ │ │ │ -000570b0: 6173 6854 6162 6c65 2020 2020 2020 2020  ashTable        
│ │ │ │ +00057090: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +000570a0: 3620 3a20 4861 7368 5461 626c 6520 2020  6 : HashTable   
│ │ │ │ +000570b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000570c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000570d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000570e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000570f0: 2020 2020 2020 207c 0a7c 2d2d 2d2d 2d2d         |.|------
│ │ │ │ +000570e0: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ +000570f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057140: 2d2d 2d2d 2d2d 2d7c 0a7c 465f 312a 2a54  -------|.|F_1**T
│ │ │ │ -00057150: 2330 2c20 5423 315f 7b31 7d29 7d2c 7b7b  #0, T#1_{1})},{{
│ │ │ │ -00057160: 302c 317d 2c20 6d61 7028 5423 322c 2046  0,1}, map(T#2, F
│ │ │ │ -00057170: 5f30 2a2a 5423 312c 2054 2331 5e7b 317d  _0**T#1, T#1^{1}
│ │ │ │ -00057180: 297d 2c20 7b7b 312c 317d 2c20 2d6d 6170  )}, {{1,1}, -map
│ │ │ │ -00057190: 2854 2332 2c20 207c 0a7c 2d2d 2d2d 2d2d  (T#2,  |.|------
│ │ │ │ +00057130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c46  ------------|.|F
│ │ │ │ +00057140: 5f31 2a2a 5423 302c 2054 2331 5f7b 317d  _1**T#0, T#1_{1}
│ │ │ │ +00057150: 297d 2c7b 7b30 2c31 7d2c 206d 6170 2854  )},{{0,1}, map(T
│ │ │ │ +00057160: 2332 2c20 465f 302a 2a54 2331 2c20 5423  #2, F_0**T#1, T#
│ │ │ │ +00057170: 315e 7b31 7d29 7d2c 207b 7b31 2c31 7d2c  1^{1})}, {{1,1},
│ │ │ │ +00057180: 202d 6d61 7028 5423 322c 2020 7c0a 7c2d   -map(T#2,  |.|-
│ │ │ │ +00057190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000571a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000571b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000571c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000571d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000571e0: 2d2d 2d2d 2d2d 2d7c 0a7c 465f 312a 2a54  -------|.|F_1**T
│ │ │ │ -000571f0: 2331 2c20 5423 315e 7b30 7d29 7d7d 2020  #1, T#1^{0})}}  
│ │ │ │ +000571d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c46  ------------|.|F
│ │ │ │ +000571e0: 5f31 2a2a 5423 312c 2054 2331 5e7b 307d  _1**T#1, T#1^{0}
│ │ │ │ +000571f0: 297d 7d20 2020 2020 2020 2020 2020 2020  )}}             
│ │ │ │  00057200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057230: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00057220: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00057230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057280: 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a 2061  -------+.|i7 : a
│ │ │ │ -00057290: 7070 6c79 286b 6579 7320 7068 692c 206b  pply(keys phi, k
│ │ │ │ -000572a0: 2d3e 6973 486f 6d6f 6765 6e65 6f75 7320  ->isHomogeneous 
│ │ │ │ -000572b0: 7068 6923 6b29 2020 2020 2020 2020 2020  phi#k)          
│ │ │ │ -000572c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000572d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00057270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00057280: 3720 3a20 6170 706c 7928 6b65 7973 2070  7 : apply(keys p
│ │ │ │ +00057290: 6869 2c20 6b2d 3e69 7348 6f6d 6f67 656e  hi, k->isHomogen
│ │ │ │ +000572a0: 656f 7573 2070 6869 236b 2920 2020 2020  eous phi#k)     
│ │ │ │ +000572b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000572c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000572d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000572e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000572f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057320: 2020 2020 2020 207c 0a7c 6f37 203d 207b         |.|o7 = {
│ │ │ │ -00057330: 7472 7565 2c20 7472 7565 2c20 7472 7565  true, true, true
│ │ │ │ -00057340: 2c20 7472 7565 7d20 2020 2020 2020 2020  , true}         
│ │ │ │ +00057310: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00057320: 3720 3d20 7b74 7275 652c 2074 7275 652c  7 = {true, true,
│ │ │ │ +00057330: 2074 7275 652c 2074 7275 657d 2020 2020   true, true}    
│ │ │ │ +00057340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057370: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00057360: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00057370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000573a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000573b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000573c0: 2020 2020 2020 207c 0a7c 6f37 203a 204c         |.|o7 : L
│ │ │ │ -000573d0: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ +000573b0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +000573c0: 3720 3a20 4c69 7374 2020 2020 2020 2020  7 : List        
│ │ │ │ +000573d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000573e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000573f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057410: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00057400: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00057410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057460: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 2058  -------+.|i8 : X
│ │ │ │ -00057470: 203d 206d 616b 654d 6f64 756c 6528 542c   = makeModule(T,
│ │ │ │ -00057480: 452c 7068 6929 2020 2020 2020 2020 2020  E,phi)          
│ │ │ │ +00057450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00057460: 3820 3a20 5820 3d20 6d61 6b65 4d6f 6475  8 : X = makeModu
│ │ │ │ +00057470: 6c65 2854 2c45 2c70 6869 2920 2020 2020  le(T,E,phi)     
│ │ │ │ +00057480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000574a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000574b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000574a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000574b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000574c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000574d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000574e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000574f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057500: 2020 2020 2020 207c 0a7c 6f38 203d 2063         |.|o8 = c
│ │ │ │ -00057510: 6f6b 6572 6e65 6c20 7b30 2c20 307d 207c  okernel {0, 0} |
│ │ │ │ -00057520: 202d 7820 3020 2030 2020 2d79 2030 2020   -x 0  0  -y 0  
│ │ │ │ -00057530: 3020 207c 2020 2020 2020 2020 2020 2020  0  |            
│ │ │ │ -00057540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057550: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00057560: 2020 2020 2020 2020 7b31 2c20 317d 207c          {1, 1} |
│ │ │ │ -00057570: 2031 2020 2d78 2030 2020 3020 202d 7920   1  -x 0  0  -y 
│ │ │ │ -00057580: 3020 207c 2020 2020 2020 2020 2020 2020  0  |            
│ │ │ │ -00057590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000575a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000575b0: 2020 2020 2020 2020 7b31 2c20 317d 207c          {1, 1} |
│ │ │ │ -000575c0: 2030 2020 3020 202d 7820 3120 2030 2020   0  0  -x 1  0  
│ │ │ │ -000575d0: 2d79 207c 2020 2020 2020 2020 2020 2020  -y |            
│ │ │ │ -000575e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000575f0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00057600: 2020 2020 2020 2020 7b32 2c20 327d 207c          {2, 2} |
│ │ │ │ -00057610: 2030 2020 3020 2031 2020 3020 202d 3120   0  0  1  0  -1 
│ │ │ │ -00057620: 3020 207c 2020 2020 2020 2020 2020 2020  0  |            
│ │ │ │ -00057630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057640: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000574f0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00057500: 3820 3d20 636f 6b65 726e 656c 207b 302c  8 = cokernel {0,
│ │ │ │ +00057510: 2030 7d20 7c20 2d78 2030 2020 3020 202d   0} | -x 0  0  -
│ │ │ │ +00057520: 7920 3020 2030 2020 7c20 2020 2020 2020  y 0  0  |       
│ │ │ │ +00057530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057540: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00057550: 2020 2020 2020 2020 2020 2020 207b 312c               {1,
│ │ │ │ +00057560: 2031 7d20 7c20 3120 202d 7820 3020 2030   1} | 1  -x 0  0
│ │ │ │ +00057570: 2020 2d79 2030 2020 7c20 2020 2020 2020    -y 0  |       
│ │ │ │ +00057580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057590: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000575a0: 2020 2020 2020 2020 2020 2020 207b 312c               {1,
│ │ │ │ +000575b0: 2031 7d20 7c20 3020 2030 2020 2d78 2031   1} | 0  0  -x 1
│ │ │ │ +000575c0: 2020 3020 202d 7920 7c20 2020 2020 2020    0  -y |       
│ │ │ │ +000575d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000575e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000575f0: 2020 2020 2020 2020 2020 2020 207b 322c               {2,
│ │ │ │ +00057600: 2032 7d20 7c20 3020 2030 2020 3120 2030   2} | 0  0  1  0
│ │ │ │ +00057610: 2020 2d31 2030 2020 7c20 2020 2020 2020    -1 0  |       
│ │ │ │ +00057620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057630: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00057640: 2020 2020 2020 2020 2020 2020 2020 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 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000576a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000576b0: 2020 2020 2020 2020 3420 2020 2020 2020          4       
│ │ │ │ +00057680: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00057690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000576a0: 2020 2020 2020 2020 2020 2020 2034 2020               4  
│ │ │ │ +000576b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000576c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000576d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000576e0: 2020 2020 2020 207c 0a7c 6f38 203a 2052         |.|o8 : R
│ │ │ │ -000576f0: 452d 6d6f 6475 6c65 2c20 7175 6f74 6965  E-module, quotie
│ │ │ │ -00057700: 6e74 206f 6620 5245 2020 2020 2020 2020  nt of RE        
│ │ │ │ +000576d0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +000576e0: 3820 3a20 5245 2d6d 6f64 756c 652c 2071  8 : RE-module, q
│ │ │ │ +000576f0: 756f 7469 656e 7420 6f66 2052 4520 2020  uotient of RE   
│ │ │ │ +00057700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057730: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00057720: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00057730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057780: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2069  -------+.|i9 : i
│ │ │ │ -00057790: 7348 6f6d 6f67 656e 656f 7573 2058 2020  sHomogeneous X  
│ │ │ │ +00057770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00057780: 3920 3a20 6973 486f 6d6f 6765 6e65 6f75  9 : isHomogeneou
│ │ │ │ +00057790: 7320 5820 2020 2020 2020 2020 2020 2020  s X             
│ │ │ │  000577a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000577b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000577c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000577d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000577c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000577d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000577e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000577f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057820: 2020 2020 2020 207c 0a7c 6f39 203d 2074         |.|o9 = t
│ │ │ │ -00057830: 7275 6520 2020 2020 2020 2020 2020 2020  rue             
│ │ │ │ +00057810: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00057820: 3920 3d20 7472 7565 2020 2020 2020 2020  9 = true        
│ │ │ │ +00057830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057870: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00057860: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00057870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000578a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000578b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000578c0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20  -------+.|i10 : 
│ │ │ │ -000578d0: 7120 3d20 6d61 7028 5a5a 2f31 3031 5b78  q = map(ZZ/101[x
│ │ │ │ -000578e0: 2c79 2c20 536b 6577 436f 6d6d 7574 6174  ,y, SkewCommutat
│ │ │ │ -000578f0: 6976 6520 3d3e 2074 7275 652c 2044 6567  ive => true, Deg
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│ │ │ │ +000578c0: 3130 203a 2071 203d 206d 6170 285a 5a2f  10 : q = map(ZZ/
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│ │ │ │ +00057910: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00057970: 2020 2020 2020 5a5a 2020 2020 2020 2020        ZZ        
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│ │ │ │ +00057970: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000579f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00057a10: 2020 2020 2031 3031 2020 2020 2020 2020       101        
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│ │ │ │ +000579d0: 302c 2078 2c20 797d 2920 2020 2020 2020  0, x, y})       
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│ │ │ │ +00057a90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00057aa0: 2020 2020 2020 2020 2020 2020 2020 5a5a                ZZ
│ │ │ │ +00057ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00057ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00057af0: 3130 203a 2052 696e 674d 6170 202d 2d2d  10 : RingMap ---
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│ │ │ │ -00057b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057b40: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00057b50: 2020 2020 2020 2020 3130 3120 2020 2020          101     
│ │ │ │ +00057b30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00057b40: 2020 2020 2020 2020 2020 2020 2031 3031               101
│ │ │ │ +00057b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00057b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00057bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  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                  
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│ │ │ │ +00057c20: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00057c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00057ca0: 5820 2020 2020 2020 2020 2020 2020 2020  X               
│ │ │ │  00057cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057cd0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00057cc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00057cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00057d30: 2020 5a5a 2020 2020 2020 2031 2020 2020    ZZ       1    
│ │ │ │ +00057d10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00057d20: 2020 2020 2020 205a 5a20 2020 2020 2020         ZZ       
│ │ │ │ +00057d30: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │  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                  
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│ │ │ │ +00057d60: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
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│ │ │ │ +00057d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057dc0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00057dd0: 2031 3031 2020 2020 2020 2020 2020 2020   101            
│ │ │ │ +00057db0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00057dc0: 2020 2020 2020 3130 3120 2020 2020 2020        101       
│ │ │ │ +00057dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00057e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057e10: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00057e00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00057e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057e60: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00057e70: 205a 5a20 2020 2020 2020 2020 2020 2020   ZZ             
│ │ │ │ +00057e50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00057e60: 2020 2020 2020 5a5a 2020 2020 2020 2020        ZZ        
│ │ │ │ +00057e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057e80: 2020 2020 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 2020 207c 0a7c 6f31 3120 3a20         |.|o11 : 
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│ │ │ │ -00057ed0: 2c20 6672 6565 2020 2020 2020 2020 2020  , free          
│ │ │ │ +00057ea0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00057eb0: 3131 203a 202d 2d2d 5b78 2e2e 795d 2d6d  11 : ---[x..y]-m
│ │ │ │ +00057ec0: 6f64 756c 652c 2066 7265 6520 2020 2020  odule, free     
│ │ │ │ +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 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00057f10: 3130 3120 2020 2020 2020 2020 2020 2020  101             
│ │ │ │ +00057ef0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00057f00: 2020 2020 2031 3031 2020 2020 2020 2020       101        
│ │ │ │ +00057f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057f50: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00057f40: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00057f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 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  ----------------
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│ │ │ │ +000580e0: 4578 7428 4d2c 6b29 206f 7220 4578 7428  Ext(M,k) or Ext(
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│ │ │ │ +00058100: 6475 6c65 206f 7665 7220 616e 2065 7874  dule over an ext
│ │ │ │ +00058110: 6572 696f 7220 616c 6765 6272 610a 0a57  erior algebra..W
│ │ │ │ +00058120: 6179 7320 746f 2075 7365 206d 616b 654d  ays to use makeM
│ │ │ │ +00058130: 6f64 756c 653a 0a3d 3d3d 3d3d 3d3d 3d3d  odule:.=========
│ │ │ │ +00058140: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ +00058150: 2020 2a20 226d 616b 654d 6f64 756c 6528    * "makeModule(
│ │ │ │ +00058160: 4861 7368 5461 626c 652c 4d61 7472 6978  HashTable,Matrix
│ │ │ │ +00058170: 2c48 6173 6854 6162 6c65 2922 0a0a 466f  ,HashTable)"..Fo
│ │ │ │ +00058180: 7220 7468 6520 7072 6f67 7261 6d6d 6572  r the programmer
│ │ │ │ +00058190: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ +000581a0: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420  ===..The object 
│ │ │ │ +000581b0: 2a6e 6f74 6520 6d61 6b65 4d6f 6475 6c65  *note makeModule
│ │ │ │ +000581c0: 3a20 6d61 6b65 4d6f 6475 6c65 2c20 6973  : makeModule, is
│ │ │ │ +000581d0: 2061 202a 6e6f 7465 206d 6574 686f 6420   a *note method 
│ │ │ │ +000581e0: 6675 6e63 7469 6f6e 3a0a 284d 6163 6175  function:.(Macau
│ │ │ │ +000581f0: 6c61 7932 446f 6329 4d65 7468 6f64 4675  lay2Doc)MethodFu
│ │ │ │ +00058200: 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d  nction,...------
│ │ │ │ +00058210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00058220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00058230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00058240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058260: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365  ----..The source
│ │ │ │ -00058270: 206f 6620 7468 6973 2064 6f63 756d 656e   of this documen
│ │ │ │ -00058280: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72  t is in./build/r
│ │ │ │ -00058290: 6570 726f 6475 6369 626c 652d 7061 7468  eproducible-path
│ │ │ │ -000582a0: 2f6d 6163 6175 6c61 7932 2d31 2e32 352e  /macaulay2-1.25.
│ │ │ │ -000582b0: 3131 2b64 732f 4d32 2f4d 6163 6175 6c61  11+ds/M2/Macaula
│ │ │ │ -000582c0: 7932 2f70 6163 6b61 6765 732f 0a43 6f6d  y2/packages/.Com
│ │ │ │ -000582d0: 706c 6574 6549 6e74 6572 7365 6374 696f  pleteIntersectio
│ │ │ │ -000582e0: 6e52 6573 6f6c 7574 696f 6e73 2e6d 323a  nResolutions.m2:
│ │ │ │ -000582f0: 3237 3539 3a30 2e0a 1f0a 4669 6c65 3a20  2759:0....File: 
│ │ │ │ -00058300: 436f 6d70 6c65 7465 496e 7465 7273 6563  CompleteIntersec
│ │ │ │ -00058310: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e  tionResolutions.
│ │ │ │ -00058320: 696e 666f 2c20 4e6f 6465 3a20 6d61 6b65  info, Node: make
│ │ │ │ -00058330: 542c 204e 6578 743a 206d 6174 7269 7846  T, Next: matrixF
│ │ │ │ -00058340: 6163 746f 7269 7a61 7469 6f6e 2c20 5072  actorization, Pr
│ │ │ │ -00058350: 6576 3a20 6d61 6b65 4d6f 6475 6c65 2c20  ev: makeModule, 
│ │ │ │ -00058360: 5570 3a20 546f 700a 0a6d 616b 6554 202d  Up: Top..makeT -
│ │ │ │ -00058370: 2d20 6d61 6b65 2074 6865 2043 4920 6f70  - make the CI op
│ │ │ │ -00058380: 6572 6174 6f72 7320 6f6e 2061 2063 6f6d  erators on a com
│ │ │ │ -00058390: 706c 6578 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a  plex.***********
│ │ │ │ +00058250: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073  ---------..The s
│ │ │ │ +00058260: 6f75 7263 6520 6f66 2074 6869 7320 646f  ource of this do
│ │ │ │ +00058270: 6375 6d65 6e74 2069 7320 696e 0a2f 6275  cument is in./bu
│ │ │ │ +00058280: 696c 642f 7265 7072 6f64 7563 6962 6c65  ild/reproducible
│ │ │ │ +00058290: 2d70 6174 682f 6d61 6361 756c 6179 322d  -path/macaulay2-
│ │ │ │ +000582a0: 312e 3235 2e31 312b 6473 2f4d 322f 4d61  1.25.11+ds/M2/Ma
│ │ │ │ +000582b0: 6361 756c 6179 322f 7061 636b 6167 6573  caulay2/packages
│ │ │ │ +000582c0: 2f0a 436f 6d70 6c65 7465 496e 7465 7273  /.CompleteInters
│ │ │ │ +000582d0: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e  ectionResolution
│ │ │ │ +000582e0: 732e 6d32 3a32 3735 393a 302e 0a1f 0a46  s.m2:2759:0....F
│ │ │ │ +000582f0: 696c 653a 2043 6f6d 706c 6574 6549 6e74  ile: CompleteInt
│ │ │ │ +00058300: 6572 7365 6374 696f 6e52 6573 6f6c 7574  ersectionResolut
│ │ │ │ +00058310: 696f 6e73 2e69 6e66 6f2c 204e 6f64 653a  ions.info, Node:
│ │ │ │ +00058320: 206d 616b 6554 2c20 4e65 7874 3a20 6d61   makeT, Next: ma
│ │ │ │ +00058330: 7472 6978 4661 6374 6f72 697a 6174 696f  trixFactorizatio
│ │ │ │ +00058340: 6e2c 2050 7265 763a 206d 616b 654d 6f64  n, Prev: makeMod
│ │ │ │ +00058350: 756c 652c 2055 703a 2054 6f70 0a0a 6d61  ule, Up: Top..ma
│ │ │ │ +00058360: 6b65 5420 2d2d 206d 616b 6520 7468 6520  keT -- make the 
│ │ │ │ +00058370: 4349 206f 7065 7261 746f 7273 206f 6e20  CI operators on 
│ │ │ │ +00058380: 6120 636f 6d70 6c65 780a 2a2a 2a2a 2a2a  a complex.******
│ │ │ │ +00058390: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000583a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000583b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000583c0: 0a0a 2020 2a20 5573 6167 653a 200a 2020  ..  * Usage: .  
│ │ │ │ -000583d0: 2020 2020 2020 5420 3d20 6d61 6b65 5428        T = makeT(
│ │ │ │ -000583e0: 6666 2c46 2c69 290a 2020 2020 2020 2020  ff,F,i).        
│ │ │ │ -000583f0: 5420 3d20 6d61 6b65 5428 6666 2c46 2c74  T = makeT(ff,F,t
│ │ │ │ -00058400: 302c 6929 0a20 202a 2049 6e70 7574 733a  0,i).  * Inputs:
│ │ │ │ -00058410: 0a20 2020 2020 202a 2066 662c 2061 202a  .      * ff, a *
│ │ │ │ -00058420: 6e6f 7465 206d 6174 7269 783a 2028 4d61  note matrix: (Ma
│ │ │ │ -00058430: 6361 756c 6179 3244 6f63 294d 6174 7269  caulay2Doc)Matri
│ │ │ │ -00058440: 782c 2c20 3178 6320 6d61 7472 6978 2077  x,, 1xc matrix w
│ │ │ │ -00058450: 686f 7365 2065 6e74 7269 6573 2061 7265  hose entries are
│ │ │ │ -00058460: 0a20 2020 2020 2020 2061 2063 6f6d 706c  .        a compl
│ │ │ │ -00058470: 6574 6520 696e 7465 7273 6563 7469 6f6e  ete intersection
│ │ │ │ -00058480: 2069 6e20 530a 2020 2020 2020 2a20 462c   in S.      * F,
│ │ │ │ -00058490: 2061 202a 6e6f 7465 2063 6f6d 706c 6578   a *note complex
│ │ │ │ -000584a0: 3a20 2843 6f6d 706c 6578 6573 2943 6f6d  : (Complexes)Com
│ │ │ │ -000584b0: 706c 6578 2c2c 206f 7665 7220 532f 6964  plex,, over S/id
│ │ │ │ -000584c0: 6561 6c20 6666 0a20 2020 2020 202a 2074  eal ff.      * t
│ │ │ │ -000584d0: 302c 2061 202a 6e6f 7465 206d 6174 7269  0, a *note matri
│ │ │ │ -000584e0: 783a 2028 4d61 6361 756c 6179 3244 6f63  x: (Macaulay2Doc
│ │ │ │ -000584f0: 294d 6174 7269 782c 2c20 4349 2d6f 7065  )Matrix,, CI-ope
│ │ │ │ -00058500: 7261 746f 7220 6f6e 2046 2066 6f72 2066  rator on F for f
│ │ │ │ -00058510: 665f 3020 746f 0a20 2020 2020 2020 2062  f_0 to.        b
│ │ │ │ -00058520: 6520 7072 6573 6572 7665 640a 2020 2020  e preserved.    
│ │ │ │ -00058530: 2020 2a20 692c 2061 6e20 2a6e 6f74 6520    * i, an *note 
│ │ │ │ -00058540: 696e 7465 6765 723a 2028 4d61 6361 756c  integer: (Macaul
│ │ │ │ -00058550: 6179 3244 6f63 295a 5a2c 2c20 6465 6669  ay2Doc)ZZ,, defi
│ │ │ │ -00058560: 6e65 2043 4920 6f70 6572 6174 6f72 7320  ne CI operators 
│ │ │ │ -00058570: 6672 6f6d 2046 5f69 0a20 2020 2020 2020  from F_i.       
│ │ │ │ -00058580: 205c 746f 2046 5f7b 692d 327d 0a20 202a   \to F_{i-2}.  *
│ │ │ │ -00058590: 204f 7574 7075 7473 3a0a 2020 2020 2020   Outputs:.      
│ │ │ │ -000585a0: 2a20 4c2c 2061 202a 6e6f 7465 206c 6973  * L, a *note lis
│ │ │ │ -000585b0: 743a 2028 4d61 6361 756c 6179 3244 6f63  t: (Macaulay2Doc
│ │ │ │ -000585c0: 294c 6973 742c 2c20 6f66 2043 4920 6f70  )List,, of CI op
│ │ │ │ -000585d0: 6572 6174 6f72 7320 465f 6920 5c74 6f20  erators F_i \to 
│ │ │ │ -000585e0: 465f 7b69 2d32 7d0a 2020 2020 2020 2020  F_{i-2}.        
│ │ │ │ -000585f0: 636f 7272 6573 706f 6e64 696e 6720 746f  corresponding to
│ │ │ │ -00058600: 2065 6e74 7269 6573 206f 6620 6666 0a0a   entries of ff..
│ │ │ │ -00058610: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
│ │ │ │ -00058620: 3d3d 3d3d 3d3d 3d0a 0a73 7562 7374 6974  =======..substit
│ │ │ │ -00058630: 7574 6520 6d61 7472 6963 6573 206f 6620  ute matrices of 
│ │ │ │ -00058640: 7477 6f20 6469 6666 6572 656e 7469 616c  two differential
│ │ │ │ -00058650: 7320 6f66 2046 2069 6e74 6f20 5320 3d20  s of F into S = 
│ │ │ │ -00058660: 7269 6e67 2066 662c 2063 6f6d 706f 7365  ring ff, compose
│ │ │ │ -00058670: 2074 6865 6d2c 0a61 6e64 2064 6976 6964   them,.and divid
│ │ │ │ -00058680: 6520 6279 2065 6e74 7269 6573 206f 6620  e by entries of 
│ │ │ │ -00058690: 6666 2c20 696e 206f 7264 6572 2e20 4966  ff, in order. If
│ │ │ │ -000586a0: 2074 6865 2073 6563 6f6e 6420 4d61 7472   the second Matr
│ │ │ │ -000586b0: 6978 2061 7267 756d 656e 7420 7430 2069  ix argument t0 i
│ │ │ │ -000586c0: 730a 7072 6573 656e 742c 2075 7365 2069  s.present, use i
│ │ │ │ -000586d0: 7420 6173 2074 6865 2066 6972 7374 2043  t as the first C
│ │ │ │ -000586e0: 4920 6f70 6572 6174 6f72 2e0a 0a54 6865  I operator...The
│ │ │ │ -000586f0: 2064 6567 7265 6573 206f 6620 7468 6520   degrees of the 
│ │ │ │ -00058700: 7461 7267 6574 7320 6f66 2074 6865 2054  targets of the T
│ │ │ │ -00058710: 5f6a 2061 7265 2063 6861 6e67 6564 2062  _j are changed b
│ │ │ │ -00058720: 7920 7468 6520 6465 6772 6565 7320 6f66  y the degrees of
│ │ │ │ -00058730: 2074 6865 2066 5f6a 2074 6f0a 6d61 6b65   the f_j to.make
│ │ │ │ -00058740: 2074 6865 2054 5f6a 2068 6f6d 6f67 656e   the T_j homogen
│ │ │ │ -00058750: 656f 7573 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  eous...+--------
│ │ │ │ +000583b0: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765  *****..  * Usage
│ │ │ │ +000583c0: 3a20 0a20 2020 2020 2020 2054 203d 206d  : .        T = m
│ │ │ │ +000583d0: 616b 6554 2866 662c 462c 6929 0a20 2020  akeT(ff,F,i).   
│ │ │ │ +000583e0: 2020 2020 2054 203d 206d 616b 6554 2866       T = makeT(f
│ │ │ │ +000583f0: 662c 462c 7430 2c69 290a 2020 2a20 496e  f,F,t0,i).  * In
│ │ │ │ +00058400: 7075 7473 3a0a 2020 2020 2020 2a20 6666  puts:.      * ff
│ │ │ │ +00058410: 2c20 6120 2a6e 6f74 6520 6d61 7472 6978  , a *note matrix
│ │ │ │ +00058420: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +00058430: 4d61 7472 6978 2c2c 2031 7863 206d 6174  Matrix,, 1xc mat
│ │ │ │ +00058440: 7269 7820 7768 6f73 6520 656e 7472 6965  rix whose entrie
│ │ │ │ +00058450: 7320 6172 650a 2020 2020 2020 2020 6120  s are.        a 
│ │ │ │ +00058460: 636f 6d70 6c65 7465 2069 6e74 6572 7365  complete interse
│ │ │ │ +00058470: 6374 696f 6e20 696e 2053 0a20 2020 2020  ction in S.     
│ │ │ │ +00058480: 202a 2046 2c20 6120 2a6e 6f74 6520 636f   * F, a *note co
│ │ │ │ +00058490: 6d70 6c65 783a 2028 436f 6d70 6c65 7865  mplex: (Complexe
│ │ │ │ +000584a0: 7329 436f 6d70 6c65 782c 2c20 6f76 6572  s)Complex,, over
│ │ │ │ +000584b0: 2053 2f69 6465 616c 2066 660a 2020 2020   S/ideal ff.    
│ │ │ │ +000584c0: 2020 2a20 7430 2c20 6120 2a6e 6f74 6520    * t0, a *note 
│ │ │ │ +000584d0: 6d61 7472 6978 3a20 284d 6163 6175 6c61  matrix: (Macaula
│ │ │ │ +000584e0: 7932 446f 6329 4d61 7472 6978 2c2c 2043  y2Doc)Matrix,, C
│ │ │ │ +000584f0: 492d 6f70 6572 6174 6f72 206f 6e20 4620  I-operator on F 
│ │ │ │ +00058500: 666f 7220 6666 5f30 2074 6f0a 2020 2020  for ff_0 to.    
│ │ │ │ +00058510: 2020 2020 6265 2070 7265 7365 7276 6564      be preserved
│ │ │ │ +00058520: 0a20 2020 2020 202a 2069 2c20 616e 202a  .      * i, an *
│ │ │ │ +00058530: 6e6f 7465 2069 6e74 6567 6572 3a20 284d  note integer: (M
│ │ │ │ +00058540: 6163 6175 6c61 7932 446f 6329 5a5a 2c2c  acaulay2Doc)ZZ,,
│ │ │ │ +00058550: 2064 6566 696e 6520 4349 206f 7065 7261   define CI opera
│ │ │ │ +00058560: 746f 7273 2066 726f 6d20 465f 690a 2020  tors from F_i.  
│ │ │ │ +00058570: 2020 2020 2020 5c74 6f20 465f 7b69 2d32        \to F_{i-2
│ │ │ │ +00058580: 7d0a 2020 2a20 4f75 7470 7574 733a 0a20  }.  * Outputs:. 
│ │ │ │ +00058590: 2020 2020 202a 204c 2c20 6120 2a6e 6f74       * L, a *not
│ │ │ │ +000585a0: 6520 6c69 7374 3a20 284d 6163 6175 6c61  e list: (Macaula
│ │ │ │ +000585b0: 7932 446f 6329 4c69 7374 2c2c 206f 6620  y2Doc)List,, of 
│ │ │ │ +000585c0: 4349 206f 7065 7261 746f 7273 2046 5f69  CI operators F_i
│ │ │ │ +000585d0: 205c 746f 2046 5f7b 692d 327d 0a20 2020   \to F_{i-2}.   
│ │ │ │ +000585e0: 2020 2020 2063 6f72 7265 7370 6f6e 6469       correspondi
│ │ │ │ +000585f0: 6e67 2074 6f20 656e 7472 6965 7320 6f66  ng to entries of
│ │ │ │ +00058600: 2066 660a 0a44 6573 6372 6970 7469 6f6e   ff..Description
│ │ │ │ +00058610: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 7375  .===========..su
│ │ │ │ +00058620: 6273 7469 7475 7465 206d 6174 7269 6365  bstitute matrice
│ │ │ │ +00058630: 7320 6f66 2074 776f 2064 6966 6665 7265  s of two differe
│ │ │ │ +00058640: 6e74 6961 6c73 206f 6620 4620 696e 746f  ntials of F into
│ │ │ │ +00058650: 2053 203d 2072 696e 6720 6666 2c20 636f   S = ring ff, co
│ │ │ │ +00058660: 6d70 6f73 6520 7468 656d 2c0a 616e 6420  mpose them,.and 
│ │ │ │ +00058670: 6469 7669 6465 2062 7920 656e 7472 6965  divide by entrie
│ │ │ │ +00058680: 7320 6f66 2066 662c 2069 6e20 6f72 6465  s of ff, in orde
│ │ │ │ +00058690: 722e 2049 6620 7468 6520 7365 636f 6e64  r. If the second
│ │ │ │ +000586a0: 204d 6174 7269 7820 6172 6775 6d65 6e74   Matrix argument
│ │ │ │ +000586b0: 2074 3020 6973 0a70 7265 7365 6e74 2c20   t0 is.present, 
│ │ │ │ +000586c0: 7573 6520 6974 2061 7320 7468 6520 6669  use it as the fi
│ │ │ │ +000586d0: 7273 7420 4349 206f 7065 7261 746f 722e  rst CI operator.
│ │ │ │ +000586e0: 0a0a 5468 6520 6465 6772 6565 7320 6f66  ..The degrees of
│ │ │ │ +000586f0: 2074 6865 2074 6172 6765 7473 206f 6620   the targets of 
│ │ │ │ +00058700: 7468 6520 545f 6a20 6172 6520 6368 616e  the T_j are chan
│ │ │ │ +00058710: 6765 6420 6279 2074 6865 2064 6567 7265  ged by the degre
│ │ │ │ +00058720: 6573 206f 6620 7468 6520 665f 6a20 746f  es of the f_j to
│ │ │ │ +00058730: 0a6d 616b 6520 7468 6520 545f 6a20 686f  .make the T_j ho
│ │ │ │ +00058740: 6d6f 6765 6e65 6f75 732e 0a0a 2b2d 2d2d  mogeneous...+---
│ │ │ │ +00058750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00058760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00058770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00058790: 3120 3a20 5320 3d20 5a5a 2f31 3031 5b78  1 : S = ZZ/101[x
│ │ │ │ -000587a0: 2c79 2c7a 5d3b 2020 2020 2020 2020 2020  ,y,z];          
│ │ │ │ -000587b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000587c0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00058780: 2d2b 0a7c 6931 203a 2053 203d 205a 5a2f  -+.|i1 : S = ZZ/
│ │ │ │ +00058790: 3130 315b 782c 792c 7a5d 3b20 2020 2020  101[x,y,z];     
│ │ │ │ +000587a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000587b0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +000587c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000587d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000587e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000587f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220  ----------+.|i2 
│ │ │ │ -00058800: 3a20 6666 203d 206d 6174 7269 7822 7833  : ff = matrix"x3
│ │ │ │ -00058810: 2c79 332c 7a33 223b 2020 2020 2020 2020  ,y3,z3";        
│ │ │ │ -00058820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058830: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000587e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +000587f0: 0a7c 6932 203a 2066 6620 3d20 6d61 7472  .|i2 : ff = matr
│ │ │ │ +00058800: 6978 2278 332c 7933 2c7a 3322 3b20 2020  ix"x3,y3,z3";   
│ │ │ │ +00058810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058820: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00058830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00058840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058860: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00058870: 2020 2020 2020 2020 3120 2020 2020 2033          1      3
│ │ │ │ +00058850: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00058860: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │ +00058870: 2020 2020 3320 2020 2020 2020 2020 2020      3           
│ │ │ │  00058880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058890: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000588a0: 0a7c 6f32 203a 204d 6174 7269 7820 5320  .|o2 : Matrix S 
│ │ │ │ -000588b0: 203c 2d2d 2053 2020 2020 2020 2020 2020   <-- S          
│ │ │ │ -000588c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000588d0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00058890: 2020 2020 7c0a 7c6f 3220 3a20 4d61 7472      |.|o2 : Matr
│ │ │ │ +000588a0: 6978 2053 2020 3c2d 2d20 5320 2020 2020  ix S  <-- S     
│ │ │ │ +000588b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000588c0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +000588d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000588e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000588f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00058910: 6933 203a 2052 203d 2053 2f69 6465 616c  i3 : R = S/ideal
│ │ │ │ -00058920: 2066 663b 2020 2020 2020 2020 2020 2020   ff;            
│ │ │ │ -00058930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058940: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00058900: 2d2d 2b0a 7c69 3320 3a20 5220 3d20 532f  --+.|i3 : R = S/
│ │ │ │ +00058910: 6964 6561 6c20 6666 3b20 2020 2020 2020  ideal ff;       
│ │ │ │ +00058920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058930: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00058940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00058950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00058960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934  -----------+.|i4
│ │ │ │ -00058980: 203a 204d 203d 2063 6f6b 6572 206d 6174   : M = coker mat
│ │ │ │ -00058990: 7269 7822 782c 792c 7a3b 792c 7a2c 7822  rix"x,y,z;y,z,x"
│ │ │ │ -000589a0: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ -000589b0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00058970: 2b0a 7c69 3420 3a20 4d20 3d20 636f 6b65  +.|i4 : M = coke
│ │ │ │ +00058980: 7220 6d61 7472 6978 2278 2c79 2c7a 3b79  r matrix"x,y,z;y
│ │ │ │ +00058990: 2c7a 2c78 223b 2020 2020 2020 2020 2020  ,z,x";          
│ │ │ │ +000589a0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +000589b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000589c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000589d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000589e0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a  ---------+.|i5 :
│ │ │ │ -000589f0: 2062 6574 7469 2028 4620 3d20 6672 6565   betti (F = free
│ │ │ │ -00058a00: 5265 736f 6c75 7469 6f6e 284d 2c20 4c65  Resolution(M, Le
│ │ │ │ -00058a10: 6e67 7468 4c69 6d69 7420 3d3e 2033 2929  ngthLimit => 3))
│ │ │ │ -00058a20: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000589d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000589e0: 7c69 3520 3a20 6265 7474 6920 2846 203d  |i5 : betti (F =
│ │ │ │ +000589f0: 2066 7265 6552 6573 6f6c 7574 696f 6e28   freeResolution(
│ │ │ │ +00058a00: 4d2c 204c 656e 6774 684c 696d 6974 203d  M, LengthLimit =
│ │ │ │ +00058a10: 3e20 3329 297c 0a7c 2020 2020 2020 2020  > 3))|.|        
│ │ │ │ +00058a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00058a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058a50: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00058a60: 2020 2020 2020 3020 3120 3220 3320 2020        0 1 2 3   
│ │ │ │ +00058a40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00058a50: 2020 2020 2020 2020 2020 2030 2031 2032             0 1 2
│ │ │ │ +00058a60: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │  00058a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058a80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00058a90: 7c6f 3520 3d20 746f 7461 6c3a 2032 2033  |o5 = total: 2 3
│ │ │ │ -00058aa0: 2035 2036 2020 2020 2020 2020 2020 2020   5 6            
│ │ │ │ -00058ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058ac0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00058ad0: 2030 3a20 3220 3320 2e20 2e20 2020 2020   0: 2 3 . .     
│ │ │ │ +00058a80: 2020 207c 0a7c 6f35 203d 2074 6f74 616c     |.|o5 = total
│ │ │ │ +00058a90: 3a20 3220 3320 3520 3620 2020 2020 2020  : 2 3 5 6       
│ │ │ │ +00058aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058ab0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00058ac0: 2020 2020 2020 303a 2032 2033 202e 202e        0: 2 3 . .
│ │ │ │ +00058ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00058ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058af0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00058b00: 2020 2020 2020 2020 313a 202e 202e 2035          1: . . 5
│ │ │ │ -00058b10: 2036 2020 2020 2020 2020 2020 2020 2020   6              
│ │ │ │ -00058b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058b30: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00058af0: 207c 0a7c 2020 2020 2020 2020 2031 3a20   |.|         1: 
│ │ │ │ +00058b00: 2e20 2e20 3520 3620 2020 2020 2020 2020  . . 5 6         
│ │ │ │ +00058b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058b20: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00058b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00058b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058b60: 2020 2020 2020 2020 2020 7c0a 7c6f 3520            |.|o5 
│ │ │ │ -00058b70: 3a20 4265 7474 6954 616c 6c79 2020 2020  : BettiTally    
│ │ │ │ +00058b50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00058b60: 0a7c 6f35 203a 2042 6574 7469 5461 6c6c  .|o5 : BettiTall
│ │ │ │ +00058b70: 7920 2020 2020 2020 2020 2020 2020 2020  y               
│ │ │ │  00058b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058ba0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00058b90: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00058ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00058bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058bd0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20  --------+.|i6 : 
│ │ │ │ -00058be0: 5420 3d20 6d61 6b65 5428 6666 2c46 2c33  T = makeT(ff,F,3
│ │ │ │ -00058bf0: 293b 2020 2020 2020 2020 2020 2020 2020  );              
│ │ │ │ -00058c00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00058c10: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00058bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00058bd0: 6936 203a 2054 203d 206d 616b 6554 2866  i6 : T = makeT(f
│ │ │ │ +00058be0: 662c 462c 3329 3b20 2020 2020 2020 2020  f,F,3);         
│ │ │ │ +00058bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058c00: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00058c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00058c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058c40: 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20 6e65  ------+.|i7 : ne
│ │ │ │ -00058c50: 744c 6973 7420 5420 2020 2020 2020 2020  tList T         
│ │ │ │ +00058c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937  -----------+.|i7
│ │ │ │ +00058c40: 203a 206e 6574 4c69 7374 2054 2020 2020   : netList T    
│ │ │ │ +00058c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00058c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058c70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00058c70: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00058c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00058c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058cb0: 2020 2020 7c0a 7c20 2020 2020 2b2d 2d2d      |.|     +---
│ │ │ │ -00058cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058cd0: 2d2d 2d2d 2d2b 2020 2020 2020 2020 2020  -----+          
│ │ │ │ -00058ce0: 2020 2020 2020 2020 2020 207c 0a7c 6f37             |.|o7
│ │ │ │ -00058cf0: 203d 207c 7b34 7d20 7c20 3020 3020 3020   = |{4} | 0 0 0 
│ │ │ │ -00058d00: 3020 2031 2030 207c 2020 2020 7c20 2020  0  1 0 |    |   
│ │ │ │ -00058d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058d20: 2020 7c0a 7c20 2020 2020 7c7b 347d 207c    |.|     |{4} |
│ │ │ │ -00058d30: 2030 2030 2030 202d 3120 3020 3020 7c20   0 0 0 -1 0 0 | 
│ │ │ │ -00058d40: 2020 207c 2020 2020 2020 2020 2020 2020     |            
│ │ │ │ -00058d50: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00058d60: 207c 7b34 7d20 7c20 3020 3020 3020 3020   |{4} | 0 0 0 0 
│ │ │ │ -00058d70: 2030 2031 207c 2020 2020 7c20 2020 2020   0 1 |    |     
│ │ │ │ -00058d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058d90: 7c0a 7c20 2020 2020 2b2d 2d2d 2d2d 2d2d  |.|     +-------
│ │ │ │ -00058da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058db0: 2d2b 2020 2020 2020 2020 2020 2020 2020  -+              
│ │ │ │ -00058dc0: 2020 2020 2020 207c 0a7c 2020 2020 207c         |.|     |
│ │ │ │ -00058dd0: 7b34 7d20 7c20 3020 3120 3020 3020 3020  {4} | 0 1 0 0 0 
│ │ │ │ -00058de0: 3020 7c20 2020 2020 7c20 2020 2020 2020  0 |     |       
│ │ │ │ -00058df0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00058e00: 7c20 2020 2020 7c7b 347d 207c 2031 2030  |     |{4} | 1 0
│ │ │ │ -00058e10: 2030 2030 2030 2030 207c 2020 2020 207c   0 0 0 0 |     |
│ │ │ │ -00058e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058e30: 2020 2020 207c 0a7c 2020 2020 207c 7b34       |.|     |{4
│ │ │ │ -00058e40: 7d20 7c20 3020 3020 3120 3020 3020 3020  } | 0 0 1 0 0 0 
│ │ │ │ -00058e50: 7c20 2020 2020 7c20 2020 2020 2020 2020  |     |         
│ │ │ │ -00058e60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00058e70: 2020 2020 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d      +-----------
│ │ │ │ -00058e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 2020  -------------+  
│ │ │ │ -00058e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058ea0: 2020 207c 0a7c 2020 2020 207c 7b34 7d20     |.|     |{4} 
│ │ │ │ -00058eb0: 7c20 3020 202d 3120 3020 2030 202d 3120  | 0  -1 0  0 -1 
│ │ │ │ -00058ec0: 3020 207c 7c20 2020 2020 2020 2020 2020  0  ||           
│ │ │ │ -00058ed0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00058ee0: 2020 7c7b 347d 207c 202d 3120 3020 2030    |{4} | -1 0  0
│ │ │ │ -00058ef0: 2020 3120 3020 2030 2020 7c7c 2020 2020    1 0  0  ||    
│ │ │ │ -00058f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058f10: 207c 0a7c 2020 2020 207c 7b34 7d20 7c20   |.|     |{4} | 
│ │ │ │ -00058f20: 3020 2030 2020 2d31 2030 2030 2020 2d31  0  0  -1 0 0  -1
│ │ │ │ -00058f30: 207c 7c20 2020 2020 2020 2020 2020 2020   ||             
│ │ │ │ -00058f40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00058f50: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -00058f60: 2d2d 2d2d 2d2d 2d2d 2d2b 2020 2020 2020  ---------+      
│ │ │ │ -00058f70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00058f80: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00058ca0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00058cb0: 202b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   +--------------
│ │ │ │ +00058cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b20 2020 2020  ----------+     
│ │ │ │ +00058cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058ce0: 7c0a 7c6f 3720 3d20 7c7b 347d 207c 2030  |.|o7 = |{4} | 0
│ │ │ │ +00058cf0: 2030 2030 2030 2020 3120 3020 7c20 2020   0 0 0  1 0 |   
│ │ │ │ +00058d00: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ +00058d10: 2020 2020 2020 207c 0a7c 2020 2020 207c         |.|     |
│ │ │ │ +00058d20: 7b34 7d20 7c20 3020 3020 3020 2d31 2030  {4} | 0 0 0 -1 0
│ │ │ │ +00058d30: 2030 207c 2020 2020 7c20 2020 2020 2020   0 |    |       
│ │ │ │ +00058d40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00058d50: 7c20 2020 2020 7c7b 347d 207c 2030 2030  |     |{4} | 0 0
│ │ │ │ +00058d60: 2030 2030 2020 3020 3120 7c20 2020 207c   0 0  0 1 |    |
│ │ │ │ +00058d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058d80: 2020 2020 207c 0a7c 2020 2020 202b 2d2d       |.|     +--
│ │ │ │ +00058d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00058da0: 2d2d 2d2d 2d2d 2b20 2020 2020 2020 2020  ------+         
│ │ │ │ +00058db0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00058dc0: 2020 2020 7c7b 347d 207c 2030 2031 2030      |{4} | 0 1 0
│ │ │ │ +00058dd0: 2030 2030 2030 207c 2020 2020 207c 2020   0 0 0 |     |  
│ │ │ │ +00058de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058df0: 2020 207c 0a7c 2020 2020 207c 7b34 7d20     |.|     |{4} 
│ │ │ │ +00058e00: 7c20 3120 3020 3020 3020 3020 3020 7c20  | 1 0 0 0 0 0 | 
│ │ │ │ +00058e10: 2020 2020 7c20 2020 2020 2020 2020 2020      |           
│ │ │ │ +00058e20: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00058e30: 2020 7c7b 347d 207c 2030 2030 2031 2030    |{4} | 0 0 1 0
│ │ │ │ +00058e40: 2030 2030 207c 2020 2020 207c 2020 2020   0 0 |     |    
│ │ │ │ +00058e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058e60: 207c 0a7c 2020 2020 202b 2d2d 2d2d 2d2d   |.|     +------
│ │ │ │ +00058e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00058e80: 2d2d 2b20 2020 2020 2020 2020 2020 2020  --+             
│ │ │ │ +00058e90: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00058ea0: 7c7b 347d 207c 2030 2020 2d31 2030 2020  |{4} | 0  -1 0  
│ │ │ │ +00058eb0: 3020 2d31 2030 2020 7c7c 2020 2020 2020  0 -1 0  ||      
│ │ │ │ +00058ec0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00058ed0: 0a7c 2020 2020 207c 7b34 7d20 7c20 2d31  .|     |{4} | -1
│ │ │ │ +00058ee0: 2030 2020 3020 2031 2030 2020 3020 207c   0  0  1 0  0  |
│ │ │ │ +00058ef0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00058f00: 2020 2020 2020 7c0a 7c20 2020 2020 7c7b        |.|     |{
│ │ │ │ +00058f10: 347d 207c 2030 2020 3020 202d 3120 3020  4} | 0  0  -1 0 
│ │ │ │ +00058f20: 3020 202d 3120 7c7c 2020 2020 2020 2020  0  -1 ||        
│ │ │ │ +00058f30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00058f40: 2020 2020 202b 2d2d 2d2d 2d2d 2d2d 2d2d       +----------
│ │ │ │ +00058f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b20  --------------+ 
│ │ │ │ +00058f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058f70: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00058f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00058f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058fb0: 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20 6973  ------+.|i8 : is
│ │ │ │ -00058fc0: 486f 6d6f 6765 6e65 6f75 7320 545f 3220  Homogeneous T_2 
│ │ │ │ +00058fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938  -----------+.|i8
│ │ │ │ +00058fb0: 203a 2069 7348 6f6d 6f67 656e 656f 7573   : isHomogeneous
│ │ │ │ +00058fc0: 2054 5f32 2020 2020 2020 2020 2020 2020   T_2            
│ │ │ │  00058fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058fe0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00058fe0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00058ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059020: 2020 2020 7c0a 7c6f 3820 3d20 7472 7565      |.|o8 = true
│ │ │ │ +00059010: 2020 2020 2020 2020 207c 0a7c 6f38 203d           |.|o8 =
│ │ │ │ +00059020: 2074 7275 6520 2020 2020 2020 2020 2020   true           
│ │ │ │  00059030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059050: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00059050: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00059060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00059070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00059080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00059470: 2020 2020 4d46 203d 206d 6174 7269 7846      MF = matrixF
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│ │ │ │ -00059a10: 6572 7469 6573 2069 6e20 7468 6520 6465  erties in the de
│ │ │ │ -00059a20: 6669 6e69 7469 6f6e 206f 6620 4d61 7472  finition of Matr
│ │ │ │ -00059a30: 6978 2046 6163 746f 7269 7a61 7469 6f6e  ix Factorization
│ │ │ │ -00059a40: 2061 7265 2076 6572 6966 6965 640a 0a54   are verified..T
│ │ │ │ -00059a50: 6865 206f 7574 7075 7420 6973 2061 206c  he output is a l
│ │ │ │ -00059a60: 6973 7420 6f66 206d 6170 7320 5c7b 642c  ist of maps \{d,
│ │ │ │ -00059a70: 685c 7d20 6f72 205c 7b64 2c68 2c67 616d  h\} or \{d,h,gam
│ │ │ │ -00059a80: 6d61 5c7d 2c20 7768 6572 6520 6761 6d6d  ma\}, where gamm
│ │ │ │ -00059a90: 6120 6973 2061 6e0a 6175 676d 656e 7461  a is an.augmenta
│ │ │ │ -00059aa0: 7469 6f6e 2c20 7468 6174 2069 732c 2061  tion, that is, a
│ │ │ │ -00059ab0: 206d 6170 2066 726f 6d20 7461 7267 6574   map from target
│ │ │ │ -00059ac0: 2064 2074 6f20 4d2e 0a0a 5468 6520 6d61   d to M...The ma
│ │ │ │ -00059ad0: 7020 6420 6973 2061 2073 7065 6369 616c  p d is a special
│ │ │ │ -00059ae0: 206c 6966 7469 6e67 2074 6f20 5320 6f66   lifting to S of
│ │ │ │ -00059af0: 2061 2070 7265 7365 6e74 6174 696f 6e20   a presentation 
│ │ │ │ -00059b00: 6f66 204d 206f 7665 7220 522e 2054 6f20  of M over R. To 
│ │ │ │ -00059b10: 6578 706c 6169 6e0a 7468 6520 636f 6e74  explain.the cont
│ │ │ │ -00059b20: 656e 7473 2c20 7765 2069 6e74 726f 6475  ents, we introdu
│ │ │ │ -00059b30: 6365 2073 6f6d 6520 6e6f 7461 7469 6f6e  ce some notation
│ │ │ │ -00059b40: 2028 6672 6f6d 2045 6973 656e 6275 6420   (from Eisenbud 
│ │ │ │ -00059b50: 616e 6420 5065 6576 612c 2022 4d69 6e69  and Peeva, "Mini
│ │ │ │ -00059b60: 6d61 6c0a 6672 6565 2072 6573 6f6c 7574  mal.free resolut
│ │ │ │ -00059b70: 696f 6e73 206f 7665 7220 636f 6d70 6c65  ions over comple
│ │ │ │ -00059b80: 7465 2069 6e74 6572 7365 6374 696f 6e73  te intersections
│ │ │ │ -00059b90: 2220 4c65 6374 7572 6520 4e6f 7465 7320  " Lecture Notes 
│ │ │ │ -00059ba0: 696e 204d 6174 6865 6d61 7469 6373 2c0a  in Mathematics,.
│ │ │ │ -00059bb0: 3231 3532 2e20 5370 7269 6e67 6572 2c20  2152. Springer, 
│ │ │ │ -00059bc0: 4368 616d 2c20 3230 3136 2e20 782b 3130  Cham, 2016. x+10
│ │ │ │ -00059bd0: 3720 7070 2e20 4953 424e 3a20 3937 382d  7 pp. ISBN: 978-
│ │ │ │ -00059be0: 332d 3331 392d 3236 3433 362d 333b 0a39  3-319-26436-3;.9
│ │ │ │ -00059bf0: 3738 2d33 2d33 3139 2d32 3634 3337 2d30  78-3-319-26437-0
│ │ │ │ -00059c00: 292e 0a0a 5228 6929 203d 2053 2f28 6666  )...R(i) = S/(ff
│ │ │ │ -00059c10: 5f30 2c2e 2e2c 6666 5f7b 692d 317d 292e  _0,..,ff_{i-1}).
│ │ │ │ -00059c20: 2048 6572 6520 303c 3d20 6920 3c3d 2063   Here 0<= i <= c
│ │ │ │ -00059c30: 2c20 616e 6420 5220 3d20 5228 6329 2061  , and R = R(c) a
│ │ │ │ -00059c40: 6e64 2053 203d 2052 2830 292e 0a0a 4228  nd S = R(0)...B(
│ │ │ │ -00059c50: 6929 203d 2074 6865 206d 6174 7269 7820  i) = the matrix 
│ │ │ │ -00059c60: 286f 7665 7220 5329 2072 6570 7265 7365  (over S) represe
│ │ │ │ -00059c70: 6e74 696e 6720 645f 693a 2042 5f31 2869  nting d_i: B_1(i
│ │ │ │ -00059c80: 2920 5c74 6f20 425f 3028 6929 0a0a 6428  ) \to B_0(i)..d(
│ │ │ │ -00059c90: 6929 3a20 415f 3128 6929 205c 746f 2041  i): A_1(i) \to A
│ │ │ │ -00059ca0: 5f30 2869 2920 7468 6520 7265 7374 7269  _0(i) the restri
│ │ │ │ -00059cb0: 6374 696f 6e20 6f66 2064 203d 2064 2863  ction of d = d(c
│ │ │ │ -00059cc0: 292e 2077 6865 7265 2041 2869 2920 3d0a  ). where A(i) =.
│ │ │ │ -00059cd0: 5c6f 706c 7573 5f7b 693d 317d 5e70 2042  \oplus_{i=1}^p B
│ │ │ │ -00059ce0: 2869 290a 0a0a 0a54 6865 206d 6170 2068  (i)....The map h
│ │ │ │ -00059cf0: 2069 7320 6120 6469 7265 6374 2073 756d   is a direct sum
│ │ │ │ -00059d00: 206f 6620 6d61 7073 2074 6172 6765 7420   of maps target 
│ │ │ │ -00059d10: 6428 7029 205c 746f 2073 6f75 7263 6520  d(p) \to source 
│ │ │ │ -00059d20: 6428 7029 2074 6861 7420 6172 650a 686f  d(p) that are.ho
│ │ │ │ -00059d30: 6d6f 746f 7069 6573 2066 6f72 2066 665f  motopies for ff_
│ │ │ │ -00059d40: 7020 6f6e 2074 6865 2072 6573 7472 6963  p on the restric
│ │ │ │ -00059d50: 7469 6f6e 2064 2870 293a 206f 7665 7220  tion d(p): over 
│ │ │ │ -00059d60: 7468 6520 7269 6e67 2052 2328 702d 3129  the ring R#(p-1)
│ │ │ │ -00059d70: 203d 0a53 2f28 6666 2331 2e2e 6666 2328   =.S/(ff#1..ff#(
│ │ │ │ -00059d80: 702d 3129 2c20 736f 2064 2870 2920 2a20  p-1), so d(p) * 
│ │ │ │ -00059d90: 6823 7020 3d20 6666 2370 206d 6f64 2028  h#p = ff#p mod (
│ │ │ │ -00059da0: 6666 2331 2e2e 6666 2328 702d 3129 2e0a  ff#1..ff#(p-1)..
│ │ │ │ -00059db0: 0a49 6e20 6164 6469 7469 6f6e 2c20 6823  .In addition, h#
│ │ │ │ -00059dc0: 7020 2a20 6428 7029 2069 6e64 7563 6573  p * d(p) induces
│ │ │ │ -00059dd0: 2066 6623 7020 6f6e 2042 3123 7020 6d6f   ff#p on B1#p mo
│ │ │ │ -00059de0: 6420 2866 6623 312e 2e66 6623 2870 2d31  d (ff#1..ff#(p-1
│ │ │ │ -00059df0: 292e 0a0a 4865 7265 2069 7320 6120 7369  )...Here is a si
│ │ │ │ -00059e00: 6d70 6c65 2065 7861 6d70 6c65 3a0a 0a2b  mple example:..+
│ │ │ │ +00059450: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20  ***..  * Usage: 
│ │ │ │ +00059460: 0a20 2020 2020 2020 204d 4620 3d20 6d61  .        MF = ma
│ │ │ │ +00059470: 7472 6978 4661 6374 6f72 697a 6174 696f  trixFactorizatio
│ │ │ │ +00059480: 6e28 6666 2c4d 290a 2020 2a20 496e 7075  n(ff,M).  * Inpu
│ │ │ │ +00059490: 7473 3a0a 2020 2020 2020 2a20 6666 2c20  ts:.      * ff, 
│ │ │ │ +000594a0: 6120 2a6e 6f74 6520 6d61 7472 6978 3a20  a *note matrix: 
│ │ │ │ +000594b0: 284d 6163 6175 6c61 7932 446f 6329 4d61  (Macaulay2Doc)Ma
│ │ │ │ +000594c0: 7472 6978 2c2c 2061 2073 7566 6669 6369  trix,, a suffici
│ │ │ │ +000594d0: 656e 746c 7920 6765 6e65 7261 6c0a 2020  ently general.  
│ │ │ │ +000594e0: 2020 2020 2020 7265 6775 6c61 7220 7365        regular se
│ │ │ │ +000594f0: 7175 656e 6365 2069 6e20 6120 7269 6e67  quence in a ring
│ │ │ │ +00059500: 2053 0a20 2020 2020 202a 204d 2c20 6120   S.      * M, a 
│ │ │ │ +00059510: 2a6e 6f74 6520 6d6f 6475 6c65 3a20 284d  *note module: (M
│ │ │ │ +00059520: 6163 6175 6c61 7932 446f 6329 4d6f 6475  acaulay2Doc)Modu
│ │ │ │ +00059530: 6c65 2c2c 2061 206d 6178 696d 616c 2043  le,, a maximal C
│ │ │ │ +00059540: 6f68 656e 2d4d 6163 6175 6c61 790a 2020  ohen-Macaulay.  
│ │ │ │ +00059550: 2020 2020 2020 6d6f 6475 6c65 206f 7665        module ove
│ │ │ │ +00059560: 7220 532f 6964 6561 6c20 6666 0a20 202a  r S/ideal ff.  *
│ │ │ │ +00059570: 202a 6e6f 7465 204f 7074 696f 6e61 6c20   *note Optional 
│ │ │ │ +00059580: 696e 7075 7473 3a20 284d 6163 6175 6c61  inputs: (Macaula
│ │ │ │ +00059590: 7932 446f 6329 7573 696e 6720 6675 6e63  y2Doc)using func
│ │ │ │ +000595a0: 7469 6f6e 7320 7769 7468 206f 7074 696f  tions with optio
│ │ │ │ +000595b0: 6e61 6c20 696e 7075 7473 2c3a 0a20 2020  nal inputs,:.   
│ │ │ │ +000595c0: 2020 202a 2041 7567 6d65 6e74 6174 696f     * Augmentatio
│ │ │ │ +000595d0: 6e20 3d3e 202e 2e2e 2c20 6465 6661 756c  n => ..., defaul
│ │ │ │ +000595e0: 7420 7661 6c75 6520 7472 7565 0a20 2020  t value true.   
│ │ │ │ +000595f0: 2020 202a 2043 6865 636b 203d 3e20 2e2e     * Check => ..
│ │ │ │ +00059600: 2e2c 2064 6566 6175 6c74 2076 616c 7565  ., default value
│ │ │ │ +00059610: 2066 616c 7365 0a20 2020 2020 202a 204c   false.      * L
│ │ │ │ +00059620: 6179 6572 6564 203d 3e20 2e2e 2e2c 2064  ayered => ..., d
│ │ │ │ +00059630: 6566 6175 6c74 2076 616c 7565 2074 7275  efault value tru
│ │ │ │ +00059640: 650a 2020 2020 2020 2a20 5665 7262 6f73  e.      * Verbos
│ │ │ │ +00059650: 6520 3d3e 202e 2e2e 2c20 6465 6661 756c  e => ..., defaul
│ │ │ │ +00059660: 7420 7661 6c75 6520 6661 6c73 650a 2020  t value false.  
│ │ │ │ +00059670: 2a20 4f75 7470 7574 733a 0a20 2020 2020  * Outputs:.     
│ │ │ │ +00059680: 202a 204d 462c 2061 202a 6e6f 7465 206c   * MF, a *note l
│ │ │ │ +00059690: 6973 743a 2028 4d61 6361 756c 6179 3244  ist: (Macaulay2D
│ │ │ │ +000596a0: 6f63 294c 6973 742c 2c20 5c7b 642c 682c  oc)List,, \{d,h,
│ │ │ │ +000596b0: 6761 6d6d 615c 7d2c 2077 6865 7265 2064  gamma\}, where d
│ │ │ │ +000596c0: 3a41 5f31 205c 746f 0a20 2020 2020 2020  :A_1 \to.       
│ │ │ │ +000596d0: 2041 5f30 2061 6e64 2068 3a20 5c6f 706c   A_0 and h: \opl
│ │ │ │ +000596e0: 7573 2041 5f30 2870 2920 5c74 6f20 415f  us A_0(p) \to A_
│ │ │ │ +000596f0: 3120 6973 2074 6865 2064 6972 6563 7420  1 is the direct 
│ │ │ │ +00059700: 7375 6d20 6f66 2070 6172 7469 616c 0a20  sum of partial. 
│ │ │ │ +00059710: 2020 2020 2020 2068 6f6d 6f74 6f70 6965         homotopie
│ │ │ │ +00059720: 732c 2061 6e64 2067 616d 6d61 3a20 415f  s, and gamma: A_
│ │ │ │ +00059730: 3020 2d3e 4d20 6973 2074 6865 2061 7567  0 ->M is the aug
│ │ │ │ +00059740: 6d65 6e74 6174 696f 6e20 2872 6574 7572  mentation (retur
│ │ │ │ +00059750: 6e65 6420 6f6e 6c79 2069 660a 2020 2020  ned only if.    
│ │ │ │ +00059760: 2020 2020 4175 676d 656e 7461 7469 6f6e      Augmentation
│ │ │ │ +00059770: 203d 3e74 7275 6529 0a0a 4465 7363 7269   =>true)..Descri
│ │ │ │ +00059780: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ +00059790: 3d0a 0a54 6865 2069 6e70 7574 206d 6f64  =..The input mod
│ │ │ │ +000597a0: 756c 6520 4d20 7368 6f75 6c64 2062 6520  ule M should be 
│ │ │ │ +000597b0: 6120 6d61 7869 6d61 6c20 436f 6865 6e2d  a maximal Cohen-
│ │ │ │ +000597c0: 4d61 6361 756c 6179 206d 6f64 756c 6520  Macaulay module 
│ │ │ │ +000597d0: 6f76 6572 2052 203d 2053 2f69 6465 616c  over R = S/ideal
│ │ │ │ +000597e0: 0a66 662e 2020 4966 204d 2069 7320 696e  .ff.  If M is in
│ │ │ │ +000597f0: 2066 6163 7420 6120 2268 6967 6820 7379   fact a "high sy
│ │ │ │ +00059800: 7a79 6779 222c 2074 6865 6e20 7468 6520  zygy", then the 
│ │ │ │ +00059810: 6675 6e63 7469 6f6e 0a6d 6174 7269 7846  function.matrixF
│ │ │ │ +00059820: 6163 746f 7269 7a61 7469 6f6e 2866 662c  actorization(ff,
│ │ │ │ +00059830: 4d2c 4c61 7965 7265 643d 3e66 616c 7365  M,Layered=>false
│ │ │ │ +00059840: 2920 7573 6573 2061 2064 6966 6665 7265  ) uses a differe
│ │ │ │ +00059850: 6e74 2c20 6661 7374 6572 2061 6c67 6f72  nt, faster algor
│ │ │ │ +00059860: 6974 686d 0a77 6869 6368 206f 6e6c 7920  ithm.which only 
│ │ │ │ +00059870: 776f 726b 7320 696e 2074 6865 2068 6967  works in the hig
│ │ │ │ +00059880: 6820 7379 7a79 6779 2063 6173 652e 0a0a  h syzygy case...
│ │ │ │ +00059890: 496e 2061 6c6c 2065 7861 6d70 6c65 7320  In all examples 
│ │ │ │ +000598a0: 7765 206b 6e6f 772c 204d 2063 616e 2062  we know, M can b
│ │ │ │ +000598b0: 6520 636f 6e73 6964 6572 6564 2061 2022  e considered a "
│ │ │ │ +000598c0: 6869 6768 2073 797a 7967 7922 2061 7320  high syzygy" as 
│ │ │ │ +000598d0: 6c6f 6e67 2061 730a 4578 745e 7b65 7665  long as.Ext^{eve
│ │ │ │ +000598e0: 6e7d 5f52 284d 2c6b 2920 616e 6420 4578  n}_R(M,k) and Ex
│ │ │ │ +000598f0: 745e 7b6f 6464 7d5f 5228 4d2c 6b29 2068  t^{odd}_R(M,k) h
│ │ │ │ +00059900: 6176 6520 6e65 6761 7469 7665 2072 6567  ave negative reg
│ │ │ │ +00059910: 756c 6172 6974 7920 6f76 6572 2074 6865  ularity over the
│ │ │ │ +00059920: 2072 696e 670a 6f66 2043 4920 6f70 6572   ring.of CI oper
│ │ │ │ +00059930: 6174 6f72 7320 2872 6567 7261 6465 6420  ators (regraded 
│ │ │ │ +00059940: 7769 7468 2076 6172 6961 626c 6573 206f  with variables o
│ │ │ │ +00059950: 6620 6465 6772 6565 2031 2e20 486f 7765  f degree 1. Howe
│ │ │ │ +00059960: 7665 722c 2074 6865 2062 6573 7420 7265  ver, the best re
│ │ │ │ +00059970: 7375 6c74 0a77 6520 6361 6e20 7072 6f76  sult.we can prov
│ │ │ │ +00059980: 6520 6973 2074 6861 7420 6974 2073 7566  e is that it suf
│ │ │ │ +00059990: 6669 6365 7320 746f 2068 6176 6520 7265  fices to have re
│ │ │ │ +000599a0: 6775 6c61 7269 7479 203c 202d 2832 2a64  gularity < -(2*d
│ │ │ │ +000599b0: 696d 2052 2b31 292e 0a0a 5768 656e 2074  im R+1)...When t
│ │ │ │ +000599c0: 6865 206f 7074 696f 6e61 6c20 696e 7075  he optional inpu
│ │ │ │ +000599d0: 7420 4368 6563 6b3d 3d74 7275 6520 2874  t Check==true (t
│ │ │ │ +000599e0: 6865 2064 6566 6175 6c74 2069 7320 4368  he default is Ch
│ │ │ │ +000599f0: 6563 6b3d 3d66 616c 7365 292c 2074 6865  eck==false), the
│ │ │ │ +00059a00: 0a70 726f 7065 7274 6965 7320 696e 2074  .properties in t
│ │ │ │ +00059a10: 6865 2064 6566 696e 6974 696f 6e20 6f66  he definition of
│ │ │ │ +00059a20: 204d 6174 7269 7820 4661 6374 6f72 697a   Matrix Factoriz
│ │ │ │ +00059a30: 6174 696f 6e20 6172 6520 7665 7269 6669  ation are verifi
│ │ │ │ +00059a40: 6564 0a0a 5468 6520 6f75 7470 7574 2069  ed..The output i
│ │ │ │ +00059a50: 7320 6120 6c69 7374 206f 6620 6d61 7073  s a list of maps
│ │ │ │ +00059a60: 205c 7b64 2c68 5c7d 206f 7220 5c7b 642c   \{d,h\} or \{d,
│ │ │ │ +00059a70: 682c 6761 6d6d 615c 7d2c 2077 6865 7265  h,gamma\}, where
│ │ │ │ +00059a80: 2067 616d 6d61 2069 7320 616e 0a61 7567   gamma is an.aug
│ │ │ │ +00059a90: 6d65 6e74 6174 696f 6e2c 2074 6861 7420  mentation, that 
│ │ │ │ +00059aa0: 6973 2c20 6120 6d61 7020 6672 6f6d 2074  is, a map from t
│ │ │ │ +00059ab0: 6172 6765 7420 6420 746f 204d 2e0a 0a54  arget d to M...T
│ │ │ │ +00059ac0: 6865 206d 6170 2064 2069 7320 6120 7370  he map d is a sp
│ │ │ │ +00059ad0: 6563 6961 6c20 6c69 6674 696e 6720 746f  ecial lifting to
│ │ │ │ +00059ae0: 2053 206f 6620 6120 7072 6573 656e 7461   S of a presenta
│ │ │ │ +00059af0: 7469 6f6e 206f 6620 4d20 6f76 6572 2052  tion of M over R
│ │ │ │ +00059b00: 2e20 546f 2065 7870 6c61 696e 0a74 6865  . To explain.the
│ │ │ │ +00059b10: 2063 6f6e 7465 6e74 732c 2077 6520 696e   contents, we in
│ │ │ │ +00059b20: 7472 6f64 7563 6520 736f 6d65 206e 6f74  troduce some not
│ │ │ │ +00059b30: 6174 696f 6e20 2866 726f 6d20 4569 7365  ation (from Eise
│ │ │ │ +00059b40: 6e62 7564 2061 6e64 2050 6565 7661 2c20  nbud and Peeva, 
│ │ │ │ +00059b50: 224d 696e 696d 616c 0a66 7265 6520 7265  "Minimal.free re
│ │ │ │ +00059b60: 736f 6c75 7469 6f6e 7320 6f76 6572 2063  solutions over c
│ │ │ │ +00059b70: 6f6d 706c 6574 6520 696e 7465 7273 6563  omplete intersec
│ │ │ │ +00059b80: 7469 6f6e 7322 204c 6563 7475 7265 204e  tions" Lecture N
│ │ │ │ +00059b90: 6f74 6573 2069 6e20 4d61 7468 656d 6174  otes in Mathemat
│ │ │ │ +00059ba0: 6963 732c 0a32 3135 322e 2053 7072 696e  ics,.2152. Sprin
│ │ │ │ +00059bb0: 6765 722c 2043 6861 6d2c 2032 3031 362e  ger, Cham, 2016.
│ │ │ │ +00059bc0: 2078 2b31 3037 2070 702e 2049 5342 4e3a   x+107 pp. ISBN:
│ │ │ │ +00059bd0: 2039 3738 2d33 2d33 3139 2d32 3634 3336   978-3-319-26436
│ │ │ │ +00059be0: 2d33 3b0a 3937 382d 332d 3331 392d 3236  -3;.978-3-319-26
│ │ │ │ +00059bf0: 3433 372d 3029 2e0a 0a52 2869 2920 3d20  437-0)...R(i) = 
│ │ │ │ +00059c00: 532f 2866 665f 302c 2e2e 2c66 665f 7b69  S/(ff_0,..,ff_{i
│ │ │ │ +00059c10: 2d31 7d29 2e20 4865 7265 2030 3c3d 2069  -1}). Here 0<= i
│ │ │ │ +00059c20: 203c 3d20 632c 2061 6e64 2052 203d 2052   <= c, and R = R
│ │ │ │ +00059c30: 2863 2920 616e 6420 5320 3d20 5228 3029  (c) and S = R(0)
│ │ │ │ +00059c40: 2e0a 0a42 2869 2920 3d20 7468 6520 6d61  ...B(i) = the ma
│ │ │ │ +00059c50: 7472 6978 2028 6f76 6572 2053 2920 7265  trix (over S) re
│ │ │ │ +00059c60: 7072 6573 656e 7469 6e67 2064 5f69 3a20  presenting d_i: 
│ │ │ │ +00059c70: 425f 3128 6929 205c 746f 2042 5f30 2869  B_1(i) \to B_0(i
│ │ │ │ +00059c80: 290a 0a64 2869 293a 2041 5f31 2869 2920  )..d(i): A_1(i) 
│ │ │ │ +00059c90: 5c74 6f20 415f 3028 6929 2074 6865 2072  \to A_0(i) the r
│ │ │ │ +00059ca0: 6573 7472 6963 7469 6f6e 206f 6620 6420  estriction of d 
│ │ │ │ +00059cb0: 3d20 6428 6329 2e20 7768 6572 6520 4128  = d(c). where A(
│ │ │ │ +00059cc0: 6929 203d 0a5c 6f70 6c75 735f 7b69 3d31  i) =.\oplus_{i=1
│ │ │ │ +00059cd0: 7d5e 7020 4228 6929 0a0a 0a0a 5468 6520  }^p B(i)....The 
│ │ │ │ +00059ce0: 6d61 7020 6820 6973 2061 2064 6972 6563  map h is a direc
│ │ │ │ +00059cf0: 7420 7375 6d20 6f66 206d 6170 7320 7461  t sum of maps ta
│ │ │ │ +00059d00: 7267 6574 2064 2870 2920 5c74 6f20 736f  rget d(p) \to so
│ │ │ │ +00059d10: 7572 6365 2064 2870 2920 7468 6174 2061  urce d(p) that a
│ │ │ │ +00059d20: 7265 0a68 6f6d 6f74 6f70 6965 7320 666f  re.homotopies fo
│ │ │ │ +00059d30: 7220 6666 5f70 206f 6e20 7468 6520 7265  r ff_p on the re
│ │ │ │ +00059d40: 7374 7269 6374 696f 6e20 6428 7029 3a20  striction d(p): 
│ │ │ │ +00059d50: 6f76 6572 2074 6865 2072 696e 6720 5223  over the ring R#
│ │ │ │ +00059d60: 2870 2d31 2920 3d0a 532f 2866 6623 312e  (p-1) =.S/(ff#1.
│ │ │ │ +00059d70: 2e66 6623 2870 2d31 292c 2073 6f20 6428  .ff#(p-1), so d(
│ │ │ │ +00059d80: 7029 202a 2068 2370 203d 2066 6623 7020  p) * h#p = ff#p 
│ │ │ │ +00059d90: 6d6f 6420 2866 6623 312e 2e66 6623 2870  mod (ff#1..ff#(p
│ │ │ │ +00059da0: 2d31 292e 0a0a 496e 2061 6464 6974 696f  -1)...In additio
│ │ │ │ +00059db0: 6e2c 2068 2370 202a 2064 2870 2920 696e  n, h#p * d(p) in
│ │ │ │ +00059dc0: 6475 6365 7320 6666 2370 206f 6e20 4231  duces ff#p on B1
│ │ │ │ +00059dd0: 2370 206d 6f64 2028 6666 2331 2e2e 6666  #p mod (ff#1..ff
│ │ │ │ +00059de0: 2328 702d 3129 2e0a 0a48 6572 6520 6973  #(p-1)...Here is
│ │ │ │ +00059df0: 2061 2073 696d 706c 6520 6578 616d 706c   a simple exampl
│ │ │ │ +00059e00: 653a 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  e:..+-----------
│ │ │ │  00059e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00059e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00059e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00059e40: 2d2d 2b0a 7c69 3120 3a20 7365 7452 616e  --+.|i1 : setRan
│ │ │ │ -00059e50: 646f 6d53 6565 6420 3020 2020 2020 2020  domSeed 0       
│ │ │ │ -00059e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059e70: 2020 2020 2020 207c 0a7c 202d 2d20 7365         |.| -- se
│ │ │ │ -00059e80: 7474 696e 6720 7261 6e64 6f6d 2073 6565  tting random see
│ │ │ │ -00059e90: 6420 746f 2030 2020 2020 2020 2020 2020  d to 0          
│ │ │ │ -00059ea0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00059e30: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2073  -------+.|i1 : s
│ │ │ │ +00059e40: 6574 5261 6e64 6f6d 5365 6564 2030 2020  etRandomSeed 0  
│ │ │ │ +00059e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00059e60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00059e70: 2d2d 2073 6574 7469 6e67 2072 616e 646f  -- setting rando
│ │ │ │ +00059e80: 6d20 7365 6564 2074 6f20 3020 2020 2020  m seed to 0     
│ │ │ │ +00059e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00059ea0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00059eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059ee0: 207c 0a7c 6f31 203d 2030 2020 2020 2020   |.|o1 = 0      
│ │ │ │ +00059ed0: 2020 2020 2020 7c0a 7c6f 3120 3d20 3020        |.|o1 = 0 
│ │ │ │ +00059ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059f10: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00059f00: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00059f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00059f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00059f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00059f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ -00059f50: 203a 206b 6b20 3d20 5a5a 2f31 3031 2020   : kk = ZZ/101  
│ │ │ │ +00059f40: 2b0a 7c69 3220 3a20 6b6b 203d 205a 5a2f  +.|i2 : kk = ZZ/
│ │ │ │ +00059f50: 3130 3120 2020 2020 2020 2020 2020 2020  101             
│ │ │ │  00059f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059f80: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00059f70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00059f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059fb0: 2020 2020 207c 0a7c 6f32 203d 206b 6b20       |.|o2 = kk 
│ │ │ │ +00059fa0: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ +00059fb0: 3d20 6b6b 2020 2020 2020 2020 2020 2020  = kk            
│ │ │ │  00059fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059fe0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00059fd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00059fe0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00059ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005a000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a010: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0005a020: 0a7c 6f32 203a 2051 756f 7469 656e 7452  .|o2 : QuotientR
│ │ │ │ -0005a030: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │ -0005a040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a050: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0005a010: 2020 2020 7c0a 7c6f 3220 3a20 5175 6f74      |.|o2 : Quot
│ │ │ │ +0005a020: 6965 6e74 5269 6e67 2020 2020 2020 2020  ientRing        
│ │ │ │ +0005a030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a040: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0005a050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005a060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005a070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005a080: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a  ---------+.|i3 :
│ │ │ │ -0005a090: 2053 203d 206b 6b5b 612c 622c 752c 765d   S = kk[a,b,u,v]
│ │ │ │ +0005a070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0005a080: 7c69 3320 3a20 5320 3d20 6b6b 5b61 2c62  |i3 : S = kk[a,b
│ │ │ │ +0005a090: 2c75 2c76 5d20 2020 2020 2020 2020 2020  ,u,v]           
│ │ │ │  0005a0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a0b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0005a0c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0005a0b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0005a0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005a0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a0f0: 2020 207c 0a7c 6f33 203d 2053 2020 2020     |.|o3 = S    
│ │ │ │ +0005a0e0: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20          |.|o3 = 
│ │ │ │ +0005a0f0: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │  0005a100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a120: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0005a110: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0005a120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005a130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005a140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a150: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0005a160: 6f33 203a 2050 6f6c 796e 6f6d 6961 6c52  o3 : PolynomialR
│ │ │ │ -0005a170: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │ -0005a180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a190: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0005a150: 2020 7c0a 7c6f 3320 3a20 506f 6c79 6e6f    |.|o3 : Polyno
│ │ │ │ +0005a160: 6d69 616c 5269 6e67 2020 2020 2020 2020  mialRing        
│ │ │ │ +0005a170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a180: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0005a190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005a1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005a1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005a1c0: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2066  -------+.|i4 : f
│ │ │ │ -0005a1d0: 6620 3d20 6d61 7472 6978 2261 752c 6276  f = matrix"au,bv
│ │ │ │ -0005a1e0: 2220 2020 2020 2020 2020 2020 2020 2020  "               
│ │ │ │ -0005a1f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0005a1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0005a1c0: 3420 3a20 6666 203d 206d 6174 7269 7822  4 : ff = matrix"
│ │ │ │ +0005a1d0: 6175 2c62 7622 2020 2020 2020 2020 2020  au,bv"          
│ │ │ │ +0005a1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a1f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0005a200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005a210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a230: 207c 0a7c 6f34 203d 207c 2061 7520 6276   |.|o4 = | au bv
│ │ │ │ -0005a240: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ -0005a250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a260: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0005a220: 2020 2020 2020 7c0a 7c6f 3420 3d20 7c20        |.|o4 = | 
│ │ │ │ +0005a230: 6175 2062 7620 7c20 2020 2020 2020 2020  au bv |         
│ │ │ │ +0005a240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a250: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0005a260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005a270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005a280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a290: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0005a2a0: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ -0005a2b0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -0005a2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a2d0: 7c0a 7c6f 3420 3a20 4d61 7472 6978 2053  |.|o4 : Matrix S
│ │ │ │ -0005a2e0: 2020 3c2d 2d20 5320 2020 2020 2020 2020    <-- S         
│ │ │ │ -0005a2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a300: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0005a290: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0005a2a0: 3120 2020 2020 2032 2020 2020 2020 2020  1      2        
│ │ │ │ +0005a2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a2c0: 2020 2020 207c 0a7c 6f34 203a 204d 6174       |.|o4 : Mat
│ │ │ │ +0005a2d0: 7269 7820 5320 203c 2d2d 2053 2020 2020  rix S  <-- S    
│ │ │ │ +0005a2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a2f0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0005a300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005a310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005a320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005a330: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520  ----------+.|i5 
│ │ │ │ -0005a340: 3a20 5220 3d20 532f 6964 6561 6c20 6666  : R = S/ideal ff
│ │ │ │ +0005a320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0005a330: 0a7c 6935 203a 2052 203d 2053 2f69 6465  .|i5 : R = S/ide
│ │ │ │ +0005a340: 616c 2066 6620 2020 2020 2020 2020 2020  al ff           
│ │ │ │  0005a350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a360: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0005a370: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0005a360: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0005a370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005a380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a3a0: 2020 2020 7c0a 7c6f 3520 3d20 5220 2020      |.|o5 = R   
│ │ │ │ +0005a390: 2020 2020 2020 2020 207c 0a7c 6f35 203d           |.|o5 =
│ │ │ │ +0005a3a0: 2052 2020 2020 2020 2020 2020 2020 2020   R              
│ │ │ │  0005a3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a3d0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005a3c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005a3d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0005a3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005a3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a400: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0005a410: 7c6f 3520 3a20 5175 6f74 6965 6e74 5269  |o5 : QuotientRi
│ │ │ │ -0005a420: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │ -0005a430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a440: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0005a400: 2020 207c 0a7c 6f35 203a 2051 756f 7469     |.|o5 : Quoti
│ │ │ │ +0005a410: 656e 7452 696e 6720 2020 2020 2020 2020  entRing         
│ │ │ │ +0005a420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a430: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0005a440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005a450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005a460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005a470: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20  --------+.|i6 : 
│ │ │ │ -0005a480: 4d30 203d 2052 5e31 2f69 6465 616c 2261  M0 = R^1/ideal"a
│ │ │ │ -0005a490: 2c62 2220 2020 2020 2020 2020 2020 2020  ,b"             
│ │ │ │ -0005a4a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0005a460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +0005a470: 6936 203a 204d 3020 3d20 525e 312f 6964  i6 : M0 = R^1/id
│ │ │ │ +0005a480: 6561 6c22 612c 6222 2020 2020 2020 2020  eal"a,b"        
│ │ │ │ +0005a490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a4a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0005a4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005a4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a4e0: 2020 7c0a 7c6f 3620 3d20 636f 6b65 726e    |.|o6 = cokern
│ │ │ │ -0005a4f0: 656c 207c 2061 2062 207c 2020 2020 2020  el | a b |      
│ │ │ │ -0005a500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a510: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005a4d0: 2020 2020 2020 207c 0a7c 6f36 203d 2063         |.|o6 = c
│ │ │ │ +0005a4e0: 6f6b 6572 6e65 6c20 7c20 6120 6220 7c20  okernel | a b | 
│ │ │ │ +0005a4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a500: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0005a510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005a520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005a530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a540: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0005a540: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0005a550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a560: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ -0005a570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a580: 207c 0a7c 6f36 203a 2052 2d6d 6f64 756c   |.|o6 : R-modul
│ │ │ │ -0005a590: 652c 2071 756f 7469 656e 7420 6f66 2052  e, quotient of R
│ │ │ │ -0005a5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a5b0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0005a560: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +0005a570: 2020 2020 2020 7c0a 7c6f 3620 3a20 522d        |.|o6 : R-
│ │ │ │ +0005a580: 6d6f 6475 6c65 2c20 7175 6f74 6965 6e74  module, quotient
│ │ │ │ +0005a590: 206f 6620 5220 2020 2020 2020 2020 2020   of R           
│ │ │ │ +0005a5a0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0005a5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005a5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005a5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005a5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937  -----------+.|i7
│ │ │ │ -0005a5f0: 203a 204d 203d 2068 6967 6853 797a 7967   : M = highSyzyg
│ │ │ │ -0005a600: 7920 4d30 2020 2020 2020 2020 2020 2020  y M0            
│ │ │ │ -0005a610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a620: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0005a5e0: 2b0a 7c69 3720 3a20 4d20 3d20 6869 6768  +.|i7 : M = high
│ │ │ │ +0005a5f0: 5379 7a79 6779 204d 3020 2020 2020 2020  Syzygy M0       
│ │ │ │ +0005a600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a610: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005a620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005a630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a650: 2020 2020 207c 0a7c 6f37 203d 2063 6f6b       |.|o7 = cok
│ │ │ │ -0005a660: 6572 6e65 6c20 7b32 7d20 7c20 6220 2d61  ernel {2} | b -a
│ │ │ │ -0005a670: 2030 2030 207c 2020 2020 2020 2020 2020   0 0 |          
│ │ │ │ -0005a680: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0005a690: 2020 2020 2020 2020 2020 207b 327d 207c             {2} |
│ │ │ │ -0005a6a0: 2030 2030 2020 6120 6220 7c20 2020 2020   0 0  a b |     
│ │ │ │ -0005a6b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0005a6c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0005a6d0: 7b32 7d20 7c20 3020 7620 2030 2075 207c  {2} | 0 v  0 u |
│ │ │ │ -0005a6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a6f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0005a640: 2020 2020 2020 2020 2020 7c0a 7c6f 3720            |.|o7 
│ │ │ │ +0005a650: 3d20 636f 6b65 726e 656c 207b 327d 207c  = cokernel {2} |
│ │ │ │ +0005a660: 2062 202d 6120 3020 3020 7c20 2020 2020   b -a 0 0 |     
│ │ │ │ +0005a670: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005a680: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0005a690: 7b32 7d20 7c20 3020 3020 2061 2062 207c  {2} | 0 0  a b |
│ │ │ │ +0005a6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a6b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0005a6c0: 2020 2020 207b 327d 207c 2030 2076 2020       {2} | 0 v  
│ │ │ │ +0005a6d0: 3020 7520 7c20 2020 2020 2020 2020 2020  0 u |           
│ │ │ │ +0005a6e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005a6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005a700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a720: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0005a730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a740: 2020 2020 2020 2020 3320 2020 2020 2020          3       
│ │ │ │ -0005a750: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0005a760: 7c6f 3720 3a20 522d 6d6f 6475 6c65 2c20  |o7 : R-module, 
│ │ │ │ -0005a770: 7175 6f74 6965 6e74 206f 6620 5220 2020  quotient of R   
│ │ │ │ -0005a780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a790: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0005a710: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005a720: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0005a730: 2020 2020 2020 2020 2020 2020 2033 2020               3  
│ │ │ │ +0005a740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a750: 2020 207c 0a7c 6f37 203a 2052 2d6d 6f64     |.|o7 : R-mod
│ │ │ │ +0005a760: 756c 652c 2071 756f 7469 656e 7420 6f66  ule, quotient of
│ │ │ │ +0005a770: 2052 2020 2020 2020 2020 2020 2020 2020   R              
│ │ │ │ +0005a780: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0005a790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005a7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005a7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005a7c0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20  --------+.|i8 : 
│ │ │ │ -0005a7d0: 4d46 203d 206d 6174 7269 7846 6163 746f  MF = matrixFacto
│ │ │ │ -0005a7e0: 7269 7a61 7469 6f6e 2866 662c 4d29 3b20  rization(ff,M); 
│ │ │ │ -0005a7f0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0005a7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +0005a7c0: 6938 203a 204d 4620 3d20 6d61 7472 6978  i8 : MF = matrix
│ │ │ │ +0005a7d0: 4661 6374 6f72 697a 6174 696f 6e28 6666  Factorization(ff
│ │ │ │ +0005a7e0: 2c4d 293b 2020 2020 2020 2020 2020 2020  ,M);            
│ │ │ │ +0005a7f0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  0005a800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005a810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005a820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005a830: 2d2d 2b0a 7c69 3920 3a20 6e65 744c 6973  --+.|i9 : netLis
│ │ │ │ -0005a840: 7420 4252 616e 6b73 204d 4620 2020 2020  t BRanks MF     
│ │ │ │ -0005a850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a860: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005a820: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 206e  -------+.|i9 : n
│ │ │ │ +0005a830: 6574 4c69 7374 2042 5261 6e6b 7320 4d46  etList BRanks MF
│ │ │ │ +0005a840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a850: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0005a860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005a870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005a880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a890: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0005a8a0: 2020 2020 2b2d 2b2d 2b20 2020 2020 2020      +-+-+       
│ │ │ │ +0005a890: 207c 0a7c 2020 2020 202b 2d2b 2d2b 2020   |.|     +-+-+  
│ │ │ │ +0005a8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005a8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a8d0: 207c 0a7c 6f39 203d 207c 327c 327c 2020   |.|o9 = |2|2|  
│ │ │ │ +0005a8c0: 2020 2020 2020 7c0a 7c6f 3920 3d20 7c32        |.|o9 = |2
│ │ │ │ +0005a8d0: 7c32 7c20 2020 2020 2020 2020 2020 2020  |2|             
│ │ │ │  0005a8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a900: 2020 2020 2020 7c0a 7c20 2020 2020 2b2d        |.|     +-
│ │ │ │ -0005a910: 2b2d 2b20 2020 2020 2020 2020 2020 2020  +-+             
│ │ │ │ +0005a8f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0005a900: 2020 202b 2d2b 2d2b 2020 2020 2020 2020     +-+-+        
│ │ │ │ +0005a910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005a920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a930: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0005a940: 2020 207c 317c 327c 2020 2020 2020 2020     |1|2|        
│ │ │ │ +0005a930: 7c0a 7c20 2020 2020 7c31 7c32 7c20 2020  |.|     |1|2|   
│ │ │ │ +0005a940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005a950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a970: 7c0a 7c20 2020 2020 2b2d 2b2d 2b20 2020  |.|     +-+-+   
│ │ │ │ +0005a960: 2020 2020 207c 0a7c 2020 2020 202b 2d2b       |.|     +-+
│ │ │ │ +0005a970: 2d2b 2020 2020 2020 2020 2020 2020 2020  -+              
│ │ │ │  0005a980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a9a0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0005a990: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0005a9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005a9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005a9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005a9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3130  ----------+.|i10
│ │ │ │ -0005a9e0: 203a 206e 6574 4c69 7374 2062 4d61 7073   : netList bMaps
│ │ │ │ -0005a9f0: 204d 4620 2020 2020 2020 2020 2020 2020   MF             
│ │ │ │ -0005aa00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0005aa10: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0005a9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0005a9d0: 0a7c 6931 3020 3a20 6e65 744c 6973 7420  .|i10 : netList 
│ │ │ │ +0005a9e0: 624d 6170 7320 4d46 2020 2020 2020 2020  bMaps MF        
│ │ │ │ +0005a9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005aa00: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0005aa10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005aa20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005aa30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005aa40: 2020 2020 7c0a 7c20 2020 2020 202b 2d2d      |.|      +--
│ │ │ │ -0005aa50: 2d2d 2d2d 2d2d 2d2d 2d2b 2020 2020 2020  ---------+      
│ │ │ │ -0005aa60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005aa70: 2020 2020 2020 2020 207c 0a7c 6f31 3020           |.|o10 
│ │ │ │ -0005aa80: 3d20 7c7b 327d 207c 2061 2062 207c 7c20  = |{2} | a b || 
│ │ │ │ +0005aa30: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005aa40: 2020 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b20    +-----------+ 
│ │ │ │ +0005aa50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005aa60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005aa70: 7c6f 3130 203d 207c 7b32 7d20 7c20 6120  |o10 = |{2} | a 
│ │ │ │ +0005aa80: 6220 7c7c 2020 2020 2020 2020 2020 2020  b ||            
│ │ │ │  0005aa90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005aaa0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0005aab0: 7c20 2020 2020 207c 7b32 7d20 7c20 3020  |      |{2} | 0 
│ │ │ │ -0005aac0: 7520 7c7c 2020 2020 2020 2020 2020 2020  u ||            
│ │ │ │ -0005aad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005aae0: 2020 207c 0a7c 2020 2020 2020 2b2d 2d2d     |.|      +---
│ │ │ │ -0005aaf0: 2d2d 2d2d 2d2d 2d2d 2b20 2020 2020 2020  --------+       
│ │ │ │ -0005ab00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ab10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0005ab20: 207c 7b32 7d20 7c20 6220 6120 7c7c 2020   |{2} | b a ||  
│ │ │ │ +0005aaa0: 2020 207c 0a7c 2020 2020 2020 7c7b 327d     |.|      |{2}
│ │ │ │ +0005aab0: 207c 2030 2075 207c 7c20 2020 2020 2020   | 0 u ||       
│ │ │ │ +0005aac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005aad0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0005aae0: 202b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 2020   +-----------+  
│ │ │ │ +0005aaf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ab00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0005ab10: 2020 2020 2020 7c7b 327d 207c 2062 2061        |{2} | b a
│ │ │ │ +0005ab20: 207c 7c20 2020 2020 2020 2020 2020 2020   ||             
│ │ │ │  0005ab30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ab40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0005ab50: 2020 2020 2020 2b2d 2d2d 2d2d 2d2d 2d2d        +---------
│ │ │ │ -0005ab60: 2d2d 2b20 2020 2020 2020 2020 2020 2020  --+             
│ │ │ │ -0005ab70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ab80: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0005ab40: 2020 7c0a 7c20 2020 2020 202b 2d2d 2d2d    |.|      +----
│ │ │ │ +0005ab50: 2d2d 2d2d 2d2d 2d2b 2020 2020 2020 2020  -------+        
│ │ │ │ +0005ab60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ab70: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0005ab80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005ab90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005aba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005abb0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20  -------+.|i11 : 
│ │ │ │ -0005abc0: 6265 7474 6920 6672 6565 5265 736f 6c75  betti freeResolu
│ │ │ │ -0005abd0: 7469 6f6e 284d 2c20 4c65 6e67 7468 4c69  tion(M, LengthLi
│ │ │ │ -0005abe0: 6d69 7420 3d3e 2037 2920 2020 7c0a 7c20  mit => 7)   |.| 
│ │ │ │ +0005aba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0005abb0: 3131 203a 2062 6574 7469 2066 7265 6552  11 : betti freeR
│ │ │ │ +0005abc0: 6573 6f6c 7574 696f 6e28 4d2c 204c 656e  esolution(M, Len
│ │ │ │ +0005abd0: 6774 684c 696d 6974 203d 3e20 3729 2020  gthLimit => 7)  
│ │ │ │ +0005abe0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0005abf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ac00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ac10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ac20: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0005ac30: 2030 2031 2032 2033 2034 2035 2036 2020   0 1 2 3 4 5 6  
│ │ │ │ -0005ac40: 3720 2020 2020 2020 2020 2020 2020 2020  7               
│ │ │ │ -0005ac50: 2020 2020 2020 7c0a 7c6f 3131 203d 2074        |.|o11 = t
│ │ │ │ -0005ac60: 6f74 616c 3a20 3320 3420 3520 3620 3720  otal: 3 4 5 6 7 
│ │ │ │ -0005ac70: 3820 3920 3130 2020 2020 2020 2020 2020  8 9 10          
│ │ │ │ -0005ac80: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0005ac90: 2020 2020 2020 2020 323a 2033 2034 2035          2: 3 4 5
│ │ │ │ -0005aca0: 2036 2037 2038 2039 2031 3020 2020 2020   6 7 8 9 10     
│ │ │ │ -0005acb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005acc0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0005ac10: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0005ac20: 2020 2020 2020 3020 3120 3220 3320 3420        0 1 2 3 4 
│ │ │ │ +0005ac30: 3520 3620 2037 2020 2020 2020 2020 2020  5 6  7          
│ │ │ │ +0005ac40: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0005ac50: 3120 3d20 746f 7461 6c3a 2033 2034 2035  1 = total: 3 4 5
│ │ │ │ +0005ac60: 2036 2037 2038 2039 2031 3020 2020 2020   6 7 8 9 10     
│ │ │ │ +0005ac70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ac80: 7c0a 7c20 2020 2020 2020 2020 2032 3a20  |.|          2: 
│ │ │ │ +0005ac90: 3320 3420 3520 3620 3720 3820 3920 3130  3 4 5 6 7 8 9 10
│ │ │ │ +0005aca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005acb0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005acc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005acd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ace0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005acf0: 2020 2020 207c 0a7c 6f31 3120 3a20 4265       |.|o11 : Be
│ │ │ │ -0005ad00: 7474 6954 616c 6c79 2020 2020 2020 2020  ttiTally        
│ │ │ │ -0005ad10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ad20: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0005ace0: 2020 2020 2020 2020 2020 7c0a 7c6f 3131            |.|o11
│ │ │ │ +0005acf0: 203a 2042 6574 7469 5461 6c6c 7920 2020   : BettiTally   
│ │ │ │ +0005ad00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ad10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005ad20: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  0005ad30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005ad40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ad50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0005ad60: 0a7c 6931 3220 3a20 696e 6669 6e69 7465  .|i12 : infinite
│ │ │ │ -0005ad70: 4265 7474 694e 756d 6265 7273 2028 4d46  BettiNumbers (MF
│ │ │ │ -0005ad80: 2c37 2920 2020 2020 2020 2020 2020 2020  ,7)             
│ │ │ │ -0005ad90: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0005ad50: 2d2d 2d2d 2b0a 7c69 3132 203a 2069 6e66  ----+.|i12 : inf
│ │ │ │ +0005ad60: 696e 6974 6542 6574 7469 4e75 6d62 6572  initeBettiNumber
│ │ │ │ +0005ad70: 7320 284d 462c 3729 2020 2020 2020 2020  s (MF,7)        
│ │ │ │ +0005ad80: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005ad90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ada0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005adb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005adc0: 2020 2020 2020 2020 207c 0a7c 6f31 3220           |.|o12 
│ │ │ │ -0005add0: 3d20 7b33 2c20 342c 2035 2c20 362c 2037  = {3, 4, 5, 6, 7
│ │ │ │ -0005ade0: 2c20 382c 2039 2c20 3130 7d20 2020 2020  , 8, 9, 10}     
│ │ │ │ -0005adf0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0005ae00: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0005adb0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005adc0: 7c6f 3132 203d 207b 332c 2034 2c20 352c  |o12 = {3, 4, 5,
│ │ │ │ +0005add0: 2036 2c20 372c 2038 2c20 392c 2031 307d   6, 7, 8, 9, 10}
│ │ │ │ +0005ade0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005adf0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0005ae00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ae10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ae20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ae30: 2020 207c 0a7c 6f31 3220 3a20 4c69 7374     |.|o12 : List
│ │ │ │ +0005ae20: 2020 2020 2020 2020 7c0a 7c6f 3132 203a          |.|o12 :
│ │ │ │ +0005ae30: 204c 6973 7420 2020 2020 2020 2020 2020   List           
│ │ │ │  0005ae40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ae50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ae60: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0005ae50: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0005ae60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005ae70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005ae80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ae90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -0005aea0: 6931 3320 3a20 6265 7474 6920 6672 6565  i13 : betti free
│ │ │ │ -0005aeb0: 5265 736f 6c75 7469 6f6e 2070 7573 6846  Resolution pushF
│ │ │ │ -0005aec0: 6f72 7761 7264 286d 6170 2852 2c53 292c  orward(map(R,S),
│ │ │ │ -0005aed0: 4d29 7c0a 7c20 2020 2020 2020 2020 2020  M)|.|           
│ │ │ │ +0005ae90: 2d2d 2b0a 7c69 3133 203a 2062 6574 7469  --+.|i13 : betti
│ │ │ │ +0005aea0: 2066 7265 6552 6573 6f6c 7574 696f 6e20   freeResolution 
│ │ │ │ +0005aeb0: 7075 7368 466f 7277 6172 6428 6d61 7028  pushForward(map(
│ │ │ │ +0005aec0: 522c 5329 2c4d 297c 0a7c 2020 2020 2020  R,S),M)|.|      
│ │ │ │ +0005aed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005aee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005aef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005af00: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0005af10: 2020 2020 2020 2030 2031 2032 2020 2020         0 1 2    
│ │ │ │ +0005aef0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0005af00: 2020 2020 2020 2020 2020 2020 3020 3120              0 1 
│ │ │ │ +0005af10: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  0005af20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005af30: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0005af40: 3133 203d 2074 6f74 616c 3a20 3320 3520  13 = total: 3 5 
│ │ │ │ -0005af50: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -0005af60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005af70: 207c 0a7c 2020 2020 2020 2020 2020 323a   |.|          2:
│ │ │ │ -0005af80: 2033 2034 202e 2020 2020 2020 2020 2020   3 4 .          
│ │ │ │ -0005af90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005afa0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0005afb0: 2020 2033 3a20 2e20 3120 3220 2020 2020     3: . 1 2     
│ │ │ │ +0005af30: 207c 0a7c 6f31 3320 3d20 746f 7461 6c3a   |.|o13 = total:
│ │ │ │ +0005af40: 2033 2035 2032 2020 2020 2020 2020 2020   3 5 2          
│ │ │ │ +0005af50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005af60: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0005af70: 2020 2032 3a20 3320 3420 2e20 2020 2020     2: 3 4 .     
│ │ │ │ +0005af80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005af90: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0005afa0: 2020 2020 2020 2020 333a 202e 2031 2032          3: . 1 2
│ │ │ │ +0005afb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005afc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005afd0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0005afd0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0005afe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005aff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005b000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005b010: 7c0a 7c6f 3133 203a 2042 6574 7469 5461  |.|o13 : BettiTa
│ │ │ │ -0005b020: 6c6c 7920 2020 2020 2020 2020 2020 2020  lly             
│ │ │ │ -0005b030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005b040: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0005b000: 2020 2020 207c 0a7c 6f31 3320 3a20 4265       |.|o13 : Be
│ │ │ │ +0005b010: 7474 6954 616c 6c79 2020 2020 2020 2020  ttiTally        
│ │ │ │ +0005b020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005b030: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0005b040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005b050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005b060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005b070: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3134  ----------+.|i14
│ │ │ │ -0005b080: 203a 2066 696e 6974 6542 6574 7469 4e75   : finiteBettiNu
│ │ │ │ -0005b090: 6d62 6572 7320 4d46 2020 2020 2020 2020  mbers MF        
│ │ │ │ -0005b0a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0005b0b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0005b060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0005b070: 0a7c 6931 3420 3a20 6669 6e69 7465 4265  .|i14 : finiteBe
│ │ │ │ +0005b080: 7474 694e 756d 6265 7273 204d 4620 2020  ttiNumbers MF   
│ │ │ │ +0005b090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005b0a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0005b0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005b0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005b0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005b0e0: 2020 2020 7c0a 7c6f 3134 203d 207b 332c      |.|o14 = {3,
│ │ │ │ -0005b0f0: 2035 2c20 327d 2020 2020 2020 2020 2020   5, 2}          
│ │ │ │ -0005b100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005b110: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005b0d0: 2020 2020 2020 2020 207c 0a7c 6f31 3420           |.|o14 
│ │ │ │ +0005b0e0: 3d20 7b33 2c20 352c 2032 7d20 2020 2020  = {3, 5, 2}     
│ │ │ │ +0005b0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005b100: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005b110: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0005b120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005b130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005b140: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0005b150: 7c6f 3134 203a 204c 6973 7420 2020 2020  |o14 : List     
│ │ │ │ +0005b140: 2020 207c 0a7c 6f31 3420 3a20 4c69 7374     |.|o14 : List
│ │ │ │ +0005b150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005b160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005b170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005b180: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0005b170: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0005b180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005b190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005b1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005b1b0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061  --------+..See a
│ │ │ │ -0005b1c0: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020  lso.========..  
│ │ │ │ -0005b1d0: 2a20 2a6e 6f74 6520 6669 6e69 7465 4265  * *note finiteBe
│ │ │ │ -0005b1e0: 7474 694e 756d 6265 7273 3a20 6669 6e69  ttiNumbers: fini
│ │ │ │ -0005b1f0: 7465 4265 7474 694e 756d 6265 7273 2c20  teBettiNumbers, 
│ │ │ │ -0005b200: 2d2d 2062 6574 7469 206e 756d 6265 7273  -- betti numbers
│ │ │ │ -0005b210: 206f 6620 6669 6e69 7465 0a20 2020 2072   of finite.    r
│ │ │ │ -0005b220: 6573 6f6c 7574 696f 6e20 636f 6d70 7574  esolution comput
│ │ │ │ -0005b230: 6564 2066 726f 6d20 6120 6d61 7472 6978  ed from a matrix
│ │ │ │ -0005b240: 2066 6163 746f 7269 7a61 7469 6f6e 0a20   factorization. 
│ │ │ │ -0005b250: 202a 202a 6e6f 7465 2069 6e66 696e 6974   * *note infinit
│ │ │ │ -0005b260: 6542 6574 7469 4e75 6d62 6572 733a 2069  eBettiNumbers: i
│ │ │ │ -0005b270: 6e66 696e 6974 6542 6574 7469 4e75 6d62  nfiniteBettiNumb
│ │ │ │ -0005b280: 6572 732c 202d 2d20 6265 7474 6920 6e75  ers, -- betti nu
│ │ │ │ -0005b290: 6d62 6572 7320 6f66 0a20 2020 2066 696e  mbers of.    fin
│ │ │ │ -0005b2a0: 6974 6520 7265 736f 6c75 7469 6f6e 2063  ite resolution c
│ │ │ │ -0005b2b0: 6f6d 7075 7465 6420 6672 6f6d 2061 206d  omputed from a m
│ │ │ │ -0005b2c0: 6174 7269 7820 6661 6374 6f72 697a 6174  atrix factorizat
│ │ │ │ -0005b2d0: 696f 6e0a 2020 2a20 2a6e 6f74 6520 6869  ion.  * *note hi
│ │ │ │ -0005b2e0: 6768 5379 7a79 6779 3a20 6869 6768 5379  ghSyzygy: highSy
│ │ │ │ -0005b2f0: 7a79 6779 2c20 2d2d 2052 6574 7572 6e73  zygy, -- Returns
│ │ │ │ -0005b300: 2061 2073 797a 7967 7920 6d6f 6475 6c65   a syzygy module
│ │ │ │ -0005b310: 206f 6e65 2062 6579 6f6e 6420 7468 650a   one beyond the.
│ │ │ │ -0005b320: 2020 2020 7265 6775 6c61 7269 7479 206f      regularity o
│ │ │ │ -0005b330: 6620 4578 7428 4d2c 6b29 0a20 202a 202a  f Ext(M,k).  * *
│ │ │ │ -0005b340: 6e6f 7465 2062 4d61 7073 3a20 624d 6170  note bMaps: bMap
│ │ │ │ -0005b350: 732c 202d 2d20 6c69 7374 2074 6865 206d  s, -- list the m
│ │ │ │ -0005b360: 6170 7320 2064 5f70 3a42 5f31 2870 292d  aps  d_p:B_1(p)-
│ │ │ │ -0005b370: 2d3e 425f 3028 7029 2069 6e20 610a 2020  ->B_0(p) in a.  
│ │ │ │ -0005b380: 2020 6d61 7472 6978 4661 6374 6f72 697a    matrixFactoriz
│ │ │ │ -0005b390: 6174 696f 6e0a 2020 2a20 2a6e 6f74 6520  ation.  * *note 
│ │ │ │ -0005b3a0: 4252 616e 6b73 3a20 4252 616e 6b73 2c20  BRanks: BRanks, 
│ │ │ │ -0005b3b0: 2d2d 2072 616e 6b73 206f 6620 7468 6520  -- ranks of the 
│ │ │ │ -0005b3c0: 6d6f 6475 6c65 7320 425f 6928 6429 2069  modules B_i(d) i
│ │ │ │ -0005b3d0: 6e20 610a 2020 2020 6d61 7472 6978 4661  n a.    matrixFa
│ │ │ │ -0005b3e0: 6374 6f72 697a 6174 696f 6e0a 0a57 6179  ctorization..Way
│ │ │ │ -0005b3f0: 7320 746f 2075 7365 206d 6174 7269 7846  s to use matrixF
│ │ │ │ -0005b400: 6163 746f 7269 7a61 7469 6f6e 3a0a 3d3d  actorization:.==
│ │ │ │ +0005b1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ +0005b1b0: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d  See also.=======
│ │ │ │ +0005b1c0: 3d0a 0a20 202a 202a 6e6f 7465 2066 696e  =..  * *note fin
│ │ │ │ +0005b1d0: 6974 6542 6574 7469 4e75 6d62 6572 733a  iteBettiNumbers:
│ │ │ │ +0005b1e0: 2066 696e 6974 6542 6574 7469 4e75 6d62   finiteBettiNumb
│ │ │ │ +0005b1f0: 6572 732c 202d 2d20 6265 7474 6920 6e75  ers, -- betti nu
│ │ │ │ +0005b200: 6d62 6572 7320 6f66 2066 696e 6974 650a  mbers of finite.
│ │ │ │ +0005b210: 2020 2020 7265 736f 6c75 7469 6f6e 2063      resolution c
│ │ │ │ +0005b220: 6f6d 7075 7465 6420 6672 6f6d 2061 206d  omputed from a m
│ │ │ │ +0005b230: 6174 7269 7820 6661 6374 6f72 697a 6174  atrix factorizat
│ │ │ │ +0005b240: 696f 6e0a 2020 2a20 2a6e 6f74 6520 696e  ion.  * *note in
│ │ │ │ +0005b250: 6669 6e69 7465 4265 7474 694e 756d 6265  finiteBettiNumbe
│ │ │ │ +0005b260: 7273 3a20 696e 6669 6e69 7465 4265 7474  rs: infiniteBett
│ │ │ │ +0005b270: 694e 756d 6265 7273 2c20 2d2d 2062 6574  iNumbers, -- bet
│ │ │ │ +0005b280: 7469 206e 756d 6265 7273 206f 660a 2020  ti numbers of.  
│ │ │ │ +0005b290: 2020 6669 6e69 7465 2072 6573 6f6c 7574    finite resolut
│ │ │ │ +0005b2a0: 696f 6e20 636f 6d70 7574 6564 2066 726f  ion computed fro
│ │ │ │ +0005b2b0: 6d20 6120 6d61 7472 6978 2066 6163 746f  m a matrix facto
│ │ │ │ +0005b2c0: 7269 7a61 7469 6f6e 0a20 202a 202a 6e6f  rization.  * *no
│ │ │ │ +0005b2d0: 7465 2068 6967 6853 797a 7967 793a 2068  te highSyzygy: h
│ │ │ │ +0005b2e0: 6967 6853 797a 7967 792c 202d 2d20 5265  ighSyzygy, -- Re
│ │ │ │ +0005b2f0: 7475 726e 7320 6120 7379 7a79 6779 206d  turns a syzygy m
│ │ │ │ +0005b300: 6f64 756c 6520 6f6e 6520 6265 796f 6e64  odule one beyond
│ │ │ │ +0005b310: 2074 6865 0a20 2020 2072 6567 756c 6172   the.    regular
│ │ │ │ +0005b320: 6974 7920 6f66 2045 7874 284d 2c6b 290a  ity of Ext(M,k).
│ │ │ │ +0005b330: 2020 2a20 2a6e 6f74 6520 624d 6170 733a    * *note bMaps:
│ │ │ │ +0005b340: 2062 4d61 7073 2c20 2d2d 206c 6973 7420   bMaps, -- list 
│ │ │ │ +0005b350: 7468 6520 6d61 7073 2020 645f 703a 425f  the maps  d_p:B_
│ │ │ │ +0005b360: 3128 7029 2d2d 3e42 5f30 2870 2920 696e  1(p)-->B_0(p) in
│ │ │ │ +0005b370: 2061 0a20 2020 206d 6174 7269 7846 6163   a.    matrixFac
│ │ │ │ +0005b380: 746f 7269 7a61 7469 6f6e 0a20 202a 202a  torization.  * *
│ │ │ │ +0005b390: 6e6f 7465 2042 5261 6e6b 733a 2042 5261  note BRanks: BRa
│ │ │ │ +0005b3a0: 6e6b 732c 202d 2d20 7261 6e6b 7320 6f66  nks, -- ranks of
│ │ │ │ +0005b3b0: 2074 6865 206d 6f64 756c 6573 2042 5f69   the modules B_i
│ │ │ │ +0005b3c0: 2864 2920 696e 2061 0a20 2020 206d 6174  (d) in a.    mat
│ │ │ │ +0005b3d0: 7269 7846 6163 746f 7269 7a61 7469 6f6e  rixFactorization
│ │ │ │ +0005b3e0: 0a0a 5761 7973 2074 6f20 7573 6520 6d61  ..Ways to use ma
│ │ │ │ +0005b3f0: 7472 6978 4661 6374 6f72 697a 6174 696f  trixFactorizatio
│ │ │ │ +0005b400: 6e3a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  n:.=============
│ │ │ │  0005b410: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0005b420: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ -0005b430: 2020 2a20 226d 6174 7269 7846 6163 746f    * "matrixFacto
│ │ │ │ -0005b440: 7269 7a61 7469 6f6e 284d 6174 7269 782c  rization(Matrix,
│ │ │ │ -0005b450: 4d6f 6475 6c65 2922 0a0a 466f 7220 7468  Module)"..For th
│ │ │ │ -0005b460: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d  e programmer.===
│ │ │ │ -0005b470: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ -0005b480: 0a54 6865 206f 626a 6563 7420 2a6e 6f74  .The object *not
│ │ │ │ -0005b490: 6520 6d61 7472 6978 4661 6374 6f72 697a  e matrixFactoriz
│ │ │ │ -0005b4a0: 6174 696f 6e3a 206d 6174 7269 7846 6163  ation: matrixFac
│ │ │ │ -0005b4b0: 746f 7269 7a61 7469 6f6e 2c20 6973 2061  torization, is a
│ │ │ │ -0005b4c0: 202a 6e6f 7465 206d 6574 686f 640a 6675   *note method.fu
│ │ │ │ -0005b4d0: 6e63 7469 6f6e 2077 6974 6820 6f70 7469  nction with opti
│ │ │ │ -0005b4e0: 6f6e 733a 2028 4d61 6361 756c 6179 3244  ons: (Macaulay2D
│ │ │ │ -0005b4f0: 6f63 294d 6574 686f 6446 756e 6374 696f  oc)MethodFunctio
│ │ │ │ -0005b500: 6e57 6974 684f 7074 696f 6e73 2c2e 0a0a  nWithOptions,...
│ │ │ │ +0005b420: 3d3d 3d0a 0a20 202a 2022 6d61 7472 6978  ===..  * "matrix
│ │ │ │ +0005b430: 4661 6374 6f72 697a 6174 696f 6e28 4d61  Factorization(Ma
│ │ │ │ +0005b440: 7472 6978 2c4d 6f64 756c 6529 220a 0a46  trix,Module)"..F
│ │ │ │ +0005b450: 6f72 2074 6865 2070 726f 6772 616d 6d65  or the programme
│ │ │ │ +0005b460: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  r.==============
│ │ │ │ +0005b470: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374  ====..The object
│ │ │ │ +0005b480: 202a 6e6f 7465 206d 6174 7269 7846 6163   *note matrixFac
│ │ │ │ +0005b490: 746f 7269 7a61 7469 6f6e 3a20 6d61 7472  torization: matr
│ │ │ │ +0005b4a0: 6978 4661 6374 6f72 697a 6174 696f 6e2c  ixFactorization,
│ │ │ │ +0005b4b0: 2069 7320 6120 2a6e 6f74 6520 6d65 7468   is a *note meth
│ │ │ │ +0005b4c0: 6f64 0a66 756e 6374 696f 6e20 7769 7468  od.function with
│ │ │ │ +0005b4d0: 206f 7074 696f 6e73 3a20 284d 6163 6175   options: (Macau
│ │ │ │ +0005b4e0: 6c61 7932 446f 6329 4d65 7468 6f64 4675  lay2Doc)MethodFu
│ │ │ │ +0005b4f0: 6e63 7469 6f6e 5769 7468 4f70 7469 6f6e  nctionWithOption
│ │ │ │ +0005b500: 732c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d  s,...-----------
│ │ │ │  0005b510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005b520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005b530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005b540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005b550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a  ---------------.
│ │ │ │ -0005b560: 0a54 6865 2073 6f75 7263 6520 6f66 2074  .The source of t
│ │ │ │ -0005b570: 6869 7320 646f 6375 6d65 6e74 2069 7320  his document is 
│ │ │ │ -0005b580: 696e 0a2f 6275 696c 642f 7265 7072 6f64  in./build/reprod
│ │ │ │ -0005b590: 7563 6962 6c65 2d70 6174 682f 6d61 6361  ucible-path/maca
│ │ │ │ -0005b5a0: 756c 6179 322d 312e 3235 2e31 312b 6473  ulay2-1.25.11+ds
│ │ │ │ -0005b5b0: 2f4d 322f 4d61 6361 756c 6179 322f 7061  /M2/Macaulay2/pa
│ │ │ │ -0005b5c0: 636b 6167 6573 2f0a 436f 6d70 6c65 7465  ckages/.Complete
│ │ │ │ -0005b5d0: 496e 7465 7273 6563 7469 6f6e 5265 736f  IntersectionReso
│ │ │ │ -0005b5e0: 6c75 7469 6f6e 732e 6d32 3a34 3033 333a  lutions.m2:4033:
│ │ │ │ -0005b5f0: 302e 0a1f 0a46 696c 653a 2043 6f6d 706c  0....File: Compl
│ │ │ │ -0005b600: 6574 6549 6e74 6572 7365 6374 696f 6e52  eteIntersectionR
│ │ │ │ -0005b610: 6573 6f6c 7574 696f 6e73 2e69 6e66 6f2c  esolutions.info,
│ │ │ │ -0005b620: 204e 6f64 653a 206d 6642 6f75 6e64 2c20   Node: mfBound, 
│ │ │ │ -0005b630: 4e65 7874 3a20 6d6f 6475 6c65 4173 4578  Next: moduleAsEx
│ │ │ │ -0005b640: 742c 2050 7265 763a 206d 6174 7269 7846  t, Prev: matrixF
│ │ │ │ -0005b650: 6163 746f 7269 7a61 7469 6f6e 2c20 5570  actorization, Up
│ │ │ │ -0005b660: 3a20 546f 700a 0a6d 6642 6f75 6e64 202d  : Top..mfBound -
│ │ │ │ -0005b670: 2d20 6465 7465 726d 696e 6573 2068 6f77  - determines how
│ │ │ │ -0005b680: 2068 6967 6820 6120 7379 7a79 6779 2074   high a syzygy t
│ │ │ │ -0005b690: 6f20 7461 6b65 2066 6f72 2022 6d61 7472  o take for "matr
│ │ │ │ -0005b6a0: 6978 4661 6374 6f72 697a 6174 696f 6e22  ixFactorization"
│ │ │ │ -0005b6b0: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  .***************
│ │ │ │ +0005b550: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365  ----..The source
│ │ │ │ +0005b560: 206f 6620 7468 6973 2064 6f63 756d 656e   of this documen
│ │ │ │ +0005b570: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72  t is in./build/r
│ │ │ │ +0005b580: 6570 726f 6475 6369 626c 652d 7061 7468  eproducible-path
│ │ │ │ +0005b590: 2f6d 6163 6175 6c61 7932 2d31 2e32 352e  /macaulay2-1.25.
│ │ │ │ +0005b5a0: 3131 2b64 732f 4d32 2f4d 6163 6175 6c61  11+ds/M2/Macaula
│ │ │ │ +0005b5b0: 7932 2f70 6163 6b61 6765 732f 0a43 6f6d  y2/packages/.Com
│ │ │ │ +0005b5c0: 706c 6574 6549 6e74 6572 7365 6374 696f  pleteIntersectio
│ │ │ │ +0005b5d0: 6e52 6573 6f6c 7574 696f 6e73 2e6d 323a  nResolutions.m2:
│ │ │ │ +0005b5e0: 3430 3333 3a30 2e0a 1f0a 4669 6c65 3a20  4033:0....File: 
│ │ │ │ +0005b5f0: 436f 6d70 6c65 7465 496e 7465 7273 6563  CompleteIntersec
│ │ │ │ +0005b600: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e  tionResolutions.
│ │ │ │ +0005b610: 696e 666f 2c20 4e6f 6465 3a20 6d66 426f  info, Node: mfBo
│ │ │ │ +0005b620: 756e 642c 204e 6578 743a 206d 6f64 756c  und, Next: modul
│ │ │ │ +0005b630: 6541 7345 7874 2c20 5072 6576 3a20 6d61  eAsExt, Prev: ma
│ │ │ │ +0005b640: 7472 6978 4661 6374 6f72 697a 6174 696f  trixFactorizatio
│ │ │ │ +0005b650: 6e2c 2055 703a 2054 6f70 0a0a 6d66 426f  n, Up: Top..mfBo
│ │ │ │ +0005b660: 756e 6420 2d2d 2064 6574 6572 6d69 6e65  und -- determine
│ │ │ │ +0005b670: 7320 686f 7720 6869 6768 2061 2073 797a  s how high a syz
│ │ │ │ +0005b680: 7967 7920 746f 2074 616b 6520 666f 7220  ygy to take for 
│ │ │ │ +0005b690: 226d 6174 7269 7846 6163 746f 7269 7a61  "matrixFactoriza
│ │ │ │ +0005b6a0: 7469 6f6e 220a 2a2a 2a2a 2a2a 2a2a 2a2a  tion".**********
│ │ │ │ +0005b6b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0005b6c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0005b6d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0005b6e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0005b6f0: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ -0005b700: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ -0005b710: 7020 3d20 6d66 426f 756e 6420 4d0a 2020  p = mfBound M.  
│ │ │ │ -0005b720: 2a20 496e 7075 7473 3a0a 2020 2020 2020  * Inputs:.      
│ │ │ │ -0005b730: 2a20 4d2c 2061 202a 6e6f 7465 206d 6f64  * M, a *note mod
│ │ │ │ -0005b740: 756c 653a 2028 4d61 6361 756c 6179 3244  ule: (Macaulay2D
│ │ │ │ -0005b750: 6f63 294d 6f64 756c 652c 2c20 6f76 6572  oc)Module,, over
│ │ │ │ -0005b760: 2061 2063 6f6d 706c 6574 6520 696e 7465   a complete inte
│ │ │ │ -0005b770: 7273 6563 7469 6f6e 0a20 202a 204f 7574  rsection.  * Out
│ │ │ │ -0005b780: 7075 7473 3a0a 2020 2020 2020 2a20 702c  puts:.      * p,
│ │ │ │ -0005b790: 2061 6e20 2a6e 6f74 6520 696e 7465 6765   an *note intege
│ │ │ │ -0005b7a0: 723a 2028 4d61 6361 756c 6179 3244 6f63  r: (Macaulay2Doc
│ │ │ │ -0005b7b0: 295a 5a2c 2c20 0a0a 4465 7363 7269 7074  )ZZ,, ..Descript
│ │ │ │ -0005b7c0: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ion.===========.
│ │ │ │ -0005b7d0: 0a49 6620 7020 3d20 6d66 426f 756e 6420  .If p = mfBound 
│ │ │ │ -0005b7e0: 4d2c 2074 6865 6e20 7468 6520 702d 7468  M, then the p-th
│ │ │ │ -0005b7f0: 2073 797a 7967 7920 6f66 204d 2c20 7768   syzygy of M, wh
│ │ │ │ -0005b800: 6963 6820 6973 2063 6f6d 7075 7465 6420  ich is computed 
│ │ │ │ -0005b810: 6279 0a68 6967 6853 797a 7967 7928 4d29  by.highSyzygy(M)
│ │ │ │ -0005b820: 2c20 7368 6f75 6c64 2028 7468 6973 2069  , should (this i
│ │ │ │ -0005b830: 7320 6120 636f 6e6a 6563 7475 7265 2920  s a conjecture) 
│ │ │ │ -0005b840: 6265 2061 2022 6869 6768 2053 797a 7967  be a "high Syzyg
│ │ │ │ -0005b850: 7922 2069 6e20 7468 6520 7365 6e73 650a  y" in the sense.
│ │ │ │ -0005b860: 7265 7175 6972 6564 2066 6f72 206d 6174  required for mat
│ │ │ │ -0005b870: 7269 7846 6163 746f 7269 7a61 7469 6f6e  rixFactorization
│ │ │ │ -0005b880: 2e20 496e 2065 7861 6d70 6c65 732c 2074  . In examples, t
│ │ │ │ -0005b890: 6865 2065 7374 696d 6174 6520 7365 656d  he estimate seem
│ │ │ │ -0005b8a0: 7320 7368 6172 7020 2865 7863 6570 740a  s sharp (except.
│ │ │ │ -0005b8b0: 7768 656e 204d 2069 7320 616c 7265 6164  when M is alread
│ │ │ │ -0005b8c0: 7920 6120 6869 6768 2073 797a 7967 7929  y a high syzygy)
│ │ │ │ -0005b8d0: 2e0a 0a54 6865 2061 6374 7561 6c20 666f  ...The actual fo
│ │ │ │ -0005b8e0: 726d 756c 6120 7573 6564 2069 733a 0a0a  rmula used is:..
│ │ │ │ -0005b8f0: 6d66 426f 756e 6420 4d20 3d20 6d61 7828  mfBound M = max(
│ │ │ │ -0005b900: 322a 725f 7b65 7665 6e7d 2c20 312b 322a  2*r_{even}, 1+2*
│ │ │ │ -0005b910: 725f 7b6f 6464 7d29 0a0a 7768 6572 6520  r_{odd})..where 
│ │ │ │ -0005b920: 725f 7b65 7665 6e7d 203d 2072 6567 756c  r_{even} = regul
│ │ │ │ -0005b930: 6172 6974 7920 6576 656e 4578 744d 6f64  arity evenExtMod
│ │ │ │ -0005b940: 756c 6520 4d20 616e 6420 725f 7b6f 6464  ule M and r_{odd
│ │ │ │ -0005b950: 7d20 3d20 7265 6775 6c61 7269 7479 0a6f  } = regularity.o
│ │ │ │ -0005b960: 6464 4578 744d 6f64 756c 6520 4d2e 2048  ddExtModule M. H
│ │ │ │ -0005b970: 6572 6520 6576 656e 4578 744d 6f64 756c  ere evenExtModul
│ │ │ │ -0005b980: 6520 4d20 6973 2074 6865 2065 7665 6e20  e M is the even 
│ │ │ │ -0005b990: 6465 6772 6565 2070 6172 7420 6f66 2045  degree part of E
│ │ │ │ -0005b9a0: 7874 284d 2c20 2872 6573 6964 7565 0a63  xt(M, (residue.c
│ │ │ │ -0005b9b0: 6c61 7373 2066 6965 6c64 2929 2e0a 0a53  lass field))...S
│ │ │ │ -0005b9c0: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d  ee also.========
│ │ │ │ -0005b9d0: 0a0a 2020 2a20 2a6e 6f74 6520 6869 6768  ..  * *note high
│ │ │ │ -0005b9e0: 5379 7a79 6779 3a20 6869 6768 5379 7a79  Syzygy: highSyzy
│ │ │ │ -0005b9f0: 6779 2c20 2d2d 2052 6574 7572 6e73 2061  gy, -- Returns a
│ │ │ │ -0005ba00: 2073 797a 7967 7920 6d6f 6475 6c65 206f   syzygy module o
│ │ │ │ -0005ba10: 6e65 2062 6579 6f6e 6420 7468 650a 2020  ne beyond the.  
│ │ │ │ -0005ba20: 2020 7265 6775 6c61 7269 7479 206f 6620    regularity of 
│ │ │ │ -0005ba30: 4578 7428 4d2c 6b29 0a20 202a 202a 6e6f  Ext(M,k).  * *no
│ │ │ │ -0005ba40: 7465 2065 7665 6e45 7874 4d6f 6475 6c65  te evenExtModule
│ │ │ │ -0005ba50: 3a20 6576 656e 4578 744d 6f64 756c 652c  : evenExtModule,
│ │ │ │ -0005ba60: 202d 2d20 6576 656e 2070 6172 7420 6f66   -- even part of
│ │ │ │ -0005ba70: 2045 7874 5e2a 284d 2c6b 2920 6f76 6572   Ext^*(M,k) over
│ │ │ │ -0005ba80: 2061 0a20 2020 2063 6f6d 706c 6574 6520   a.    complete 
│ │ │ │ -0005ba90: 696e 7465 7273 6563 7469 6f6e 2061 7320  intersection as 
│ │ │ │ -0005baa0: 6d6f 6475 6c65 206f 7665 7220 4349 206f  module over CI o
│ │ │ │ -0005bab0: 7065 7261 746f 7220 7269 6e67 0a20 202a  perator ring.  *
│ │ │ │ -0005bac0: 202a 6e6f 7465 206f 6464 4578 744d 6f64   *note oddExtMod
│ │ │ │ -0005bad0: 756c 653a 206f 6464 4578 744d 6f64 756c  ule: oddExtModul
│ │ │ │ -0005bae0: 652c 202d 2d20 6f64 6420 7061 7274 206f  e, -- odd part o
│ │ │ │ -0005baf0: 6620 4578 745e 2a28 4d2c 6b29 206f 7665  f Ext^*(M,k) ove
│ │ │ │ -0005bb00: 7220 6120 636f 6d70 6c65 7465 0a20 2020  r a complete.   
│ │ │ │ -0005bb10: 2069 6e74 6572 7365 6374 696f 6e20 6173   intersection as
│ │ │ │ -0005bb20: 206d 6f64 756c 6520 6f76 6572 2043 4920   module over CI 
│ │ │ │ -0005bb30: 6f70 6572 6174 6f72 2072 696e 670a 2020  operator ring.  
│ │ │ │ -0005bb40: 2a20 2a6e 6f74 6520 6d61 7472 6978 4661  * *note matrixFa
│ │ │ │ -0005bb50: 6374 6f72 697a 6174 696f 6e3a 206d 6174  ctorization: mat
│ │ │ │ -0005bb60: 7269 7846 6163 746f 7269 7a61 7469 6f6e  rixFactorization
│ │ │ │ -0005bb70: 2c20 2d2d 204d 6170 7320 696e 2061 2068  , -- Maps in a h
│ │ │ │ -0005bb80: 6967 6865 720a 2020 2020 636f 6469 6d65  igher.    codime
│ │ │ │ -0005bb90: 6e73 696f 6e20 6d61 7472 6978 2066 6163  nsion matrix fac
│ │ │ │ -0005bba0: 746f 7269 7a61 7469 6f6e 0a0a 5761 7973  torization..Ways
│ │ │ │ -0005bbb0: 2074 6f20 7573 6520 6d66 426f 756e 643a   to use mfBound:
│ │ │ │ -0005bbc0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ -0005bbd0: 3d3d 3d3d 3d0a 0a20 202a 2022 6d66 426f  =====..  * "mfBo
│ │ │ │ -0005bbe0: 756e 6428 4d6f 6475 6c65 2922 0a0a 466f  und(Module)"..Fo
│ │ │ │ -0005bbf0: 7220 7468 6520 7072 6f67 7261 6d6d 6572  r the programmer
│ │ │ │ -0005bc00: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ -0005bc10: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420  ===..The object 
│ │ │ │ -0005bc20: 2a6e 6f74 6520 6d66 426f 756e 643a 206d  *note mfBound: m
│ │ │ │ -0005bc30: 6642 6f75 6e64 2c20 6973 2061 202a 6e6f  fBound, is a *no
│ │ │ │ -0005bc40: 7465 206d 6574 686f 6420 6675 6e63 7469  te method functi
│ │ │ │ -0005bc50: 6f6e 3a0a 284d 6163 6175 6c61 7932 446f  on:.(Macaulay2Do
│ │ │ │ -0005bc60: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e  c)MethodFunction
│ │ │ │ -0005bc70: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ,...------------
│ │ │ │ +0005b6e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a  ***************.
│ │ │ │ +0005b6f0: 0a20 202a 2055 7361 6765 3a20 0a20 2020  .  * Usage: .   
│ │ │ │ +0005b700: 2020 2020 2070 203d 206d 6642 6f75 6e64       p = mfBound
│ │ │ │ +0005b710: 204d 0a20 202a 2049 6e70 7574 733a 0a20   M.  * Inputs:. 
│ │ │ │ +0005b720: 2020 2020 202a 204d 2c20 6120 2a6e 6f74       * M, a *not
│ │ │ │ +0005b730: 6520 6d6f 6475 6c65 3a20 284d 6163 6175  e module: (Macau
│ │ │ │ +0005b740: 6c61 7932 446f 6329 4d6f 6475 6c65 2c2c  lay2Doc)Module,,
│ │ │ │ +0005b750: 206f 7665 7220 6120 636f 6d70 6c65 7465   over a complete
│ │ │ │ +0005b760: 2069 6e74 6572 7365 6374 696f 6e0a 2020   intersection.  
│ │ │ │ +0005b770: 2a20 4f75 7470 7574 733a 0a20 2020 2020  * Outputs:.     
│ │ │ │ +0005b780: 202a 2070 2c20 616e 202a 6e6f 7465 2069   * p, an *note i
│ │ │ │ +0005b790: 6e74 6567 6572 3a20 284d 6163 6175 6c61  nteger: (Macaula
│ │ │ │ +0005b7a0: 7932 446f 6329 5a5a 2c2c 200a 0a44 6573  y2Doc)ZZ,, ..Des
│ │ │ │ +0005b7b0: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ +0005b7c0: 3d3d 3d3d 0a0a 4966 2070 203d 206d 6642  ====..If p = mfB
│ │ │ │ +0005b7d0: 6f75 6e64 204d 2c20 7468 656e 2074 6865  ound M, then the
│ │ │ │ +0005b7e0: 2070 2d74 6820 7379 7a79 6779 206f 6620   p-th syzygy of 
│ │ │ │ +0005b7f0: 4d2c 2077 6869 6368 2069 7320 636f 6d70  M, which is comp
│ │ │ │ +0005b800: 7574 6564 2062 790a 6869 6768 5379 7a79  uted by.highSyzy
│ │ │ │ +0005b810: 6779 284d 292c 2073 686f 756c 6420 2874  gy(M), should (t
│ │ │ │ +0005b820: 6869 7320 6973 2061 2063 6f6e 6a65 6374  his is a conject
│ │ │ │ +0005b830: 7572 6529 2062 6520 6120 2268 6967 6820  ure) be a "high 
│ │ │ │ +0005b840: 5379 7a79 6779 2220 696e 2074 6865 2073  Syzygy" in the s
│ │ │ │ +0005b850: 656e 7365 0a72 6571 7569 7265 6420 666f  ense.required fo
│ │ │ │ +0005b860: 7220 6d61 7472 6978 4661 6374 6f72 697a  r matrixFactoriz
│ │ │ │ +0005b870: 6174 696f 6e2e 2049 6e20 6578 616d 706c  ation. In exampl
│ │ │ │ +0005b880: 6573 2c20 7468 6520 6573 7469 6d61 7465  es, the estimate
│ │ │ │ +0005b890: 2073 6565 6d73 2073 6861 7270 2028 6578   seems sharp (ex
│ │ │ │ +0005b8a0: 6365 7074 0a77 6865 6e20 4d20 6973 2061  cept.when M is a
│ │ │ │ +0005b8b0: 6c72 6561 6479 2061 2068 6967 6820 7379  lready a high sy
│ │ │ │ +0005b8c0: 7a79 6779 292e 0a0a 5468 6520 6163 7475  zygy)...The actu
│ │ │ │ +0005b8d0: 616c 2066 6f72 6d75 6c61 2075 7365 6420  al formula used 
│ │ │ │ +0005b8e0: 6973 3a0a 0a6d 6642 6f75 6e64 204d 203d  is:..mfBound M =
│ │ │ │ +0005b8f0: 206d 6178 2832 2a72 5f7b 6576 656e 7d2c   max(2*r_{even},
│ │ │ │ +0005b900: 2031 2b32 2a72 5f7b 6f64 647d 290a 0a77   1+2*r_{odd})..w
│ │ │ │ +0005b910: 6865 7265 2072 5f7b 6576 656e 7d20 3d20  here r_{even} = 
│ │ │ │ +0005b920: 7265 6775 6c61 7269 7479 2065 7665 6e45  regularity evenE
│ │ │ │ +0005b930: 7874 4d6f 6475 6c65 204d 2061 6e64 2072  xtModule M and r
│ │ │ │ +0005b940: 5f7b 6f64 647d 203d 2072 6567 756c 6172  _{odd} = regular
│ │ │ │ +0005b950: 6974 790a 6f64 6445 7874 4d6f 6475 6c65  ity.oddExtModule
│ │ │ │ +0005b960: 204d 2e20 4865 7265 2065 7665 6e45 7874   M. Here evenExt
│ │ │ │ +0005b970: 4d6f 6475 6c65 204d 2069 7320 7468 6520  Module M is the 
│ │ │ │ +0005b980: 6576 656e 2064 6567 7265 6520 7061 7274  even degree part
│ │ │ │ +0005b990: 206f 6620 4578 7428 4d2c 2028 7265 7369   of Ext(M, (resi
│ │ │ │ +0005b9a0: 6475 650a 636c 6173 7320 6669 656c 6429  due.class field)
│ │ │ │ +0005b9b0: 292e 0a0a 5365 6520 616c 736f 0a3d 3d3d  )...See also.===
│ │ │ │ +0005b9c0: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
│ │ │ │ +0005b9d0: 2068 6967 6853 797a 7967 793a 2068 6967   highSyzygy: hig
│ │ │ │ +0005b9e0: 6853 797a 7967 792c 202d 2d20 5265 7475  hSyzygy, -- Retu
│ │ │ │ +0005b9f0: 726e 7320 6120 7379 7a79 6779 206d 6f64  rns a syzygy mod
│ │ │ │ +0005ba00: 756c 6520 6f6e 6520 6265 796f 6e64 2074  ule one beyond t
│ │ │ │ +0005ba10: 6865 0a20 2020 2072 6567 756c 6172 6974  he.    regularit
│ │ │ │ +0005ba20: 7920 6f66 2045 7874 284d 2c6b 290a 2020  y of Ext(M,k).  
│ │ │ │ +0005ba30: 2a20 2a6e 6f74 6520 6576 656e 4578 744d  * *note evenExtM
│ │ │ │ +0005ba40: 6f64 756c 653a 2065 7665 6e45 7874 4d6f  odule: evenExtMo
│ │ │ │ +0005ba50: 6475 6c65 2c20 2d2d 2065 7665 6e20 7061  dule, -- even pa
│ │ │ │ +0005ba60: 7274 206f 6620 4578 745e 2a28 4d2c 6b29  rt of Ext^*(M,k)
│ │ │ │ +0005ba70: 206f 7665 7220 610a 2020 2020 636f 6d70   over a.    comp
│ │ │ │ +0005ba80: 6c65 7465 2069 6e74 6572 7365 6374 696f  lete intersectio
│ │ │ │ +0005ba90: 6e20 6173 206d 6f64 756c 6520 6f76 6572  n as module over
│ │ │ │ +0005baa0: 2043 4920 6f70 6572 6174 6f72 2072 696e   CI operator rin
│ │ │ │ +0005bab0: 670a 2020 2a20 2a6e 6f74 6520 6f64 6445  g.  * *note oddE
│ │ │ │ +0005bac0: 7874 4d6f 6475 6c65 3a20 6f64 6445 7874  xtModule: oddExt
│ │ │ │ +0005bad0: 4d6f 6475 6c65 2c20 2d2d 206f 6464 2070  Module, -- odd p
│ │ │ │ +0005bae0: 6172 7420 6f66 2045 7874 5e2a 284d 2c6b  art of Ext^*(M,k
│ │ │ │ +0005baf0: 2920 6f76 6572 2061 2063 6f6d 706c 6574  ) over a complet
│ │ │ │ +0005bb00: 650a 2020 2020 696e 7465 7273 6563 7469  e.    intersecti
│ │ │ │ +0005bb10: 6f6e 2061 7320 6d6f 6475 6c65 206f 7665  on as module ove
│ │ │ │ +0005bb20: 7220 4349 206f 7065 7261 746f 7220 7269  r CI operator ri
│ │ │ │ +0005bb30: 6e67 0a20 202a 202a 6e6f 7465 206d 6174  ng.  * *note mat
│ │ │ │ +0005bb40: 7269 7846 6163 746f 7269 7a61 7469 6f6e  rixFactorization
│ │ │ │ +0005bb50: 3a20 6d61 7472 6978 4661 6374 6f72 697a  : matrixFactoriz
│ │ │ │ +0005bb60: 6174 696f 6e2c 202d 2d20 4d61 7073 2069  ation, -- Maps i
│ │ │ │ +0005bb70: 6e20 6120 6869 6768 6572 0a20 2020 2063  n a higher.    c
│ │ │ │ +0005bb80: 6f64 696d 656e 7369 6f6e 206d 6174 7269  odimension matri
│ │ │ │ +0005bb90: 7820 6661 6374 6f72 697a 6174 696f 6e0a  x factorization.
│ │ │ │ +0005bba0: 0a57 6179 7320 746f 2075 7365 206d 6642  .Ways to use mfB
│ │ │ │ +0005bbb0: 6f75 6e64 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d  ound:.==========
│ │ │ │ +0005bbc0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ +0005bbd0: 226d 6642 6f75 6e64 284d 6f64 756c 6529  "mfBound(Module)
│ │ │ │ +0005bbe0: 220a 0a46 6f72 2074 6865 2070 726f 6772  "..For the progr
│ │ │ │ +0005bbf0: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d  ammer.==========
│ │ │ │ +0005bc00: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62  ========..The ob
│ │ │ │ +0005bc10: 6a65 6374 202a 6e6f 7465 206d 6642 6f75  ject *note mfBou
│ │ │ │ +0005bc20: 6e64 3a20 6d66 426f 756e 642c 2069 7320  nd: mfBound, is 
│ │ │ │ +0005bc30: 6120 2a6e 6f74 6520 6d65 7468 6f64 2066  a *note method f
│ │ │ │ +0005bc40: 756e 6374 696f 6e3a 0a28 4d61 6361 756c  unction:.(Macaul
│ │ │ │ +0005bc50: 6179 3244 6f63 294d 6574 686f 6446 756e  ay2Doc)MethodFun
│ │ │ │ +0005bc60: 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d  ction,...-------
│ │ │ │ +0005bc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005bc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005bc90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005bca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005bcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005bcc0: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520  ---..The source 
│ │ │ │ -0005bcd0: 6f66 2074 6869 7320 646f 6375 6d65 6e74  of this document
│ │ │ │ -0005bce0: 2069 7320 696e 0a2f 6275 696c 642f 7265   is in./build/re
│ │ │ │ -0005bcf0: 7072 6f64 7563 6962 6c65 2d70 6174 682f  producible-path/
│ │ │ │ -0005bd00: 6d61 6361 756c 6179 322d 312e 3235 2e31  macaulay2-1.25.1
│ │ │ │ -0005bd10: 312b 6473 2f4d 322f 4d61 6361 756c 6179  1+ds/M2/Macaulay
│ │ │ │ -0005bd20: 322f 7061 636b 6167 6573 2f0a 436f 6d70  2/packages/.Comp
│ │ │ │ -0005bd30: 6c65 7465 496e 7465 7273 6563 7469 6f6e  leteIntersection
│ │ │ │ -0005bd40: 5265 736f 6c75 7469 6f6e 732e 6d32 3a33  Resolutions.m2:3
│ │ │ │ -0005bd50: 3334 393a 302e 0a1f 0a46 696c 653a 2043  349:0....File: C
│ │ │ │ -0005bd60: 6f6d 706c 6574 6549 6e74 6572 7365 6374  ompleteIntersect
│ │ │ │ -0005bd70: 696f 6e52 6573 6f6c 7574 696f 6e73 2e69  ionResolutions.i
│ │ │ │ -0005bd80: 6e66 6f2c 204e 6f64 653a 206d 6f64 756c  nfo, Node: modul
│ │ │ │ -0005bd90: 6541 7345 7874 2c20 4e65 7874 3a20 6e65  eAsExt, Next: ne
│ │ │ │ -0005bda0: 7745 7874 2c20 5072 6576 3a20 6d66 426f  wExt, Prev: mfBo
│ │ │ │ -0005bdb0: 756e 642c 2055 703a 2054 6f70 0a0a 6d6f  und, Up: Top..mo
│ │ │ │ -0005bdc0: 6475 6c65 4173 4578 7420 2d2d 2046 696e  duleAsExt -- Fin
│ │ │ │ -0005bdd0: 6420 6120 6d6f 6475 6c65 2077 6974 6820  d a module with 
│ │ │ │ -0005bde0: 6769 7665 6e20 6173 796d 7074 6f74 6963  given asymptotic
│ │ │ │ -0005bdf0: 2072 6573 6f6c 7574 696f 6e0a 2a2a 2a2a   resolution.****
│ │ │ │ +0005bcb0: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f  --------..The so
│ │ │ │ +0005bcc0: 7572 6365 206f 6620 7468 6973 2064 6f63  urce of this doc
│ │ │ │ +0005bcd0: 756d 656e 7420 6973 2069 6e0a 2f62 7569  ument is in./bui
│ │ │ │ +0005bce0: 6c64 2f72 6570 726f 6475 6369 626c 652d  ld/reproducible-
│ │ │ │ +0005bcf0: 7061 7468 2f6d 6163 6175 6c61 7932 2d31  path/macaulay2-1
│ │ │ │ +0005bd00: 2e32 352e 3131 2b64 732f 4d32 2f4d 6163  .25.11+ds/M2/Mac
│ │ │ │ +0005bd10: 6175 6c61 7932 2f70 6163 6b61 6765 732f  aulay2/packages/
│ │ │ │ +0005bd20: 0a43 6f6d 706c 6574 6549 6e74 6572 7365  .CompleteInterse
│ │ │ │ +0005bd30: 6374 696f 6e52 6573 6f6c 7574 696f 6e73  ctionResolutions
│ │ │ │ +0005bd40: 2e6d 323a 3333 3439 3a30 2e0a 1f0a 4669  .m2:3349:0....Fi
│ │ │ │ +0005bd50: 6c65 3a20 436f 6d70 6c65 7465 496e 7465  le: CompleteInte
│ │ │ │ +0005bd60: 7273 6563 7469 6f6e 5265 736f 6c75 7469  rsectionResoluti
│ │ │ │ +0005bd70: 6f6e 732e 696e 666f 2c20 4e6f 6465 3a20  ons.info, Node: 
│ │ │ │ +0005bd80: 6d6f 6475 6c65 4173 4578 742c 204e 6578  moduleAsExt, Nex
│ │ │ │ +0005bd90: 743a 206e 6577 4578 742c 2050 7265 763a  t: newExt, Prev:
│ │ │ │ +0005bda0: 206d 6642 6f75 6e64 2c20 5570 3a20 546f   mfBound, Up: To
│ │ │ │ +0005bdb0: 700a 0a6d 6f64 756c 6541 7345 7874 202d  p..moduleAsExt -
│ │ │ │ +0005bdc0: 2d20 4669 6e64 2061 206d 6f64 756c 6520  - Find a module 
│ │ │ │ +0005bdd0: 7769 7468 2067 6976 656e 2061 7379 6d70  with given asymp
│ │ │ │ +0005bde0: 746f 7469 6320 7265 736f 6c75 7469 6f6e  totic resolution
│ │ │ │ +0005bdf0: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  .***************
│ │ │ │  0005be00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0005be10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0005be20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0005be30: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055  *********..  * U
│ │ │ │ -0005be40: 7361 6765 3a20 0a20 2020 2020 2020 204d  sage: .        M
│ │ │ │ -0005be50: 203d 206d 6f64 756c 6541 7345 7874 284d   = moduleAsExt(M
│ │ │ │ -0005be60: 4d2c 5229 0a20 202a 2049 6e70 7574 733a  M,R).  * Inputs:
│ │ │ │ -0005be70: 0a20 2020 2020 202a 204d 2c20 6120 2a6e  .      * M, a *n
│ │ │ │ -0005be80: 6f74 6520 6d6f 6475 6c65 3a20 284d 6163  ote module: (Mac
│ │ │ │ -0005be90: 6175 6c61 7932 446f 6329 4d6f 6475 6c65  aulay2Doc)Module
│ │ │ │ -0005bea0: 2c2c 206d 6f64 756c 6520 6f76 6572 2070  ,, module over p
│ │ │ │ -0005beb0: 6f6c 796e 6f6d 6961 6c20 7269 6e67 0a20  olynomial ring. 
│ │ │ │ -0005bec0: 2020 2020 2020 2077 6974 6820 6320 7661         with c va
│ │ │ │ -0005bed0: 7269 6162 6c65 730a 2020 2020 2020 2a20  riables.      * 
│ │ │ │ -0005bee0: 522c 2061 202a 6e6f 7465 2072 696e 673a  R, a *note ring:
│ │ │ │ -0005bef0: 2028 4d61 6361 756c 6179 3244 6f63 2952   (Macaulay2Doc)R
│ │ │ │ -0005bf00: 696e 672c 2c20 2867 7261 6465 6429 2063  ing,, (graded) c
│ │ │ │ -0005bf10: 6f6d 706c 6574 6520 696e 7465 7273 6563  omplete intersec
│ │ │ │ -0005bf20: 7469 6f6e 0a20 2020 2020 2020 2072 696e  tion.        rin
│ │ │ │ -0005bf30: 6720 6f66 2063 6f64 696d 656e 7369 6f6e  g of codimension
│ │ │ │ -0005bf40: 2063 2c20 656d 6265 6464 696e 6720 6469   c, embedding di
│ │ │ │ -0005bf50: 6d65 6e73 696f 6e20 6e0a 2020 2a20 4f75  mension n.  * Ou
│ │ │ │ -0005bf60: 7470 7574 733a 0a20 2020 2020 202a 204e  tputs:.      * N
│ │ │ │ -0005bf70: 2c20 6120 2a6e 6f74 6520 6d6f 6475 6c65  , a *note module
│ │ │ │ -0005bf80: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ -0005bf90: 4d6f 6475 6c65 2c2c 206d 6f64 756c 6520  Module,, module 
│ │ │ │ -0005bfa0: 6f76 6572 2052 2073 7563 6820 7468 6174  over R such that
│ │ │ │ -0005bfb0: 0a20 2020 2020 2020 2045 7874 5f52 284e  .        Ext_R(N
│ │ │ │ -0005bfc0: 2c6b 2920 3d20 4d5c 6f74 696d 6573 205c  ,k) = M\otimes \
│ │ │ │ -0005bfd0: 7765 6467 6528 6b5e 6e29 2069 6e20 6c61  wedge(k^n) in la
│ │ │ │ -0005bfe0: 7267 6520 686f 6d6f 6c6f 6769 6361 6c20  rge homological 
│ │ │ │ -0005bff0: 6465 6772 6565 2e0a 0a44 6573 6372 6970  degree...Descrip
│ │ │ │ -0005c000: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  tion.===========
│ │ │ │ -0005c010: 0a0a 5468 6520 726f 7574 696e 6520 6060  ..The routine ``
│ │ │ │ -0005c020: 6d6f 6475 6c65 4173 4578 7427 2720 6973  moduleAsExt'' is
│ │ │ │ -0005c030: 2061 2070 6172 7469 616c 2069 6e76 6572   a partial inver
│ │ │ │ -0005c040: 7365 2074 6f20 7468 6520 726f 7574 696e  se to the routin
│ │ │ │ -0005c050: 6520 4578 744d 6f64 756c 652c 0a63 6f6d  e ExtModule,.com
│ │ │ │ -0005c060: 7075 7465 6420 666f 6c6c 6f77 696e 6720  puted following 
│ │ │ │ -0005c070: 6964 6561 7320 6f66 2041 7672 616d 6f76  ideas of Avramov
│ │ │ │ -0005c080: 2061 6e64 204a 6f72 6765 6e73 656e 3a20   and Jorgensen: 
│ │ │ │ -0005c090: 6769 7665 6e20 6120 6d6f 6475 6c65 2045  given a module E
│ │ │ │ -0005c0a0: 206f 7665 7220 610a 706f 6c79 6e6f 6d69   over a.polynomi
│ │ │ │ -0005c0b0: 616c 2072 696e 6720 6b5b 785f 312e 2e78  al ring k[x_1..x
│ │ │ │ -0005c0c0: 5f63 5d2c 2069 7420 7072 6f76 6964 6573  _c], it provides
│ │ │ │ -0005c0d0: 2061 206d 6f64 756c 6520 4e20 6f76 6572   a module N over
│ │ │ │ -0005c0e0: 2061 2073 7065 6369 6669 6564 2070 6f6c   a specified pol
│ │ │ │ -0005c0f0: 796e 6f6d 6961 6c0a 7269 6e67 2069 6e20  ynomial.ring in 
│ │ │ │ -0005c100: 6e20 7661 7269 6162 6c65 7320 7375 6368  n variables such
│ │ │ │ -0005c110: 2074 6861 7420 4578 7428 4e2c 6b29 2061   that Ext(N,k) a
│ │ │ │ -0005c120: 6772 6565 7320 7769 7468 2024 4527 3d45  grees with $E'=E
│ │ │ │ -0005c130: 5c6f 7469 6d65 7320 5c77 6564 6765 286b  \otimes \wedge(k
│ │ │ │ -0005c140: 5e6e 2924 0a61 6674 6572 2074 7275 6e63  ^n)$.after trunc
│ │ │ │ -0005c150: 6174 696f 6e2e 2048 6572 6520 7468 6520  ation. Here the 
│ │ │ │ -0005c160: 6772 6164 696e 6720 6f6e 2045 2069 7320  grading on E is 
│ │ │ │ -0005c170: 7461 6b65 6e20 746f 2062 6520 6576 656e  taken to be even
│ │ │ │ -0005c180: 2c20 7768 696c 650a 245c 7765 6467 6528  , while.$\wedge(
│ │ │ │ -0005c190: 6b5e 6e29 2420 6861 7320 6765 6e65 7261  k^n)$ has genera
│ │ │ │ -0005c1a0: 746f 7273 2069 6e20 6465 6772 6565 2031  tors in degree 1
│ │ │ │ -0005c1b0: 2e20 5468 6520 726f 7574 696e 6520 6866  . The routine hf
│ │ │ │ -0005c1c0: 4d6f 6475 6c65 4173 4578 7420 636f 6d70  ModuleAsExt comp
│ │ │ │ -0005c1d0: 7574 6573 0a74 6865 2072 6573 756c 7469  utes.the resulti
│ │ │ │ -0005c1e0: 6e67 2068 696c 6265 7274 2066 756e 6374  ng hilbert funct
│ │ │ │ -0005c1f0: 696f 6e20 666f 7220 4527 2e20 5468 6973  ion for E'. This
│ │ │ │ -0005c200: 2075 7365 7320 6964 6561 7320 6f66 2041   uses ideas of A
│ │ │ │ -0005c210: 7672 616d 6f76 2061 6e64 0a4a 6f72 6765  vramov and.Jorge
│ │ │ │ -0005c220: 6e73 656e 2e20 4e6f 7465 2074 6861 7420  nsen. Note that 
│ │ │ │ -0005c230: 7468 6520 6d6f 6475 6c65 2045 7874 284e  the module Ext(N
│ │ │ │ -0005c240: 2c6b 2920 2874 7275 6e63 6174 6564 2920  ,k) (truncated) 
│ │ │ │ -0005c250: 7769 6c6c 2061 7574 6f6d 6174 6963 616c  will automatical
│ │ │ │ -0005c260: 6c79 2062 6520 6672 6565 0a6f 7665 7220  ly be free.over 
│ │ │ │ -0005c270: 7468 6520 6578 7465 7269 6f72 2061 6c67  the exterior alg
│ │ │ │ -0005c280: 6562 7261 2024 5c77 6564 6765 286b 5e6e  ebra $\wedge(k^n
│ │ │ │ -0005c290: 2924 2067 656e 6572 6174 6564 2062 7920  )$ generated by 
│ │ │ │ -0005c2a0: 4578 745e 3128 6b2c 6b29 3b20 6e6f 7420  Ext^1(k,k); not 
│ │ │ │ -0005c2b0: 6120 7479 7069 6361 6c0a 4578 7420 6d6f  a typical.Ext mo
│ │ │ │ -0005c2c0: 6475 6c65 2e0a 0a4d 6f72 6520 7072 6563  dule...More prec
│ │ │ │ -0005c2d0: 6973 656c 793a 0a0a 5375 7070 6f73 6520  isely:..Suppose 
│ │ │ │ -0005c2e0: 7468 6174 2024 5220 3d20 6b5b 615f 312c  that $R = k[a_1,
│ │ │ │ -0005c2f0: 5c64 6f74 732c 2061 5f6e 5d2f 2866 5f31  \dots, a_n]/(f_1
│ │ │ │ -0005c300: 2c5c 646f 7473 2c66 5f63 2924 206c 6574  ,\dots,f_c)$ let
│ │ │ │ -0005c310: 2024 4b4b 203d 0a6b 5b78 5f31 2c5c 646f   $KK =.k[x_1,\do
│ │ │ │ -0005c320: 7473 2c78 5f63 5d24 2c20 616e 6420 6c65  ts,x_c]$, and le
│ │ │ │ -0005c330: 7420 245c 4c61 6d62 6461 203d 205c 7765  t $\Lambda = \we
│ │ │ │ -0005c340: 6467 6520 6b5e 6e24 2e20 2445 203d 204b  dge k^n$. $E = K
│ │ │ │ -0005c350: 4b5c 6f74 696d 6573 5c4c 616d 6264 6124  K\otimes\Lambda$
│ │ │ │ -0005c360: 2c20 736f 0a74 6861 7420 7468 6520 6d69  , so.that the mi
│ │ │ │ -0005c370: 6e69 6d61 6c20 2452 242d 6672 6565 2072  nimal $R$-free r
│ │ │ │ -0005c380: 6573 6f6c 7574 696f 6e20 6f66 2024 6b24  esolution of $k$
│ │ │ │ -0005c390: 2068 6173 2075 6e64 6572 6c79 696e 6720   has underlying 
│ │ │ │ -0005c3a0: 6d6f 6475 6c65 2024 525c 6f74 696d 6573  module $R\otimes
│ │ │ │ -0005c3b0: 0a45 5e2a 242c 2077 6865 7265 2024 455e  .E^*$, where $E^
│ │ │ │ -0005c3c0: 2a24 2069 7320 7468 6520 6772 6164 6564  *$ is the graded
│ │ │ │ -0005c3d0: 2076 6563 746f 7220 7370 6163 6520 6475   vector space du
│ │ │ │ -0005c3e0: 616c 206f 6620 2445 242e 0a0a 4c65 7420  al of $E$...Let 
│ │ │ │ -0005c3f0: 4d4d 2062 6520 7468 6520 7265 7375 6c74  MM be the result
│ │ │ │ -0005c400: 206f 6620 7472 756e 6361 7469 6e67 204d   of truncating M
│ │ │ │ -0005c410: 2061 7420 6974 7320 7265 6775 6c61 7269   at its regulari
│ │ │ │ -0005c420: 7479 2061 6e64 2073 6869 6674 696e 6720  ty and shifting 
│ │ │ │ -0005c430: 6974 2073 6f20 7468 6174 0a69 7420 6973  it so that.it is
│ │ │ │ -0005c440: 2067 656e 6572 6174 6564 2069 6e20 6465   generated in de
│ │ │ │ -0005c450: 6772 6565 2030 2e20 4c65 7420 2446 2420  gree 0. Let $F$ 
│ │ │ │ -0005c460: 6265 2061 2024 4b4b 242d 6672 6565 2072  be a $KK$-free r
│ │ │ │ -0005c470: 6573 6f6c 7574 696f 6e20 6f66 2024 4d4d  esolution of $MM
│ │ │ │ -0005c480: 242c 2061 6e64 0a77 7269 7465 2024 465f  $, and.write $F_
│ │ │ │ -0005c490: 6920 3d20 4b4b 5c6f 7469 6d65 7320 565f  i = KK\otimes V_
│ │ │ │ -0005c4a0: 692e 2420 5369 6e63 6520 6c69 6e65 6172  i.$ Since linear
│ │ │ │ -0005c4b0: 2066 6f72 6d73 206f 7665 7220 244b 4b24   forms over $KK$
│ │ │ │ -0005c4c0: 2063 6f72 7265 7370 6f6e 6420 746f 2043   correspond to C
│ │ │ │ -0005c4d0: 490a 6f70 6572 6174 6f72 7320 6f66 2064  I.operators of d
│ │ │ │ -0005c4e0: 6567 7265 6520 2d32 206f 6e20 7468 6520  egree -2 on the 
│ │ │ │ -0005c4f0: 7265 736f 6c75 7469 6f6e 2047 206f 6620  resolution G of 
│ │ │ │ -0005c500: 6b20 6f76 6572 2052 2c20 7765 206d 6179  k over R, we may
│ │ │ │ -0005c510: 2066 6f72 6d20 6120 6d61 7020 2424 0a64   form a map $$.d
│ │ │ │ -0005c520: 5f31 2b64 5f32 3a20 5c73 756d 5f7b 693d  _1+d_2: \sum_{i=
│ │ │ │ -0005c530: 307d 5e6d 2047 5f7b 692b 317d 5c6f 7469  0}^m G_{i+1}\oti
│ │ │ │ -0005c540: 6d65 7320 565f 7b6d 2d69 7d5e 2a20 5c74  mes V_{m-i}^* \t
│ │ │ │ -0005c550: 6f20 5c73 756d 5f7b 693d 307d 5e6d 2047  o \sum_{i=0}^m G
│ │ │ │ -0005c560: 5f69 5c6f 7469 6d65 730a 565f 7b6d 2d69  _i\otimes.V_{m-i
│ │ │ │ -0005c570: 7d5e 2a20 2424 2077 6865 7265 2024 645f  }^* $$ where $d_
│ │ │ │ -0005c580: 3124 2069 7320 7468 6520 6469 7265 6374  1$ is the direct
│ │ │ │ -0005c590: 2073 756d 206f 6620 7468 6520 6469 6666   sum of the diff
│ │ │ │ -0005c5a0: 6572 656e 7469 616c 7320 2428 475f 7b69  erentials $(G_{i
│ │ │ │ -0005c5b0: 2b31 7d5c 746f 0a47 5f69 295c 6f74 696d  +1}\to.G_i)\otim
│ │ │ │ -0005c5c0: 6573 2056 5f69 5e2a 2420 616e 6420 2464  es V_i^*$ and $d
│ │ │ │ -0005c5d0: 5f32 2420 6973 2074 6865 2064 6972 6563  _2$ is the direc
│ │ │ │ -0005c5e0: 7420 7375 6d20 6f66 2074 6865 206d 6170  t sum of the map
│ │ │ │ -0005c5f0: 7320 245c 7068 695f 6924 2064 6566 696e  s $\phi_i$ defin
│ │ │ │ -0005c600: 6564 0a66 726f 6d20 7468 6520 6469 6666  ed.from the diff
│ │ │ │ -0005c610: 6572 656e 7469 616c 7320 6f66 2024 4624  erentials of $F$
│ │ │ │ -0005c620: 2062 7920 7375 6273 7469 7475 7469 6e67   by substituting
│ │ │ │ -0005c630: 2043 4920 6f70 6572 6174 6f72 7320 666f   CI operators fo
│ │ │ │ -0005c640: 7220 6c69 6e65 6172 2066 6f72 6d73 2c0a  r linear forms,.
│ │ │ │ -0005c650: 245c 7068 695f 693a 2047 5f7b 692b 317d  $\phi_i: G_{i+1}
│ │ │ │ -0005c660: 5c6f 7469 6d65 7320 565f 6920 5c74 6f20  \otimes V_i \to 
│ │ │ │ -0005c670: 475f 7b69 2d31 7d5c 6f74 696d 6573 2056  G_{i-1}\otimes V
│ │ │ │ -0005c680: 5f7b 692d 317d 242e 2054 6865 2073 6372  _{i-1}$. The scr
│ │ │ │ -0005c690: 6970 7420 7265 7475 726e 7320 7468 650a  ipt returns the.
│ │ │ │ -0005c6a0: 6d6f 6475 6c65 204e 2074 6861 7420 6973  module N that is
│ │ │ │ -0005c6b0: 2074 6865 2063 6f6b 6572 6e65 6c20 6f66   the cokernel of
│ │ │ │ -0005c6c0: 2024 645f 312b 645f 3224 2e0a 0a54 6865   $d_1+d_2$...The
│ │ │ │ -0005c6d0: 206d 6f64 756c 6520 2445 7874 5f52 284e   module $Ext_R(N
│ │ │ │ -0005c6e0: 2c6b 2924 2061 6772 6565 732c 2061 6674  ,k)$ agrees, aft
│ │ │ │ -0005c6f0: 6572 2061 2066 6577 2073 7465 7073 2c20  er a few steps, 
│ │ │ │ -0005c700: 7769 7468 2074 6865 206d 6f64 756c 6520  with the module 
│ │ │ │ -0005c710: 6465 7269 7665 6420 6672 6f6d 0a24 4d4d  derived from.$MM
│ │ │ │ -0005c720: 2420 6279 2074 656e 736f 7269 6e67 2069  $ by tensoring i
│ │ │ │ -0005c730: 7420 7769 7468 2024 5c4c 616d 6264 6124  t with $\Lambda$
│ │ │ │ -0005c740: 2c20 7468 6174 2069 732c 2077 6974 6820  , that is, with 
│ │ │ │ -0005c750: 7468 6520 6d6f 6475 6c65 c39f 2024 2420  the module.. $$ 
│ │ │ │ -0005c760: 4d4d 2720 3d20 5c73 756d 5f6a 0a28 4d4d  MM' = \sum_j.(MM
│ │ │ │ -0005c770: 2728 6a29 5c6f 7469 6d65 7320 5c4c 616d  '(j)\otimes \Lam
│ │ │ │ -0005c780: 6264 615f 6a29 2024 2420 736f 2074 6861  bda_j) $$ so tha
│ │ │ │ -0005c790: 7420 244d 4d27 5f70 203d 2028 4d4d 5f70  t $MM'_p = (MM_p
│ │ │ │ -0005c7a0: 5c6f 7469 6d65 7320 4c61 6d62 6461 5f30  \otimes Lambda_0
│ │ │ │ -0005c7b0: 2920 5c6f 706c 7573 0a28 4d4d 5f7b 702d  ) \oplus.(MM_{p-
│ │ │ │ -0005c7c0: 317d 5c6f 7469 6d65 7320 4c61 6d62 6461  1}\otimes Lambda
│ │ │ │ -0005c7d0: 5f31 2920 5c6f 706c 7573 5c63 646f 7473  _1) \oplus\cdots
│ │ │ │ -0005c7e0: 242e 0a0a 5468 6520 6675 6e63 7469 6f6e  $...The function
│ │ │ │ -0005c7f0: 2068 664d 6f64 756c 6541 7345 7874 2063   hfModuleAsExt c
│ │ │ │ -0005c800: 6f6d 7075 7465 7320 7468 6520 4869 6c62  omputes the Hilb
│ │ │ │ -0005c810: 6572 7420 6675 6e63 7469 6f6e 206f 6620  ert function of 
│ │ │ │ -0005c820: 4d4d 2720 6e75 6d65 7269 6361 6c6c 790a  MM' numerically.
│ │ │ │ -0005c830: 6672 6f6d 2074 6861 7420 6f66 204d 4d2e  from that of MM.
│ │ │ │ -0005c840: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..+-------------
│ │ │ │ +0005be20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a  **************..
│ │ │ │ +0005be30: 2020 2a20 5573 6167 653a 200a 2020 2020    * Usage: .    
│ │ │ │ +0005be40: 2020 2020 4d20 3d20 6d6f 6475 6c65 4173      M = moduleAs
│ │ │ │ +0005be50: 4578 7428 4d4d 2c52 290a 2020 2a20 496e  Ext(MM,R).  * In
│ │ │ │ +0005be60: 7075 7473 3a0a 2020 2020 2020 2a20 4d2c  puts:.      * M,
│ │ │ │ +0005be70: 2061 202a 6e6f 7465 206d 6f64 756c 653a   a *note module:
│ │ │ │ +0005be80: 2028 4d61 6361 756c 6179 3244 6f63 294d   (Macaulay2Doc)M
│ │ │ │ +0005be90: 6f64 756c 652c 2c20 6d6f 6475 6c65 206f  odule,, module o
│ │ │ │ +0005bea0: 7665 7220 706f 6c79 6e6f 6d69 616c 2072  ver polynomial r
│ │ │ │ +0005beb0: 696e 670a 2020 2020 2020 2020 7769 7468  ing.        with
│ │ │ │ +0005bec0: 2063 2076 6172 6961 626c 6573 0a20 2020   c variables.   
│ │ │ │ +0005bed0: 2020 202a 2052 2c20 6120 2a6e 6f74 6520     * R, a *note 
│ │ │ │ +0005bee0: 7269 6e67 3a20 284d 6163 6175 6c61 7932  ring: (Macaulay2
│ │ │ │ +0005bef0: 446f 6329 5269 6e67 2c2c 2028 6772 6164  Doc)Ring,, (grad
│ │ │ │ +0005bf00: 6564 2920 636f 6d70 6c65 7465 2069 6e74  ed) complete int
│ │ │ │ +0005bf10: 6572 7365 6374 696f 6e0a 2020 2020 2020  ersection.      
│ │ │ │ +0005bf20: 2020 7269 6e67 206f 6620 636f 6469 6d65    ring of codime
│ │ │ │ +0005bf30: 6e73 696f 6e20 632c 2065 6d62 6564 6469  nsion c, embeddi
│ │ │ │ +0005bf40: 6e67 2064 696d 656e 7369 6f6e 206e 0a20  ng dimension n. 
│ │ │ │ +0005bf50: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ +0005bf60: 2020 2a20 4e2c 2061 202a 6e6f 7465 206d    * N, a *note m
│ │ │ │ +0005bf70: 6f64 756c 653a 2028 4d61 6361 756c 6179  odule: (Macaulay
│ │ │ │ +0005bf80: 3244 6f63 294d 6f64 756c 652c 2c20 6d6f  2Doc)Module,, mo
│ │ │ │ +0005bf90: 6475 6c65 206f 7665 7220 5220 7375 6368  dule over R such
│ │ │ │ +0005bfa0: 2074 6861 740a 2020 2020 2020 2020 4578   that.        Ex
│ │ │ │ +0005bfb0: 745f 5228 4e2c 6b29 203d 204d 5c6f 7469  t_R(N,k) = M\oti
│ │ │ │ +0005bfc0: 6d65 7320 5c77 6564 6765 286b 5e6e 2920  mes \wedge(k^n) 
│ │ │ │ +0005bfd0: 696e 206c 6172 6765 2068 6f6d 6f6c 6f67  in large homolog
│ │ │ │ +0005bfe0: 6963 616c 2064 6567 7265 652e 0a0a 4465  ical degree...De
│ │ │ │ +0005bff0: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d  scription.======
│ │ │ │ +0005c000: 3d3d 3d3d 3d0a 0a54 6865 2072 6f75 7469  =====..The routi
│ │ │ │ +0005c010: 6e65 2060 606d 6f64 756c 6541 7345 7874  ne ``moduleAsExt
│ │ │ │ +0005c020: 2727 2069 7320 6120 7061 7274 6961 6c20  '' is a partial 
│ │ │ │ +0005c030: 696e 7665 7273 6520 746f 2074 6865 2072  inverse to the r
│ │ │ │ +0005c040: 6f75 7469 6e65 2045 7874 4d6f 6475 6c65  outine ExtModule
│ │ │ │ +0005c050: 2c0a 636f 6d70 7574 6564 2066 6f6c 6c6f  ,.computed follo
│ │ │ │ +0005c060: 7769 6e67 2069 6465 6173 206f 6620 4176  wing ideas of Av
│ │ │ │ +0005c070: 7261 6d6f 7620 616e 6420 4a6f 7267 656e  ramov and Jorgen
│ │ │ │ +0005c080: 7365 6e3a 2067 6976 656e 2061 206d 6f64  sen: given a mod
│ │ │ │ +0005c090: 756c 6520 4520 6f76 6572 2061 0a70 6f6c  ule E over a.pol
│ │ │ │ +0005c0a0: 796e 6f6d 6961 6c20 7269 6e67 206b 5b78  ynomial ring k[x
│ │ │ │ +0005c0b0: 5f31 2e2e 785f 635d 2c20 6974 2070 726f  _1..x_c], it pro
│ │ │ │ +0005c0c0: 7669 6465 7320 6120 6d6f 6475 6c65 204e  vides a module N
│ │ │ │ +0005c0d0: 206f 7665 7220 6120 7370 6563 6966 6965   over a specifie
│ │ │ │ +0005c0e0: 6420 706f 6c79 6e6f 6d69 616c 0a72 696e  d polynomial.rin
│ │ │ │ +0005c0f0: 6720 696e 206e 2076 6172 6961 626c 6573  g in n variables
│ │ │ │ +0005c100: 2073 7563 6820 7468 6174 2045 7874 284e   such that Ext(N
│ │ │ │ +0005c110: 2c6b 2920 6167 7265 6573 2077 6974 6820  ,k) agrees with 
│ │ │ │ +0005c120: 2445 273d 455c 6f74 696d 6573 205c 7765  $E'=E\otimes \we
│ │ │ │ +0005c130: 6467 6528 6b5e 6e29 240a 6166 7465 7220  dge(k^n)$.after 
│ │ │ │ +0005c140: 7472 756e 6361 7469 6f6e 2e20 4865 7265  truncation. Here
│ │ │ │ +0005c150: 2074 6865 2067 7261 6469 6e67 206f 6e20   the grading on 
│ │ │ │ +0005c160: 4520 6973 2074 616b 656e 2074 6f20 6265  E is taken to be
│ │ │ │ +0005c170: 2065 7665 6e2c 2077 6869 6c65 0a24 5c77   even, while.$\w
│ │ │ │ +0005c180: 6564 6765 286b 5e6e 2924 2068 6173 2067  edge(k^n)$ has g
│ │ │ │ +0005c190: 656e 6572 6174 6f72 7320 696e 2064 6567  enerators in deg
│ │ │ │ +0005c1a0: 7265 6520 312e 2054 6865 2072 6f75 7469  ree 1. The routi
│ │ │ │ +0005c1b0: 6e65 2068 664d 6f64 756c 6541 7345 7874  ne hfModuleAsExt
│ │ │ │ +0005c1c0: 2063 6f6d 7075 7465 730a 7468 6520 7265   computes.the re
│ │ │ │ +0005c1d0: 7375 6c74 696e 6720 6869 6c62 6572 7420  sulting hilbert 
│ │ │ │ +0005c1e0: 6675 6e63 7469 6f6e 2066 6f72 2045 272e  function for E'.
│ │ │ │ +0005c1f0: 2054 6869 7320 7573 6573 2069 6465 6173   This uses ideas
│ │ │ │ +0005c200: 206f 6620 4176 7261 6d6f 7620 616e 640a   of Avramov and.
│ │ │ │ +0005c210: 4a6f 7267 656e 7365 6e2e 204e 6f74 6520  Jorgensen. Note 
│ │ │ │ +0005c220: 7468 6174 2074 6865 206d 6f64 756c 6520  that the module 
│ │ │ │ +0005c230: 4578 7428 4e2c 6b29 2028 7472 756e 6361  Ext(N,k) (trunca
│ │ │ │ +0005c240: 7465 6429 2077 696c 6c20 6175 746f 6d61  ted) will automa
│ │ │ │ +0005c250: 7469 6361 6c6c 7920 6265 2066 7265 650a  tically be free.
│ │ │ │ +0005c260: 6f76 6572 2074 6865 2065 7874 6572 696f  over the exterio
│ │ │ │ +0005c270: 7220 616c 6765 6272 6120 245c 7765 6467  r algebra $\wedg
│ │ │ │ +0005c280: 6528 6b5e 6e29 2420 6765 6e65 7261 7465  e(k^n)$ generate
│ │ │ │ +0005c290: 6420 6279 2045 7874 5e31 286b 2c6b 293b  d by Ext^1(k,k);
│ │ │ │ +0005c2a0: 206e 6f74 2061 2074 7970 6963 616c 0a45   not a typical.E
│ │ │ │ +0005c2b0: 7874 206d 6f64 756c 652e 0a0a 4d6f 7265  xt module...More
│ │ │ │ +0005c2c0: 2070 7265 6369 7365 6c79 3a0a 0a53 7570   precisely:..Sup
│ │ │ │ +0005c2d0: 706f 7365 2074 6861 7420 2452 203d 206b  pose that $R = k
│ │ │ │ +0005c2e0: 5b61 5f31 2c5c 646f 7473 2c20 615f 6e5d  [a_1,\dots, a_n]
│ │ │ │ +0005c2f0: 2f28 665f 312c 5c64 6f74 732c 665f 6329  /(f_1,\dots,f_c)
│ │ │ │ +0005c300: 2420 6c65 7420 244b 4b20 3d0a 6b5b 785f  $ let $KK =.k[x_
│ │ │ │ +0005c310: 312c 5c64 6f74 732c 785f 635d 242c 2061  1,\dots,x_c]$, a
│ │ │ │ +0005c320: 6e64 206c 6574 2024 5c4c 616d 6264 6120  nd let $\Lambda 
│ │ │ │ +0005c330: 3d20 5c77 6564 6765 206b 5e6e 242e 2024  = \wedge k^n$. $
│ │ │ │ +0005c340: 4520 3d20 4b4b 5c6f 7469 6d65 735c 4c61  E = KK\otimes\La
│ │ │ │ +0005c350: 6d62 6461 242c 2073 6f0a 7468 6174 2074  mbda$, so.that t
│ │ │ │ +0005c360: 6865 206d 696e 696d 616c 2024 5224 2d66  he minimal $R$-f
│ │ │ │ +0005c370: 7265 6520 7265 736f 6c75 7469 6f6e 206f  ree resolution o
│ │ │ │ +0005c380: 6620 246b 2420 6861 7320 756e 6465 726c  f $k$ has underl
│ │ │ │ +0005c390: 7969 6e67 206d 6f64 756c 6520 2452 5c6f  ying module $R\o
│ │ │ │ +0005c3a0: 7469 6d65 730a 455e 2a24 2c20 7768 6572  times.E^*$, wher
│ │ │ │ +0005c3b0: 6520 2445 5e2a 2420 6973 2074 6865 2067  e $E^*$ is the g
│ │ │ │ +0005c3c0: 7261 6465 6420 7665 6374 6f72 2073 7061  raded vector spa
│ │ │ │ +0005c3d0: 6365 2064 7561 6c20 6f66 2024 4524 2e0a  ce dual of $E$..
│ │ │ │ +0005c3e0: 0a4c 6574 204d 4d20 6265 2074 6865 2072  .Let MM be the r
│ │ │ │ +0005c3f0: 6573 756c 7420 6f66 2074 7275 6e63 6174  esult of truncat
│ │ │ │ +0005c400: 696e 6720 4d20 6174 2069 7473 2072 6567  ing M at its reg
│ │ │ │ +0005c410: 756c 6172 6974 7920 616e 6420 7368 6966  ularity and shif
│ │ │ │ +0005c420: 7469 6e67 2069 7420 736f 2074 6861 740a  ting it so that.
│ │ │ │ +0005c430: 6974 2069 7320 6765 6e65 7261 7465 6420  it is generated 
│ │ │ │ +0005c440: 696e 2064 6567 7265 6520 302e 204c 6574  in degree 0. Let
│ │ │ │ +0005c450: 2024 4624 2062 6520 6120 244b 4b24 2d66   $F$ be a $KK$-f
│ │ │ │ +0005c460: 7265 6520 7265 736f 6c75 7469 6f6e 206f  ree resolution o
│ │ │ │ +0005c470: 6620 244d 4d24 2c20 616e 640a 7772 6974  f $MM$, and.writ
│ │ │ │ +0005c480: 6520 2446 5f69 203d 204b 4b5c 6f74 696d  e $F_i = KK\otim
│ │ │ │ +0005c490: 6573 2056 5f69 2e24 2053 696e 6365 206c  es V_i.$ Since l
│ │ │ │ +0005c4a0: 696e 6561 7220 666f 726d 7320 6f76 6572  inear forms over
│ │ │ │ +0005c4b0: 2024 4b4b 2420 636f 7272 6573 706f 6e64   $KK$ correspond
│ │ │ │ +0005c4c0: 2074 6f20 4349 0a6f 7065 7261 746f 7273   to CI.operators
│ │ │ │ +0005c4d0: 206f 6620 6465 6772 6565 202d 3220 6f6e   of degree -2 on
│ │ │ │ +0005c4e0: 2074 6865 2072 6573 6f6c 7574 696f 6e20   the resolution 
│ │ │ │ +0005c4f0: 4720 6f66 206b 206f 7665 7220 522c 2077  G of k over R, w
│ │ │ │ +0005c500: 6520 6d61 7920 666f 726d 2061 206d 6170  e may form a map
│ │ │ │ +0005c510: 2024 240a 645f 312b 645f 323a 205c 7375   $$.d_1+d_2: \su
│ │ │ │ +0005c520: 6d5f 7b69 3d30 7d5e 6d20 475f 7b69 2b31  m_{i=0}^m G_{i+1
│ │ │ │ +0005c530: 7d5c 6f74 696d 6573 2056 5f7b 6d2d 697d  }\otimes V_{m-i}
│ │ │ │ +0005c540: 5e2a 205c 746f 205c 7375 6d5f 7b69 3d30  ^* \to \sum_{i=0
│ │ │ │ +0005c550: 7d5e 6d20 475f 695c 6f74 696d 6573 0a56  }^m G_i\otimes.V
│ │ │ │ +0005c560: 5f7b 6d2d 697d 5e2a 2024 2420 7768 6572  _{m-i}^* $$ wher
│ │ │ │ +0005c570: 6520 2464 5f31 2420 6973 2074 6865 2064  e $d_1$ is the d
│ │ │ │ +0005c580: 6972 6563 7420 7375 6d20 6f66 2074 6865  irect sum of the
│ │ │ │ +0005c590: 2064 6966 6665 7265 6e74 6961 6c73 2024   differentials $
│ │ │ │ +0005c5a0: 2847 5f7b 692b 317d 5c74 6f0a 475f 6929  (G_{i+1}\to.G_i)
│ │ │ │ +0005c5b0: 5c6f 7469 6d65 7320 565f 695e 2a24 2061  \otimes V_i^*$ a
│ │ │ │ +0005c5c0: 6e64 2024 645f 3224 2069 7320 7468 6520  nd $d_2$ is the 
│ │ │ │ +0005c5d0: 6469 7265 6374 2073 756d 206f 6620 7468  direct sum of th
│ │ │ │ +0005c5e0: 6520 6d61 7073 2024 5c70 6869 5f69 2420  e maps $\phi_i$ 
│ │ │ │ +0005c5f0: 6465 6669 6e65 640a 6672 6f6d 2074 6865  defined.from the
│ │ │ │ +0005c600: 2064 6966 6665 7265 6e74 6961 6c73 206f   differentials o
│ │ │ │ +0005c610: 6620 2446 2420 6279 2073 7562 7374 6974  f $F$ by substit
│ │ │ │ +0005c620: 7574 696e 6720 4349 206f 7065 7261 746f  uting CI operato
│ │ │ │ +0005c630: 7273 2066 6f72 206c 696e 6561 7220 666f  rs for linear fo
│ │ │ │ +0005c640: 726d 732c 0a24 5c70 6869 5f69 3a20 475f  rms,.$\phi_i: G_
│ │ │ │ +0005c650: 7b69 2b31 7d5c 6f74 696d 6573 2056 5f69  {i+1}\otimes V_i
│ │ │ │ +0005c660: 205c 746f 2047 5f7b 692d 317d 5c6f 7469   \to G_{i-1}\oti
│ │ │ │ +0005c670: 6d65 7320 565f 7b69 2d31 7d24 2e20 5468  mes V_{i-1}$. Th
│ │ │ │ +0005c680: 6520 7363 7269 7074 2072 6574 7572 6e73  e script returns
│ │ │ │ +0005c690: 2074 6865 0a6d 6f64 756c 6520 4e20 7468   the.module N th
│ │ │ │ +0005c6a0: 6174 2069 7320 7468 6520 636f 6b65 726e  at is the cokern
│ │ │ │ +0005c6b0: 656c 206f 6620 2464 5f31 2b64 5f32 242e  el of $d_1+d_2$.
│ │ │ │ +0005c6c0: 0a0a 5468 6520 6d6f 6475 6c65 2024 4578  ..The module $Ex
│ │ │ │ +0005c6d0: 745f 5228 4e2c 6b29 2420 6167 7265 6573  t_R(N,k)$ agrees
│ │ │ │ +0005c6e0: 2c20 6166 7465 7220 6120 6665 7720 7374  , after a few st
│ │ │ │ +0005c6f0: 6570 732c 2077 6974 6820 7468 6520 6d6f  eps, with the mo
│ │ │ │ +0005c700: 6475 6c65 2064 6572 6976 6564 2066 726f  dule derived fro
│ │ │ │ +0005c710: 6d0a 244d 4d24 2062 7920 7465 6e73 6f72  m.$MM$ by tensor
│ │ │ │ +0005c720: 696e 6720 6974 2077 6974 6820 245c 4c61  ing it with $\La
│ │ │ │ +0005c730: 6d62 6461 242c 2074 6861 7420 6973 2c20  mbda$, that is, 
│ │ │ │ +0005c740: 7769 7468 2074 6865 206d 6f64 756c 65c3  with the module.
│ │ │ │ +0005c750: 9f20 2424 204d 4d27 203d 205c 7375 6d5f  . $$ MM' = \sum_
│ │ │ │ +0005c760: 6a0a 284d 4d27 286a 295c 6f74 696d 6573  j.(MM'(j)\otimes
│ │ │ │ +0005c770: 205c 4c61 6d62 6461 5f6a 2920 2424 2073   \Lambda_j) $$ s
│ │ │ │ +0005c780: 6f20 7468 6174 2024 4d4d 275f 7020 3d20  o that $MM'_p = 
│ │ │ │ +0005c790: 284d 4d5f 705c 6f74 696d 6573 204c 616d  (MM_p\otimes Lam
│ │ │ │ +0005c7a0: 6264 615f 3029 205c 6f70 6c75 730a 284d  bda_0) \oplus.(M
│ │ │ │ +0005c7b0: 4d5f 7b70 2d31 7d5c 6f74 696d 6573 204c  M_{p-1}\otimes L
│ │ │ │ +0005c7c0: 616d 6264 615f 3129 205c 6f70 6c75 735c  ambda_1) \oplus\
│ │ │ │ +0005c7d0: 6364 6f74 7324 2e0a 0a54 6865 2066 756e  cdots$...The fun
│ │ │ │ +0005c7e0: 6374 696f 6e20 6866 4d6f 6475 6c65 4173  ction hfModuleAs
│ │ │ │ +0005c7f0: 4578 7420 636f 6d70 7574 6573 2074 6865  Ext computes the
│ │ │ │ +0005c800: 2048 696c 6265 7274 2066 756e 6374 696f   Hilbert functio
│ │ │ │ +0005c810: 6e20 6f66 204d 4d27 206e 756d 6572 6963  n of MM' numeric
│ │ │ │ +0005c820: 616c 6c79 0a66 726f 6d20 7468 6174 206f  ally.from that o
│ │ │ │ +0005c830: 6620 4d4d 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  f MM...+--------
│ │ │ │ +0005c840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005c850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c870: 2d2d 2d2b 0a7c 6931 203a 206b 6b20 3d20  ---+.|i1 : kk = 
│ │ │ │ -0005c880: 5a5a 2f31 3031 3b20 2020 2020 2020 2020  ZZ/101;         
│ │ │ │ -0005c890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c8a0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0005c860: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20  --------+.|i1 : 
│ │ │ │ +0005c870: 6b6b 203d 205a 5a2f 3130 313b 2020 2020  kk = ZZ/101;    
│ │ │ │ +0005c880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c890: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0005c8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005c8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c8d0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a  ---------+.|i2 :
│ │ │ │ -0005c8e0: 2053 203d 206b 6b5b 612c 622c 635d 3b20   S = kk[a,b,c]; 
│ │ │ │ +0005c8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0005c8d0: 7c69 3220 3a20 5320 3d20 6b6b 5b61 2c62  |i2 : S = kk[a,b
│ │ │ │ +0005c8e0: 2c63 5d3b 2020 2020 2020 2020 2020 2020  ,c];            
│ │ │ │  0005c8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c900: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0005c900: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0005c910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005c920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0005c940: 0a7c 6933 203a 2066 6620 3d20 6d61 7472  .|i3 : ff = matr
│ │ │ │ -0005c950: 6978 7b7b 615e 342c 2062 5e34 2c63 5e34  ix{{a^4, b^4,c^4
│ │ │ │ -0005c960: 7d7d 3b20 2020 2020 2020 2020 2020 2020  }};             
│ │ │ │ -0005c970: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0005c930: 2d2d 2d2d 2b0a 7c69 3320 3a20 6666 203d  ----+.|i3 : ff =
│ │ │ │ +0005c940: 206d 6174 7269 787b 7b61 5e34 2c20 625e   matrix{{a^4, b^
│ │ │ │ +0005c950: 342c 635e 347d 7d3b 2020 2020 2020 2020  4,c^4}};        
│ │ │ │ +0005c960: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005c970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c9a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0005c9b0: 2020 2020 2031 2020 2020 2020 3320 2020       1      3   
│ │ │ │ -0005c9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c9d0: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20          |.|o3 : 
│ │ │ │ -0005c9e0: 4d61 7472 6978 2053 2020 3c2d 2d20 5320  Matrix S  <-- S 
│ │ │ │ +0005c990: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0005c9a0: 2020 2020 2020 2020 2020 3120 2020 2020            1     
│ │ │ │ +0005c9b0: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ +0005c9c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0005c9d0: 6f33 203a 204d 6174 7269 7820 5320 203c  o3 : Matrix S  <
│ │ │ │ +0005c9e0: 2d2d 2053 2020 2020 2020 2020 2020 2020  -- S            
│ │ │ │  0005c9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ca00: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0005ca00: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  0005ca10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005ca20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ca30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0005ca40: 7c69 3420 3a20 5220 3d20 532f 6964 6561  |i4 : R = S/idea
│ │ │ │ -0005ca50: 6c20 6666 3b20 2020 2020 2020 2020 2020  l ff;           
│ │ │ │ -0005ca60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ca70: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0005ca30: 2d2d 2d2b 0a7c 6934 203a 2052 203d 2053  ---+.|i4 : R = S
│ │ │ │ +0005ca40: 2f69 6465 616c 2066 663b 2020 2020 2020  /ideal ff;      
│ │ │ │ +0005ca50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ca60: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0005ca70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005ca80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ca90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005caa0: 2d2d 2d2d 2b0a 7c69 3520 3a20 4f70 7320  ----+.|i5 : Ops 
│ │ │ │ -0005cab0: 3d20 6b6b 5b78 5f31 2c78 5f32 2c78 5f33  = kk[x_1,x_2,x_3
│ │ │ │ -0005cac0: 5d3b 2020 2020 2020 2020 2020 2020 2020  ];              
│ │ │ │ -0005cad0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0005ca90: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a  ---------+.|i5 :
│ │ │ │ +0005caa0: 204f 7073 203d 206b 6b5b 785f 312c 785f   Ops = kk[x_1,x_
│ │ │ │ +0005cab0: 322c 785f 335d 3b20 2020 2020 2020 2020  2,x_3];         
│ │ │ │ +0005cac0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0005cad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005cae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005caf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005cb00: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620  ----------+.|i6 
│ │ │ │ -0005cb10: 3a20 4d4d 203d 204f 7073 5e31 2f28 785f  : MM = Ops^1/(x_
│ │ │ │ -0005cb20: 312a 6964 6561 6c28 785f 325e 322c 785f  1*ideal(x_2^2,x_
│ │ │ │ -0005cb30: 3329 293b 2020 2020 2020 2020 207c 0a2b  3));         |.+
│ │ │ │ +0005caf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0005cb00: 0a7c 6936 203a 204d 4d20 3d20 4f70 735e  .|i6 : MM = Ops^
│ │ │ │ +0005cb10: 312f 2878 5f31 2a69 6465 616c 2878 5f32  1/(x_1*ideal(x_2
│ │ │ │ +0005cb20: 5e32 2c78 5f33 2929 3b20 2020 2020 2020  ^2,x_3));       
│ │ │ │ +0005cb30: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  0005cb40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005cb50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005cb60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005cb70: 2b0a 7c69 3720 3a20 4e20 3d20 6d6f 6475  +.|i7 : N = modu
│ │ │ │ -0005cb80: 6c65 4173 4578 7428 4d4d 2c52 293b 2020  leAsExt(MM,R);  
│ │ │ │ -0005cb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005cba0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0005cb60: 2d2d 2d2d 2d2b 0a7c 6937 203a 204e 203d  -----+.|i7 : N =
│ │ │ │ +0005cb70: 206d 6f64 756c 6541 7345 7874 284d 4d2c   moduleAsExt(MM,
│ │ │ │ +0005cb80: 5229 3b20 2020 2020 2020 2020 2020 2020  R);             
│ │ │ │ +0005cb90: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0005cba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005cbb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005cbc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005cbd0: 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20 6265  ------+.|i8 : be
│ │ │ │ -0005cbe0: 7474 6920 6672 6565 5265 736f 6c75 7469  tti freeResoluti
│ │ │ │ -0005cbf0: 6f6e 2820 4e2c 204c 656e 6774 684c 696d  on( N, LengthLim
│ │ │ │ -0005cc00: 6974 203d 3e20 3130 297c 0a7c 2020 2020  it => 10)|.|    
│ │ │ │ +0005cbc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938  -----------+.|i8
│ │ │ │ +0005cbd0: 203a 2062 6574 7469 2066 7265 6552 6573   : betti freeRes
│ │ │ │ +0005cbe0: 6f6c 7574 696f 6e28 204e 2c20 4c65 6e67  olution( N, Leng
│ │ │ │ +0005cbf0: 7468 4c69 6d69 7420 3d3e 2031 3029 7c0a  thLimit => 10)|.
│ │ │ │ +0005cc00: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0005cc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005cc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005cc30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0005cc40: 2020 2020 2020 2020 2020 2020 3020 2031              0  1
│ │ │ │ -0005cc50: 2020 3220 2033 2020 3420 2035 2020 3620    2  3  4  5  6 
│ │ │ │ -0005cc60: 2037 2020 3820 2039 2031 3020 2020 207c   7  8  9 10    |
│ │ │ │ -0005cc70: 0a7c 6f38 203d 2074 6f74 616c 3a20 3336  .|o8 = total: 36
│ │ │ │ -0005cc80: 2032 3720 3239 2033 3120 3333 2033 3520   27 29 31 33 35 
│ │ │ │ -0005cc90: 3337 2033 3920 3431 2034 3320 3435 2020  37 39 41 43 45  
│ │ │ │ -0005cca0: 2020 7c0a 7c20 2020 2020 2020 202d 363a    |.|        -6:
│ │ │ │ -0005ccb0: 2031 3820 2036 2020 2e20 202e 2020 2e20   18  6  .  .  . 
│ │ │ │ -0005ccc0: 202e 2020 2e20 202e 2020 2e20 202e 2020   .  .  .  .  .  
│ │ │ │ -0005ccd0: 2e20 2020 207c 0a7c 2020 2020 2020 2020  .    |.|        
│ │ │ │ -0005cce0: 2d35 3a20 202e 2020 2e20 202e 2020 2e20  -5:  .  .  .  . 
│ │ │ │ -0005ccf0: 202e 2020 2e20 202e 2020 2e20 202e 2020   .  .  .  .  .  
│ │ │ │ -0005cd00: 2e20 202e 2020 2020 7c0a 7c20 2020 2020  .  .    |.|     
│ │ │ │ -0005cd10: 2020 202d 343a 2031 3820 3231 2032 3120     -4: 18 21 21 
│ │ │ │ -0005cd20: 2037 2020 2e20 202e 2020 2e20 202e 2020   7  .  .  .  .  
│ │ │ │ -0005cd30: 2e20 202e 2020 2e20 2020 207c 0a7c 2020  .  .  .    |.|  
│ │ │ │ -0005cd40: 2020 2020 2020 2d33 3a20 202e 2020 2e20        -3:  .  . 
│ │ │ │ -0005cd50: 202e 2020 2e20 202e 2020 2e20 202e 2020   .  .  .  .  .  
│ │ │ │ -0005cd60: 2e20 202e 2020 2e20 202e 2020 2020 7c0a  .  .  .  .    |.
│ │ │ │ -0005cd70: 7c20 2020 2020 2020 202d 323a 2020 2e20  |        -2:  . 
│ │ │ │ -0005cd80: 202e 2020 3820 3234 2032 3420 2038 2020   .  8 24 24  8  
│ │ │ │ -0005cd90: 2e20 202e 2020 2e20 202e 2020 2e20 2020  .  .  .  .  .   
│ │ │ │ -0005cda0: 207c 0a7c 2020 2020 2020 2020 2d31 3a20   |.|        -1: 
│ │ │ │ -0005cdb0: 202e 2020 2e20 202e 2020 2e20 202e 2020   .  .  .  .  .  
│ │ │ │ -0005cdc0: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ -0005cdd0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0005cde0: 303a 2020 2e20 202e 2020 2e20 202e 2020  0:  .  .  .  .  
│ │ │ │ -0005cdf0: 3920 3237 2032 3720 2039 2020 2e20 202e  9 27 27  9  .  .
│ │ │ │ -0005ce00: 2020 2e20 2020 207c 0a7c 2020 2020 2020    .    |.|      
│ │ │ │ -0005ce10: 2020 2031 3a20 202e 2020 2e20 202e 2020     1:  .  .  .  
│ │ │ │ -0005ce20: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ -0005ce30: 2020 2e20 202e 2020 2020 7c0a 7c20 2020    .  .    |.|   
│ │ │ │ -0005ce40: 2020 2020 2020 323a 2020 2e20 202e 2020        2:  .  .  
│ │ │ │ -0005ce50: 2e20 202e 2020 2e20 202e 2031 3020 3330  .  .  .  . 10 30
│ │ │ │ -0005ce60: 2033 3020 3130 2020 2e20 2020 207c 0a7c   30 10  .    |.|
│ │ │ │ -0005ce70: 2020 2020 2020 2020 2033 3a20 202e 2020           3:  .  
│ │ │ │ -0005ce80: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ -0005ce90: 2020 2e20 202e 2020 2e20 202e 2020 2020    .  .  .  .    
│ │ │ │ -0005cea0: 7c0a 7c20 2020 2020 2020 2020 343a 2020  |.|         4:  
│ │ │ │ -0005ceb0: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ -0005cec0: 2020 2e20 202e 2031 3120 3333 2033 3320    .  . 11 33 33 
│ │ │ │ -0005ced0: 2020 207c 0a7c 2020 2020 2020 2020 2035     |.|         5
│ │ │ │ -0005cee0: 3a20 202e 2020 2e20 202e 2020 2e20 202e  :  .  .  .  .  .
│ │ │ │ -0005cef0: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ -0005cf00: 202e 2020 2020 7c0a 7c20 2020 2020 2020   .    |.|       
│ │ │ │ -0005cf10: 2020 363a 2020 2e20 202e 2020 2e20 202e    6:  .  .  .  .
│ │ │ │ -0005cf20: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ -0005cf30: 202e 2031 3220 2020 207c 0a7c 2020 2020   . 12    |.|    
│ │ │ │ +0005cc30: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0005cc40: 2030 2020 3120 2032 2020 3320 2034 2020   0  1  2  3  4  
│ │ │ │ +0005cc50: 3520 2036 2020 3720 2038 2020 3920 3130  5  6  7  8  9 10
│ │ │ │ +0005cc60: 2020 2020 7c0a 7c6f 3820 3d20 746f 7461      |.|o8 = tota
│ │ │ │ +0005cc70: 6c3a 2033 3620 3237 2032 3920 3331 2033  l: 36 27 29 31 3
│ │ │ │ +0005cc80: 3320 3335 2033 3720 3339 2034 3120 3433  3 35 37 39 41 43
│ │ │ │ +0005cc90: 2034 3520 2020 207c 0a7c 2020 2020 2020   45    |.|      
│ │ │ │ +0005cca0: 2020 2d36 3a20 3138 2020 3620 202e 2020    -6: 18  6  .  
│ │ │ │ +0005ccb0: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ +0005ccc0: 2020 2e20 202e 2020 2020 7c0a 7c20 2020    .  .    |.|   
│ │ │ │ +0005ccd0: 2020 2020 202d 353a 2020 2e20 202e 2020       -5:  .  .  
│ │ │ │ +0005cce0: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ +0005ccf0: 2020 2e20 202e 2020 2e20 2020 207c 0a7c    .  .  .    |.|
│ │ │ │ +0005cd00: 2020 2020 2020 2020 2d34 3a20 3138 2032          -4: 18 2
│ │ │ │ +0005cd10: 3120 3231 2020 3720 202e 2020 2e20 202e  1 21  7  .  .  .
│ │ │ │ +0005cd20: 2020 2e20 202e 2020 2e20 202e 2020 2020    .  .  .  .    
│ │ │ │ +0005cd30: 7c0a 7c20 2020 2020 2020 202d 333a 2020  |.|        -3:  
│ │ │ │ +0005cd40: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ +0005cd50: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ +0005cd60: 2020 207c 0a7c 2020 2020 2020 2020 2d32     |.|        -2
│ │ │ │ +0005cd70: 3a20 202e 2020 2e20 2038 2032 3420 3234  :  .  .  8 24 24
│ │ │ │ +0005cd80: 2020 3820 202e 2020 2e20 202e 2020 2e20    8  .  .  .  . 
│ │ │ │ +0005cd90: 202e 2020 2020 7c0a 7c20 2020 2020 2020   .    |.|       
│ │ │ │ +0005cda0: 202d 313a 2020 2e20 202e 2020 2e20 202e   -1:  .  .  .  .
│ │ │ │ +0005cdb0: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ +0005cdc0: 202e 2020 2e20 2020 207c 0a7c 2020 2020   .  .    |.|    
│ │ │ │ +0005cdd0: 2020 2020 2030 3a20 202e 2020 2e20 202e       0:  .  .  .
│ │ │ │ +0005cde0: 2020 2e20 2039 2032 3720 3237 2020 3920    .  9 27 27  9 
│ │ │ │ +0005cdf0: 202e 2020 2e20 202e 2020 2020 7c0a 7c20   .  .  .    |.| 
│ │ │ │ +0005ce00: 2020 2020 2020 2020 313a 2020 2e20 202e          1:  .  .
│ │ │ │ +0005ce10: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ +0005ce20: 202e 2020 2e20 202e 2020 2e20 2020 207c   .  .  .  .    |
│ │ │ │ +0005ce30: 0a7c 2020 2020 2020 2020 2032 3a20 202e  .|         2:  .
│ │ │ │ +0005ce40: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ +0005ce50: 3130 2033 3020 3330 2031 3020 202e 2020  10 30 30 10  .  
│ │ │ │ +0005ce60: 2020 7c0a 7c20 2020 2020 2020 2020 333a    |.|         3:
│ │ │ │ +0005ce70: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ +0005ce80: 202e 2020 2e20 202e 2020 2e20 202e 2020   .  .  .  .  .  
│ │ │ │ +0005ce90: 2e20 2020 207c 0a7c 2020 2020 2020 2020  .    |.|        
│ │ │ │ +0005cea0: 2034 3a20 202e 2020 2e20 202e 2020 2e20   4:  .  .  .  . 
│ │ │ │ +0005ceb0: 202e 2020 2e20 202e 2020 2e20 3131 2033   .  .  .  . 11 3
│ │ │ │ +0005cec0: 3320 3333 2020 2020 7c0a 7c20 2020 2020  3 33    |.|     
│ │ │ │ +0005ced0: 2020 2020 353a 2020 2e20 202e 2020 2e20      5:  .  .  . 
│ │ │ │ +0005cee0: 202e 2020 2e20 202e 2020 2e20 202e 2020   .  .  .  .  .  
│ │ │ │ +0005cef0: 2e20 202e 2020 2e20 2020 207c 0a7c 2020  .  .  .    |.|  
│ │ │ │ +0005cf00: 2020 2020 2020 2036 3a20 202e 2020 2e20         6:  .  . 
│ │ │ │ +0005cf10: 202e 2020 2e20 202e 2020 2e20 202e 2020   .  .  .  .  .  
│ │ │ │ +0005cf20: 2e20 202e 2020 2e20 3132 2020 2020 7c0a  .  .  . 12    |.
│ │ │ │ +0005cf30: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0005cf40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005cf50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005cf60: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0005cf70: 3820 3a20 4265 7474 6954 616c 6c79 2020  8 : BettiTally  
│ │ │ │ +0005cf60: 207c 0a7c 6f38 203a 2042 6574 7469 5461   |.|o8 : BettiTa
│ │ │ │ +0005cf70: 6c6c 7920 2020 2020 2020 2020 2020 2020  lly             
│ │ │ │  0005cf80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005cf90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0005cfa0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0005cf90: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0005cfa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005cfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005cfc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005cfd0: 2d2d 2b0a 7c69 3920 3a20 6866 4d6f 6475  --+.|i9 : hfModu
│ │ │ │ -0005cfe0: 6c65 4173 4578 7428 3132 2c4d 4d2c 3329  leAsExt(12,MM,3)
│ │ │ │ -0005cff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d000: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005cfc0: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2068  -------+.|i9 : h
│ │ │ │ +0005cfd0: 664d 6f64 756c 6541 7345 7874 2831 322c  fModuleAsExt(12,
│ │ │ │ +0005cfe0: 4d4d 2c33 2920 2020 2020 2020 2020 2020  MM,3)           
│ │ │ │ +0005cff0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0005d000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d030: 2020 2020 2020 2020 7c0a 7c6f 3920 3d20          |.|o9 = 
│ │ │ │ -0005d040: 2832 332c 2032 352c 2032 372c 2032 392c  (23, 25, 27, 29,
│ │ │ │ -0005d050: 2033 312c 2033 332c 2033 352c 2033 372c   31, 33, 35, 37,
│ │ │ │ -0005d060: 2033 392c 2034 3129 2020 207c 0a7c 2020   39, 41)   |.|  
│ │ │ │ +0005d020: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0005d030: 6f39 203d 2028 3233 2c20 3235 2c20 3237  o9 = (23, 25, 27
│ │ │ │ +0005d040: 2c20 3239 2c20 3331 2c20 3333 2c20 3335  , 29, 31, 33, 35
│ │ │ │ +0005d050: 2c20 3337 2c20 3339 2c20 3431 2920 2020  , 37, 39, 41)   
│ │ │ │ +0005d060: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0005d070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d090: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0005d0a0: 7c6f 3920 3a20 5365 7175 656e 6365 2020  |o9 : Sequence  
│ │ │ │ +0005d090: 2020 207c 0a7c 6f39 203a 2053 6571 7565     |.|o9 : Seque
│ │ │ │ +0005d0a0: 6e63 6520 2020 2020 2020 2020 2020 2020  nce             
│ │ │ │  0005d0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d0d0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0005d0c0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0005d0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005d0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005d0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005d100: 2d2d 2d2d 2b0a 0a43 6176 6561 740a 3d3d  ----+..Caveat.==
│ │ │ │ -0005d110: 3d3d 3d3d 0a0a 5468 6520 656c 656d 656e  ====..The elemen
│ │ │ │ -0005d120: 7473 2066 5f31 2e2e 665f 6320 6d75 7374  ts f_1..f_c must
│ │ │ │ -0005d130: 2062 6520 686f 6d6f 6765 6e65 6f75 7320   be homogeneous 
│ │ │ │ -0005d140: 6f66 2074 6865 2073 616d 6520 6465 6772  of the same degr
│ │ │ │ -0005d150: 6565 2e20 5468 6520 7363 7269 7074 2063  ee. The script c
│ │ │ │ -0005d160: 6f75 6c64 0a62 6520 7265 7772 6974 7465  ould.be rewritte
│ │ │ │ -0005d170: 6e20 746f 2061 6363 6f6d 6d6f 6461 7465  n to accommodate
│ │ │ │ -0005d180: 2064 6966 6665 7265 6e74 2064 6567 7265   different degre
│ │ │ │ -0005d190: 6573 2c20 6275 7420 6f6e 6c79 2062 7920  es, but only by 
│ │ │ │ -0005d1a0: 676f 696e 6720 746f 2074 6865 206c 6f63  going to the loc
│ │ │ │ -0005d1b0: 616c 0a63 6174 6567 6f72 790a 0a53 6565  al.category..See
│ │ │ │ -0005d1c0: 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a   also.========..
│ │ │ │ -0005d1d0: 2020 2a20 2a6e 6f74 6520 4578 744d 6f64    * *note ExtMod
│ │ │ │ -0005d1e0: 756c 653a 2045 7874 4d6f 6475 6c65 2c20  ule: ExtModule, 
│ │ │ │ -0005d1f0: 2d2d 2045 7874 5e2a 284d 2c6b 2920 6f76  -- Ext^*(M,k) ov
│ │ │ │ -0005d200: 6572 2061 2063 6f6d 706c 6574 6520 696e  er a complete in
│ │ │ │ -0005d210: 7465 7273 6563 7469 6f6e 2061 730a 2020  tersection as.  
│ │ │ │ -0005d220: 2020 6d6f 6475 6c65 206f 7665 7220 4349    module over CI
│ │ │ │ -0005d230: 206f 7065 7261 746f 7220 7269 6e67 0a20   operator ring. 
│ │ │ │ -0005d240: 202a 202a 6e6f 7465 2065 7665 6e45 7874   * *note evenExt
│ │ │ │ -0005d250: 4d6f 6475 6c65 3a20 6576 656e 4578 744d  Module: evenExtM
│ │ │ │ -0005d260: 6f64 756c 652c 202d 2d20 6576 656e 2070  odule, -- even p
│ │ │ │ -0005d270: 6172 7420 6f66 2045 7874 5e2a 284d 2c6b  art of Ext^*(M,k
│ │ │ │ -0005d280: 2920 6f76 6572 2061 0a20 2020 2063 6f6d  ) over a.    com
│ │ │ │ -0005d290: 706c 6574 6520 696e 7465 7273 6563 7469  plete intersecti
│ │ │ │ -0005d2a0: 6f6e 2061 7320 6d6f 6475 6c65 206f 7665  on as module ove
│ │ │ │ -0005d2b0: 7220 4349 206f 7065 7261 746f 7220 7269  r CI operator ri
│ │ │ │ -0005d2c0: 6e67 0a20 202a 202a 6e6f 7465 206f 6464  ng.  * *note odd
│ │ │ │ -0005d2d0: 4578 744d 6f64 756c 653a 206f 6464 4578  ExtModule: oddEx
│ │ │ │ -0005d2e0: 744d 6f64 756c 652c 202d 2d20 6f64 6420  tModule, -- odd 
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│ │ │ │ -0005da60: 6275 696c 742d 696e 2067 6c6f 6261 6c20  built-in global 
│ │ │ │ -0005da70: 4578 740a 7772 6974 7465 6e20 6279 2041  Ext.written by A
│ │ │ │ -0005da80: 7672 616d 6f76 2061 6e64 2047 7261 7973  vramov and Grays
│ │ │ │ -0005da90: 6f6e 2028 6275 7420 6e6f 7465 2074 6865  on (but note the
│ │ │ │ -0005daa0: 2064 6966 6665 7265 6e63 652c 2065 7870   difference, exp
│ │ │ │ -0005dab0: 6c61 696e 6564 2062 656c 6f77 292e 0a0a  lained below)...
│ │ │ │ -0005dac0: 4966 204c 6966 7420 3d3e 2066 616c 7365  If Lift => false
│ │ │ │ -0005dad0: 2074 6865 2072 6573 756c 7420 6973 2072   the result is r
│ │ │ │ -0005dae0: 6574 7572 6e65 6420 6f76 6572 2061 6e64  eturned over and
│ │ │ │ -0005daf0: 2065 7874 656e 7369 6f6e 206f 6620 5262   extension of Rb
│ │ │ │ -0005db00: 6172 3b20 6966 204c 6966 7420 3d3e 0a74  ar; if Lift =>.t
│ │ │ │ -0005db10: 7275 6520 7468 6520 7265 7375 6c74 2069  rue the result i
│ │ │ │ -0005db20: 7320 7265 7475 726e 6564 206f 7665 7220  s returned over 
│ │ │ │ -0005db30: 616e 6420 6578 7465 6e73 696f 6e20 6f66  and extension of
│ │ │ │ -0005db40: 2052 2e0a 0a49 6620 4772 6164 696e 6720   R...If Grading 
│ │ │ │ -0005db50: 3d3e 2032 2c20 7468 6520 6465 6661 756c  => 2, the defaul
│ │ │ │ -0005db60: 742c 2074 6865 6e20 7468 6520 7265 7375  t, then the resu
│ │ │ │ -0005db70: 6c74 2069 7320 6269 6772 6164 6564 2028  lt is bigraded (
│ │ │ │ -0005db80: 7468 6973 2069 7320 6e65 6365 7373 6172  this is necessar
│ │ │ │ -0005db90: 790a 7768 656e 2043 6865 636b 3d3e 7472  y.when Check=>tr
│ │ │ │ -0005dba0: 7565 0a0a 5468 6520 6465 6661 756c 7420  ue..The default 
│ │ │ │ -0005dbb0: 5661 7269 6162 6c65 7320 3d3e 2073 796d  Variables => sym
│ │ │ │ -0005dbc0: 626f 6c20 2273 2220 6769 7665 7320 7468  bol "s" gives th
│ │ │ │ -0005dbd0: 6520 6e65 7720 7661 7269 6162 6c65 7320  e new variables 
│ │ │ │ -0005dbe0: 7468 6520 6e61 6d65 2073 5f69 2c0a 693d  the name s_i,.i=
│ │ │ │ -0005dbf0: 302e 2e63 2d31 2e20 286e 6f74 6520 7468  0..c-1. (note th
│ │ │ │ -0005dc00: 6174 2074 6865 2062 7569 6c74 696e 2045  at the builtin E
│ │ │ │ -0005dc10: 7874 2075 7365 7320 585f 312e 2e58 5f63  xt uses X_1..X_c
│ │ │ │ -0005dc20: 2e0a 0a4f 6e20 536f 6d65 2065 7861 6d70  ...On Some examp
│ │ │ │ -0005dc30: 6c65 7320 6e65 7745 7874 2069 7320 6661  les newExt is fa
│ │ │ │ -0005dc40: 7374 6572 2074 6861 6e20 4578 743b 206f  ster than Ext; o
│ │ │ │ -0005dc50: 6e20 6f74 6865 7273 2069 7427 7320 736c  n others it's sl
│ │ │ │ -0005dc60: 6f77 6572 2e0a 0a41 2073 696d 706c 6520  ower...A simple 
│ │ │ │ -0005dc70: 6578 616d 706c 653a 2069 6620 5220 3d20  example: if R = 
│ │ │ │ -0005dc80: 6b5b 785f 312e 2e78 5f6e 5d20 616e 6420  k[x_1..x_n] and 
│ │ │ │ -0005dc90: 4920 6973 2063 6f6e 7461 696e 6564 2069  I is contained i
│ │ │ │ -0005dca0: 6e20 7468 6520 6375 6265 206f 6620 7468  n the cube of th
│ │ │ │ -0005dcb0: 650a 6d61 7869 6d61 6c20 6964 6561 6c2c  e.maximal ideal,
│ │ │ │ -0005dcc0: 2074 6865 6e20 4578 7428 6b2c 6b29 2069   then Ext(k,k) i
│ │ │ │ -0005dcd0: 7320 6120 6672 6565 2053 2f28 785f 312e  s a free S/(x_1.
│ │ │ │ -0005dce0: 2e78 5f6e 2920 3d20 6b5b 735f 302e 2e73  .x_n) = k[s_0..s
│ │ │ │ -0005dcf0: 5f28 632d 3129 5d2d 206d 6f64 756c 650a  _(c-1)]- module.
│ │ │ │ -0005dd00: 7769 7468 2062 696e 6f6d 6961 6c28 6e2c  with binomial(n,
│ │ │ │ -0005dd10: 6929 2067 656e 6572 6174 6f72 7320 696e  i) generators in
│ │ │ │ -0005dd20: 2064 6567 7265 6520 690a 0a2b 2d2d 2d2d   degree i..+----
│ │ │ │ +0005d690: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a  ***********..  *
│ │ │ │ +0005d6a0: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ +0005d6b0: 2045 203d 206e 6577 4578 7428 4d2c 4e29   E = newExt(M,N)
│ │ │ │ +0005d6c0: 0a20 202a 2049 6e70 7574 733a 0a20 2020  .  * Inputs:.   
│ │ │ │ +0005d6d0: 2020 202a 204d 2c20 6120 2a6e 6f74 6520     * M, a *note 
│ │ │ │ +0005d6e0: 6d6f 6475 6c65 3a20 284d 6163 6175 6c61  module: (Macaula
│ │ │ │ +0005d6f0: 7932 446f 6329 4d6f 6475 6c65 2c2c 206f  y2Doc)Module,, o
│ │ │ │ +0005d700: 7665 7220 6120 636f 6d70 6c65 7465 2069  ver a complete i
│ │ │ │ +0005d710: 6e74 6572 7365 6374 696f 6e0a 2020 2020  ntersection.    
│ │ │ │ +0005d720: 2020 2020 5262 6172 0a20 2020 2020 202a      Rbar.      *
│ │ │ │ +0005d730: 204e 2c20 6120 2a6e 6f74 6520 6d6f 6475   N, a *note modu
│ │ │ │ +0005d740: 6c65 3a20 284d 6163 6175 6c61 7932 446f  le: (Macaulay2Do
│ │ │ │ +0005d750: 6329 4d6f 6475 6c65 2c2c 206f 7665 7220  c)Module,, over 
│ │ │ │ +0005d760: 5262 6172 0a20 202a 202a 6e6f 7465 204f  Rbar.  * *note O
│ │ │ │ +0005d770: 7074 696f 6e61 6c20 696e 7075 7473 3a20  ptional inputs: 
│ │ │ │ +0005d780: 284d 6163 6175 6c61 7932 446f 6329 7573  (Macaulay2Doc)us
│ │ │ │ +0005d790: 696e 6720 6675 6e63 7469 6f6e 7320 7769  ing functions wi
│ │ │ │ +0005d7a0: 7468 206f 7074 696f 6e61 6c20 696e 7075  th optional inpu
│ │ │ │ +0005d7b0: 7473 2c3a 0a20 2020 2020 202a 2043 6865  ts,:.      * Che
│ │ │ │ +0005d7c0: 636b 203d 3e20 2e2e 2e2c 2064 6566 6175  ck => ..., defau
│ │ │ │ +0005d7d0: 6c74 2076 616c 7565 2066 616c 7365 0a20  lt value false. 
│ │ │ │ +0005d7e0: 2020 2020 202a 2047 7261 6469 6e67 203d       * Grading =
│ │ │ │ +0005d7f0: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076  > ..., default v
│ │ │ │ +0005d800: 616c 7565 2032 0a20 2020 2020 202a 204c  alue 2.      * L
│ │ │ │ +0005d810: 6966 7420 3d3e 202e 2e2e 2c20 6465 6661  ift => ..., defa
│ │ │ │ +0005d820: 756c 7420 7661 6c75 6520 6661 6c73 650a  ult value false.
│ │ │ │ +0005d830: 2020 2020 2020 2a20 5661 7269 6162 6c65        * Variable
│ │ │ │ +0005d840: 7320 3d3e 202e 2e2e 2c20 6465 6661 756c  s => ..., defaul
│ │ │ │ +0005d850: 7420 7661 6c75 6520 730a 2020 2a20 4f75  t value s.  * Ou
│ │ │ │ +0005d860: 7470 7574 733a 0a20 2020 2020 202a 2045  tputs:.      * E
│ │ │ │ +0005d870: 2c20 6120 2a6e 6f74 6520 6d6f 6475 6c65  , a *note module
│ │ │ │ +0005d880: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +0005d890: 4d6f 6475 6c65 2c2c 206f 7665 7220 6120  Module,, over a 
│ │ │ │ +0005d8a0: 7269 6e67 2053 206d 6164 6520 6672 6f6d  ring S made from
│ │ │ │ +0005d8b0: 2072 696e 670a 2020 2020 2020 2020 7072   ring.        pr
│ │ │ │ +0005d8c0: 6573 656e 7461 7469 6f6e 2052 6261 7220  esentation Rbar 
│ │ │ │ +0005d8d0: 7769 7468 2063 6f64 696d 2052 6261 7220  with codim Rbar 
│ │ │ │ +0005d8e0: 6e65 7720 7661 7269 6162 6c65 730a 0a44  new variables..D
│ │ │ │ +0005d8f0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ +0005d900: 3d3d 3d3d 3d3d 0a0a 4c65 7420 5262 6172  ======..Let Rbar
│ │ │ │ +0005d910: 203d 2052 2f28 6631 2e2e 6663 292c 2061   = R/(f1..fc), a
│ │ │ │ +0005d920: 2063 6f6d 706c 6574 6520 696e 7465 7273   complete inters
│ │ │ │ +0005d930: 6563 7469 6f6e 206f 6620 636f 6469 6d65  ection of codime
│ │ │ │ +0005d940: 6e73 696f 6e20 632c 2061 6e64 206c 6574  nsion c, and let
│ │ │ │ +0005d950: 204d 2c4e 2062 650a 5262 6172 2d6d 6f64   M,N be.Rbar-mod
│ │ │ │ +0005d960: 756c 6573 2e20 5765 2061 7373 756d 6520  ules. We assume 
│ │ │ │ +0005d970: 7468 6174 2074 6865 2070 7573 6846 6f72  that the pushFor
│ │ │ │ +0005d980: 7761 7264 206f 6620 4d20 746f 2052 2068  ward of M to R h
│ │ │ │ +0005d990: 6173 2066 696e 6974 6520 6672 6565 0a72  as finite free.r
│ │ │ │ +0005d9a0: 6573 6f6c 7574 696f 6e2e 2054 6865 2073  esolution. The s
│ │ │ │ +0005d9b0: 6372 6970 7420 7468 656e 2063 6f6d 7075  cript then compu
│ │ │ │ +0005d9c0: 7465 7320 7468 6520 746f 7461 6c20 4578  tes the total Ex
│ │ │ │ +0005d9d0: 7428 4d2c 4e29 2061 7320 6120 6d6f 6475  t(M,N) as a modu
│ │ │ │ +0005d9e0: 6c65 206f 7665 7220 5320 3d0a 6b6b 2873  le over S =.kk(s
│ │ │ │ +0005d9f0: 5f31 2e2e 735f 632c 6765 6e73 2052 292c  _1..s_c,gens R),
│ │ │ │ +0005da00: 2075 7369 6e67 2045 6973 656e 6275 6453   using EisenbudS
│ │ │ │ +0005da10: 6861 6d61 7368 546f 7461 6c2e 0a0a 4966  hamashTotal...If
│ │ │ │ +0005da20: 2043 6865 636b 203d 3e20 7472 7565 2c20   Check => true, 
│ │ │ │ +0005da30: 7468 656e 2074 6865 2072 6573 756c 7420  then the result 
│ │ │ │ +0005da40: 6973 2063 6f6d 7061 7265 6420 7769 7468  is compared with
│ │ │ │ +0005da50: 2074 6865 2062 7569 6c74 2d69 6e20 676c   the built-in gl
│ │ │ │ +0005da60: 6f62 616c 2045 7874 0a77 7269 7474 656e  obal Ext.written
│ │ │ │ +0005da70: 2062 7920 4176 7261 6d6f 7620 616e 6420   by Avramov and 
│ │ │ │ +0005da80: 4772 6179 736f 6e20 2862 7574 206e 6f74  Grayson (but not
│ │ │ │ +0005da90: 6520 7468 6520 6469 6666 6572 656e 6365  e the difference
│ │ │ │ +0005daa0: 2c20 6578 706c 6169 6e65 6420 6265 6c6f  , explained belo
│ │ │ │ +0005dab0: 7729 2e0a 0a49 6620 4c69 6674 203d 3e20  w)...If Lift => 
│ │ │ │ +0005dac0: 6661 6c73 6520 7468 6520 7265 7375 6c74  false the result
│ │ │ │ +0005dad0: 2069 7320 7265 7475 726e 6564 206f 7665   is returned ove
│ │ │ │ +0005dae0: 7220 616e 6420 6578 7465 6e73 696f 6e20  r and extension 
│ │ │ │ +0005daf0: 6f66 2052 6261 723b 2069 6620 4c69 6674  of Rbar; if Lift
│ │ │ │ +0005db00: 203d 3e0a 7472 7565 2074 6865 2072 6573   =>.true the res
│ │ │ │ +0005db10: 756c 7420 6973 2072 6574 7572 6e65 6420  ult is returned 
│ │ │ │ +0005db20: 6f76 6572 2061 6e64 2065 7874 656e 7369  over and extensi
│ │ │ │ +0005db30: 6f6e 206f 6620 522e 0a0a 4966 2047 7261  on of R...If Gra
│ │ │ │ +0005db40: 6469 6e67 203d 3e20 322c 2074 6865 2064  ding => 2, the d
│ │ │ │ +0005db50: 6566 6175 6c74 2c20 7468 656e 2074 6865  efault, then the
│ │ │ │ +0005db60: 2072 6573 756c 7420 6973 2062 6967 7261   result is bigra
│ │ │ │ +0005db70: 6465 6420 2874 6869 7320 6973 206e 6563  ded (this is nec
│ │ │ │ +0005db80: 6573 7361 7279 0a77 6865 6e20 4368 6563  essary.when Chec
│ │ │ │ +0005db90: 6b3d 3e74 7275 650a 0a54 6865 2064 6566  k=>true..The def
│ │ │ │ +0005dba0: 6175 6c74 2056 6172 6961 626c 6573 203d  ault Variables =
│ │ │ │ +0005dbb0: 3e20 7379 6d62 6f6c 2022 7322 2067 6976  > symbol "s" giv
│ │ │ │ +0005dbc0: 6573 2074 6865 206e 6577 2076 6172 6961  es the new varia
│ │ │ │ +0005dbd0: 626c 6573 2074 6865 206e 616d 6520 735f  bles the name s_
│ │ │ │ +0005dbe0: 692c 0a69 3d30 2e2e 632d 312e 2028 6e6f  i,.i=0..c-1. (no
│ │ │ │ +0005dbf0: 7465 2074 6861 7420 7468 6520 6275 696c  te that the buil
│ │ │ │ +0005dc00: 7469 6e20 4578 7420 7573 6573 2058 5f31  tin Ext uses X_1
│ │ │ │ +0005dc10: 2e2e 585f 632e 0a0a 4f6e 2053 6f6d 6520  ..X_c...On Some 
│ │ │ │ +0005dc20: 6578 616d 706c 6573 206e 6577 4578 7420  examples newExt 
│ │ │ │ +0005dc30: 6973 2066 6173 7465 7220 7468 616e 2045  is faster than E
│ │ │ │ +0005dc40: 7874 3b20 6f6e 206f 7468 6572 7320 6974  xt; on others it
│ │ │ │ +0005dc50: 2773 2073 6c6f 7765 722e 0a0a 4120 7369  's slower...A si
│ │ │ │ +0005dc60: 6d70 6c65 2065 7861 6d70 6c65 3a20 6966  mple example: if
│ │ │ │ +0005dc70: 2052 203d 206b 5b78 5f31 2e2e 785f 6e5d   R = k[x_1..x_n]
│ │ │ │ +0005dc80: 2061 6e64 2049 2069 7320 636f 6e74 6169   and I is contai
│ │ │ │ +0005dc90: 6e65 6420 696e 2074 6865 2063 7562 6520  ned in the cube 
│ │ │ │ +0005dca0: 6f66 2074 6865 0a6d 6178 696d 616c 2069  of the.maximal i
│ │ │ │ +0005dcb0: 6465 616c 2c20 7468 656e 2045 7874 286b  deal, then Ext(k
│ │ │ │ +0005dcc0: 2c6b 2920 6973 2061 2066 7265 6520 532f  ,k) is a free S/
│ │ │ │ +0005dcd0: 2878 5f31 2e2e 785f 6e29 203d 206b 5b73  (x_1..x_n) = k[s
│ │ │ │ +0005dce0: 5f30 2e2e 735f 2863 2d31 295d 2d20 6d6f  _0..s_(c-1)]- mo
│ │ │ │ +0005dcf0: 6475 6c65 0a77 6974 6820 6269 6e6f 6d69  dule.with binomi
│ │ │ │ +0005dd00: 616c 286e 2c69 2920 6765 6e65 7261 746f  al(n,i) generato
│ │ │ │ +0005dd10: 7273 2069 6e20 6465 6772 6565 2069 0a0a  rs in degree i..
│ │ │ │ +0005dd20: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  0005dd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005dd40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005dd50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005dd60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005dd70: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a  ---------+.|i1 :
│ │ │ │ -0005dd80: 206e 203d 2033 3b63 3d32 3b20 2020 2020   n = 3;c=2;     
│ │ │ │ +0005dd60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0005dd70: 7c69 3120 3a20 6e20 3d20 333b 633d 323b  |i1 : n = 3;c=2;
│ │ │ │ +0005dd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005dd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005dda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ddb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ddc0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0005ddb0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005ddc0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  0005ddd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005dde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005ddf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005de00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005de10: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a  ---------+.|i3 :
│ │ │ │ -0005de20: 2052 203d 205a 5a2f 3130 315b 785f 302e   R = ZZ/101[x_0.
│ │ │ │ -0005de30: 2e78 5f28 6e2d 3129 5d20 2020 2020 2020  .x_(n-1)]       
│ │ │ │ +0005de00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0005de10: 7c69 3320 3a20 5220 3d20 5a5a 2f31 3031  |i3 : R = ZZ/101
│ │ │ │ +0005de20: 5b78 5f30 2e2e 785f 286e 2d31 295d 2020  [x_0..x_(n-1)]  
│ │ │ │ +0005de30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005de40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005de50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005de60: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005de50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005de60: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0005de70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005de80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005de90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005dea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005deb0: 2020 2020 2020 2020 207c 0a7c 6f33 203d           |.|o3 =
│ │ │ │ -0005dec0: 2052 2020 2020 2020 2020 2020 2020 2020   R              
│ │ │ │ +0005dea0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005deb0: 7c6f 3320 3d20 5220 2020 2020 2020 2020  |o3 = R         
│ │ │ │ +0005dec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ded0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005dee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005def0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005df00: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005def0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005df00: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0005df10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005df20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005df30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005df40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005df50: 2020 2020 2020 2020 207c 0a7c 6f33 203a           |.|o3 :
│ │ │ │ -0005df60: 2050 6f6c 796e 6f6d 6961 6c52 696e 6720   PolynomialRing 
│ │ │ │ +0005df40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005df50: 7c6f 3320 3a20 506f 6c79 6e6f 6d69 616c  |o3 : Polynomial
│ │ │ │ +0005df60: 5269 6e67 2020 2020 2020 2020 2020 2020  Ring            
│ │ │ │  0005df70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005df80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005df90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005dfa0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0005df90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005dfa0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  0005dfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005dfc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005dfd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005dfe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005dff0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a  ---------+.|i4 :
│ │ │ │ -0005e000: 2052 6261 7220 3d20 522f 2869 6465 616c   Rbar = R/(ideal
│ │ │ │ -0005e010: 2061 7070 6c79 2863 2c20 692d 3e20 525f   apply(c, i-> R_
│ │ │ │ -0005e020: 695e 3329 2920 2020 2020 2020 2020 2020  i^3))           
│ │ │ │ -0005e030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e040: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005dfe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0005dff0: 7c69 3420 3a20 5262 6172 203d 2052 2f28  |i4 : Rbar = R/(
│ │ │ │ +0005e000: 6964 6561 6c20 6170 706c 7928 632c 2069  ideal apply(c, i
│ │ │ │ +0005e010: 2d3e 2052 5f69 5e33 2929 2020 2020 2020  -> R_i^3))      
│ │ │ │ +0005e020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005e030: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005e040: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0005e050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e090: 2020 2020 2020 2020 207c 0a7c 6f34 203d           |.|o4 =
│ │ │ │ -0005e0a0: 2052 6261 7220 2020 2020 2020 2020 2020   Rbar           
│ │ │ │ +0005e080: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005e090: 7c6f 3420 3d20 5262 6172 2020 2020 2020  |o4 = Rbar      
│ │ │ │ +0005e0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e0e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005e0d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005e0e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0005e0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e130: 2020 2020 2020 2020 207c 0a7c 6f34 203a           |.|o4 :
│ │ │ │ -0005e140: 2051 756f 7469 656e 7452 696e 6720 2020   QuotientRing   
│ │ │ │ +0005e120: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005e130: 7c6f 3420 3a20 5175 6f74 6965 6e74 5269  |o4 : QuotientRi
│ │ │ │ +0005e140: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │  0005e150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e180: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0005e170: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005e180: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  0005e190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005e1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005e1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005e1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005e1d0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a  ---------+.|i5 :
│ │ │ │ -0005e1e0: 204d 6261 7220 3d20 4e62 6172 203d 2063   Mbar = Nbar = c
│ │ │ │ -0005e1f0: 6f6b 6572 2076 6172 7320 5262 6172 2020  oker vars Rbar  
│ │ │ │ +0005e1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0005e1d0: 7c69 3520 3a20 4d62 6172 203d 204e 6261  |i5 : Mbar = Nba
│ │ │ │ +0005e1e0: 7220 3d20 636f 6b65 7220 7661 7273 2052  r = coker vars R
│ │ │ │ +0005e1f0: 6261 7220 2020 2020 2020 2020 2020 2020  bar             
│ │ │ │  0005e200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e220: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005e210: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005e220: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0005e230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e270: 2020 2020 2020 2020 207c 0a7c 6f35 203d           |.|o5 =
│ │ │ │ -0005e280: 2063 6f6b 6572 6e65 6c20 7c20 785f 3020   cokernel | x_0 
│ │ │ │ -0005e290: 785f 3120 785f 3220 7c20 2020 2020 2020  x_1 x_2 |       
│ │ │ │ +0005e260: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005e270: 7c6f 3520 3d20 636f 6b65 726e 656c 207c  |o5 = cokernel |
│ │ │ │ +0005e280: 2078 5f30 2078 5f31 2078 5f32 207c 2020   x_0 x_1 x_2 |  
│ │ │ │ +0005e290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e2c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005e2b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005e2c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0005e2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e310: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005e300: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005e310: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0005e320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e330: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ +0005e330: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │  0005e340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e360: 2020 2020 2020 2020 207c 0a7c 6f35 203a           |.|o5 :
│ │ │ │ -0005e370: 2052 6261 722d 6d6f 6475 6c65 2c20 7175   Rbar-module, qu
│ │ │ │ -0005e380: 6f74 6965 6e74 206f 6620 5262 6172 2020  otient of Rbar  
│ │ │ │ +0005e350: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005e360: 7c6f 3520 3a20 5262 6172 2d6d 6f64 756c  |o5 : Rbar-modul
│ │ │ │ +0005e370: 652c 2071 756f 7469 656e 7420 6f66 2052  e, quotient of R
│ │ │ │ +0005e380: 6261 7220 2020 2020 2020 2020 2020 2020  bar             
│ │ │ │  0005e390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e3b0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0005e3a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005e3b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  0005e3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005e3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005e3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005e3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005e400: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a  ---------+.|i6 :
│ │ │ │ -0005e410: 2045 203d 206e 6577 4578 7428 4d62 6172   E = newExt(Mbar
│ │ │ │ -0005e420: 2c4e 6261 7229 2020 2020 2020 2020 2020  ,Nbar)          
│ │ │ │ +0005e3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0005e400: 7c69 3620 3a20 4520 3d20 6e65 7745 7874  |i6 : E = newExt
│ │ │ │ +0005e410: 284d 6261 722c 4e62 6172 2920 2020 2020  (Mbar,Nbar)     
│ │ │ │ +0005e420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e450: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005e440: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005e450: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0005e460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e4a0: 2020 2020 2020 2020 207c 0a7c 6f36 203d           |.|o6 =
│ │ │ │ -0005e4b0: 2063 6f6b 6572 6e65 6c20 7b30 2c20 307d   cokernel {0, 0}
│ │ │ │ -0005e4c0: 2020 207c 2078 5f32 2078 5f31 2078 5f30     | x_2 x_1 x_0
│ │ │ │ -0005e4d0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005e4e0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005e4f0: 2030 2020 2030 2020 207c 0a7c 2020 2020   0   0   |.|    
│ │ │ │ -0005e500: 2020 2020 2020 2020 2020 7b2d 322c 202d            {-2, -
│ │ │ │ -0005e510: 327d 207c 2030 2020 2030 2020 2030 2020  2} | 0   0   0  
│ │ │ │ -0005e520: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005e530: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005e540: 2030 2020 2078 5f32 207c 0a7c 2020 2020   0   x_2 |.|    
│ │ │ │ -0005e550: 2020 2020 2020 2020 2020 7b2d 322c 202d            {-2, -
│ │ │ │ -0005e560: 327d 207c 2030 2020 2030 2020 2030 2020  2} | 0   0   0  
│ │ │ │ -0005e570: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005e580: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005e590: 2030 2020 2030 2020 207c 0a7c 2020 2020   0   0   |.|    
│ │ │ │ -0005e5a0: 2020 2020 2020 2020 2020 7b2d 322c 202d            {-2, -
│ │ │ │ -0005e5b0: 327d 207c 2030 2020 2030 2020 2030 2020  2} | 0   0   0  
│ │ │ │ -0005e5c0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005e5d0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005e5e0: 2030 2020 2030 2020 207c 0a7c 2020 2020   0   0   |.|    
│ │ │ │ -0005e5f0: 2020 2020 2020 2020 2020 7b2d 312c 202d            {-1, -
│ │ │ │ -0005e600: 317d 207c 2030 2020 2030 2020 2030 2020  1} | 0   0   0  
│ │ │ │ -0005e610: 2078 5f32 2078 5f31 2078 5f30 2030 2020   x_2 x_1 x_0 0  
│ │ │ │ -0005e620: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005e630: 2030 2020 2030 2020 207c 0a7c 2020 2020   0   0   |.|    
│ │ │ │ -0005e640: 2020 2020 2020 2020 2020 7b2d 312c 202d            {-1, -
│ │ │ │ -0005e650: 317d 207c 2030 2020 2030 2020 2030 2020  1} | 0   0   0  
│ │ │ │ -0005e660: 2030 2020 2030 2020 2030 2020 2078 5f32   0   0   0   x_2
│ │ │ │ -0005e670: 2078 5f31 2078 5f30 2030 2020 2030 2020   x_1 x_0 0   0  
│ │ │ │ -0005e680: 2030 2020 2030 2020 207c 0a7c 2020 2020   0   0   |.|    
│ │ │ │ -0005e690: 2020 2020 2020 2020 2020 7b2d 312c 202d            {-1, -
│ │ │ │ -0005e6a0: 317d 207c 2030 2020 2030 2020 2030 2020  1} | 0   0   0  
│ │ │ │ -0005e6b0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005e6c0: 2030 2020 2030 2020 2078 5f32 2078 5f31   0   0   x_2 x_1
│ │ │ │ -0005e6d0: 2078 5f30 2030 2020 207c 0a7c 2020 2020   x_0 0   |.|    
│ │ │ │ -0005e6e0: 2020 2020 2020 2020 2020 7b2d 332c 202d            {-3, -
│ │ │ │ -0005e6f0: 337d 207c 2030 2020 2030 2020 2030 2020  3} | 0   0   0  
│ │ │ │ -0005e700: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005e710: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005e720: 2030 2020 2030 2020 207c 0a7c 2020 2020   0   0   |.|    
│ │ │ │ +0005e490: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005e4a0: 7c6f 3620 3d20 636f 6b65 726e 656c 207b  |o6 = cokernel {
│ │ │ │ +0005e4b0: 302c 2030 7d20 2020 7c20 785f 3220 785f  0, 0}   | x_2 x_
│ │ │ │ +0005e4c0: 3120 785f 3020 3020 2020 3020 2020 3020  1 x_0 0   0   0 
│ │ │ │ +0005e4d0: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +0005e4e0: 2020 3020 2020 3020 2020 3020 2020 7c0a    0   0   0   |.
│ │ │ │ +0005e4f0: 7c20 2020 2020 2020 2020 2020 2020 207b  |              {
│ │ │ │ +0005e500: 2d32 2c20 2d32 7d20 7c20 3020 2020 3020  -2, -2} | 0   0 
│ │ │ │ +0005e510: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +0005e520: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +0005e530: 2020 3020 2020 3020 2020 785f 3220 7c0a    0   0   x_2 |.
│ │ │ │ +0005e540: 7c20 2020 2020 2020 2020 2020 2020 207b  |              {
│ │ │ │ +0005e550: 2d32 2c20 2d32 7d20 7c20 3020 2020 3020  -2, -2} | 0   0 
│ │ │ │ +0005e560: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +0005e570: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +0005e580: 2020 3020 2020 3020 2020 3020 2020 7c0a    0   0   0   |.
│ │ │ │ +0005e590: 7c20 2020 2020 2020 2020 2020 2020 207b  |              {
│ │ │ │ +0005e5a0: 2d32 2c20 2d32 7d20 7c20 3020 2020 3020  -2, -2} | 0   0 
│ │ │ │ +0005e5b0: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +0005e5c0: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +0005e5d0: 2020 3020 2020 3020 2020 3020 2020 7c0a    0   0   0   |.
│ │ │ │ +0005e5e0: 7c20 2020 2020 2020 2020 2020 2020 207b  |              {
│ │ │ │ +0005e5f0: 2d31 2c20 2d31 7d20 7c20 3020 2020 3020  -1, -1} | 0   0 
│ │ │ │ +0005e600: 2020 3020 2020 785f 3220 785f 3120 785f    0   x_2 x_1 x_
│ │ │ │ +0005e610: 3020 3020 2020 3020 2020 3020 2020 3020  0 0   0   0   0 
│ │ │ │ +0005e620: 2020 3020 2020 3020 2020 3020 2020 7c0a    0   0   0   |.
│ │ │ │ +0005e630: 7c20 2020 2020 2020 2020 2020 2020 207b  |              {
│ │ │ │ +0005e640: 2d31 2c20 2d31 7d20 7c20 3020 2020 3020  -1, -1} | 0   0 
│ │ │ │ +0005e650: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +0005e660: 2020 785f 3220 785f 3120 785f 3020 3020    x_2 x_1 x_0 0 
│ │ │ │ +0005e670: 2020 3020 2020 3020 2020 3020 2020 7c0a    0   0   0   |.
│ │ │ │ +0005e680: 7c20 2020 2020 2020 2020 2020 2020 207b  |              {
│ │ │ │ +0005e690: 2d31 2c20 2d31 7d20 7c20 3020 2020 3020  -1, -1} | 0   0 
│ │ │ │ +0005e6a0: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +0005e6b0: 2020 3020 2020 3020 2020 3020 2020 785f    0   0   0   x_
│ │ │ │ +0005e6c0: 3220 785f 3120 785f 3020 3020 2020 7c0a  2 x_1 x_0 0   |.
│ │ │ │ +0005e6d0: 7c20 2020 2020 2020 2020 2020 2020 207b  |              {
│ │ │ │ +0005e6e0: 2d33 2c20 2d33 7d20 7c20 3020 2020 3020  -3, -3} | 0   0 
│ │ │ │ +0005e6f0: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +0005e700: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +0005e710: 2020 3020 2020 3020 2020 3020 2020 7c0a    0   0   0   |.
│ │ │ │ +0005e720: 7c20 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e770: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0005e780: 2020 5a5a 2020 2020 2020 2020 2020 2020    ZZ            
│ │ │ │ -0005e790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e7a0: 2020 2020 2020 2020 202f 205a 5a20 2020           / ZZ   
│ │ │ │ -0005e7b0: 2020 2020 2020 2020 2020 2020 205c 2020               \  
│ │ │ │ -0005e7c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0005e7d0: 202d 2d2d 5b73 202e 2e73 202c 2078 202e   ---[s ..s , x .
│ │ │ │ -0005e7e0: 2e78 205d 2020 2020 2020 2020 2020 2020  .x ]            
│ │ │ │ -0005e7f0: 2020 2020 2020 2020 207c 2d2d 2d5b 7320           |---[s 
│ │ │ │ -0005e800: 2e2e 7320 2c20 7820 2e2e 7820 5d7c 2020  ..s , x ..x ]|  
│ │ │ │ -0005e810: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0005e820: 2031 3031 2020 3020 2020 3120 2020 3020   101  0   1   0 
│ │ │ │ -0005e830: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -0005e840: 2020 2020 2020 2020 207c 3130 3120 2030           |101  0
│ │ │ │ -0005e850: 2020 2031 2020 2030 2020 2032 207c 3820     1   0   2 |8 
│ │ │ │ -0005e860: 2020 2020 2020 2020 207c 0a7c 6f36 203a           |.|o6 :
│ │ │ │ -0005e870: 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   ---------------
│ │ │ │ -0005e880: 2d2d 2d2d 2d6d 6f64 756c 652c 2071 756f  -----module, quo
│ │ │ │ -0005e890: 7469 656e 7420 6f66 207c 2d2d 2d2d 2d2d  tient of |------
│ │ │ │ -0005e8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 2020  -------------|  
│ │ │ │ -0005e8b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0005e8c0: 2020 2020 2020 2020 2033 2020 2033 2020           3   3  
│ │ │ │ -0005e8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e8e0: 2020 2020 2020 2020 207c 2020 2020 2020           |      
│ │ │ │ -0005e8f0: 2020 3320 2020 3320 2020 2020 207c 2020    3   3      |  
│ │ │ │ -0005e900: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0005e910: 2020 2020 2020 2028 7820 2c20 7820 2920         (x , x ) 
│ │ │ │ -0005e920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e930: 2020 2020 2020 2020 207c 2020 2020 2020           |      
│ │ │ │ -0005e940: 2878 202c 2078 2029 2020 2020 207c 2020  (x , x )     |  
│ │ │ │ -0005e950: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0005e960: 2020 2020 2020 2020 2030 2020 2031 2020           0   1  
│ │ │ │ -0005e970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e980: 2020 2020 2020 2020 205c 2020 2020 2020           \      
│ │ │ │ -0005e990: 2020 3020 2020 3120 2020 2020 202f 2020    0   1      /  
│ │ │ │ -0005e9a0: 2020 2020 2020 2020 207c 0a7c 2d2d 2d2d           |.|----
│ │ │ │ +0005e760: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005e770: 7c20 2020 2020 205a 5a20 2020 2020 2020  |      ZZ       
│ │ │ │ +0005e780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005e790: 2020 2020 2020 2020 2020 2020 2020 2f20                / 
│ │ │ │ +0005e7a0: 5a5a 2020 2020 2020 2020 2020 2020 2020  ZZ              
│ │ │ │ +0005e7b0: 2020 5c20 2020 2020 2020 2020 2020 7c0a    \           |.
│ │ │ │ +0005e7c0: 7c20 2020 2020 2d2d 2d5b 7320 2e2e 7320  |     ---[s ..s 
│ │ │ │ +0005e7d0: 2c20 7820 2e2e 7820 5d20 2020 2020 2020  , x ..x ]       
│ │ │ │ +0005e7e0: 2020 2020 2020 2020 2020 2020 2020 7c2d                |-
│ │ │ │ +0005e7f0: 2d2d 5b73 202e 2e73 202c 2078 202e 2e78  --[s ..s , x ..x
│ │ │ │ +0005e800: 205d 7c20 2020 2020 2020 2020 2020 7c0a   ]|           |.
│ │ │ │ +0005e810: 7c20 2020 2020 3130 3120 2030 2020 2031  |     101  0   1
│ │ │ │ +0005e820: 2020 2030 2020 2032 2020 2020 2020 2020     0   2        
│ │ │ │ +0005e830: 2020 2020 2020 2020 2020 2020 2020 7c31                |1
│ │ │ │ +0005e840: 3031 2020 3020 2020 3120 2020 3020 2020  01  0   1   0   
│ │ │ │ +0005e850: 3220 7c38 2020 2020 2020 2020 2020 7c0a  2 |8          |.
│ │ │ │ +0005e860: 7c6f 3620 3a20 2d2d 2d2d 2d2d 2d2d 2d2d  |o6 : ----------
│ │ │ │ +0005e870: 2d2d 2d2d 2d2d 2d2d 2d2d 6d6f 6475 6c65  ----------module
│ │ │ │ +0005e880: 2c20 7175 6f74 6965 6e74 206f 6620 7c2d  , quotient of |-
│ │ │ │ +0005e890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005e8a0: 2d2d 7c20 2020 2020 2020 2020 2020 7c0a  --|           |.
│ │ │ │ +0005e8b0: 7c20 2020 2020 2020 2020 2020 2020 3320  |             3 
│ │ │ │ +0005e8c0: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │ +0005e8d0: 2020 2020 2020 2020 2020 2020 2020 7c20                | 
│ │ │ │ +0005e8e0: 2020 2020 2020 2033 2020 2033 2020 2020         3   3    
│ │ │ │ +0005e8f0: 2020 7c20 2020 2020 2020 2020 2020 7c0a    |           |.
│ │ │ │ +0005e900: 7c20 2020 2020 2020 2020 2020 2878 202c  |           (x ,
│ │ │ │ +0005e910: 2078 2029 2020 2020 2020 2020 2020 2020   x )            
│ │ │ │ +0005e920: 2020 2020 2020 2020 2020 2020 2020 7c20                | 
│ │ │ │ +0005e930: 2020 2020 2028 7820 2c20 7820 2920 2020       (x , x )   
│ │ │ │ +0005e940: 2020 7c20 2020 2020 2020 2020 2020 7c0a    |           |.
│ │ │ │ +0005e950: 7c20 2020 2020 2020 2020 2020 2020 3020  |             0 
│ │ │ │ +0005e960: 2020 3120 2020 2020 2020 2020 2020 2020    1             
│ │ │ │ +0005e970: 2020 2020 2020 2020 2020 2020 2020 5c20                \ 
│ │ │ │ +0005e980: 2020 2020 2020 2030 2020 2031 2020 2020         0   1    
│ │ │ │ +0005e990: 2020 2f20 2020 2020 2020 2020 2020 7c0a    /           |.
│ │ │ │ +0005e9a0: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │  0005e9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005e9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005e9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005e9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005e9f0: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 3020 2020  ---------|.|0   
│ │ │ │ -0005ea00: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ -0005ea10: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ -0005ea20: 3020 2020 3020 2020 7c20 2020 2020 2020  0   0   |       
│ │ │ │ -0005ea30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ea40: 2020 2020 2020 2020 207c 0a7c 785f 3120           |.|x_1 
│ │ │ │ -0005ea50: 785f 3020 3020 2020 3020 2020 3020 2020  x_0 0   0   0   
│ │ │ │ -0005ea60: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ -0005ea70: 3020 2020 3020 2020 7c20 2020 2020 2020  0   0   |       
│ │ │ │ -0005ea80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ea90: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
│ │ │ │ -0005eaa0: 3020 2020 785f 3220 785f 3120 785f 3020  0   x_2 x_1 x_0 
│ │ │ │ -0005eab0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ -0005eac0: 3020 2020 3020 2020 7c20 2020 2020 2020  0   0   |       
│ │ │ │ -0005ead0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005eae0: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
│ │ │ │ -0005eaf0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ -0005eb00: 785f 3220 785f 3120 785f 3020 3020 2020  x_2 x_1 x_0 0   
│ │ │ │ -0005eb10: 3020 2020 3020 2020 7c20 2020 2020 2020  0   0   |       
│ │ │ │ -0005eb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005eb30: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
│ │ │ │ -0005eb40: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ -0005eb50: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ -0005eb60: 3020 2020 3020 2020 7c20 2020 2020 2020  0   0   |       
│ │ │ │ -0005eb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005eb80: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
│ │ │ │ -0005eb90: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ -0005eba0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ -0005ebb0: 3020 2020 3020 2020 7c20 2020 2020 2020  0   0   |       
│ │ │ │ -0005ebc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ebd0: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
│ │ │ │ -0005ebe0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ -0005ebf0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ -0005ec00: 3020 2020 3020 2020 7c20 2020 2020 2020  0   0   |       
│ │ │ │ -0005ec10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ec20: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
│ │ │ │ -0005ec30: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ -0005ec40: 3020 2020 3020 2020 3020 2020 785f 3220  0   0   0   x_2 
│ │ │ │ -0005ec50: 785f 3120 785f 3020 7c20 2020 2020 2020  x_1 x_0 |       
│ │ │ │ -0005ec60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ec70: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0005e9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +0005e9f0: 7c30 2020 2030 2020 2030 2020 2030 2020  |0   0   0   0  
│ │ │ │ +0005ea00: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ +0005ea10: 2030 2020 2030 2020 2030 2020 207c 2020   0   0   0   |  
│ │ │ │ +0005ea20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ea30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005ea40: 7c78 5f31 2078 5f30 2030 2020 2030 2020  |x_1 x_0 0   0  
│ │ │ │ +0005ea50: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ +0005ea60: 2030 2020 2030 2020 2030 2020 207c 2020   0   0   0   |  
│ │ │ │ +0005ea70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ea80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005ea90: 7c30 2020 2030 2020 2078 5f32 2078 5f31  |0   0   x_2 x_1
│ │ │ │ +0005eaa0: 2078 5f30 2030 2020 2030 2020 2030 2020   x_0 0   0   0  
│ │ │ │ +0005eab0: 2030 2020 2030 2020 2030 2020 207c 2020   0   0   0   |  
│ │ │ │ +0005eac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ead0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005eae0: 7c30 2020 2030 2020 2030 2020 2030 2020  |0   0   0   0  
│ │ │ │ +0005eaf0: 2030 2020 2078 5f32 2078 5f31 2078 5f30   0   x_2 x_1 x_0
│ │ │ │ +0005eb00: 2030 2020 2030 2020 2030 2020 207c 2020   0   0   0   |  
│ │ │ │ +0005eb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005eb20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005eb30: 7c30 2020 2030 2020 2030 2020 2030 2020  |0   0   0   0  
│ │ │ │ +0005eb40: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ +0005eb50: 2030 2020 2030 2020 2030 2020 207c 2020   0   0   0   |  
│ │ │ │ +0005eb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005eb70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005eb80: 7c30 2020 2030 2020 2030 2020 2030 2020  |0   0   0   0  
│ │ │ │ +0005eb90: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ +0005eba0: 2030 2020 2030 2020 2030 2020 207c 2020   0   0   0   |  
│ │ │ │ +0005ebb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ebc0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005ebd0: 7c30 2020 2030 2020 2030 2020 2030 2020  |0   0   0   0  
│ │ │ │ +0005ebe0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ +0005ebf0: 2030 2020 2030 2020 2030 2020 207c 2020   0   0   0   |  
│ │ │ │ +0005ec00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ec10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005ec20: 7c30 2020 2030 2020 2030 2020 2030 2020  |0   0   0   0  
│ │ │ │ +0005ec30: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ +0005ec40: 2078 5f32 2078 5f31 2078 5f30 207c 2020   x_2 x_1 x_0 |  
│ │ │ │ +0005ec50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ec60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005ec70: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  0005ec80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005ec90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005eca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ecb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ecc0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a  ---------+.|i7 :
│ │ │ │ -0005ecd0: 2074 616c 6c79 2064 6567 7265 6573 2045   tally degrees E
│ │ │ │ +0005ecb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0005ecc0: 7c69 3720 3a20 7461 6c6c 7920 6465 6772  |i7 : tally degr
│ │ │ │ +0005ecd0: 6565 7320 4520 2020 2020 2020 2020 2020  ees E           
│ │ │ │  0005ece0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ecf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ed00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ed10: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005ed00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005ed10: 7c20 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ed60: 2020 2020 2020 2020 207c 0a7c 6f37 203d           |.|o7 =
│ │ │ │ -0005ed70: 2054 616c 6c79 7b7b 2d31 2c20 2d31 7d20   Tally{{-1, -1} 
│ │ │ │ -0005ed80: 3d3e 2033 7d20 2020 2020 2020 2020 2020  => 3}           
│ │ │ │ +0005ed50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005ed60: 7c6f 3720 3d20 5461 6c6c 797b 7b2d 312c  |o7 = Tally{{-1,
│ │ │ │ +0005ed70: 202d 317d 203d 3e20 337d 2020 2020 2020   -1} => 3}      
│ │ │ │ +0005ed80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ed90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005eda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005edb0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0005edc0: 2020 2020 2020 207b 2d32 2c20 2d32 7d20         {-2, -2} 
│ │ │ │ -0005edd0: 3d3e 2033 2020 2020 2020 2020 2020 2020  => 3            
│ │ │ │ +0005eda0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005edb0: 7c20 2020 2020 2020 2020 2020 7b2d 322c  |           {-2,
│ │ │ │ +0005edc0: 202d 327d 203d 3e20 3320 2020 2020 2020   -2} => 3       
│ │ │ │ +0005edd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ede0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005edf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ee00: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0005ee10: 2020 2020 2020 207b 2d33 2c20 2d33 7d20         {-3, -3} 
│ │ │ │ -0005ee20: 3d3e 2031 2020 2020 2020 2020 2020 2020  => 1            
│ │ │ │ +0005edf0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005ee00: 7c20 2020 2020 2020 2020 2020 7b2d 332c  |           {-3,
│ │ │ │ +0005ee10: 202d 337d 203d 3e20 3120 2020 2020 2020   -3} => 1       
│ │ │ │ +0005ee20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ee30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ee40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ee50: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0005ee60: 2020 2020 2020 207b 302c 2030 7d20 3d3e         {0, 0} =>
│ │ │ │ -0005ee70: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │ +0005ee40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005ee50: 7c20 2020 2020 2020 2020 2020 7b30 2c20  |           {0, 
│ │ │ │ +0005ee60: 307d 203d 3e20 3120 2020 2020 2020 2020  0} => 1         
│ │ │ │ +0005ee70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ee80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ee90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005eea0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005ee90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005eea0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0005eeb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005eec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005eed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005eee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005eef0: 2020 2020 2020 2020 207c 0a7c 6f37 203a           |.|o7 :
│ │ │ │ -0005ef00: 2054 616c 6c79 2020 2020 2020 2020 2020   Tally          
│ │ │ │ +0005eee0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005eef0: 7c6f 3720 3a20 5461 6c6c 7920 2020 2020  |o7 : Tally     
│ │ │ │ +0005ef00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ef10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ef20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ef30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ef40: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0005ef30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005ef40: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  0005ef50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005ef60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005ef70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ef80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ef90: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a  ---------+.|i8 :
│ │ │ │ -0005efa0: 2061 6e6e 6968 696c 6174 6f72 2045 2020   annihilator E  
│ │ │ │ +0005ef80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0005ef90: 7c69 3820 3a20 616e 6e69 6869 6c61 746f  |i8 : annihilato
│ │ │ │ +0005efa0: 7220 4520 2020 2020 2020 2020 2020 2020  r E             
│ │ │ │  0005efb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005efc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005efd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005efe0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005efd0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005efe0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0005eff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f030: 2020 2020 2020 2020 207c 0a7c 6f38 203d           |.|o8 =
│ │ │ │ -0005f040: 2069 6465 616c 2028 7820 2c20 7820 2c20   ideal (x , x , 
│ │ │ │ -0005f050: 7820 2920 2020 2020 2020 2020 2020 2020  x )             
│ │ │ │ +0005f020: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005f030: 7c6f 3820 3d20 6964 6561 6c20 2878 202c  |o8 = ideal (x ,
│ │ │ │ +0005f040: 2078 202c 2078 2029 2020 2020 2020 2020   x , x )        
│ │ │ │ +0005f050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f080: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0005f090: 2020 2020 2020 2020 2032 2020 2031 2020           2   1  
│ │ │ │ -0005f0a0: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ +0005f070: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005f080: 7c20 2020 2020 2020 2020 2020 2020 3220  |             2 
│ │ │ │ +0005f090: 2020 3120 2020 3020 2020 2020 2020 2020    1   0         
│ │ │ │ +0005f0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f0d0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005f0c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005f0d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0005f0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f120: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0005f130: 2020 2020 2020 2020 2020 205a 5a20 2020             ZZ   
│ │ │ │ +0005f110: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005f120: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0005f130: 5a5a 2020 2020 2020 2020 2020 2020 2020  ZZ              
│ │ │ │  0005f140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f170: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0005f180: 2020 2020 2020 2020 2020 2d2d 2d5b 7320            ---[s 
│ │ │ │ -0005f190: 2e2e 7320 2c20 7820 2e2e 7820 5d20 2020  ..s , x ..x ]   
│ │ │ │ +0005f160: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005f170: 7c20 2020 2020 2020 2020 2020 2020 202d  |              -
│ │ │ │ +0005f180: 2d2d 5b73 202e 2e73 202c 2078 202e 2e78  --[s ..s , x ..x
│ │ │ │ +0005f190: 205d 2020 2020 2020 2020 2020 2020 2020   ]              
│ │ │ │  0005f1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f1c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0005f1d0: 2020 2020 2020 2020 2020 3130 3120 2030            101  0
│ │ │ │ -0005f1e0: 2020 2031 2020 2030 2020 2032 2020 2020     1   0   2    
│ │ │ │ +0005f1b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005f1c0: 7c20 2020 2020 2020 2020 2020 2020 2031  |              1
│ │ │ │ +0005f1d0: 3031 2020 3020 2020 3120 2020 3020 2020  01  0   1   0   
│ │ │ │ +0005f1e0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  0005f1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f210: 2020 2020 2020 2020 207c 0a7c 6f38 203a           |.|o8 :
│ │ │ │ -0005f220: 2049 6465 616c 206f 6620 2d2d 2d2d 2d2d   Ideal of ------
│ │ │ │ -0005f230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d20 2020  -------------   
│ │ │ │ +0005f200: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005f210: 7c6f 3820 3a20 4964 6561 6c20 6f66 202d  |o8 : Ideal of -
│ │ │ │ +0005f220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005f230: 2d2d 2020 2020 2020 2020 2020 2020 2020  --              
│ │ │ │  0005f240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f260: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0005f270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f280: 2020 3320 2020 3320 2020 2020 2020 2020    3   3         
│ │ │ │ +0005f250: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005f260: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0005f270: 2020 2020 2020 2033 2020 2033 2020 2020         3   3    
│ │ │ │ +0005f280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f2b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0005f2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f2d0: 2878 202c 2078 2029 2020 2020 2020 2020  (x , x )        
│ │ │ │ +0005f2a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005f2b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0005f2c0: 2020 2020 2028 7820 2c20 7820 2920 2020       (x , x )   
│ │ │ │ +0005f2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f300: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0005f310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f320: 2020 3020 2020 3120 2020 2020 2020 2020    0   1         
│ │ │ │ +0005f2f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005f300: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0005f310: 2020 2020 2020 2030 2020 2031 2020 2020         0   1    
│ │ │ │ +0005f320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f350: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0005f340: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005f350: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  0005f360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005f370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005f380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005f390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005f3a0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 416e 2065  ---------+..An e
│ │ │ │ -0005f3b0: 7861 6d70 6c65 2077 6865 7265 2074 6865  xample where the
│ │ │ │ -0005f3c0: 2062 7569 6c74 2d69 6e20 676c 6f62 616c   built-in global
│ │ │ │ -0005f3d0: 2045 7874 2069 7320 6861 7264 2074 6f20   Ext is hard to 
│ │ │ │ -0005f3e0: 636f 6d70 6172 6520 6469 7265 6374 6c79  compare directly
│ │ │ │ -0005f3f0: 2077 6974 6820 6f75 720a 6d65 7468 6f64   with our.method
│ │ │ │ -0005f400: 206f 6620 636f 6d70 7574 6174 696f 6e3a   of computation:
│ │ │ │ -0005f410: 2049 202a 6775 6573 732a 2074 6861 7420   I *guess* that 
│ │ │ │ -0005f420: 7468 6520 7369 676e 2063 686f 6963 6573  the sign choices
│ │ │ │ -0005f430: 2069 6e20 7468 6520 6275 696c 742d 696e   in the built-in
│ │ │ │ -0005f440: 2061 6d6f 756e 740a 6573 7365 6e74 6961   amount.essentia
│ │ │ │ -0005f450: 6c6c 7920 746f 2061 2063 6861 6e67 6520  lly to a change 
│ │ │ │ -0005f460: 6f66 2076 6172 6961 626c 6520 696e 2074  of variable in t
│ │ │ │ -0005f470: 6865 206e 6577 2076 6172 6961 626c 6573  he new variables
│ │ │ │ -0005f480: 2c20 616e 6420 7370 6f69 6c20 616e 2065  , and spoil an e
│ │ │ │ -0005f490: 6173 790a 636f 6d70 6172 6973 6f6e 2e20  asy.comparison. 
│ │ │ │ -0005f4a0: 4275 7420 666f 7220 6578 616d 706c 6520  But for example 
│ │ │ │ -0005f4b0: 7468 6520 6269 2d67 7261 6465 6420 4265  the bi-graded Be
│ │ │ │ -0005f4c0: 7474 6920 6e75 6d62 6572 7320 6172 6520  tti numbers are 
│ │ │ │ -0005f4d0: 6571 7561 6c2e 2074 6869 7320 7365 656d  equal. this seem
│ │ │ │ -0005f4e0: 730a 746f 2073 7461 7274 2077 6974 6820  s.to start with 
│ │ │ │ -0005f4f0: 633d 332e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d  c=3...+---------
│ │ │ │ +0005f390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0005f3a0: 0a41 6e20 6578 616d 706c 6520 7768 6572  .An example wher
│ │ │ │ +0005f3b0: 6520 7468 6520 6275 696c 742d 696e 2067  e the built-in g
│ │ │ │ +0005f3c0: 6c6f 6261 6c20 4578 7420 6973 2068 6172  lobal Ext is har
│ │ │ │ +0005f3d0: 6420 746f 2063 6f6d 7061 7265 2064 6972  d to compare dir
│ │ │ │ +0005f3e0: 6563 746c 7920 7769 7468 206f 7572 0a6d  ectly with our.m
│ │ │ │ +0005f3f0: 6574 686f 6420 6f66 2063 6f6d 7075 7461  ethod of computa
│ │ │ │ +0005f400: 7469 6f6e 3a20 4920 2a67 7565 7373 2a20  tion: I *guess* 
│ │ │ │ +0005f410: 7468 6174 2074 6865 2073 6967 6e20 6368  that the sign ch
│ │ │ │ +0005f420: 6f69 6365 7320 696e 2074 6865 2062 7569  oices in the bui
│ │ │ │ +0005f430: 6c74 2d69 6e20 616d 6f75 6e74 0a65 7373  lt-in amount.ess
│ │ │ │ +0005f440: 656e 7469 616c 6c79 2074 6f20 6120 6368  entially to a ch
│ │ │ │ +0005f450: 616e 6765 206f 6620 7661 7269 6162 6c65  ange of variable
│ │ │ │ +0005f460: 2069 6e20 7468 6520 6e65 7720 7661 7269   in the new vari
│ │ │ │ +0005f470: 6162 6c65 732c 2061 6e64 2073 706f 696c  ables, and spoil
│ │ │ │ +0005f480: 2061 6e20 6561 7379 0a63 6f6d 7061 7269   an easy.compari
│ │ │ │ +0005f490: 736f 6e2e 2042 7574 2066 6f72 2065 7861  son. But for exa
│ │ │ │ +0005f4a0: 6d70 6c65 2074 6865 2062 692d 6772 6164  mple the bi-grad
│ │ │ │ +0005f4b0: 6564 2042 6574 7469 206e 756d 6265 7273  ed Betti numbers
│ │ │ │ +0005f4c0: 2061 7265 2065 7175 616c 2e20 7468 6973   are equal. this
│ │ │ │ +0005f4d0: 2073 6565 6d73 0a74 6f20 7374 6172 7420   seems.to start 
│ │ │ │ +0005f4e0: 7769 7468 2063 3d33 2e0a 0a2b 2d2d 2d2d  with c=3...+----
│ │ │ │ +0005f4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005f500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005f510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005f520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005f530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005f540: 2d2d 2d2d 2b0a 7c69 3920 3a20 7365 7452  ----+.|i9 : setR
│ │ │ │ -0005f550: 616e 646f 6d53 6565 6420 3020 2020 2020  andomSeed 0     
│ │ │ │ +0005f530: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a  ---------+.|i9 :
│ │ │ │ +0005f540: 2073 6574 5261 6e64 6f6d 5365 6564 2030   setRandomSeed 0
│ │ │ │ +0005f550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f590: 2020 2020 7c0a 7c20 2d2d 2073 6574 7469      |.| -- setti
│ │ │ │ -0005f5a0: 6e67 2072 616e 646f 6d20 7365 6564 2074  ng random seed t
│ │ │ │ -0005f5b0: 6f20 3020 2020 2020 2020 2020 2020 2020  o 0             
│ │ │ │ +0005f580: 2020 2020 2020 2020 207c 0a7c 202d 2d20           |.| -- 
│ │ │ │ +0005f590: 7365 7474 696e 6720 7261 6e64 6f6d 2073  setting random s
│ │ │ │ +0005f5a0: 6565 6420 746f 2030 2020 2020 2020 2020  eed to 0        
│ │ │ │ +0005f5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f5e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0005f5d0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005f5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f630: 2020 2020 7c0a 7c6f 3920 3d20 3020 2020      |.|o9 = 0   
│ │ │ │ +0005f620: 2020 2020 2020 2020 207c 0a7c 6f39 203d           |.|o9 =
│ │ │ │ +0005f630: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │  0005f640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f680: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0005f670: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0005f680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005f690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005f6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005f6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005f6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005f6d0: 2d2d 2d2d 2b0a 7c69 3130 203a 206e 203d  ----+.|i10 : n =
│ │ │ │ -0005f6e0: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ +0005f6c0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020  ---------+.|i10 
│ │ │ │ +0005f6d0: 3a20 6e20 3d20 3320 2020 2020 2020 2020  : n = 3         
│ │ │ │ +0005f6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f720: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0005f710: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005f720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f770: 2020 2020 7c0a 7c6f 3130 203d 2033 2020      |.|o10 = 3  
│ │ │ │ +0005f760: 2020 2020 2020 2020 207c 0a7c 6f31 3020           |.|o10 
│ │ │ │ +0005f770: 3d20 3320 2020 2020 2020 2020 2020 2020  = 3             
│ │ │ │  0005f780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f7c0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0005f7b0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0005f7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005f7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005f7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005f7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005f800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005f810: 2d2d 2d2d 2b0a 7c69 3131 203a 2063 203d  ----+.|i11 : c =
│ │ │ │ -0005f820: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ +0005f800: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120  ---------+.|i11 
│ │ │ │ +0005f810: 3a20 6320 3d20 3320 2020 2020 2020 2020  : c = 3         
│ │ │ │ +0005f820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f860: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0005f850: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005f860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f8b0: 2020 2020 7c0a 7c6f 3131 203d 2033 2020      |.|o11 = 3  
│ │ │ │ +0005f8a0: 2020 2020 2020 2020 207c 0a7c 6f31 3120           |.|o11 
│ │ │ │ +0005f8b0: 3d20 3320 2020 2020 2020 2020 2020 2020  = 3             
│ │ │ │  0005f8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f900: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0005f8f0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0005f900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005f910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005f920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005f930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005f940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005f950: 2d2d 2d2d 2b0a 7c69 3132 203a 206b 6b20  ----+.|i12 : kk 
│ │ │ │ -0005f960: 3d20 5a5a 2f31 3031 2020 2020 2020 2020  = ZZ/101        
│ │ │ │ +0005f940: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220  ---------+.|i12 
│ │ │ │ +0005f950: 3a20 6b6b 203d 205a 5a2f 3130 3120 2020  : kk = ZZ/101   
│ │ │ │ +0005f960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f9a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0005f990: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005f9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f9f0: 2020 2020 7c0a 7c6f 3132 203d 206b 6b20      |.|o12 = kk 
│ │ │ │ +0005f9e0: 2020 2020 2020 2020 207c 0a7c 6f31 3220           |.|o12 
│ │ │ │ +0005f9f0: 3d20 6b6b 2020 2020 2020 2020 2020 2020  = kk            
│ │ │ │  0005fa00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fa10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fa20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fa30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fa40: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0005fa30: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005fa40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fa50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fa60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fa70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fa80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fa90: 2020 2020 7c0a 7c6f 3132 203a 2051 756f      |.|o12 : Quo
│ │ │ │ -0005faa0: 7469 656e 7452 696e 6720 2020 2020 2020  tientRing       
│ │ │ │ +0005fa80: 2020 2020 2020 2020 207c 0a7c 6f31 3220           |.|o12 
│ │ │ │ +0005fa90: 3a20 5175 6f74 6965 6e74 5269 6e67 2020  : QuotientRing  
│ │ │ │ +0005faa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fae0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0005fad0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0005fae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005faf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005fb00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005fb10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005fb20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005fb30: 2d2d 2d2d 2b0a 7c69 3133 203a 2052 203d  ----+.|i13 : R =
│ │ │ │ -0005fb40: 206b 6b5b 785f 302e 2e78 5f28 6e2d 3129   kk[x_0..x_(n-1)
│ │ │ │ -0005fb50: 5d20 2020 2020 2020 2020 2020 2020 2020  ]               
│ │ │ │ +0005fb20: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3320  ---------+.|i13 
│ │ │ │ +0005fb30: 3a20 5220 3d20 6b6b 5b78 5f30 2e2e 785f  : R = kk[x_0..x_
│ │ │ │ +0005fb40: 286e 2d31 295d 2020 2020 2020 2020 2020  (n-1)]          
│ │ │ │ +0005fb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fb80: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0005fb70: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005fb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fbd0: 2020 2020 7c0a 7c6f 3133 203d 2052 2020      |.|o13 = R  
│ │ │ │ +0005fbc0: 2020 2020 2020 2020 207c 0a7c 6f31 3320           |.|o13 
│ │ │ │ +0005fbd0: 3d20 5220 2020 2020 2020 2020 2020 2020  = R             
│ │ │ │  0005fbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fc20: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0005fc10: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005fc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fc70: 2020 2020 7c0a 7c6f 3133 203a 2050 6f6c      |.|o13 : Pol
│ │ │ │ -0005fc80: 796e 6f6d 6961 6c52 696e 6720 2020 2020  ynomialRing     
│ │ │ │ +0005fc60: 2020 2020 2020 2020 207c 0a7c 6f31 3320           |.|o13 
│ │ │ │ +0005fc70: 3a20 506f 6c79 6e6f 6d69 616c 5269 6e67  : PolynomialRing
│ │ │ │ +0005fc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fcb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fcc0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0005fcb0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0005fcc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005fcd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005fce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005fcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005fd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005fd10: 2d2d 2d2d 2b0a 7c69 3134 203a 2049 203d  ----+.|i14 : I =
│ │ │ │ -0005fd20: 2069 6465 616c 2061 7070 6c79 2863 2c20   ideal apply(c, 
│ │ │ │ -0005fd30: 692d 3e52 5f69 5e32 2920 2020 2020 2020  i->R_i^2)       
│ │ │ │ +0005fd00: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3420  ---------+.|i14 
│ │ │ │ +0005fd10: 3a20 4920 3d20 6964 6561 6c20 6170 706c  : I = ideal appl
│ │ │ │ +0005fd20: 7928 632c 2069 2d3e 525f 695e 3229 2020  y(c, i->R_i^2)  
│ │ │ │ +0005fd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fd60: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0005fd50: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005fd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fdb0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0005fdc0: 2020 2020 2032 2020 2032 2020 2032 2020       2   2   2  
│ │ │ │ +0005fda0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005fdb0: 2020 2020 2020 2020 2020 3220 2020 3220            2   2 
│ │ │ │ +0005fdc0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │  0005fdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fe00: 2020 2020 7c0a 7c6f 3134 203d 2069 6465      |.|o14 = ide
│ │ │ │ -0005fe10: 616c 2028 7820 2c20 7820 2c20 7820 2920  al (x , x , x ) 
│ │ │ │ +0005fdf0: 2020 2020 2020 2020 207c 0a7c 6f31 3420           |.|o14 
│ │ │ │ +0005fe00: 3d20 6964 6561 6c20 2878 202c 2078 202c  = ideal (x , x ,
│ │ │ │ +0005fe10: 2078 2029 2020 2020 2020 2020 2020 2020   x )            
│ │ │ │  0005fe20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fe30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fe40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fe50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0005fe60: 2020 2020 2030 2020 2031 2020 2032 2020       0   1   2  
│ │ │ │ +0005fe40: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005fe50: 2020 2020 2020 2020 2020 3020 2020 3120            0   1 
│ │ │ │ +0005fe60: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │  0005fe70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fe80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fe90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fea0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0005fe90: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005fea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005feb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fef0: 2020 2020 7c0a 7c6f 3134 203a 2049 6465      |.|o14 : Ide
│ │ │ │ -0005ff00: 616c 206f 6620 5220 2020 2020 2020 2020  al of R         
│ │ │ │ +0005fee0: 2020 2020 2020 2020 207c 0a7c 6f31 3420           |.|o14 
│ │ │ │ +0005fef0: 3a20 4964 6561 6c20 6f66 2052 2020 2020  : Ideal of R    
│ │ │ │ +0005ff00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ff10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ff20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ff30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ff40: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0005ff30: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0005ff40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005ff50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005ff60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005ff70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ff80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ff90: 2d2d 2d2d 2b0a 7c69 3135 203a 2066 6620  ----+.|i15 : ff 
│ │ │ │ -0005ffa0: 3d20 6765 6e73 2049 2020 2020 2020 2020  = gens I        
│ │ │ │ +0005ff80: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3520  ---------+.|i15 
│ │ │ │ +0005ff90: 3a20 6666 203d 2067 656e 7320 4920 2020  : ff = gens I   
│ │ │ │ +0005ffa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ffb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ffc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ffd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ffe0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0005ffd0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005ffe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060030: 2020 2020 7c0a 7c6f 3135 203d 207c 2078      |.|o15 = | x
│ │ │ │ -00060040: 5f30 5e32 2078 5f31 5e32 2078 5f32 5e32  _0^2 x_1^2 x_2^2
│ │ │ │ -00060050: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ +00060020: 2020 2020 2020 2020 207c 0a7c 6f31 3520           |.|o15 
│ │ │ │ +00060030: 3d20 7c20 785f 305e 3220 785f 315e 3220  = | x_0^2 x_1^2 
│ │ │ │ +00060040: 785f 325e 3220 7c20 2020 2020 2020 2020  x_2^2 |         
│ │ │ │ +00060050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060080: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00060070: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00060080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000600a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000600b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000600c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000600d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000600e0: 2020 2020 2031 2020 2020 2020 3320 2020       1      3   
│ │ │ │ +000600c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000600d0: 2020 2020 2020 2020 2020 3120 2020 2020            1     
│ │ │ │ +000600e0: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │  000600f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060120: 2020 2020 7c0a 7c6f 3135 203a 204d 6174      |.|o15 : Mat
│ │ │ │ -00060130: 7269 7820 5220 203c 2d2d 2052 2020 2020  rix R  <-- R    
│ │ │ │ +00060110: 2020 2020 2020 2020 207c 0a7c 6f31 3520           |.|o15 
│ │ │ │ +00060120: 3a20 4d61 7472 6978 2052 2020 3c2d 2d20  : Matrix R  <-- 
│ │ │ │ +00060130: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │  00060140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060170: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00060160: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00060170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00060180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00060190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000601a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000601b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000601c0: 2d2d 2d2d 2b0a 7c69 3136 203a 2052 6261  ----+.|i16 : Rba
│ │ │ │ -000601d0: 7220 3d20 522f 4920 2020 2020 2020 2020  r = R/I         
│ │ │ │ +000601b0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3620  ---------+.|i16 
│ │ │ │ +000601c0: 3a20 5262 6172 203d 2052 2f49 2020 2020  : Rbar = R/I    
│ │ │ │ +000601d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000601e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000601f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060210: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00060200: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00060210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060260: 2020 2020 7c0a 7c6f 3136 203d 2052 6261      |.|o16 = Rba
│ │ │ │ -00060270: 7220 2020 2020 2020 2020 2020 2020 2020  r               
│ │ │ │ +00060250: 2020 2020 2020 2020 207c 0a7c 6f31 3620           |.|o16 
│ │ │ │ +00060260: 3d20 5262 6172 2020 2020 2020 2020 2020  = Rbar          
│ │ │ │ +00060270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000602a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000602b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000602a0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000602b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000602c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000602d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000602e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000602f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060300: 2020 2020 7c0a 7c6f 3136 203a 2051 756f      |.|o16 : Quo
│ │ │ │ -00060310: 7469 656e 7452 696e 6720 2020 2020 2020  tientRing       
│ │ │ │ +000602f0: 2020 2020 2020 2020 207c 0a7c 6f31 3620           |.|o16 
│ │ │ │ +00060300: 3a20 5175 6f74 6965 6e74 5269 6e67 2020  : QuotientRing  
│ │ │ │ +00060310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060350: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00060340: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00060350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00060360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00060370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00060380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00060390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000603a0: 2d2d 2d2d 2b0a 7c69 3137 203a 2062 6172  ----+.|i17 : bar
│ │ │ │ -000603b0: 203d 206d 6170 2852 6261 722c 2052 2920   = map(Rbar, R) 
│ │ │ │ +00060390: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3720  ---------+.|i17 
│ │ │ │ +000603a0: 3a20 6261 7220 3d20 6d61 7028 5262 6172  : bar = map(Rbar
│ │ │ │ +000603b0: 2c20 5229 2020 2020 2020 2020 2020 2020  , R)            
│ │ │ │  000603c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000603d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000603e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000603f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000603e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000603f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060440: 2020 2020 7c0a 7c6f 3137 203d 206d 6170      |.|o17 = map
│ │ │ │ -00060450: 2028 5262 6172 2c20 522c 207b 7820 2c20   (Rbar, R, {x , 
│ │ │ │ -00060460: 7820 2c20 7820 7d29 2020 2020 2020 2020  x , x })        
│ │ │ │ +00060430: 2020 2020 2020 2020 207c 0a7c 6f31 3720           |.|o17 
│ │ │ │ +00060440: 3d20 6d61 7020 2852 6261 722c 2052 2c20  = map (Rbar, R, 
│ │ │ │ +00060450: 7b78 202c 2078 202c 2078 207d 2920 2020  {x , x , x })   
│ │ │ │ +00060460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060490: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000604a0: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -000604b0: 2031 2020 2032 2020 2020 2020 2020 2020   1   2          
│ │ │ │ +00060480: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00060490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000604a0: 2020 3020 2020 3120 2020 3220 2020 2020    0   1   2     
│ │ │ │ +000604b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000604c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000604d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000604e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000604d0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000604e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000604f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060530: 2020 2020 7c0a 7c6f 3137 203a 2052 696e      |.|o17 : Rin
│ │ │ │ -00060540: 674d 6170 2052 6261 7220 3c2d 2d20 5220  gMap Rbar <-- R 
│ │ │ │ +00060520: 2020 2020 2020 2020 207c 0a7c 6f31 3720           |.|o17 
│ │ │ │ +00060530: 3a20 5269 6e67 4d61 7020 5262 6172 203c  : RingMap Rbar <
│ │ │ │ +00060540: 2d2d 2052 2020 2020 2020 2020 2020 2020  -- R            
│ │ │ │  00060550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060580: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00060570: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00060580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00060590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000605a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000605b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000605c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000605d0: 2d2d 2d2d 2b0a 7c69 3138 203a 204b 203d  ----+.|i18 : K =
│ │ │ │ -000605e0: 2063 6f6b 6572 2076 6172 7320 5262 6172   coker vars Rbar
│ │ │ │ +000605c0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3820  ---------+.|i18 
│ │ │ │ +000605d0: 3a20 4b20 3d20 636f 6b65 7220 7661 7273  : K = coker vars
│ │ │ │ +000605e0: 2052 6261 7220 2020 2020 2020 2020 2020   Rbar           
│ │ │ │  000605f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060620: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00060610: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00060620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060670: 2020 2020 7c0a 7c6f 3138 203d 2063 6f6b      |.|o18 = cok
│ │ │ │ -00060680: 6572 6e65 6c20 7c20 785f 3020 785f 3120  ernel | x_0 x_1 
│ │ │ │ -00060690: 785f 3220 7c20 2020 2020 2020 2020 2020  x_2 |           
│ │ │ │ +00060660: 2020 2020 2020 2020 207c 0a7c 6f31 3820           |.|o18 
│ │ │ │ +00060670: 3d20 636f 6b65 726e 656c 207c 2078 5f30  = cokernel | x_0
│ │ │ │ +00060680: 2078 5f31 2078 5f32 207c 2020 2020 2020   x_1 x_2 |      
│ │ │ │ +00060690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000606a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000606b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000606c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000606b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000606c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000606d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000606e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000606f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060710: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00060720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060730: 2020 2020 2020 2020 2020 3120 2020 2020            1     
│ │ │ │ +00060700: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00060710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00060720: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ +00060730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060760: 2020 2020 7c0a 7c6f 3138 203a 2052 6261      |.|o18 : Rba
│ │ │ │ -00060770: 722d 6d6f 6475 6c65 2c20 7175 6f74 6965  r-module, quotie
│ │ │ │ -00060780: 6e74 206f 6620 5262 6172 2020 2020 2020  nt of Rbar      
│ │ │ │ +00060750: 2020 2020 2020 2020 207c 0a7c 6f31 3820           |.|o18 
│ │ │ │ +00060760: 3a20 5262 6172 2d6d 6f64 756c 652c 2071  : Rbar-module, q
│ │ │ │ +00060770: 756f 7469 656e 7420 6f66 2052 6261 7220  uotient of Rbar 
│ │ │ │ +00060780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000607a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000607b0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +000607a0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +000607b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000607c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000607d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000607e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000607f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00060800: 2d2d 2d2d 2b0a 7c69 3139 203a 204d 6261  ----+.|i19 : Mba
│ │ │ │ -00060810: 7220 3d20 7072 756e 6520 636f 6b65 7220  r = prune coker 
│ │ │ │ -00060820: 7261 6e64 6f6d 2852 6261 725e 322c 2052  random(Rbar^2, R
│ │ │ │ -00060830: 6261 725e 7b2d 322c 2d32 7d29 2020 2020  bar^{-2,-2})    
│ │ │ │ -00060840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060850: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000607f0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3920  ---------+.|i19 
│ │ │ │ +00060800: 3a20 4d62 6172 203d 2070 7275 6e65 2063  : Mbar = prune c
│ │ │ │ +00060810: 6f6b 6572 2072 616e 646f 6d28 5262 6172  oker random(Rbar
│ │ │ │ +00060820: 5e32 2c20 5262 6172 5e7b 2d32 2c2d 327d  ^2, Rbar^{-2,-2}
│ │ │ │ +00060830: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +00060840: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00060850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000608a0: 2020 2020 7c0a 7c6f 3139 203d 2063 6f6b      |.|o19 = cok
│ │ │ │ -000608b0: 6572 6e65 6c20 7c20 785f 3078 5f31 2b31  ernel | x_0x_1+1
│ │ │ │ -000608c0: 3578 5f30 785f 322b 3338 785f 3178 5f32  5x_0x_2+38x_1x_2
│ │ │ │ -000608d0: 2034 3578 5f30 785f 322b 3239 785f 3178   45x_0x_2+29x_1x
│ │ │ │ -000608e0: 5f32 2020 2020 2020 2020 7c20 2020 2020  _2        |     
│ │ │ │ -000608f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00060900: 2020 2020 2020 7c20 3335 785f 3078 5f32        | 35x_0x_2
│ │ │ │ -00060910: 2d33 3078 5f31 785f 3220 2020 2020 2020  -30x_1x_2       
│ │ │ │ -00060920: 2078 5f30 785f 312d 3130 785f 3078 5f32   x_0x_1-10x_0x_2
│ │ │ │ -00060930: 2d32 3278 5f31 785f 3220 7c20 2020 2020  -22x_1x_2 |     
│ │ │ │ -00060940: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00060890: 2020 2020 2020 2020 207c 0a7c 6f31 3920           |.|o19 
│ │ │ │ +000608a0: 3d20 636f 6b65 726e 656c 207c 2078 5f30  = cokernel | x_0
│ │ │ │ +000608b0: 785f 312b 3135 785f 3078 5f32 2b33 3878  x_1+15x_0x_2+38x
│ │ │ │ +000608c0: 5f31 785f 3220 3435 785f 3078 5f32 2b32  _1x_2 45x_0x_2+2
│ │ │ │ +000608d0: 3978 5f31 785f 3220 2020 2020 2020 207c  9x_1x_2        |
│ │ │ │ +000608e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000608f0: 2020 2020 2020 2020 2020 207c 2033 3578             | 35x
│ │ │ │ +00060900: 5f30 785f 322d 3330 785f 3178 5f32 2020  _0x_2-30x_1x_2  
│ │ │ │ +00060910: 2020 2020 2020 785f 3078 5f31 2d31 3078        x_0x_1-10x
│ │ │ │ +00060920: 5f30 785f 322d 3232 785f 3178 5f32 207c  _0x_2-22x_1x_2 |
│ │ │ │ +00060930: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00060940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060990: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000609a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000609b0: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +00060980: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00060990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000609a0: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +000609b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000609c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000609d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000609e0: 2020 2020 7c0a 7c6f 3139 203a 2052 6261      |.|o19 : Rba
│ │ │ │ -000609f0: 722d 6d6f 6475 6c65 2c20 7175 6f74 6965  r-module, quotie
│ │ │ │ -00060a00: 6e74 206f 6620 5262 6172 2020 2020 2020  nt of Rbar      
│ │ │ │ +000609d0: 2020 2020 2020 2020 207c 0a7c 6f31 3920           |.|o19 
│ │ │ │ +000609e0: 3a20 5262 6172 2d6d 6f64 756c 652c 2071  : Rbar-module, q
│ │ │ │ +000609f0: 756f 7469 656e 7420 6f66 2052 6261 7220  uotient of Rbar 
│ │ │ │ +00060a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060a30: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00060a20: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00060a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00060a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00060a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00060a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00060a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00060a80: 2d2d 2d2d 2b0a 7c69 3230 203a 2045 5320  ----+.|i20 : ES 
│ │ │ │ -00060a90: 3d20 6e65 7745 7874 284d 6261 722c 4b2c  = newExt(Mbar,K,
│ │ │ │ -00060aa0: 4c69 6674 203d 3e20 7472 7565 2920 2020  Lift => true)   
│ │ │ │ +00060a70: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3020  ---------+.|i20 
│ │ │ │ +00060a80: 3a20 4553 203d 206e 6577 4578 7428 4d62  : ES = newExt(Mb
│ │ │ │ +00060a90: 6172 2c4b 2c4c 6966 7420 3d3e 2074 7275  ar,K,Lift => tru
│ │ │ │ +00060aa0: 6529 2020 2020 2020 2020 2020 2020 2020  e)              
│ │ │ │  00060ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060ad0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00060ac0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00060ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060b20: 2020 2020 7c0a 7c6f 3230 203d 2063 6f6b      |.|o20 = cok
│ │ │ │ -00060b30: 6572 6e65 6c20 7b30 2c20 307d 2020 207c  ernel {0, 0}   |
│ │ │ │ -00060b40: 2078 5f32 2078 5f31 2078 5f30 2030 2020   x_2 x_1 x_0 0  
│ │ │ │ -00060b50: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00060b60: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00060b70: 2073 5f32 7c0a 7c20 2020 2020 2020 2020   s_2|.|         
│ │ │ │ -00060b80: 2020 2020 2020 7b30 2c20 307d 2020 207c        {0, 0}   |
│ │ │ │ -00060b90: 2030 2020 2030 2020 2030 2020 2078 5f32   0   0   0   x_2
│ │ │ │ -00060ba0: 2078 5f31 2078 5f30 2030 2020 2030 2020   x_1 x_0 0   0  
│ │ │ │ -00060bb0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00060bc0: 2030 2020 7c0a 7c20 2020 2020 2020 2020   0  |.|         
│ │ │ │ -00060bd0: 2020 2020 2020 7b2d 322c 202d 337d 207c        {-2, -3} |
│ │ │ │ -00060be0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00060bf0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00060c00: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00060c10: 2030 2020 7c0a 7c20 2020 2020 2020 2020   0  |.|         
│ │ │ │ -00060c20: 2020 2020 2020 7b2d 322c 202d 337d 207c        {-2, -3} |
│ │ │ │ -00060c30: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00060c40: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00060c50: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00060c60: 2030 2020 7c0a 7c20 2020 2020 2020 2020   0  |.|         
│ │ │ │ -00060c70: 2020 2020 2020 7b2d 322c 202d 337d 207c        {-2, -3} |
│ │ │ │ -00060c80: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00060c90: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00060ca0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00060cb0: 2030 2020 7c0a 7c20 2020 2020 2020 2020   0  |.|         
│ │ │ │ -00060cc0: 2020 2020 2020 7b2d 322c 202d 337d 207c        {-2, -3} |
│ │ │ │ -00060cd0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00060ce0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00060cf0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00060d00: 2030 2020 7c0a 7c20 2020 2020 2020 2020   0  |.|         
│ │ │ │ -00060d10: 2020 2020 2020 7b2d 312c 202d 327d 207c        {-1, -2} |
│ │ │ │ -00060d20: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00060d30: 2030 2020 2030 2020 2078 5f32 2078 5f31   0   0   x_2 x_1
│ │ │ │ -00060d40: 2078 5f30 2030 2020 2030 2020 2030 2020   x_0 0   0   0  
│ │ │ │ -00060d50: 2030 2020 7c0a 7c20 2020 2020 2020 2020   0  |.|         
│ │ │ │ -00060d60: 2020 2020 2020 7b2d 312c 202d 327d 207c        {-1, -2} |
│ │ │ │ -00060d70: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00060d80: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00060d90: 2030 2020 2078 5f32 2078 5f31 2078 5f30   0   x_2 x_1 x_0
│ │ │ │ -00060da0: 2030 2020 7c0a 7c20 2020 2020 2020 2020   0  |.|         
│ │ │ │ +00060b10: 2020 2020 2020 2020 207c 0a7c 6f32 3020           |.|o20 
│ │ │ │ +00060b20: 3d20 636f 6b65 726e 656c 207b 302c 2030  = cokernel {0, 0
│ │ │ │ +00060b30: 7d20 2020 7c20 785f 3220 785f 3120 785f  }   | x_2 x_1 x_
│ │ │ │ +00060b40: 3020 3020 2020 3020 2020 3020 2020 3020  0 0   0   0   0 
│ │ │ │ +00060b50: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00060b60: 2020 3020 2020 735f 327c 0a7c 2020 2020    0   s_2|.|    
│ │ │ │ +00060b70: 2020 2020 2020 2020 2020 207b 302c 2030             {0, 0
│ │ │ │ +00060b80: 7d20 2020 7c20 3020 2020 3020 2020 3020  }   | 0   0   0 
│ │ │ │ +00060b90: 2020 785f 3220 785f 3120 785f 3020 3020    x_2 x_1 x_0 0 
│ │ │ │ +00060ba0: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00060bb0: 2020 3020 2020 3020 207c 0a7c 2020 2020    0   0  |.|    
│ │ │ │ +00060bc0: 2020 2020 2020 2020 2020 207b 2d32 2c20             {-2, 
│ │ │ │ +00060bd0: 2d33 7d20 7c20 3020 2020 3020 2020 3020  -3} | 0   0   0 
│ │ │ │ +00060be0: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00060bf0: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00060c00: 2020 3020 2020 3020 207c 0a7c 2020 2020    0   0  |.|    
│ │ │ │ +00060c10: 2020 2020 2020 2020 2020 207b 2d32 2c20             {-2, 
│ │ │ │ +00060c20: 2d33 7d20 7c20 3020 2020 3020 2020 3020  -3} | 0   0   0 
│ │ │ │ +00060c30: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00060c40: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00060c50: 2020 3020 2020 3020 207c 0a7c 2020 2020    0   0  |.|    
│ │ │ │ +00060c60: 2020 2020 2020 2020 2020 207b 2d32 2c20             {-2, 
│ │ │ │ +00060c70: 2d33 7d20 7c20 3020 2020 3020 2020 3020  -3} | 0   0   0 
│ │ │ │ +00060c80: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00060c90: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00060ca0: 2020 3020 2020 3020 207c 0a7c 2020 2020    0   0  |.|    
│ │ │ │ +00060cb0: 2020 2020 2020 2020 2020 207b 2d32 2c20             {-2, 
│ │ │ │ +00060cc0: 2d33 7d20 7c20 3020 2020 3020 2020 3020  -3} | 0   0   0 
│ │ │ │ +00060cd0: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00060ce0: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00060cf0: 2020 3020 2020 3020 207c 0a7c 2020 2020    0   0  |.|    
│ │ │ │ +00060d00: 2020 2020 2020 2020 2020 207b 2d31 2c20             {-1, 
│ │ │ │ +00060d10: 2d32 7d20 7c20 3020 2020 3020 2020 3020  -2} | 0   0   0 
│ │ │ │ +00060d20: 2020 3020 2020 3020 2020 3020 2020 785f    0   0   0   x_
│ │ │ │ +00060d30: 3220 785f 3120 785f 3020 3020 2020 3020  2 x_1 x_0 0   0 
│ │ │ │ +00060d40: 2020 3020 2020 3020 207c 0a7c 2020 2020    0   0  |.|    
│ │ │ │ +00060d50: 2020 2020 2020 2020 2020 207b 2d31 2c20             {-1, 
│ │ │ │ +00060d60: 2d32 7d20 7c20 3020 2020 3020 2020 3020  -2} | 0   0   0 
│ │ │ │ +00060d70: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00060d80: 2020 3020 2020 3020 2020 785f 3220 785f    0   0   x_2 x_
│ │ │ │ +00060d90: 3120 785f 3020 3020 207c 0a7c 2020 2020  1 x_0 0  |.|    
│ │ │ │ +00060da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060df0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00060de0: 2020 2020 2020 2020 207c 0a7c 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060e30: 2020 2020 2020 2020 3820 2020 2020 2020          8       
│ │ │ │ -00060e40: 2020 2020 7c0a 7c6f 3230 203a 206b 6b5b      |.|o20 : kk[
│ │ │ │ -00060e50: 7320 2e2e 7320 2c20 7820 2e2e 7820 5d2d  s ..s , x ..x ]-
│ │ │ │ -00060e60: 6d6f 6475 6c65 2c20 7175 6f74 6965 6e74  module, quotient
│ │ │ │ -00060e70: 206f 6620 286b 6b5b 7320 2e2e 7320 2c20   of (kk[s ..s , 
│ │ │ │ -00060e80: 7820 2e2e 7820 5d29 2020 2020 2020 2020  x ..x ])        
│ │ │ │ -00060e90: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00060ea0: 2030 2020 2032 2020 2030 2020 2032 2020   0   2   0   2  
│ │ │ │ -00060eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060ec0: 2020 2020 2020 2020 2030 2020 2032 2020           0   2  
│ │ │ │ -00060ed0: 2030 2020 2032 2020 2020 2020 2020 2020   0   2          
│ │ │ │ -00060ee0: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
│ │ │ │ +00060e20: 2020 2020 2020 2020 2020 2020 2038 2020               8  
│ │ │ │ +00060e30: 2020 2020 2020 2020 207c 0a7c 6f32 3020           |.|o20 
│ │ │ │ +00060e40: 3a20 6b6b 5b73 202e 2e73 202c 2078 202e  : kk[s ..s , x .
│ │ │ │ +00060e50: 2e78 205d 2d6d 6f64 756c 652c 2071 756f  .x ]-module, quo
│ │ │ │ +00060e60: 7469 656e 7420 6f66 2028 6b6b 5b73 202e  tient of (kk[s .
│ │ │ │ +00060e70: 2e73 202c 2078 202e 2e78 205d 2920 2020  .s , x ..x ])   
│ │ │ │ +00060e80: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00060e90: 2020 2020 2020 3020 2020 3220 2020 3020        0   2   0 
│ │ │ │ +00060ea0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +00060eb0: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ +00060ec0: 2020 3220 2020 3020 2020 3220 2020 2020    2   0   2     
│ │ │ │ +00060ed0: 2020 2020 2020 2020 207c 0a7c 2d2d 2d2d           |.|----
│ │ │ │ +00060ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00060ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00060f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00060f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00060f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00060f30: 2d2d 2d2d 7c0a 7c73 5f31 2073 5f30 2030  ----|.|s_1 s_0 0
│ │ │ │ -00060f40: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00060f50: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00060f60: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00060f70: 2020 2030 2020 2030 2020 2020 2020 2020     0   0        
│ │ │ │ -00060f80: 2020 2020 7c0a 7c30 2020 2030 2020 2073      |.|0   0   s
│ │ │ │ -00060f90: 5f32 2073 5f31 2073 5f30 2030 2020 2030  _2 s_1 s_0 0   0
│ │ │ │ -00060fa0: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00060fb0: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00060fc0: 2020 2030 2020 2030 2020 2020 2020 2020     0   0        
│ │ │ │ -00060fd0: 2020 2020 7c0a 7c30 2020 2030 2020 2030      |.|0   0   0
│ │ │ │ -00060fe0: 2020 2030 2020 2030 2020 2078 5f32 2078     0   0   x_2 x
│ │ │ │ -00060ff0: 5f31 2078 5f30 2030 2020 2030 2020 2030  _1 x_0 0   0   0
│ │ │ │ -00061000: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00061010: 2020 2030 2020 2030 2020 2020 2020 2020     0   0        
│ │ │ │ -00061020: 2020 2020 7c0a 7c30 2020 2030 2020 2030      |.|0   0   0
│ │ │ │ -00061030: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00061040: 2020 2030 2020 2078 5f32 2078 5f31 2078     0   x_2 x_1 x
│ │ │ │ -00061050: 5f30 2030 2020 2030 2020 2030 2020 2030  _0 0   0   0   0
│ │ │ │ -00061060: 2020 2030 2020 2030 2020 2020 2020 2020     0   0        
│ │ │ │ -00061070: 2020 2020 7c0a 7c30 2020 2030 2020 2030      |.|0   0   0
│ │ │ │ -00061080: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00061090: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -000610a0: 2020 2078 5f32 2078 5f31 2078 5f30 2030     x_2 x_1 x_0 0
│ │ │ │ -000610b0: 2020 2030 2020 2030 2020 2020 2020 2020     0   0        
│ │ │ │ -000610c0: 2020 2020 7c0a 7c30 2020 2030 2020 2030      |.|0   0   0
│ │ │ │ -000610d0: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -000610e0: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -000610f0: 2020 2030 2020 2030 2020 2030 2020 2078     0   0   0   x
│ │ │ │ -00061100: 5f32 2078 5f31 2078 5f30 2020 2020 2020  _2 x_1 x_0      
│ │ │ │ -00061110: 2020 2020 7c0a 7c30 2020 2030 2020 2030      |.|0   0   0
│ │ │ │ -00061120: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00061130: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00061140: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00061150: 2020 2030 2020 2030 2020 2020 2020 2020     0   0        
│ │ │ │ -00061160: 2020 2020 7c0a 7c30 2020 2030 2020 2030      |.|0   0   0
│ │ │ │ -00061170: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00061180: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00061190: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -000611a0: 2020 2030 2020 2030 2020 2020 2020 2020     0   0        
│ │ │ │ -000611b0: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
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│ │ │ │ +00060f30: 735f 3020 3020 2020 3020 2020 3020 2020  s_0 0   0   0   
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│ │ │ │ +00060fc0: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
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│ │ │ │ +00060fe0: 785f 3220 785f 3120 785f 3020 3020 2020  x_2 x_1 x_0 0   
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│ │ │ │ +00061000: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00061010: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
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│ │ │ │ +00061040: 785f 3120 785f 3020 3020 2020 3020 2020  x_1 x_0 0   0   
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│ │ │ │ +00061100: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
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│ │ │ │ +00061150: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
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│ │ │ │ -000611f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00061200: 2d2d 2d2d 7c0a 7c30 2020 2020 2020 2020  ----|.|0        
│ │ │ │ -00061210: 2020 2020 2020 2030 2020 2020 2020 2020         0        
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│ │ │ │ -00061240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061250: 2020 2020 7c0a 7c30 2020 2020 2020 2020      |.|0        
│ │ │ │ -00061260: 2020 2020 2020 2030 2020 2020 2020 2020         0        
│ │ │ │ -00061270: 2020 2020 2020 2030 2020 2020 2020 2020         0        
│ │ │ │ -00061280: 2020 2020 2020 3020 2020 2020 2020 2020        0         
│ │ │ │ -00061290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000612a0: 2020 2020 7c0a 7c73 5f30 2d31 3173 5f31      |.|s_0-11s_1
│ │ │ │ -000612b0: 2d34 3073 5f32 202d 735f 3120 2020 2020  -40s_2 -s_1     
│ │ │ │ -000612c0: 2020 2020 2020 2039 735f 312d 3233 735f         9s_1-23s_
│ │ │ │ -000612d0: 3220 2020 2020 3137 735f 3120 2020 2020  2     17s_1     
│ │ │ │ -000612e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000612f0: 2020 2020 7c0a 7c34 3573 5f31 2d33 3573      |.|45s_1-35s
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│ │ │ │ -00061310: 2020 2020 2020 2073 5f30 2d37 735f 312b         s_0-7s_1+
│ │ │ │ -00061320: 3135 735f 3220 735f 3120 2020 2020 2020  15s_2 s_1       
│ │ │ │ -00061330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061340: 2020 2020 7c0a 7c2d 3130 735f 312d 3236      |.|-10s_1-26
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│ │ │ │ -00061360: 2d34 3073 5f32 2033 3473 5f31 2b34 735f  -40s_2 34s_1+4s_
│ │ │ │ -00061370: 3220 2020 2020 3973 5f31 2d32 3373 5f32  2     9s_1-23s_2
│ │ │ │ -00061380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061390: 2020 2020 7c0a 7c35 3073 5f31 2d73 5f32      |.|50s_1-s_2
│ │ │ │ -000613a0: 2020 2020 2020 2034 3573 5f31 2d33 3573         45s_1-35s
│ │ │ │ -000613b0: 5f32 2020 2020 2031 3073 5f31 2b32 3673  _2     10s_1+26s
│ │ │ │ -000613c0: 5f32 2020 2020 735f 302d 3438 735f 312b  _2    s_0-48s_1+
│ │ │ │ -000613d0: 3135 735f 3220 2020 2020 2020 2020 2020  15s_2           
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│ │ │ │ -000613f0: 2020 2020 2020 2030 2020 2020 2020 2020         0        
│ │ │ │ -00061400: 2020 2020 2020 2030 2020 2020 2020 2020         0        
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│ │ │ │ -00061440: 2020 2020 2020 2030 2020 2020 2020 2020         0        
│ │ │ │ -00061450: 2020 2020 2020 2030 2020 2020 2020 2020         0        
│ │ │ │ -00061460: 2020 2020 2020 3020 2020 2020 2020 2020        0         
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│ │ │ │ +000612b0: 2020 2020 2020 2020 2020 2020 3973 5f31              9s_1
│ │ │ │ +000612c0: 2d32 3373 5f32 2020 2020 2031 3773 5f31  -23s_2     17s_1
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│ │ │ │ +000612e0: 2020 2020 2020 2020 207c 0a7c 3435 735f           |.|45s_
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│ │ │ │ +00061320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061330: 2020 2020 2020 2020 207c 0a7c 2d31 3073           |.|-10s
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│ │ │ │ +00061350: 3439 735f 312d 3430 735f 3220 3334 735f  49s_1-40s_2 34s_
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│ │ │ │ +00061380: 2020 2020 2020 2020 207c 0a7c 3530 735f           |.|50s_
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│ │ │ │ +000613b0: 312b 3236 735f 3220 2020 2073 5f30 2d34  1+26s_2    s_0-4
│ │ │ │ +000613c0: 3873 5f31 2b31 3573 5f32 2020 2020 2020  8s_1+15s_2      
│ │ │ │ +000613d0: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
│ │ │ │ +000613e0: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ +000613f0: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ +00061400: 2020 2020 2020 2020 2020 2030 2020 2020             0    
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│ │ │ │  00061500: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00061580: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000615b0: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
│ │ │ │ +000615c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000615d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000615e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000615f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061600: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00061630: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00061660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061680: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000616b0: 2020 2020 7c0a 7c73 5f30 5e32 2b34 3273      |.|s_0^2+42s
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│ │ │ │ -000616d0: 735f 3073 5f32 2d33 3573 5f31 735f 322b  s_0s_2-35s_1s_2+
│ │ │ │ -000616e0: 3973 5f32 5e32 2020 2020 2020 2020 2020  9s_2^2          
│ │ │ │ -000616f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061700: 2020 2020 7c0a 7c30 2020 2020 2020 2020      |.|0        
│ │ │ │ +000616a0: 2020 2020 2020 2020 207c 0a7c 735f 305e           |.|s_0^
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│ │ │ │ +000616c0: 5e32 2d32 3573 5f30 735f 322d 3335 735f  ^2-25s_0s_2-35s_
│ │ │ │ +000616d0: 3173 5f32 2b39 735f 325e 3220 2020 2020  1s_2+9s_2^2     
│ │ │ │ +000616e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000616f0: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
│ │ │ │ +00061700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061710: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00061780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00061790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  000617b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000617c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000617d0: 2020 2020 2020 207c 2020 2020 2020 2020         |        
│ │ │ │ -000617e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000617f0: 2020 2020 7c0a 7c30 2020 2020 2020 2020      |.|0        
│ │ │ │ +000617c0: 2020 2020 2020 2020 2020 2020 7c20 2020              |   
│ │ │ │ +000617d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000617e0: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
│ │ │ │ +000617f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061810: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00061830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061840: 2020 2020 7c0a 7c30 2020 2020 2020 2020      |.|0        
│ │ │ │ +00061810: 2020 2020 2020 2020 2020 2020 7c20 2020              |   
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│ │ │ │ +00061840: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00061860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061870: 2020 2020 2020 207c 2020 2020 2020 2020         |        
│ │ │ │ -00061880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061890: 2020 2020 7c0a 7c30 2020 2020 2020 2020      |.|0        
│ │ │ │ +00061860: 2020 2020 2020 2020 2020 2020 7c20 2020              |   
│ │ │ │ +00061870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061880: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
│ │ │ │ +00061890: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000618b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000618c0: 2020 2020 2020 207c 2020 2020 2020 2020         |        
│ │ │ │ -000618d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000618e0: 2020 2020 7c0a 7c30 2020 2020 2020 2020      |.|0        
│ │ │ │ +000618b0: 2020 2020 2020 2020 2020 2020 7c20 2020              |   
│ │ │ │ +000618c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000618d0: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
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│ │ │ │  000618f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061900: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00061920: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00061900: 2020 2020 2020 2020 2020 2020 7c20 2020              |   
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│ │ │ │ -00061950: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00061970: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00061950: 2020 2020 2020 2020 2020 2020 7c20 2020              |   
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│ │ │ │  00061990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000619a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000619c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000619d0: 2020 2020 7c0a 7c73 5f30 5e32 2b34 3273      |.|s_0^2+42s
│ │ │ │ -000619e0: 5f30 735f 312d 3330 735f 315e 322d 3235  _0s_1-30s_1^2-25
│ │ │ │ -000619f0: 735f 3073 5f32 2d33 3573 5f31 735f 322b  s_0s_2-35s_1s_2+
│ │ │ │ -00061a00: 3973 5f32 5e32 207c 2020 2020 2020 2020  9s_2^2 |        
│ │ │ │ -00061a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061a20: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +000619a0: 2020 2020 2020 2020 2020 2020 7c20 2020              |   
│ │ │ │ +000619b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000619c0: 2020 2020 2020 2020 207c 0a7c 735f 305e           |.|s_0^
│ │ │ │ +000619d0: 322b 3432 735f 3073 5f31 2d33 3073 5f31  2+42s_0s_1-30s_1
│ │ │ │ +000619e0: 5e32 2d32 3573 5f30 735f 322d 3335 735f  ^2-25s_0s_2-35s_
│ │ │ │ +000619f0: 3173 5f32 2b39 735f 325e 3220 7c20 2020  1s_2+9s_2^2 |   
│ │ │ │ +00061a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061a10: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00061a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00061a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00061a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00061a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00061a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00061a70: 2d2d 2d2d 2b0a 7c69 3231 203a 2053 203d  ----+.|i21 : S =
│ │ │ │ -00061a80: 2072 696e 6720 4553 2020 2020 2020 2020   ring ES        
│ │ │ │ +00061a60: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3120  ---------+.|i21 
│ │ │ │ +00061a70: 3a20 5320 3d20 7269 6e67 2045 5320 2020  : S = ring ES   
│ │ │ │ +00061a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061ac0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00061ab0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00061ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061b10: 2020 2020 7c0a 7c6f 3231 203d 2053 2020      |.|o21 = S  
│ │ │ │ +00061b00: 2020 2020 2020 2020 207c 0a7c 6f32 3120           |.|o21 
│ │ │ │ +00061b10: 3d20 5320 2020 2020 2020 2020 2020 2020  = S             
│ │ │ │  00061b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061b60: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00061b50: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00061b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061bb0: 2020 2020 7c0a 7c6f 3231 203a 2050 6f6c      |.|o21 : Pol
│ │ │ │ -00061bc0: 796e 6f6d 6961 6c52 696e 6720 2020 2020  ynomialRing     
│ │ │ │ +00061ba0: 2020 2020 2020 2020 207c 0a7c 6f32 3120           |.|o21 
│ │ │ │ +00061bb0: 3a20 506f 6c79 6e6f 6d69 616c 5269 6e67  : PolynomialRing
│ │ │ │ +00061bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061c00: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00061bf0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00061c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00061c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00061c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00061c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00061c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00061c50: 2d2d 2d2d 2b0a 0a63 6f6d 7061 7265 2077  ----+..compare w
│ │ │ │ -00061c60: 6974 6820 7468 6520 6275 696c 742d 696e  ith the built-in
│ │ │ │ -00061c70: 2045 7874 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d   Ext..+---------
│ │ │ │ +00061c40: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 636f 6d70  ---------+..comp
│ │ │ │ +00061c50: 6172 6520 7769 7468 2074 6865 2062 7569  are with the bui
│ │ │ │ +00061c60: 6c74 2d69 6e20 4578 740a 0a2b 2d2d 2d2d  lt-in Ext..+----
│ │ │ │ +00061c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00061c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00061c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00061ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00061cb0: 2b0a 7c69 3232 203a 2045 4520 3d20 4578  +.|i22 : EE = Ex
│ │ │ │ -00061cc0: 7428 4d62 6172 2c4b 293b 2020 2020 2020  t(Mbar,K);      
│ │ │ │ +00061ca0: 2d2d 2d2d 2d2b 0a7c 6932 3220 3a20 4545  -----+.|i22 : EE
│ │ │ │ +00061cb0: 203d 2045 7874 284d 6261 722c 4b29 3b20   = Ext(Mbar,K); 
│ │ │ │ +00061cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061ce0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00061ce0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  00061cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00061d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00061d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00061d20: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3233 203a  --------+.|i23 :
│ │ │ │ -00061d30: 2053 2720 3d20 7269 6e67 2045 4520 2d2d   S' = ring EE --
│ │ │ │ -00061d40: 206e 6f74 6520 7468 6174 2053 2720 6973   note that S' is
│ │ │ │ -00061d50: 2074 6865 2070 6f6c 796e 6f6d 6961 6c20   the polynomial 
│ │ │ │ -00061d60: 7269 6e67 7c0a 7c20 2020 2020 2020 2020  ring|.|         
│ │ │ │ +00061d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00061d20: 6932 3320 3a20 5327 203d 2072 696e 6720  i23 : S' = ring 
│ │ │ │ +00061d30: 4545 202d 2d20 6e6f 7465 2074 6861 7420  EE -- note that 
│ │ │ │ +00061d40: 5327 2069 7320 7468 6520 706f 6c79 6e6f  S' is the polyno
│ │ │ │ +00061d50: 6d69 616c 2072 696e 677c 0a7c 2020 2020  mial ring|.|    
│ │ │ │ +00061d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061da0: 7c0a 7c6f 3233 203d 2053 2720 2020 2020  |.|o23 = S'     
│ │ │ │ +00061d90: 2020 2020 207c 0a7c 6f32 3320 3d20 5327       |.|o23 = S'
│ │ │ │ +00061da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061dd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00061dd0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00061de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061e10: 2020 2020 2020 2020 7c0a 7c6f 3233 203a          |.|o23 :
│ │ │ │ -00061e20: 2050 6f6c 796e 6f6d 6961 6c52 696e 6720   PolynomialRing 
│ │ │ │ +00061e00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00061e10: 6f32 3320 3a20 506f 6c79 6e6f 6d69 616c  o23 : Polynomial
│ │ │ │ +00061e20: 5269 6e67 2020 2020 2020 2020 2020 2020  Ring            
│ │ │ │  00061e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061e50: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00061e40: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00061e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00061e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00061e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00061e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00061e90: 2b0a 0a54 6865 2074 776f 2076 6572 7369  +..The two versi
│ │ │ │ -00061ea0: 6f6e 7320 6f66 2045 7874 2061 7070 6561  ons of Ext appea
│ │ │ │ -00061eb0: 7220 746f 2062 6520 7468 6520 7361 6d65  r to be the same
│ │ │ │ -00061ec0: 2075 7020 746f 2063 6861 6e67 6520 6f66   up to change of
│ │ │ │ -00061ed0: 2076 6172 6961 626c 6573 3a0a 0a2b 2d2d   variables:..+--
│ │ │ │ +00061e80: 2d2d 2d2d 2d2b 0a0a 5468 6520 7477 6f20  -----+..The two 
│ │ │ │ +00061e90: 7665 7273 696f 6e73 206f 6620 4578 7420  versions of Ext 
│ │ │ │ +00061ea0: 6170 7065 6172 2074 6f20 6265 2074 6865  appear to be the
│ │ │ │ +00061eb0: 2073 616d 6520 7570 2074 6f20 6368 616e   same up to chan
│ │ │ │ +00061ec0: 6765 206f 6620 7661 7269 6162 6c65 733a  ge of variables:
│ │ │ │ +00061ed0: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..+-------------
│ │ │ │  00061ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00061ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00061f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00061f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00061f20: 6932 3420 3a20 4120 3d20 6672 6565 5265  i24 : A = freeRe
│ │ │ │ -00061f30: 736f 6c75 7469 6f6e 2045 5320 2020 2020  solution ES     
│ │ │ │ +00061f10: 2d2d 2b0a 7c69 3234 203a 2041 203d 2066  --+.|i24 : A = f
│ │ │ │ +00061f20: 7265 6552 6573 6f6c 7574 696f 6e20 4553  reeResolution ES
│ │ │ │ +00061f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061f50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00061f60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00061f50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00061f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061fa0: 207c 0a7c 2020 2020 2020 2038 2020 2020   |.|       8    
│ │ │ │ -00061fb0: 2020 3336 2020 2020 2020 3636 2020 2020    36      66    
│ │ │ │ -00061fc0: 2020 3634 2020 2020 2020 3336 2020 2020    64      36    
│ │ │ │ -00061fd0: 2020 3132 2020 2020 2020 3220 2020 2020    12      2     
│ │ │ │ -00061fe0: 2020 207c 0a7c 6f32 3420 3d20 5320 203c     |.|o24 = S  <
│ │ │ │ -00061ff0: 2d2d 2053 2020 203c 2d2d 2053 2020 203c  -- S   <-- S   <
│ │ │ │ -00062000: 2d2d 2053 2020 203c 2d2d 2053 2020 203c  -- S   <-- S   <
│ │ │ │ -00062010: 2d2d 2053 2020 203c 2d2d 2053 2020 2020  -- S   <-- S    
│ │ │ │ -00062020: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00061f90: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00061fa0: 3820 2020 2020 2033 3620 2020 2020 2036  8      36      6
│ │ │ │ +00061fb0: 3620 2020 2020 2036 3420 2020 2020 2033  6      64      3
│ │ │ │ +00061fc0: 3620 2020 2020 2031 3220 2020 2020 2032  6      12      2
│ │ │ │ +00061fd0: 2020 2020 2020 2020 7c0a 7c6f 3234 203d          |.|o24 =
│ │ │ │ +00061fe0: 2053 2020 3c2d 2d20 5320 2020 3c2d 2d20   S  <-- S   <-- 
│ │ │ │ +00061ff0: 5320 2020 3c2d 2d20 5320 2020 3c2d 2d20  S   <-- S   <-- 
│ │ │ │ +00062000: 5320 2020 3c2d 2d20 5320 2020 3c2d 2d20  S   <-- S   <-- 
│ │ │ │ +00062010: 5320 2020 2020 2020 2020 7c0a 7c20 2020  S         |.|   
│ │ │ │ +00062020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062060: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00062070: 3020 2020 2020 2031 2020 2020 2020 2032  0      1       2
│ │ │ │ -00062080: 2020 2020 2020 2033 2020 2020 2020 2034         3       4
│ │ │ │ -00062090: 2020 2020 2020 2035 2020 2020 2020 2036         5       6
│ │ │ │ -000620a0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00062050: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00062060: 2020 2020 2030 2020 2020 2020 3120 2020       0      1   
│ │ │ │ +00062070: 2020 2020 3220 2020 2020 2020 3320 2020      2       3   
│ │ │ │ +00062080: 2020 2020 3420 2020 2020 2020 3520 2020      4       5   
│ │ │ │ +00062090: 2020 2020 3620 2020 2020 2020 2020 7c0a      6         |.
│ │ │ │ +000620a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000620b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000620c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000620d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000620e0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ -000620f0: 3420 3a20 436f 6d70 6c65 7820 2020 2020  4 : Complex     
│ │ │ │ +000620e0: 7c0a 7c6f 3234 203a 2043 6f6d 706c 6578  |.|o24 : Complex
│ │ │ │ +000620f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062120: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00062120: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00062130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00062160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00062170: 0a7c 6932 3520 3a20 4220 3d20 6672 6565  .|i25 : B = free
│ │ │ │ -00062180: 5265 736f 6c75 7469 6f6e 2045 4520 2020  Resolution EE   
│ │ │ │ +00062160: 2d2d 2d2d 2b0a 7c69 3235 203a 2042 203d  ----+.|i25 : B =
│ │ │ │ +00062170: 2066 7265 6552 6573 6f6c 7574 696f 6e20   freeResolution 
│ │ │ │ +00062180: 4545 2020 2020 2020 2020 2020 2020 2020  EE              
│ │ │ │  00062190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000621a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000621b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000621a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000621b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000621c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000621d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000621e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000621f0: 2020 207c 0a7c 2020 2020 2020 2020 3820     |.|        8 
│ │ │ │ -00062200: 2020 2020 2020 3336 2020 2020 2020 2036        36       6
│ │ │ │ -00062210: 3620 2020 2020 2020 3634 2020 2020 2020  6       64      
│ │ │ │ -00062220: 2033 3620 2020 2020 2020 3132 2020 2020   36       12    
│ │ │ │ -00062230: 2020 2032 207c 0a7c 6f32 3520 3d20 5327     2 |.|o25 = S'
│ │ │ │ -00062240: 2020 3c2d 2d20 5327 2020 203c 2d2d 2053    <-- S'   <-- S
│ │ │ │ -00062250: 2720 2020 3c2d 2d20 5327 2020 203c 2d2d  '   <-- S'   <--
│ │ │ │ -00062260: 2053 2720 2020 3c2d 2d20 5327 2020 203c   S'   <-- S'   <
│ │ │ │ -00062270: 2d2d 2053 2720 207c 0a7c 2020 2020 2020  -- S'  |.|      
│ │ │ │ +000621e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000621f0: 2020 2038 2020 2020 2020 2033 3620 2020     8       36   
│ │ │ │ +00062200: 2020 2020 3636 2020 2020 2020 2036 3420      66       64 
│ │ │ │ +00062210: 2020 2020 2020 3336 2020 2020 2020 2031        36       1
│ │ │ │ +00062220: 3220 2020 2020 2020 3220 7c0a 7c6f 3235  2       2 |.|o25
│ │ │ │ +00062230: 203d 2053 2720 203c 2d2d 2053 2720 2020   = S'  <-- S'   
│ │ │ │ +00062240: 3c2d 2d20 5327 2020 203c 2d2d 2053 2720  <-- S'   <-- S' 
│ │ │ │ +00062250: 2020 3c2d 2d20 5327 2020 203c 2d2d 2053    <-- S'   <-- S
│ │ │ │ +00062260: 2720 2020 3c2d 2d20 5327 2020 7c0a 7c20  '   <-- S'  |.| 
│ │ │ │ +00062270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000622a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000622b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000622c0: 2020 3020 2020 2020 2020 3120 2020 2020    0       1     
│ │ │ │ -000622d0: 2020 2032 2020 2020 2020 2020 3320 2020     2        3   
│ │ │ │ -000622e0: 2020 2020 2034 2020 2020 2020 2020 3520       4        5 
│ │ │ │ -000622f0: 2020 2020 2020 2036 2020 207c 0a7c 2020         6   |.|  
│ │ │ │ +000622a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000622b0: 7c20 2020 2020 2030 2020 2020 2020 2031  |      0       1
│ │ │ │ +000622c0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +000622d0: 2033 2020 2020 2020 2020 3420 2020 2020   3        4     
│ │ │ │ +000622e0: 2020 2035 2020 2020 2020 2020 3620 2020     5        6   
│ │ │ │ +000622f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00062300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062330: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00062340: 6f32 3520 3a20 436f 6d70 6c65 7820 2020  o25 : Complex   
│ │ │ │ +00062330: 2020 7c0a 7c6f 3235 203a 2043 6f6d 706c    |.|o25 : Compl
│ │ │ │ +00062340: 6578 2020 2020 2020 2020 2020 2020 2020  ex              
│ │ │ │  00062350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062370: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00062380: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00062370: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00062380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000623a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000623b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000623c0: 2d2b 0a7c 6932 3620 3a20 616c 6c28 6c65  -+.|i26 : all(le
│ │ │ │ -000623d0: 6e67 7468 2041 2b31 2c20 692d 3e20 736f  ngth A+1, i-> so
│ │ │ │ -000623e0: 7274 2064 6567 7265 6573 2041 5f69 203d  rt degrees A_i =
│ │ │ │ -000623f0: 3d20 736f 7274 2064 6567 7265 6573 2042  = sort degrees B
│ │ │ │ -00062400: 5f69 297c 0a7c 2020 2020 2020 2020 2020  _i)|.|          
│ │ │ │ +000623b0: 2d2d 2d2d 2d2d 2b0a 7c69 3236 203a 2061  ------+.|i26 : a
│ │ │ │ +000623c0: 6c6c 286c 656e 6774 6820 412b 312c 2069  ll(length A+1, i
│ │ │ │ +000623d0: 2d3e 2073 6f72 7420 6465 6772 6565 7320  -> sort degrees 
│ │ │ │ +000623e0: 415f 6920 3d3d 2073 6f72 7420 6465 6772  A_i == sort degr
│ │ │ │ +000623f0: 6565 7320 425f 6929 7c0a 7c20 2020 2020  ees B_i)|.|     
│ │ │ │ +00062400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062440: 2020 2020 207c 0a7c 6f32 3620 3d20 7472       |.|o26 = tr
│ │ │ │ -00062450: 7565 2020 2020 2020 2020 2020 2020 2020  ue              
│ │ │ │ +00062430: 2020 2020 2020 2020 2020 7c0a 7c6f 3236            |.|o26
│ │ │ │ +00062440: 203d 2074 7275 6520 2020 2020 2020 2020   = true         
│ │ │ │ +00062450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062480: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00062470: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00062480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000624a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000624b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000624c0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 6275 7420  ---------+..but 
│ │ │ │ -000624d0: 7468 6579 2068 6176 6520 6170 7061 7265  they have appare
│ │ │ │ -000624e0: 6e74 6c79 2064 6966 6665 7265 6e74 2061  ntly different a
│ │ │ │ -000624f0: 6e6e 6968 696c 6174 6f72 730a 0a2b 2d2d  nnihilators..+--
│ │ │ │ +000624b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000624c0: 0a62 7574 2074 6865 7920 6861 7665 2061  .but they have a
│ │ │ │ +000624d0: 7070 6172 656e 746c 7920 6469 6666 6572  pparently differ
│ │ │ │ +000624e0: 656e 7420 616e 6e69 6869 6c61 746f 7273  ent annihilators
│ │ │ │ +000624f0: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..+-------------
│ │ │ │  00062500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00062530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00062540: 2d2d 2d2b 0a7c 6932 3720 3a20 616e 6e20  ---+.|i27 : ann 
│ │ │ │ -00062550: 4545 2020 2020 2020 2020 2020 2020 2020  EE              
│ │ │ │ +00062530: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3237 203a  --------+.|i27 :
│ │ │ │ +00062540: 2061 6e6e 2045 4520 2020 2020 2020 2020   ann EE         
│ │ │ │ +00062550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062580: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00062580: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00062590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000625a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000625b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000625c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000625d0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -000625e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000625f0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -00062600: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -00062610: 2020 2020 2020 2020 2032 207c 0a7c 6f32           2 |.|o2
│ │ │ │ -00062620: 3720 3d20 6964 6561 6c20 2878 202c 2078  7 = ideal (x , x
│ │ │ │ -00062630: 202c 2078 202c 2058 2020 2b20 3431 5820   , x , X  + 41X 
│ │ │ │ -00062640: 5820 202d 2033 3758 2020 2d20 3134 5820  X  - 37X  - 14X 
│ │ │ │ -00062650: 5820 202d 2032 3958 2058 2020 2b20 3435  X  - 29X X  + 45
│ │ │ │ -00062660: 5820 297c 0a7c 2020 2020 2020 2020 2020  X )|.|          
│ │ │ │ -00062670: 2020 2020 3220 2020 3120 2020 3020 2020      2   1   0   
│ │ │ │ -00062680: 3120 2020 2020 2031 2032 2020 2020 2020  1      1 2      
│ │ │ │ -00062690: 3220 2020 2020 2031 2033 2020 2020 2020  2      1 3      
│ │ │ │ -000626a0: 3220 3320 2020 2020 2033 207c 0a7c 2020  2 3      3 |.|  
│ │ │ │ +000625c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000625d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000625e0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +000625f0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +00062600: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +00062610: 7c0a 7c6f 3237 203d 2069 6465 616c 2028  |.|o27 = ideal (
│ │ │ │ +00062620: 7820 2c20 7820 2c20 7820 2c20 5820 202b  x , x , x , X  +
│ │ │ │ +00062630: 2034 3158 2058 2020 2d20 3337 5820 202d   41X X  - 37X  -
│ │ │ │ +00062640: 2031 3458 2058 2020 2d20 3239 5820 5820   14X X  - 29X X 
│ │ │ │ +00062650: 202b 2034 3558 2029 7c0a 7c20 2020 2020   + 45X )|.|     
│ │ │ │ +00062660: 2020 2020 2020 2020 2032 2020 2031 2020           2   1  
│ │ │ │ +00062670: 2030 2020 2031 2020 2020 2020 3120 3220   0   1      1 2 
│ │ │ │ +00062680: 2020 2020 2032 2020 2020 2020 3120 3320       2      1 3 
│ │ │ │ +00062690: 2020 2020 2032 2033 2020 2020 2020 3320       2 3      3 
│ │ │ │ +000626a0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000626b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000626c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000626d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000626e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000626f0: 2020 207c 0a7c 6f32 3720 3a20 4964 6561     |.|o27 : Idea
│ │ │ │ -00062700: 6c20 6f66 2053 2720 2020 2020 2020 2020  l of S'         
│ │ │ │ +000626e0: 2020 2020 2020 2020 7c0a 7c6f 3237 203a          |.|o27 :
│ │ │ │ +000626f0: 2049 6465 616c 206f 6620 5327 2020 2020   Ideal of S'    
│ │ │ │ +00062700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062730: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00062730: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00062740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00062770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00062780: 2d2d 2d2b 0a7c 6932 3820 3a20 616e 6e20  ---+.|i28 : ann 
│ │ │ │ -00062790: 4553 2020 2020 2020 2020 2020 2020 2020  ES              
│ │ │ │ +00062770: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3238 203a  --------+.|i28 :
│ │ │ │ +00062780: 2061 6e6e 2045 5320 2020 2020 2020 2020   ann ES         
│ │ │ │ +00062790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000627a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000627b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000627c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000627c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000627d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000627e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000627f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062810: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00062820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062830: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -00062840: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -00062850: 2020 2020 2020 2020 3220 207c 0a7c 6f32          2  |.|o2
│ │ │ │ -00062860: 3820 3d20 6964 6561 6c20 2878 202c 2078  8 = ideal (x , x
│ │ │ │ -00062870: 202c 2078 202c 2073 2020 2b20 3432 7320   , x , s  + 42s 
│ │ │ │ -00062880: 7320 202d 2033 3073 2020 2d20 3235 7320  s  - 30s  - 25s 
│ │ │ │ -00062890: 7320 202d 2033 3573 2073 2020 2b20 3973  s  - 35s s  + 9s
│ │ │ │ -000628a0: 2029 207c 0a7c 2020 2020 2020 2020 2020   ) |.|          
│ │ │ │ -000628b0: 2020 2020 3220 2020 3120 2020 3020 2020      2   1   0   
│ │ │ │ -000628c0: 3020 2020 2020 2030 2031 2020 2020 2020  0      0 1      
│ │ │ │ -000628d0: 3120 2020 2020 2030 2032 2020 2020 2020  1      0 2      
│ │ │ │ -000628e0: 3120 3220 2020 2020 3220 207c 0a7c 2020  1 2     2  |.|  
│ │ │ │ +00062800: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00062810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062820: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +00062830: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +00062840: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +00062850: 7c0a 7c6f 3238 203d 2069 6465 616c 2028  |.|o28 = ideal (
│ │ │ │ +00062860: 7820 2c20 7820 2c20 7820 2c20 7320 202b  x , x , x , s  +
│ │ │ │ +00062870: 2034 3273 2073 2020 2d20 3330 7320 202d   42s s  - 30s  -
│ │ │ │ +00062880: 2032 3573 2073 2020 2d20 3335 7320 7320   25s s  - 35s s 
│ │ │ │ +00062890: 202b 2039 7320 2920 7c0a 7c20 2020 2020   + 9s ) |.|     
│ │ │ │ +000628a0: 2020 2020 2020 2020 2032 2020 2031 2020           2   1  
│ │ │ │ +000628b0: 2030 2020 2030 2020 2020 2020 3020 3120   0   0      0 1 
│ │ │ │ +000628c0: 2020 2020 2031 2020 2020 2020 3020 3220       1      0 2 
│ │ │ │ +000628d0: 2020 2020 2031 2032 2020 2020 2032 2020       1 2     2  
│ │ │ │ +000628e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000628f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062930: 2020 207c 0a7c 6f32 3820 3a20 4964 6561     |.|o28 : Idea
│ │ │ │ -00062940: 6c20 6f66 2053 2020 2020 2020 2020 2020  l of S          
│ │ │ │ +00062920: 2020 2020 2020 2020 7c0a 7c6f 3238 203a          |.|o28 :
│ │ │ │ +00062930: 2049 6465 616c 206f 6620 5320 2020 2020   Ideal of S     
│ │ │ │ +00062940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062970: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00062970: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00062980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000629a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000629b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000629c0: 2d2d 2d2b 0a0a 616e 6420 696e 2066 6163  ---+..and in fac
│ │ │ │ -000629d0: 7420 7468 6579 2061 7265 206e 6f74 2069  t they are not i
│ │ │ │ -000629e0: 736f 6d6f 7270 6869 633a 0a0a 2b2d 2d2d  somorphic:..+---
│ │ │ │ +000629b0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a61 6e64 2069  --------+..and i
│ │ │ │ +000629c0: 6e20 6661 6374 2074 6865 7920 6172 6520  n fact they are 
│ │ │ │ +000629d0: 6e6f 7420 6973 6f6d 6f72 7068 6963 3a0a  not isomorphic:.
│ │ │ │ +000629e0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  000629f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00062a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00062a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3239  ----------+.|i29
│ │ │ │ -00062a40: 203a 2045 4574 6f45 5320 3d20 6d61 7028   : EEtoES = map(
│ │ │ │ -00062a50: 7269 6e67 2045 532c 7269 6e67 2045 452c  ring ES,ring EE,
│ │ │ │ -00062a60: 2067 656e 7320 7269 6e67 2045 5329 2020   gens ring ES)  
│ │ │ │ -00062a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062a80: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00062a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00062a30: 0a7c 6932 3920 3a20 4545 746f 4553 203d  .|i29 : EEtoES =
│ │ │ │ +00062a40: 206d 6170 2872 696e 6720 4553 2c72 696e   map(ring ES,rin
│ │ │ │ +00062a50: 6720 4545 2c20 6765 6e73 2072 696e 6720  g EE, gens ring 
│ │ │ │ +00062a60: 4553 2920 2020 2020 2020 2020 2020 2020  ES)             
│ │ │ │ +00062a70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00062a80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00062a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c6f 3239            |.|o29
│ │ │ │ -00062ae0: 203d 206d 6170 2028 532c 2053 272c 207b   = map (S, S', {
│ │ │ │ -00062af0: 7320 2c20 7320 2c20 7320 2c20 7820 2c20  s , s , s , x , 
│ │ │ │ -00062b00: 7820 2c20 7820 7d29 2020 2020 2020 2020  x , x })        
│ │ │ │ -00062b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062b20: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00062b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062b40: 2030 2020 2031 2020 2032 2020 2030 2020   0   1   2   0  
│ │ │ │ -00062b50: 2031 2020 2032 2020 2020 2020 2020 2020   1   2          
│ │ │ │ -00062b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062b70: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00062ac0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00062ad0: 0a7c 6f32 3920 3d20 6d61 7020 2853 2c20  .|o29 = map (S, 
│ │ │ │ +00062ae0: 5327 2c20 7b73 202c 2073 202c 2073 202c  S', {s , s , s ,
│ │ │ │ +00062af0: 2078 202c 2078 202c 2078 207d 2920 2020   x , x , x })   
│ │ │ │ +00062b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062b10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00062b20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00062b30: 2020 2020 2020 3020 2020 3120 2020 3220        0   1   2 
│ │ │ │ +00062b40: 2020 3020 2020 3120 2020 3220 2020 2020    0   1   2     
│ │ │ │ +00062b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062b60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00062b70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00062b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062bc0: 2020 2020 2020 2020 2020 7c0a 7c6f 3239            |.|o29
│ │ │ │ -00062bd0: 203a 2052 696e 674d 6170 2053 203c 2d2d   : RingMap S <--
│ │ │ │ -00062be0: 2053 2720 2020 2020 2020 2020 2020 2020   S'             
│ │ │ │ +00062bb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00062bc0: 0a7c 6f32 3920 3a20 5269 6e67 4d61 7020  .|o29 : RingMap 
│ │ │ │ +00062bd0: 5320 3c2d 2d20 5327 2020 2020 2020 2020  S <-- S'        
│ │ │ │ +00062be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062c10: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00062c00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00062c10: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00062c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00062c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00062c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3330  ----------+.|i30
│ │ │ │ -00062c70: 203a 2045 4527 203d 2063 6f6b 6572 2045   : EE' = coker E
│ │ │ │ -00062c80: 4574 6f45 5320 7072 6573 656e 7461 7469  EtoES presentati
│ │ │ │ -00062c90: 6f6e 2045 4520 2020 2020 2020 2020 2020  on EE           
│ │ │ │ -00062ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062cb0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00062c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00062c60: 0a7c 6933 3020 3a20 4545 2720 3d20 636f  .|i30 : EE' = co
│ │ │ │ +00062c70: 6b65 7220 4545 746f 4553 2070 7265 7365  ker EEtoES prese
│ │ │ │ +00062c80: 6e74 6174 696f 6e20 4545 2020 2020 2020  ntation EE      
│ │ │ │ +00062c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062ca0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00062cb0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00062cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062d00: 2020 2020 2020 2020 2020 7c0a 7c6f 3330            |.|o30
│ │ │ │ -00062d10: 203d 2063 6f6b 6572 6e65 6c20 7b30 2c20   = cokernel {0, 
│ │ │ │ -00062d20: 307d 2020 207c 2078 5f32 2078 5f31 2078  0}   | x_2 x_1 x
│ │ │ │ -00062d30: 5f30 2030 2020 2030 2020 2030 2020 2030  _0 0   0   0   0
│ │ │ │ -00062d40: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00062d50: 2020 2030 2020 2073 5f32 7c0a 7c20 2020     0   s_2|.|   
│ │ │ │ -00062d60: 2020 2020 2020 2020 2020 2020 7b30 2c20              {0, 
│ │ │ │ -00062d70: 307d 2020 207c 2030 2020 2030 2020 2030  0}   | 0   0   0
│ │ │ │ -00062d80: 2020 2078 5f32 2078 5f31 2078 5f30 2030     x_2 x_1 x_0 0
│ │ │ │ -00062d90: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00062da0: 2020 2030 2020 2030 2020 7c0a 7c20 2020     0   0  |.|   
│ │ │ │ -00062db0: 2020 2020 2020 2020 2020 2020 7b2d 312c              {-1,
│ │ │ │ -00062dc0: 202d 327d 207c 2030 2020 2030 2020 2030   -2} | 0   0   0
│ │ │ │ -00062dd0: 2020 2030 2020 2030 2020 2030 2020 2078     0   0   0   x
│ │ │ │ -00062de0: 5f32 2078 5f31 2078 5f30 2030 2020 2030  _2 x_1 x_0 0   0
│ │ │ │ -00062df0: 2020 2030 2020 2030 2020 7c0a 7c20 2020     0   0  |.|   
│ │ │ │ -00062e00: 2020 2020 2020 2020 2020 2020 7b2d 312c              {-1,
│ │ │ │ -00062e10: 202d 327d 207c 2030 2020 2030 2020 2030   -2} | 0   0   0
│ │ │ │ -00062e20: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00062e30: 2020 2030 2020 2030 2020 2078 5f32 2078     0   0   x_2 x
│ │ │ │ -00062e40: 5f31 2078 5f30 2030 2020 7c0a 7c20 2020  _1 x_0 0  |.|   
│ │ │ │ -00062e50: 2020 2020 2020 2020 2020 2020 7b2d 322c              {-2,
│ │ │ │ -00062e60: 202d 337d 207c 2030 2020 2030 2020 2030   -3} | 0   0   0
│ │ │ │ -00062e70: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00062e80: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00062e90: 2020 2030 2020 2030 2020 7c0a 7c20 2020     0   0  |.|   
│ │ │ │ -00062ea0: 2020 2020 2020 2020 2020 2020 7b2d 322c              {-2,
│ │ │ │ -00062eb0: 202d 337d 207c 2030 2020 2030 2020 2030   -3} | 0   0   0
│ │ │ │ -00062ec0: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00062ed0: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00062ee0: 2020 2030 2020 2030 2020 7c0a 7c20 2020     0   0  |.|   
│ │ │ │ -00062ef0: 2020 2020 2020 2020 2020 2020 7b2d 322c              {-2,
│ │ │ │ -00062f00: 202d 337d 207c 2030 2020 2030 2020 2030   -3} | 0   0   0
│ │ │ │ -00062f10: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00062f20: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00062f30: 2020 2030 2020 2030 2020 7c0a 7c20 2020     0   0  |.|   
│ │ │ │ -00062f40: 2020 2020 2020 2020 2020 2020 7b2d 322c              {-2,
│ │ │ │ -00062f50: 202d 337d 207c 2030 2020 2030 2020 2030   -3} | 0   0   0
│ │ │ │ -00062f60: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00062f70: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00062f80: 2020 2030 2020 2030 2020 7c0a 7c20 2020     0   0  |.|   
│ │ │ │ +00062cf0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00062d00: 0a7c 6f33 3020 3d20 636f 6b65 726e 656c  .|o30 = cokernel
│ │ │ │ +00062d10: 207b 302c 2030 7d20 2020 7c20 785f 3220   {0, 0}   | x_2 
│ │ │ │ +00062d20: 785f 3120 785f 3020 3020 2020 3020 2020  x_1 x_0 0   0   
│ │ │ │ +00062d30: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00062d40: 3020 2020 3020 2020 3020 2020 735f 327c  0   0   0   s_2|
│ │ │ │ +00062d50: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00062d60: 207b 302c 2030 7d20 2020 7c20 3020 2020   {0, 0}   | 0   
│ │ │ │ +00062d70: 3020 2020 3020 2020 785f 3220 785f 3120  0   0   x_2 x_1 
│ │ │ │ +00062d80: 785f 3020 3020 2020 3020 2020 3020 2020  x_0 0   0   0   
│ │ │ │ +00062d90: 3020 2020 3020 2020 3020 2020 3020 207c  0   0   0   0  |
│ │ │ │ +00062da0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00062db0: 207b 2d31 2c20 2d32 7d20 7c20 3020 2020   {-1, -2} | 0   
│ │ │ │ +00062dc0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00062dd0: 3020 2020 785f 3220 785f 3120 785f 3020  0   x_2 x_1 x_0 
│ │ │ │ +00062de0: 3020 2020 3020 2020 3020 2020 3020 207c  0   0   0   0  |
│ │ │ │ +00062df0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00062e00: 207b 2d31 2c20 2d32 7d20 7c20 3020 2020   {-1, -2} | 0   
│ │ │ │ +00062e10: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00062e20: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00062e30: 785f 3220 785f 3120 785f 3020 3020 207c  x_2 x_1 x_0 0  |
│ │ │ │ +00062e40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00062e50: 207b 2d32 2c20 2d33 7d20 7c20 3020 2020   {-2, -3} | 0   
│ │ │ │ +00062e60: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00062e70: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00062e80: 3020 2020 3020 2020 3020 2020 3020 207c  0   0   0   0  |
│ │ │ │ +00062e90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00062ea0: 207b 2d32 2c20 2d33 7d20 7c20 3020 2020   {-2, -3} | 0   
│ │ │ │ +00062eb0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00062ec0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00062ed0: 3020 2020 3020 2020 3020 2020 3020 207c  0   0   0   0  |
│ │ │ │ +00062ee0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00062ef0: 207b 2d32 2c20 2d33 7d20 7c20 3020 2020   {-2, -3} | 0   
│ │ │ │ +00062f00: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00062f10: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00062f20: 3020 2020 3020 2020 3020 2020 3020 207c  0   0   0   0  |
│ │ │ │ +00062f30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00062f40: 207b 2d32 2c20 2d33 7d20 7c20 3020 2020   {-2, -3} | 0   
│ │ │ │ +00062f50: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00062f60: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00062f70: 3020 2020 3020 2020 3020 2020 3020 207c  0   0   0   0  |
│ │ │ │ +00062f80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00062f90: 2020 2020 2020 2020 2020 2020 2020 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 7c0a 7c20 2020            |.|   
│ │ │ │ -00062fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062ff0: 2020 2020 2020 2020 2020 3820 2020 2020            8     
│ │ │ │ +00062fc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00062fd0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00062fe0: 2020 2020 2020 2020 2020 2020 2020 2038                 8
│ │ │ │ +00062ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063020: 2020 2020 2020 2020 2020 7c0a 7c6f 3330            |.|o30
│ │ │ │ -00063030: 203a 2053 2d6d 6f64 756c 652c 2071 756f   : S-module, quo
│ │ │ │ -00063040: 7469 656e 7420 6f66 2053 2020 2020 2020  tient of S      
│ │ │ │ +00063010: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00063020: 0a7c 6f33 3020 3a20 532d 6d6f 6475 6c65  .|o30 : S-module
│ │ │ │ +00063030: 2c20 7175 6f74 6965 6e74 206f 6620 5320  , quotient of S 
│ │ │ │ +00063040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063070: 2020 2020 2020 2020 2020 7c0a 7c2d 2d2d            |.|---
│ │ │ │ +00063060: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00063070: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  00063080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00063090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000630a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000630b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000630c0: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c73 5f31  ----------|.|s_1
│ │ │ │ -000630d0: 2073 5f30 2030 2020 2030 2020 2030 2020   s_0 0   0   0  
│ │ │ │ -000630e0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -000630f0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00063100: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00063110: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ -00063120: 2030 2020 2073 5f32 2073 5f31 2073 5f30   0   s_2 s_1 s_0
│ │ │ │ -00063130: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00063140: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00063150: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00063160: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ -00063170: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00063180: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00063190: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -000631a0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -000631b0: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ -000631c0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -000631d0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -000631e0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -000631f0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00063200: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ -00063210: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00063220: 2078 5f32 2078 5f31 2078 5f30 2030 2020   x_2 x_1 x_0 0  
│ │ │ │ -00063230: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00063240: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00063250: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ -00063260: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00063270: 2030 2020 2030 2020 2030 2020 2078 5f32   0   0   0   x_2
│ │ │ │ -00063280: 2078 5f31 2078 5f30 2030 2020 2030 2020   x_1 x_0 0   0  
│ │ │ │ -00063290: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -000632a0: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ -000632b0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -000632c0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -000632d0: 2030 2020 2030 2020 2078 5f32 2078 5f31   0   0   x_2 x_1
│ │ │ │ -000632e0: 2078 5f30 2030 2020 2030 2020 2030 2020   x_0 0   0   0  
│ │ │ │ -000632f0: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ -00063300: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00063310: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00063320: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00063330: 2030 2020 2078 5f32 2078 5f31 2078 5f30   0   x_2 x_1 x_0
│ │ │ │ -00063340: 2020 2020 2020 2020 2020 7c0a 7c2d 2d2d            |.|---
│ │ │ │ +000630b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +000630c0: 0a7c 735f 3120 735f 3020 3020 2020 3020  .|s_1 s_0 0   0 
│ │ │ │ +000630d0: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +000630e0: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +000630f0: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00063100: 2020 3020 2020 2020 2020 2020 2020 207c    0            |
│ │ │ │ +00063110: 0a7c 3020 2020 3020 2020 735f 3220 735f  .|0   0   s_2 s_
│ │ │ │ +00063120: 3120 735f 3020 3020 2020 3020 2020 3020  1 s_0 0   0   0 
│ │ │ │ +00063130: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00063140: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00063150: 2020 3020 2020 2020 2020 2020 2020 207c    0            |
│ │ │ │ +00063160: 0a7c 3020 2020 3020 2020 3020 2020 3020  .|0   0   0   0 
│ │ │ │ +00063170: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00063180: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00063190: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +000631a0: 2020 3020 2020 2020 2020 2020 2020 207c    0            |
│ │ │ │ +000631b0: 0a7c 3020 2020 3020 2020 3020 2020 3020  .|0   0   0   0 
│ │ │ │ +000631c0: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +000631d0: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +000631e0: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +000631f0: 2020 3020 2020 2020 2020 2020 2020 207c    0            |
│ │ │ │ +00063200: 0a7c 3020 2020 3020 2020 3020 2020 3020  .|0   0   0   0 
│ │ │ │ +00063210: 2020 3020 2020 785f 3220 785f 3120 785f    0   x_2 x_1 x_
│ │ │ │ +00063220: 3020 3020 2020 3020 2020 3020 2020 3020  0 0   0   0   0 
│ │ │ │ +00063230: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00063240: 2020 3020 2020 2020 2020 2020 2020 207c    0            |
│ │ │ │ +00063250: 0a7c 3020 2020 3020 2020 3020 2020 3020  .|0   0   0   0 
│ │ │ │ +00063260: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00063270: 2020 785f 3220 785f 3120 785f 3020 3020    x_2 x_1 x_0 0 
│ │ │ │ +00063280: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00063290: 2020 3020 2020 2020 2020 2020 2020 207c    0            |
│ │ │ │ +000632a0: 0a7c 3020 2020 3020 2020 3020 2020 3020  .|0   0   0   0 
│ │ │ │ +000632b0: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +000632c0: 2020 3020 2020 3020 2020 3020 2020 785f    0   0   0   x_
│ │ │ │ +000632d0: 3220 785f 3120 785f 3020 3020 2020 3020  2 x_1 x_0 0   0 
│ │ │ │ +000632e0: 2020 3020 2020 2020 2020 2020 2020 207c    0            |
│ │ │ │ +000632f0: 0a7c 3020 2020 3020 2020 3020 2020 3020  .|0   0   0   0 
│ │ │ │ +00063300: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00063310: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00063320: 2020 3020 2020 3020 2020 785f 3220 785f    0   0   x_2 x_
│ │ │ │ +00063330: 3120 785f 3020 2020 2020 2020 2020 207c  1 x_0          |
│ │ │ │ +00063340: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  00063350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00063360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00063370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00063380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00063390: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c30 2020  ----------|.|0  
│ │ │ │ -000633a0: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -000633b0: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -000633c0: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -000633d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000633e0: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ -000633f0: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -00063400: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -00063410: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -00063420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063430: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ -00063440: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -00063450: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -00063460: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -00063470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063480: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ -00063490: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -000634a0: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -000634b0: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -000634c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000634d0: 2020 2020 2020 2020 2020 7c0a 7c73 5f30            |.|s_0
│ │ │ │ -000634e0: 2d31 3873 5f31 2d33 3273 5f32 202d 3237  -18s_1-32s_2 -27
│ │ │ │ -000634f0: 735f 312b 3235 735f 3220 2020 2034 3273  s_1+25s_2    42s
│ │ │ │ -00063500: 5f31 2020 2020 2020 2020 2020 202d 3232  _1           -22
│ │ │ │ -00063510: 735f 3120 2020 2020 2020 2020 2020 2020  s_1             
│ │ │ │ -00063520: 2020 2020 2020 2020 2020 7c0a 7c32 3373            |.|23s
│ │ │ │ -00063530: 5f31 2d34 3173 5f32 2020 2020 2073 5f30  _1-41s_2     s_0
│ │ │ │ -00063540: 2d34 3273 5f31 2b31 3873 5f32 202d 3435  -42s_1+18s_2 -45
│ │ │ │ -00063550: 735f 3120 2020 2020 2020 2020 202d 3432  s_1          -42
│ │ │ │ -00063560: 735f 3120 2020 2020 2020 2020 2020 2020  s_1             
│ │ │ │ -00063570: 2020 2020 2020 2020 2020 7c0a 7c2d 3432            |.|-42
│ │ │ │ -00063580: 735f 3220 2020 2020 2020 2020 2032 3273  s_2          22s
│ │ │ │ -00063590: 5f32 2020 2020 2020 2020 2020 2073 5f30  _2           s_0
│ │ │ │ -000635a0: 2d31 3873 5f31 2d33 3273 5f32 202d 3237  -18s_1-32s_2 -27
│ │ │ │ -000635b0: 735f 312b 3235 735f 3220 2020 2020 2020  s_1+25s_2       
│ │ │ │ -000635c0: 2020 2020 2020 2020 2020 7c0a 7c34 3573            |.|45s
│ │ │ │ -000635d0: 5f32 2020 2020 2020 2020 2020 2034 3273  _2           42s
│ │ │ │ -000635e0: 5f32 2020 2020 2020 2020 2020 2032 3373  _2           23s
│ │ │ │ -000635f0: 5f31 2d34 3173 5f32 2020 2020 2073 5f30  _1-41s_2     s_0
│ │ │ │ -00063600: 2d34 3273 5f31 2b31 3873 5f32 2020 2020  -42s_1+18s_2    
│ │ │ │ -00063610: 2020 2020 2020 2020 2020 7c0a 7c2d 2d2d            |.|---
│ │ │ │ +00063380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00063390: 0a7c 3020 2020 2020 2020 2020 2020 2020  .|0             
│ │ │ │ +000633a0: 2020 3020 2020 2020 2020 2020 2020 2020    0             
│ │ │ │ +000633b0: 2020 3020 2020 2020 2020 2020 2020 2020    0             
│ │ │ │ +000633c0: 2020 3020 2020 2020 2020 2020 2020 2020    0             
│ │ │ │ +000633d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000633e0: 0a7c 3020 2020 2020 2020 2020 2020 2020  .|0             
│ │ │ │ +000633f0: 2020 3020 2020 2020 2020 2020 2020 2020    0             
│ │ │ │ +00063400: 2020 3020 2020 2020 2020 2020 2020 2020    0             
│ │ │ │ +00063410: 2020 3020 2020 2020 2020 2020 2020 2020    0             
│ │ │ │ +00063420: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00063430: 0a7c 3020 2020 2020 2020 2020 2020 2020  .|0             
│ │ │ │ +00063440: 2020 3020 2020 2020 2020 2020 2020 2020    0             
│ │ │ │ +00063450: 2020 3020 2020 2020 2020 2020 2020 2020    0             
│ │ │ │ +00063460: 2020 3020 2020 2020 2020 2020 2020 2020    0             
│ │ │ │ +00063470: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00063480: 0a7c 3020 2020 2020 2020 2020 2020 2020  .|0             
│ │ │ │ +00063490: 2020 3020 2020 2020 2020 2020 2020 2020    0             
│ │ │ │ +000634a0: 2020 3020 2020 2020 2020 2020 2020 2020    0             
│ │ │ │ +000634b0: 2020 3020 2020 2020 2020 2020 2020 2020    0             
│ │ │ │ +000634c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000634d0: 0a7c 735f 302d 3138 735f 312d 3332 735f  .|s_0-18s_1-32s_
│ │ │ │ +000634e0: 3220 2d32 3773 5f31 2b32 3573 5f32 2020  2 -27s_1+25s_2  
│ │ │ │ +000634f0: 2020 3432 735f 3120 2020 2020 2020 2020    42s_1         
│ │ │ │ +00063500: 2020 2d32 3273 5f31 2020 2020 2020 2020    -22s_1        
│ │ │ │ +00063510: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00063520: 0a7c 3233 735f 312d 3431 735f 3220 2020  .|23s_1-41s_2   
│ │ │ │ +00063530: 2020 735f 302d 3432 735f 312b 3138 735f    s_0-42s_1+18s_
│ │ │ │ +00063540: 3220 2d34 3573 5f31 2020 2020 2020 2020  2 -45s_1        
│ │ │ │ +00063550: 2020 2d34 3273 5f31 2020 2020 2020 2020    -42s_1        
│ │ │ │ +00063560: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00063570: 0a7c 2d34 3273 5f32 2020 2020 2020 2020  .|-42s_2        
│ │ │ │ +00063580: 2020 3232 735f 3220 2020 2020 2020 2020    22s_2         
│ │ │ │ +00063590: 2020 735f 302d 3138 735f 312d 3332 735f    s_0-18s_1-32s_
│ │ │ │ +000635a0: 3220 2d32 3773 5f31 2b32 3573 5f32 2020  2 -27s_1+25s_2  
│ │ │ │ +000635b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000635c0: 0a7c 3435 735f 3220 2020 2020 2020 2020  .|45s_2         
│ │ │ │ +000635d0: 2020 3432 735f 3220 2020 2020 2020 2020    42s_2         
│ │ │ │ +000635e0: 2020 3233 735f 312d 3431 735f 3220 2020    23s_1-41s_2   
│ │ │ │ +000635f0: 2020 735f 302d 3432 735f 312b 3138 735f    s_0-42s_1+18s_
│ │ │ │ +00063600: 3220 2020 2020 2020 2020 2020 2020 207c  2              |
│ │ │ │ +00063610: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  00063620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00063630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00063640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00063650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00063660: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c30 2020  ----------|.|0  
│ │ │ │ +00063650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00063660: 0a7c 3020 2020 2020 2020 2020 2020 2020  .|0             
│ │ │ │  00063670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000636a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000636b0: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ +000636a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000636b0: 0a7c 3020 2020 2020 2020 2020 2020 2020  .|0             
│ │ │ │  000636c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000636d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000636e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000636f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063700: 2020 2020 2020 2020 2020 7c0a 7c73 5f30            |.|s_0
│ │ │ │ -00063710: 5e32 2b34 3173 5f30 735f 312d 3337 735f  ^2+41s_0s_1-37s_
│ │ │ │ -00063720: 315e 322d 3134 735f 3073 5f32 2d32 3973  1^2-14s_0s_2-29s
│ │ │ │ -00063730: 5f31 735f 322b 3435 735f 325e 3220 2020  _1s_2+45s_2^2   
│ │ │ │ -00063740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063750: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ +000636f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00063700: 0a7c 735f 305e 322b 3431 735f 3073 5f31  .|s_0^2+41s_0s_1
│ │ │ │ +00063710: 2d33 3773 5f31 5e32 2d31 3473 5f30 735f  -37s_1^2-14s_0s_
│ │ │ │ +00063720: 322d 3239 735f 3173 5f32 2b34 3573 5f32  2-29s_1s_2+45s_2
│ │ │ │ +00063730: 5e32 2020 2020 2020 2020 2020 2020 2020  ^2              
│ │ │ │ +00063740: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00063750: 0a7c 3020 2020 2020 2020 2020 2020 2020  .|0             
│ │ │ │  00063760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000637a0: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ +00063790: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000637a0: 0a7c 3020 2020 2020 2020 2020 2020 2020  .|0             
│ │ │ │  000637b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000637c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000637d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000637e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000637f0: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ +000637e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000637f0: 0a7c 3020 2020 2020 2020 2020 2020 2020  .|0             
│ │ │ │  00063800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063840: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ +00063830: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00063840: 0a7c 3020 2020 2020 2020 2020 2020 2020  .|0             
│ │ │ │  00063850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c30 2020            |.|0  
│ │ │ │ +00063880: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00063890: 0a7c 3020 2020 2020 2020 2020 2020 2020  .|0             
│ │ │ │  000638a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000638b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000638c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000638d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000638e0: 2020 2020 2020 2020 2020 7c0a 7c2d 2d2d            |.|---
│ │ │ │ +000638d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000638e0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  000638f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00063900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00063910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00063920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00063930: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c30 2020  ----------|.|0  
│ │ │ │ +00063920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00063930: 0a7c 3020 2020 2020 2020 2020 2020 2020  .|0             
│ │ │ │  00063940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063960: 2020 2020 2020 2020 2020 2020 2020 7c20                | 
│ │ │ │ -00063970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063980: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ +00063960: 2020 207c 2020 2020 2020 2020 2020 2020     |            
│ │ │ │ +00063970: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00063980: 0a7c 3020 2020 2020 2020 2020 2020 2020  .|0             
│ │ │ │  00063990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000639a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000639b0: 2020 2020 2020 2020 2020 2020 2020 7c20                | 
│ │ │ │ -000639c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000639d0: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ +000639b0: 2020 207c 2020 2020 2020 2020 2020 2020     |            
│ │ │ │ +000639c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000639d0: 0a7c 3020 2020 2020 2020 2020 2020 2020  .|0             
│ │ │ │  000639e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000639f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063a00: 2020 2020 2020 2020 2020 2020 2020 7c20                | 
│ │ │ │ -00063a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063a20: 2020 2020 2020 2020 2020 7c0a 7c73 5f30            |.|s_0
│ │ │ │ -00063a30: 5e32 2b34 3173 5f30 735f 312d 3337 735f  ^2+41s_0s_1-37s_
│ │ │ │ -00063a40: 315e 322d 3134 735f 3073 5f32 2d32 3973  1^2-14s_0s_2-29s
│ │ │ │ -00063a50: 5f31 735f 322b 3435 735f 325e 3220 7c20  _1s_2+45s_2^2 | 
│ │ │ │ -00063a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063a70: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ +00063a00: 2020 207c 2020 2020 2020 2020 2020 2020     |            
│ │ │ │ +00063a10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00063a20: 0a7c 735f 305e 322b 3431 735f 3073 5f31  .|s_0^2+41s_0s_1
│ │ │ │ +00063a30: 2d33 3773 5f31 5e32 2d31 3473 5f30 735f  -37s_1^2-14s_0s_
│ │ │ │ +00063a40: 322d 3239 735f 3173 5f32 2b34 3573 5f32  2-29s_1s_2+45s_2
│ │ │ │ +00063a50: 5e32 207c 2020 2020 2020 2020 2020 2020  ^2 |            
│ │ │ │ +00063a60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00063a70: 0a7c 3020 2020 2020 2020 2020 2020 2020  .|0             
│ │ │ │  00063a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063aa0: 2020 2020 2020 2020 2020 2020 2020 7c20                | 
│ │ │ │ -00063ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063ac0: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ +00063aa0: 2020 207c 2020 2020 2020 2020 2020 2020     |            
│ │ │ │ +00063ab0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00063ac0: 0a7c 3020 2020 2020 2020 2020 2020 2020  .|0             
│ │ │ │  00063ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063af0: 2020 2020 2020 2020 2020 2020 2020 7c20                | 
│ │ │ │ -00063b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063b10: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ +00063af0: 2020 207c 2020 2020 2020 2020 2020 2020     |            
│ │ │ │ +00063b00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00063b10: 0a7c 3020 2020 2020 2020 2020 2020 2020  .|0             
│ │ │ │  00063b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063b40: 2020 2020 2020 2020 2020 2020 2020 7c20                | 
│ │ │ │ -00063b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063b60: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ +00063b40: 2020 207c 2020 2020 2020 2020 2020 2020     |            
│ │ │ │ +00063b50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00063b60: 0a7c 3020 2020 2020 2020 2020 2020 2020  .|0             
│ │ │ │  00063b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063b90: 2020 2020 2020 2020 2020 2020 2020 7c20                | 
│ │ │ │ -00063ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063bb0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00063b90: 2020 207c 2020 2020 2020 2020 2020 2020     |            
│ │ │ │ +00063ba0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00063bb0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00063bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00063bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00063be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00063bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00063c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3331  ----------+.|i31
│ │ │ │ -00063c10: 203a 2048 203d 2048 6f6d 2845 4527 2c45   : H = Hom(EE',E
│ │ │ │ -00063c20: 5329 3b20 2020 2020 2020 2020 2020 2020  S);             
│ │ │ │ +00063bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00063c00: 0a7c 6933 3120 3a20 4820 3d20 486f 6d28  .|i31 : H = Hom(
│ │ │ │ +00063c10: 4545 272c 4553 293b 2020 2020 2020 2020  EE',ES);        
│ │ │ │ +00063c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063c50: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00063c40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00063c50: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00063c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00063c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00063c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00063c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00063ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3332  ----------+.|i32
│ │ │ │ -00063cb0: 203a 2051 203d 2070 6f73 6974 696f 6e73   : Q = positions
│ │ │ │ -00063cc0: 2864 6567 7265 6573 2074 6172 6765 7420  (degrees target 
│ │ │ │ -00063cd0: 7072 6573 656e 7461 7469 6f6e 2048 2c20  presentation H, 
│ │ │ │ -00063ce0: 692d 3e20 6920 3d3d 207b 302c 307d 2920  i-> i == {0,0}) 
│ │ │ │ -00063cf0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00063c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00063ca0: 0a7c 6933 3220 3a20 5120 3d20 706f 7369  .|i32 : Q = posi
│ │ │ │ +00063cb0: 7469 6f6e 7328 6465 6772 6565 7320 7461  tions(degrees ta
│ │ │ │ +00063cc0: 7267 6574 2070 7265 7365 6e74 6174 696f  rget presentatio
│ │ │ │ +00063cd0: 6e20 482c 2069 2d3e 2069 203d 3d20 7b30  n H, i-> i == {0
│ │ │ │ +00063ce0: 2c30 7d29 2020 2020 2020 2020 2020 207c  ,0})           |
│ │ │ │ +00063cf0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00063d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063d40: 2020 2020 2020 2020 2020 7c0a 7c6f 3332            |.|o32
│ │ │ │ -00063d50: 203d 207b 382c 2039 2c20 3130 2c20 3131   = {8, 9, 10, 11
│ │ │ │ -00063d60: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +00063d30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00063d40: 0a7c 6f33 3220 3d20 7b38 2c20 392c 2031  .|o32 = {8, 9, 1
│ │ │ │ +00063d50: 302c 2031 317d 2020 2020 2020 2020 2020  0, 11}          
│ │ │ │ +00063d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063d90: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00063d80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00063d90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00063da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063de0: 2020 2020 2020 2020 2020 7c0a 7c6f 3332            |.|o32
│ │ │ │ -00063df0: 203a 204c 6973 7420 2020 2020 2020 2020   : List         
│ │ │ │ +00063dd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00063de0: 0a7c 6f33 3220 3a20 4c69 7374 2020 2020  .|o32 : List    
│ │ │ │ +00063df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063e30: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00063e20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00063e30: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00063e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00063e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00063e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00063e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00063e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3333  ----------+.|i33
│ │ │ │ -00063e90: 203a 2066 203d 2073 756d 2851 2c20 702d   : f = sum(Q, p-
│ │ │ │ -00063ea0: 3e20 7261 6e64 6f6d 2028 535e 312c 2053  > random (S^1, S
│ │ │ │ -00063eb0: 5e31 292a 2a68 6f6d 6f6d 6f72 7068 6973  ^1)**homomorphis
│ │ │ │ -00063ec0: 6d20 485f 7b70 7d29 2020 2020 2020 2020  m H_{p})        
│ │ │ │ -00063ed0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00063e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00063e80: 0a7c 6933 3320 3a20 6620 3d20 7375 6d28  .|i33 : f = sum(
│ │ │ │ +00063e90: 512c 2070 2d3e 2072 616e 646f 6d20 2853  Q, p-> random (S
│ │ │ │ +00063ea0: 5e31 2c20 535e 3129 2a2a 686f 6d6f 6d6f  ^1, S^1)**homomo
│ │ │ │ +00063eb0: 7270 6869 736d 2048 5f7b 707d 2920 2020  rphism H_{p})   
│ │ │ │ +00063ec0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00063ed0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00063ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063f20: 2020 2020 2020 2020 2020 7c0a 7c6f 3333            |.|o33
│ │ │ │ -00063f30: 203d 207b 302c 2030 7d20 2020 7c20 2d33   = {0, 0}   | -3
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│ │ │ │ -000646d0: 2020 2020 2a20 4d2c 2061 202a 6e6f 7465      * M, a *note
│ │ │ │ -000646e0: 206d 6f64 756c 653a 2028 4d61 6361 756c   module: (Macaul
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│ │ │ │ -00064710: 696e 7465 7273 6563 7469 6f6e 0a20 2020  intersection.   
│ │ │ │ -00064720: 2020 2020 2072 696e 670a 2020 2a20 2a6e       ring.  * *n
│ │ │ │ -00064730: 6f74 6520 4f70 7469 6f6e 616c 2069 6e70  ote Optional inp
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│ │ │ │ -00064750: 6f63 2975 7369 6e67 2066 756e 6374 696f  oc)using functio
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│ │ │ │ -00064780: 2a20 4f75 7452 696e 6720 3d3e 202e 2e2e  * OutRing => ...
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│ │ │ │ -000647a0: 300a 2020 2a20 4f75 7470 7574 733a 0a20  0.  * Outputs:. 
│ │ │ │ -000647b0: 2020 2020 202a 2045 2c20 6120 2a6e 6f74       * E, a *not
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│ │ │ │ -000647d0: 6c61 7932 446f 6329 4d6f 6475 6c65 2c2c  lay2Doc)Module,,
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│ │ │ │ -00064800: 2020 2020 2067 656e 7320 696e 2064 6567       gens in deg
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│ │ │ │ -00064830: 4578 7472 6163 7473 2074 6865 206f 6464  Extracts the odd
│ │ │ │ -00064840: 2064 6567 7265 6520 7061 7274 2066 726f   degree part fro
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│ │ │ │ -00064860: 6620 7468 6520 6f70 7469 6f6e 616c 2061  f the optional a
│ │ │ │ -00064870: 7267 756d 656e 7420 4f75 7452 696e 670a  rgument OutRing.
│ │ │ │ -00064880: 3d3e 2054 2069 7320 6769 7665 6e2c 2061  => T is given, a
│ │ │ │ -00064890: 6e64 2063 6c61 7373 2054 203d 3d3d 2050  nd class T === P
│ │ │ │ -000648a0: 6f6c 796e 6f6d 6961 6c52 696e 672c 2074  olynomialRing, t
│ │ │ │ -000648b0: 6865 6e20 7468 6520 6f75 7470 7574 2077  hen the output w
│ │ │ │ -000648c0: 696c 6c20 6265 2061 206d 6f64 756c 650a  ill be a module.
│ │ │ │ -000648d0: 6f76 6572 2054 2e0a 0a2b 2d2d 2d2d 2d2d  over T...+------
│ │ │ │ +00064680: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a  **************..
│ │ │ │ +00064690: 2020 2a20 5573 6167 653a 200a 2020 2020    * Usage: .    
│ │ │ │ +000646a0: 2020 2020 4520 3d20 6f64 6445 7874 4d6f      E = oddExtMo
│ │ │ │ +000646b0: 6475 6c65 204d 0a20 202a 2049 6e70 7574  dule M.  * Input
│ │ │ │ +000646c0: 733a 0a20 2020 2020 202a 204d 2c20 6120  s:.      * M, a 
│ │ │ │ +000646d0: 2a6e 6f74 6520 6d6f 6475 6c65 3a20 284d  *note module: (M
│ │ │ │ +000646e0: 6163 6175 6c61 7932 446f 6329 4d6f 6475  acaulay2Doc)Modu
│ │ │ │ +000646f0: 6c65 2c2c 206f 7665 7220 6120 636f 6d70  le,, over a comp
│ │ │ │ +00064700: 6c65 7465 2069 6e74 6572 7365 6374 696f  lete intersectio
│ │ │ │ +00064710: 6e0a 2020 2020 2020 2020 7269 6e67 0a20  n.        ring. 
│ │ │ │ +00064720: 202a 202a 6e6f 7465 204f 7074 696f 6e61   * *note Optiona
│ │ │ │ +00064730: 6c20 696e 7075 7473 3a20 284d 6163 6175  l inputs: (Macau
│ │ │ │ +00064740: 6c61 7932 446f 6329 7573 696e 6720 6675  lay2Doc)using fu
│ │ │ │ +00064750: 6e63 7469 6f6e 7320 7769 7468 206f 7074  nctions with opt
│ │ │ │ +00064760: 696f 6e61 6c20 696e 7075 7473 2c3a 0a20  ional inputs,:. 
│ │ │ │ +00064770: 2020 2020 202a 204f 7574 5269 6e67 203d       * OutRing =
│ │ │ │ +00064780: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076  > ..., default v
│ │ │ │ +00064790: 616c 7565 2030 0a20 202a 204f 7574 7075  alue 0.  * Outpu
│ │ │ │ +000647a0: 7473 3a0a 2020 2020 2020 2a20 452c 2061  ts:.      * E, a
│ │ │ │ +000647b0: 202a 6e6f 7465 206d 6f64 756c 653a 2028   *note module: (
│ │ │ │ +000647c0: 4d61 6361 756c 6179 3244 6f63 294d 6f64  Macaulay2Doc)Mod
│ │ │ │ +000647d0: 756c 652c 2c20 6f76 6572 2061 2070 6f6c  ule,, over a pol
│ │ │ │ +000647e0: 796e 6f6d 6961 6c20 7269 6e67 2077 6974  ynomial ring wit
│ │ │ │ +000647f0: 680a 2020 2020 2020 2020 6765 6e73 2069  h.        gens i
│ │ │ │ +00064800: 6e20 6465 6772 6565 2031 0a0a 4465 7363  n degree 1..Desc
│ │ │ │ +00064810: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
│ │ │ │ +00064820: 3d3d 3d0a 0a45 7874 7261 6374 7320 7468  ===..Extracts th
│ │ │ │ +00064830: 6520 6f64 6420 6465 6772 6565 2070 6172  e odd degree par
│ │ │ │ +00064840: 7420 6672 6f6d 2045 7874 4d6f 6475 6c65  t from ExtModule
│ │ │ │ +00064850: 204d 2e20 4966 2074 6865 206f 7074 696f   M. If the optio
│ │ │ │ +00064860: 6e61 6c20 6172 6775 6d65 6e74 204f 7574  nal argument Out
│ │ │ │ +00064870: 5269 6e67 0a3d 3e20 5420 6973 2067 6976  Ring.=> T is giv
│ │ │ │ +00064880: 656e 2c20 616e 6420 636c 6173 7320 5420  en, and class T 
│ │ │ │ +00064890: 3d3d 3d20 506f 6c79 6e6f 6d69 616c 5269  === PolynomialRi
│ │ │ │ +000648a0: 6e67 2c20 7468 656e 2074 6865 206f 7574  ng, then the out
│ │ │ │ +000648b0: 7075 7420 7769 6c6c 2062 6520 6120 6d6f  put will be a mo
│ │ │ │ +000648c0: 6475 6c65 0a6f 7665 7220 542e 0a0a 2b2d  dule.over T...+-
│ │ │ │ +000648d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000648e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000648f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00064900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00064910: 2d2d 2d2b 0a7c 6931 203a 206b 6b3d 205a  ---+.|i1 : kk= Z
│ │ │ │ -00064920: 5a2f 3130 3120 2020 2020 2020 2020 2020  Z/101           
│ │ │ │ +00064900: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20  --------+.|i1 : 
│ │ │ │ +00064910: 6b6b 3d20 5a5a 2f31 3031 2020 2020 2020  kk= ZZ/101      
│ │ │ │ +00064920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00064930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064940: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00064950: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00064940: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00064950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00064960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00064970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064980: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -00064990: 203d 206b 6b20 2020 2020 2020 2020 2020   = kk           
│ │ │ │ +00064980: 7c0a 7c6f 3120 3d20 6b6b 2020 2020 2020  |.|o1 = kk      
│ │ │ │ +00064990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000649a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000649b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000649c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000649b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000649c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000649d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000649e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000649f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064a00: 2020 207c 0a7c 6f31 203a 2051 756f 7469     |.|o1 : Quoti
│ │ │ │ -00064a10: 656e 7452 696e 6720 2020 2020 2020 2020  entRing         
│ │ │ │ +000649f0: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20          |.|o1 : 
│ │ │ │ +00064a00: 5175 6f74 6965 6e74 5269 6e67 2020 2020  QuotientRing    
│ │ │ │ +00064a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00064a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064a30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00064a40: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00064a30: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00064a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00064a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00064a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00064a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ -00064a80: 203a 2053 203d 206b 6b5b 782c 792c 7a5d   : S = kk[x,y,z]
│ │ │ │ +00064a70: 2b0a 7c69 3220 3a20 5320 3d20 6b6b 5b78  +.|i2 : S = kk[x
│ │ │ │ +00064a80: 2c79 2c7a 5d20 2020 2020 2020 2020 2020  ,y,z]           
│ │ │ │  00064a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064ab0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00064aa0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00064ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00064ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00064ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064af0: 2020 207c 0a7c 6f32 203d 2053 2020 2020     |.|o2 = S    
│ │ │ │ +00064ae0: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20          |.|o2 = 
│ │ │ │ +00064af0: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │  00064b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00064b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064b20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00064b30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00064b20: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00064b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00064b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00064b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064b60: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ -00064b70: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e   : PolynomialRin
│ │ │ │ -00064b80: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ -00064b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064ba0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00064b60: 7c0a 7c6f 3220 3a20 506f 6c79 6e6f 6d69  |.|o2 : Polynomi
│ │ │ │ +00064b70: 616c 5269 6e67 2020 2020 2020 2020 2020  alRing          
│ │ │ │ +00064b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064b90: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00064ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00064bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00064bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00064bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00064be0: 2d2d 2d2b 0a7c 6933 203a 2049 3220 3d20  ---+.|i3 : I2 = 
│ │ │ │ -00064bf0: 6964 6561 6c22 7833 2c79 7a22 2020 2020  ideal"x3,yz"    
│ │ │ │ +00064bd0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │ +00064be0: 4932 203d 2069 6465 616c 2278 332c 797a  I2 = ideal"x3,yz
│ │ │ │ +00064bf0: 2220 2020 2020 2020 2020 2020 2020 2020  "               
│ │ │ │  00064c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064c10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00064c20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00064c10: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00064c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00064c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00064c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064c50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00064c60: 2020 2020 2020 2020 2020 2033 2020 2020             3    
│ │ │ │ +00064c50: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00064c60: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │  00064c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064c90: 2020 2020 2020 207c 0a7c 6f33 203d 2069         |.|o3 = i
│ │ │ │ -00064ca0: 6465 616c 2028 7820 2c20 792a 7a29 2020  deal (x , y*z)  
│ │ │ │ +00064c80: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00064c90: 3320 3d20 6964 6561 6c20 2878 202c 2079  3 = ideal (x , y
│ │ │ │ +00064ca0: 2a7a 2920 2020 2020 2020 2020 2020 2020  *z)             
│ │ │ │  00064cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064cd0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00064cc0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00064cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00064ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00064cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064d00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00064d10: 0a7c 6f33 203a 2049 6465 616c 206f 6620  .|o3 : Ideal of 
│ │ │ │ -00064d20: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │ +00064d00: 2020 2020 7c0a 7c6f 3320 3a20 4964 6561      |.|o3 : Idea
│ │ │ │ +00064d10: 6c20 6f66 2053 2020 2020 2020 2020 2020  l of S          
│ │ │ │ +00064d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00064d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064d40: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00064d40: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00064d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00064d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00064d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00064d80: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2052  -------+.|i4 : R
│ │ │ │ -00064d90: 3220 3d20 532f 4932 2020 2020 2020 2020  2 = S/I2        
│ │ │ │ +00064d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00064d80: 3420 3a20 5232 203d 2053 2f49 3220 2020  4 : R2 = S/I2   
│ │ │ │ +00064d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00064da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064dc0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00064db0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00064dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00064dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00064de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064df0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00064e00: 0a7c 6f34 203d 2052 3220 2020 2020 2020  .|o4 = R2       
│ │ │ │ +00064df0: 2020 2020 7c0a 7c6f 3420 3d20 5232 2020      |.|o4 = R2  
│ │ │ │ +00064e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00064e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00064e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064e30: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00064e30: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00064e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00064e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064e70: 2020 2020 2020 207c 0a7c 6f34 203a 2051         |.|o4 : Q
│ │ │ │ -00064e80: 756f 7469 656e 7452 696e 6720 2020 2020  uotientRing     
│ │ │ │ +00064e60: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00064e70: 3420 3a20 5175 6f74 6965 6e74 5269 6e67  4 : QuotientRing
│ │ │ │ +00064e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00064e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064eb0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00064ea0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00064eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00064ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00064ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00064ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00064ef0: 0a7c 6935 203a 204d 3220 3d20 5232 5e31  .|i5 : M2 = R2^1
│ │ │ │ -00064f00: 2f69 6465 616c 2278 322c 792c 7a22 2020  /ideal"x2,y,z"  
│ │ │ │ +00064ee0: 2d2d 2d2d 2b0a 7c69 3520 3a20 4d32 203d  ----+.|i5 : M2 =
│ │ │ │ +00064ef0: 2052 325e 312f 6964 6561 6c22 7832 2c79   R2^1/ideal"x2,y
│ │ │ │ +00064f00: 2c7a 2220 2020 2020 2020 2020 2020 2020  ,z"             
│ │ │ │  00064f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064f20: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00064f20: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00064f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00064f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064f60: 2020 2020 2020 207c 0a7c 6f35 203d 2063         |.|o5 = c
│ │ │ │ -00064f70: 6f6b 6572 6e65 6c20 7c20 7832 2079 207a  okernel | x2 y z
│ │ │ │ -00064f80: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ -00064f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064fa0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00064f50: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00064f60: 3520 3d20 636f 6b65 726e 656c 207c 2078  5 = cokernel | x
│ │ │ │ +00064f70: 3220 7920 7a20 7c20 2020 2020 2020 2020  2 y z |         
│ │ │ │ +00064f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064f90: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00064fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00064fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00064fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064fd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00064fe0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00064ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065000: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ -00065010: 2020 2020 2020 2020 2020 207c 0a7c 6f35             |.|o5
│ │ │ │ -00065020: 203a 2052 322d 6d6f 6475 6c65 2c20 7175   : R2-module, qu
│ │ │ │ -00065030: 6f74 6965 6e74 206f 6620 5232 2020 2020  otient of R2    
│ │ │ │ -00065040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065050: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00064fd0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00064fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064ff0: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ +00065000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00065010: 7c0a 7c6f 3520 3a20 5232 2d6d 6f64 756c  |.|o5 : R2-modul
│ │ │ │ +00065020: 652c 2071 756f 7469 656e 7420 6f66 2052  e, quotient of R
│ │ │ │ +00065030: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00065040: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00065050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00065060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00065070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00065080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00065090: 2d2d 2d2b 0a7c 6936 203a 2062 6574 7469  ---+.|i6 : betti
│ │ │ │ -000650a0: 2066 7265 6552 6573 6f6c 7574 696f 6e20   freeResolution 
│ │ │ │ -000650b0: 284d 322c 204c 656e 6774 684c 696d 6974  (M2, LengthLimit
│ │ │ │ -000650c0: 203d 3e31 3029 2020 2020 2020 2020 207c   =>10)         |
│ │ │ │ -000650d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00065080: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20  --------+.|i6 : 
│ │ │ │ +00065090: 6265 7474 6920 6672 6565 5265 736f 6c75  betti freeResolu
│ │ │ │ +000650a0: 7469 6f6e 2028 4d32 2c20 4c65 6e67 7468  tion (M2, Length
│ │ │ │ +000650b0: 4c69 6d69 7420 3d3e 3130 2920 2020 2020  Limit =>10)     
│ │ │ │ +000650c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000650d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000650e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000650f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065100: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00065110: 2020 2020 2020 2020 2020 3020 3120 3220            0 1 2 
│ │ │ │ -00065120: 3320 3420 2035 2020 3620 2037 2020 3820  3 4  5  6  7  8 
│ │ │ │ -00065130: 2039 2031 3020 2020 2020 2020 2020 2020   9 10           
│ │ │ │ -00065140: 2020 2020 2020 207c 0a7c 6f36 203d 2074         |.|o6 = t
│ │ │ │ -00065150: 6f74 616c 3a20 3120 3320 3520 3720 3920  otal: 1 3 5 7 9 
│ │ │ │ -00065160: 3131 2031 3320 3135 2031 3720 3139 2032  11 13 15 17 19 2
│ │ │ │ -00065170: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ -00065180: 2020 207c 0a7c 2020 2020 2020 2020 2030     |.|         0
│ │ │ │ -00065190: 3a20 3120 3220 3220 3220 3220 2032 2020  : 1 2 2 2 2  2  
│ │ │ │ -000651a0: 3220 2032 2020 3220 2032 2020 3220 2020  2  2  2  2  2   
│ │ │ │ -000651b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000651c0: 0a7c 2020 2020 2020 2020 2031 3a20 2e20  .|         1: . 
│ │ │ │ -000651d0: 3120 3320 3420 3420 2034 2020 3420 2034  1 3 4 4  4  4  4
│ │ │ │ -000651e0: 2020 3420 2034 2020 3420 2020 2020 2020    4  4  4       
│ │ │ │ -000651f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00065200: 2020 2020 2020 2032 3a20 2e20 2e20 2e20         2: . . . 
│ │ │ │ -00065210: 3120 3320 2034 2020 3420 2034 2020 3420  1 3  4  4  4  4 
│ │ │ │ -00065220: 2034 2020 3420 2020 2020 2020 2020 2020   4  4           
│ │ │ │ -00065230: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00065240: 2020 2033 3a20 2e20 2e20 2e20 2e20 2e20     3: . . . . . 
│ │ │ │ -00065250: 2031 2020 3320 2034 2020 3420 2034 2020   1  3  4  4  4  
│ │ │ │ -00065260: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ -00065270: 2020 207c 0a7c 2020 2020 2020 2020 2034     |.|         4
│ │ │ │ -00065280: 3a20 2e20 2e20 2e20 2e20 2e20 202e 2020  : . . . . .  .  
│ │ │ │ -00065290: 2e20 2031 2020 3320 2034 2020 3420 2020  .  1  3  4  4   
│ │ │ │ -000652a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000652b0: 0a7c 2020 2020 2020 2020 2035 3a20 2e20  .|         5: . 
│ │ │ │ -000652c0: 2e20 2e20 2e20 2e20 202e 2020 2e20 202e  . . . .  .  .  .
│ │ │ │ -000652d0: 2020 2e20 2031 2020 3320 2020 2020 2020    .  1  3       
│ │ │ │ -000652e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00065100: 7c0a 7c20 2020 2020 2020 2020 2020 2030  |.|            0
│ │ │ │ +00065110: 2031 2032 2033 2034 2020 3520 2036 2020   1 2 3 4  5  6  
│ │ │ │ +00065120: 3720 2038 2020 3920 3130 2020 2020 2020  7  8  9 10      
│ │ │ │ +00065130: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00065140: 3620 3d20 746f 7461 6c3a 2031 2033 2035  6 = total: 1 3 5
│ │ │ │ +00065150: 2037 2039 2031 3120 3133 2031 3520 3137   7 9 11 13 15 17
│ │ │ │ +00065160: 2031 3920 3231 2020 2020 2020 2020 2020   19 21          
│ │ │ │ +00065170: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00065180: 2020 2020 303a 2031 2032 2032 2032 2032      0: 1 2 2 2 2
│ │ │ │ +00065190: 2020 3220 2032 2020 3220 2032 2020 3220    2  2  2  2  2 
│ │ │ │ +000651a0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +000651b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000651c0: 313a 202e 2031 2033 2034 2034 2020 3420  1: . 1 3 4 4  4 
│ │ │ │ +000651d0: 2034 2020 3420 2034 2020 3420 2034 2020   4  4  4  4  4  
│ │ │ │ +000651e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000651f0: 7c0a 7c20 2020 2020 2020 2020 323a 202e  |.|         2: .
│ │ │ │ +00065200: 202e 202e 2031 2033 2020 3420 2034 2020   . . 1 3  4  4  
│ │ │ │ +00065210: 3420 2034 2020 3420 2034 2020 2020 2020  4  4  4  4      
│ │ │ │ +00065220: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00065230: 2020 2020 2020 2020 333a 202e 202e 202e          3: . . .
│ │ │ │ +00065240: 202e 202e 2020 3120 2033 2020 3420 2034   . .  1  3  4  4
│ │ │ │ +00065250: 2020 3420 2034 2020 2020 2020 2020 2020    4  4          
│ │ │ │ +00065260: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00065270: 2020 2020 343a 202e 202e 202e 202e 202e      4: . . . . .
│ │ │ │ +00065280: 2020 2e20 202e 2020 3120 2033 2020 3420    .  .  1  3  4 
│ │ │ │ +00065290: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ +000652a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000652b0: 353a 202e 202e 202e 202e 202e 2020 2e20  5: . . . . .  . 
│ │ │ │ +000652c0: 202e 2020 2e20 202e 2020 3120 2033 2020   .  .  .  1  3  
│ │ │ │ +000652d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000652e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000652f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00065300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065320: 2020 2020 2020 207c 0a7c 6f36 203a 2042         |.|o6 : B
│ │ │ │ -00065330: 6574 7469 5461 6c6c 7920 2020 2020 2020  ettiTally       
│ │ │ │ +00065310: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00065320: 3620 3a20 4265 7474 6954 616c 6c79 2020  6 : BettiTally  
│ │ │ │ +00065330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00065340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065360: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00065350: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00065360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00065370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00065380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00065390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000653a0: 0a7c 6937 203a 2045 203d 2045 7874 4d6f  .|i7 : E = ExtMo
│ │ │ │ -000653b0: 6475 6c65 204d 3220 2020 2020 2020 2020  dule M2         
│ │ │ │ +00065390: 2d2d 2d2d 2b0a 7c69 3720 3a20 4520 3d20  ----+.|i7 : E = 
│ │ │ │ +000653a0: 4578 744d 6f64 756c 6520 4d32 2020 2020  ExtModule M2    
│ │ │ │ +000653b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000653c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000653d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000653d0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000653e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000653f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065410: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00065420: 2020 2020 2020 2020 2020 2038 2020 2020             8    
│ │ │ │ +00065400: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00065410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00065420: 3820 2020 2020 2020 2020 2020 2020 2020  8               
│ │ │ │  00065430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065450: 2020 207c 0a7c 6f37 203d 2028 6b6b 5b58     |.|o7 = (kk[X
│ │ │ │ -00065460: 202e 2e58 205d 2920 2020 2020 2020 2020   ..X ])         
│ │ │ │ +00065440: 2020 2020 2020 2020 7c0a 7c6f 3720 3d20          |.|o7 = 
│ │ │ │ +00065450: 286b 6b5b 5820 2e2e 5820 5d29 2020 2020  (kk[X ..X ])    
│ │ │ │ +00065460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00065470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065480: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00065490: 0a7c 2020 2020 2020 2020 2020 3020 2020  .|          0   
│ │ │ │ -000654a0: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +00065480: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00065490: 2030 2020 2031 2020 2020 2020 2020 2020   0   1          
│ │ │ │ +000654a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000654b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000654c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000654c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000654d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000654e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000654f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065500: 2020 2020 2020 207c 0a7c 6f37 203a 206b         |.|o7 : k
│ │ │ │ -00065510: 6b5b 5820 2e2e 5820 5d2d 6d6f 6475 6c65  k[X ..X ]-module
│ │ │ │ -00065520: 2c20 6672 6565 2c20 6465 6772 6565 7320  , free, degrees 
│ │ │ │ -00065530: 7b30 2e2e 312c 2032 3a31 2c20 333a 322c  {0..1, 2:1, 3:2,
│ │ │ │ -00065540: 2033 7d7c 0a7c 2020 2020 2020 2020 2030   3}|.|         0
│ │ │ │ -00065550: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │ +000654f0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00065500: 3720 3a20 6b6b 5b58 202e 2e58 205d 2d6d  7 : kk[X ..X ]-m
│ │ │ │ +00065510: 6f64 756c 652c 2066 7265 652c 2064 6567  odule, free, deg
│ │ │ │ +00065520: 7265 6573 207b 302e 2e31 2c20 323a 312c  rees {0..1, 2:1,
│ │ │ │ +00065530: 2033 3a32 2c20 337d 7c0a 7c20 2020 2020   3:2, 3}|.|     
│ │ │ │ +00065540: 2020 2020 3020 2020 3120 2020 2020 2020      0   1       
│ │ │ │ +00065550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00065560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065570: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00065580: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00065570: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00065580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00065590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000655a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000655b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938  -----------+.|i8
│ │ │ │ -000655c0: 203a 2061 7070 6c79 2874 6f4c 6973 7428   : apply(toList(
│ │ │ │ -000655d0: 302e 2e31 3029 2c20 692d 3e68 696c 6265  0..10), i->hilbe
│ │ │ │ -000655e0: 7274 4675 6e63 7469 6f6e 2869 2c20 4529  rtFunction(i, E)
│ │ │ │ -000655f0: 2920 2020 2020 207c 0a7c 2020 2020 2020  )      |.|      
│ │ │ │ +000655b0: 2b0a 7c69 3820 3a20 6170 706c 7928 746f  +.|i8 : apply(to
│ │ │ │ +000655c0: 4c69 7374 2830 2e2e 3130 292c 2069 2d3e  List(0..10), i->
│ │ │ │ +000655d0: 6869 6c62 6572 7446 756e 6374 696f 6e28  hilbertFunction(
│ │ │ │ +000655e0: 692c 2045 2929 2020 2020 2020 7c0a 7c20  i, E))      |.| 
│ │ │ │ +000655f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00065600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00065610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065630: 2020 207c 0a7c 6f38 203d 207b 312c 2033     |.|o8 = {1, 3
│ │ │ │ -00065640: 2c20 352c 2037 2c20 392c 2031 312c 2031  , 5, 7, 9, 11, 1
│ │ │ │ -00065650: 332c 2031 352c 2031 372c 2031 392c 2032  3, 15, 17, 19, 2
│ │ │ │ -00065660: 317d 2020 2020 2020 2020 2020 2020 207c  1}             |
│ │ │ │ -00065670: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00065620: 2020 2020 2020 2020 7c0a 7c6f 3820 3d20          |.|o8 = 
│ │ │ │ +00065630: 7b31 2c20 332c 2035 2c20 372c 2039 2c20  {1, 3, 5, 7, 9, 
│ │ │ │ +00065640: 3131 2c20 3133 2c20 3135 2c20 3137 2c20  11, 13, 15, 17, 
│ │ │ │ +00065650: 3139 2c20 3231 7d20 2020 2020 2020 2020  19, 21}         
│ │ │ │ +00065660: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00065670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00065680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00065690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000656a0: 2020 2020 2020 2020 2020 207c 0a7c 6f38             |.|o8
│ │ │ │ -000656b0: 203a 204c 6973 7420 2020 2020 2020 2020   : List         
│ │ │ │ +000656a0: 7c0a 7c6f 3820 3a20 4c69 7374 2020 2020  |.|o8 : List    
│ │ │ │ +000656b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000656c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000656d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000656e0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +000656d0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +000656e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000656f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00065700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00065710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00065720: 2d2d 2d2b 0a7c 6939 203a 2045 6f64 6420  ---+.|i9 : Eodd 
│ │ │ │ -00065730: 3d20 6f64 6445 7874 4d6f 6475 6c65 204d  = oddExtModule M
│ │ │ │ -00065740: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -00065750: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00065760: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00065710: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3920 3a20  --------+.|i9 : 
│ │ │ │ +00065720: 456f 6464 203d 206f 6464 4578 744d 6f64  Eodd = oddExtMod
│ │ │ │ +00065730: 756c 6520 4d32 2020 2020 2020 2020 2020  ule M2          
│ │ │ │ +00065740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00065750: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00065760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00065770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00065780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065790: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -000657a0: 2020 2020 2020 2020 2020 2020 2020 2034                 4
│ │ │ │ +00065790: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000657a0: 2020 2020 3420 2020 2020 2020 2020 2020      4           
│ │ │ │  000657b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000657c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000657d0: 2020 2020 2020 207c 0a7c 6f39 203d 2028         |.|o9 = (
│ │ │ │ -000657e0: 6b6b 5b58 202e 2e58 205d 2920 2020 2020  kk[X ..X ])     
│ │ │ │ +000657c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +000657d0: 3920 3d20 286b 6b5b 5820 2e2e 5820 5d29  9 = (kk[X ..X ])
│ │ │ │ +000657e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000657f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065810: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00065820: 3020 2020 3120 2020 2020 2020 2020 2020  0   1           
│ │ │ │ +00065800: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00065810: 2020 2020 2030 2020 2031 2020 2020 2020       0   1      
│ │ │ │ +00065820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00065830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065840: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00065850: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00065840: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00065850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00065860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00065870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065880: 2020 2020 2020 2020 2020 207c 0a7c 6f39             |.|o9
│ │ │ │ -00065890: 203a 206b 6b5b 5820 2e2e 5820 5d2d 6d6f   : kk[X ..X ]-mo
│ │ │ │ -000658a0: 6475 6c65 2c20 6672 6565 2c20 6465 6772  dule, free, degr
│ │ │ │ -000658b0: 6565 7320 7b33 3a30 2c20 317d 2020 2020  ees {3:0, 1}    
│ │ │ │ -000658c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000658d0: 2020 2030 2020 2031 2020 2020 2020 2020     0   1        
│ │ │ │ +00065880: 7c0a 7c6f 3920 3a20 6b6b 5b58 202e 2e58  |.|o9 : kk[X ..X
│ │ │ │ +00065890: 205d 2d6d 6f64 756c 652c 2066 7265 652c   ]-module, free,
│ │ │ │ +000658a0: 2064 6567 7265 6573 207b 333a 302c 2031   degrees {3:0, 1
│ │ │ │ +000658b0: 7d20 2020 2020 2020 2020 2020 7c0a 7c20  }           |.| 
│ │ │ │ +000658c0: 2020 2020 2020 2020 3020 2020 3120 2020          0   1   
│ │ │ │ +000658d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000658e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000658f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065900: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +000658f0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00065900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00065910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00065920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00065930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00065940: 0a7c 6931 3020 3a20 6170 706c 7928 746f  .|i10 : apply(to
│ │ │ │ -00065950: 4c69 7374 2830 2e2e 3529 2c20 692d 3e68  List(0..5), i->h
│ │ │ │ -00065960: 696c 6265 7274 4675 6e63 7469 6f6e 2869  ilbertFunction(i
│ │ │ │ -00065970: 2c20 456f 6464 2929 2020 207c 0a7c 2020  , Eodd))   |.|  
│ │ │ │ +00065930: 2d2d 2d2d 2b0a 7c69 3130 203a 2061 7070  ----+.|i10 : app
│ │ │ │ +00065940: 6c79 2874 6f4c 6973 7428 302e 2e35 292c  ly(toList(0..5),
│ │ │ │ +00065950: 2069 2d3e 6869 6c62 6572 7446 756e 6374   i->hilbertFunct
│ │ │ │ +00065960: 696f 6e28 692c 2045 6f64 6429 2920 2020  ion(i, Eodd))   
│ │ │ │ +00065970: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00065980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00065990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000659a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000659b0: 2020 2020 2020 207c 0a7c 6f31 3020 3d20         |.|o10 = 
│ │ │ │ -000659c0: 7b33 2c20 372c 2031 312c 2031 352c 2031  {3, 7, 11, 15, 1
│ │ │ │ -000659d0: 392c 2032 337d 2020 2020 2020 2020 2020  9, 23}          
│ │ │ │ -000659e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000659f0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000659a0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +000659b0: 3130 203d 207b 332c 2037 2c20 3131 2c20  10 = {3, 7, 11, 
│ │ │ │ +000659c0: 3135 2c20 3139 2c20 3233 7d20 2020 2020  15, 19, 23}     
│ │ │ │ +000659d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000659e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000659f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00065a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00065a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065a20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00065a30: 0a7c 6f31 3020 3a20 4c69 7374 2020 2020  .|o10 : List    
│ │ │ │ +00065a20: 2020 2020 7c0a 7c6f 3130 203a 204c 6973      |.|o10 : Lis
│ │ │ │ +00065a30: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │  00065a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00065a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065a60: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00065a60: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00065a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00065a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00065a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00065aa0: 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 616c  -------+..See al
│ │ │ │ -00065ab0: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  so.========..  *
│ │ │ │ -00065ac0: 202a 6e6f 7465 2045 7874 4d6f 6475 6c65   *note ExtModule
│ │ │ │ -00065ad0: 3a20 4578 744d 6f64 756c 652c 202d 2d20  : ExtModule, -- 
│ │ │ │ -00065ae0: 4578 745e 2a28 4d2c 6b29 206f 7665 7220  Ext^*(M,k) over 
│ │ │ │ -00065af0: 6120 636f 6d70 6c65 7465 2069 6e74 6572  a complete inter
│ │ │ │ -00065b00: 7365 6374 696f 6e20 6173 0a20 2020 206d  section as.    m
│ │ │ │ -00065b10: 6f64 756c 6520 6f76 6572 2043 4920 6f70  odule over CI op
│ │ │ │ -00065b20: 6572 6174 6f72 2072 696e 670a 2020 2a20  erator ring.  * 
│ │ │ │ -00065b30: 2a6e 6f74 6520 6576 656e 4578 744d 6f64  *note evenExtMod
│ │ │ │ -00065b40: 756c 653a 2065 7665 6e45 7874 4d6f 6475  ule: evenExtModu
│ │ │ │ -00065b50: 6c65 2c20 2d2d 2065 7665 6e20 7061 7274  le, -- even part
│ │ │ │ -00065b60: 206f 6620 4578 745e 2a28 4d2c 6b29 206f   of Ext^*(M,k) o
│ │ │ │ -00065b70: 7665 7220 610a 2020 2020 636f 6d70 6c65  ver a.    comple
│ │ │ │ -00065b80: 7465 2069 6e74 6572 7365 6374 696f 6e20  te intersection 
│ │ │ │ -00065b90: 6173 206d 6f64 756c 6520 6f76 6572 2043  as module over C
│ │ │ │ -00065ba0: 4920 6f70 6572 6174 6f72 2072 696e 670a  I operator ring.
│ │ │ │ -00065bb0: 2020 2a20 2a6e 6f74 6520 4f75 7452 696e    * *note OutRin
│ │ │ │ -00065bc0: 673a 204f 7574 5269 6e67 2c20 2d2d 204f  g: OutRing, -- O
│ │ │ │ -00065bd0: 7074 696f 6e20 616c 6c6f 7769 6e67 2073  ption allowing s
│ │ │ │ -00065be0: 7065 6369 6669 6361 7469 6f6e 206f 6620  pecification of 
│ │ │ │ -00065bf0: 7468 6520 7269 6e67 206f 7665 720a 2020  the ring over.  
│ │ │ │ -00065c00: 2020 7768 6963 6820 7468 6520 6f75 7470    which the outp
│ │ │ │ -00065c10: 7574 2069 7320 6465 6669 6e65 640a 0a57  ut is defined..W
│ │ │ │ -00065c20: 6179 7320 746f 2075 7365 206f 6464 4578  ays to use oddEx
│ │ │ │ -00065c30: 744d 6f64 756c 653a 0a3d 3d3d 3d3d 3d3d  tModule:.=======
│ │ │ │ -00065c40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00065c50: 3d3d 0a0a 2020 2a20 226f 6464 4578 744d  ==..  * "oddExtM
│ │ │ │ -00065c60: 6f64 756c 6528 4d6f 6475 6c65 2922 0a0a  odule(Module)"..
│ │ │ │ -00065c70: 466f 7220 7468 6520 7072 6f67 7261 6d6d  For the programm
│ │ │ │ -00065c80: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  er.=============
│ │ │ │ -00065c90: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563  =====..The objec
│ │ │ │ -00065ca0: 7420 2a6e 6f74 6520 6f64 6445 7874 4d6f  t *note oddExtMo
│ │ │ │ -00065cb0: 6475 6c65 3a20 6f64 6445 7874 4d6f 6475  dule: oddExtModu
│ │ │ │ -00065cc0: 6c65 2c20 6973 2061 202a 6e6f 7465 206d  le, is a *note m
│ │ │ │ -00065cd0: 6574 686f 6420 6675 6e63 7469 6f6e 2077  ethod function w
│ │ │ │ -00065ce0: 6974 680a 6f70 7469 6f6e 733a 2028 4d61  ith.options: (Ma
│ │ │ │ -00065cf0: 6361 756c 6179 3244 6f63 294d 6574 686f  caulay2Doc)Metho
│ │ │ │ -00065d00: 6446 756e 6374 696f 6e57 6974 684f 7074  dFunctionWithOpt
│ │ │ │ -00065d10: 696f 6e73 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d  ions,...--------
│ │ │ │ +00065a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53  ------------+..S
│ │ │ │ +00065aa0: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d  ee also.========
│ │ │ │ +00065ab0: 0a0a 2020 2a20 2a6e 6f74 6520 4578 744d  ..  * *note ExtM
│ │ │ │ +00065ac0: 6f64 756c 653a 2045 7874 4d6f 6475 6c65  odule: ExtModule
│ │ │ │ +00065ad0: 2c20 2d2d 2045 7874 5e2a 284d 2c6b 2920  , -- Ext^*(M,k) 
│ │ │ │ +00065ae0: 6f76 6572 2061 2063 6f6d 706c 6574 6520  over a complete 
│ │ │ │ +00065af0: 696e 7465 7273 6563 7469 6f6e 2061 730a  intersection as.
│ │ │ │ +00065b00: 2020 2020 6d6f 6475 6c65 206f 7665 7220      module over 
│ │ │ │ +00065b10: 4349 206f 7065 7261 746f 7220 7269 6e67  CI operator ring
│ │ │ │ +00065b20: 0a20 202a 202a 6e6f 7465 2065 7665 6e45  .  * *note evenE
│ │ │ │ +00065b30: 7874 4d6f 6475 6c65 3a20 6576 656e 4578  xtModule: evenEx
│ │ │ │ +00065b40: 744d 6f64 756c 652c 202d 2d20 6576 656e  tModule, -- even
│ │ │ │ +00065b50: 2070 6172 7420 6f66 2045 7874 5e2a 284d   part of Ext^*(M
│ │ │ │ +00065b60: 2c6b 2920 6f76 6572 2061 0a20 2020 2063  ,k) over a.    c
│ │ │ │ +00065b70: 6f6d 706c 6574 6520 696e 7465 7273 6563  omplete intersec
│ │ │ │ +00065b80: 7469 6f6e 2061 7320 6d6f 6475 6c65 206f  tion as module o
│ │ │ │ +00065b90: 7665 7220 4349 206f 7065 7261 746f 7220  ver CI operator 
│ │ │ │ +00065ba0: 7269 6e67 0a20 202a 202a 6e6f 7465 204f  ring.  * *note O
│ │ │ │ +00065bb0: 7574 5269 6e67 3a20 4f75 7452 696e 672c  utRing: OutRing,
│ │ │ │ +00065bc0: 202d 2d20 4f70 7469 6f6e 2061 6c6c 6f77   -- Option allow
│ │ │ │ +00065bd0: 696e 6720 7370 6563 6966 6963 6174 696f  ing specificatio
│ │ │ │ +00065be0: 6e20 6f66 2074 6865 2072 696e 6720 6f76  n of the ring ov
│ │ │ │ +00065bf0: 6572 0a20 2020 2077 6869 6368 2074 6865  er.    which the
│ │ │ │ +00065c00: 206f 7574 7075 7420 6973 2064 6566 696e   output is defin
│ │ │ │ +00065c10: 6564 0a0a 5761 7973 2074 6f20 7573 6520  ed..Ways to use 
│ │ │ │ +00065c20: 6f64 6445 7874 4d6f 6475 6c65 3a0a 3d3d  oddExtModule:.==
│ │ │ │ +00065c30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00065c40: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 6f64  =======..  * "od
│ │ │ │ +00065c50: 6445 7874 4d6f 6475 6c65 284d 6f64 756c  dExtModule(Modul
│ │ │ │ +00065c60: 6529 220a 0a46 6f72 2074 6865 2070 726f  e)"..For the pro
│ │ │ │ +00065c70: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d  grammer.========
│ │ │ │ +00065c80: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520  ==========..The 
│ │ │ │ +00065c90: 6f62 6a65 6374 202a 6e6f 7465 206f 6464  object *note odd
│ │ │ │ +00065ca0: 4578 744d 6f64 756c 653a 206f 6464 4578  ExtModule: oddEx
│ │ │ │ +00065cb0: 744d 6f64 756c 652c 2069 7320 6120 2a6e  tModule, is a *n
│ │ │ │ +00065cc0: 6f74 6520 6d65 7468 6f64 2066 756e 6374  ote method funct
│ │ │ │ +00065cd0: 696f 6e20 7769 7468 0a6f 7074 696f 6e73  ion with.options
│ │ │ │ +00065ce0: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +00065cf0: 4d65 7468 6f64 4675 6e63 7469 6f6e 5769  MethodFunctionWi
│ │ │ │ +00065d00: 7468 4f70 7469 6f6e 732c 2e0a 0a2d 2d2d  thOptions,...---
│ │ │ │ +00065d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00065d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00065d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00065d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00065d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00065d60: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75  -------..The sou
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│ │ │ │ -00065d80: 6d65 6e74 2069 7320 696e 0a2f 6275 696c  ment is in./buil
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│ │ │ │ -00065dc0: 756c 6179 322f 7061 636b 6167 6573 2f0a  ulay2/packages/.
│ │ │ │ -00065dd0: 436f 6d70 6c65 7465 496e 7465 7273 6563  CompleteIntersec
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│ │ │ │ -00065e20: 6e73 2e69 6e66 6f2c 204e 6f64 653a 204f  ns.info, Node: O
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│ │ │ │ -00065e50: 6445 7874 4d6f 6475 6c65 2c20 5570 3a20  dExtModule, Up: 
│ │ │ │ -00065e60: 546f 700a 0a4f 7074 696d 6973 6d20 2d2d  Top..Optimism --
│ │ │ │ -00065e70: 204f 7074 696f 6e20 746f 2068 6967 6853   Option to highS
│ │ │ │ -00065e80: 797a 7967 790a 2a2a 2a2a 2a2a 2a2a 2a2a  yzygy.**********
│ │ │ │ -00065e90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ -00065ec0: 5379 7a79 6779 284d 2c20 4f70 7469 6d69  Syzygy(M, Optimi
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│ │ │ │ -00065ee0: 7574 733a 0a20 2020 2020 202a 204f 7074  uts:.      * Opt
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│ │ │ │ -00065f00: 696e 7465 6765 723a 2028 4d61 6361 756c  integer: (Macaul
│ │ │ │ -00065f10: 6179 3244 6f63 295a 5a2c 2c20 0a0a 4465  ay2Doc)ZZ,, ..De
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│ │ │ │ -00065f50: 7468 6520 702d 7468 2073 797a 7967 792c  the p-th syzygy,
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│ │ │ │ -00065f70: 284d 2c4f 7074 696d 6973 6d3d 3e72 290a  (M,Optimism=>r).
│ │ │ │ -00065f80: 6368 6f6f 7365 7320 7468 6520 2870 2d72  chooses the (p-r
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│ │ │ │ -00065fa0: 7369 7469 7665 204f 7074 696d 6973 6d20  sitive Optimism 
│ │ │ │ -00065fb0: 6368 6f6f 7365 7320 6120 6c6f 7765 7220  chooses a lower 
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│ │ │ │ -00065fe0: 2061 2068 6967 6865 7220 2268 6967 6822   a higher "high"
│ │ │ │ -00065ff0: 2073 797a 7967 792e 0a0a 4361 7665 6174   syzygy...Caveat
│ │ │ │ -00066000: 0a3d 3d3d 3d3d 3d0a 0a41 7265 2074 6865  .======..Are the
│ │ │ │ -00066010: 7265 2063 6173 6573 2077 6865 6e20 706f  re cases when po
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│ │ │ │ -00066030: 6973 206a 7573 7469 6669 6564 3f0a 0a53  is justified?..S
│ │ │ │ -00066040: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d  ee also.========
│ │ │ │ -00066050: 0a0a 2020 2a20 2a6e 6f74 6520 6d66 426f  ..  * *note mfBo
│ │ │ │ -00066060: 756e 643a 206d 6642 6f75 6e64 2c20 2d2d  und: mfBound, --
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│ │ │ │ -00066090: 2074 616b 6520 666f 720a 2020 2020 226d   take for.    "m
│ │ │ │ -000660a0: 6174 7269 7846 6163 746f 7269 7a61 7469  atrixFactorizati
│ │ │ │ -000660b0: 6f6e 220a 2020 2a20 2a6e 6f74 6520 6869  on".  * *note hi
│ │ │ │ -000660c0: 6768 5379 7a79 6779 3a20 6869 6768 5379  ghSyzygy: highSy
│ │ │ │ -000660d0: 7a79 6779 2c20 2d2d 2052 6574 7572 6e73  zygy, -- Returns
│ │ │ │ -000660e0: 2061 2073 797a 7967 7920 6d6f 6475 6c65   a syzygy module
│ │ │ │ -000660f0: 206f 6e65 2062 6579 6f6e 6420 7468 650a   one beyond the.
│ │ │ │ -00066100: 2020 2020 7265 6775 6c61 7269 7479 206f      regularity o
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│ │ │ │ -00066130: 6e61 6c20 6172 6775 6d65 6e74 206e 616d  nal argument nam
│ │ │ │ -00066140: 6564 204f 7074 696d 6973 6d3a 0a3d 3d3d  ed Optimism:.===
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│ │ │ │ +00065d70: 2064 6f63 756d 656e 7420 6973 2069 6e0a   document is in.
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│ │ │ │ +00065e20: 6465 3a20 4f70 7469 6d69 736d 2c20 4e65  de: Optimism, Ne
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│ │ │ │ +00065e40: 763a 206f 6464 4578 744d 6f64 756c 652c  v: oddExtModule,
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│ │ │ │ +00065e80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00065e90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a  ***********..  *
│ │ │ │ +00065ea0: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ +00065eb0: 2068 6967 6853 797a 7967 7928 4d2c 204f   highSyzygy(M, O
│ │ │ │ +00065ec0: 7074 696d 6973 6d20 3d3e 2031 290a 2020  ptimism => 1).  
│ │ │ │ +00065ed0: 2a20 496e 7075 7473 3a0a 2020 2020 2020  * Inputs:.      
│ │ │ │ +00065ee0: 2a20 4f70 7469 6d69 736d 2c20 616e 202a  * Optimism, an *
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│ │ │ │ +00065f10: 200a 0a44 6573 6372 6970 7469 6f6e 0a3d   ..Description.=
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│ │ │ │ +00065f40: 6f73 6573 2074 6865 2070 2d74 6820 7379  oses the p-th sy
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│ │ │ │ +00065f60: 797a 7967 7928 4d2c 4f70 7469 6d69 736d  yzygy(M,Optimism
│ │ │ │ +00065f70: 3d3e 7229 0a63 686f 6f73 6573 2074 6865  =>r).chooses the
│ │ │ │ +00065f80: 2028 702d 7229 2d74 6820 7379 7a79 6779   (p-r)-th syzygy
│ │ │ │ +00065f90: 2e20 2850 6f73 6974 6976 6520 4f70 7469  . (Positive Opti
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│ │ │ │ +00065fd0: 696d 6973 6d20 6120 6869 6768 6572 2022  imism a higher "
│ │ │ │ +00065fe0: 6869 6768 2220 7379 7a79 6779 2e0a 0a43  high" syzygy...C
│ │ │ │ +00065ff0: 6176 6561 740a 3d3d 3d3d 3d3d 0a0a 4172  aveat.======..Ar
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│ │ │ │ +00066040: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
│ │ │ │ +00066050: 206d 6642 6f75 6e64 3a20 6d66 426f 756e   mfBound: mfBoun
│ │ │ │ +00066060: 642c 202d 2d20 6465 7465 726d 696e 6573  d, -- determines
│ │ │ │ +00066070: 2068 6f77 2068 6967 6820 6120 7379 7a79   how high a syzy
│ │ │ │ +00066080: 6779 2074 6f20 7461 6b65 2066 6f72 0a20  gy to take for. 
│ │ │ │ +00066090: 2020 2022 6d61 7472 6978 4661 6374 6f72     "matrixFactor
│ │ │ │ +000660a0: 697a 6174 696f 6e22 0a20 202a 202a 6e6f  ization".  * *no
│ │ │ │ +000660b0: 7465 2068 6967 6853 797a 7967 793a 2068  te highSyzygy: h
│ │ │ │ +000660c0: 6967 6853 797a 7967 792c 202d 2d20 5265  ighSyzygy, -- Re
│ │ │ │ +000660d0: 7475 726e 7320 6120 7379 7a79 6779 206d  turns a syzygy m
│ │ │ │ +000660e0: 6f64 756c 6520 6f6e 6520 6265 796f 6e64  odule one beyond
│ │ │ │ +000660f0: 2074 6865 0a20 2020 2072 6567 756c 6172   the.    regular
│ │ │ │ +00066100: 6974 7920 6f66 2045 7874 284d 2c6b 290a  ity of Ext(M,k).
│ │ │ │ +00066110: 0a46 756e 6374 696f 6e73 2077 6974 6820  .Functions with 
│ │ │ │ +00066120: 6f70 7469 6f6e 616c 2061 7267 756d 656e  optional argumen
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│ │ │ │ +00066140: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  :.==============
│ │ │ │  00066150: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │  00066160: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00066170: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20  =============.. 
│ │ │ │ -00066180: 202a 2022 6869 6768 5379 7a79 6779 282e   * "highSyzygy(.
│ │ │ │ -00066190: 2e2e 2c4f 7074 696d 6973 6d3d 3e2e 2e2e  ..,Optimism=>...
│ │ │ │ -000661a0: 2922 202d 2d20 7365 6520 2a6e 6f74 6520  )" -- see *note 
│ │ │ │ -000661b0: 6869 6768 5379 7a79 6779 3a20 6869 6768  highSyzygy: high
│ │ │ │ -000661c0: 5379 7a79 6779 2c20 2d2d 0a20 2020 2052  Syzygy, --.    R
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│ │ │ │ -000661e0: 6d6f 6475 6c65 206f 6e65 2062 6579 6f6e  module one beyon
│ │ │ │ -000661f0: 6420 7468 6520 7265 6775 6c61 7269 7479  d the regularity
│ │ │ │ -00066200: 206f 6620 4578 7428 4d2c 6b29 0a20 202a   of Ext(M,k).  *
│ │ │ │ -00066210: 2022 7477 6f4d 6f6e 6f6d 6961 6c73 282e   "twoMonomials(.
│ │ │ │ -00066220: 2e2e 2c4f 7074 696d 6973 6d3d 3e2e 2e2e  ..,Optimism=>...
│ │ │ │ -00066230: 2922 202d 2d20 7365 6520 2a6e 6f74 6520  )" -- see *note 
│ │ │ │ -00066240: 7477 6f4d 6f6e 6f6d 6961 6c73 3a20 7477  twoMonomials: tw
│ │ │ │ -00066250: 6f4d 6f6e 6f6d 6961 6c73 2c0a 2020 2020  oMonomials,.    
│ │ │ │ -00066260: 2d2d 2074 616c 6c79 2074 6865 2073 6571  -- tally the seq
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│ │ │ │ -00066280: 2066 6f72 2063 6572 7461 696e 2065 7861   for certain exa
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│ │ │ │ -000662b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  ============..Th
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│ │ │ │ +00066170: 3d3d 0a0a 2020 2a20 2268 6967 6853 797a  ==..  * "highSyz
│ │ │ │ +00066180: 7967 7928 2e2e 2e2c 4f70 7469 6d69 736d  ygy(...,Optimism
│ │ │ │ +00066190: 3d3e 2e2e 2e29 2220 2d2d 2073 6565 202a  =>...)" -- see *
│ │ │ │ +000661a0: 6e6f 7465 2068 6967 6853 797a 7967 793a  note highSyzygy:
│ │ │ │ +000661b0: 2068 6967 6853 797a 7967 792c 202d 2d0a   highSyzygy, --.
│ │ │ │ +000661c0: 2020 2020 5265 7475 726e 7320 6120 7379      Returns a sy
│ │ │ │ +000661d0: 7a79 6779 206d 6f64 756c 6520 6f6e 6520  zygy module one 
│ │ │ │ +000661e0: 6265 796f 6e64 2074 6865 2072 6567 756c  beyond the regul
│ │ │ │ +000661f0: 6172 6974 7920 6f66 2045 7874 284d 2c6b  arity of Ext(M,k
│ │ │ │ +00066200: 290a 2020 2a20 2274 776f 4d6f 6e6f 6d69  ).  * "twoMonomi
│ │ │ │ +00066210: 616c 7328 2e2e 2e2c 4f70 7469 6d69 736d  als(...,Optimism
│ │ │ │ +00066220: 3d3e 2e2e 2e29 2220 2d2d 2073 6565 202a  =>...)" -- see *
│ │ │ │ +00066230: 6e6f 7465 2074 776f 4d6f 6e6f 6d69 616c  note twoMonomial
│ │ │ │ +00066240: 733a 2074 776f 4d6f 6e6f 6d69 616c 732c  s: twoMonomials,
│ │ │ │ +00066250: 0a20 2020 202d 2d20 7461 6c6c 7920 7468  .    -- tally th
│ │ │ │ +00066260: 6520 7365 7175 656e 6365 7320 6f66 2042  e sequences of B
│ │ │ │ +00066270: 5261 6e6b 7320 666f 7220 6365 7274 6169  Ranks for certai
│ │ │ │ +00066280: 6e20 6578 616d 706c 6573 0a0a 466f 7220  n examples..For 
│ │ │ │ +00066290: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d  the programmer.=
│ │ │ │ +000662a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +000662b0: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e  =..The object *n
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│ │ │ │  00066320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  00066340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00066350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54  -------------..T
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│ │ │ │ -00066460: 204f 7074 696f 6e20 616c 6c6f 7769 6e67   Option allowing
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│ │ │ │ +00066390: 6163 6175 6c61 7932 2d31 2e32 352e 3131  acaulay2-1.25.11
│ │ │ │ +000663a0: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932  +ds/M2/Macaulay2
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│ │ │ │ +000663d0: 6573 6f6c 7574 696f 6e73 2e6d 323a 3331  esolutions.m2:31
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│ │ │ │ +00066410: 666f 2c20 4e6f 6465 3a20 4f75 7452 696e  fo, Node: OutRin
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│ │ │ │ +00066440: 2c20 5570 3a20 546f 700a 0a4f 7574 5269  , Up: Top..OutRi
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│ │ │ │  000664b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000664c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000664d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000664e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000664f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00066500: 2a0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d  *..See also.====
│ │ │ │ -00066510: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520  ====..  * *note 
│ │ │ │ -00066520: 6576 656e 4578 744d 6f64 756c 653a 2065  evenExtModule: e
│ │ │ │ -00066530: 7665 6e45 7874 4d6f 6475 6c65 2c20 2d2d  venExtModule, --
│ │ │ │ -00066540: 2065 7665 6e20 7061 7274 206f 6620 4578   even part of Ex
│ │ │ │ -00066550: 745e 2a28 4d2c 6b29 206f 7665 7220 610a  t^*(M,k) over a.
│ │ │ │ -00066560: 2020 2020 636f 6d70 6c65 7465 2069 6e74      complete int
│ │ │ │ -00066570: 6572 7365 6374 696f 6e20 6173 206d 6f64  ersection as mod
│ │ │ │ -00066580: 756c 6520 6f76 6572 2043 4920 6f70 6572  ule over CI oper
│ │ │ │ -00066590: 6174 6f72 2072 696e 670a 2020 2a20 2a6e  ator ring.  * *n
│ │ │ │ -000665a0: 6f74 6520 6f64 6445 7874 4d6f 6475 6c65  ote oddExtModule
│ │ │ │ -000665b0: 3a20 6f64 6445 7874 4d6f 6475 6c65 2c20  : oddExtModule, 
│ │ │ │ -000665c0: 2d2d 206f 6464 2070 6172 7420 6f66 2045  -- odd part of E
│ │ │ │ -000665d0: 7874 5e2a 284d 2c6b 2920 6f76 6572 2061  xt^*(M,k) over a
│ │ │ │ -000665e0: 2063 6f6d 706c 6574 650a 2020 2020 696e   complete.    in
│ │ │ │ -000665f0: 7465 7273 6563 7469 6f6e 2061 7320 6d6f  tersection as mo
│ │ │ │ -00066600: 6475 6c65 206f 7665 7220 4349 206f 7065  dule over CI ope
│ │ │ │ -00066610: 7261 746f 7220 7269 6e67 0a0a 4675 6e63  rator ring..Func
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│ │ │ │ -00066630: 6e61 6c20 6172 6775 6d65 6e74 206e 616d  nal argument nam
│ │ │ │ -00066640: 6564 204f 7574 5269 6e67 3a0a 3d3d 3d3d  ed OutRing:.====
│ │ │ │ +000664f0: 2a2a 2a2a 2a2a 0a0a 5365 6520 616c 736f  ******..See also
│ │ │ │ +00066500: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a  .========..  * *
│ │ │ │ +00066510: 6e6f 7465 2065 7665 6e45 7874 4d6f 6475  note evenExtModu
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│ │ │ │ +00066540: 6f66 2045 7874 5e2a 284d 2c6b 2920 6f76  of Ext^*(M,k) ov
│ │ │ │ +00066550: 6572 2061 0a20 2020 2063 6f6d 706c 6574  er a.    complet
│ │ │ │ +00066560: 6520 696e 7465 7273 6563 7469 6f6e 2061  e intersection a
│ │ │ │ +00066570: 7320 6d6f 6475 6c65 206f 7665 7220 4349  s module over CI
│ │ │ │ +00066580: 206f 7065 7261 746f 7220 7269 6e67 0a20   operator ring. 
│ │ │ │ +00066590: 202a 202a 6e6f 7465 206f 6464 4578 744d   * *note oddExtM
│ │ │ │ +000665a0: 6f64 756c 653a 206f 6464 4578 744d 6f64  odule: oddExtMod
│ │ │ │ +000665b0: 756c 652c 202d 2d20 6f64 6420 7061 7274  ule, -- odd part
│ │ │ │ +000665c0: 206f 6620 4578 745e 2a28 4d2c 6b29 206f   of Ext^*(M,k) o
│ │ │ │ +000665d0: 7665 7220 6120 636f 6d70 6c65 7465 0a20  ver a complete. 
│ │ │ │ +000665e0: 2020 2069 6e74 6572 7365 6374 696f 6e20     intersection 
│ │ │ │ +000665f0: 6173 206d 6f64 756c 6520 6f76 6572 2043  as module over C
│ │ │ │ +00066600: 4920 6f70 6572 6174 6f72 2072 696e 670a  I operator ring.
│ │ │ │ +00066610: 0a46 756e 6374 696f 6e73 2077 6974 6820  .Functions with 
│ │ │ │ +00066620: 6f70 7469 6f6e 616c 2061 7267 756d 656e  optional argumen
│ │ │ │ +00066630: 7420 6e61 6d65 6420 4f75 7452 696e 673a  t named OutRing:
│ │ │ │ +00066640: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │  00066650: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │  00066660: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00066670: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ -00066680: 2022 6576 656e 4578 744d 6f64 756c 6528   "evenExtModule(
│ │ │ │ -00066690: 2e2e 2e2c 4f75 7452 696e 673d 3e2e 2e2e  ...,OutRing=>...
│ │ │ │ -000666a0: 2922 202d 2d20 7365 6520 2a6e 6f74 6520  )" -- see *note 
│ │ │ │ -000666b0: 6576 656e 4578 744d 6f64 756c 653a 0a20  evenExtModule:. 
│ │ │ │ -000666c0: 2020 2065 7665 6e45 7874 4d6f 6475 6c65     evenExtModule
│ │ │ │ -000666d0: 2c20 2d2d 2065 7665 6e20 7061 7274 206f  , -- even part o
│ │ │ │ -000666e0: 6620 4578 745e 2a28 4d2c 6b29 206f 7665  f Ext^*(M,k) ove
│ │ │ │ -000666f0: 7220 6120 636f 6d70 6c65 7465 2069 6e74  r a complete int
│ │ │ │ -00066700: 6572 7365 6374 696f 6e20 6173 0a20 2020  ersection as.   
│ │ │ │ -00066710: 206d 6f64 756c 6520 6f76 6572 2043 4920   module over CI 
│ │ │ │ -00066720: 6f70 6572 6174 6f72 2072 696e 670a 2020  operator ring.  
│ │ │ │ -00066730: 2a20 226f 6464 4578 744d 6f64 756c 6528  * "oddExtModule(
│ │ │ │ -00066740: 2e2e 2e2c 4f75 7452 696e 673d 3e2e 2e2e  ...,OutRing=>...
│ │ │ │ -00066750: 2922 202d 2d20 7365 6520 2a6e 6f74 6520  )" -- see *note 
│ │ │ │ -00066760: 6f64 6445 7874 4d6f 6475 6c65 3a20 6f64  oddExtModule: od
│ │ │ │ -00066770: 6445 7874 4d6f 6475 6c65 2c0a 2020 2020  dExtModule,.    
│ │ │ │ -00066780: 2d2d 206f 6464 2070 6172 7420 6f66 2045  -- odd part of E
│ │ │ │ -00066790: 7874 5e2a 284d 2c6b 2920 6f76 6572 2061  xt^*(M,k) over a
│ │ │ │ -000667a0: 2063 6f6d 706c 6574 6520 696e 7465 7273   complete inters
│ │ │ │ -000667b0: 6563 7469 6f6e 2061 7320 6d6f 6475 6c65  ection as module
│ │ │ │ -000667c0: 206f 7665 7220 4349 0a20 2020 206f 7065   over CI.    ope
│ │ │ │ -000667d0: 7261 746f 7220 7269 6e67 0a0a 466f 7220  rator ring..For 
│ │ │ │ -000667e0: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d  the programmer.=
│ │ │ │ -000667f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00066800: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e  =..The object *n
│ │ │ │ -00066810: 6f74 6520 4f75 7452 696e 673a 204f 7574  ote OutRing: Out
│ │ │ │ -00066820: 5269 6e67 2c20 6973 2061 202a 6e6f 7465  Ring, is a *note
│ │ │ │ -00066830: 2073 796d 626f 6c3a 2028 4d61 6361 756c   symbol: (Macaul
│ │ │ │ -00066840: 6179 3244 6f63 2953 796d 626f 6c2c 2e0a  ay2Doc)Symbol,..
│ │ │ │ -00066850: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .---------------
│ │ │ │ +00066670: 0a0a 2020 2a20 2265 7665 6e45 7874 4d6f  ..  * "evenExtMo
│ │ │ │ +00066680: 6475 6c65 282e 2e2e 2c4f 7574 5269 6e67  dule(...,OutRing
│ │ │ │ +00066690: 3d3e 2e2e 2e29 2220 2d2d 2073 6565 202a  =>...)" -- see *
│ │ │ │ +000666a0: 6e6f 7465 2065 7665 6e45 7874 4d6f 6475  note evenExtModu
│ │ │ │ +000666b0: 6c65 3a0a 2020 2020 6576 656e 4578 744d  le:.    evenExtM
│ │ │ │ +000666c0: 6f64 756c 652c 202d 2d20 6576 656e 2070  odule, -- even p
│ │ │ │ +000666d0: 6172 7420 6f66 2045 7874 5e2a 284d 2c6b  art of Ext^*(M,k
│ │ │ │ +000666e0: 2920 6f76 6572 2061 2063 6f6d 706c 6574  ) over a complet
│ │ │ │ +000666f0: 6520 696e 7465 7273 6563 7469 6f6e 2061  e intersection a
│ │ │ │ +00066700: 730a 2020 2020 6d6f 6475 6c65 206f 7665  s.    module ove
│ │ │ │ +00066710: 7220 4349 206f 7065 7261 746f 7220 7269  r CI operator ri
│ │ │ │ +00066720: 6e67 0a20 202a 2022 6f64 6445 7874 4d6f  ng.  * "oddExtMo
│ │ │ │ +00066730: 6475 6c65 282e 2e2e 2c4f 7574 5269 6e67  dule(...,OutRing
│ │ │ │ +00066740: 3d3e 2e2e 2e29 2220 2d2d 2073 6565 202a  =>...)" -- see *
│ │ │ │ +00066750: 6e6f 7465 206f 6464 4578 744d 6f64 756c  note oddExtModul
│ │ │ │ +00066760: 653a 206f 6464 4578 744d 6f64 756c 652c  e: oddExtModule,
│ │ │ │ +00066770: 0a20 2020 202d 2d20 6f64 6420 7061 7274  .    -- odd part
│ │ │ │ +00066780: 206f 6620 4578 745e 2a28 4d2c 6b29 206f   of Ext^*(M,k) o
│ │ │ │ +00066790: 7665 7220 6120 636f 6d70 6c65 7465 2069  ver a complete i
│ │ │ │ +000667a0: 6e74 6572 7365 6374 696f 6e20 6173 206d  ntersection as m
│ │ │ │ +000667b0: 6f64 756c 6520 6f76 6572 2043 490a 2020  odule over CI.  
│ │ │ │ +000667c0: 2020 6f70 6572 6174 6f72 2072 696e 670a    operator ring.
│ │ │ │ +000667d0: 0a46 6f72 2074 6865 2070 726f 6772 616d  .For the program
│ │ │ │ +000667e0: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  mer.============
│ │ │ │ +000667f0: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65  ======..The obje
│ │ │ │ +00066800: 6374 202a 6e6f 7465 204f 7574 5269 6e67  ct *note OutRing
│ │ │ │ +00066810: 3a20 4f75 7452 696e 672c 2069 7320 6120  : OutRing, is a 
│ │ │ │ +00066820: 2a6e 6f74 6520 7379 6d62 6f6c 3a20 284d  *note symbol: (M
│ │ │ │ +00066830: 6163 6175 6c61 7932 446f 6329 5379 6d62  acaulay2Doc)Symb
│ │ │ │ +00066840: 6f6c 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d  ol,...----------
│ │ │ │ +00066850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00066860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00066870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00066880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00066890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000668a0: 0a0a 5468 6520 736f 7572 6365 206f 6620  ..The source of 
│ │ │ │ -000668b0: 7468 6973 2064 6f63 756d 656e 7420 6973  this document is
│ │ │ │ -000668c0: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f   in./build/repro
│ │ │ │ -000668d0: 6475 6369 626c 652d 7061 7468 2f6d 6163  ducible-path/mac
│ │ │ │ -000668e0: 6175 6c61 7932 2d31 2e32 352e 3131 2b64  aulay2-1.25.11+d
│ │ │ │ -000668f0: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70  s/M2/Macaulay2/p
│ │ │ │ -00066900: 6163 6b61 6765 732f 0a43 6f6d 706c 6574  ackages/.Complet
│ │ │ │ -00066910: 6549 6e74 6572 7365 6374 696f 6e52 6573  eIntersectionRes
│ │ │ │ -00066920: 6f6c 7574 696f 6e73 2e6d 323a 3336 3035  olutions.m2:3605
│ │ │ │ -00066930: 3a30 2e0a 1f0a 4669 6c65 3a20 436f 6d70  :0....File: Comp
│ │ │ │ -00066940: 6c65 7465 496e 7465 7273 6563 7469 6f6e  leteIntersection
│ │ │ │ -00066950: 5265 736f 6c75 7469 6f6e 732e 696e 666f  Resolutions.info
│ │ │ │ -00066960: 2c20 4e6f 6465 3a20 7073 694d 6170 732c  , Node: psiMaps,
│ │ │ │ -00066970: 204e 6578 743a 2072 6567 756c 6172 6974   Next: regularit
│ │ │ │ -00066980: 7953 6571 7565 6e63 652c 2050 7265 763a  ySequence, Prev:
│ │ │ │ -00066990: 204f 7574 5269 6e67 2c20 5570 3a20 546f   OutRing, Up: To
│ │ │ │ -000669a0: 700a 0a70 7369 4d61 7073 202d 2d20 6c69  p..psiMaps -- li
│ │ │ │ -000669b0: 7374 2074 6865 206d 6170 7320 2070 7369  st the maps  psi
│ │ │ │ -000669c0: 2870 293a 2042 5f31 2870 2920 2d2d 3e20  (p): B_1(p) --> 
│ │ │ │ -000669d0: 415f 3028 702d 3129 2069 6e20 6120 6d61  A_0(p-1) in a ma
│ │ │ │ -000669e0: 7472 6978 4661 6374 6f72 697a 6174 696f  trixFactorizatio
│ │ │ │ -000669f0: 6e0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  n.**************
│ │ │ │ +00066890: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263  -----..The sourc
│ │ │ │ +000668a0: 6520 6f66 2074 6869 7320 646f 6375 6d65  e of this docume
│ │ │ │ +000668b0: 6e74 2069 7320 696e 0a2f 6275 696c 642f  nt is in./build/
│ │ │ │ +000668c0: 7265 7072 6f64 7563 6962 6c65 2d70 6174  reproducible-pat
│ │ │ │ +000668d0: 682f 6d61 6361 756c 6179 322d 312e 3235  h/macaulay2-1.25
│ │ │ │ +000668e0: 2e31 312b 6473 2f4d 322f 4d61 6361 756c  .11+ds/M2/Macaul
│ │ │ │ +000668f0: 6179 322f 7061 636b 6167 6573 2f0a 436f  ay2/packages/.Co
│ │ │ │ +00066900: 6d70 6c65 7465 496e 7465 7273 6563 7469  mpleteIntersecti
│ │ │ │ +00066910: 6f6e 5265 736f 6c75 7469 6f6e 732e 6d32  onResolutions.m2
│ │ │ │ +00066920: 3a33 3630 353a 302e 0a1f 0a46 696c 653a  :3605:0....File:
│ │ │ │ +00066930: 2043 6f6d 706c 6574 6549 6e74 6572 7365   CompleteInterse
│ │ │ │ +00066940: 6374 696f 6e52 6573 6f6c 7574 696f 6e73  ctionResolutions
│ │ │ │ +00066950: 2e69 6e66 6f2c 204e 6f64 653a 2070 7369  .info, Node: psi
│ │ │ │ +00066960: 4d61 7073 2c20 4e65 7874 3a20 7265 6775  Maps, Next: regu
│ │ │ │ +00066970: 6c61 7269 7479 5365 7175 656e 6365 2c20  laritySequence, 
│ │ │ │ +00066980: 5072 6576 3a20 4f75 7452 696e 672c 2055  Prev: OutRing, U
│ │ │ │ +00066990: 703a 2054 6f70 0a0a 7073 694d 6170 7320  p: Top..psiMaps 
│ │ │ │ +000669a0: 2d2d 206c 6973 7420 7468 6520 6d61 7073  -- list the maps
│ │ │ │ +000669b0: 2020 7073 6928 7029 3a20 425f 3128 7029    psi(p): B_1(p)
│ │ │ │ +000669c0: 202d 2d3e 2041 5f30 2870 2d31 2920 696e   --> A_0(p-1) in
│ │ │ │ +000669d0: 2061 206d 6174 7269 7846 6163 746f 7269   a matrixFactori
│ │ │ │ +000669e0: 7a61 7469 6f6e 0a2a 2a2a 2a2a 2a2a 2a2a  zation.*********
│ │ │ │ +000669f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00066a00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00066a10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00066a20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00066a30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00066a40: 0a0a 2020 2a20 5573 6167 653a 200a 2020  ..  * Usage: .  
│ │ │ │ -00066a50: 2020 2020 2020 7073 6d61 7073 203d 2070        psmaps = p
│ │ │ │ -00066a60: 7369 4d61 7073 206d 660a 2020 2a20 496e  siMaps mf.  * In
│ │ │ │ -00066a70: 7075 7473 3a0a 2020 2020 2020 2a20 6d66  puts:.      * mf
│ │ │ │ -00066a80: 2c20 6120 2a6e 6f74 6520 6c69 7374 3a20  , a *note list: 
│ │ │ │ -00066a90: 284d 6163 6175 6c61 7932 446f 6329 4c69  (Macaulay2Doc)Li
│ │ │ │ -00066aa0: 7374 2c2c 206f 7574 7075 7420 6f66 2061  st,, output of a
│ │ │ │ -00066ab0: 206d 6174 7269 7846 6163 746f 7269 7a61   matrixFactoriza
│ │ │ │ -00066ac0: 7469 6f6e 0a20 2020 2020 2020 2063 6f6d  tion.        com
│ │ │ │ -00066ad0: 7075 7461 7469 6f6e 0a20 202a 204f 7574  putation.  * Out
│ │ │ │ -00066ae0: 7075 7473 3a0a 2020 2020 2020 2a20 7073  puts:.      * ps
│ │ │ │ -00066af0: 6d61 7073 2c20 6120 2a6e 6f74 6520 6c69  maps, a *note li
│ │ │ │ -00066b00: 7374 3a20 284d 6163 6175 6c61 7932 446f  st: (Macaulay2Do
│ │ │ │ -00066b10: 6329 4c69 7374 2c2c 206c 6973 7420 6d61  c)List,, list ma
│ │ │ │ -00066b20: 7472 6963 6573 2024 645f 703a 0a20 2020  trices $d_p:.   
│ │ │ │ -00066b30: 2020 2020 2042 5f31 2870 295c 746f 2041       B_1(p)\to A
│ │ │ │ -00066b40: 5f30 2870 2d31 2924 0a0a 4465 7363 7269  _0(p-1)$..Descri
│ │ │ │ -00066b50: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ -00066b60: 3d0a 0a53 6565 2074 6865 2064 6f63 756d  =..See the docum
│ │ │ │ -00066b70: 656e 7461 7469 6f6e 2066 6f72 206d 6174  entation for mat
│ │ │ │ -00066b80: 7269 7846 6163 746f 7269 7a61 7469 6f6e  rixFactorization
│ │ │ │ -00066b90: 2066 6f72 2061 6e20 6578 616d 706c 652e   for an example.
│ │ │ │ -00066ba0: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d  ..See also.=====
│ │ │ │ -00066bb0: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 206d  ===..  * *note m
│ │ │ │ -00066bc0: 6174 7269 7846 6163 746f 7269 7a61 7469  atrixFactorizati
│ │ │ │ -00066bd0: 6f6e 3a20 6d61 7472 6978 4661 6374 6f72  on: matrixFactor
│ │ │ │ -00066be0: 697a 6174 696f 6e2c 202d 2d20 4d61 7073  ization, -- Maps
│ │ │ │ -00066bf0: 2069 6e20 6120 6869 6768 6572 0a20 2020   in a higher.   
│ │ │ │ -00066c00: 2063 6f64 696d 656e 7369 6f6e 206d 6174   codimension mat
│ │ │ │ -00066c10: 7269 7820 6661 6374 6f72 697a 6174 696f  rix factorizatio
│ │ │ │ -00066c20: 6e0a 2020 2a20 2a6e 6f74 6520 4252 616e  n.  * *note BRan
│ │ │ │ -00066c30: 6b73 3a20 4252 616e 6b73 2c20 2d2d 2072  ks: BRanks, -- r
│ │ │ │ -00066c40: 616e 6b73 206f 6620 7468 6520 6d6f 6475  anks of the modu
│ │ │ │ -00066c50: 6c65 7320 425f 6928 6429 2069 6e20 610a  les B_i(d) in a.
│ │ │ │ -00066c60: 2020 2020 6d61 7472 6978 4661 6374 6f72      matrixFactor
│ │ │ │ -00066c70: 697a 6174 696f 6e0a 2020 2a20 2a6e 6f74  ization.  * *not
│ │ │ │ -00066c80: 6520 624d 6170 733a 2062 4d61 7073 2c20  e bMaps: bMaps, 
│ │ │ │ -00066c90: 2d2d 206c 6973 7420 7468 6520 6d61 7073  -- list the maps
│ │ │ │ -00066ca0: 2020 645f 703a 425f 3128 7029 2d2d 3e42    d_p:B_1(p)-->B
│ │ │ │ -00066cb0: 5f30 2870 2920 696e 2061 0a20 2020 206d  _0(p) in a.    m
│ │ │ │ -00066cc0: 6174 7269 7846 6163 746f 7269 7a61 7469  atrixFactorizati
│ │ │ │ -00066cd0: 6f6e 0a20 202a 202a 6e6f 7465 2064 4d61  on.  * *note dMa
│ │ │ │ -00066ce0: 7073 3a20 644d 6170 732c 202d 2d20 6c69  ps: dMaps, -- li
│ │ │ │ -00066cf0: 7374 2074 6865 206d 6170 7320 2064 2870  st the maps  d(p
│ │ │ │ -00066d00: 293a 415f 3128 7029 2d2d 3e20 415f 3028  ):A_1(p)--> A_0(
│ │ │ │ -00066d10: 7029 2069 6e20 610a 2020 2020 6d61 7472  p) in a.    matr
│ │ │ │ -00066d20: 6978 4661 6374 6f72 697a 6174 696f 6e0a  ixFactorization.
│ │ │ │ -00066d30: 2020 2a20 2a6e 6f74 6520 684d 6170 733a    * *note hMaps:
│ │ │ │ -00066d40: 2068 4d61 7073 2c20 2d2d 206c 6973 7420   hMaps, -- list 
│ │ │ │ -00066d50: 7468 6520 6d61 7073 2020 6828 7029 3a20  the maps  h(p): 
│ │ │ │ -00066d60: 415f 3028 7029 2d2d 3e20 415f 3128 7029  A_0(p)--> A_1(p)
│ │ │ │ -00066d70: 2069 6e20 610a 2020 2020 6d61 7472 6978   in a.    matrix
│ │ │ │ -00066d80: 4661 6374 6f72 697a 6174 696f 6e0a 0a57  Factorization..W
│ │ │ │ -00066d90: 6179 7320 746f 2075 7365 2070 7369 4d61  ays to use psiMa
│ │ │ │ -00066da0: 7073 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ps:.============
│ │ │ │ -00066db0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2270  ========..  * "p
│ │ │ │ -00066dc0: 7369 4d61 7073 284c 6973 7429 220a 0a46  siMaps(List)"..F
│ │ │ │ -00066dd0: 6f72 2074 6865 2070 726f 6772 616d 6d65  or the programme
│ │ │ │ -00066de0: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  r.==============
│ │ │ │ -00066df0: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374  ====..The object
│ │ │ │ -00066e00: 202a 6e6f 7465 2070 7369 4d61 7073 3a20   *note psiMaps: 
│ │ │ │ -00066e10: 7073 694d 6170 732c 2069 7320 6120 2a6e  psiMaps, is a *n
│ │ │ │ -00066e20: 6f74 6520 6d65 7468 6f64 2066 756e 6374  ote method funct
│ │ │ │ -00066e30: 696f 6e3a 0a28 4d61 6361 756c 6179 3244  ion:.(Macaulay2D
│ │ │ │ -00066e40: 6f63 294d 6574 686f 6446 756e 6374 696f  oc)MethodFunctio
│ │ │ │ -00066e50: 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d  n,...-----------
│ │ │ │ +00066a30: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765  *****..  * Usage
│ │ │ │ +00066a40: 3a20 0a20 2020 2020 2020 2070 736d 6170  : .        psmap
│ │ │ │ +00066a50: 7320 3d20 7073 694d 6170 7320 6d66 0a20  s = psiMaps mf. 
│ │ │ │ +00066a60: 202a 2049 6e70 7574 733a 0a20 2020 2020   * Inputs:.     
│ │ │ │ +00066a70: 202a 206d 662c 2061 202a 6e6f 7465 206c   * mf, a *note l
│ │ │ │ +00066a80: 6973 743a 2028 4d61 6361 756c 6179 3244  ist: (Macaulay2D
│ │ │ │ +00066a90: 6f63 294c 6973 742c 2c20 6f75 7470 7574  oc)List,, output
│ │ │ │ +00066aa0: 206f 6620 6120 6d61 7472 6978 4661 6374   of a matrixFact
│ │ │ │ +00066ab0: 6f72 697a 6174 696f 6e0a 2020 2020 2020  orization.      
│ │ │ │ +00066ac0: 2020 636f 6d70 7574 6174 696f 6e0a 2020    computation.  
│ │ │ │ +00066ad0: 2a20 4f75 7470 7574 733a 0a20 2020 2020  * Outputs:.     
│ │ │ │ +00066ae0: 202a 2070 736d 6170 732c 2061 202a 6e6f   * psmaps, a *no
│ │ │ │ +00066af0: 7465 206c 6973 743a 2028 4d61 6361 756c  te list: (Macaul
│ │ │ │ +00066b00: 6179 3244 6f63 294c 6973 742c 2c20 6c69  ay2Doc)List,, li
│ │ │ │ +00066b10: 7374 206d 6174 7269 6365 7320 2464 5f70  st matrices $d_p
│ │ │ │ +00066b20: 3a0a 2020 2020 2020 2020 425f 3128 7029  :.        B_1(p)
│ │ │ │ +00066b30: 5c74 6f20 415f 3028 702d 3129 240a 0a44  \to A_0(p-1)$..D
│ │ │ │ +00066b40: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ +00066b50: 3d3d 3d3d 3d3d 0a0a 5365 6520 7468 6520  ======..See the 
│ │ │ │ +00066b60: 646f 6375 6d65 6e74 6174 696f 6e20 666f  documentation fo
│ │ │ │ +00066b70: 7220 6d61 7472 6978 4661 6374 6f72 697a  r matrixFactoriz
│ │ │ │ +00066b80: 6174 696f 6e20 666f 7220 616e 2065 7861  ation for an exa
│ │ │ │ +00066b90: 6d70 6c65 2e0a 0a53 6565 2061 6c73 6f0a  mple...See also.
│ │ │ │ +00066ba0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ +00066bb0: 6f74 6520 6d61 7472 6978 4661 6374 6f72  ote matrixFactor
│ │ │ │ +00066bc0: 697a 6174 696f 6e3a 206d 6174 7269 7846  ization: matrixF
│ │ │ │ +00066bd0: 6163 746f 7269 7a61 7469 6f6e 2c20 2d2d  actorization, --
│ │ │ │ +00066be0: 204d 6170 7320 696e 2061 2068 6967 6865   Maps in a highe
│ │ │ │ +00066bf0: 720a 2020 2020 636f 6469 6d65 6e73 696f  r.    codimensio
│ │ │ │ +00066c00: 6e20 6d61 7472 6978 2066 6163 746f 7269  n matrix factori
│ │ │ │ +00066c10: 7a61 7469 6f6e 0a20 202a 202a 6e6f 7465  zation.  * *note
│ │ │ │ +00066c20: 2042 5261 6e6b 733a 2042 5261 6e6b 732c   BRanks: BRanks,
│ │ │ │ +00066c30: 202d 2d20 7261 6e6b 7320 6f66 2074 6865   -- ranks of the
│ │ │ │ +00066c40: 206d 6f64 756c 6573 2042 5f69 2864 2920   modules B_i(d) 
│ │ │ │ +00066c50: 696e 2061 0a20 2020 206d 6174 7269 7846  in a.    matrixF
│ │ │ │ +00066c60: 6163 746f 7269 7a61 7469 6f6e 0a20 202a  actorization.  *
│ │ │ │ +00066c70: 202a 6e6f 7465 2062 4d61 7073 3a20 624d   *note bMaps: bM
│ │ │ │ +00066c80: 6170 732c 202d 2d20 6c69 7374 2074 6865  aps, -- list the
│ │ │ │ +00066c90: 206d 6170 7320 2064 5f70 3a42 5f31 2870   maps  d_p:B_1(p
│ │ │ │ +00066ca0: 292d 2d3e 425f 3028 7029 2069 6e20 610a  )-->B_0(p) in a.
│ │ │ │ +00066cb0: 2020 2020 6d61 7472 6978 4661 6374 6f72      matrixFactor
│ │ │ │ +00066cc0: 697a 6174 696f 6e0a 2020 2a20 2a6e 6f74  ization.  * *not
│ │ │ │ +00066cd0: 6520 644d 6170 733a 2064 4d61 7073 2c20  e dMaps: dMaps, 
│ │ │ │ +00066ce0: 2d2d 206c 6973 7420 7468 6520 6d61 7073  -- list the maps
│ │ │ │ +00066cf0: 2020 6428 7029 3a41 5f31 2870 292d 2d3e    d(p):A_1(p)-->
│ │ │ │ +00066d00: 2041 5f30 2870 2920 696e 2061 0a20 2020   A_0(p) in a.   
│ │ │ │ +00066d10: 206d 6174 7269 7846 6163 746f 7269 7a61   matrixFactoriza
│ │ │ │ +00066d20: 7469 6f6e 0a20 202a 202a 6e6f 7465 2068  tion.  * *note h
│ │ │ │ +00066d30: 4d61 7073 3a20 684d 6170 732c 202d 2d20  Maps: hMaps, -- 
│ │ │ │ +00066d40: 6c69 7374 2074 6865 206d 6170 7320 2068  list the maps  h
│ │ │ │ +00066d50: 2870 293a 2041 5f30 2870 292d 2d3e 2041  (p): A_0(p)--> A
│ │ │ │ +00066d60: 5f31 2870 2920 696e 2061 0a20 2020 206d  _1(p) in a.    m
│ │ │ │ +00066d70: 6174 7269 7846 6163 746f 7269 7a61 7469  atrixFactorizati
│ │ │ │ +00066d80: 6f6e 0a0a 5761 7973 2074 6f20 7573 6520  on..Ways to use 
│ │ │ │ +00066d90: 7073 694d 6170 733a 0a3d 3d3d 3d3d 3d3d  psiMaps:.=======
│ │ │ │ +00066da0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20  =============.. 
│ │ │ │ +00066db0: 202a 2022 7073 694d 6170 7328 4c69 7374   * "psiMaps(List
│ │ │ │ +00066dc0: 2922 0a0a 466f 7220 7468 6520 7072 6f67  )"..For the prog
│ │ │ │ +00066dd0: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d  rammer.=========
│ │ │ │ +00066de0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f  =========..The o
│ │ │ │ +00066df0: 626a 6563 7420 2a6e 6f74 6520 7073 694d  bject *note psiM
│ │ │ │ +00066e00: 6170 733a 2070 7369 4d61 7073 2c20 6973  aps: psiMaps, is
│ │ │ │ +00066e10: 2061 202a 6e6f 7465 206d 6574 686f 6420   a *note method 
│ │ │ │ +00066e20: 6675 6e63 7469 6f6e 3a0a 284d 6163 6175  function:.(Macau
│ │ │ │ +00066e30: 6c61 7932 446f 6329 4d65 7468 6f64 4675  lay2Doc)MethodFu
│ │ │ │ +00066e40: 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d  nction,...------
│ │ │ │ +00066e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00066e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00066e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00066e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00066e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00066ea0: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365  ----..The source
│ │ │ │ -00066eb0: 206f 6620 7468 6973 2064 6f63 756d 656e   of this documen
│ │ │ │ -00066ec0: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72  t is in./build/r
│ │ │ │ -00066ed0: 6570 726f 6475 6369 626c 652d 7061 7468  eproducible-path
│ │ │ │ -00066ee0: 2f6d 6163 6175 6c61 7932 2d31 2e32 352e  /macaulay2-1.25.
│ │ │ │ -00066ef0: 3131 2b64 732f 4d32 2f4d 6163 6175 6c61  11+ds/M2/Macaula
│ │ │ │ -00066f00: 7932 2f70 6163 6b61 6765 732f 0a43 6f6d  y2/packages/.Com
│ │ │ │ -00066f10: 706c 6574 6549 6e74 6572 7365 6374 696f  pleteIntersectio
│ │ │ │ -00066f20: 6e52 6573 6f6c 7574 696f 6e73 2e6d 323a  nResolutions.m2:
│ │ │ │ -00066f30: 3434 3832 3a30 2e0a 1f0a 4669 6c65 3a20  4482:0....File: 
│ │ │ │ -00066f40: 436f 6d70 6c65 7465 496e 7465 7273 6563  CompleteIntersec
│ │ │ │ -00066f50: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e  tionResolutions.
│ │ │ │ -00066f60: 696e 666f 2c20 4e6f 6465 3a20 7265 6775  info, Node: regu
│ │ │ │ -00066f70: 6c61 7269 7479 5365 7175 656e 6365 2c20  laritySequence, 
│ │ │ │ -00066f80: 4e65 7874 3a20 5332 2c20 5072 6576 3a20  Next: S2, Prev: 
│ │ │ │ -00066f90: 7073 694d 6170 732c 2055 703a 2054 6f70  psiMaps, Up: Top
│ │ │ │ -00066fa0: 0a0a 7265 6775 6c61 7269 7479 5365 7175  ..regularitySequ
│ │ │ │ -00066fb0: 656e 6365 202d 2d20 7265 6775 6c61 7269  ence -- regulari
│ │ │ │ -00066fc0: 7479 206f 6620 4578 7420 6d6f 6475 6c65  ty of Ext module
│ │ │ │ -00066fd0: 7320 666f 7220 6120 7365 7175 656e 6365  s for a sequence
│ │ │ │ -00066fe0: 206f 6620 4d43 4d20 6170 7072 6f78 696d   of MCM approxim
│ │ │ │ -00066ff0: 6174 696f 6e73 0a2a 2a2a 2a2a 2a2a 2a2a  ations.*********
│ │ │ │ +00066e90: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073  ---------..The s
│ │ │ │ +00066ea0: 6f75 7263 6520 6f66 2074 6869 7320 646f  ource of this do
│ │ │ │ +00066eb0: 6375 6d65 6e74 2069 7320 696e 0a2f 6275  cument is in./bu
│ │ │ │ +00066ec0: 696c 642f 7265 7072 6f64 7563 6962 6c65  ild/reproducible
│ │ │ │ +00066ed0: 2d70 6174 682f 6d61 6361 756c 6179 322d  -path/macaulay2-
│ │ │ │ +00066ee0: 312e 3235 2e31 312b 6473 2f4d 322f 4d61  1.25.11+ds/M2/Ma
│ │ │ │ +00066ef0: 6361 756c 6179 322f 7061 636b 6167 6573  caulay2/packages
│ │ │ │ +00066f00: 2f0a 436f 6d70 6c65 7465 496e 7465 7273  /.CompleteInters
│ │ │ │ +00066f10: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e  ectionResolution
│ │ │ │ +00066f20: 732e 6d32 3a34 3438 323a 302e 0a1f 0a46  s.m2:4482:0....F
│ │ │ │ +00066f30: 696c 653a 2043 6f6d 706c 6574 6549 6e74  ile: CompleteInt
│ │ │ │ +00066f40: 6572 7365 6374 696f 6e52 6573 6f6c 7574  ersectionResolut
│ │ │ │ +00066f50: 696f 6e73 2e69 6e66 6f2c 204e 6f64 653a  ions.info, Node:
│ │ │ │ +00066f60: 2072 6567 756c 6172 6974 7953 6571 7565   regularitySeque
│ │ │ │ +00066f70: 6e63 652c 204e 6578 743a 2053 322c 2050  nce, Next: S2, P
│ │ │ │ +00066f80: 7265 763a 2070 7369 4d61 7073 2c20 5570  rev: psiMaps, Up
│ │ │ │ +00066f90: 3a20 546f 700a 0a72 6567 756c 6172 6974  : Top..regularit
│ │ │ │ +00066fa0: 7953 6571 7565 6e63 6520 2d2d 2072 6567  ySequence -- reg
│ │ │ │ +00066fb0: 756c 6172 6974 7920 6f66 2045 7874 206d  ularity of Ext m
│ │ │ │ +00066fc0: 6f64 756c 6573 2066 6f72 2061 2073 6571  odules for a seq
│ │ │ │ +00066fd0: 7565 6e63 6520 6f66 204d 434d 2061 7070  uence of MCM app
│ │ │ │ +00066fe0: 726f 7869 6d61 7469 6f6e 730a 2a2a 2a2a  roximations.****
│ │ │ │ +00066ff0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00067000: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00067010: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00067020: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00067030: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00067040: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a  ***********..  *
│ │ │ │ -00067050: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ -00067060: 204c 203d 2072 6567 756c 6172 6974 7953   L = regularityS
│ │ │ │ -00067070: 6571 7565 6e63 6520 2852 2c4d 290a 2020  equence (R,M).  
│ │ │ │ -00067080: 2a20 496e 7075 7473 3a0a 2020 2020 2020  * Inputs:.      
│ │ │ │ -00067090: 2a20 522c 2061 202a 6e6f 7465 206c 6973  * R, a *note lis
│ │ │ │ -000670a0: 743a 2028 4d61 6361 756c 6179 3244 6f63  t: (Macaulay2Doc
│ │ │ │ -000670b0: 294c 6973 742c 2c20 6c69 7374 206f 6620  )List,, list of 
│ │ │ │ -000670c0: 7269 6e67 7320 525f 6920 3d0a 2020 2020  rings R_i =.    
│ │ │ │ -000670d0: 2020 2020 532f 2866 5f30 2e2e 665f 7b28      S/(f_0..f_{(
│ │ │ │ -000670e0: 692d 3129 7d29 2c20 636f 6d70 6c65 7465  i-1)}), complete
│ │ │ │ -000670f0: 2069 6e74 6572 7365 6374 696f 6e73 0a20   intersections. 
│ │ │ │ -00067100: 2020 2020 202a 204d 2c20 6120 2a6e 6f74       * M, a *not
│ │ │ │ -00067110: 6520 6d6f 6475 6c65 3a20 284d 6163 6175  e module: (Macau
│ │ │ │ -00067120: 6c61 7932 446f 6329 4d6f 6475 6c65 2c2c  lay2Doc)Module,,
│ │ │ │ -00067130: 206d 6f64 756c 6520 6f76 6572 2052 5f63   module over R_c
│ │ │ │ -00067140: 2077 6865 7265 2063 203d 0a20 2020 2020   where c =.     
│ │ │ │ -00067150: 2020 206c 656e 6774 6820 5220 2d20 312e     length R - 1.
│ │ │ │ -00067160: 0a20 202a 204f 7574 7075 7473 3a0a 2020  .  * Outputs:.  
│ │ │ │ -00067170: 2020 2020 2a20 4c2c 2061 202a 6e6f 7465      * L, a *note
│ │ │ │ -00067180: 206c 6973 743a 2028 4d61 6361 756c 6179   list: (Macaulay
│ │ │ │ -00067190: 3244 6f63 294c 6973 742c 2c20 4c69 7374  2Doc)List,, List
│ │ │ │ -000671a0: 206f 6620 7061 6972 7320 7b72 6567 756c   of pairs {regul
│ │ │ │ -000671b0: 6172 6974 790a 2020 2020 2020 2020 6576  arity.        ev
│ │ │ │ -000671c0: 656e 4578 744d 6f64 756c 6520 4d5f 692c  enExtModule M_i,
│ │ │ │ -000671d0: 2072 6567 756c 6172 6974 7920 6f64 6445   regularity oddE
│ │ │ │ -000671e0: 7874 4d6f 6475 6c65 204d 5f69 290a 0a44  xtModule M_i)..D
│ │ │ │ -000671f0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ -00067200: 3d3d 3d3d 3d3d 0a0a 436f 6d70 7574 6573  ======..Computes
│ │ │ │ -00067210: 2074 6865 206e 6f6e 2d66 7265 6520 7061   the non-free pa
│ │ │ │ -00067220: 7274 7320 4d5f 6920 6f66 2074 6865 204d  rts M_i of the M
│ │ │ │ -00067230: 434d 2061 7070 726f 7869 6d61 7469 6f6e  CM approximation
│ │ │ │ -00067240: 2074 6f20 4d20 6f76 6572 2052 5f69 2c0a   to M over R_i,.
│ │ │ │ -00067250: 7374 6f70 7069 6e67 2077 6865 6e20 4d5f  stopping when M_
│ │ │ │ -00067260: 6920 6265 636f 6d65 7320 6672 6565 2c20  i becomes free, 
│ │ │ │ -00067270: 616e 6420 7265 7475 726e 7320 7468 6520  and returns the 
│ │ │ │ -00067280: 6c69 7374 2077 686f 7365 2065 6c65 6d65  list whose eleme
│ │ │ │ -00067290: 6e74 7320 6172 6520 7468 650a 7061 6972  nts are the.pair
│ │ │ │ -000672a0: 7320 6f66 2072 6567 756c 6172 6974 6965  s of regularitie
│ │ │ │ -000672b0: 732c 2073 7461 7274 696e 6720 7769 7468  s, starting with
│ │ │ │ -000672c0: 204d 5f7b 2863 2d31 297d 204e 6f74 6520   M_{(c-1)} Note 
│ │ │ │ -000672d0: 7468 6174 2074 6865 2066 6972 7374 2070  that the first p
│ │ │ │ -000672e0: 6169 7220 6973 2066 6f72 0a74 6865 0a0a  air is for.the..
│ │ │ │ -000672f0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00067040: 0a0a 2020 2a20 5573 6167 653a 200a 2020  ..  * Usage: .  
│ │ │ │ +00067050: 2020 2020 2020 4c20 3d20 7265 6775 6c61        L = regula
│ │ │ │ +00067060: 7269 7479 5365 7175 656e 6365 2028 522c  ritySequence (R,
│ │ │ │ +00067070: 4d29 0a20 202a 2049 6e70 7574 733a 0a20  M).  * Inputs:. 
│ │ │ │ +00067080: 2020 2020 202a 2052 2c20 6120 2a6e 6f74       * R, a *not
│ │ │ │ +00067090: 6520 6c69 7374 3a20 284d 6163 6175 6c61  e list: (Macaula
│ │ │ │ +000670a0: 7932 446f 6329 4c69 7374 2c2c 206c 6973  y2Doc)List,, lis
│ │ │ │ +000670b0: 7420 6f66 2072 696e 6773 2052 5f69 203d  t of rings R_i =
│ │ │ │ +000670c0: 0a20 2020 2020 2020 2053 2f28 665f 302e  .        S/(f_0.
│ │ │ │ +000670d0: 2e66 5f7b 2869 2d31 297d 292c 2063 6f6d  .f_{(i-1)}), com
│ │ │ │ +000670e0: 706c 6574 6520 696e 7465 7273 6563 7469  plete intersecti
│ │ │ │ +000670f0: 6f6e 730a 2020 2020 2020 2a20 4d2c 2061  ons.      * M, a
│ │ │ │ +00067100: 202a 6e6f 7465 206d 6f64 756c 653a 2028   *note module: (
│ │ │ │ +00067110: 4d61 6361 756c 6179 3244 6f63 294d 6f64  Macaulay2Doc)Mod
│ │ │ │ +00067120: 756c 652c 2c20 6d6f 6475 6c65 206f 7665  ule,, module ove
│ │ │ │ +00067130: 7220 525f 6320 7768 6572 6520 6320 3d0a  r R_c where c =.
│ │ │ │ +00067140: 2020 2020 2020 2020 6c65 6e67 7468 2052          length R
│ │ │ │ +00067150: 202d 2031 2e0a 2020 2a20 4f75 7470 7574   - 1..  * Output
│ │ │ │ +00067160: 733a 0a20 2020 2020 202a 204c 2c20 6120  s:.      * L, a 
│ │ │ │ +00067170: 2a6e 6f74 6520 6c69 7374 3a20 284d 6163  *note list: (Mac
│ │ │ │ +00067180: 6175 6c61 7932 446f 6329 4c69 7374 2c2c  aulay2Doc)List,,
│ │ │ │ +00067190: 204c 6973 7420 6f66 2070 6169 7273 207b   List of pairs {
│ │ │ │ +000671a0: 7265 6775 6c61 7269 7479 0a20 2020 2020  regularity.     
│ │ │ │ +000671b0: 2020 2065 7665 6e45 7874 4d6f 6475 6c65     evenExtModule
│ │ │ │ +000671c0: 204d 5f69 2c20 7265 6775 6c61 7269 7479   M_i, regularity
│ │ │ │ +000671d0: 206f 6464 4578 744d 6f64 756c 6520 4d5f   oddExtModule M_
│ │ │ │ +000671e0: 6929 0a0a 4465 7363 7269 7074 696f 6e0a  i)..Description.
│ │ │ │ +000671f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a43 6f6d  ===========..Com
│ │ │ │ +00067200: 7075 7465 7320 7468 6520 6e6f 6e2d 6672  putes the non-fr
│ │ │ │ +00067210: 6565 2070 6172 7473 204d 5f69 206f 6620  ee parts M_i of 
│ │ │ │ +00067220: 7468 6520 4d43 4d20 6170 7072 6f78 696d  the MCM approxim
│ │ │ │ +00067230: 6174 696f 6e20 746f 204d 206f 7665 7220  ation to M over 
│ │ │ │ +00067240: 525f 692c 0a73 746f 7070 696e 6720 7768  R_i,.stopping wh
│ │ │ │ +00067250: 656e 204d 5f69 2062 6563 6f6d 6573 2066  en M_i becomes f
│ │ │ │ +00067260: 7265 652c 2061 6e64 2072 6574 7572 6e73  ree, and returns
│ │ │ │ +00067270: 2074 6865 206c 6973 7420 7768 6f73 6520   the list whose 
│ │ │ │ +00067280: 656c 656d 656e 7473 2061 7265 2074 6865  elements are the
│ │ │ │ +00067290: 0a70 6169 7273 206f 6620 7265 6775 6c61  .pairs of regula
│ │ │ │ +000672a0: 7269 7469 6573 2c20 7374 6172 7469 6e67  rities, starting
│ │ │ │ +000672b0: 2077 6974 6820 4d5f 7b28 632d 3129 7d20   with M_{(c-1)} 
│ │ │ │ +000672c0: 4e6f 7465 2074 6861 7420 7468 6520 6669  Note that the fi
│ │ │ │ +000672d0: 7273 7420 7061 6972 2069 7320 666f 720a  rst pair is for.
│ │ │ │ +000672e0: 7468 650a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  the..+----------
│ │ │ │ +000672f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00067300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00067310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00067330: 0a7c 6931 203a 2063 203d 2033 3b64 3d32  .|i1 : c = 3;d=2
│ │ │ │ +00067320: 2d2d 2d2d 2b0a 7c69 3120 3a20 6320 3d20  ----+.|i1 : c = 
│ │ │ │ +00067330: 333b 643d 3220 2020 2020 2020 2020 2020  3;d=2           
│ │ │ │  00067340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067370: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00067360: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00067370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000673a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000673b0: 207c 0a7c 6f32 203d 2032 2020 2020 2020   |.|o2 = 2      
│ │ │ │ +000673a0: 2020 2020 2020 7c0a 7c6f 3220 3d20 3220        |.|o2 = 2 
│ │ │ │ +000673b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000673c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000673d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000673e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000673f0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +000673e0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +000673f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00067400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00067410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067430: 2d2d 2d2b 0a7c 6933 203a 2052 203d 2073  ---+.|i3 : R = s
│ │ │ │ -00067440: 6574 7570 5269 6e67 7328 632c 6429 3b20  etupRings(c,d); 
│ │ │ │ +00067420: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │ +00067430: 5220 3d20 7365 7475 7052 696e 6773 2863  R = setupRings(c
│ │ │ │ +00067440: 2c64 293b 2020 2020 2020 2020 2020 2020  ,d);            
│ │ │ │  00067450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067470: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00067460: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00067470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00067480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00067490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000674a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000674b0: 2d2d 2d2d 2d2b 0a7c 6934 203a 2052 6320  -----+.|i4 : Rc 
│ │ │ │ -000674c0: 3d20 525f 6320 2020 2020 2020 2020 2020  = R_c           
│ │ │ │ +000674a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420  ----------+.|i4 
│ │ │ │ +000674b0: 3a20 5263 203d 2052 5f63 2020 2020 2020  : Rc = R_c      
│ │ │ │ +000674c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000674d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000674e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000674f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000674e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000674f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067530: 2020 2020 2020 207c 0a7c 6f34 203d 2052         |.|o4 = R
│ │ │ │ -00067540: 6320 2020 2020 2020 2020 2020 2020 2020  c               
│ │ │ │ +00067520: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00067530: 3420 3d20 5263 2020 2020 2020 2020 2020  4 = Rc          
│ │ │ │ +00067540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067570: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00067560: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00067570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000675a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000675b0: 2020 2020 2020 2020 207c 0a7c 6f34 203a           |.|o4 :
│ │ │ │ -000675c0: 2051 756f 7469 656e 7452 696e 6720 2020   QuotientRing   
│ │ │ │ +000675a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000675b0: 7c6f 3420 3a20 5175 6f74 6965 6e74 5269  |o4 : QuotientRi
│ │ │ │ +000675c0: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │  000675d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000675e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000675f0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +000675e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000675f0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00067600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00067610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00067620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935  -----------+.|i5
│ │ │ │ -00067640: 203a 204d 203d 2063 6f6b 6572 206d 6174   : M = coker mat
│ │ │ │ -00067650: 7269 787b 7b52 635f 302c 5263 5f31 2c52  rix{{Rc_0,Rc_1,R
│ │ │ │ -00067660: 635f 327d 2c7b 5263 5f31 2c52 635f 322c  c_2},{Rc_1,Rc_2,
│ │ │ │ -00067670: 5263 5f30 7d7d 2020 2020 2020 7c0a 7c20  Rc_0}}      |.| 
│ │ │ │ +00067630: 2b0a 7c69 3520 3a20 4d20 3d20 636f 6b65  +.|i5 : M = coke
│ │ │ │ +00067640: 7220 6d61 7472 6978 7b7b 5263 5f30 2c52  r matrix{{Rc_0,R
│ │ │ │ +00067650: 635f 312c 5263 5f32 7d2c 7b52 635f 312c  c_1,Rc_2},{Rc_1,
│ │ │ │ +00067660: 5263 5f32 2c52 635f 307d 7d20 2020 2020  Rc_2,Rc_0}}     
│ │ │ │ +00067670: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00067680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000676a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000676b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000676c0: 6f35 203d 2063 6f6b 6572 6e65 6c20 7c20  o5 = cokernel | 
│ │ │ │ -000676d0: 785f 3020 785f 3120 785f 3220 7c20 2020  x_0 x_1 x_2 |   
│ │ │ │ +000676b0: 2020 7c0a 7c6f 3520 3d20 636f 6b65 726e    |.|o5 = cokern
│ │ │ │ +000676c0: 656c 207c 2078 5f30 2078 5f31 2078 5f32  el | x_0 x_1 x_2
│ │ │ │ +000676d0: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │  000676e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000676f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00067700: 7c20 2020 2020 2020 2020 2020 2020 207c  |              |
│ │ │ │ -00067710: 2078 5f31 2078 5f32 2078 5f30 207c 2020   x_1 x_2 x_0 |  
│ │ │ │ +000676f0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00067700: 2020 2020 7c20 785f 3120 785f 3220 785f      | x_1 x_2 x_
│ │ │ │ +00067710: 3020 7c20 2020 2020 2020 2020 2020 2020  0 |             
│ │ │ │  00067720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067730: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00067740: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00067730: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00067740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067780: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00067790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000677a0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -000677b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000677c0: 207c 0a7c 6f35 203a 2052 632d 6d6f 6475   |.|o5 : Rc-modu
│ │ │ │ -000677d0: 6c65 2c20 7175 6f74 6965 6e74 206f 6620  le, quotient of 
│ │ │ │ -000677e0: 5263 2020 2020 2020 2020 2020 2020 2020  Rc              
│ │ │ │ -000677f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067800: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00067770: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00067780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00067790: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +000677a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000677b0: 2020 2020 2020 7c0a 7c6f 3520 3a20 5263        |.|o5 : Rc
│ │ │ │ +000677c0: 2d6d 6f64 756c 652c 2071 756f 7469 656e  -module, quotien
│ │ │ │ +000677d0: 7420 6f66 2052 6320 2020 2020 2020 2020  t of Rc         
│ │ │ │ +000677e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000677f0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00067800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00067810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00067820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067840: 2d2d 2d2b 0a7c 6936 203a 2072 6567 756c  ---+.|i6 : regul
│ │ │ │ -00067850: 6172 6974 7953 6571 7565 6e63 6528 522c  aritySequence(R,
│ │ │ │ -00067860: 4d29 2020 2020 2020 2020 2020 2020 2020  M)              
│ │ │ │ -00067870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067880: 2020 2020 7c0a 7c72 6567 2065 7665 6e20      |.|reg even 
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│ │ │ │ -000678c0: 6420 6578 747c 0a7c 2020 2020 2020 2020  d ext|.|        
│ │ │ │ +00067830: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20  --------+.|i6 : 
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│ │ │ │ +00067850: 6365 2852 2c4d 2920 2020 2020 2020 2020  ce(R,M)         
│ │ │ │ +00067860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00067870: 2020 2020 2020 2020 207c 0a7c 7265 6720           |.|reg 
│ │ │ │ +00067880: 6576 656e 2065 7874 2c20 736f 6320 6465  even ext, soc de
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│ │ │ │ +000678a0: 206f 6464 2065 7874 2c20 736f 6320 6465   odd ext, soc de
│ │ │ │ +000678b0: 6773 206f 6464 2065 7874 7c0a 7c20 2020  gs odd ext|.|   
│ │ │ │ +000678c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000678d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000678e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000678f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067900: 2020 2020 2020 7c0a 7c7b 332c 207b 312c        |.|{3, {1,
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│ │ │ │ -00067930: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00067970: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000678f0: 2020 2020 2020 2020 2020 207c 0a7c 7b33             |.|{3
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│ │ │ │ +00067920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00067930: 2020 2020 2020 2020 2020 2020 7c0a 7c7b              |.|{
│ │ │ │ +00067940: 322c 207b 302c 2030 2c20 302c 2031 7d2c  2, {0, 0, 0, 1},
│ │ │ │ +00067950: 2032 2c20 7b30 2c20 302c 2030 7d7d 2020   2, {0, 0, 0}}  
│ │ │ │ +00067960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00067970: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00067980: 7b30 2c20 7b7d 2c20 302c 207b 7d7d 2020  {0, {}, 0, {}}  
│ │ │ │ +00067990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000679a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000679b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000679c0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +000679b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000679c0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  000679d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000679e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000679f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00067b20: 204d 434d 2061 7070 726f 7869 6d61 7469   MCM approximati
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│ │ │ │ -00067b40: 7265 6775 6c61 7269 7479 5365 7175 656e  regularitySequen
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│ │ │ │ -00067b60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ -00067bb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ +00067a00: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d  ..See also.=====
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│ │ │ │ +00067a50: 2d2d 2072 6574 7572 6e73 2070 6169 7220  -- returns pair 
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│ │ │ │ +00067aa0: 6520 6175 736c 616e 6465 7249 6e76 6172  e auslanderInvar
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│ │ │ │ +00067b50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ +00067b80: 6528 4c69 7374 2c4d 6f64 756c 6529 220a  e(List,Module)".
│ │ │ │ +00067b90: 0a46 6f72 2074 6865 2070 726f 6772 616d  .For the program
│ │ │ │ +00067ba0: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  mer.============
│ │ │ │ +00067bb0: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65  ======..The obje
│ │ │ │ +00067bc0: 6374 202a 6e6f 7465 2072 6567 756c 6172  ct *note regular
│ │ │ │ +00067bd0: 6974 7953 6571 7565 6e63 653a 2072 6567  itySequence: reg
│ │ │ │ +00067be0: 756c 6172 6974 7953 6571 7565 6e63 652c  ularitySequence,
│ │ │ │ +00067bf0: 2069 7320 6120 2a6e 6f74 6520 6d65 7468   is a *note meth
│ │ │ │ +00067c00: 6f64 0a66 756e 6374 696f 6e3a 2028 4d61  od.function: (Ma
│ │ │ │ +00067c10: 6361 756c 6179 3244 6f63 294d 6574 686f  caulay2Doc)Metho
│ │ │ │ +00067c20: 6446 756e 6374 696f 6e2c 2e0a 0a2d 2d2d  dFunction,...---
│ │ │ │ +00067c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00067c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00067c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00067c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067c80: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75  -------..The sou
│ │ │ │ -00067c90: 7263 6520 6f66 2074 6869 7320 646f 6375  rce of this docu
│ │ │ │ -00067ca0: 6d65 6e74 2069 7320 696e 0a2f 6275 696c  ment is in./buil
│ │ │ │ -00067cb0: 642f 7265 7072 6f64 7563 6962 6c65 2d70  d/reproducible-p
│ │ │ │ -00067cc0: 6174 682f 6d61 6361 756c 6179 322d 312e  ath/macaulay2-1.
│ │ │ │ -00067cd0: 3235 2e31 312b 6473 2f4d 322f 4d61 6361  25.11+ds/M2/Maca
│ │ │ │ -00067ce0: 756c 6179 322f 7061 636b 6167 6573 2f0a  ulay2/packages/.
│ │ │ │ -00067cf0: 436f 6d70 6c65 7465 496e 7465 7273 6563  CompleteIntersec
│ │ │ │ -00067d00: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e  tionResolutions.
│ │ │ │ -00067d10: 6d32 3a32 3630 303a 302e 0a1f 0a46 696c  m2:2600:0....Fil
│ │ │ │ -00067d20: 653a 2043 6f6d 706c 6574 6549 6e74 6572  e: CompleteInter
│ │ │ │ -00067d30: 7365 6374 696f 6e52 6573 6f6c 7574 696f  sectionResolutio
│ │ │ │ -00067d40: 6e73 2e69 6e66 6f2c 204e 6f64 653a 2053  ns.info, Node: S
│ │ │ │ -00067d50: 322c 204e 6578 743a 2053 6861 6d61 7368  2, Next: Shamash
│ │ │ │ -00067d60: 2c20 5072 6576 3a20 7265 6775 6c61 7269  , Prev: regulari
│ │ │ │ -00067d70: 7479 5365 7175 656e 6365 2c20 5570 3a20  tySequence, Up: 
│ │ │ │ -00067d80: 546f 700a 0a53 3220 2d2d 2055 6e69 7665  Top..S2 -- Unive
│ │ │ │ -00067d90: 7273 616c 206d 6170 2074 6f20 6120 6d6f  rsal map to a mo
│ │ │ │ -00067da0: 6475 6c65 2073 6174 6973 6679 696e 6720  dule satisfying 
│ │ │ │ -00067db0: 5365 7272 6527 7320 636f 6e64 6974 696f  Serre's conditio
│ │ │ │ -00067dc0: 6e20 5332 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a  n S2.***********
│ │ │ │ +00067c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468  ------------..Th
│ │ │ │ +00067c80: 6520 736f 7572 6365 206f 6620 7468 6973  e source of this
│ │ │ │ +00067c90: 2064 6f63 756d 656e 7420 6973 2069 6e0a   document is in.
│ │ │ │ +00067ca0: 2f62 7569 6c64 2f72 6570 726f 6475 6369  /build/reproduci
│ │ │ │ +00067cb0: 626c 652d 7061 7468 2f6d 6163 6175 6c61  ble-path/macaula
│ │ │ │ +00067cc0: 7932 2d31 2e32 352e 3131 2b64 732f 4d32  y2-1.25.11+ds/M2
│ │ │ │ +00067cd0: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61  /Macaulay2/packa
│ │ │ │ +00067ce0: 6765 732f 0a43 6f6d 706c 6574 6549 6e74  ges/.CompleteInt
│ │ │ │ +00067cf0: 6572 7365 6374 696f 6e52 6573 6f6c 7574  ersectionResolut
│ │ │ │ +00067d00: 696f 6e73 2e6d 323a 3236 3030 3a30 2e0a  ions.m2:2600:0..
│ │ │ │ +00067d10: 1f0a 4669 6c65 3a20 436f 6d70 6c65 7465  ..File: Complete
│ │ │ │ +00067d20: 496e 7465 7273 6563 7469 6f6e 5265 736f  IntersectionReso
│ │ │ │ +00067d30: 6c75 7469 6f6e 732e 696e 666f 2c20 4e6f  lutions.info, No
│ │ │ │ +00067d40: 6465 3a20 5332 2c20 4e65 7874 3a20 5368  de: S2, Next: Sh
│ │ │ │ +00067d50: 616d 6173 682c 2050 7265 763a 2072 6567  amash, Prev: reg
│ │ │ │ +00067d60: 756c 6172 6974 7953 6571 7565 6e63 652c  ularitySequence,
│ │ │ │ +00067d70: 2055 703a 2054 6f70 0a0a 5332 202d 2d20   Up: Top..S2 -- 
│ │ │ │ +00067d80: 556e 6976 6572 7361 6c20 6d61 7020 746f  Universal map to
│ │ │ │ +00067d90: 2061 206d 6f64 756c 6520 7361 7469 7366   a module satisf
│ │ │ │ +00067da0: 7969 6e67 2053 6572 7265 2773 2063 6f6e  ying Serre's con
│ │ │ │ +00067db0: 6469 7469 6f6e 2053 320a 2a2a 2a2a 2a2a  dition S2.******
│ │ │ │ +00067dc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00067dd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00067de0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00067df0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00067e00: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a  ****..  * Usage:
│ │ │ │ -00067e10: 200a 2020 2020 2020 2020 6620 3d20 5332   .        f = S2
│ │ │ │ -00067e20: 2862 2c4d 290a 2020 2a20 496e 7075 7473  (b,M).  * Inputs
│ │ │ │ -00067e30: 3a0a 2020 2020 2020 2a20 622c 2061 6e20  :.      * b, an 
│ │ │ │ -00067e40: 2a6e 6f74 6520 696e 7465 6765 723a 2028  *note integer: (
│ │ │ │ -00067e50: 4d61 6361 756c 6179 3244 6f63 295a 5a2c  Macaulay2Doc)ZZ,
│ │ │ │ -00067e60: 2c20 6465 6772 6565 2062 6f75 6e64 2074  , degree bound t
│ │ │ │ -00067e70: 6f20 7768 6963 6820 746f 2063 6172 7279  o which to carry
│ │ │ │ -00067e80: 0a20 2020 2020 2020 2074 6865 2063 6f6d  .        the com
│ │ │ │ -00067e90: 7075 7461 7469 6f6e 0a20 2020 2020 202a  putation.      *
│ │ │ │ -00067ea0: 204d 2c20 6120 2a6e 6f74 6520 6d6f 6475   M, a *note modu
│ │ │ │ -00067eb0: 6c65 3a20 284d 6163 6175 6c61 7932 446f  le: (Macaulay2Do
│ │ │ │ -00067ec0: 6329 4d6f 6475 6c65 2c2c 200a 2020 2a20  c)Module,, .  * 
│ │ │ │ -00067ed0: 4f75 7470 7574 733a 0a20 2020 2020 202a  Outputs:.      *
│ │ │ │ -00067ee0: 2066 2c20 6120 2a6e 6f74 6520 6d61 7472   f, a *note matr
│ │ │ │ -00067ef0: 6978 3a20 284d 6163 6175 6c61 7932 446f  ix: (Macaulay2Do
│ │ │ │ -00067f00: 6329 4d61 7472 6978 2c2c 2064 6566 696e  c)Matrix,, defin
│ │ │ │ -00067f10: 696e 6720 6120 6d61 7020 4d2d 2d3e 4d27  ing a map M-->M'
│ │ │ │ -00067f20: 2074 6861 740a 2020 2020 2020 2020 6167   that.        ag
│ │ │ │ -00067f30: 7265 6573 2077 6974 6820 7468 6520 5332  rees with the S2
│ │ │ │ -00067f40: 2d69 6669 6361 7469 6f6e 206f 6620 4d20  -ification of M 
│ │ │ │ -00067f50: 696e 2064 6567 7265 6573 2024 5c67 6571  in degrees $\geq
│ │ │ │ -00067f60: 2062 240a 0a44 6573 6372 6970 7469 6f6e   b$..Description
│ │ │ │ -00067f70: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 4966  .===========..If
│ │ │ │ -00067f80: 204d 2069 7320 6120 6772 6164 6564 206d   M is a graded m
│ │ │ │ -00067f90: 6f64 756c 6520 6f76 6572 2061 2072 696e  odule over a rin
│ │ │ │ -00067fa0: 6720 532c 2074 6865 6e20 7468 6520 5332  g S, then the S2
│ │ │ │ -00067fb0: 2d69 6669 6361 7469 6f6e 206f 6620 4d20  -ification of M 
│ │ │ │ -00067fc0: 6973 205c 7375 6d5f 7b64 0a5c 696e 205a  is \sum_{d.\in Z
│ │ │ │ -00067fd0: 5a7d 2048 5e30 2828 7368 6561 6620 4d29  Z} H^0((sheaf M)
│ │ │ │ -00067fe0: 2864 2929 2c20 7768 6963 6820 6d61 7920  (d)), which may 
│ │ │ │ -00067ff0: 6265 2063 6f6d 7075 7465 6420 6173 206c  be computed as l
│ │ │ │ -00068000: 696d 5f7b 642d 3e5c 696e 6674 797d 2048  im_{d->\infty} H
│ │ │ │ -00068010: 6f6d 2849 5f64 2c4d 292c 0a77 6865 7265  om(I_d,M),.where
│ │ │ │ -00068020: 2049 5f64 2069 7320 616e 7920 7365 7175   I_d is any sequ
│ │ │ │ -00068030: 656e 6365 206f 6620 6964 6561 6c73 2063  ence of ideals c
│ │ │ │ -00068040: 6f6e 7461 696e 6564 2069 6e20 6869 6768  ontained in high
│ │ │ │ -00068050: 6572 2061 6e64 2068 6967 6865 7220 706f  er and higher po
│ │ │ │ -00068060: 7765 7273 206f 660a 535f 2b2e 2054 6865  wers of.S_+. The
│ │ │ │ -00068070: 7265 2069 7320 6120 6e61 7475 7261 6c20  re is a natural 
│ │ │ │ -00068080: 7265 7374 7269 6374 696f 6e20 6d61 7020  restriction map 
│ │ │ │ -00068090: 663a 204d 203d 2048 6f6d 2853 2c4d 2920  f: M = Hom(S,M) 
│ │ │ │ -000680a0: 5c74 6f20 486f 6d28 495f 642c 4d29 2e20  \to Hom(I_d,M). 
│ │ │ │ -000680b0: 5765 0a63 6f6d 7075 7465 2061 6c6c 2074  We.compute all t
│ │ │ │ -000680c0: 6869 7320 7573 696e 6720 7468 6520 6964  his using the id
│ │ │ │ -000680d0: 6561 6c73 2049 5f64 2067 656e 6572 6174  eals I_d generat
│ │ │ │ -000680e0: 6564 2062 7920 7468 6520 642d 7468 2070  ed by the d-th p
│ │ │ │ -000680f0: 6f77 6572 7320 6f66 2074 6865 0a76 6172  owers of the.var
│ │ │ │ -00068100: 6961 626c 6573 2069 6e20 532e 0a0a 5369  iables in S...Si
│ │ │ │ -00068110: 6e63 6520 7468 6520 7265 7375 6c74 206d  nce the result m
│ │ │ │ -00068120: 6179 206e 6f74 2062 6520 6669 6e69 7465  ay not be finite
│ │ │ │ -00068130: 6c79 2067 656e 6572 6174 6564 2028 7468  ly generated (th
│ │ │ │ -00068140: 6973 2068 6170 7065 6e73 2069 6620 616e  is happens if an
│ │ │ │ -00068150: 6420 6f6e 6c79 2069 6620 4d0a 6861 7320  d only if M.has 
│ │ │ │ -00068160: 616e 2061 7373 6f63 6961 7465 6420 7072  an associated pr
│ │ │ │ -00068170: 696d 6520 6f66 2064 696d 656e 7369 6f6e  ime of dimension
│ │ │ │ -00068180: 2031 292c 2077 6520 636f 6d70 7574 6520   1), we compute 
│ │ │ │ -00068190: 6f6e 6c79 2075 7020 746f 2061 2073 7065  only up to a spe
│ │ │ │ -000681a0: 6369 6669 6564 0a64 6567 7265 6520 626f  cified.degree bo
│ │ │ │ -000681b0: 756e 6420 622e 2046 6f72 2074 6865 2072  und b. For the r
│ │ │ │ -000681c0: 6573 756c 7420 746f 2062 6520 636f 7272  esult to be corr
│ │ │ │ -000681d0: 6563 7420 646f 776e 2074 6f20 6465 6772  ect down to degr
│ │ │ │ -000681e0: 6565 2062 2c20 6974 2069 7320 7375 6666  ee b, it is suff
│ │ │ │ -000681f0: 6963 6965 6e74 0a74 6f20 636f 6d70 7574  icient.to comput
│ │ │ │ -00068200: 6520 486f 6d28 492c 4d29 2077 6865 7265  e Hom(I,M) where
│ │ │ │ -00068210: 2049 205c 7375 6273 6574 2028 535f 2b29   I \subset (S_+)
│ │ │ │ -00068220: 5e7b 722d 627d 2e0a 0a2b 2d2d 2d2d 2d2d  ^{r-b}...+------
│ │ │ │ +00067df0: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055  *********..  * U
│ │ │ │ +00067e00: 7361 6765 3a20 0a20 2020 2020 2020 2066  sage: .        f
│ │ │ │ +00067e10: 203d 2053 3228 622c 4d29 0a20 202a 2049   = S2(b,M).  * I
│ │ │ │ +00067e20: 6e70 7574 733a 0a20 2020 2020 202a 2062  nputs:.      * b
│ │ │ │ +00067e30: 2c20 616e 202a 6e6f 7465 2069 6e74 6567  , an *note integ
│ │ │ │ +00067e40: 6572 3a20 284d 6163 6175 6c61 7932 446f  er: (Macaulay2Do
│ │ │ │ +00067e50: 6329 5a5a 2c2c 2064 6567 7265 6520 626f  c)ZZ,, degree bo
│ │ │ │ +00067e60: 756e 6420 746f 2077 6869 6368 2074 6f20  und to which to 
│ │ │ │ +00067e70: 6361 7272 790a 2020 2020 2020 2020 7468  carry.        th
│ │ │ │ +00067e80: 6520 636f 6d70 7574 6174 696f 6e0a 2020  e computation.  
│ │ │ │ +00067e90: 2020 2020 2a20 4d2c 2061 202a 6e6f 7465      * M, a *note
│ │ │ │ +00067ea0: 206d 6f64 756c 653a 2028 4d61 6361 756c   module: (Macaul
│ │ │ │ +00067eb0: 6179 3244 6f63 294d 6f64 756c 652c 2c20  ay2Doc)Module,, 
│ │ │ │ +00067ec0: 0a20 202a 204f 7574 7075 7473 3a0a 2020  .  * Outputs:.  
│ │ │ │ +00067ed0: 2020 2020 2a20 662c 2061 202a 6e6f 7465      * f, a *note
│ │ │ │ +00067ee0: 206d 6174 7269 783a 2028 4d61 6361 756c   matrix: (Macaul
│ │ │ │ +00067ef0: 6179 3244 6f63 294d 6174 7269 782c 2c20  ay2Doc)Matrix,, 
│ │ │ │ +00067f00: 6465 6669 6e69 6e67 2061 206d 6170 204d  defining a map M
│ │ │ │ +00067f10: 2d2d 3e4d 2720 7468 6174 0a20 2020 2020  -->M' that.     
│ │ │ │ +00067f20: 2020 2061 6772 6565 7320 7769 7468 2074     agrees with t
│ │ │ │ +00067f30: 6865 2053 322d 6966 6963 6174 696f 6e20  he S2-ification 
│ │ │ │ +00067f40: 6f66 204d 2069 6e20 6465 6772 6565 7320  of M in degrees 
│ │ │ │ +00067f50: 245c 6765 7120 6224 0a0a 4465 7363 7269  $\geq b$..Descri
│ │ │ │ +00067f60: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ +00067f70: 3d0a 0a49 6620 4d20 6973 2061 2067 7261  =..If M is a gra
│ │ │ │ +00067f80: 6465 6420 6d6f 6475 6c65 206f 7665 7220  ded module over 
│ │ │ │ +00067f90: 6120 7269 6e67 2053 2c20 7468 656e 2074  a ring S, then t
│ │ │ │ +00067fa0: 6865 2053 322d 6966 6963 6174 696f 6e20  he S2-ification 
│ │ │ │ +00067fb0: 6f66 204d 2069 7320 5c73 756d 5f7b 640a  of M is \sum_{d.
│ │ │ │ +00067fc0: 5c69 6e20 5a5a 7d20 485e 3028 2873 6865  \in ZZ} H^0((she
│ │ │ │ +00067fd0: 6166 204d 2928 6429 292c 2077 6869 6368  af M)(d)), which
│ │ │ │ +00067fe0: 206d 6179 2062 6520 636f 6d70 7574 6564   may be computed
│ │ │ │ +00067ff0: 2061 7320 6c69 6d5f 7b64 2d3e 5c69 6e66   as lim_{d->\inf
│ │ │ │ +00068000: 7479 7d20 486f 6d28 495f 642c 4d29 2c0a  ty} Hom(I_d,M),.
│ │ │ │ +00068010: 7768 6572 6520 495f 6420 6973 2061 6e79  where I_d is any
│ │ │ │ +00068020: 2073 6571 7565 6e63 6520 6f66 2069 6465   sequence of ide
│ │ │ │ +00068030: 616c 7320 636f 6e74 6169 6e65 6420 696e  als contained in
│ │ │ │ +00068040: 2068 6967 6865 7220 616e 6420 6869 6768   higher and high
│ │ │ │ +00068050: 6572 2070 6f77 6572 7320 6f66 0a53 5f2b  er powers of.S_+
│ │ │ │ +00068060: 2e20 5468 6572 6520 6973 2061 206e 6174  . There is a nat
│ │ │ │ +00068070: 7572 616c 2072 6573 7472 6963 7469 6f6e  ural restriction
│ │ │ │ +00068080: 206d 6170 2066 3a20 4d20 3d20 486f 6d28   map f: M = Hom(
│ │ │ │ +00068090: 532c 4d29 205c 746f 2048 6f6d 2849 5f64  S,M) \to Hom(I_d
│ │ │ │ +000680a0: 2c4d 292e 2057 650a 636f 6d70 7574 6520  ,M). We.compute 
│ │ │ │ +000680b0: 616c 6c20 7468 6973 2075 7369 6e67 2074  all this using t
│ │ │ │ +000680c0: 6865 2069 6465 616c 7320 495f 6420 6765  he ideals I_d ge
│ │ │ │ +000680d0: 6e65 7261 7465 6420 6279 2074 6865 2064  nerated by the d
│ │ │ │ +000680e0: 2d74 6820 706f 7765 7273 206f 6620 7468  -th powers of th
│ │ │ │ +000680f0: 650a 7661 7269 6162 6c65 7320 696e 2053  e.variables in S
│ │ │ │ +00068100: 2e0a 0a53 696e 6365 2074 6865 2072 6573  ...Since the res
│ │ │ │ +00068110: 756c 7420 6d61 7920 6e6f 7420 6265 2066  ult may not be f
│ │ │ │ +00068120: 696e 6974 656c 7920 6765 6e65 7261 7465  initely generate
│ │ │ │ +00068130: 6420 2874 6869 7320 6861 7070 656e 7320  d (this happens 
│ │ │ │ +00068140: 6966 2061 6e64 206f 6e6c 7920 6966 204d  if and only if M
│ │ │ │ +00068150: 0a68 6173 2061 6e20 6173 736f 6369 6174  .has an associat
│ │ │ │ +00068160: 6564 2070 7269 6d65 206f 6620 6469 6d65  ed prime of dime
│ │ │ │ +00068170: 6e73 696f 6e20 3129 2c20 7765 2063 6f6d  nsion 1), we com
│ │ │ │ +00068180: 7075 7465 206f 6e6c 7920 7570 2074 6f20  pute only up to 
│ │ │ │ +00068190: 6120 7370 6563 6966 6965 640a 6465 6772  a specified.degr
│ │ │ │ +000681a0: 6565 2062 6f75 6e64 2062 2e20 466f 7220  ee bound b. For 
│ │ │ │ +000681b0: 7468 6520 7265 7375 6c74 2074 6f20 6265  the result to be
│ │ │ │ +000681c0: 2063 6f72 7265 6374 2064 6f77 6e20 746f   correct down to
│ │ │ │ +000681d0: 2064 6567 7265 6520 622c 2069 7420 6973   degree b, it is
│ │ │ │ +000681e0: 2073 7566 6669 6369 656e 740a 746f 2063   sufficient.to c
│ │ │ │ +000681f0: 6f6d 7075 7465 2048 6f6d 2849 2c4d 2920  ompute Hom(I,M) 
│ │ │ │ +00068200: 7768 6572 6520 4920 5c73 7562 7365 7420  where I \subset 
│ │ │ │ +00068210: 2853 5f2b 295e 7b72 2d62 7d2e 0a0a 2b2d  (S_+)^{r-b}...+-
│ │ │ │ +00068220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00068230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00068240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00068250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00068260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00068270: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 206b  -------+.|i1 : k
│ │ │ │ -00068280: 6b3d 5a5a 2f31 3031 2020 2020 2020 2020  k=ZZ/101        
│ │ │ │ +00068260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00068270: 3120 3a20 6b6b 3d5a 5a2f 3130 3120 2020  1 : kk=ZZ/101   
│ │ │ │ +00068280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000682a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000682b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000682c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000682b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000682c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000682d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000682e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000682f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068310: 2020 2020 2020 207c 0a7c 6f31 203d 206b         |.|o1 = k
│ │ │ │ -00068320: 6b20 2020 2020 2020 2020 2020 2020 2020  k               
│ │ │ │ +00068300: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00068310: 3120 3d20 6b6b 2020 2020 2020 2020 2020  1 = kk          
│ │ │ │ +00068320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068360: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00068350: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00068360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068370: 2020 2020 2020 2020 2020 2020 2020 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 207c 0a7c 6f31 203a 2051         |.|o1 : Q
│ │ │ │ -000683c0: 756f 7469 656e 7452 696e 6720 2020 2020  uotientRing     
│ │ │ │ +000683a0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +000683b0: 3120 3a20 5175 6f74 6965 6e74 5269 6e67  1 : QuotientRing
│ │ │ │ +000683c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000683d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000683e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000683f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068400: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +000683f0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00068400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00068410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00068420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00068430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00068440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00068450: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2053  -------+.|i2 : S
│ │ │ │ -00068460: 203d 206b 6b5b 612c 622c 632c 645d 2020   = kk[a,b,c,d]  
│ │ │ │ +00068440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00068450: 3220 3a20 5320 3d20 6b6b 5b61 2c62 2c63  2 : S = kk[a,b,c
│ │ │ │ +00068460: 2c64 5d20 2020 2020 2020 2020 2020 2020  ,d]             
│ │ │ │  00068470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000684a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00068490: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000684a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000684b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000684c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000684d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000684e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000684f0: 2020 2020 2020 207c 0a7c 6f32 203d 2053         |.|o2 = S
│ │ │ │ +000684e0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +000684f0: 3220 3d20 5320 2020 2020 2020 2020 2020  2 = S           
│ │ │ │  00068500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068540: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00068530: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00068540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068590: 2020 2020 2020 207c 0a7c 6f32 203a 2050         |.|o2 : P
│ │ │ │ -000685a0: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020  olynomialRing   
│ │ │ │ +00068580: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00068590: 3220 3a20 506f 6c79 6e6f 6d69 616c 5269  2 : PolynomialRi
│ │ │ │ +000685a0: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │  000685b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000685c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000685d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000685e0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +000685d0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +000685e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000685f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00068600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00068610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00068620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00068630: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 204d  -------+.|i3 : M
│ │ │ │ -00068640: 203d 2074 7275 6e63 6174 6528 332c 535e   = truncate(3,S^
│ │ │ │ -00068650: 3129 2020 2020 2020 2020 2020 2020 2020  1)              
│ │ │ │ +00068620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00068630: 3320 3a20 4d20 3d20 7472 756e 6361 7465  3 : M = truncate
│ │ │ │ +00068640: 2833 2c53 5e31 2920 2020 2020 2020 2020  (3,S^1)         
│ │ │ │ +00068650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068680: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00068670: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00068680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000686a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000686b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000686c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000686d0: 2020 2020 2020 207c 0a7c 6f33 203d 2069         |.|o3 = i
│ │ │ │ -000686e0: 6d61 6765 207c 2064 3320 6364 3220 6264  mage | d3 cd2 bd
│ │ │ │ -000686f0: 3220 6164 3220 6332 6420 6263 6420 6163  2 ad2 c2d bcd ac
│ │ │ │ -00068700: 6420 6232 6420 6162 6420 6132 6420 6333  d b2d abd a2d c3
│ │ │ │ -00068710: 2062 6332 2061 6332 2062 3263 2061 6263   bc2 ac2 b2c abc
│ │ │ │ -00068720: 2061 3263 2062 337c 0a7c 2020 2020 2020   a2c b3|.|      
│ │ │ │ +000686c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +000686d0: 3320 3d20 696d 6167 6520 7c20 6433 2063  3 = image | d3 c
│ │ │ │ +000686e0: 6432 2062 6432 2061 6432 2063 3264 2062  d2 bd2 ad2 c2d b
│ │ │ │ +000686f0: 6364 2061 6364 2062 3264 2061 6264 2061  cd acd b2d abd a
│ │ │ │ +00068700: 3264 2063 3320 6263 3220 6163 3220 6232  2d c3 bc2 ac2 b2
│ │ │ │ +00068710: 6320 6162 6320 6132 6320 6233 7c0a 7c20  c abc a2c b3|.| 
│ │ │ │ +00068720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068770: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00068780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068790: 2020 2020 2020 2031 2020 2020 2020 2020         1        
│ │ │ │ +00068760: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00068770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068780: 2020 2020 2020 2020 2020 2020 3120 2020              1   
│ │ │ │ +00068790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000687a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000687b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000687c0: 2020 2020 2020 207c 0a7c 6f33 203a 2053         |.|o3 : S
│ │ │ │ -000687d0: 2d6d 6f64 756c 652c 2073 7562 6d6f 6475  -module, submodu
│ │ │ │ -000687e0: 6c65 206f 6620 5320 2020 2020 2020 2020  le of S         
│ │ │ │ +000687b0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +000687c0: 3320 3a20 532d 6d6f 6475 6c65 2c20 7375  3 : S-module, su
│ │ │ │ +000687d0: 626d 6f64 756c 6520 6f66 2053 2020 2020  bmodule of S    
│ │ │ │ +000687e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000687f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068810: 2020 2020 2020 207c 0a7c 2d2d 2d2d 2d2d         |.|------
│ │ │ │ +00068800: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ +00068810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00068820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00068830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00068840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00068850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00068860: 2d2d 2d2d 2d2d 2d7c 0a7c 6162 3220 6132  -------|.|ab2 a2
│ │ │ │ -00068870: 6220 6133 207c 2020 2020 2020 2020 2020  b a3 |          
│ │ │ │ +00068850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c61  ------------|.|a
│ │ │ │ +00068860: 6232 2061 3262 2061 3320 7c20 2020 2020  b2 a2b a3 |     
│ │ │ │ +00068870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000688a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000688b0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +000688a0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +000688b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000688c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000688d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000688e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000688f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00068900: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2062  -------+.|i4 : b
│ │ │ │ -00068910: 6574 7469 206d 6174 7269 7820 5332 2830  etti matrix S2(0
│ │ │ │ -00068920: 2c4d 2920 2020 2020 2020 2020 2020 2020  ,M)             
│ │ │ │ +000688f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00068900: 3420 3a20 6265 7474 6920 6d61 7472 6978  4 : betti matrix
│ │ │ │ +00068910: 2053 3228 302c 4d29 2020 2020 2020 2020   S2(0,M)        
│ │ │ │ +00068920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068950: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00068940: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00068950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000689a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000689b0: 2020 2020 2020 3020 2031 2020 2020 2020        0  1      
│ │ │ │ +00068990: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000689a0: 2020 2020 2020 2020 2020 2030 2020 3120             0  1 
│ │ │ │ +000689b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000689c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000689d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000689e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000689f0: 2020 2020 2020 207c 0a7c 6f34 203d 2074         |.|o4 = t
│ │ │ │ -00068a00: 6f74 616c 3a20 3120 3230 2020 2020 2020  otal: 1 20      
│ │ │ │ +000689e0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +000689f0: 3420 3d20 746f 7461 6c3a 2031 2032 3020  4 = total: 1 20 
│ │ │ │ +00068a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068a40: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00068a50: 2020 2030 3a20 3120 202e 2020 2020 2020     0: 1  .      
│ │ │ │ +00068a30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00068a40: 2020 2020 2020 2020 303a 2031 2020 2e20          0: 1  . 
│ │ │ │ +00068a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068a90: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00068aa0: 2020 2031 3a20 2e20 202e 2020 2020 2020     1: .  .      
│ │ │ │ +00068a80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00068a90: 2020 2020 2020 2020 313a 202e 2020 2e20          1: .  . 
│ │ │ │ +00068aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068ae0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00068af0: 2020 2032 3a20 2e20 3230 2020 2020 2020     2: . 20      
│ │ │ │ +00068ad0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00068ae0: 2020 2020 2020 2020 323a 202e 2032 3020          2: . 20 
│ │ │ │ +00068af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068b30: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00068b20: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00068b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068b80: 2020 2020 2020 207c 0a7c 6f34 203a 2042         |.|o4 : B
│ │ │ │ -00068b90: 6574 7469 5461 6c6c 7920 2020 2020 2020  ettiTally       
│ │ │ │ +00068b70: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00068b80: 3420 3a20 4265 7474 6954 616c 6c79 2020  4 : BettiTally  
│ │ │ │ +00068b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068bd0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00068bc0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00068bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00068be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00068bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00068c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00068c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00068c20: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2062  -------+.|i5 : b
│ │ │ │ -00068c30: 6574 7469 206d 6174 7269 7820 5332 2831  etti matrix S2(1
│ │ │ │ -00068c40: 2c4d 2920 2020 2020 2020 2020 2020 2020  ,M)             
│ │ │ │ +00068c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00068c20: 3520 3a20 6265 7474 6920 6d61 7472 6978  5 : betti matrix
│ │ │ │ +00068c30: 2053 3228 312c 4d29 2020 2020 2020 2020   S2(1,M)        
│ │ │ │ +00068c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068c70: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00068c60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00068c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068cc0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00068cd0: 2020 2020 2020 3020 2031 2020 2020 2020        0  1      
│ │ │ │ +00068cb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00068cc0: 2020 2020 2020 2020 2020 2030 2020 3120             0  1 
│ │ │ │ +00068cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068d10: 2020 2020 2020 207c 0a7c 6f35 203d 2074         |.|o5 = t
│ │ │ │ -00068d20: 6f74 616c 3a20 3120 3230 2020 2020 2020  otal: 1 20      
│ │ │ │ +00068d00: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00068d10: 3520 3d20 746f 7461 6c3a 2031 2032 3020  5 = total: 1 20 
│ │ │ │ +00068d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068d60: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00068d70: 2020 2030 3a20 3120 202e 2020 2020 2020     0: 1  .      
│ │ │ │ +00068d50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00068d60: 2020 2020 2020 2020 303a 2031 2020 2e20          0: 1  . 
│ │ │ │ +00068d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068db0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00068dc0: 2020 2031 3a20 2e20 202e 2020 2020 2020     1: .  .      
│ │ │ │ +00068da0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00068db0: 2020 2020 2020 2020 313a 202e 2020 2e20          1: .  . 
│ │ │ │ +00068dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068e00: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00068e10: 2020 2032 3a20 2e20 3230 2020 2020 2020     2: . 20      
│ │ │ │ +00068df0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00068e00: 2020 2020 2020 2020 323a 202e 2032 3020          2: . 20 
│ │ │ │ +00068e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068e50: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00068e40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00068e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068ea0: 2020 2020 2020 207c 0a7c 6f35 203a 2042         |.|o5 : B
│ │ │ │ -00068eb0: 6574 7469 5461 6c6c 7920 2020 2020 2020  ettiTally       
│ │ │ │ +00068e90: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00068ea0: 3520 3a20 4265 7474 6954 616c 6c79 2020  5 : BettiTally  
│ │ │ │ +00068eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068ef0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00068ee0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00068ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00068f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00068f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00068f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00068f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00068f40: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 204d  -------+.|i6 : M
│ │ │ │ -00068f50: 203d 2053 5e31 2f69 6e74 6572 7365 6374   = S^1/intersect
│ │ │ │ -00068f60: 2869 6465 616c 2261 2c62 2c63 222c 2069  (ideal"a,b,c", i
│ │ │ │ -00068f70: 6465 616c 2262 2c63 2c64 222c 6964 6561  deal"b,c,d",idea
│ │ │ │ -00068f80: 6c22 632c 642c 6122 2c69 6465 616c 2264  l"c,d,a",ideal"d
│ │ │ │ -00068f90: 2c61 2c62 2229 207c 0a7c 2020 2020 2020  ,a,b") |.|      
│ │ │ │ +00068f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00068f40: 3620 3a20 4d20 3d20 535e 312f 696e 7465  6 : M = S^1/inte
│ │ │ │ +00068f50: 7273 6563 7428 6964 6561 6c22 612c 622c  rsect(ideal"a,b,
│ │ │ │ +00068f60: 6322 2c20 6964 6561 6c22 622c 632c 6422  c", ideal"b,c,d"
│ │ │ │ +00068f70: 2c69 6465 616c 2263 2c64 2c61 222c 6964  ,ideal"c,d,a",id
│ │ │ │ +00068f80: 6561 6c22 642c 612c 6222 2920 7c0a 7c20  eal"d,a,b") |.| 
│ │ │ │ +00068f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068fa0: 2020 2020 2020 2020 2020 2020 2020 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 207c 0a7c 6f36 203d 2063         |.|o6 = c
│ │ │ │ -00068ff0: 6f6b 6572 6e65 6c20 7c20 6364 2062 6420  okernel | cd bd 
│ │ │ │ -00069000: 6164 2062 6320 6163 2061 6220 7c20 2020  ad bc ac ab |   
│ │ │ │ +00068fd0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00068fe0: 3620 3d20 636f 6b65 726e 656c 207c 2063  6 = cokernel | c
│ │ │ │ +00068ff0: 6420 6264 2061 6420 6263 2061 6320 6162  d bd ad bc ac ab
│ │ │ │ +00069000: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │  00069010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069030: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00069020: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00069030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069080: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00069090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000690a0: 2020 2020 2020 3120 2020 2020 2020 2020        1         
│ │ │ │ +00069070: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00069080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069090: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ +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 207c 0a7c 6f36 203a 2053         |.|o6 : S
│ │ │ │ -000690e0: 2d6d 6f64 756c 652c 2071 756f 7469 656e  -module, quotien
│ │ │ │ -000690f0: 7420 6f66 2053 2020 2020 2020 2020 2020  t of S          
│ │ │ │ +000690c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +000690d0: 3620 3a20 532d 6d6f 6475 6c65 2c20 7175  6 : S-module, qu
│ │ │ │ +000690e0: 6f74 6965 6e74 206f 6620 5320 2020 2020  otient of S     
│ │ │ │ +000690f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069120: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00069110: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00069120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00069130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00069140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00069150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069170: 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a 2070  -------+.|i7 : p
│ │ │ │ -00069180: 7275 6e65 2073 6f75 7263 6520 5332 2830  rune source S2(0
│ │ │ │ -00069190: 2c4d 2920 2020 2020 2020 2020 2020 2020  ,M)             
│ │ │ │ +00069160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00069170: 3720 3a20 7072 756e 6520 736f 7572 6365  7 : prune source
│ │ │ │ +00069180: 2053 3228 302c 4d29 2020 2020 2020 2020   S2(0,M)        
│ │ │ │ +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 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000691b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000691c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000691d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000691e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000691f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069210: 2020 2020 2020 207c 0a7c 6f37 203d 2063         |.|o7 = c
│ │ │ │ -00069220: 6f6b 6572 6e65 6c20 7c20 6364 2062 6420  okernel | cd bd 
│ │ │ │ -00069230: 6164 2062 6320 6163 2061 6220 7c20 2020  ad bc ac ab |   
│ │ │ │ +00069200: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00069210: 3720 3d20 636f 6b65 726e 656c 207c 2063  7 = cokernel | c
│ │ │ │ +00069220: 6420 6264 2061 6420 6263 2061 6320 6162  d bd ad bc ac ab
│ │ │ │ +00069230: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │  00069240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069260: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00069250: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00069260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069270: 2020 2020 2020 2020 2020 2020 2020 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 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000692c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000692d0: 2020 2020 2020 3120 2020 2020 2020 2020        1         
│ │ │ │ +000692a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000692b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000692c0: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ +000692d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000692e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000692f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069300: 2020 2020 2020 207c 0a7c 6f37 203a 2053         |.|o7 : S
│ │ │ │ -00069310: 2d6d 6f64 756c 652c 2071 756f 7469 656e  -module, quotien
│ │ │ │ -00069320: 7420 6f66 2053 2020 2020 2020 2020 2020  t of S          
│ │ │ │ +000692f0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00069300: 3720 3a20 532d 6d6f 6475 6c65 2c20 7175  7 : S-module, qu
│ │ │ │ +00069310: 6f74 6965 6e74 206f 6620 5320 2020 2020  otient of S     
│ │ │ │ +00069320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069350: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00069340: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00069350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00069360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00069370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00069380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000693a0: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 2070  -------+.|i8 : p
│ │ │ │ -000693b0: 7275 6e65 2074 6172 6765 7420 5332 2830  rune target S2(0
│ │ │ │ -000693c0: 2c4d 2920 2020 2020 2020 2020 2020 2020  ,M)             
│ │ │ │ +00069390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +000693a0: 3820 3a20 7072 756e 6520 7461 7267 6574  8 : prune target
│ │ │ │ +000693b0: 2053 3228 302c 4d29 2020 2020 2020 2020   S2(0,M)        
│ │ │ │ +000693c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000693d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000693e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000693f0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000693e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000693f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069440: 2020 2020 2020 207c 0a7c 6f38 203d 2063         |.|o8 = c
│ │ │ │ -00069450: 6f6b 6572 6e65 6c20 7b2d 317d 207c 2064  okernel {-1} | d
│ │ │ │ -00069460: 2063 2062 2030 2030 2030 2030 2030 2030   c b 0 0 0 0 0 0
│ │ │ │ -00069470: 2030 2030 2030 207c 2020 2020 2020 2020   0 0 0 |        
│ │ │ │ -00069480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069490: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000694a0: 2020 2020 2020 2020 7b2d 317d 207c 2030          {-1} | 0
│ │ │ │ -000694b0: 2030 2030 2064 2063 2061 2030 2030 2030   0 0 d c a 0 0 0
│ │ │ │ -000694c0: 2030 2030 2030 207c 2020 2020 2020 2020   0 0 0 |        
│ │ │ │ -000694d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000694e0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000694f0: 2020 2020 2020 2020 7b2d 317d 207c 2030          {-1} | 0
│ │ │ │ -00069500: 2030 2030 2030 2030 2030 2064 2062 2061   0 0 0 0 0 d b a
│ │ │ │ -00069510: 2030 2030 2030 207c 2020 2020 2020 2020   0 0 0 |        
│ │ │ │ -00069520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069530: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00069540: 2020 2020 2020 2020 7b2d 317d 207c 2030          {-1} | 0
│ │ │ │ -00069550: 2030 2030 2030 2030 2030 2030 2030 2030   0 0 0 0 0 0 0 0
│ │ │ │ -00069560: 2063 2062 2061 207c 2020 2020 2020 2020   c b a |        
│ │ │ │ -00069570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069580: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00069430: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00069440: 3820 3d20 636f 6b65 726e 656c 207b 2d31  8 = cokernel {-1
│ │ │ │ +00069450: 7d20 7c20 6420 6320 6220 3020 3020 3020  } | d c b 0 0 0 
│ │ │ │ +00069460: 3020 3020 3020 3020 3020 3020 7c20 2020  0 0 0 0 0 0 |   
│ │ │ │ +00069470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069480: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00069490: 2020 2020 2020 2020 2020 2020 207b 2d31               {-1
│ │ │ │ +000694a0: 7d20 7c20 3020 3020 3020 6420 6320 6120  } | 0 0 0 d c a 
│ │ │ │ +000694b0: 3020 3020 3020 3020 3020 3020 7c20 2020  0 0 0 0 0 0 |   
│ │ │ │ +000694c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000694d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000694e0: 2020 2020 2020 2020 2020 2020 207b 2d31               {-1
│ │ │ │ +000694f0: 7d20 7c20 3020 3020 3020 3020 3020 3020  } | 0 0 0 0 0 0 
│ │ │ │ +00069500: 6420 6220 6120 3020 3020 3020 7c20 2020  d b a 0 0 0 |   
│ │ │ │ +00069510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069520: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00069530: 2020 2020 2020 2020 2020 2020 207b 2d31               {-1
│ │ │ │ +00069540: 7d20 7c20 3020 3020 3020 3020 3020 3020  } | 0 0 0 0 0 0 
│ │ │ │ +00069550: 3020 3020 3020 6320 6220 6120 7c20 2020  0 0 0 c b a |   
│ │ │ │ +00069560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069570: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00069580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069590: 2020 2020 2020 2020 2020 2020 2020 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 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000695e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000695f0: 2020 2020 2020 3420 2020 2020 2020 2020        4         
│ │ │ │ +000695c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000695d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000695e0: 2020 2020 2020 2020 2020 2034 2020 2020             4    
│ │ │ │ +000695f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069620: 2020 2020 2020 207c 0a7c 6f38 203a 2053         |.|o8 : S
│ │ │ │ -00069630: 2d6d 6f64 756c 652c 2071 756f 7469 656e  -module, quotien
│ │ │ │ -00069640: 7420 6f66 2053 2020 2020 2020 2020 2020  t of S          
│ │ │ │ +00069610: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00069620: 3820 3a20 532d 6d6f 6475 6c65 2c20 7175  8 : S-module, qu
│ │ │ │ +00069630: 6f74 6965 6e74 206f 6620 5320 2020 2020  otient of S     
│ │ │ │ +00069640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069670: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00069660: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00069670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00069680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00069690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000696a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000696b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000696c0: 2d2d 2d2d 2d2d 2d2b 0a0a 4174 206f 6e65  -------+..At one
│ │ │ │ -000696d0: 2074 696d 6520 4445 2068 6f70 6564 2074   time DE hoped t
│ │ │ │ -000696e0: 6861 742c 2069 6620 4d20 7765 7265 2061  hat, if M were a
│ │ │ │ -000696f0: 206d 6f64 756c 6520 6f76 6572 2074 6865   module over the
│ │ │ │ -00069700: 2063 6f6d 706c 6574 6520 696e 7465 7273   complete inters
│ │ │ │ -00069710: 6563 7469 6f6e 2052 0a77 6974 6820 7265  ection R.with re
│ │ │ │ -00069720: 7369 6475 6520 6669 656c 6420 6b2c 2074  sidue field k, t
│ │ │ │ -00069730: 6865 6e20 7468 6520 6e61 7475 7261 6c20  hen the natural 
│ │ │ │ -00069740: 6d61 7020 6672 6f6d 2022 636f 6d70 6c65  map from "comple
│ │ │ │ -00069750: 7465 2220 4578 7420 6d6f 6475 6c65 2022  te" Ext module "
│ │ │ │ -00069760: 2877 6964 6568 6174 0a45 7874 295f 5228  (widehat.Ext)_R(
│ │ │ │ -00069770: 4d2c 6b29 2220 746f 2074 6865 2053 322d  M,k)" to the S2-
│ │ │ │ -00069780: 6966 6963 6174 696f 6e20 6f66 2045 7874  ification of Ext
│ │ │ │ -00069790: 5f52 284d 2c6b 2920 776f 756c 6420 6265  _R(M,k) would be
│ │ │ │ -000697a0: 2073 7572 6a65 6374 6976 653b 0a65 7175   surjective;.equ
│ │ │ │ -000697b0: 6976 616c 656e 746c 792c 2069 6620 4e20  ivalently, if N 
│ │ │ │ -000697c0: 7765 7265 2061 2073 7566 6669 6369 656e  were a sufficien
│ │ │ │ -000697d0: 746c 7920 6e65 6761 7469 7665 2073 797a  tly negative syz
│ │ │ │ -000697e0: 7967 7920 6f66 204d 2c20 7468 656e 2074  ygy of M, then t
│ │ │ │ -000697f0: 6865 2066 6972 7374 0a6c 6f63 616c 2063  he first.local c
│ │ │ │ -00069800: 6f68 6f6d 6f6c 6f67 7920 6d6f 6475 6c65  ohomology module
│ │ │ │ -00069810: 206f 6620 4578 745f 5228 4d2c 6b29 2077   of Ext_R(M,k) w
│ │ │ │ -00069820: 6f75 6c64 2062 6520 7a65 726f 2e20 5468  ould be zero. Th
│ │ │ │ -00069830: 6973 2069 7320 6661 6c73 652c 2061 7320  is is false, as 
│ │ │ │ -00069840: 7368 6f77 6e20 6279 0a74 6865 2066 6f6c  shown by.the fol
│ │ │ │ -00069850: 6c6f 7769 6e67 2065 7861 6d70 6c65 3a0a  lowing example:.
│ │ │ │ -00069860: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000696b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a41  ------------+..A
│ │ │ │ +000696c0: 7420 6f6e 6520 7469 6d65 2044 4520 686f  t one time DE ho
│ │ │ │ +000696d0: 7065 6420 7468 6174 2c20 6966 204d 2077  ped that, if M w
│ │ │ │ +000696e0: 6572 6520 6120 6d6f 6475 6c65 206f 7665  ere a module ove
│ │ │ │ +000696f0: 7220 7468 6520 636f 6d70 6c65 7465 2069  r the complete i
│ │ │ │ +00069700: 6e74 6572 7365 6374 696f 6e20 520a 7769  ntersection R.wi
│ │ │ │ +00069710: 7468 2072 6573 6964 7565 2066 6965 6c64  th residue field
│ │ │ │ +00069720: 206b 2c20 7468 656e 2074 6865 206e 6174   k, then the nat
│ │ │ │ +00069730: 7572 616c 206d 6170 2066 726f 6d20 2263  ural map from "c
│ │ │ │ +00069740: 6f6d 706c 6574 6522 2045 7874 206d 6f64  omplete" Ext mod
│ │ │ │ +00069750: 756c 6520 2228 7769 6465 6861 740a 4578  ule "(widehat.Ex
│ │ │ │ +00069760: 7429 5f52 284d 2c6b 2922 2074 6f20 7468  t)_R(M,k)" to th
│ │ │ │ +00069770: 6520 5332 2d69 6669 6361 7469 6f6e 206f  e S2-ification o
│ │ │ │ +00069780: 6620 4578 745f 5228 4d2c 6b29 2077 6f75  f Ext_R(M,k) wou
│ │ │ │ +00069790: 6c64 2062 6520 7375 726a 6563 7469 7665  ld be surjective
│ │ │ │ +000697a0: 3b0a 6571 7569 7661 6c65 6e74 6c79 2c20  ;.equivalently, 
│ │ │ │ +000697b0: 6966 204e 2077 6572 6520 6120 7375 6666  if N were a suff
│ │ │ │ +000697c0: 6963 6965 6e74 6c79 206e 6567 6174 6976  iciently negativ
│ │ │ │ +000697d0: 6520 7379 7a79 6779 206f 6620 4d2c 2074  e syzygy of M, t
│ │ │ │ +000697e0: 6865 6e20 7468 6520 6669 7273 740a 6c6f  hen the first.lo
│ │ │ │ +000697f0: 6361 6c20 636f 686f 6d6f 6c6f 6779 206d  cal cohomology m
│ │ │ │ +00069800: 6f64 756c 6520 6f66 2045 7874 5f52 284d  odule of Ext_R(M
│ │ │ │ +00069810: 2c6b 2920 776f 756c 6420 6265 207a 6572  ,k) would be zer
│ │ │ │ +00069820: 6f2e 2054 6869 7320 6973 2066 616c 7365  o. This is false
│ │ │ │ +00069830: 2c20 6173 2073 686f 776e 2062 790a 7468  , as shown by.th
│ │ │ │ +00069840: 6520 666f 6c6c 6f77 696e 6720 6578 616d  e following exam
│ │ │ │ +00069850: 706c 653a 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d  ple:..+---------
│ │ │ │ +00069860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00069870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069890: 2d2d 2d2b 0a7c 6939 203a 2053 203d 205a  ---+.|i9 : S = Z
│ │ │ │ -000698a0: 5a2f 3130 315b 785f 302e 2e78 5f32 5d3b  Z/101[x_0..x_2];
│ │ │ │ -000698b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000698c0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00069880: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3920 3a20  --------+.|i9 : 
│ │ │ │ +00069890: 5320 3d20 5a5a 2f31 3031 5b78 5f30 2e2e  S = ZZ/101[x_0..
│ │ │ │ +000698a0: 785f 325d 3b20 2020 2020 2020 2020 2020  x_2];           
│ │ │ │ +000698b0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +000698c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000698d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000698e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000698f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -00069900: 3020 3a20 6666 203d 2061 7070 6c79 2833  0 : ff = apply(3
│ │ │ │ -00069910: 2c20 692d 3e78 5f69 5e32 293b 2020 2020  , i->x_i^2);    
│ │ │ │ -00069920: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00069930: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000698f0: 2b0a 7c69 3130 203a 2066 6620 3d20 6170  +.|i10 : ff = ap
│ │ │ │ +00069900: 706c 7928 332c 2069 2d3e 785f 695e 3229  ply(3, i->x_i^2)
│ │ │ │ +00069910: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ +00069920: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00069930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00069940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069960: 2d2d 2d2b 0a7c 6931 3120 3a20 5220 3d20  ---+.|i11 : R = 
│ │ │ │ -00069970: 532f 6964 6561 6c20 6666 3b20 2020 2020  S/ideal ff;     
│ │ │ │ -00069980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069990: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00069950: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131 203a  --------+.|i11 :
│ │ │ │ +00069960: 2052 203d 2053 2f69 6465 616c 2066 663b   R = S/ideal ff;
│ │ │ │ +00069970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069980: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00069990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000699a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000699b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000699c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -000699d0: 3220 3a20 4d20 3d20 636f 6b65 726e 656c  2 : M = cokernel
│ │ │ │ -000699e0: 206d 6174 7269 7820 7b7b 785f 302c 2078   matrix {{x_0, x
│ │ │ │ -000699f0: 5f31 2a78 5f32 7d7d 3b20 2020 2020 207c  _1*x_2}};      |
│ │ │ │ -00069a00: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000699c0: 2b0a 7c69 3132 203a 204d 203d 2063 6f6b  +.|i12 : M = cok
│ │ │ │ +000699d0: 6572 6e65 6c20 6d61 7472 6978 207b 7b78  ernel matrix {{x
│ │ │ │ +000699e0: 5f30 2c20 785f 312a 785f 327d 7d3b 2020  _0, x_1*x_2}};  
│ │ │ │ +000699f0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00069a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00069a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069a30: 2d2d 2d2b 0a7c 6931 3320 3a20 6220 3d20  ---+.|i13 : b = 
│ │ │ │ -00069a40: 353b 2020 2020 2020 2020 2020 2020 2020  5;              
│ │ │ │ -00069a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069a60: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00069a20: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3133 203a  --------+.|i13 :
│ │ │ │ +00069a30: 2062 203d 2035 3b20 2020 2020 2020 2020   b = 5;         
│ │ │ │ +00069a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069a50: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00069a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00069a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00069a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -00069aa0: 3420 3a20 4d62 203d 2070 7275 6e65 2073  4 : Mb = prune s
│ │ │ │ -00069ab0: 797a 7967 794d 6f64 756c 6528 2d62 2c4d  yzygyModule(-b,M
│ │ │ │ -00069ac0: 293b 2020 2020 2020 2020 2020 2020 207c  );             |
│ │ │ │ -00069ad0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00069a90: 2b0a 7c69 3134 203a 204d 6220 3d20 7072  +.|i14 : Mb = pr
│ │ │ │ +00069aa0: 756e 6520 7379 7a79 6779 4d6f 6475 6c65  une syzygyModule
│ │ │ │ +00069ab0: 282d 622c 4d29 3b20 2020 2020 2020 2020  (-b,M);         
│ │ │ │ +00069ac0: 2020 2020 7c0a 2b2d 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  ----------------
│ │ │ │ -00069b00: 2d2d 2d2b 0a7c 6931 3520 3a20 4520 3d20  ---+.|i15 : E = 
│ │ │ │ -00069b10: 7072 756e 6520 6576 656e 4578 744d 6f64  prune evenExtMod
│ │ │ │ -00069b20: 756c 6520 4d62 3b20 2020 2020 2020 2020  ule Mb;         
│ │ │ │ -00069b30: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00069af0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3135 203a  --------+.|i15 :
│ │ │ │ +00069b00: 2045 203d 2070 7275 6e65 2065 7665 6e45   E = prune evenE
│ │ │ │ +00069b10: 7874 4d6f 6475 6c65 204d 623b 2020 2020  xtModule Mb;    
│ │ │ │ +00069b20: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00069b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00069b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00069b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -00069b70: 3620 3a20 5332 6d61 7020 3d20 5332 2830  6 : S2map = S2(0
│ │ │ │ -00069b80: 2c45 293b 2020 2020 2020 2020 2020 2020  ,E);            
│ │ │ │ -00069b90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00069ba0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00069b60: 2b0a 7c69 3136 203a 2053 326d 6170 203d  +.|i16 : S2map =
│ │ │ │ +00069b70: 2053 3228 302c 4529 3b20 2020 2020 2020   S2(0,E);       
│ │ │ │ +00069b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069b90: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00069ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069bd0: 2020 207c 0a7c 6f31 3620 3a20 4d61 7472     |.|o16 : Matr
│ │ │ │ -00069be0: 6978 2020 2020 2020 2020 2020 2020 2020  ix              
│ │ │ │ -00069bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069c00: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00069bc0: 2020 2020 2020 2020 7c0a 7c6f 3136 203a          |.|o16 :
│ │ │ │ +00069bd0: 204d 6174 7269 7820 2020 2020 2020 2020   Matrix         
│ │ │ │ +00069be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069bf0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00069c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00069c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00069c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -00069c40: 3720 3a20 5345 203d 2070 7275 6e65 2074  7 : SE = prune t
│ │ │ │ -00069c50: 6172 6765 7420 5332 6d61 703b 2020 2020  arget S2map;    
│ │ │ │ -00069c60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00069c70: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00069c30: 2b0a 7c69 3137 203a 2053 4520 3d20 7072  +.|i17 : SE = pr
│ │ │ │ +00069c40: 756e 6520 7461 7267 6574 2053 326d 6170  une target S2map
│ │ │ │ +00069c50: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ +00069c60: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00069c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00069c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069ca0: 2d2d 2d2b 0a7c 6931 3820 3a20 6578 7472  ---+.|i18 : extr
│ │ │ │ -00069cb0: 6120 3d20 7072 756e 6520 636f 6b65 7220  a = prune coker 
│ │ │ │ -00069cc0: 5332 6d61 703b 2020 2020 2020 2020 2020  S2map;          
│ │ │ │ -00069cd0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00069c90: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3138 203a  --------+.|i18 :
│ │ │ │ +00069ca0: 2065 7874 7261 203d 2070 7275 6e65 2063   extra = prune c
│ │ │ │ +00069cb0: 6f6b 6572 2053 326d 6170 3b20 2020 2020  oker S2map;     
│ │ │ │ +00069cc0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00069cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00069ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00069cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -00069d10: 3920 3a20 4b45 203d 2070 7275 6e65 206b  9 : KE = prune k
│ │ │ │ -00069d20: 6572 2053 326d 6170 3b20 2020 2020 2020  er S2map;       
│ │ │ │ -00069d30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00069d40: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00069d00: 2b0a 7c69 3139 203a 204b 4520 3d20 7072  +.|i19 : KE = pr
│ │ │ │ +00069d10: 756e 6520 6b65 7220 5332 6d61 703b 2020  une ker S2map;  
│ │ │ │ +00069d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069d30: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00069d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00069d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069d70: 2d2d 2d2b 0a7c 6932 3020 3a20 6265 7474  ---+.|i20 : bett
│ │ │ │ -00069d80: 6920 6672 6565 5265 736f 6c75 7469 6f6e  i freeResolution
│ │ │ │ -00069d90: 284d 622c 204c 656e 6774 684c 696d 6974  (Mb, LengthLimit
│ │ │ │ -00069da0: 203d 3e20 3130 297c 0a7c 2020 2020 2020   => 10)|.|      
│ │ │ │ +00069d60: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3230 203a  --------+.|i20 :
│ │ │ │ +00069d70: 2062 6574 7469 2066 7265 6552 6573 6f6c   betti freeResol
│ │ │ │ +00069d80: 7574 696f 6e28 4d62 2c20 4c65 6e67 7468  ution(Mb, Length
│ │ │ │ +00069d90: 4c69 6d69 7420 3d3e 2031 3029 7c0a 7c20  Limit => 10)|.| 
│ │ │ │ +00069da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069dd0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00069de0: 2020 2020 2020 2020 2020 2020 3020 2031              0  1
│ │ │ │ -00069df0: 2032 2033 2034 2035 2036 2037 2038 2020   2 3 4 5 6 7 8  
│ │ │ │ -00069e00: 3920 3130 2020 2020 2020 2020 2020 207c  9 10           |
│ │ │ │ -00069e10: 0a7c 6f32 3020 3d20 746f 7461 6c3a 2032  .|o20 = total: 2
│ │ │ │ -00069e20: 3020 3134 2039 2035 2032 2031 2032 2034  0 14 9 5 2 1 2 4
│ │ │ │ -00069e30: 2037 2031 3120 3136 2020 2020 2020 2020   7 11 16        
│ │ │ │ -00069e40: 2020 207c 0a7c 2020 2020 2020 2020 202d     |.|         -
│ │ │ │ -00069e50: 363a 2032 3020 3134 2039 2035 2032 202e  6: 20 14 9 5 2 .
│ │ │ │ -00069e60: 202e 202e 202e 2020 2e20 202e 2020 2020   . . .  .  .    
│ │ │ │ -00069e70: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00069e80: 2020 202d 353a 2020 2e20 202e 202e 202e     -5:  .  . . .
│ │ │ │ -00069e90: 202e 2031 2031 2031 2031 2020 3120 2031   . 1 1 1 1  1  1
│ │ │ │ -00069ea0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00069eb0: 2020 2020 2020 202d 343a 2020 2e20 202e         -4:  .  .
│ │ │ │ -00069ec0: 202e 202e 202e 202e 2031 2033 2036 2031   . . . . 1 3 6 1
│ │ │ │ -00069ed0: 3020 3135 2020 2020 2020 2020 2020 207c  0 15           |
│ │ │ │ -00069ee0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00069dd0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00069de0: 2030 2020 3120 3220 3320 3420 3520 3620   0  1 2 3 4 5 6 
│ │ │ │ +00069df0: 3720 3820 2039 2031 3020 2020 2020 2020  7 8  9 10       
│ │ │ │ +00069e00: 2020 2020 7c0a 7c6f 3230 203d 2074 6f74      |.|o20 = tot
│ │ │ │ +00069e10: 616c 3a20 3230 2031 3420 3920 3520 3220  al: 20 14 9 5 2 
│ │ │ │ +00069e20: 3120 3220 3420 3720 3131 2031 3620 2020  1 2 4 7 11 16   
│ │ │ │ +00069e30: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00069e40: 2020 2020 2d36 3a20 3230 2031 3420 3920      -6: 20 14 9 
│ │ │ │ +00069e50: 3520 3220 2e20 2e20 2e20 2e20 202e 2020  5 2 . . . .  .  
│ │ │ │ +00069e60: 2e20 2020 2020 2020 2020 2020 7c0a 7c20  .           |.| 
│ │ │ │ +00069e70: 2020 2020 2020 2020 2d35 3a20 202e 2020          -5:  .  
│ │ │ │ +00069e80: 2e20 2e20 2e20 2e20 3120 3120 3120 3120  . . . . 1 1 1 1 
│ │ │ │ +00069e90: 2031 2020 3120 2020 2020 2020 2020 2020   1  1           
│ │ │ │ +00069ea0: 7c0a 7c20 2020 2020 2020 2020 2d34 3a20  |.|         -4: 
│ │ │ │ +00069eb0: 202e 2020 2e20 2e20 2e20 2e20 2e20 3120   .  . . . . . 1 
│ │ │ │ +00069ec0: 3320 3620 3130 2031 3520 2020 2020 2020  3 6 10 15       
│ │ │ │ +00069ed0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00069ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069f10: 2020 207c 0a7c 6f32 3020 3a20 4265 7474     |.|o20 : Bett
│ │ │ │ -00069f20: 6954 616c 6c79 2020 2020 2020 2020 2020  iTally          
│ │ │ │ -00069f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069f40: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00069f00: 2020 2020 2020 2020 7c0a 7c6f 3230 203a          |.|o20 :
│ │ │ │ +00069f10: 2042 6574 7469 5461 6c6c 7920 2020 2020   BettiTally     
│ │ │ │ +00069f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069f30: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00069f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00069f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00069f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ -00069f80: 3120 3a20 6170 706c 7920 2835 2c20 692d  1 : apply (5, i-
│ │ │ │ -00069f90: 3e20 6869 6c62 6572 7446 756e 6374 696f  > hilbertFunctio
│ │ │ │ -00069fa0: 6e28 692c 204b 4529 2920 2020 2020 207c  n(i, KE))      |
│ │ │ │ -00069fb0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00069f70: 2b0a 7c69 3231 203a 2061 7070 6c79 2028  +.|i21 : apply (
│ │ │ │ +00069f80: 352c 2069 2d3e 2068 696c 6265 7274 4675  5, i-> hilbertFu
│ │ │ │ +00069f90: 6e63 7469 6f6e 2869 2c20 4b45 2929 2020  nction(i, KE))  
│ │ │ │ +00069fa0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00069fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069fe0: 2020 207c 0a7c 6f32 3120 3d20 7b32 302c     |.|o21 = {20,
│ │ │ │ -00069ff0: 2039 2c20 322c 2030 2c20 307d 2020 2020   9, 2, 0, 0}    
│ │ │ │ -0006a000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a010: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00069fd0: 2020 2020 2020 2020 7c0a 7c6f 3231 203d          |.|o21 =
│ │ │ │ +00069fe0: 207b 3230 2c20 392c 2032 2c20 302c 2030   {20, 9, 2, 0, 0
│ │ │ │ +00069ff0: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +0006a000: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0006a010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a040: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ -0006a050: 3120 3a20 4c69 7374 2020 2020 2020 2020  1 : List        
│ │ │ │ +0006a040: 7c0a 7c6f 3231 203a 204c 6973 7420 2020  |.|o21 : List   
│ │ │ │ +0006a050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a070: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0006a080: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0006a070: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0006a080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006a090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006a0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006a0b0: 2d2d 2d2b 0a7c 6932 3220 3a20 6170 706c  ---+.|i22 : appl
│ │ │ │ -0006a0c0: 7920 2835 2c20 692d 3e20 6869 6c62 6572  y (5, i-> hilber
│ │ │ │ -0006a0d0: 7446 756e 6374 696f 6e28 692c 2045 2929  tFunction(i, E))
│ │ │ │ -0006a0e0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0006a0a0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3232 203a  --------+.|i22 :
│ │ │ │ +0006a0b0: 2061 7070 6c79 2028 352c 2069 2d3e 2068   apply (5, i-> h
│ │ │ │ +0006a0c0: 696c 6265 7274 4675 6e63 7469 6f6e 2869  ilbertFunction(i
│ │ │ │ +0006a0d0: 2c20 4529 2920 2020 2020 2020 7c0a 7c20  , E))       |.| 
│ │ │ │ +0006a0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a110: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ -0006a120: 3220 3d20 7b32 302c 2039 2c20 322c 2032  2 = {20, 9, 2, 2
│ │ │ │ -0006a130: 2c20 377d 2020 2020 2020 2020 2020 2020  , 7}            
│ │ │ │ -0006a140: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0006a150: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0006a110: 7c0a 7c6f 3232 203d 207b 3230 2c20 392c  |.|o22 = {20, 9,
│ │ │ │ +0006a120: 2032 2c20 322c 2037 7d20 2020 2020 2020   2, 2, 7}       
│ │ │ │ +0006a130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a140: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0006a150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a180: 2020 207c 0a7c 6f32 3220 3a20 4c69 7374     |.|o22 : List
│ │ │ │ +0006a170: 2020 2020 2020 2020 7c0a 7c6f 3232 203a          |.|o22 :
│ │ │ │ +0006a180: 204c 6973 7420 2020 2020 2020 2020 2020   List           
│ │ │ │  0006a190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a1b0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0006a1a0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0006a1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006a1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006a1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006a1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ -0006a1f0: 3320 3a20 6170 706c 7920 2835 2c20 692d  3 : apply (5, i-
│ │ │ │ -0006a200: 3e20 6869 6c62 6572 7446 756e 6374 696f  > hilbertFunctio
│ │ │ │ -0006a210: 6e28 692c 2053 4529 2920 2020 2020 207c  n(i, SE))      |
│ │ │ │ -0006a220: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0006a1e0: 2b0a 7c69 3233 203a 2061 7070 6c79 2028  +.|i23 : apply (
│ │ │ │ +0006a1f0: 352c 2069 2d3e 2068 696c 6265 7274 4675  5, i-> hilbertFu
│ │ │ │ +0006a200: 6e63 7469 6f6e 2869 2c20 5345 2929 2020  nction(i, SE))  
│ │ │ │ +0006a210: 2020 2020 7c0a 7c20 2020 2020 2020 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 207c 0a7c 6f32 3320 3d20 7b31 2c20     |.|o23 = {1, 
│ │ │ │ -0006a260: 312c 2031 2c20 322c 2037 7d20 2020 2020  1, 1, 2, 7}     
│ │ │ │ -0006a270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a280: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0006a240: 2020 2020 2020 2020 7c0a 7c6f 3233 203d          |.|o23 =
│ │ │ │ +0006a250: 207b 312c 2031 2c20 312c 2032 2c20 377d   {1, 1, 1, 2, 7}
│ │ │ │ +0006a260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a270: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0006a280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a2b0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ -0006a2c0: 3320 3a20 4c69 7374 2020 2020 2020 2020  3 : List        
│ │ │ │ +0006a2b0: 7c0a 7c6f 3233 203a 204c 6973 7420 2020  |.|o23 : List   
│ │ │ │ +0006a2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a2e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0006a2f0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0006a2e0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0006a2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006a300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006a310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006a320: 2d2d 2d2b 0a7c 6932 3420 3a20 6170 706c  ---+.|i24 : appl
│ │ │ │ -0006a330: 7920 2835 2c20 692d 3e20 6869 6c62 6572  y (5, i-> hilber
│ │ │ │ -0006a340: 7446 756e 6374 696f 6e28 692c 2065 7874  tFunction(i, ext
│ │ │ │ -0006a350: 7261 2929 2020 207c 0a7c 2020 2020 2020  ra))   |.|      
│ │ │ │ +0006a310: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3234 203a  --------+.|i24 :
│ │ │ │ +0006a320: 2061 7070 6c79 2028 352c 2069 2d3e 2068   apply (5, i-> h
│ │ │ │ +0006a330: 696c 6265 7274 4675 6e63 7469 6f6e 2869  ilbertFunction(i
│ │ │ │ +0006a340: 2c20 6578 7472 6129 2920 2020 7c0a 7c20  , extra))   |.| 
│ │ │ │ +0006a350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a380: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ -0006a390: 3420 3d20 7b31 2c20 312c 2031 2c20 302c  4 = {1, 1, 1, 0,
│ │ │ │ -0006a3a0: 2030 7d20 2020 2020 2020 2020 2020 2020   0}             
│ │ │ │ -0006a3b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0006a3c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0006a380: 7c0a 7c6f 3234 203d 207b 312c 2031 2c20  |.|o24 = {1, 1, 
│ │ │ │ +0006a390: 312c 2030 2c20 307d 2020 2020 2020 2020  1, 0, 0}        
│ │ │ │ +0006a3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a3b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0006a3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a3f0: 2020 207c 0a7c 6f32 3420 3a20 4c69 7374     |.|o24 : List
│ │ │ │ +0006a3e0: 2020 2020 2020 2020 7c0a 7c6f 3234 203a          |.|o24 :
│ │ │ │ +0006a3f0: 204c 6973 7420 2020 2020 2020 2020 2020   List           
│ │ │ │  0006a400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a420: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0006a410: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0006a420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006a430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006a440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006a450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4361  -----------+..Ca
│ │ │ │ -0006a460: 7665 6174 0a3d 3d3d 3d3d 3d0a 0a54 6578  veat.======..Tex
│ │ │ │ -0006a470: 7420 5332 2d69 6669 6361 7469 6f6e 2069  t S2-ification i
│ │ │ │ -0006a480: 7320 7265 6c61 7465 6420 746f 2063 6f6d  s related to com
│ │ │ │ -0006a490: 7075 7469 6e67 2063 6f68 6f6d 6f6c 6f67  puting cohomolog
│ │ │ │ -0006a4a0: 7920 616e 6420 746f 2063 6f6d 7075 7469  y and to computi
│ │ │ │ -0006a4b0: 6e67 2069 6e74 6567 7261 6c0a 636c 6f73  ng integral.clos
│ │ │ │ -0006a4c0: 7572 653b 2074 6865 7265 2061 7265 2073  ure; there are s
│ │ │ │ -0006a4d0: 6372 6970 7473 2069 6e20 7468 6f73 6520  cripts in those 
│ │ │ │ -0006a4e0: 7061 636b 6167 6573 2074 6861 7420 7072  packages that pr
│ │ │ │ -0006a4f0: 6f64 7563 6520 616e 2053 322d 6966 6963  oduce an S2-ific
│ │ │ │ -0006a500: 6174 696f 6e2c 2062 7574 0a6f 6e65 2074  ation, but.one t
│ │ │ │ -0006a510: 616b 6573 2061 2072 696e 6720 6173 2061  akes a ring as a
│ │ │ │ -0006a520: 7267 756d 656e 7420 616e 6420 7468 6520  rgument and the 
│ │ │ │ -0006a530: 6f74 6865 7220 646f 6573 6e27 7420 7072  other doesn't pr
│ │ │ │ -0006a540: 6f64 7563 6520 7468 6520 636f 6d70 6172  oduce the compar
│ │ │ │ -0006a550: 6973 6f6e 206d 6170 2e0a 0a53 6565 2061  ison map...See a
│ │ │ │ -0006a560: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020  lso.========..  
│ │ │ │ -0006a570: 2a20 2a6e 6f74 6520 496e 7465 6772 616c  * *note Integral
│ │ │ │ -0006a580: 436c 6f73 7572 653a 2028 496e 7465 6772  Closure: (Integr
│ │ │ │ -0006a590: 616c 436c 6f73 7572 6529 546f 702c 202d  alClosure)Top, -
│ │ │ │ -0006a5a0: 2d20 726f 7574 696e 6573 2066 6f72 2069  - routines for i
│ │ │ │ -0006a5b0: 6e74 6567 7261 6c0a 2020 2020 636c 6f73  ntegral.    clos
│ │ │ │ -0006a5c0: 7572 6520 6f66 2061 6666 696e 6520 646f  ure of affine do
│ │ │ │ -0006a5d0: 6d61 696e 7320 616e 6420 6964 6561 6c73  mains and ideals
│ │ │ │ -0006a5e0: 0a20 202a 202a 6e6f 7465 206d 616b 6553  .  * *note makeS
│ │ │ │ -0006a5f0: 323a 2028 496e 7465 6772 616c 436c 6f73  2: (IntegralClos
│ │ │ │ -0006a600: 7572 6529 6d61 6b65 5332 2c20 2d2d 2063  ure)makeS2, -- c
│ │ │ │ -0006a610: 6f6d 7075 7465 2074 6865 2053 3269 6669  ompute the S2ifi
│ │ │ │ -0006a620: 6361 7469 6f6e 206f 6620 610a 2020 2020  cation of a.    
│ │ │ │ -0006a630: 7265 6475 6365 6420 7269 6e67 0a20 202a  reduced ring.  *
│ │ │ │ -0006a640: 202a 6e6f 7465 2042 4747 3a20 2842 4747   *note BGG: (BGG
│ │ │ │ -0006a650: 2954 6f70 2c20 2d2d 2042 6572 6e73 7465  )Top, -- Bernste
│ │ │ │ -0006a660: 696e 2d47 656c 2766 616e 642d 4765 6c27  in-Gel'fand-Gel'
│ │ │ │ -0006a670: 6661 6e64 2063 6f72 7265 7370 6f6e 6465  fand corresponde
│ │ │ │ -0006a680: 6e63 650a 2020 2a20 2a6e 6f74 6520 636f  nce.  * *note co
│ │ │ │ -0006a690: 686f 6d6f 6c6f 6779 3a20 284d 6163 6175  homology: (Macau
│ │ │ │ -0006a6a0: 6c61 7932 446f 6329 636f 686f 6d6f 6c6f  lay2Doc)cohomolo
│ │ │ │ -0006a6b0: 6779 2c20 2d2d 2067 656e 6572 616c 2063  gy, -- general c
│ │ │ │ -0006a6c0: 6f68 6f6d 6f6c 6f67 7920 6675 6e63 746f  ohomology functo
│ │ │ │ -0006a6d0: 720a 2020 2a20 4848 5e5a 5a20 5375 6d4f  r.  * HH^ZZ SumO
│ │ │ │ -0006a6e0: 6654 7769 7374 7320 286d 6973 7369 6e67  fTwists (missing
│ │ │ │ -0006a6f0: 2064 6f63 756d 656e 7461 7469 6f6e 290a   documentation).
│ │ │ │ -0006a700: 0a57 6179 7320 746f 2075 7365 2053 323a  .Ways to use S2:
│ │ │ │ -0006a710: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ -0006a720: 0a0a 2020 2a20 2253 3228 5a5a 2c4d 6f64  ..  * "S2(ZZ,Mod
│ │ │ │ -0006a730: 756c 6529 220a 0a46 6f72 2074 6865 2070  ule)"..For the p
│ │ │ │ -0006a740: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d  rogrammer.======
│ │ │ │ -0006a750: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  ============..Th
│ │ │ │ -0006a760: 6520 6f62 6a65 6374 202a 6e6f 7465 2053  e object *note S
│ │ │ │ -0006a770: 323a 2053 322c 2069 7320 6120 2a6e 6f74  2: S2, is a *not
│ │ │ │ -0006a780: 6520 6d65 7468 6f64 2066 756e 6374 696f  e method functio
│ │ │ │ -0006a790: 6e3a 0a28 4d61 6361 756c 6179 3244 6f63  n:.(Macaulay2Doc
│ │ │ │ -0006a7a0: 294d 6574 686f 6446 756e 6374 696f 6e2c  )MethodFunction,
│ │ │ │ -0006a7b0: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...-------------
│ │ │ │ +0006a450: 2b0a 0a43 6176 6561 740a 3d3d 3d3d 3d3d  +..Caveat.======
│ │ │ │ +0006a460: 0a0a 5465 7874 2053 322d 6966 6963 6174  ..Text S2-ificat
│ │ │ │ +0006a470: 696f 6e20 6973 2072 656c 6174 6564 2074  ion is related t
│ │ │ │ +0006a480: 6f20 636f 6d70 7574 696e 6720 636f 686f  o computing coho
│ │ │ │ +0006a490: 6d6f 6c6f 6779 2061 6e64 2074 6f20 636f  mology and to co
│ │ │ │ +0006a4a0: 6d70 7574 696e 6720 696e 7465 6772 616c  mputing integral
│ │ │ │ +0006a4b0: 0a63 6c6f 7375 7265 3b20 7468 6572 6520  .closure; there 
│ │ │ │ +0006a4c0: 6172 6520 7363 7269 7074 7320 696e 2074  are scripts in t
│ │ │ │ +0006a4d0: 686f 7365 2070 6163 6b61 6765 7320 7468  hose packages th
│ │ │ │ +0006a4e0: 6174 2070 726f 6475 6365 2061 6e20 5332  at produce an S2
│ │ │ │ +0006a4f0: 2d69 6669 6361 7469 6f6e 2c20 6275 740a  -ification, but.
│ │ │ │ +0006a500: 6f6e 6520 7461 6b65 7320 6120 7269 6e67  one takes a ring
│ │ │ │ +0006a510: 2061 7320 6172 6775 6d65 6e74 2061 6e64   as argument and
│ │ │ │ +0006a520: 2074 6865 206f 7468 6572 2064 6f65 736e   the other doesn
│ │ │ │ +0006a530: 2774 2070 726f 6475 6365 2074 6865 2063  't produce the c
│ │ │ │ +0006a540: 6f6d 7061 7269 736f 6e20 6d61 702e 0a0a  omparison map...
│ │ │ │ +0006a550: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d  See also.=======
│ │ │ │ +0006a560: 3d0a 0a20 202a 202a 6e6f 7465 2049 6e74  =..  * *note Int
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│ │ │ │ +0006a5a0: 666f 7220 696e 7465 6772 616c 0a20 2020  for integral.   
│ │ │ │ +0006a5b0: 2063 6c6f 7375 7265 206f 6620 6166 6669   closure of affi
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│ │ │ │ +0006a5d0: 6465 616c 730a 2020 2a20 2a6e 6f74 6520  deals.  * *note 
│ │ │ │ +0006a5e0: 6d61 6b65 5332 3a20 2849 6e74 6567 7261  makeS2: (Integra
│ │ │ │ +0006a5f0: 6c43 6c6f 7375 7265 296d 616b 6553 322c  lClosure)makeS2,
│ │ │ │ +0006a600: 202d 2d20 636f 6d70 7574 6520 7468 6520   -- compute the 
│ │ │ │ +0006a610: 5332 6966 6963 6174 696f 6e20 6f66 2061  S2ification of a
│ │ │ │ +0006a620: 0a20 2020 2072 6564 7563 6564 2072 696e  .    reduced rin
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│ │ │ │ +0006a640: 2028 4247 4729 546f 702c 202d 2d20 4265   (BGG)Top, -- Be
│ │ │ │ +0006a650: 726e 7374 6569 6e2d 4765 6c27 6661 6e64  rnstein-Gel'fand
│ │ │ │ +0006a660: 2d47 656c 2766 616e 6420 636f 7272 6573  -Gel'fand corres
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│ │ │ │ +0006a680: 7465 2063 6f68 6f6d 6f6c 6f67 793a 2028  te cohomology: (
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│ │ │ │ +0006a6c0: 756e 6374 6f72 0a20 202a 2048 485e 5a5a  unctor.  * HH^ZZ
│ │ │ │ +0006a6d0: 2053 756d 4f66 5477 6973 7473 2028 6d69   SumOfTwists (mi
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│ │ │ │ +0006a7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006a7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006a7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006a7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006a7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006a800: 2d2d 0a0a 5468 6520 736f 7572 6365 206f  --..The source o
│ │ │ │ -0006a810: 6620 7468 6973 2064 6f63 756d 656e 7420  f this document 
│ │ │ │ -0006a820: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570  is in./build/rep
│ │ │ │ -0006a830: 726f 6475 6369 626c 652d 7061 7468 2f6d  roducible-path/m
│ │ │ │ -0006a840: 6163 6175 6c61 7932 2d31 2e32 352e 3131  acaulay2-1.25.11
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│ │ │ │ -0006a890: 3833 3a30 2e0a 1f0a 4669 6c65 3a20 436f  83:0....File: Co
│ │ │ │ -0006a8a0: 6d70 6c65 7465 496e 7465 7273 6563 7469  mpleteIntersecti
│ │ │ │ -0006a8b0: 6f6e 5265 736f 6c75 7469 6f6e 732e 696e  onResolutions.in
│ │ │ │ -0006a8c0: 666f 2c20 4e6f 6465 3a20 5368 616d 6173  fo, Node: Shamas
│ │ │ │ -0006a8d0: 682c 204e 6578 743a 2073 706c 6974 7469  h, Next: splitti
│ │ │ │ -0006a8e0: 6e67 732c 2050 7265 763a 2053 322c 2055  ngs, Prev: S2, U
│ │ │ │ -0006a8f0: 703a 2054 6f70 0a0a 5368 616d 6173 6820  p: Top..Shamash 
│ │ │ │ -0006a900: 2d2d 2043 6f6d 7075 7465 7320 7468 6520  -- Computes the 
│ │ │ │ -0006a910: 5368 616d 6173 6820 436f 6d70 6c65 780a  Shamash Complex.
│ │ │ │ +0006a7f0: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75  -------..The sou
│ │ │ │ +0006a800: 7263 6520 6f66 2074 6869 7320 646f 6375  rce of this docu
│ │ │ │ +0006a810: 6d65 6e74 2069 7320 696e 0a2f 6275 696c  ment is in./buil
│ │ │ │ +0006a820: 642f 7265 7072 6f64 7563 6962 6c65 2d70  d/reproducible-p
│ │ │ │ +0006a830: 6174 682f 6d61 6361 756c 6179 322d 312e  ath/macaulay2-1.
│ │ │ │ +0006a840: 3235 2e31 312b 6473 2f4d 322f 4d61 6361  25.11+ds/M2/Maca
│ │ │ │ +0006a850: 756c 6179 322f 7061 636b 6167 6573 2f0a  ulay2/packages/.
│ │ │ │ +0006a860: 436f 6d70 6c65 7465 496e 7465 7273 6563  CompleteIntersec
│ │ │ │ +0006a870: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e  tionResolutions.
│ │ │ │ +0006a880: 6d32 3a33 3838 333a 302e 0a1f 0a46 696c  m2:3883:0....Fil
│ │ │ │ +0006a890: 653a 2043 6f6d 706c 6574 6549 6e74 6572  e: CompleteInter
│ │ │ │ +0006a8a0: 7365 6374 696f 6e52 6573 6f6c 7574 696f  sectionResolutio
│ │ │ │ +0006a8b0: 6e73 2e69 6e66 6f2c 204e 6f64 653a 2053  ns.info, Node: S
│ │ │ │ +0006a8c0: 6861 6d61 7368 2c20 4e65 7874 3a20 7370  hamash, Next: sp
│ │ │ │ +0006a8d0: 6c69 7474 696e 6773 2c20 5072 6576 3a20  littings, Prev: 
│ │ │ │ +0006a8e0: 5332 2c20 5570 3a20 546f 700a 0a53 6861  S2, Up: Top..Sha
│ │ │ │ +0006a8f0: 6d61 7368 202d 2d20 436f 6d70 7574 6573  mash -- Computes
│ │ │ │ +0006a900: 2074 6865 2053 6861 6d61 7368 2043 6f6d   the Shamash Com
│ │ │ │ +0006a910: 706c 6578 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a  plex.***********
│ │ │ │  0006a920: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0006a930: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0006a940: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 7361  *******..  * Usa
│ │ │ │ -0006a950: 6765 3a20 0a20 2020 2020 2020 2046 4620  ge: .        FF 
│ │ │ │ -0006a960: 3d20 5368 616d 6173 6828 6666 2c46 2c6c  = Shamash(ff,F,l
│ │ │ │ -0006a970: 656e 290a 2020 2020 2020 2020 4646 203d  en).        FF =
│ │ │ │ -0006a980: 2053 6861 6d61 7368 2852 6261 722c 462c   Shamash(Rbar,F,
│ │ │ │ -0006a990: 6c65 6e29 0a20 202a 2049 6e70 7574 733a  len).  * Inputs:
│ │ │ │ -0006a9a0: 0a20 2020 2020 202a 2066 662c 2061 202a  .      * ff, a *
│ │ │ │ -0006a9b0: 6e6f 7465 206d 6174 7269 783a 2028 4d61  note matrix: (Ma
│ │ │ │ -0006a9c0: 6361 756c 6179 3244 6f63 294d 6174 7269  caulay2Doc)Matri
│ │ │ │ -0006a9d0: 782c 2c20 3120 7820 3120 4d61 7472 6978  x,, 1 x 1 Matrix
│ │ │ │ -0006a9e0: 206f 7665 7220 7269 6e67 2046 2e0a 2020   over ring F..  
│ │ │ │ -0006a9f0: 2020 2020 2a20 5262 6172 2c20 6120 2a6e      * Rbar, a *n
│ │ │ │ -0006aa00: 6f74 6520 7269 6e67 3a20 284d 6163 6175  ote ring: (Macau
│ │ │ │ -0006aa10: 6c61 7932 446f 6329 5269 6e67 2c2c 2072  lay2Doc)Ring,, r
│ │ │ │ -0006aa20: 696e 6720 4620 6d6f 6420 6964 6561 6c20  ing F mod ideal 
│ │ │ │ -0006aa30: 6666 0a20 2020 2020 202a 2046 2c20 6120  ff.      * F, a 
│ │ │ │ -0006aa40: 2a6e 6f74 6520 636f 6d70 6c65 783a 2028  *note complex: (
│ │ │ │ -0006aa50: 436f 6d70 6c65 7865 7329 436f 6d70 6c65  Complexes)Comple
│ │ │ │ -0006aa60: 782c 2c20 6465 6669 6e65 6420 6f76 6572  x,, defined over
│ │ │ │ -0006aa70: 2072 696e 6720 6666 0a20 2020 2020 202a   ring ff.      *
│ │ │ │ -0006aa80: 206c 656e 2c20 616e 202a 6e6f 7465 2069   len, an *note i
│ │ │ │ -0006aa90: 6e74 6567 6572 3a20 284d 6163 6175 6c61  nteger: (Macaula
│ │ │ │ -0006aaa0: 7932 446f 6329 5a5a 2c2c 200a 2020 2a20  y2Doc)ZZ,, .  * 
│ │ │ │ -0006aab0: 4f75 7470 7574 733a 0a20 2020 2020 202a  Outputs:.      *
│ │ │ │ -0006aac0: 2046 462c 2061 202a 6e6f 7465 2063 6f6d   FF, a *note com
│ │ │ │ -0006aad0: 706c 6578 3a20 2843 6f6d 706c 6578 6573  plex: (Complexes
│ │ │ │ -0006aae0: 2943 6f6d 706c 6578 2c2c 2063 6861 696e  )Complex,, chain
│ │ │ │ -0006aaf0: 2063 6f6d 706c 6578 206f 7665 7220 2872   complex over (r
│ │ │ │ -0006ab00: 696e 670a 2020 2020 2020 2020 4629 2f28  ing.        F)/(
│ │ │ │ -0006ab10: 6964 6561 6c20 6666 290a 0a44 6573 6372  ideal ff)..Descr
│ │ │ │ -0006ab20: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d  iption.=========
│ │ │ │ -0006ab30: 3d3d 0a0a 4c65 7420 5220 3d20 7269 6e67  ==..Let R = ring
│ │ │ │ -0006ab40: 2046 203d 2072 696e 6720 6666 2c20 616e   F = ring ff, an
│ │ │ │ -0006ab50: 6420 5262 6172 203d 2052 2f28 6964 6561  d Rbar = R/(idea
│ │ │ │ -0006ab60: 6c20 6629 2c20 7768 6572 6520 6666 203d  l f), where ff =
│ │ │ │ -0006ab70: 206d 6174 7269 787b 7b66 7d7d 2069 7320   matrix{{f}} is 
│ │ │ │ -0006ab80: 610a 3178 3120 6d61 7472 6978 2077 686f  a.1x1 matrix who
│ │ │ │ -0006ab90: 7365 2065 6e74 7279 2069 7320 6120 6e6f  se entry is a no
│ │ │ │ -0006aba0: 6e7a 6572 6f64 6976 6973 6f72 2069 6e20  nzerodivisor in 
│ │ │ │ -0006abb0: 522e 2054 6865 2063 6f6d 706c 6578 2046  R. The complex F
│ │ │ │ -0006abc0: 2073 686f 756c 6420 6164 6d69 7420 610a   should admit a.
│ │ │ │ -0006abd0: 7379 7374 656d 206f 6620 6869 6768 6572  system of higher
│ │ │ │ -0006abe0: 2068 6f6d 6f74 6f70 6965 7320 666f 7220   homotopies for 
│ │ │ │ -0006abf0: 7468 6520 656e 7472 7920 6f66 2066 662c  the entry of ff,
│ │ │ │ -0006ac00: 2072 6574 7572 6e65 6420 6279 2074 6865   returned by the
│ │ │ │ -0006ac10: 2063 616c 6c0a 6d61 6b65 486f 6d6f 746f   call.makeHomoto
│ │ │ │ -0006ac20: 7069 6573 2866 662c 4629 2e0a 0a54 6865  pies(ff,F)...The
│ │ │ │ -0006ac30: 2063 6f6d 706c 6578 2046 4620 6861 7320   complex FF has 
│ │ │ │ -0006ac40: 7465 726d 730a 0a46 465f 7b32 2a69 7d20  terms..FF_{2*i} 
│ │ │ │ -0006ac50: 3d20 5262 6172 2a2a 2846 5f30 202b 2b20  = Rbar**(F_0 ++ 
│ │ │ │ -0006ac60: 465f 3220 2b2b 202e 2e20 2b2b 2046 5f69  F_2 ++ .. ++ F_i
│ │ │ │ -0006ac70: 290a 0a46 465f 7b32 2a69 2b31 7d20 3d20  )..FF_{2*i+1} = 
│ │ │ │ -0006ac80: 5262 6172 2a2a 2846 5f31 202b 2b20 465f  Rbar**(F_1 ++ F_
│ │ │ │ -0006ac90: 3320 2b2b 2e2e 2b2b 465f 7b32 2a69 2b31  3 ++..++F_{2*i+1
│ │ │ │ -0006aca0: 7d29 0a0a 616e 6420 6d61 7073 206d 6164  })..and maps mad
│ │ │ │ -0006acb0: 6520 6672 6f6d 2074 6865 2068 6967 6865  e from the highe
│ │ │ │ -0006acc0: 7220 686f 6d6f 746f 7069 6573 2e0a 0a46  r homotopies...F
│ │ │ │ -0006acd0: 6f72 2074 6865 2063 6173 6520 6f66 2061  or the case of a
│ │ │ │ -0006ace0: 2063 6f6d 706c 6574 6520 696e 7465 7273   complete inters
│ │ │ │ -0006acf0: 6563 7469 6f6e 206f 6620 6869 6768 6572  ection of higher
│ │ │ │ -0006ad00: 2063 6f64 696d 656e 7369 6f6e 2c20 6f72   codimension, or
│ │ │ │ -0006ad10: 2074 6f20 7365 6520 7468 650a 636f 6d70   to see the.comp
│ │ │ │ -0006ad20: 6f6e 656e 7473 206f 6620 7468 6520 7265  onents of the re
│ │ │ │ -0006ad30: 736f 6c75 7469 6f6e 2061 7320 7375 6d6d  solution as summ
│ │ │ │ -0006ad40: 616e 6473 206f 6620 4646 5f6a 2c20 7573  ands of FF_j, us
│ │ │ │ -0006ad50: 6520 7468 6520 726f 7574 696e 650a 4569  e the routine.Ei
│ │ │ │ -0006ad60: 7365 6e62 7564 5368 616d 6173 6820 696e  senbudShamash in
│ │ │ │ -0006ad70: 7374 6561 642e 0a0a 2b2d 2d2d 2d2d 2d2d  stead...+-------
│ │ │ │ +0006a930: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020  ************..  
│ │ │ │ +0006a940: 2a20 5573 6167 653a 200a 2020 2020 2020  * Usage: .      
│ │ │ │ +0006a950: 2020 4646 203d 2053 6861 6d61 7368 2866    FF = Shamash(f
│ │ │ │ +0006a960: 662c 462c 6c65 6e29 0a20 2020 2020 2020  f,F,len).       
│ │ │ │ +0006a970: 2046 4620 3d20 5368 616d 6173 6828 5262   FF = Shamash(Rb
│ │ │ │ +0006a980: 6172 2c46 2c6c 656e 290a 2020 2a20 496e  ar,F,len).  * In
│ │ │ │ +0006a990: 7075 7473 3a0a 2020 2020 2020 2a20 6666  puts:.      * ff
│ │ │ │ +0006a9a0: 2c20 6120 2a6e 6f74 6520 6d61 7472 6978  , a *note matrix
│ │ │ │ +0006a9b0: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +0006a9c0: 4d61 7472 6978 2c2c 2031 2078 2031 204d  Matrix,, 1 x 1 M
│ │ │ │ +0006a9d0: 6174 7269 7820 6f76 6572 2072 696e 6720  atrix over ring 
│ │ │ │ +0006a9e0: 462e 0a20 2020 2020 202a 2052 6261 722c  F..      * Rbar,
│ │ │ │ +0006a9f0: 2061 202a 6e6f 7465 2072 696e 673a 2028   a *note ring: (
│ │ │ │ +0006aa00: 4d61 6361 756c 6179 3244 6f63 2952 696e  Macaulay2Doc)Rin
│ │ │ │ +0006aa10: 672c 2c20 7269 6e67 2046 206d 6f64 2069  g,, ring F mod i
│ │ │ │ +0006aa20: 6465 616c 2066 660a 2020 2020 2020 2a20  deal ff.      * 
│ │ │ │ +0006aa30: 462c 2061 202a 6e6f 7465 2063 6f6d 706c  F, a *note compl
│ │ │ │ +0006aa40: 6578 3a20 2843 6f6d 706c 6578 6573 2943  ex: (Complexes)C
│ │ │ │ +0006aa50: 6f6d 706c 6578 2c2c 2064 6566 696e 6564  omplex,, defined
│ │ │ │ +0006aa60: 206f 7665 7220 7269 6e67 2066 660a 2020   over ring ff.  
│ │ │ │ +0006aa70: 2020 2020 2a20 6c65 6e2c 2061 6e20 2a6e      * len, an *n
│ │ │ │ +0006aa80: 6f74 6520 696e 7465 6765 723a 2028 4d61  ote integer: (Ma
│ │ │ │ +0006aa90: 6361 756c 6179 3244 6f63 295a 5a2c 2c20  caulay2Doc)ZZ,, 
│ │ │ │ +0006aaa0: 0a20 202a 204f 7574 7075 7473 3a0a 2020  .  * Outputs:.  
│ │ │ │ +0006aab0: 2020 2020 2a20 4646 2c20 6120 2a6e 6f74      * FF, a *not
│ │ │ │ +0006aac0: 6520 636f 6d70 6c65 783a 2028 436f 6d70  e complex: (Comp
│ │ │ │ +0006aad0: 6c65 7865 7329 436f 6d70 6c65 782c 2c20  lexes)Complex,, 
│ │ │ │ +0006aae0: 6368 6169 6e20 636f 6d70 6c65 7820 6f76  chain complex ov
│ │ │ │ +0006aaf0: 6572 2028 7269 6e67 0a20 2020 2020 2020  er (ring.       
│ │ │ │ +0006ab00: 2046 292f 2869 6465 616c 2066 6629 0a0a   F)/(ideal ff)..
│ │ │ │ +0006ab10: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
│ │ │ │ +0006ab20: 3d3d 3d3d 3d3d 3d0a 0a4c 6574 2052 203d  =======..Let R =
│ │ │ │ +0006ab30: 2072 696e 6720 4620 3d20 7269 6e67 2066   ring F = ring f
│ │ │ │ +0006ab40: 662c 2061 6e64 2052 6261 7220 3d20 522f  f, and Rbar = R/
│ │ │ │ +0006ab50: 2869 6465 616c 2066 292c 2077 6865 7265  (ideal f), where
│ │ │ │ +0006ab60: 2066 6620 3d20 6d61 7472 6978 7b7b 667d   ff = matrix{{f}
│ │ │ │ +0006ab70: 7d20 6973 2061 0a31 7831 206d 6174 7269  } is a.1x1 matri
│ │ │ │ +0006ab80: 7820 7768 6f73 6520 656e 7472 7920 6973  x whose entry is
│ │ │ │ +0006ab90: 2061 206e 6f6e 7a65 726f 6469 7669 736f   a nonzerodiviso
│ │ │ │ +0006aba0: 7220 696e 2052 2e20 5468 6520 636f 6d70  r in R. The comp
│ │ │ │ +0006abb0: 6c65 7820 4620 7368 6f75 6c64 2061 646d  lex F should adm
│ │ │ │ +0006abc0: 6974 2061 0a73 7973 7465 6d20 6f66 2068  it a.system of h
│ │ │ │ +0006abd0: 6967 6865 7220 686f 6d6f 746f 7069 6573  igher homotopies
│ │ │ │ +0006abe0: 2066 6f72 2074 6865 2065 6e74 7279 206f   for the entry o
│ │ │ │ +0006abf0: 6620 6666 2c20 7265 7475 726e 6564 2062  f ff, returned b
│ │ │ │ +0006ac00: 7920 7468 6520 6361 6c6c 0a6d 616b 6548  y the call.makeH
│ │ │ │ +0006ac10: 6f6d 6f74 6f70 6965 7328 6666 2c46 292e  omotopies(ff,F).
│ │ │ │ +0006ac20: 0a0a 5468 6520 636f 6d70 6c65 7820 4646  ..The complex FF
│ │ │ │ +0006ac30: 2068 6173 2074 6572 6d73 0a0a 4646 5f7b   has terms..FF_{
│ │ │ │ +0006ac40: 322a 697d 203d 2052 6261 722a 2a28 465f  2*i} = Rbar**(F_
│ │ │ │ +0006ac50: 3020 2b2b 2046 5f32 202b 2b20 2e2e 202b  0 ++ F_2 ++ .. +
│ │ │ │ +0006ac60: 2b20 465f 6929 0a0a 4646 5f7b 322a 692b  + F_i)..FF_{2*i+
│ │ │ │ +0006ac70: 317d 203d 2052 6261 722a 2a28 465f 3120  1} = Rbar**(F_1 
│ │ │ │ +0006ac80: 2b2b 2046 5f33 202b 2b2e 2e2b 2b46 5f7b  ++ F_3 ++..++F_{
│ │ │ │ +0006ac90: 322a 692b 317d 290a 0a61 6e64 206d 6170  2*i+1})..and map
│ │ │ │ +0006aca0: 7320 6d61 6465 2066 726f 6d20 7468 6520  s made from the 
│ │ │ │ +0006acb0: 6869 6768 6572 2068 6f6d 6f74 6f70 6965  higher homotopie
│ │ │ │ +0006acc0: 732e 0a0a 466f 7220 7468 6520 6361 7365  s...For the case
│ │ │ │ +0006acd0: 206f 6620 6120 636f 6d70 6c65 7465 2069   of a complete i
│ │ │ │ +0006ace0: 6e74 6572 7365 6374 696f 6e20 6f66 2068  ntersection of h
│ │ │ │ +0006acf0: 6967 6865 7220 636f 6469 6d65 6e73 696f  igher codimensio
│ │ │ │ +0006ad00: 6e2c 206f 7220 746f 2073 6565 2074 6865  n, or to see the
│ │ │ │ +0006ad10: 0a63 6f6d 706f 6e65 6e74 7320 6f66 2074  .components of t
│ │ │ │ +0006ad20: 6865 2072 6573 6f6c 7574 696f 6e20 6173  he resolution as
│ │ │ │ +0006ad30: 2073 756d 6d61 6e64 7320 6f66 2046 465f   summands of FF_
│ │ │ │ +0006ad40: 6a2c 2075 7365 2074 6865 2072 6f75 7469  j, use the routi
│ │ │ │ +0006ad50: 6e65 0a45 6973 656e 6275 6453 6861 6d61  ne.EisenbudShama
│ │ │ │ +0006ad60: 7368 2069 6e73 7465 6164 2e0a 0a2b 2d2d  sh instead...+--
│ │ │ │ +0006ad70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006ad80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006ad90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006ada0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -0006adb0: 6931 203a 2053 203d 205a 5a2f 3130 315b  i1 : S = ZZ/101[
│ │ │ │ -0006adc0: 782c 792c 7a5d 2020 2020 2020 2020 2020  x,y,z]          
│ │ │ │ -0006add0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ade0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0006ada0: 2d2d 2b0a 7c69 3120 3a20 5320 3d20 5a5a  --+.|i1 : S = ZZ
│ │ │ │ +0006adb0: 2f31 3031 5b78 2c79 2c7a 5d20 2020 2020  /101[x,y,z]     
│ │ │ │ +0006adc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006add0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0006ade0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006adf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ae00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ae10: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -0006ae20: 203d 2053 2020 2020 2020 2020 2020 2020   = S            
│ │ │ │ +0006ae10: 7c0a 7c6f 3120 3d20 5320 2020 2020 2020  |.|o1 = S       
│ │ │ │ +0006ae20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ae30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ae40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ae50: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0006ae40: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0006ae50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ae60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ae70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ae80: 2020 2020 2020 2020 207c 0a7c 6f31 203a           |.|o1 :
│ │ │ │ -0006ae90: 2050 6f6c 796e 6f6d 6961 6c52 696e 6720   PolynomialRing 
│ │ │ │ +0006ae70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0006ae80: 7c6f 3120 3a20 506f 6c79 6e6f 6d69 616c  |o1 : Polynomial
│ │ │ │ +0006ae90: 5269 6e67 2020 2020 2020 2020 2020 2020  Ring            
│ │ │ │  0006aea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006aeb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006aec0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0006aeb0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0006aec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006aed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006aee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006aef0: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2052  -------+.|i2 : R
│ │ │ │ -0006af00: 203d 2053 2f69 6465 616c 2278 332c 7933   = S/ideal"x3,y3
│ │ │ │ -0006af10: 2220 2020 2020 2020 2020 2020 2020 2020  "               
│ │ │ │ -0006af20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0006af30: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0006aee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0006aef0: 3220 3a20 5220 3d20 532f 6964 6561 6c22  2 : R = S/ideal"
│ │ │ │ +0006af00: 7833 2c79 3322 2020 2020 2020 2020 2020  x3,y3"          
│ │ │ │ +0006af10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006af20: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006af30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006af40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006af50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006af60: 2020 2020 207c 0a7c 6f32 203d 2052 2020       |.|o2 = R  
│ │ │ │ +0006af50: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ +0006af60: 3d20 5220 2020 2020 2020 2020 2020 2020  = R             
│ │ │ │  0006af70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006af80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006af90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0006af90: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0006afa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006afb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006afc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006afd0: 2020 207c 0a7c 6f32 203a 2051 756f 7469     |.|o2 : Quoti
│ │ │ │ -0006afe0: 656e 7452 696e 6720 2020 2020 2020 2020  entRing         
│ │ │ │ -0006aff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b000: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0006afc0: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20          |.|o2 : 
│ │ │ │ +0006afd0: 5175 6f74 6965 6e74 5269 6e67 2020 2020  QuotientRing    
│ │ │ │ +0006afe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006aff0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006b000: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  0006b010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006b020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006b030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006b040: 2d2b 0a7c 6933 203a 204d 203d 2052 5e31  -+.|i3 : M = R^1
│ │ │ │ -0006b050: 2f69 6465 616c 2878 2c79 2c7a 2920 2020  /ideal(x,y,z)   
│ │ │ │ -0006b060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b070: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0006b030: 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 4d20  ------+.|i3 : M 
│ │ │ │ +0006b040: 3d20 525e 312f 6964 6561 6c28 782c 792c  = R^1/ideal(x,y,
│ │ │ │ +0006b050: 7a29 2020 2020 2020 2020 2020 2020 2020  z)              
│ │ │ │ +0006b060: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0006b070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b0a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0006b0b0: 0a7c 6f33 203d 2063 6f6b 6572 6e65 6c20  .|o3 = cokernel 
│ │ │ │ -0006b0c0: 7c20 7820 7920 7a20 7c20 2020 2020 2020  | x y z |       
│ │ │ │ -0006b0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b0e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0006b0a0: 2020 2020 7c0a 7c6f 3320 3d20 636f 6b65      |.|o3 = coke
│ │ │ │ +0006b0b0: 726e 656c 207c 2078 2079 207a 207c 2020  rnel | x y z |  
│ │ │ │ +0006b0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b0d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0006b0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b110: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0006b110: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0006b120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b130: 2020 2020 2020 2020 2020 2020 3120 2020              1   
│ │ │ │ -0006b140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b150: 2020 2020 7c0a 7c6f 3320 3a20 522d 6d6f      |.|o3 : R-mo
│ │ │ │ -0006b160: 6475 6c65 2c20 7175 6f74 6965 6e74 206f  dule, quotient o
│ │ │ │ -0006b170: 6620 5220 2020 2020 2020 2020 2020 2020  f R             
│ │ │ │ -0006b180: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0006b130: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │ +0006b140: 2020 2020 2020 2020 207c 0a7c 6f33 203a           |.|o3 :
│ │ │ │ +0006b150: 2052 2d6d 6f64 756c 652c 2071 756f 7469   R-module, quoti
│ │ │ │ +0006b160: 656e 7420 6f66 2052 2020 2020 2020 2020  ent of R        
│ │ │ │ +0006b170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b180: 7c0a 2b2d 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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006b1c0: 2d2d 2b0a 7c69 3420 3a20 4620 3d20 6672  --+.|i4 : F = fr
│ │ │ │ -0006b1d0: 6565 5265 736f 6c75 7469 6f6e 284d 2c20  eeResolution(M, 
│ │ │ │ -0006b1e0: 4c65 6e67 7468 4c69 6d69 7420 3d3e 2034  LengthLimit => 4
│ │ │ │ -0006b1f0: 2920 2020 2020 2020 207c 0a7c 2020 2020  )        |.|    
│ │ │ │ +0006b1b0: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2046  -------+.|i4 : F
│ │ │ │ +0006b1c0: 203d 2066 7265 6552 6573 6f6c 7574 696f   = freeResolutio
│ │ │ │ +0006b1d0: 6e28 4d2c 204c 656e 6774 684c 696d 6974  n(M, LengthLimit
│ │ │ │ +0006b1e0: 203d 3e20 3429 2020 2020 2020 2020 7c0a   => 4)        |.
│ │ │ │ +0006b1f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0006b200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b230: 7c0a 7c20 2020 2020 2031 2020 2020 2020  |.|      1      
│ │ │ │ -0006b240: 3320 2020 2020 2035 2020 2020 2020 3720  3      5      7 
│ │ │ │ -0006b250: 2020 2020 2039 2020 2020 2020 2020 2020       9          
│ │ │ │ -0006b260: 2020 2020 2020 207c 0a7c 6f34 203d 2052         |.|o4 = R
│ │ │ │ -0006b270: 2020 3c2d 2d20 5220 203c 2d2d 2052 2020    <-- R  <-- R  
│ │ │ │ -0006b280: 3c2d 2d20 5220 203c 2d2d 2052 2020 2020  <-- R  <-- R    
│ │ │ │ -0006b290: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0006b2a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0006b220: 2020 2020 207c 0a7c 2020 2020 2020 3120       |.|      1 
│ │ │ │ +0006b230: 2020 2020 2033 2020 2020 2020 3520 2020       3      5   
│ │ │ │ +0006b240: 2020 2037 2020 2020 2020 3920 2020 2020     7      9     
│ │ │ │ +0006b250: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0006b260: 3420 3d20 5220 203c 2d2d 2052 2020 3c2d  4 = R  <-- R  <-
│ │ │ │ +0006b270: 2d20 5220 203c 2d2d 2052 2020 3c2d 2d20  - R  <-- R  <-- 
│ │ │ │ +0006b280: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │ +0006b290: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006b2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b2d0: 2020 2020 207c 0a7c 2020 2020 2030 2020       |.|     0  
│ │ │ │ -0006b2e0: 2020 2020 3120 2020 2020 2032 2020 2020      1      2    
│ │ │ │ -0006b2f0: 2020 3320 2020 2020 2034 2020 2020 2020    3      4      
│ │ │ │ -0006b300: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0006b2c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0006b2d0: 2020 3020 2020 2020 2031 2020 2020 2020    0      1      
│ │ │ │ +0006b2e0: 3220 2020 2020 2033 2020 2020 2020 3420  2      3      4 
│ │ │ │ +0006b2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b300: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0006b310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b340: 2020 207c 0a7c 6f34 203a 2043 6f6d 706c     |.|o4 : Compl
│ │ │ │ -0006b350: 6578 2020 2020 2020 2020 2020 2020 2020  ex              
│ │ │ │ -0006b360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b370: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0006b330: 2020 2020 2020 2020 7c0a 7c6f 3420 3a20          |.|o4 : 
│ │ │ │ +0006b340: 436f 6d70 6c65 7820 2020 2020 2020 2020  Complex         
│ │ │ │ +0006b350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b360: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006b370: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  0006b380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006b390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006b3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006b3b0: 2d2b 0a7c 6935 203a 2066 6620 3d20 6d61  -+.|i5 : ff = ma
│ │ │ │ -0006b3c0: 7472 6978 7b7b 7a5e 337d 7d20 2020 2020  trix{{z^3}}     
│ │ │ │ -0006b3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b3e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0006b3a0: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 6666  ------+.|i5 : ff
│ │ │ │ +0006b3b0: 203d 206d 6174 7269 787b 7b7a 5e33 7d7d   = matrix{{z^3}}
│ │ │ │ +0006b3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b3d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0006b3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b410: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0006b420: 0a7c 6f35 203d 207c 207a 3320 7c20 2020  .|o5 = | z3 |   
│ │ │ │ +0006b410: 2020 2020 7c0a 7c6f 3520 3d20 7c20 7a33      |.|o5 = | z3
│ │ │ │ +0006b420: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │  0006b430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b450: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0006b440: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0006b450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b480: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0006b490: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │ -0006b4a0: 2020 2020 3120 2020 2020 2020 2020 2020      1           
│ │ │ │ -0006b4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b4c0: 2020 2020 7c0a 7c6f 3520 3a20 4d61 7472      |.|o5 : Matr
│ │ │ │ -0006b4d0: 6978 2052 2020 3c2d 2d20 5220 2020 2020  ix R  <-- R     
│ │ │ │ +0006b480: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0006b490: 2020 3120 2020 2020 2031 2020 2020 2020    1      1      
│ │ │ │ +0006b4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b4b0: 2020 2020 2020 2020 207c 0a7c 6f35 203a           |.|o5 :
│ │ │ │ +0006b4c0: 204d 6174 7269 7820 5220 203c 2d2d 2052   Matrix R  <-- R
│ │ │ │ +0006b4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b4f0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0006b4f0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  0006b500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006b510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006b520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006b530: 2d2d 2b0a 7c69 3620 3a20 5231 203d 2052  --+.|i6 : R1 = R
│ │ │ │ -0006b540: 2f69 6465 616c 2066 6620 2020 2020 2020  /ideal ff       
│ │ │ │ -0006b550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b560: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0006b520: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 2052  -------+.|i6 : R
│ │ │ │ +0006b530: 3120 3d20 522f 6964 6561 6c20 6666 2020  1 = R/ideal ff  
│ │ │ │ +0006b540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b550: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0006b560: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0006b570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b5a0: 7c0a 7c6f 3620 3d20 5231 2020 2020 2020  |.|o6 = R1      
│ │ │ │ +0006b590: 2020 2020 207c 0a7c 6f36 203d 2052 3120       |.|o6 = R1 
│ │ │ │ +0006b5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b5d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0006b5c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0006b5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b600: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0006b610: 7c6f 3620 3a20 5175 6f74 6965 6e74 5269  |o6 : QuotientRi
│ │ │ │ -0006b620: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │ -0006b630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b640: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0006b600: 2020 207c 0a7c 6f36 203a 2051 756f 7469     |.|o6 : Quoti
│ │ │ │ +0006b610: 656e 7452 696e 6720 2020 2020 2020 2020  entRing         
│ │ │ │ +0006b620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b630: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0006b640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006b650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006b660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006b670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0006b680: 3720 3a20 6265 7474 6920 4620 2020 2020  7 : betti F     
│ │ │ │ +0006b670: 2d2b 0a7c 6937 203a 2062 6574 7469 2046  -+.|i7 : betti F
│ │ │ │ +0006b680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b6b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006b6a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0006b6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b6e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0006b6f0: 2020 2020 2020 2020 2030 2031 2032 2033           0 1 2 3
│ │ │ │ -0006b700: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ -0006b710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b720: 207c 0a7c 6f37 203d 2074 6f74 616c 3a20   |.|o7 = total: 
│ │ │ │ -0006b730: 3120 3320 3520 3720 3920 2020 2020 2020  1 3 5 7 9       
│ │ │ │ -0006b740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b750: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0006b760: 2020 2020 303a 2031 2033 2033 2031 202e      0: 1 3 3 1 .
│ │ │ │ +0006b6d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006b6e0: 0a7c 2020 2020 2020 2020 2020 2020 3020  .|            0 
│ │ │ │ +0006b6f0: 3120 3220 3320 3420 2020 2020 2020 2020  1 2 3 4         
│ │ │ │ +0006b700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b710: 2020 2020 2020 7c0a 7c6f 3720 3d20 746f        |.|o7 = to
│ │ │ │ +0006b720: 7461 6c3a 2031 2033 2035 2037 2039 2020  tal: 1 3 5 7 9  
│ │ │ │ +0006b730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b740: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0006b750: 2020 2020 2020 2020 2030 3a20 3120 3320           0: 1 3 
│ │ │ │ +0006b760: 3320 3120 2e20 2020 2020 2020 2020 2020  3 1 .           
│ │ │ │  0006b770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b780: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0006b790: 0a7c 2020 2020 2020 2020 2031 3a20 2e20  .|         1: . 
│ │ │ │ -0006b7a0: 2e20 3220 3620 3620 2020 2020 2020 2020  . 2 6 6         
│ │ │ │ -0006b7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b7c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0006b7d0: 2020 323a 202e 202e 202e 202e 2033 2020    2: . . . . 3  
│ │ │ │ +0006b780: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0006b790: 313a 202e 202e 2032 2036 2036 2020 2020  1: . . 2 6 6    
│ │ │ │ +0006b7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b7b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0006b7c0: 2020 2020 2020 2032 3a20 2e20 2e20 2e20         2: . . . 
│ │ │ │ +0006b7d0: 2e20 3320 2020 2020 2020 2020 2020 2020  . 3             
│ │ │ │  0006b7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b7f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0006b7f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0006b800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b830: 2020 2020 7c0a 7c6f 3720 3a20 4265 7474      |.|o7 : Bett
│ │ │ │ -0006b840: 6954 616c 6c79 2020 2020 2020 2020 2020  iTally          
│ │ │ │ +0006b820: 2020 2020 2020 2020 207c 0a7c 6f37 203a           |.|o7 :
│ │ │ │ +0006b830: 2042 6574 7469 5461 6c6c 7920 2020 2020   BettiTally     
│ │ │ │ +0006b840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b860: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0006b860: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  0006b870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006b880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006b890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006b8a0: 2d2d 2b0a 7c69 3820 3a20 4646 203d 2053  --+.|i8 : FF = S
│ │ │ │ -0006b8b0: 6861 6d61 7368 2866 662c 462c 3429 2020  hamash(ff,F,4)  
│ │ │ │ -0006b8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b8d0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0006b890: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 2046  -------+.|i8 : F
│ │ │ │ +0006b8a0: 4620 3d20 5368 616d 6173 6828 6666 2c46  F = Shamash(ff,F
│ │ │ │ +0006b8b0: 2c34 2920 2020 2020 2020 2020 2020 2020  ,4)             
│ │ │ │ +0006b8c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0006b8d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0006b8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b910: 7c0a 7c20 2020 2020 2f20 525c 3120 2020  |.|     / R\1   
│ │ │ │ -0006b920: 2020 2f20 525c 3320 2020 2020 2f20 525c    / R\3     / R\
│ │ │ │ -0006b930: 3620 2020 2020 2f20 525c 3130 2020 2020  6     / R\10    
│ │ │ │ -0006b940: 202f 2052 5c31 357c 0a7c 6f38 203d 207c   / R\15|.|o8 = |
│ │ │ │ -0006b950: 2d2d 7c20 203c 2d2d 207c 2d2d 7c20 203c  --|  <-- |--|  <
│ │ │ │ -0006b960: 2d2d 207c 2d2d 7c20 203c 2d2d 207c 2d2d  -- |--|  <-- |--
│ │ │ │ -0006b970: 7c20 2020 3c2d 2d20 7c2d 2d7c 2020 7c0a  |   <-- |--|  |.
│ │ │ │ -0006b980: 7c20 2020 2020 7c20 337c 2020 2020 2020  |     | 3|      
│ │ │ │ -0006b990: 7c20 337c 2020 2020 2020 7c20 337c 2020  | 3|      | 3|  
│ │ │ │ -0006b9a0: 2020 2020 7c20 337c 2020 2020 2020 207c      | 3|       |
│ │ │ │ -0006b9b0: 2033 7c20 207c 0a7c 2020 2020 205c 7a20   3|  |.|     \z 
│ │ │ │ -0006b9c0: 2f20 2020 2020 205c 7a20 2f20 2020 2020  /      \z /     
│ │ │ │ -0006b9d0: 205c 7a20 2f20 2020 2020 205c 7a20 2f20   \z /      \z / 
│ │ │ │ -0006b9e0: 2020 2020 2020 5c7a 202f 2020 7c0a 7c20        \z /  |.| 
│ │ │ │ +0006b900: 2020 2020 207c 0a7c 2020 2020 202f 2052       |.|     / R
│ │ │ │ +0006b910: 5c31 2020 2020 202f 2052 5c33 2020 2020  \1     / R\3    
│ │ │ │ +0006b920: 202f 2052 5c36 2020 2020 202f 2052 5c31   / R\6     / R\1
│ │ │ │ +0006b930: 3020 2020 2020 2f20 525c 3135 7c0a 7c6f  0     / R\15|.|o
│ │ │ │ +0006b940: 3820 3d20 7c2d 2d7c 2020 3c2d 2d20 7c2d  8 = |--|  <-- |-
│ │ │ │ +0006b950: 2d7c 2020 3c2d 2d20 7c2d 2d7c 2020 3c2d  -|  <-- |--|  <-
│ │ │ │ +0006b960: 2d20 7c2d 2d7c 2020 203c 2d2d 207c 2d2d  - |--|   <-- |--
│ │ │ │ +0006b970: 7c20 207c 0a7c 2020 2020 207c 2033 7c20  |  |.|     | 3| 
│ │ │ │ +0006b980: 2020 2020 207c 2033 7c20 2020 2020 207c       | 3|      |
│ │ │ │ +0006b990: 2033 7c20 2020 2020 207c 2033 7c20 2020   3|      | 3|   
│ │ │ │ +0006b9a0: 2020 2020 7c20 337c 2020 7c0a 7c20 2020      | 3|  |.|   
│ │ │ │ +0006b9b0: 2020 5c7a 202f 2020 2020 2020 5c7a 202f    \z /      \z /
│ │ │ │ +0006b9c0: 2020 2020 2020 5c7a 202f 2020 2020 2020        \z /      
│ │ │ │ +0006b9d0: 5c7a 202f 2020 2020 2020 205c 7a20 2f20  \z /       \z / 
│ │ │ │ +0006b9e0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0006b9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ba00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ba10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ba20: 2020 207c 0a7c 2020 2020 2030 2020 2020     |.|     0    
│ │ │ │ -0006ba30: 2020 2020 2031 2020 2020 2020 2020 2032       1         2
│ │ │ │ -0006ba40: 2020 2020 2020 2020 2033 2020 2020 2020           3      
│ │ │ │ -0006ba50: 2020 2020 3420 2020 2020 7c0a 7c20 2020      4     |.|   
│ │ │ │ +0006ba10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0006ba20: 3020 2020 2020 2020 2020 3120 2020 2020  0         1     
│ │ │ │ +0006ba30: 2020 2020 3220 2020 2020 2020 2020 3320      2         3 
│ │ │ │ +0006ba40: 2020 2020 2020 2020 2034 2020 2020 207c           4     |
│ │ │ │ +0006ba50: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0006ba60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ba70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ba80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ba90: 207c 0a7c 6f38 203a 2043 6f6d 706c 6578   |.|o8 : Complex
│ │ │ │ +0006ba80: 2020 2020 2020 7c0a 7c6f 3820 3a20 436f        |.|o8 : Co
│ │ │ │ +0006ba90: 6d70 6c65 7820 2020 2020 2020 2020 2020  mplex           
│ │ │ │  0006baa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bac0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0006bab0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0006bac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006bad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006bae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006baf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0006bb00: 0a7c 6939 203a 2047 4720 3d20 5368 616d  .|i9 : GG = Sham
│ │ │ │ -0006bb10: 6173 6828 5231 2c46 2c34 2920 2020 2020  ash(R1,F,4)     
│ │ │ │ -0006bb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bb30: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0006baf0: 2d2d 2d2d 2b0a 7c69 3920 3a20 4747 203d  ----+.|i9 : GG =
│ │ │ │ +0006bb00: 2053 6861 6d61 7368 2852 312c 462c 3429   Shamash(R1,F,4)
│ │ │ │ +0006bb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006bb20: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0006bb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006bb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006bb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bb60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0006bb70: 2020 2020 2020 2031 2020 2020 2020 2033         1       3
│ │ │ │ -0006bb80: 2020 2020 2020 2036 2020 2020 2020 2031         6       1
│ │ │ │ -0006bb90: 3020 2020 2020 2020 3135 2020 2020 2020  0       15      
│ │ │ │ -0006bba0: 2020 2020 7c0a 7c6f 3920 3d20 5231 2020      |.|o9 = R1  
│ │ │ │ -0006bbb0: 3c2d 2d20 5231 2020 3c2d 2d20 5231 2020  <-- R1  <-- R1  
│ │ │ │ -0006bbc0: 3c2d 2d20 5231 2020 203c 2d2d 2052 3120  <-- R1   <-- R1 
│ │ │ │ -0006bbd0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0006bb60: 2020 7c0a 7c20 2020 2020 2020 3120 2020    |.|       1   
│ │ │ │ +0006bb70: 2020 2020 3320 2020 2020 2020 3620 2020      3       6   
│ │ │ │ +0006bb80: 2020 2020 3130 2020 2020 2020 2031 3520      10       15 
│ │ │ │ +0006bb90: 2020 2020 2020 2020 207c 0a7c 6f39 203d           |.|o9 =
│ │ │ │ +0006bba0: 2052 3120 203c 2d2d 2052 3120 203c 2d2d   R1  <-- R1  <--
│ │ │ │ +0006bbb0: 2052 3120 203c 2d2d 2052 3120 2020 3c2d   R1  <-- R1   <-
│ │ │ │ +0006bbc0: 2d20 5231 2020 2020 2020 2020 2020 2020  - R1            
│ │ │ │ +0006bbd0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0006bbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006bbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bc10: 2020 7c0a 7c20 2020 2020 3020 2020 2020    |.|     0     
│ │ │ │ -0006bc20: 2020 3120 2020 2020 2020 3220 2020 2020    1       2     
│ │ │ │ -0006bc30: 2020 3320 2020 2020 2020 2034 2020 2020    3        4    
│ │ │ │ -0006bc40: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0006bc00: 2020 2020 2020 207c 0a7c 2020 2020 2030         |.|     0
│ │ │ │ +0006bc10: 2020 2020 2020 2031 2020 2020 2020 2032         1       2
│ │ │ │ +0006bc20: 2020 2020 2020 2033 2020 2020 2020 2020         3        
│ │ │ │ +0006bc30: 3420 2020 2020 2020 2020 2020 2020 7c0a  4             |.
│ │ │ │ +0006bc40: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0006bc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006bc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bc80: 7c0a 7c6f 3920 3a20 436f 6d70 6c65 7820  |.|o9 : Complex 
│ │ │ │ +0006bc70: 2020 2020 207c 0a7c 6f39 203a 2043 6f6d       |.|o9 : Com
│ │ │ │ +0006bc80: 706c 6578 2020 2020 2020 2020 2020 2020  plex            
│ │ │ │  0006bc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bcb0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0006bca0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0006bcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006bcc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006bcd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006bce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0006bcf0: 7c69 3130 203a 2062 6574 7469 2046 4620  |i10 : betti FF 
│ │ │ │ +0006bce0: 2d2d 2d2b 0a7c 6931 3020 3a20 6265 7474  ---+.|i10 : bett
│ │ │ │ +0006bcf0: 6920 4646 2020 2020 2020 2020 2020 2020  i FF            
│ │ │ │  0006bd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bd20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006bd10: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0006bd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006bd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006bd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bd50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0006bd60: 2020 2020 2020 2020 2020 2020 3020 3120              0 1 
│ │ │ │ -0006bd70: 3220 2033 2020 3420 2020 2020 2020 2020  2  3  4         
│ │ │ │ -0006bd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bd90: 2020 207c 0a7c 6f31 3020 3d20 746f 7461     |.|o10 = tota
│ │ │ │ -0006bda0: 6c3a 2031 2033 2036 2031 3020 3135 2020  l: 1 3 6 10 15  
│ │ │ │ -0006bdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bdc0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0006bdd0: 2020 2020 2020 2030 3a20 3120 3320 3320         0: 1 3 3 
│ │ │ │ -0006bde0: 2031 2020 2e20 2020 2020 2020 2020 2020   1  .           
│ │ │ │ -0006bdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006be00: 207c 0a7c 2020 2020 2020 2020 2020 313a   |.|          1:
│ │ │ │ -0006be10: 202e 202e 2033 2020 3920 2039 2020 2020   . . 3  9  9    
│ │ │ │ -0006be20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006be30: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0006be40: 2020 2020 2032 3a20 2e20 2e20 2e20 202e       2: . . .  .
│ │ │ │ -0006be50: 2020 3620 2020 2020 2020 2020 2020 2020    6             
│ │ │ │ -0006be60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0006be70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0006bd50: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0006bd60: 2030 2031 2032 2020 3320 2034 2020 2020   0 1 2  3  4    
│ │ │ │ +0006bd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006bd80: 2020 2020 2020 2020 7c0a 7c6f 3130 203d          |.|o10 =
│ │ │ │ +0006bd90: 2074 6f74 616c 3a20 3120 3320 3620 3130   total: 1 3 6 10
│ │ │ │ +0006bda0: 2031 3520 2020 2020 2020 2020 2020 2020   15             
│ │ │ │ +0006bdb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006bdc0: 0a7c 2020 2020 2020 2020 2020 303a 2031  .|          0: 1
│ │ │ │ +0006bdd0: 2033 2033 2020 3120 202e 2020 2020 2020   3 3  1  .      
│ │ │ │ +0006bde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006bdf0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0006be00: 2020 2031 3a20 2e20 2e20 3320 2039 2020     1: . . 3  9  
│ │ │ │ +0006be10: 3920 2020 2020 2020 2020 2020 2020 2020  9               
│ │ │ │ +0006be20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0006be30: 2020 2020 2020 2020 2020 323a 202e 202e            2: . .
│ │ │ │ +0006be40: 202e 2020 2e20 2036 2020 2020 2020 2020   .  .  6        
│ │ │ │ +0006be50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006be60: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0006be70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006be80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006be90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bea0: 2020 2020 2020 7c0a 7c6f 3130 203a 2042        |.|o10 : B
│ │ │ │ -0006beb0: 6574 7469 5461 6c6c 7920 2020 2020 2020  ettiTally       
│ │ │ │ +0006be90: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0006bea0: 3020 3a20 4265 7474 6954 616c 6c79 2020  0 : BettiTally  
│ │ │ │ +0006beb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006bec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bed0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0006bed0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  0006bee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006bef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006bf00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006bf10: 2d2d 2d2d 2b0a 7c69 3131 203a 2062 6574  ----+.|i11 : bet
│ │ │ │ -0006bf20: 7469 2047 4720 2020 2020 2020 2020 2020  ti GG           
│ │ │ │ +0006bf00: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120  ---------+.|i11 
│ │ │ │ +0006bf10: 3a20 6265 7474 6920 4747 2020 2020 2020  : betti GG      
│ │ │ │ +0006bf20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006bf30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bf40: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0006bf40: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0006bf50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006bf60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bf70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bf80: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0006bf90: 2020 3020 3120 3220 2033 2020 3420 2020    0 1 2  3  4   
│ │ │ │ -0006bfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bfb0: 2020 2020 2020 2020 207c 0a7c 6f31 3120           |.|o11 
│ │ │ │ -0006bfc0: 3d20 746f 7461 6c3a 2031 2033 2036 2031  = total: 1 3 6 1
│ │ │ │ -0006bfd0: 3020 3135 2020 2020 2020 2020 2020 2020  0 15            
│ │ │ │ -0006bfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bff0: 7c0a 7c20 2020 2020 2020 2020 2030 3a20  |.|          0: 
│ │ │ │ -0006c000: 3120 3320 3320 2031 2020 2e20 2020 2020  1 3 3  1  .     
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│ │ │ │ +0006c770: 6e63 650a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  nce.************
│ │ │ │  0006c780: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0006c790: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0006c7a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0006c7b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0006c7c0: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20  ***..  * Usage: 
│ │ │ │ -0006c7d0: 0a20 2020 2020 2020 2078 203d 2073 706c  .        x = spl
│ │ │ │ -0006c7e0: 6974 7469 6e67 7328 612c 6229 0a20 202a  ittings(a,b).  *
│ │ │ │ -0006c7f0: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ -0006c800: 2061 2c20 6120 2a6e 6f74 6520 6d61 7472   a, a *note matr
│ │ │ │ -0006c810: 6978 3a20 284d 6163 6175 6c61 7932 446f  ix: (Macaulay2Do
│ │ │ │ -0006c820: 6329 4d61 7472 6978 2c2c 206d 6170 7320  c)Matrix,, maps 
│ │ │ │ -0006c830: 696e 746f 2074 6865 206b 6572 6e65 6c20  into the kernel 
│ │ │ │ -0006c840: 6f66 2062 0a20 2020 2020 202a 2062 2c20  of b.      * b, 
│ │ │ │ -0006c850: 6120 2a6e 6f74 6520 6d61 7472 6978 3a20  a *note matrix: 
│ │ │ │ -0006c860: 284d 6163 6175 6c61 7932 446f 6329 4d61  (Macaulay2Doc)Ma
│ │ │ │ -0006c870: 7472 6978 2c2c 2072 6570 7265 7365 6e74  trix,, represent
│ │ │ │ -0006c880: 696e 6720 6120 7375 726a 6563 7469 6f6e  ing a surjection
│ │ │ │ -0006c890: 0a20 2020 2020 2020 2066 726f 6d20 7461  .        from ta
│ │ │ │ -0006c8a0: 7267 6574 2061 2074 6f20 6120 6672 6565  rget a to a free
│ │ │ │ -0006c8b0: 206d 6f64 756c 650a 2020 2a20 4f75 7470   module.  * Outp
│ │ │ │ -0006c8c0: 7574 733a 0a20 2020 2020 202a 204c 2c20  uts:.      * L, 
│ │ │ │ -0006c8d0: 6120 2a6e 6f74 6520 6c69 7374 3a20 284d  a *note list: (M
│ │ │ │ -0006c8e0: 6163 6175 6c61 7932 446f 6329 4c69 7374  acaulay2Doc)List
│ │ │ │ -0006c8f0: 2c2c 204c 203d 205c 7b73 6967 6d61 2c74  ,, L = \{sigma,t
│ │ │ │ -0006c900: 6175 5c7d 2c20 7370 6c69 7474 696e 6773  au\}, splittings
│ │ │ │ -0006c910: 206f 660a 2020 2020 2020 2020 612c 6220   of.        a,b 
│ │ │ │ -0006c920: 7265 7370 6563 7469 7665 6c79 0a0a 4465  respectively..De
│ │ │ │ -0006c930: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d  scription.======
│ │ │ │ -0006c940: 3d3d 3d3d 3d0a 0a41 7373 756d 696e 6720  =====..Assuming 
│ │ │ │ -0006c950: 7468 6174 2028 612c 6229 2061 7265 2074  that (a,b) are t
│ │ │ │ -0006c960: 6865 206d 6170 7320 6f66 2061 2072 6967  he maps of a rig
│ │ │ │ -0006c970: 6874 2065 7861 6374 2073 6571 7565 6e63  ht exact sequenc
│ │ │ │ -0006c980: 650a 0a24 305c 746f 2041 5c74 6f20 425c  e..$0\to A\to B\
│ │ │ │ -0006c990: 746f 2043 205c 746f 2030 240a 0a77 6974  to C \to 0$..wit
│ │ │ │ -0006c9a0: 6820 422c 2043 2066 7265 652c 2074 6865  h B, C free, the
│ │ │ │ -0006c9b0: 2073 6372 6970 7420 7072 6f64 7563 6573   script produces
│ │ │ │ -0006c9c0: 2061 2070 6169 7220 6f66 206d 6170 7320   a pair of maps 
│ │ │ │ -0006c9d0: 7369 676d 612c 2074 6175 2077 6974 6820  sigma, tau with 
│ │ │ │ -0006c9e0: 2474 6175 3a20 4320 5c74 6f0a 4224 2061  $tau: C \to.B$ a
│ │ │ │ -0006c9f0: 2073 706c 6974 7469 6e67 206f 6620 6220   splitting of b 
│ │ │ │ -0006ca00: 616e 6420 2473 6967 6d61 3a20 4220 5c74  and $sigma: B \t
│ │ │ │ -0006ca10: 6f20 4124 2061 2073 706c 6974 7469 6e67  o A$ a splitting
│ │ │ │ -0006ca20: 206f 6620 613b 2074 6861 7420 6973 2c0a   of a; that is,.
│ │ │ │ -0006ca30: 0a24 612a 7369 676d 612b 7461 752a 6220  .$a*sigma+tau*b 
│ │ │ │ -0006ca40: 3d20 315f 4224 0a0a 2473 6967 6d61 2a61  = 1_B$..$sigma*a
│ │ │ │ -0006ca50: 203d 2031 5f41 240a 0a24 622a 7461 7520   = 1_A$..$b*tau 
│ │ │ │ -0006ca60: 3d20 315f 4324 0a0a 2b2d 2d2d 2d2d 2d2d  = 1_C$..+-------
│ │ │ │ +0006c7b0: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573  ********..  * Us
│ │ │ │ +0006c7c0: 6167 653a 200a 2020 2020 2020 2020 7820  age: .        x 
│ │ │ │ +0006c7d0: 3d20 7370 6c69 7474 696e 6773 2861 2c62  = splittings(a,b
│ │ │ │ +0006c7e0: 290a 2020 2a20 496e 7075 7473 3a0a 2020  ).  * Inputs:.  
│ │ │ │ +0006c7f0: 2020 2020 2a20 612c 2061 202a 6e6f 7465      * a, a *note
│ │ │ │ +0006c800: 206d 6174 7269 783a 2028 4d61 6361 756c   matrix: (Macaul
│ │ │ │ +0006c810: 6179 3244 6f63 294d 6174 7269 782c 2c20  ay2Doc)Matrix,, 
│ │ │ │ +0006c820: 6d61 7073 2069 6e74 6f20 7468 6520 6b65  maps into the ke
│ │ │ │ +0006c830: 726e 656c 206f 6620 620a 2020 2020 2020  rnel of b.      
│ │ │ │ +0006c840: 2a20 622c 2061 202a 6e6f 7465 206d 6174  * b, a *note mat
│ │ │ │ +0006c850: 7269 783a 2028 4d61 6361 756c 6179 3244  rix: (Macaulay2D
│ │ │ │ +0006c860: 6f63 294d 6174 7269 782c 2c20 7265 7072  oc)Matrix,, repr
│ │ │ │ +0006c870: 6573 656e 7469 6e67 2061 2073 7572 6a65  esenting a surje
│ │ │ │ +0006c880: 6374 696f 6e0a 2020 2020 2020 2020 6672  ction.        fr
│ │ │ │ +0006c890: 6f6d 2074 6172 6765 7420 6120 746f 2061  om target a to a
│ │ │ │ +0006c8a0: 2066 7265 6520 6d6f 6475 6c65 0a20 202a   free module.  *
│ │ │ │ +0006c8b0: 204f 7574 7075 7473 3a0a 2020 2020 2020   Outputs:.      
│ │ │ │ +0006c8c0: 2a20 4c2c 2061 202a 6e6f 7465 206c 6973  * L, a *note lis
│ │ │ │ +0006c8d0: 743a 2028 4d61 6361 756c 6179 3244 6f63  t: (Macaulay2Doc
│ │ │ │ +0006c8e0: 294c 6973 742c 2c20 4c20 3d20 5c7b 7369  )List,, L = \{si
│ │ │ │ +0006c8f0: 676d 612c 7461 755c 7d2c 2073 706c 6974  gma,tau\}, split
│ │ │ │ +0006c900: 7469 6e67 7320 6f66 0a20 2020 2020 2020  tings of.       
│ │ │ │ +0006c910: 2061 2c62 2072 6573 7065 6374 6976 656c   a,b respectivel
│ │ │ │ +0006c920: 790a 0a44 6573 6372 6970 7469 6f6e 0a3d  y..Description.=
│ │ │ │ +0006c930: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 4173 7375  ==========..Assu
│ │ │ │ +0006c940: 6d69 6e67 2074 6861 7420 2861 2c62 2920  ming that (a,b) 
│ │ │ │ +0006c950: 6172 6520 7468 6520 6d61 7073 206f 6620  are the maps of 
│ │ │ │ +0006c960: 6120 7269 6768 7420 6578 6163 7420 7365  a right exact se
│ │ │ │ +0006c970: 7175 656e 6365 0a0a 2430 5c74 6f20 415c  quence..$0\to A\
│ │ │ │ +0006c980: 746f 2042 5c74 6f20 4320 5c74 6f20 3024  to B\to C \to 0$
│ │ │ │ +0006c990: 0a0a 7769 7468 2042 2c20 4320 6672 6565  ..with B, C free
│ │ │ │ +0006c9a0: 2c20 7468 6520 7363 7269 7074 2070 726f  , the script pro
│ │ │ │ +0006c9b0: 6475 6365 7320 6120 7061 6972 206f 6620  duces a pair of 
│ │ │ │ +0006c9c0: 6d61 7073 2073 6967 6d61 2c20 7461 7520  maps sigma, tau 
│ │ │ │ +0006c9d0: 7769 7468 2024 7461 753a 2043 205c 746f  with $tau: C \to
│ │ │ │ +0006c9e0: 0a42 2420 6120 7370 6c69 7474 696e 6720  .B$ a splitting 
│ │ │ │ +0006c9f0: 6f66 2062 2061 6e64 2024 7369 676d 613a  of b and $sigma:
│ │ │ │ +0006ca00: 2042 205c 746f 2041 2420 6120 7370 6c69   B \to A$ a spli
│ │ │ │ +0006ca10: 7474 696e 6720 6f66 2061 3b20 7468 6174  tting of a; that
│ │ │ │ +0006ca20: 2069 732c 0a0a 2461 2a73 6967 6d61 2b74   is,..$a*sigma+t
│ │ │ │ +0006ca30: 6175 2a62 203d 2031 5f42 240a 0a24 7369  au*b = 1_B$..$si
│ │ │ │ +0006ca40: 676d 612a 6120 3d20 315f 4124 0a0a 2462  gma*a = 1_A$..$b
│ │ │ │ +0006ca50: 2a74 6175 203d 2031 5f43 240a 0a2b 2d2d  *tau = 1_C$..+--
│ │ │ │ +0006ca60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006ca70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006ca80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006ca90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006caa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006cab0: 2d2d 2d2b 0a7c 6931 203a 206b 6b3d 205a  ---+.|i1 : kk= Z
│ │ │ │ -0006cac0: 5a2f 3130 3120 2020 2020 2020 2020 2020  Z/101           
│ │ │ │ +0006caa0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20  --------+.|i1 : 
│ │ │ │ +0006cab0: 6b6b 3d20 5a5a 2f31 3031 2020 2020 2020  kk= ZZ/101      
│ │ │ │ +0006cac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006caf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cb00: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0006caf0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006cb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cb40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0006cb50: 6f31 203d 206b 6b20 2020 2020 2020 2020  o1 = kk         
│ │ │ │ +0006cb40: 2020 7c0a 7c6f 3120 3d20 6b6b 2020 2020    |.|o1 = kk    
│ │ │ │ +0006cb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cb90: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0006cb80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006cb90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0006cba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cbe0: 2020 2020 2020 207c 0a7c 6f31 203a 2051         |.|o1 : Q
│ │ │ │ -0006cbf0: 756f 7469 656e 7452 696e 6720 2020 2020  uotientRing     
│ │ │ │ +0006cbd0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0006cbe0: 3120 3a20 5175 6f74 6965 6e74 5269 6e67  1 : QuotientRing
│ │ │ │ +0006cbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cc30: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0006cc20: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0006cc30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006cc40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006cc50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006cc60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006cc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006cc80: 2d2b 0a7c 6932 203a 2053 203d 206b 6b5b  -+.|i2 : S = kk[
│ │ │ │ -0006cc90: 782c 792c 7a5d 2020 2020 2020 2020 2020  x,y,z]          
│ │ │ │ +0006cc70: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 5320  ------+.|i2 : S 
│ │ │ │ +0006cc80: 3d20 6b6b 5b78 2c79 2c7a 5d20 2020 2020  = kk[x,y,z]     
│ │ │ │ +0006cc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ccb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ccc0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0006ccd0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0006ccc0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006ccd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ccf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cd10: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ -0006cd20: 203d 2053 2020 2020 2020 2020 2020 2020   = S            
│ │ │ │ +0006cd10: 7c0a 7c6f 3220 3d20 5320 2020 2020 2020  |.|o2 = S       
│ │ │ │ +0006cd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cd60: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0006cd50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0006cd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cdb0: 2020 2020 207c 0a7c 6f32 203a 2050 6f6c       |.|o2 : Pol
│ │ │ │ -0006cdc0: 796e 6f6d 6961 6c52 696e 6720 2020 2020  ynomialRing     
│ │ │ │ +0006cda0: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ +0006cdb0: 3a20 506f 6c79 6e6f 6d69 616c 5269 6e67  : PolynomialRing
│ │ │ │ +0006cdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ce00: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0006cdf0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0006ce00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006ce10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006ce20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006ce30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006ce40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0006ce50: 0a7c 6933 203a 2073 6574 5261 6e64 6f6d  .|i3 : setRandom
│ │ │ │ -0006ce60: 5365 6564 2030 2020 2020 2020 2020 2020  Seed 0          
│ │ │ │ +0006ce40: 2d2d 2d2d 2b0a 7c69 3320 3a20 7365 7452  ----+.|i3 : setR
│ │ │ │ +0006ce50: 616e 646f 6d53 6565 6420 3020 2020 2020  andomSeed 0     
│ │ │ │ +0006ce60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ce70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ce80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ce90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0006cea0: 2d2d 2073 6574 7469 6e67 2072 616e 646f  -- setting rando
│ │ │ │ -0006ceb0: 6d20 7365 6564 2074 6f20 3020 2020 2020  m seed to 0     
│ │ │ │ +0006ce90: 207c 0a7c 202d 2d20 7365 7474 696e 6720   |.| -- setting 
│ │ │ │ +0006cea0: 7261 6e64 6f6d 2073 6565 6420 746f 2030  random seed to 0
│ │ │ │ +0006ceb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ced0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cee0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0006ced0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0006cee0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0006cef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cf00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cf10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cf20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cf30: 2020 2020 2020 7c0a 7c6f 3320 3d20 3020        |.|o3 = 0 
│ │ │ │ +0006cf20: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ +0006cf30: 203d 2030 2020 2020 2020 2020 2020 2020   = 0            
│ │ │ │  0006cf40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cf50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cf60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cf70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cf80: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0006cf70: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0006cf80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006cf90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006cfa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006cfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006cfc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006cfd0: 2b0a 7c69 3420 3a20 7420 3d20 7261 6e64  +.|i4 : t = rand
│ │ │ │ -0006cfe0: 6f6d 2853 5e7b 323a 2d31 2c32 3a2d 327d  om(S^{2:-1,2:-2}
│ │ │ │ -0006cff0: 2c20 535e 7b33 3a2d 312c 343a 2d32 7d29  , S^{3:-1,4:-2})
│ │ │ │ +0006cfc0: 2d2d 2d2d 2d2b 0a7c 6934 203a 2074 203d  -----+.|i4 : t =
│ │ │ │ +0006cfd0: 2072 616e 646f 6d28 535e 7b32 3a2d 312c   random(S^{2:-1,
│ │ │ │ +0006cfe0: 323a 2d32 7d2c 2053 5e7b 333a 2d31 2c34  2:-2}, S^{3:-1,4
│ │ │ │ +0006cff0: 3a2d 327d 2920 2020 2020 2020 2020 2020  :-2})           
│ │ │ │  0006d000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d010: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0006d010: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0006d020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d060: 2020 2020 2020 2020 2020 7c0a 7c6f 3420            |.|o4 
│ │ │ │ -0006d070: 3d20 7b31 7d20 7c20 3234 2020 2d33 3620  = {1} | 24  -36 
│ │ │ │ -0006d080: 2d33 3020 3339 782d 3433 792b 3435 7a20  -30 39x-43y+45z 
│ │ │ │ -0006d090: 2032 3178 2d31 3579 2d33 347a 2033 3478   21x-15y-34z 34x
│ │ │ │ -0006d0a0: 2d32 3879 2d34 387a 2020 3139 782d 3437  -28y-48z  19x-47
│ │ │ │ -0006d0b0: 792d 3437 7a20 7c7c 0a7c 2020 2020 207b  y-47z ||.|     {
│ │ │ │ -0006d0c0: 317d 207c 202d 3239 2031 3920 2031 3920  1} | -29 19  19 
│ │ │ │ -0006d0d0: 202d 3437 782b 3338 792b 3437 7a20 2d33   -47x+38y+47z -3
│ │ │ │ -0006d0e0: 3978 2b32 792b 3139 7a20 2d31 3878 2b31  9x+2y+19z -18x+1
│ │ │ │ -0006d0f0: 3679 2d31 367a 202d 3133 782b 3232 792b  6y-16z -13x+22y+
│ │ │ │ -0006d100: 377a 207c 7c0a 7c20 2020 2020 7b32 7d20  7z ||.|     {2} 
│ │ │ │ -0006d110: 7c20 3020 2020 3020 2020 3020 2020 2d31  | 0   0   0   -1
│ │ │ │ -0006d120: 3020 2020 2020 2020 2020 202d 3239 2020  0          -29  
│ │ │ │ -0006d130: 2020 2020 2020 202d 3820 2020 2020 2020         -8       
│ │ │ │ -0006d140: 2020 2020 2d32 3220 2020 2020 2020 2020      -22         
│ │ │ │ -0006d150: 7c7c 0a7c 2020 2020 207b 327d 207c 2030  ||.|     {2} | 0
│ │ │ │ -0006d160: 2020 2030 2020 2030 2020 202d 3239 2020     0   0   -29  
│ │ │ │ -0006d170: 2020 2020 2020 2020 2d32 3420 2020 2020          -24     
│ │ │ │ -0006d180: 2020 2020 2d33 3820 2020 2020 2020 2020      -38         
│ │ │ │ -0006d190: 202d 3136 2020 2020 2020 2020 207c 7c0a   -16         ||.
│ │ │ │ -0006d1a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0006d050: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006d060: 0a7c 6f34 203d 207b 317d 207c 2032 3420  .|o4 = {1} | 24 
│ │ │ │ +0006d070: 202d 3336 202d 3330 2033 3978 2d34 3379   -36 -30 39x-43y
│ │ │ │ +0006d080: 2b34 357a 2020 3231 782d 3135 792d 3334  +45z  21x-15y-34
│ │ │ │ +0006d090: 7a20 3334 782d 3238 792d 3438 7a20 2031  z 34x-28y-48z  1
│ │ │ │ +0006d0a0: 3978 2d34 3779 2d34 377a 207c 7c0a 7c20  9x-47y-47z ||.| 
│ │ │ │ +0006d0b0: 2020 2020 7b31 7d20 7c20 2d32 3920 3139      {1} | -29 19
│ │ │ │ +0006d0c0: 2020 3139 2020 2d34 3778 2b33 3879 2b34    19  -47x+38y+4
│ │ │ │ +0006d0d0: 377a 202d 3339 782b 3279 2b31 397a 202d  7z -39x+2y+19z -
│ │ │ │ +0006d0e0: 3138 782b 3136 792d 3136 7a20 2d31 3378  18x+16y-16z -13x
│ │ │ │ +0006d0f0: 2b32 3279 2b37 7a20 7c7c 0a7c 2020 2020  +22y+7z ||.|    
│ │ │ │ +0006d100: 207b 327d 207c 2030 2020 2030 2020 2030   {2} | 0   0   0
│ │ │ │ +0006d110: 2020 202d 3130 2020 2020 2020 2020 2020     -10          
│ │ │ │ +0006d120: 2d32 3920 2020 2020 2020 2020 2d38 2020  -29         -8  
│ │ │ │ +0006d130: 2020 2020 2020 2020 202d 3232 2020 2020           -22    
│ │ │ │ +0006d140: 2020 2020 207c 7c0a 7c20 2020 2020 7b32       ||.|     {2
│ │ │ │ +0006d150: 7d20 7c20 3020 2020 3020 2020 3020 2020  } | 0   0   0   
│ │ │ │ +0006d160: 2d32 3920 2020 2020 2020 2020 202d 3234  -29          -24
│ │ │ │ +0006d170: 2020 2020 2020 2020 202d 3338 2020 2020           -38    
│ │ │ │ +0006d180: 2020 2020 2020 2d31 3620 2020 2020 2020        -16       
│ │ │ │ +0006d190: 2020 7c7c 0a7c 2020 2020 2020 2020 2020    ||.|          
│ │ │ │ +0006d1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d1e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0006d1f0: 2020 2020 2020 2020 2020 2034 2020 2020             4    
│ │ │ │ -0006d200: 2020 3720 2020 2020 2020 2020 2020 2020    7             
│ │ │ │ +0006d1e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0006d1f0: 3420 2020 2020 2037 2020 2020 2020 2020  4      7        
│ │ │ │ +0006d200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d230: 2020 2020 2020 2020 7c0a 7c6f 3420 3a20          |.|o4 : 
│ │ │ │ -0006d240: 4d61 7472 6978 2053 2020 3c2d 2d20 5320  Matrix S  <-- S 
│ │ │ │ +0006d220: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0006d230: 6f34 203a 204d 6174 7269 7820 5320 203c  o4 : Matrix S  <
│ │ │ │ +0006d240: 2d2d 2053 2020 2020 2020 2020 2020 2020  -- S            
│ │ │ │  0006d250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d280: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0006d270: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0006d280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006d290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006d2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006d2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006d2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006d2d0: 2d2d 2b0a 7c69 3520 3a20 7373 203d 2073  --+.|i5 : ss = s
│ │ │ │ -0006d2e0: 706c 6974 7469 6e67 7328 7379 7a20 742c  plittings(syz t,
│ │ │ │ -0006d2f0: 2074 2920 2020 2020 2020 2020 2020 2020   t)             
│ │ │ │ +0006d2c0: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2073  -------+.|i5 : s
│ │ │ │ +0006d2d0: 7320 3d20 7370 6c69 7474 696e 6773 2873  s = splittings(s
│ │ │ │ +0006d2e0: 797a 2074 2c20 7429 2020 2020 2020 2020  yz t, t)        
│ │ │ │ +0006d2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d310: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0006d320: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0006d310: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0006d320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d360: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0006d370: 3520 3d20 7b7b 317d 207c 2030 2030 2031  5 = {{1} | 0 0 1
│ │ │ │ -0006d380: 2030 2030 2030 2020 2030 2020 7c2c 207b   0 0 0   0  |, {
│ │ │ │ -0006d390: 317d 207c 202d 3237 2032 2020 3133 782d  1} | -27 2  13x-
│ │ │ │ -0006d3a0: 3130 792b 3433 7a20 3530 782d 3334 792d  10y+43z 50x-34y-
│ │ │ │ -0006d3b0: 3530 7a20 7c7d 2020 207c 0a7c 2020 2020  50z |}   |.|    
│ │ │ │ -0006d3c0: 2020 7b32 7d20 7c20 3020 3020 3020 3020    {2} | 0 0 0 0 
│ │ │ │ -0006d3d0: 3020 2d33 3120 2d36 207c 2020 7b31 7d20  0 -31 -6 |  {1} 
│ │ │ │ -0006d3e0: 7c20 2d34 2020 3335 2032 3278 2b33 3279  | -4  35 22x+32y
│ │ │ │ -0006d3f0: 2b34 337a 202d 3778 2d38 792d 3237 7a20  +43z -7x-8y-27z 
│ │ │ │ -0006d400: 207c 2020 2020 7c0a 7c20 2020 2020 207b   |    |.|      {
│ │ │ │ -0006d410: 327d 207c 2030 2030 2030 2030 2030 2032  2} | 0 0 0 0 0 2
│ │ │ │ -0006d420: 3920 2039 2020 7c20 207b 317d 207c 2030  9  9  |  {1} | 0
│ │ │ │ -0006d430: 2020 2030 2020 3020 2020 2020 2020 2020     0  0         
│ │ │ │ -0006d440: 2020 3020 2020 2020 2020 2020 2020 7c20    0           | 
│ │ │ │ -0006d450: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0006d460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d470: 2020 2020 2020 7b32 7d20 7c20 3020 2020        {2} | 0   
│ │ │ │ -0006d480: 3020 202d 3235 2020 2020 2020 2020 2032  0  -25         2
│ │ │ │ -0006d490: 3620 2020 2020 2020 2020 207c 2020 2020  6          |    
│ │ │ │ -0006d4a0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0006d4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d4c0: 2020 207b 327d 207c 2030 2020 2030 2020     {2} | 0   0  
│ │ │ │ -0006d4d0: 3236 2020 2020 2020 2020 2020 2d32 2020  26          -2  
│ │ │ │ -0006d4e0: 2020 2020 2020 2020 7c20 2020 207c 0a7c          |    |.|
│ │ │ │ +0006d360: 207c 0a7c 6f35 203d 207b 7b31 7d20 7c20   |.|o5 = {{1} | 
│ │ │ │ +0006d370: 3020 3020 3120 3020 3020 3020 2020 3020  0 0 1 0 0 0   0 
│ │ │ │ +0006d380: 207c 2c20 7b31 7d20 7c20 2d32 3720 3220   |, {1} | -27 2 
│ │ │ │ +0006d390: 2031 3378 2d31 3079 2b34 337a 2035 3078   13x-10y+43z 50x
│ │ │ │ +0006d3a0: 2d33 3479 2d35 307a 207c 7d20 2020 7c0a  -34y-50z |}   |.
│ │ │ │ +0006d3b0: 7c20 2020 2020 207b 327d 207c 2030 2030  |      {2} | 0 0
│ │ │ │ +0006d3c0: 2030 2030 2030 202d 3331 202d 3620 7c20   0 0 0 -31 -6 | 
│ │ │ │ +0006d3d0: 207b 317d 207c 202d 3420 2033 3520 3232   {1} | -4  35 22
│ │ │ │ +0006d3e0: 782b 3332 792b 3433 7a20 2d37 782d 3879  x+32y+43z -7x-8y
│ │ │ │ +0006d3f0: 2d32 377a 2020 7c20 2020 207c 0a7c 2020  -27z  |    |.|  
│ │ │ │ +0006d400: 2020 2020 7b32 7d20 7c20 3020 3020 3020      {2} | 0 0 0 
│ │ │ │ +0006d410: 3020 3020 3239 2020 3920 207c 2020 7b31  0 0 29  9  |  {1
│ │ │ │ +0006d420: 7d20 7c20 3020 2020 3020 2030 2020 2020  } | 0   0  0    
│ │ │ │ +0006d430: 2020 2020 2020 2030 2020 2020 2020 2020         0        
│ │ │ │ +0006d440: 2020 207c 2020 2020 7c0a 7c20 2020 2020     |    |.|     
│ │ │ │ +0006d450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006d460: 2020 2020 2020 2020 2020 207b 327d 207c             {2} |
│ │ │ │ +0006d470: 2030 2020 2030 2020 2d32 3520 2020 2020   0   0  -25     
│ │ │ │ +0006d480: 2020 2020 3236 2020 2020 2020 2020 2020      26          
│ │ │ │ +0006d490: 7c20 2020 207c 0a7c 2020 2020 2020 2020  |    |.|        
│ │ │ │ +0006d4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006d4b0: 2020 2020 2020 2020 7b32 7d20 7c20 3020          {2} | 0 
│ │ │ │ +0006d4c0: 2020 3020 2032 3620 2020 2020 2020 2020    0  26         
│ │ │ │ +0006d4d0: 202d 3220 2020 2020 2020 2020 207c 2020   -2          |  
│ │ │ │ +0006d4e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0006d4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d510: 7b32 7d20 7c20 3020 2020 3020 2030 2020  {2} | 0   0  0  
│ │ │ │ -0006d520: 2020 2020 2020 2020 2030 2020 2020 2020           0      
│ │ │ │ -0006d530: 2020 2020 207c 2020 2020 7c0a 7c20 2020       |    |.|   
│ │ │ │ +0006d500: 2020 2020 207b 327d 207c 2030 2020 2030       {2} | 0   0
│ │ │ │ +0006d510: 2020 3020 2020 2020 2020 2020 2020 3020    0           0 
│ │ │ │ +0006d520: 2020 2020 2020 2020 2020 7c20 2020 207c            |    |
│ │ │ │ +0006d530: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0006d540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d550: 2020 2020 2020 2020 2020 2020 207b 327d               {2}
│ │ │ │ -0006d560: 207c 2030 2020 2030 2020 3020 2020 2020   | 0   0  0     
│ │ │ │ -0006d570: 2020 2020 2020 3020 2020 2020 2020 2020        0         
│ │ │ │ -0006d580: 2020 7c20 2020 207c 0a7c 2020 2020 2020    |    |.|      
│ │ │ │ +0006d550: 2020 7b32 7d20 7c20 3020 2020 3020 2030    {2} | 0   0  0
│ │ │ │ +0006d560: 2020 2020 2020 2020 2020 2030 2020 2020             0    
│ │ │ │ +0006d570: 2020 2020 2020 207c 2020 2020 7c0a 7c20         |    |.| 
│ │ │ │ +0006d580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d5d0: 2020 2020 7c0a 7c6f 3520 3a20 4c69 7374      |.|o5 : List
│ │ │ │ +0006d5c0: 2020 2020 2020 2020 207c 0a7c 6f35 203a           |.|o5 :
│ │ │ │ +0006d5d0: 204c 6973 7420 2020 2020 2020 2020 2020   List           
│ │ │ │  0006d5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d620: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0006d610: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0006d620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006d630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006d640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006d650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006d660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0006d670: 7c69 3620 3a20 7373 2f62 6574 7469 2020  |i6 : ss/betti  
│ │ │ │ +0006d660: 2d2d 2d2b 0a7c 6936 203a 2073 732f 6265  ---+.|i6 : ss/be
│ │ │ │ +0006d670: 7474 6920 2020 2020 2020 2020 2020 2020  tti             
│ │ │ │  0006d680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d6b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0006d6b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0006d6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d700: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0006d710: 2020 2020 2020 2020 3020 3120 2020 2020          0 1     
│ │ │ │ -0006d720: 2020 2020 3020 3120 2020 2020 2020 2020      0 1         
│ │ │ │ +0006d6f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0006d700: 2020 2020 2020 2020 2020 2020 2030 2031               0 1
│ │ │ │ +0006d710: 2020 2020 2020 2020 2030 2031 2020 2020           0 1    
│ │ │ │ +0006d720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d750: 2020 2020 207c 0a7c 6f36 203d 207b 746f       |.|o6 = {to
│ │ │ │ -0006d760: 7461 6c3a 2033 2037 2c20 746f 7461 6c3a  tal: 3 7, total:
│ │ │ │ -0006d770: 2037 2034 7d20 2020 2020 2020 2020 2020   7 4}           
│ │ │ │ +0006d740: 2020 2020 2020 2020 2020 7c0a 7c6f 3620            |.|o6 
│ │ │ │ +0006d750: 3d20 7b74 6f74 616c 3a20 3320 372c 2074  = {total: 3 7, t
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│ │ │ │ +0006d770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d7a0: 2020 7c0a 7c20 2020 2020 2020 2020 2030    |.|          0
│ │ │ │ -0006d7b0: 3a20 2e20 3320 2020 2020 2030 3a20 2e20  : . 3      0: . 
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│ │ │ │ +0006d7a0: 2020 2020 303a 202e 2033 2020 2020 2020      0: . 3      
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│ │ │ │ +0006d7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0006d7e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0006d7f0: 2031 3a20 3120 3420 2020 2020 2031 3a20   1: 1 4      1: 
│ │ │ │ +0006d800: 3320 3220 2020 2020 2020 2020 2020 2020  3 2             
│ │ │ │  0006d810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d830: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
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│ │ │ │ +0006d850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d880: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0006d870: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0006d880: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0006d890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d8d0: 2020 2020 2020 7c0a 7c6f 3620 3a20 4c69        |.|o6 : Li
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│ │ │ │ +0006d8c0: 2020 2020 2020 2020 2020 207c 0a7c 6f36             |.|o6
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│ │ │ │ +0006d8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0006d900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d920: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0006d910: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0006d920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006d930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006d940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006d950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006d960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006d970: 2b0a 0a57 6179 7320 746f 2075 7365 2073  +..Ways to use s
│ │ │ │ -0006d980: 706c 6974 7469 6e67 733a 0a3d 3d3d 3d3d  plittings:.=====
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│ │ │ │ -0006d9a0: 3d3d 0a0a 2020 2a20 2273 706c 6974 7469  ==..  * "splitti
│ │ │ │ -0006d9b0: 6e67 7328 4d61 7472 6978 2c4d 6174 7269  ngs(Matrix,Matri
│ │ │ │ -0006d9c0: 7829 220a 0a46 6f72 2074 6865 2070 726f  x)"..For the pro
│ │ │ │ -0006d9d0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d  grammer.========
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│ │ │ │ -0006d9f0: 6f62 6a65 6374 202a 6e6f 7465 2073 706c  object *note spl
│ │ │ │ -0006da00: 6974 7469 6e67 733a 2073 706c 6974 7469  ittings: splitti
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│ │ │ │ +0006d990: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 7370  =======..  * "sp
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│ │ │ │ +0006d9b0: 4d61 7472 6978 2922 0a0a 466f 7220 7468  Matrix)"..For th
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│ │ │ │ +0006d9d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ +0006d9e0: 0a54 6865 206f 626a 6563 7420 2a6e 6f74  .The object *not
│ │ │ │ +0006d9f0: 6520 7370 6c69 7474 696e 6773 3a20 7370  e splittings: sp
│ │ │ │ +0006da00: 6c69 7474 696e 6773 2c20 6973 2061 202a  littings, is a *
│ │ │ │ +0006da10: 6e6f 7465 206d 6574 686f 6420 6675 6e63  note method func
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│ │ │ │ +0006da30: 446f 6329 4d65 7468 6f64 4675 6e63 7469  Doc)MethodFuncti
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│ │ │ │  0006da60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006da70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006da80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006da90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006daa0: 0a0a 5468 6520 736f 7572 6365 206f 6620  ..The source of 
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│ │ │ │ -0006dad0: 6475 6369 626c 652d 7061 7468 2f6d 6163  ducible-path/mac
│ │ │ │ -0006dae0: 6175 6c61 7932 2d31 2e32 352e 3131 2b64  aulay2-1.25.11+d
│ │ │ │ -0006daf0: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70  s/M2/Macaulay2/p
│ │ │ │ -0006db00: 6163 6b61 6765 732f 0a43 6f6d 706c 6574  ackages/.Complet
│ │ │ │ -0006db10: 6549 6e74 6572 7365 6374 696f 6e52 6573  eIntersectionRes
│ │ │ │ -0006db20: 6f6c 7574 696f 6e73 2e6d 323a 3339 3235  olutions.m2:3925
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│ │ │ │ -0006db50: 5265 736f 6c75 7469 6f6e 732e 696e 666f  Resolutions.info
│ │ │ │ -0006db60: 2c20 4e6f 6465 3a20 7374 6162 6c65 486f  , Node: stableHo
│ │ │ │ -0006db70: 6d2c 204e 6578 743a 2073 756d 5477 6f4d  m, Next: sumTwoM
│ │ │ │ -0006db80: 6f6e 6f6d 6961 6c73 2c20 5072 6576 3a20  onomials, Prev: 
│ │ │ │ -0006db90: 7370 6c69 7474 696e 6773 2c20 5570 3a20  splittings, Up: 
│ │ │ │ -0006dba0: 546f 700a 0a73 7461 626c 6548 6f6d 202d  Top..stableHom -
│ │ │ │ -0006dbb0: 2d20 6d61 7020 6672 6f6d 2048 6f6d 284d  - map from Hom(M
│ │ │ │ -0006dbc0: 2c4e 2920 746f 2074 6865 2073 7461 626c  ,N) to the stabl
│ │ │ │ -0006dbd0: 6520 486f 6d20 6d6f 6475 6c65 0a2a 2a2a  e Hom module.***
│ │ │ │ +0006da90: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263  -----..The sourc
│ │ │ │ +0006daa0: 6520 6f66 2074 6869 7320 646f 6375 6d65  e of this docume
│ │ │ │ +0006dab0: 6e74 2069 7320 696e 0a2f 6275 696c 642f  nt is in./build/
│ │ │ │ +0006dac0: 7265 7072 6f64 7563 6962 6c65 2d70 6174  reproducible-pat
│ │ │ │ +0006dad0: 682f 6d61 6361 756c 6179 322d 312e 3235  h/macaulay2-1.25
│ │ │ │ +0006dae0: 2e31 312b 6473 2f4d 322f 4d61 6361 756c  .11+ds/M2/Macaul
│ │ │ │ +0006daf0: 6179 322f 7061 636b 6167 6573 2f0a 436f  ay2/packages/.Co
│ │ │ │ +0006db00: 6d70 6c65 7465 496e 7465 7273 6563 7469  mpleteIntersecti
│ │ │ │ +0006db10: 6f6e 5265 736f 6c75 7469 6f6e 732e 6d32  onResolutions.m2
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│ │ │ │ +0006db30: 2043 6f6d 706c 6574 6549 6e74 6572 7365   CompleteInterse
│ │ │ │ +0006db40: 6374 696f 6e52 6573 6f6c 7574 696f 6e73  ctionResolutions
│ │ │ │ +0006db50: 2e69 6e66 6f2c 204e 6f64 653a 2073 7461  .info, Node: sta
│ │ │ │ +0006db60: 626c 6548 6f6d 2c20 4e65 7874 3a20 7375  bleHom, Next: su
│ │ │ │ +0006db70: 6d54 776f 4d6f 6e6f 6d69 616c 732c 2050  mTwoMonomials, P
│ │ │ │ +0006db80: 7265 763a 2073 706c 6974 7469 6e67 732c  rev: splittings,
│ │ │ │ +0006db90: 2055 703a 2054 6f70 0a0a 7374 6162 6c65   Up: Top..stable
│ │ │ │ +0006dba0: 486f 6d20 2d2d 206d 6170 2066 726f 6d20  Hom -- map from 
│ │ │ │ +0006dbb0: 486f 6d28 4d2c 4e29 2074 6f20 7468 6520  Hom(M,N) to the 
│ │ │ │ +0006dbc0: 7374 6162 6c65 2048 6f6d 206d 6f64 756c  stable Hom modul
│ │ │ │ +0006dbd0: 650a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  e.**************
│ │ │ │  0006dbe0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0006dbf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0006dc00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0006dc10: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a  ****..  * Usage:
│ │ │ │ -0006dc20: 200a 2020 2020 2020 2020 7020 3d20 7374   .        p = st
│ │ │ │ -0006dc30: 6162 6c65 486f 6d28 4d2c 4e29 0a20 202a  ableHom(M,N).  *
│ │ │ │ -0006dc40: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ -0006dc50: 204d 2c20 6120 2a6e 6f74 6520 6d6f 6475   M, a *note modu
│ │ │ │ -0006dc60: 6c65 3a20 284d 6163 6175 6c61 7932 446f  le: (Macaulay2Do
│ │ │ │ -0006dc70: 6329 4d6f 6475 6c65 2c2c 200a 2020 2020  c)Module,, .    
│ │ │ │ -0006dc80: 2020 2a20 4e2c 2061 202a 6e6f 7465 206d    * N, a *note m
│ │ │ │ -0006dc90: 6f64 756c 653a 2028 4d61 6361 756c 6179  odule: (Macaulay
│ │ │ │ -0006dca0: 3244 6f63 294d 6f64 756c 652c 2c20 0a20  2Doc)Module,, . 
│ │ │ │ -0006dcb0: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ -0006dcc0: 2020 2a20 702c 2061 202a 6e6f 7465 206d    * p, a *note m
│ │ │ │ -0006dcd0: 6174 7269 783a 2028 4d61 6361 756c 6179  atrix: (Macaulay
│ │ │ │ -0006dce0: 3244 6f63 294d 6174 7269 782c 2c20 7072  2Doc)Matrix,, pr
│ │ │ │ -0006dcf0: 6f6a 6563 7469 6f6e 2066 726f 6d20 486f  ojection from Ho
│ │ │ │ -0006dd00: 6d28 4d2c 4e29 2074 6f0a 2020 2020 2020  m(M,N) to.      
│ │ │ │ -0006dd10: 2020 7468 6520 7374 6162 6c65 2048 6f6d    the stable Hom
│ │ │ │ -0006dd20: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d  ..Description.==
│ │ │ │ -0006dd30: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 2073  =========..The s
│ │ │ │ -0006dd40: 7461 626c 6520 486f 6d20 6973 2048 6f6d  table Hom is Hom
│ │ │ │ -0006dd50: 284d 2c4e 292f 5420 7768 6572 6520 5420  (M,N)/T where T 
│ │ │ │ -0006dd60: 6973 2074 6865 2073 7562 6d6f 6475 6c65  is the submodule
│ │ │ │ -0006dd70: 206f 6620 686f 6d6f 6d6f 7270 6869 736d   of homomorphism
│ │ │ │ -0006dd80: 7320 7468 6174 0a66 6163 746f 7220 7468  s that.factor th
│ │ │ │ -0006dd90: 726f 7567 6820 6120 6672 6565 2063 6f76  rough a free cov
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│ │ │ │ -0006ddb0: 6976 616c 656e 746c 792c 2074 6872 6f75  ivalently, throu
│ │ │ │ -0006ddc0: 6768 2061 6e79 2070 726f 6a65 6374 6976  gh any projectiv
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│ │ │ │ -0006dde0: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
│ │ │ │ -0006ddf0: 2069 7353 7461 626c 7954 7269 7669 616c   isStablyTrivial
│ │ │ │ -0006de00: 3a20 6973 5374 6162 6c79 5472 6976 6961  : isStablyTrivia
│ │ │ │ -0006de10: 6c2c 202d 2d20 7265 7475 726e 7320 7472  l, -- returns tr
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│ │ │ │ -0006de30: 6573 2074 6f0a 2020 2020 3020 756e 6465  es to.    0 unde
│ │ │ │ -0006de40: 7220 7374 6162 6c65 486f 6d0a 0a57 6179  r stableHom..Way
│ │ │ │ -0006de50: 7320 746f 2075 7365 2073 7461 626c 6548  s to use stableH
│ │ │ │ -0006de60: 6f6d 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  om:.============
│ │ │ │ -0006de70: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ -0006de80: 2273 7461 626c 6548 6f6d 284d 6f64 756c  "stableHom(Modul
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│ │ │ │ -0006deb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0006dec0: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e  =..The object *n
│ │ │ │ -0006ded0: 6f74 6520 7374 6162 6c65 486f 6d3a 2073  ote stableHom: s
│ │ │ │ -0006dee0: 7461 626c 6548 6f6d 2c20 6973 2061 202a  tableHom, is a *
│ │ │ │ -0006def0: 6e6f 7465 206d 6574 686f 6420 6675 6e63  note method func
│ │ │ │ -0006df00: 7469 6f6e 3a0a 284d 6163 6175 6c61 7932  tion:.(Macaulay2
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│ │ │ │ +0006dcc0: 6f74 6520 6d61 7472 6978 3a20 284d 6163  ote matrix: (Mac
│ │ │ │ +0006dcd0: 6175 6c61 7932 446f 6329 4d61 7472 6978  aulay2Doc)Matrix
│ │ │ │ +0006dce0: 2c2c 2070 726f 6a65 6374 696f 6e20 6672  ,, projection fr
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│ │ │ │ +0006dd00: 2020 2020 2020 2074 6865 2073 7461 626c         the stabl
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│ │ │ │ +0006dd30: 5468 6520 7374 6162 6c65 2048 6f6d 2069  The stable Hom i
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│ │ │ │ +0006ddb0: 7468 726f 7567 6820 616e 7920 7072 6f6a  through any proj
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│ │ │ │ +0006dde0: 2a6e 6f74 6520 6973 5374 6162 6c79 5472  *note isStablyTr
│ │ │ │ +0006ddf0: 6976 6961 6c3a 2069 7353 7461 626c 7954  ivial: isStablyT
│ │ │ │ +0006de00: 7269 7669 616c 2c20 2d2d 2072 6574 7572  rivial, -- retur
│ │ │ │ +0006de10: 6e73 2074 7275 6520 6966 2074 6865 206d  ns true if the m
│ │ │ │ +0006de20: 6170 2067 6f65 7320 746f 0a20 2020 2030  ap goes to.    0
│ │ │ │ +0006de30: 2075 6e64 6572 2073 7461 626c 6548 6f6d   under stableHom
│ │ │ │ +0006de40: 0a0a 5761 7973 2074 6f20 7573 6520 7374  ..Ways to use st
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│ │ │ │ +0006de80: 4d6f 6475 6c65 2c4d 6f64 756c 6529 220a  Module,Module)".
│ │ │ │ +0006de90: 0a46 6f72 2074 6865 2070 726f 6772 616d  .For the program
│ │ │ │ +0006dea0: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  mer.============
│ │ │ │ +0006deb0: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65  ======..The obje
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│ │ │ │ +0006ded0: 6f6d 3a20 7374 6162 6c65 486f 6d2c 2069  om: stableHom, i
│ │ │ │ +0006dee0: 7320 6120 2a6e 6f74 6520 6d65 7468 6f64  s a *note method
│ │ │ │ +0006def0: 2066 756e 6374 696f 6e3a 0a28 4d61 6361   function:.(Maca
│ │ │ │ +0006df00: 756c 6179 3244 6f63 294d 6574 686f 6446  ulay2Doc)MethodF
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│ │ │ │  0006df30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006df40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006df50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006df60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006df70: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263  -----..The sourc
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│ │ │ │ -0006dfa0: 7265 7072 6f64 7563 6962 6c65 2d70 6174  reproducible-pat
│ │ │ │ -0006dfb0: 682f 6d61 6361 756c 6179 322d 312e 3235  h/macaulay2-1.25
│ │ │ │ -0006dfc0: 2e31 312b 6473 2f4d 322f 4d61 6361 756c  .11+ds/M2/Macaul
│ │ │ │ -0006dfd0: 6179 322f 7061 636b 6167 6573 2f0a 436f  ay2/packages/.Co
│ │ │ │ -0006dfe0: 6d70 6c65 7465 496e 7465 7273 6563 7469  mpleteIntersecti
│ │ │ │ -0006dff0: 6f6e 5265 736f 6c75 7469 6f6e 732e 6d32  onResolutions.m2
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│ │ │ │ -0006e010: 2043 6f6d 706c 6574 6549 6e74 6572 7365   CompleteInterse
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│ │ │ │ -0006e030: 2e69 6e66 6f2c 204e 6f64 653a 2073 756d  .info, Node: sum
│ │ │ │ -0006e040: 5477 6f4d 6f6e 6f6d 6961 6c73 2c20 4e65  TwoMonomials, Ne
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│ │ │ │ -0006e060: 6f6e 2c20 5072 6576 3a20 7374 6162 6c65  on, Prev: stable
│ │ │ │ -0006e070: 486f 6d2c 2055 703a 2054 6f70 0a0a 7375  Hom, Up: Top..su
│ │ │ │ -0006e080: 6d54 776f 4d6f 6e6f 6d69 616c 7320 2d2d  mTwoMonomials --
│ │ │ │ -0006e090: 2074 616c 6c79 2074 6865 2073 6571 7565   tally the seque
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│ │ │ │ +0006df60: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520  ----------..The 
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│ │ │ │ +0006df80: 6f63 756d 656e 7420 6973 2069 6e0a 2f62  ocument is in./b
│ │ │ │ +0006df90: 7569 6c64 2f72 6570 726f 6475 6369 626c  uild/reproducibl
│ │ │ │ +0006dfa0: 652d 7061 7468 2f6d 6163 6175 6c61 7932  e-path/macaulay2
│ │ │ │ +0006dfb0: 2d31 2e32 352e 3131 2b64 732f 4d32 2f4d  -1.25.11+ds/M2/M
│ │ │ │ +0006dfc0: 6163 6175 6c61 7932 2f70 6163 6b61 6765  acaulay2/package
│ │ │ │ +0006dfd0: 732f 0a43 6f6d 706c 6574 6549 6e74 6572  s/.CompleteInter
│ │ │ │ +0006dfe0: 7365 6374 696f 6e52 6573 6f6c 7574 696f  sectionResolutio
│ │ │ │ +0006dff0: 6e73 2e6d 323a 3436 3439 3a30 2e0a 1f0a  ns.m2:4649:0....
│ │ │ │ +0006e000: 4669 6c65 3a20 436f 6d70 6c65 7465 496e  File: CompleteIn
│ │ │ │ +0006e010: 7465 7273 6563 7469 6f6e 5265 736f 6c75  tersectionResolu
│ │ │ │ +0006e020: 7469 6f6e 732e 696e 666f 2c20 4e6f 6465  tions.info, Node
│ │ │ │ +0006e030: 3a20 7375 6d54 776f 4d6f 6e6f 6d69 616c  : sumTwoMonomial
│ │ │ │ +0006e040: 732c 204e 6578 743a 2054 6174 6552 6573  s, Next: TateRes
│ │ │ │ +0006e050: 6f6c 7574 696f 6e2c 2050 7265 763a 2073  olution, Prev: s
│ │ │ │ +0006e060: 7461 626c 6548 6f6d 2c20 5570 3a20 546f  tableHom, Up: To
│ │ │ │ +0006e070: 700a 0a73 756d 5477 6f4d 6f6e 6f6d 6961  p..sumTwoMonomia
│ │ │ │ +0006e080: 6c73 202d 2d20 7461 6c6c 7920 7468 6520  ls -- tally the 
│ │ │ │ +0006e090: 7365 7175 656e 6365 7320 6f66 2042 5261  sequences of BRa
│ │ │ │ +0006e0a0: 6e6b 7320 666f 7220 6365 7274 6169 6e20  nks for certain 
│ │ │ │ +0006e0b0: 6578 616d 706c 6573 0a2a 2a2a 2a2a 2a2a  examples.*******
│ │ │ │ +0006e0c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0006e0d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0006e0e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0006e0f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0006e100: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055  *********..  * U
│ │ │ │ -0006e110: 7361 6765 3a20 0a20 2020 2020 2020 2073  sage: .        s
│ │ │ │ -0006e120: 756d 5477 6f4d 6f6e 6f6d 6961 6c73 2863  umTwoMonomials(c
│ │ │ │ -0006e130: 2c64 290a 2020 2a20 496e 7075 7473 3a0a  ,d).  * Inputs:.
│ │ │ │ -0006e140: 2020 2020 2020 2a20 632c 2061 6e20 2a6e        * c, an *n
│ │ │ │ -0006e150: 6f74 6520 696e 7465 6765 723a 2028 4d61  ote integer: (Ma
│ │ │ │ -0006e160: 6361 756c 6179 3244 6f63 295a 5a2c 2c20  caulay2Doc)ZZ,, 
│ │ │ │ -0006e170: 636f 6469 6d65 6e73 696f 6e20 696e 2077  codimension in w
│ │ │ │ -0006e180: 6869 6368 2074 6f20 776f 726b 0a20 2020  hich to work.   
│ │ │ │ -0006e190: 2020 202a 2064 2c20 616e 202a 6e6f 7465     * d, an *note
│ │ │ │ -0006e1a0: 2069 6e74 6567 6572 3a20 284d 6163 6175   integer: (Macau
│ │ │ │ -0006e1b0: 6c61 7932 446f 6329 5a5a 2c2c 2064 6567  lay2Doc)ZZ,, deg
│ │ │ │ -0006e1c0: 7265 6520 6f66 2074 6865 206d 6f6e 6f6d  ree of the monom
│ │ │ │ -0006e1d0: 6961 6c73 2074 6f20 7461 6b65 0a20 202a  ials to take.  *
│ │ │ │ -0006e1e0: 204f 7574 7075 7473 3a0a 2020 2020 2020   Outputs:.      
│ │ │ │ -0006e1f0: 2a20 542c 2061 202a 6e6f 7465 2074 616c  * T, a *note tal
│ │ │ │ -0006e200: 6c79 3a20 284d 6163 6175 6c61 7932 446f  ly: (Macaulay2Do
│ │ │ │ -0006e210: 6329 5461 6c6c 792c 2c20 0a0a 4465 7363  c)Tally,, ..Desc
│ │ │ │ -0006e220: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
│ │ │ │ -0006e230: 3d3d 3d0a 0a74 616c 6c69 6573 2074 6865  ===..tallies the
│ │ │ │ -0006e240: 2073 6571 7565 6e63 6573 206f 6620 422d   sequences of B-
│ │ │ │ -0006e250: 7261 6e6b 7320 7468 6174 206f 6363 7572  ranks that occur
│ │ │ │ -0006e260: 2066 6f72 2073 756d 7320 6f66 2070 6169   for sums of pai
│ │ │ │ -0006e270: 7273 206f 6620 6d6f 6e6f 6d69 616c 7320  rs of monomials 
│ │ │ │ -0006e280: 696e 2052 0a3d 2053 2f28 642d 7468 2070  in R.= S/(d-th p
│ │ │ │ -0006e290: 6f77 6572 7320 6f66 2074 6865 2076 6172  owers of the var
│ │ │ │ -0006e2a0: 6961 626c 6573 292c 2077 6974 6820 6675  iables), with fu
│ │ │ │ -0006e2b0: 6c6c 2063 6f6d 706c 6578 6974 7920 283d  ll complexity (=
│ │ │ │ -0006e2c0: 6329 3b20 7468 6174 2069 732c 2066 6f72  c); that is, for
│ │ │ │ -0006e2d0: 2061 6e0a 6170 7072 6f70 7269 6174 6520   an.appropriate 
│ │ │ │ -0006e2e0: 7379 7a79 6779 204d 206f 6620 4d30 203d  syzygy M of M0 =
│ │ │ │ -0006e2f0: 2052 2f28 6d31 2b6d 3229 2077 6865 7265   R/(m1+m2) where
│ │ │ │ -0006e300: 206d 3120 616e 6420 6d32 2061 7265 206d   m1 and m2 are m
│ │ │ │ -0006e310: 6f6e 6f6d 6961 6c73 206f 6620 7468 650a  onomials of the.
│ │ │ │ -0006e320: 7361 6d65 2064 6567 7265 652e 0a0a 2b2d  same degree...+-
│ │ │ │ +0006e0f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a  **************..
│ │ │ │ +0006e100: 2020 2a20 5573 6167 653a 200a 2020 2020    * Usage: .    
│ │ │ │ +0006e110: 2020 2020 7375 6d54 776f 4d6f 6e6f 6d69      sumTwoMonomi
│ │ │ │ +0006e120: 616c 7328 632c 6429 0a20 202a 2049 6e70  als(c,d).  * Inp
│ │ │ │ +0006e130: 7574 733a 0a20 2020 2020 202a 2063 2c20  uts:.      * c, 
│ │ │ │ +0006e140: 616e 202a 6e6f 7465 2069 6e74 6567 6572  an *note integer
│ │ │ │ +0006e150: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +0006e160: 5a5a 2c2c 2063 6f64 696d 656e 7369 6f6e  ZZ,, codimension
│ │ │ │ +0006e170: 2069 6e20 7768 6963 6820 746f 2077 6f72   in which to wor
│ │ │ │ +0006e180: 6b0a 2020 2020 2020 2a20 642c 2061 6e20  k.      * d, an 
│ │ │ │ +0006e190: 2a6e 6f74 6520 696e 7465 6765 723a 2028  *note integer: (
│ │ │ │ +0006e1a0: 4d61 6361 756c 6179 3244 6f63 295a 5a2c  Macaulay2Doc)ZZ,
│ │ │ │ +0006e1b0: 2c20 6465 6772 6565 206f 6620 7468 6520  , degree of the 
│ │ │ │ +0006e1c0: 6d6f 6e6f 6d69 616c 7320 746f 2074 616b  monomials to tak
│ │ │ │ +0006e1d0: 650a 2020 2a20 4f75 7470 7574 733a 0a20  e.  * Outputs:. 
│ │ │ │ +0006e1e0: 2020 2020 202a 2054 2c20 6120 2a6e 6f74       * T, a *not
│ │ │ │ +0006e1f0: 6520 7461 6c6c 793a 2028 4d61 6361 756c  e tally: (Macaul
│ │ │ │ +0006e200: 6179 3244 6f63 2954 616c 6c79 2c2c 200a  ay2Doc)Tally,, .
│ │ │ │ +0006e210: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d  .Description.===
│ │ │ │ +0006e220: 3d3d 3d3d 3d3d 3d3d 0a0a 7461 6c6c 6965  ========..tallie
│ │ │ │ +0006e230: 7320 7468 6520 7365 7175 656e 6365 7320  s the sequences 
│ │ │ │ +0006e240: 6f66 2042 2d72 616e 6b73 2074 6861 7420  of B-ranks that 
│ │ │ │ +0006e250: 6f63 6375 7220 666f 7220 7375 6d73 206f  occur for sums o
│ │ │ │ +0006e260: 6620 7061 6972 7320 6f66 206d 6f6e 6f6d  f pairs of monom
│ │ │ │ +0006e270: 6961 6c73 2069 6e20 520a 3d20 532f 2864  ials in R.= S/(d
│ │ │ │ +0006e280: 2d74 6820 706f 7765 7273 206f 6620 7468  -th powers of th
│ │ │ │ +0006e290: 6520 7661 7269 6162 6c65 7329 2c20 7769  e variables), wi
│ │ │ │ +0006e2a0: 7468 2066 756c 6c20 636f 6d70 6c65 7869  th full complexi
│ │ │ │ +0006e2b0: 7479 2028 3d63 293b 2074 6861 7420 6973  ty (=c); that is
│ │ │ │ +0006e2c0: 2c20 666f 7220 616e 0a61 7070 726f 7072  , for an.appropr
│ │ │ │ +0006e2d0: 6961 7465 2073 797a 7967 7920 4d20 6f66  iate syzygy M of
│ │ │ │ +0006e2e0: 204d 3020 3d20 522f 286d 312b 6d32 2920   M0 = R/(m1+m2) 
│ │ │ │ +0006e2f0: 7768 6572 6520 6d31 2061 6e64 206d 3220  where m1 and m2 
│ │ │ │ +0006e300: 6172 6520 6d6f 6e6f 6d69 616c 7320 6f66  are monomials of
│ │ │ │ +0006e310: 2074 6865 0a73 616d 6520 6465 6772 6565   the.same degree
│ │ │ │ +0006e320: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...+------------
│ │ │ │  0006e330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006e340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006e350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006e360: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 7365  ------+.|i1 : se
│ │ │ │ -0006e370: 7452 616e 646f 6d53 6565 6420 3020 2020  tRandomSeed 0   
│ │ │ │ +0006e350: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120  ----------+.|i1 
│ │ │ │ +0006e360: 3a20 7365 7452 616e 646f 6d53 6565 6420  : setRandomSeed 
│ │ │ │ +0006e370: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │  0006e380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e3a0: 7c0a 7c20 2d2d 2073 6574 7469 6e67 2072  |.| -- setting r
│ │ │ │ -0006e3b0: 616e 646f 6d20 7365 6564 2074 6f20 3020  andom seed to 0 
│ │ │ │ -0006e3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e3d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0006e390: 2020 207c 0a7c 202d 2d20 7365 7474 696e     |.| -- settin
│ │ │ │ +0006e3a0: 6720 7261 6e64 6f6d 2073 6565 6420 746f  g random seed to
│ │ │ │ +0006e3b0: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ +0006e3c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0006e3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e410: 2020 2020 7c0a 7c6f 3120 3d20 3020 2020      |.|o1 = 0   
│ │ │ │ +0006e400: 2020 2020 207c 0a7c 6f31 203d 2030 2020       |.|o1 = 0  
│ │ │ │ +0006e410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e440: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0006e450: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0006e430: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0006e440: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0006e450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006e460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006e470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006e480: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20  --------+.|i2 : 
│ │ │ │ -0006e490: 7375 6d54 776f 4d6f 6e6f 6d69 616c 7328  sumTwoMonomials(
│ │ │ │ -0006e4a0: 322c 3329 2020 2020 2020 2020 2020 2020  2,3)            
│ │ │ │ -0006e4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e4c0: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e    |.| -- used 0.
│ │ │ │ -0006e4d0: 3430 3430 3532 7320 2863 7075 293b 2030  404052s (cpu); 0
│ │ │ │ -0006e4e0: 2e33 3333 3438 3873 2028 7468 7265 6164  .333488s (thread
│ │ │ │ -0006e4f0: 293b 2030 7320 2867 6329 2020 7c0a 7c32  ); 0s (gc)  |.|2
│ │ │ │ +0006e470: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2073  -------+.|i2 : s
│ │ │ │ +0006e480: 756d 5477 6f4d 6f6e 6f6d 6961 6c73 2832  umTwoMonomials(2
│ │ │ │ +0006e490: 2c33 2920 2020 2020 2020 2020 2020 2020  ,3)             
│ │ │ │ +0006e4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006e4b0: 7c0a 7c20 2d2d 2075 7365 6420 302e 3731  |.| -- used 0.71
│ │ │ │ +0006e4c0: 3333 3834 7320 2863 7075 293b 2030 2e34  3384s (cpu); 0.4
│ │ │ │ +0006e4d0: 3635 3135 7320 2874 6872 6561 6429 3b20  6515s (thread); 
│ │ │ │ +0006e4e0: 3073 2028 6763 2920 207c 0a7c 3220 2020  0s (gc)  |.|2   
│ │ │ │ +0006e4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e530: 2020 2020 2020 7c0a 7c54 616c 6c79 7b7b        |.|Tally{{
│ │ │ │ -0006e540: 7b32 2c20 327d 2c20 7b31 2c20 327d 7d20  {2, 2}, {1, 2}} 
│ │ │ │ -0006e550: 3d3e 2033 7d20 2020 2020 2020 2020 2020  => 3}           
│ │ │ │ +0006e520: 2020 7c0a 7c54 616c 6c79 7b7b 7b32 2c20    |.|Tally{{{2, 
│ │ │ │ +0006e530: 327d 2c20 7b31 2c20 327d 7d20 3d3e 2033  2}, {1, 2}} => 3
│ │ │ │ +0006e540: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +0006e550: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0006e560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e570: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0006e570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e5a0: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d            |.| --
│ │ │ │ -0006e5b0: 2075 7365 6420 302e 3230 3032 3632 7320   used 0.200262s 
│ │ │ │ -0006e5c0: 2863 7075 293b 2030 2e31 3234 3331 3173  (cpu); 0.124311s
│ │ │ │ -0006e5d0: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │ -0006e5e0: 6329 2020 7c0a 7c33 2020 2020 2020 2020  c)  |.|3        
│ │ │ │ +0006e590: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ +0006e5a0: 302e 3330 3030 3338 7320 2863 7075 293b  0.300038s (cpu);
│ │ │ │ +0006e5b0: 2030 2e31 3536 3434 3773 2028 7468 7265   0.156447s (thre
│ │ │ │ +0006e5c0: 6164 293b 2030 7320 2867 6329 207c 0a7c  ad); 0s (gc) |.|
│ │ │ │ +0006e5d0: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ +0006e5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e610: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0006e620: 7c54 616c 6c79 7b7b 7b32 2c20 327d 2c20  |Tally{{{2, 2}, 
│ │ │ │ -0006e630: 7b31 2c20 327d 7d20 3d3e 2031 7d20 2020  {1, 2}} => 1}   
│ │ │ │ -0006e640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e650: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0006e600: 2020 2020 2020 7c0a 7c54 616c 6c79 7b7b        |.|Tally{{
│ │ │ │ +0006e610: 7b32 2c20 327d 2c20 7b31 2c20 327d 7d20  {2, 2}, {1, 2}} 
│ │ │ │ +0006e620: 3d3e 2031 7d20 2020 2020 2020 2020 2020  => 1}           
│ │ │ │ +0006e630: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006e640: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0006e650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e690: 2020 7c0a 7c20 2d2d 2075 7365 6420 332e    |.| -- used 3.
│ │ │ │ -0006e6a0: 3630 3765 2d30 3673 2028 6370 7529 3b20  607e-06s (cpu); 
│ │ │ │ -0006e6b0: 332e 3133 3565 2d30 3673 2028 7468 7265  3.135e-06s (thre
│ │ │ │ -0006e6c0: 6164 293b 2030 7320 2867 6329 7c0a 7c34  ad); 0s (gc)|.|4
│ │ │ │ +0006e670: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075          |.| -- u
│ │ │ │ +0006e680: 7365 6420 352e 3538 652d 3036 7320 2863  sed 5.58e-06s (c
│ │ │ │ +0006e690: 7075 293b 2033 2e31 3132 652d 3036 7320  pu); 3.112e-06s 
│ │ │ │ +0006e6a0: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │ +0006e6b0: 297c 0a7c 3420 2020 2020 2020 2020 2020  )|.|4           
│ │ │ │ +0006e6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e700: 2020 2020 2020 7c0a 7c54 616c 6c79 7b7d        |.|Tally{}
│ │ │ │ +0006e6e0: 2020 2020 2020 2020 2020 7c0a 7c54 616c            |.|Tal
│ │ │ │ +0006e6f0: 6c79 7b7d 2020 2020 2020 2020 2020 2020  ly{}            
│ │ │ │ +0006e700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e740: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ -0006e750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006e760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006e770: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565  ----------+..See
│ │ │ │ -0006e780: 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a   also.========..
│ │ │ │ -0006e790: 2020 2a20 2a6e 6f74 6520 7477 6f4d 6f6e    * *note twoMon
│ │ │ │ -0006e7a0: 6f6d 6961 6c73 3a20 7477 6f4d 6f6e 6f6d  omials: twoMonom
│ │ │ │ -0006e7b0: 6961 6c73 2c20 2d2d 2074 616c 6c79 2074  ials, -- tally t
│ │ │ │ -0006e7c0: 6865 2073 6571 7565 6e63 6573 206f 6620  he sequences of 
│ │ │ │ -0006e7d0: 4252 616e 6b73 2066 6f72 0a20 2020 2063  BRanks for.    c
│ │ │ │ -0006e7e0: 6572 7461 696e 2065 7861 6d70 6c65 730a  ertain examples.
│ │ │ │ -0006e7f0: 0a57 6179 7320 746f 2075 7365 2073 756d  .Ways to use sum
│ │ │ │ -0006e800: 5477 6f4d 6f6e 6f6d 6961 6c73 3a0a 3d3d  TwoMonomials:.==
│ │ │ │ -0006e810: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0006e820: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ -0006e830: 2273 756d 5477 6f4d 6f6e 6f6d 6961 6c73  "sumTwoMonomials
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│ │ │ │ +0006ee70: 2020 2020 2020 207c 0a7c 6f31 203a 2050         |.|o1 : P
│ │ │ │ +0006ee80: 6f6c 796e 6f6d 6961 6c52 696e 672c 2033  olynomialRing, 3
│ │ │ │ +0006ee90: 2073 6b65 7720 636f 6d6d 7574 6174 6976   skew commutativ
│ │ │ │ +0006eea0: 6520 7661 7269 6162 6c65 2873 2920 2020  e variable(s)   
│ │ │ │ +0006eeb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006eec0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0006eed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006eee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006eef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006ef00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006ef10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006ef20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006ef30: 2d2d 2d2d 2d2b 0a7c 6932 203a 204d 203d  -----+.|i2 : M =
│ │ │ │ -0006ef40: 2063 6f6b 6572 206d 6170 2845 5e32 2c20   coker map(E^2, 
│ │ │ │ -0006ef50: 455e 7b2d 317d 2c20 6d61 7472 6978 2261  E^{-1}, matrix"a
│ │ │ │ -0006ef60: 623b 6263 2229 2020 2020 2020 2020 2020  b;bc")          
│ │ │ │ +0006ef10: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 204d  -------+.|i2 : M
│ │ │ │ +0006ef20: 203d 2063 6f6b 6572 206d 6170 2845 5e32   = coker map(E^2
│ │ │ │ +0006ef30: 2c20 455e 7b2d 317d 2c20 6d61 7472 6978  , E^{-1}, matrix
│ │ │ │ +0006ef40: 2261 623b 6263 2229 2020 2020 2020 2020  "ab;bc")        
│ │ │ │ +0006ef50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006ef60: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0006ef70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ef80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006ef80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ef90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006efa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006efb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006efc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006efd0: 2020 2020 207c 0a7c 6f32 203d 2063 6f6b       |.|o2 = cok
│ │ │ │ -0006efe0: 6572 6e65 6c20 7c20 6162 207c 2020 2020  ernel | ab |    
│ │ │ │ +0006efb0: 2020 2020 2020 207c 0a7c 6f32 203d 2063         |.|o2 = c
│ │ │ │ +0006efc0: 6f6b 6572 6e65 6c20 7c20 6162 207c 2020  okernel | ab |  
│ │ │ │ +0006efd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006efe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006eff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f020: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0006f030: 2020 2020 2020 7c20 6263 207c 2020 2020        | bc |    
│ │ │ │ +0006f000: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0006f010: 2020 2020 2020 2020 7c20 6263 207c 2020          | bc |  
│ │ │ │ +0006f020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f050: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0006f060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f070: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006f070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f0a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0006f0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f0c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006f0c0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │  0006f0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f0e0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -0006f0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f110: 2020 2020 207c 0a7c 6f32 203a 2045 2d6d       |.|o2 : E-m
│ │ │ │ -0006f120: 6f64 756c 652c 2071 756f 7469 656e 7420  odule, quotient 
│ │ │ │ -0006f130: 6f66 2045 2020 2020 2020 2020 2020 2020  of E            
│ │ │ │ -0006f140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f160: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0006f0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f0f0: 2020 2020 2020 207c 0a7c 6f32 203a 2045         |.|o2 : E
│ │ │ │ +0006f100: 2d6d 6f64 756c 652c 2071 756f 7469 656e  -module, quotien
│ │ │ │ +0006f110: 7420 6f66 2045 2020 2020 2020 2020 2020  t of E          
│ │ │ │ +0006f120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f140: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0006f150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006f160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006f170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006f180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006f190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006f1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006f1b0: 2d2d 2d2d 2d2b 0a7c 6933 203a 2070 7265  -----+.|i3 : pre
│ │ │ │ -0006f1c0: 7365 6e74 6174 696f 6e20 4d20 2020 2020  sentation M     
│ │ │ │ +0006f190: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2070  -------+.|i3 : p
│ │ │ │ +0006f1a0: 7265 7365 6e74 6174 696f 6e20 4d20 2020  resentation M   
│ │ │ │ +0006f1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f1e0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0006f1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f200: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006f200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f250: 2020 2020 207c 0a7c 6f33 203d 207c 2061       |.|o3 = | a
│ │ │ │ -0006f260: 6220 7c20 2020 2020 2020 2020 2020 2020  b |             
│ │ │ │ +0006f230: 2020 2020 2020 207c 0a7c 6f33 203d 207c         |.|o3 = |
│ │ │ │ +0006f240: 2061 6220 7c20 2020 2020 2020 2020 2020   ab |           
│ │ │ │ +0006f250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f2a0: 2020 2020 207c 0a7c 2020 2020 207c 2062       |.|     | b
│ │ │ │ -0006f2b0: 6320 7c20 2020 2020 2020 2020 2020 2020  c |             
│ │ │ │ +0006f280: 2020 2020 2020 207c 0a7c 2020 2020 207c         |.|     |
│ │ │ │ +0006f290: 2062 6320 7c20 2020 2020 2020 2020 2020   bc |           
│ │ │ │ +0006f2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f2d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0006f2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f2f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006f2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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                  
│ │ │ │ -0006f340: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0006f350: 2020 2020 2032 2020 2020 2020 3120 2020       2      1   
│ │ │ │ +0006f320: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0006f330: 2020 2020 2020 2032 2020 2020 2020 3120         2      1 
│ │ │ │ +0006f340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f390: 2020 2020 207c 0a7c 6f33 203a 204d 6174       |.|o3 : Mat
│ │ │ │ -0006f3a0: 7269 7820 4520 203c 2d2d 2045 2020 2020  rix E  <-- E    
│ │ │ │ +0006f370: 2020 2020 2020 207c 0a7c 6f33 203a 204d         |.|o3 : M
│ │ │ │ +0006f380: 6174 7269 7820 4520 203c 2d2d 2045 2020  atrix E  <-- E  
│ │ │ │ +0006f390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f3e0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0006f3c0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0006f3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006f3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006f3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006f400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006f410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006f420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006f430: 2d2d 2d2d 2d2b 0a7c 6934 203a 2054 6174  -----+.|i4 : Tat
│ │ │ │ -0006f440: 6552 6573 6f6c 7574 696f 6e28 4d2c 2d32  eResolution(M,-2
│ │ │ │ -0006f450: 2c37 2920 2020 2020 2020 2020 2020 2020  ,7)             
│ │ │ │ -0006f460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f410: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2054  -------+.|i4 : T
│ │ │ │ +0006f420: 6174 6552 6573 6f6c 7574 696f 6e28 4d2c  ateResolution(M,
│ │ │ │ +0006f430: 2d32 2c37 2920 2020 2020 2020 2020 2020  -2,7)           
│ │ │ │ +0006f440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f460: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0006f470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f480: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006f480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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                  
│ │ │ │ -0006f4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f4d0: 2020 2020 207c 0a7c 2020 2020 2020 3920       |.|      9 
│ │ │ │ -0006f4e0: 2020 2020 2035 2020 2020 2020 3220 2020       5      2   
│ │ │ │ -0006f4f0: 2020 2031 2020 2020 2020 3220 2020 2020     1      2     
│ │ │ │ -0006f500: 2034 2020 2020 2020 3720 2020 2020 2031   4      7      1
│ │ │ │ -0006f510: 3120 2020 2020 2031 3620 2020 2020 2032  1      16      2
│ │ │ │ -0006f520: 3220 2020 207c 0a7c 6f34 203d 2045 2020  2    |.|o4 = E  
│ │ │ │ -0006f530: 3c2d 2d20 4520 203c 2d2d 2045 2020 3c2d  <-- E  <-- E  <-
│ │ │ │ -0006f540: 2d20 4520 203c 2d2d 2045 2020 3c2d 2d20  - E  <-- E  <-- 
│ │ │ │ -0006f550: 4520 203c 2d2d 2045 2020 3c2d 2d20 4520  E  <-- E  <-- E 
│ │ │ │ -0006f560: 2020 3c2d 2d20 4520 2020 3c2d 2d20 4520    <-- E   <-- E 
│ │ │ │ -0006f570: 2020 3c2d 2d7c 0a7c 2020 2020 2020 2020    <--|.|        
│ │ │ │ +0006f4b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0006f4c0: 3920 2020 2020 2035 2020 2020 2020 3220  9      5      2 
│ │ │ │ +0006f4d0: 2020 2020 2031 2020 2020 2020 3220 2020       1      2   
│ │ │ │ +0006f4e0: 2020 2034 2020 2020 2020 3720 2020 2020     4      7     
│ │ │ │ +0006f4f0: 2031 3120 2020 2020 2031 3620 2020 2020   11      16     
│ │ │ │ +0006f500: 2032 3220 2020 207c 0a7c 6f34 203d 2045   22    |.|o4 = E
│ │ │ │ +0006f510: 2020 3c2d 2d20 4520 203c 2d2d 2045 2020    <-- E  <-- E  
│ │ │ │ +0006f520: 3c2d 2d20 4520 203c 2d2d 2045 2020 3c2d  <-- E  <-- E  <-
│ │ │ │ +0006f530: 2d20 4520 203c 2d2d 2045 2020 3c2d 2d20  - E  <-- E  <-- 
│ │ │ │ +0006f540: 4520 2020 3c2d 2d20 4520 2020 3c2d 2d20  E   <-- E   <-- 
│ │ │ │ +0006f550: 4520 2020 3c2d 2d7c 0a7c 2020 2020 2020  E   <--|.|      
│ │ │ │ +0006f560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f5c0: 2020 2020 207c 0a7c 2020 2020 202d 3220       |.|     -2 
│ │ │ │ -0006f5d0: 2020 2020 2d31 2020 2020 2030 2020 2020      -1     0    
│ │ │ │ -0006f5e0: 2020 3120 2020 2020 2032 2020 2020 2020    1      2      
│ │ │ │ -0006f5f0: 3320 2020 2020 2034 2020 2020 2020 3520  3      4      5 
│ │ │ │ -0006f600: 2020 2020 2020 3620 2020 2020 2020 3720        6       7 
│ │ │ │ -0006f610: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006f5a0: 2020 2020 2020 207c 0a7c 2020 2020 202d         |.|     -
│ │ │ │ +0006f5b0: 3220 2020 2020 2d31 2020 2020 2030 2020  2     -1     0  
│ │ │ │ +0006f5c0: 2020 2020 3120 2020 2020 2032 2020 2020      1      2    
│ │ │ │ +0006f5d0: 2020 3320 2020 2020 2034 2020 2020 2020    3      4      
│ │ │ │ +0006f5e0: 3520 2020 2020 2020 3620 2020 2020 2020  5       6       
│ │ │ │ +0006f5f0: 3720 2020 2020 207c 0a7c 2020 2020 2020  7      |.|      
│ │ │ │ +0006f600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f610: 2020 2020 2020 2020 2020 2020 2020 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 6f34 203a 2043 6f6d       |.|o4 : Com
│ │ │ │ -0006f670: 706c 6578 2020 2020 2020 2020 2020 2020  plex            
│ │ │ │ +0006f640: 2020 2020 2020 207c 0a7c 6f34 203a 2043         |.|o4 : C
│ │ │ │ +0006f650: 6f6d 706c 6578 2020 2020 2020 2020 2020  omplex          
│ │ │ │ +0006f660: 2020 2020 2020 2020 2020 2020 2020 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 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +0006f690: 2020 2020 2020 207c 0a7c 2d2d 2d2d 2d2d         |.|------
│ │ │ │ +0006f6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006f6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006f6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006f6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006f6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006f6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006f700: 2d2d 2d2d 2d7c 0a7c 3020 2020 2020 2020  -----|.|0       
│ │ │ │ +0006f6e0: 2d2d 2d2d 2d2d 2d7c 0a7c 3020 2020 2020  -------|.|0     
│ │ │ │ +0006f6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f730: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0006f740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f750: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006f750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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                  
│ │ │ │ +0006f780: 2020 2020 2020 207c 0a7c 3820 2020 2020         |.|8     
│ │ │ │  0006f790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f7a0: 2020 2020 207c 0a7c 3820 2020 2020 2020       |.|8       
│ │ │ │ +0006f7a0: 2020 2020 2020 2020 2020 2020 2020 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 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0006f7d0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0006f7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006f7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006f800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006f810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006f820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006f830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006f840: 2d2d 2d2d 2d2b 0a0a 4361 7665 6174 0a3d  -----+..Caveat.=
│ │ │ │ -0006f850: 3d3d 3d3d 3d0a 0a49 6e20 6120 7072 6576  =====..In a prev
│ │ │ │ -0006f860: 696f 7573 2076 6572 7369 6f6e 206f 6620  ious version of 
│ │ │ │ -0006f870: 7468 6973 2073 6372 6970 742c 2074 6869  this script, thi
│ │ │ │ -0006f880: 7320 636f 6d6d 616e 6420 7265 7475 726e  s command return
│ │ │ │ -0006f890: 6564 2061 2062 6574 7469 2074 6162 6c65  ed a betti table
│ │ │ │ -0006f8a0: 3b20 6e6f 770a 7573 6520 2262 6574 7469  ; now.use "betti
│ │ │ │ -0006f8b0: 2054 6174 6552 6573 6f6c 7574 696f 6e22   TateResolution"
│ │ │ │ -0006f8c0: 2069 6e73 7465 6164 2e0a 0a57 6179 7320   instead...Ways 
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│ │ │ │ -000705f0: 6175 6c61 7932 446f 6329 7573 696e 6720  aulay2Doc)using 
│ │ │ │ -00070600: 6675 6e63 7469 6f6e 7320 7769 7468 206f  functions with o
│ │ │ │ -00070610: 7074 696f 6e61 6c20 696e 7075 7473 2c3a  ptional inputs,:
│ │ │ │ -00070620: 0a20 2020 2020 202a 204f 7074 696d 6973  .      * Optimis
│ │ │ │ -00070630: 6d20 3d3e 202e 2e2e 2c20 6465 6661 756c  m => ..., defaul
│ │ │ │ -00070640: 7420 7661 6c75 6520 300a 2020 2a20 4f75  t value 0.  * Ou
│ │ │ │ -00070650: 7470 7574 733a 0a20 2020 2020 202a 2054  tputs:.      * T
│ │ │ │ -00070660: 2c20 6120 2a6e 6f74 6520 7461 6c6c 793a  , a *note tally:
│ │ │ │ -00070670: 2028 4d61 6361 756c 6179 3244 6f63 2954   (Macaulay2Doc)T
│ │ │ │ -00070680: 616c 6c79 2c2c 200a 0a44 6573 6372 6970  ally,, ..Descrip
│ │ │ │ -00070690: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  tion.===========
│ │ │ │ -000706a0: 0a0a 7461 6c6c 6965 7320 7468 6520 7365  ..tallies the se
│ │ │ │ -000706b0: 7175 656e 6365 7320 6f66 2042 2d72 616e  quences of B-ran
│ │ │ │ -000706c0: 6b73 2074 6861 7420 6f63 6375 7220 666f  ks that occur fo
│ │ │ │ -000706d0: 7220 6964 6561 6c73 2067 656e 6572 6174  r ideals generat
│ │ │ │ -000706e0: 6564 2062 7920 7061 6972 7320 6f66 0a6d  ed by pairs of.m
│ │ │ │ -000706f0: 6f6e 6f6d 6961 6c73 2069 6e20 5220 3d20  onomials in R = 
│ │ │ │ -00070700: 532f 2864 2d74 6820 706f 7765 7273 206f  S/(d-th powers o
│ │ │ │ -00070710: 6620 7468 6520 7661 7269 6162 6c65 7329  f the variables)
│ │ │ │ -00070720: 2c20 7769 7468 2066 756c 6c20 636f 6d70  , with full comp
│ │ │ │ -00070730: 6c65 7869 7479 2028 3d63 293b 0a74 6861  lexity (=c);.tha
│ │ │ │ -00070740: 7420 6973 2c20 666f 7220 616e 2061 7070  t is, for an app
│ │ │ │ -00070750: 726f 7072 6961 7465 2073 797a 7967 7920  ropriate syzygy 
│ │ │ │ -00070760: 4d20 6f66 204d 3020 3d20 522f 286d 312c  M of M0 = R/(m1,
│ │ │ │ -00070770: 206d 3229 2077 6865 7265 206d 3120 616e   m2) where m1 an
│ │ │ │ -00070780: 6420 6d32 2061 7265 0a6d 6f6e 6f6d 6961  d m2 are.monomia
│ │ │ │ -00070790: 6c73 206f 6620 7468 6520 7361 6d65 2064  ls of the same d
│ │ │ │ -000707a0: 6567 7265 652e 0a0a 2b2d 2d2d 2d2d 2d2d  egree...+-------
│ │ │ │ +000704d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a  **************..
│ │ │ │ +000704e0: 2020 2a20 5573 6167 653a 200a 2020 2020    * Usage: .    
│ │ │ │ +000704f0: 2020 2020 5420 3d20 5477 6f4d 6f6e 6f6d      T = TwoMonom
│ │ │ │ +00070500: 6961 6c73 2863 2c64 290a 2020 2a20 496e  ials(c,d).  * In
│ │ │ │ +00070510: 7075 7473 3a0a 2020 2020 2020 2a20 632c  puts:.      * c,
│ │ │ │ +00070520: 2061 6e20 2a6e 6f74 6520 696e 7465 6765   an *note intege
│ │ │ │ +00070530: 723a 2028 4d61 6361 756c 6179 3244 6f63  r: (Macaulay2Doc
│ │ │ │ +00070540: 295a 5a2c 2c20 636f 6469 6d65 6e73 696f  )ZZ,, codimensio
│ │ │ │ +00070550: 6e20 696e 2077 6869 6368 2074 6f20 776f  n in which to wo
│ │ │ │ +00070560: 726b 0a20 2020 2020 202a 2064 2c20 616e  rk.      * d, an
│ │ │ │ +00070570: 202a 6e6f 7465 2069 6e74 6567 6572 3a20   *note integer: 
│ │ │ │ +00070580: 284d 6163 6175 6c61 7932 446f 6329 5a5a  (Macaulay2Doc)ZZ
│ │ │ │ +00070590: 2c2c 2064 6567 7265 6520 6f66 2074 6865  ,, degree of the
│ │ │ │ +000705a0: 206d 6f6e 6f6d 6961 6c73 2074 6f20 7461   monomials to ta
│ │ │ │ +000705b0: 6b65 0a20 202a 202a 6e6f 7465 204f 7074  ke.  * *note Opt
│ │ │ │ +000705c0: 696f 6e61 6c20 696e 7075 7473 3a20 284d  ional inputs: (M
│ │ │ │ +000705d0: 6163 6175 6c61 7932 446f 6329 7573 696e  acaulay2Doc)usin
│ │ │ │ +000705e0: 6720 6675 6e63 7469 6f6e 7320 7769 7468  g functions with
│ │ │ │ +000705f0: 206f 7074 696f 6e61 6c20 696e 7075 7473   optional inputs
│ │ │ │ +00070600: 2c3a 0a20 2020 2020 202a 204f 7074 696d  ,:.      * Optim
│ │ │ │ +00070610: 6973 6d20 3d3e 202e 2e2e 2c20 6465 6661  ism => ..., defa
│ │ │ │ +00070620: 756c 7420 7661 6c75 6520 300a 2020 2a20  ult value 0.  * 
│ │ │ │ +00070630: 4f75 7470 7574 733a 0a20 2020 2020 202a  Outputs:.      *
│ │ │ │ +00070640: 2054 2c20 6120 2a6e 6f74 6520 7461 6c6c   T, a *note tall
│ │ │ │ +00070650: 793a 2028 4d61 6361 756c 6179 3244 6f63  y: (Macaulay2Doc
│ │ │ │ +00070660: 2954 616c 6c79 2c2c 200a 0a44 6573 6372  )Tally,, ..Descr
│ │ │ │ +00070670: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d  iption.=========
│ │ │ │ +00070680: 3d3d 0a0a 7461 6c6c 6965 7320 7468 6520  ==..tallies the 
│ │ │ │ +00070690: 7365 7175 656e 6365 7320 6f66 2042 2d72  sequences of B-r
│ │ │ │ +000706a0: 616e 6b73 2074 6861 7420 6f63 6375 7220  anks that occur 
│ │ │ │ +000706b0: 666f 7220 6964 6561 6c73 2067 656e 6572  for ideals gener
│ │ │ │ +000706c0: 6174 6564 2062 7920 7061 6972 7320 6f66  ated by pairs of
│ │ │ │ +000706d0: 0a6d 6f6e 6f6d 6961 6c73 2069 6e20 5220  .monomials in R 
│ │ │ │ +000706e0: 3d20 532f 2864 2d74 6820 706f 7765 7273  = S/(d-th powers
│ │ │ │ +000706f0: 206f 6620 7468 6520 7661 7269 6162 6c65   of the variable
│ │ │ │ +00070700: 7329 2c20 7769 7468 2066 756c 6c20 636f  s), with full co
│ │ │ │ +00070710: 6d70 6c65 7869 7479 2028 3d63 293b 0a74  mplexity (=c);.t
│ │ │ │ +00070720: 6861 7420 6973 2c20 666f 7220 616e 2061  hat is, for an a
│ │ │ │ +00070730: 7070 726f 7072 6961 7465 2073 797a 7967  ppropriate syzyg
│ │ │ │ +00070740: 7920 4d20 6f66 204d 3020 3d20 522f 286d  y M of M0 = R/(m
│ │ │ │ +00070750: 312c 206d 3229 2077 6865 7265 206d 3120  1, m2) where m1 
│ │ │ │ +00070760: 616e 6420 6d32 2061 7265 0a6d 6f6e 6f6d  and m2 are.monom
│ │ │ │ +00070770: 6961 6c73 206f 6620 7468 6520 7361 6d65  ials of the same
│ │ │ │ +00070780: 2064 6567 7265 652e 0a0a 2b2d 2d2d 2d2d   degree...+-----
│ │ │ │ +00070790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000707a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000707b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000707c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000707d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -000707e0: 7c69 3120 3a20 7365 7452 616e 646f 6d53  |i1 : setRandomS
│ │ │ │ -000707f0: 6565 6420 3020 2020 2020 2020 2020 2020  eed 0           
│ │ │ │ -00070800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070810: 2020 2020 2020 7c0a 7c20 2d2d 2073 6574        |.| -- set
│ │ │ │ -00070820: 7469 6e67 2072 616e 646f 6d20 7365 6564  ting random seed
│ │ │ │ -00070830: 2074 6f20 3020 2020 2020 2020 2020 2020   to 0           
│ │ │ │ -00070840: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00070850: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00070860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070880: 2020 2020 2020 7c0a 7c6f 3120 3d20 3020        |.|o1 = 0 
│ │ │ │ +000707c0: 2b0a 7c69 3120 3a20 7365 7452 616e 646f  +.|i1 : setRando
│ │ │ │ +000707d0: 6d53 6565 6420 3020 2020 2020 2020 2020  mSeed 0         
│ │ │ │ +000707e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000707f0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2073          |.| -- s
│ │ │ │ +00070800: 6574 7469 6e67 2072 616e 646f 6d20 7365  etting random se
│ │ │ │ +00070810: 6564 2074 6f20 3020 2020 2020 2020 2020  ed to 0         
│ │ │ │ +00070820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00070830: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00070840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00070850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00070860: 2020 2020 2020 2020 7c0a 7c6f 3120 3d20          |.|o1 = 
│ │ │ │ +00070870: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ +00070880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000708a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000708b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000708c0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -000708d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000708e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000708f0: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 7477  ------+.|i2 : tw
│ │ │ │ -00070900: 6f4d 6f6e 6f6d 6961 6c73 2832 2c33 2920  oMonomials(2,3) 
│ │ │ │ -00070910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070920: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00070930: 7c20 2d2d 2075 7365 6420 302e 3832 3933  | -- used 0.8293
│ │ │ │ -00070940: 3131 7320 2863 7075 293b 2030 2e36 3133  11s (cpu); 0.613
│ │ │ │ -00070950: 3932 3873 2028 7468 7265 6164 293b 2030  928s (thread); 0
│ │ │ │ -00070960: 7320 2867 6329 7c0a 7c32 2020 2020 2020  s (gc)|.|2      
│ │ │ │ +000708a0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +000708b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000708c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000708d0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20  --------+.|i2 : 
│ │ │ │ +000708e0: 7477 6f4d 6f6e 6f6d 6961 6c73 2832 2c33  twoMonomials(2,3
│ │ │ │ +000708f0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +00070900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00070910: 7c0a 7c20 2d2d 2075 7365 6420 312e 3135  |.| -- used 1.15
│ │ │ │ +00070920: 3832 3273 2028 6370 7529 3b20 302e 3733  822s (cpu); 0.73
│ │ │ │ +00070930: 3530 3835 7320 2874 6872 6561 6429 3b20  5085s (thread); 
│ │ │ │ +00070940: 3073 2028 6763 2920 7c0a 7c32 2020 2020  0s (gc) |.|2    
│ │ │ │ +00070950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00070960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070990: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000709a0: 7c54 616c 6c79 7b7b 7b31 2c20 317d 7d20  |Tally{{{1, 1}} 
│ │ │ │ -000709b0: 3d3e 2032 2020 2020 2020 2020 7d20 2020  => 2        }   
│ │ │ │ -000709c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000709d0: 2020 2020 2020 7c0a 7c20 2020 2020 207b        |.|      {
│ │ │ │ -000709e0: 7b32 2c20 327d 2c20 7b31 2c20 327d 7d20  {2, 2}, {1, 2}} 
│ │ │ │ -000709f0: 3d3e 2034 2020 2020 2020 2020 2020 2020  => 4            
│ │ │ │ -00070a00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00070a10: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00070a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070a40: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365        |.| -- use
│ │ │ │ -00070a50: 6420 302e 3431 3132 3973 2028 6370 7529  d 0.41129s (cpu)
│ │ │ │ -00070a60: 3b20 302e 3333 3734 3634 7320 2874 6872  ; 0.337464s (thr
│ │ │ │ -00070a70: 6561 6429 3b20 3073 2028 6763 2920 7c0a  ead); 0s (gc) |.
│ │ │ │ -00070a80: 7c33 2020 2020 2020 2020 2020 2020 2020  |3              
│ │ │ │ -00070a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070ab0: 2020 2020 2020 7c0a 7c54 616c 6c79 7b7b        |.|Tally{{
│ │ │ │ -00070ac0: 7b32 2c20 327d 2c20 7b31 2c20 327d 7d20  {2, 2}, {1, 2}} 
│ │ │ │ -00070ad0: 3d3e 2032 7d20 2020 2020 2020 2020 2020  => 2}           
│ │ │ │ -00070ae0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00070af0: 7c20 2020 2020 207b 7b33 2c20 337d 2c20  |      {{3, 3}, 
│ │ │ │ -00070b00: 7b32 2c20 337d 7d20 3d3e 2031 2020 2020  {2, 3}} => 1    
│ │ │ │ +00070980: 7c0a 7c54 616c 6c79 7b7b 7b31 2c20 317d  |.|Tally{{{1, 1}
│ │ │ │ +00070990: 7d20 3d3e 2032 2020 2020 2020 2020 7d20  } => 2        } 
│ │ │ │ +000709a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000709b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000709c0: 207b 7b32 2c20 327d 2c20 7b31 2c20 327d   {{2, 2}, {1, 2}
│ │ │ │ +000709d0: 7d20 3d3e 2034 2020 2020 2020 2020 2020  } => 4          
│ │ │ │ +000709e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000709f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00070a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00070a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00070a20: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075          |.| -- u
│ │ │ │ +00070a30: 7365 6420 302e 3631 3635 3138 7320 2863  sed 0.616518s (c
│ │ │ │ +00070a40: 7075 293b 2030 2e34 3530 3435 3673 2028  pu); 0.450456s (
│ │ │ │ +00070a50: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 0s (gc)
│ │ │ │ +00070a60: 7c0a 7c33 2020 2020 2020 2020 2020 2020  |.|3            
│ │ │ │ +00070a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00070a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00070a90: 2020 2020 2020 2020 7c0a 7c54 616c 6c79          |.|Tally
│ │ │ │ +00070aa0: 7b7b 7b32 2c20 327d 2c20 7b31 2c20 327d  {{{2, 2}, {1, 2}
│ │ │ │ +00070ab0: 7d20 3d3e 2032 7d20 2020 2020 2020 2020  } => 2}         
│ │ │ │ +00070ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00070ad0: 7c0a 7c20 2020 2020 207b 7b33 2c20 337d  |.|      {{3, 3}
│ │ │ │ +00070ae0: 2c20 7b32 2c20 337d 7d20 3d3e 2031 2020  , {2, 3}} => 1  
│ │ │ │ +00070af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00070b00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  00070b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070b20: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00070b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070b50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00070b60: 7c20 2d2d 2075 7365 6420 302e 3139 3331  | -- used 0.1931
│ │ │ │ -00070b70: 3735 7320 2863 7075 293b 2030 2e31 3336  75s (cpu); 0.136
│ │ │ │ -00070b80: 3030 3273 2028 7468 7265 6164 293b 2030  002s (thread); 0
│ │ │ │ -00070b90: 7320 2867 6329 7c0a 7c34 2020 2020 2020  s (gc)|.|4      
│ │ │ │ +00070b40: 7c0a 7c20 2d2d 2075 7365 6420 302e 3235  |.| -- used 0.25
│ │ │ │ +00070b50: 3132 3235 7320 2863 7075 293b 2030 2e31  1225s (cpu); 0.1
│ │ │ │ +00070b60: 3535 3939 3873 2028 7468 7265 6164 293b  55998s (thread);
│ │ │ │ +00070b70: 2030 7320 2867 6329 7c0a 7c34 2020 2020   0s (gc)|.|4    
│ │ │ │ +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 2020 2020 2020 2020 2020                  
│ │ │ │ -00070bc0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00070bd0: 7c54 616c 6c79 7b7b 7b32 2c20 327d 2c20  |Tally{{{2, 2}, 
│ │ │ │ -00070be0: 7b31 2c20 327d 7d20 3d3e 2031 7d20 2020  {1, 2}} => 1}   
│ │ │ │ -00070bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070c00: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00070bb0: 7c0a 7c54 616c 6c79 7b7b 7b32 2c20 327d  |.|Tally{{{2, 2}
│ │ │ │ +00070bc0: 2c20 7b31 2c20 327d 7d20 3d3e 2031 7d20  , {1, 2}} => 1} 
│ │ │ │ +00070bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00070be0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00070bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00070c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  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 2b0a  --------------+.
│ │ │ │ -00070c40: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d  .See also.======
│ │ │ │ -00070c50: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 7477  ==..  * *note tw
│ │ │ │ -00070c60: 6f4d 6f6e 6f6d 6961 6c73 3a20 7477 6f4d  oMonomials: twoM
│ │ │ │ -00070c70: 6f6e 6f6d 6961 6c73 2c20 2d2d 2074 616c  onomials, -- tal
│ │ │ │ -00070c80: 6c79 2074 6865 2073 6571 7565 6e63 6573  ly the sequences
│ │ │ │ -00070c90: 206f 6620 4252 616e 6b73 2066 6f72 0a20   of BRanks for. 
│ │ │ │ -00070ca0: 2020 2063 6572 7461 696e 2065 7861 6d70     certain examp
│ │ │ │ -00070cb0: 6c65 730a 0a57 6179 7320 746f 2075 7365  les..Ways to use
│ │ │ │ -00070cc0: 2074 776f 4d6f 6e6f 6d69 616c 733a 0a3d   twoMonomials:.=
│ │ │ │ -00070cd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00070ce0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2274  ========..  * "t
│ │ │ │ -00070cf0: 776f 4d6f 6e6f 6d69 616c 7328 5a5a 2c5a  woMonomials(ZZ,Z
│ │ │ │ -00070d00: 5a29 220a 0a46 6f72 2074 6865 2070 726f  Z)"..For the pro
│ │ │ │ -00070d10: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d  grammer.========
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│ │ │ │ +000710a0: 2066 696e 6974 6542 6574 7469 4e75 6d62   finiteBettiNumb
│ │ │ │ +000710b0: 6572 737f 3136 3030 3333 0a4e 6f64 653a  ers.160033.Node:
│ │ │ │ +000710c0: 2066 7265 6545 7874 6572 696f 7253 756d   freeExteriorSum
│ │ │ │ +000710d0: 6d61 6e64 7f31 3635 3835 310a 4e6f 6465  mand.165851.Node
│ │ │ │ +000710e0: 3a20 4772 6164 696e 677f 3136 3837 3034  : Grading.168704
│ │ │ │ +000710f0: 0a4e 6f64 653a 2068 667f 3137 3030 3539  .Node: hf.170059
│ │ │ │ +00071100: 0a4e 6f64 653a 2068 664d 6f64 756c 6541  .Node: hfModuleA
│ │ │ │ +00071110: 7345 7874 7f31 3730 3934 310a 4e6f 6465  sExt.170941.Node
│ │ │ │ +00071120: 3a20 6869 6768 5379 7a79 6779 7f31 3734  : highSyzygy.174
│ │ │ │ +00071130: 3837 300a 4e6f 6465 3a20 684d 6170 737f  870.Node: hMaps.
│ │ │ │ +00071140: 3138 3233 3730 0a4e 6f64 653a 2048 6f6d  182370.Node: Hom
│ │ │ │ +00071150: 5769 7468 436f 6d70 6f6e 656e 7473 7f31  WithComponents.1
│ │ │ │ +00071160: 3833 3936 340a 4e6f 6465 3a20 696e 6669  83964.Node: infi
│ │ │ │ +00071170: 6e69 7465 4265 7474 694e 756d 6265 7273  niteBettiNumbers
│ │ │ │ +00071180: 7f31 3835 3337 350a 4e6f 6465 3a20 6973  .185375.Node: is
│ │ │ │ +00071190: 4c69 6e65 6172 7f31 3931 3331 360a 4e6f  Linear.191316.No
│ │ │ │ +000711a0: 6465 3a20 6973 5175 6173 6952 6567 756c  de: isQuasiRegul
│ │ │ │ +000711b0: 6172 7f31 3932 3333 380a 4e6f 6465 3a20  ar.192338.Node: 
│ │ │ │ +000711c0: 6973 5374 6162 6c79 5472 6976 6961 6c7f  isStablyTrivial.
│ │ │ │ +000711d0: 3139 3535 3735 0a4e 6f64 653a 206b 6f73  195575.Node: kos
│ │ │ │ +000711e0: 7a75 6c45 7874 656e 7369 6f6e 7f32 3033  zulExtension.203
│ │ │ │ +000711f0: 3032 380a 4e6f 6465 3a20 4c61 7965 7265  028.Node: Layere
│ │ │ │ +00071200: 647f 3230 3435 3833 0a4e 6f64 653a 206c  d.204583.Node: l
│ │ │ │ +00071210: 6179 6572 6564 5265 736f 6c75 7469 6f6e  ayeredResolution
│ │ │ │ +00071220: 7f32 3035 3935 310a 4e6f 6465 3a20 4c69  .205951.Node: Li
│ │ │ │ +00071230: 6674 7f32 3238 3631 310a 4e6f 6465 3a20  ft.228611.Node: 
│ │ │ │ +00071240: 6d61 6b65 4669 6e69 7465 5265 736f 6c75  makeFiniteResolu
│ │ │ │ +00071250: 7469 6f6e 7f32 3239 3633 340a 4e6f 6465  tion.229634.Node
│ │ │ │ +00071260: 3a20 6d61 6b65 4669 6e69 7465 5265 736f  : makeFiniteReso
│ │ │ │ +00071270: 6c75 7469 6f6e 436f 6469 6d32 7f32 3530  lutionCodim2.250
│ │ │ │ +00071280: 3337 310a 4e6f 6465 3a20 6d61 6b65 486f  371.Node: makeHo
│ │ │ │ +00071290: 6d6f 746f 7069 6573 7f32 3539 3435 390a  motopies.259459.
│ │ │ │ +000712a0: 4e6f 6465 3a20 6d61 6b65 486f 6d6f 746f  Node: makeHomoto
│ │ │ │ +000712b0: 7069 6573 317f 3334 3732 3230 0a4e 6f64  pies1.347220.Nod
│ │ │ │ +000712c0: 653a 206d 616b 6548 6f6d 6f74 6f70 6965  e: makeHomotopie
│ │ │ │ +000712d0: 734f 6e48 6f6d 6f6c 6f67 797f 3334 3838  sOnHomology.3488
│ │ │ │ +000712e0: 3534 0a4e 6f64 653a 206d 616b 654d 6f64  54.Node: makeMod
│ │ │ │ +000712f0: 756c 657f 3335 3034 3939 0a4e 6f64 653a  ule.350499.Node:
│ │ │ │ +00071300: 206d 616b 6554 7f33 3631 3139 370a 4e6f   makeT.361197.No
│ │ │ │ +00071310: 6465 3a20 6d61 7472 6978 4661 6374 6f72  de: matrixFactor
│ │ │ │ +00071320: 697a 6174 696f 6e7f 3336 3533 3936 0a4e  ization.365396.N
│ │ │ │ +00071330: 6f64 653a 206d 6642 6f75 6e64 7f33 3734  ode: mfBound.374
│ │ │ │ +00071340: 3234 380a 4e6f 6465 3a20 6d6f 6475 6c65  248.Node: module
│ │ │ │ +00071350: 4173 4578 747f 3337 3631 3430 0a4e 6f64  AsExt.376140.Nod
│ │ │ │ +00071360: 653a 206e 6577 4578 747f 3338 3233 3838  e: newExt.382388
│ │ │ │ +00071370: 0a4e 6f64 653a 206f 6464 4578 744d 6f64  .Node: oddExtMod
│ │ │ │ +00071380: 756c 657f 3431 3039 3734 0a4e 6f64 653a  ule.410974.Node:
│ │ │ │ +00071390: 204f 7074 696d 6973 6d7f 3431 3732 3634   Optimism.417264
│ │ │ │ +000713a0: 0a4e 6f64 653a 204f 7574 5269 6e67 7f34  .Node: OutRing.4
│ │ │ │ +000713b0: 3138 3739 300a 4e6f 6465 3a20 7073 694d  18790.Node: psiM
│ │ │ │ +000713c0: 6170 737f 3432 3031 3337 0a4e 6f64 653a  aps.420137.Node:
│ │ │ │ +000713d0: 2072 6567 756c 6172 6974 7953 6571 7565   regularitySeque
│ │ │ │ +000713e0: 6e63 657f 3432 3136 3737 0a4e 6f64 653a  nce.421677.Node:
│ │ │ │ +000713f0: 2053 327f 3432 3532 3332 0a4e 6f64 653a   S2.425232.Node:
│ │ │ │ +00071400: 2053 6861 6d61 7368 7f34 3336 3336 330a   Shamash.436363.
│ │ │ │ +00071410: 4e6f 6465 3a20 7370 6c69 7474 696e 6773  Node: splittings
│ │ │ │ +00071420: 7f34 3434 3130 320a 4e6f 6465 3a20 7374  .444102.Node: st
│ │ │ │ +00071430: 6162 6c65 486f 6d7f 3434 3933 3231 0a4e  ableHom.449321.N
│ │ │ │ +00071440: 6f64 653a 2073 756d 5477 6f4d 6f6e 6f6d  ode: sumTwoMonom
│ │ │ │ +00071450: 6961 6c73 7f34 3530 3535 380a 4e6f 6465  ials.450558.Node
│ │ │ │ +00071460: 3a20 5461 7465 5265 736f 6c75 7469 6f6e  : TateResolution
│ │ │ │ +00071470: 7f34 3533 3032 390a 4e6f 6465 3a20 7465  .453029.Node: te
│ │ │ │ +00071480: 6e73 6f72 5769 7468 436f 6d70 6f6e 656e  nsorWithComponen
│ │ │ │ +00071490: 7473 7f34 3537 3430 360a 4e6f 6465 3a20  ts.457406.Node: 
│ │ │ │ +000714a0: 746f 4172 7261 797f 3435 3838 3533 0a4e  toArray.458853.N
│ │ │ │ +000714b0: 6f64 653a 2074 776f 4d6f 6e6f 6d69 616c  ode: twoMonomial
│ │ │ │ +000714c0: 737f 3435 3937 3735 0a1f 0a45 6e64 2054  s.459775...End T
│ │ │ │ +000714d0: 6167 2054 6162 6c65 0a                   ag Table.
│ │ ├── ./usr/share/info/ConnectionMatrices.info.gz
│ │ │ ├── ConnectionMatrices.info
│ │ │ │ @@ -2415,30 +2415,30 @@
│ │ │ │  000096e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000096f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  00009700: 7c69 3920 3a20 656c 6170 7365 6454 696d  |i9 : elapsedTim
│ │ │ │  00009710: 6520 4120 3d20 636f 6e6e 6563 7469 6f6e  e A = connection
│ │ │ │  00009720: 4d61 7472 6963 6573 2049 3b20 2020 2020  Matrices I;     
│ │ │ │  00009730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009740: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00009750: 7c20 2d2d 2032 2e38 3635 3836 7320 656c  | -- 2.86586s el
│ │ │ │ +00009750: 7c20 2d2d 2032 2e34 3330 3934 7320 656c  | -- 2.43094s el
│ │ │ │  00009760: 6170 7365 6420 2020 2020 2020 2020 2020  apsed           
│ │ │ │  00009770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009790: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  000097a0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  000097b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000097c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000097d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000097e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  000097f0: 7c69 3130 203a 2065 6c61 7073 6564 5469  |i10 : elapsedTi
│ │ │ │  00009800: 6d65 2061 7373 6572 7420 6973 496e 7465  me assert isInte
│ │ │ │  00009810: 6772 6162 6c65 2041 2020 2020 2020 2020  grable A        
│ │ │ │  00009820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009830: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00009840: 7c20 2d2d 2035 2e39 3335 3437 7320 656c  | -- 5.93547s el
│ │ │ │ +00009840: 7c20 2d2d 2034 2e32 3237 3839 7320 656c  | -- 4.22789s el
│ │ │ │  00009850: 6170 7365 6420 2020 2020 2020 2020 2020  apsed           
│ │ │ │  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 7c0a                |.
│ │ │ │  00009890: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  000098a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000098b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ @@ -4559,15 +4559,15 @@
│ │ │ │  00011ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011d00: 2b0a 7c69 3134 203a 2065 6c61 7073 6564  +.|i14 : elapsed
│ │ │ │  00011d10: 5469 6d65 2067 203d 2067 6175 6765 4d61  Time g = gaugeMa
│ │ │ │  00011d20: 7472 6978 2849 2c20 4229 3b20 2020 2020  trix(I, B);     
│ │ │ │  00011d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011d50: 7c0a 7c20 2d2d 202e 3735 3832 3437 7320  |.| -- .758247s 
│ │ │ │ +00011d50: 7c0a 7c20 2d2d 202e 3534 3238 3434 7320  |.| -- .542844s 
│ │ │ │  00011d60: 656c 6170 7365 6420 2020 2020 2020 2020  elapsed         
│ │ │ │  00011d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011da0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00011db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -4589,30 +4589,30 @@
│ │ │ │  00011ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011ee0: 2b0a 7c69 3135 203a 2065 6c61 7073 6564  +.|i15 : elapsed
│ │ │ │  00011ef0: 5469 6d65 2041 3120 3d20 6761 7567 6554  Time A1 = gaugeT
│ │ │ │  00011f00: 7261 6e73 666f 726d 2867 2c20 4129 3b20  ransform(g, A); 
│ │ │ │  00011f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011f30: 7c0a 7c20 2d2d 2031 2e35 3139 3335 7320  |.| -- 1.51935s 
│ │ │ │ +00011f30: 7c0a 7c20 2d2d 2031 2e31 3933 3233 7320  |.| -- 1.19323s 
│ │ │ │  00011f40: 656c 6170 7365 6420 2020 2020 2020 2020  elapsed         
│ │ │ │  00011f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011f80: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00011f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011fd0: 2b0a 7c69 3136 203a 2065 6c61 7073 6564  +.|i16 : elapsed
│ │ │ │  00011fe0: 5469 6d65 2061 7373 6572 7420 6973 496e  Time assert isIn
│ │ │ │  00011ff0: 7465 6772 6162 6c65 2041 3120 2020 2020  tegrable A1     
│ │ │ │  00012000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012020: 7c0a 7c20 2d2d 202e 3833 3731 3437 7320  |.| -- .837147s 
│ │ │ │ +00012020: 7c0a 7c20 2d2d 202e 3930 3135 3031 7320  |.| -- .901501s 
│ │ │ │  00012030: 656c 6170 7365 6420 2020 2020 2020 2020  elapsed         
│ │ │ │  00012040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012070: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00012080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00012090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ @@ -5030,31 +5030,31 @@
│ │ │ │  00013a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013a70: 2d2d 2d2d 2d2b 0a7c 6931 3920 3a20 656c  -----+.|i19 : el
│ │ │ │  00013a80: 6170 7365 6454 696d 6520 4132 203d 2067  apsedTime A2 = g
│ │ │ │  00013a90: 6175 6765 5472 616e 7366 6f72 6d28 6368  augeTransform(ch
│ │ │ │  00013aa0: 616e 6765 4570 732c 2041 3129 3b20 2020  angeEps, A1);   
│ │ │ │  00013ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013ac0: 2020 2020 207c 0a7c 202d 2d20 2e34 3237       |.| -- .427
│ │ │ │ -00013ad0: 3934 3873 2065 6c61 7073 6564 2020 2020  948s elapsed    
│ │ │ │ +00013ac0: 2020 2020 207c 0a7c 202d 2d20 2e33 3234       |.| -- .324
│ │ │ │ +00013ad0: 3630 3273 2065 6c61 7073 6564 2020 2020  602s elapsed    
│ │ │ │  00013ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013b10: 2020 2020 207c 0a2b 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 2d2d 2d2d  ----------------
│ │ │ │  00013b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013b60: 2d2d 2d2d 2d2b 0a7c 6932 3020 3a20 656c  -----+.|i20 : el
│ │ │ │  00013b70: 6170 7365 6454 696d 6520 6173 7365 7274  apsedTime assert
│ │ │ │  00013b80: 2069 7349 6e74 6567 7261 626c 6520 4132   isIntegrable A2
│ │ │ │  00013b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013bb0: 2020 2020 207c 0a7c 202d 2d20 2e37 3135       |.| -- .715
│ │ │ │ -00013bc0: 3438 3773 2065 6c61 7073 6564 2020 2020  487s elapsed    
│ │ │ │ +00013bb0: 2020 2020 207c 0a7c 202d 2d20 2e36 3338       |.| -- .638
│ │ │ │ +00013bc0: 3930 3473 2065 6c61 7073 6564 2020 2020  904s elapsed    
│ │ │ │  00013bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013c00: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  00013c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ @@ -5440,30 +5440,30 @@
│ │ │ │  000153f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │  00015410: 6937 203a 2065 6c61 7073 6564 5469 6d65  i7 : elapsedTime
│ │ │ │  00015420: 2041 203d 2063 6f6e 6e65 6374 696f 6e4d   A = connectionM
│ │ │ │  00015430: 6174 7269 6365 7320 493b 2020 2020 2020  atrices I;      
│ │ │ │  00015440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015450: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00015460: 202d 2d20 2e32 3236 3131 3273 2065 6c61   -- .226112s ela
│ │ │ │ -00015470: 7073 6564 2020 2020 2020 2020 2020 2020  psed            
│ │ │ │ +00015460: 202d 2d20 2e31 3835 3131 7320 656c 6170   -- .18511s elap
│ │ │ │ +00015470: 7365 6420 2020 2020 2020 2020 2020 2020  sed             
│ │ │ │  00015480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000154a0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │  000154b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000154c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000154d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000154e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000154f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │  00015500: 6938 203a 2065 6c61 7073 6564 5469 6d65  i8 : elapsedTime
│ │ │ │  00015510: 2061 7373 6572 7420 6973 496e 7465 6772   assert isIntegr
│ │ │ │  00015520: 6162 6c65 2041 2020 2020 2020 2020 2020  able A          
│ │ │ │  00015530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015540: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00015550: 202d 2d20 2e31 3532 3638 3373 2065 6c61   -- .152683s ela
│ │ │ │ +00015550: 202d 2d20 2e31 3730 3735 3373 2065 6c61   -- .170753s ela
│ │ │ │  00015560: 7073 6564 2020 2020 2020 2020 2020 2020  psed            
│ │ │ │  00015570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015590: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │  000155a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000155b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000155c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ ├── ./usr/share/info/Cremona.info.gz
│ │ │ ├── Cremona.info
│ │ │ │ @@ -147,16 +147,16 @@
│ │ │ │  00000920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00000930: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2074  -------+.|i2 : t
│ │ │ │  00000940: 696d 6520 7068 6920 3d20 746f 4d61 7020  ime phi = toMap 
│ │ │ │  00000950: 6d69 6e6f 7273 2833 2c6d 6174 7269 787b  minors(3,matrix{
│ │ │ │  00000960: 7b74 5f30 2e2e 745f 347d 2c7b 745f 312e  {t_0..t_4},{t_1.
│ │ │ │  00000970: 2e74 5f35 7d2c 7b74 5f32 2e2e 745f 367d  .t_5},{t_2..t_6}
│ │ │ │  00000980: 7d29 2020 2020 207c 0a7c 202d 2d20 7573  })     |.| -- us
│ │ │ │ -00000990: 6564 2030 2e30 3034 3434 3635 3173 2028  ed 0.00444651s (
│ │ │ │ -000009a0: 6370 7529 3b20 302e 3030 3434 3431 3831  cpu); 0.00444181
│ │ │ │ +00000990: 6564 2030 2e30 3035 3633 3634 3973 2028  ed 0.00563649s (
│ │ │ │ +000009a0: 6370 7529 3b20 302e 3030 3536 3333 3633  cpu); 0.00563363
│ │ │ │  000009b0: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │  000009c0: 6763 2920 2020 2020 2020 2020 2020 2020  gc)             
│ │ │ │  000009d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  000009e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000009f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00000a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00000a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -322,16 +322,16 @@
│ │ │ │  00001410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00001420: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2074  -------+.|i3 : t
│ │ │ │  00001430: 696d 6520 4a20 3d20 6b65 726e 656c 2870  ime J = kernel(p
│ │ │ │  00001440: 6869 2c32 2920 2020 2020 2020 2020 2020  hi,2)           
│ │ │ │  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
│ │ │ │ -00001480: 6564 2030 2e31 3339 3938 3273 2028 6370  ed 0.139982s (cp
│ │ │ │ -00001490: 7529 3b20 302e 3037 3039 3037 3473 2028  u); 0.0709074s (
│ │ │ │ +00001480: 6564 2030 2e31 3833 3933 3673 2028 6370  ed 0.183936s (cp
│ │ │ │ +00001490: 7529 3b20 302e 3038 3438 3231 3973 2028  u); 0.0848219s (
│ │ │ │  000014a0: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 0s (gc)
│ │ │ │  000014b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000014c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  000014d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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                  
│ │ │ │ @@ -387,16 +387,16 @@
│ │ │ │  00001820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00001830: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2074  -------+.|i4 : t
│ │ │ │  00001840: 696d 6520 6465 6772 6565 4d61 7020 7068  ime degreeMap ph
│ │ │ │  00001850: 6920 2020 2020 2020 2020 2020 2020 2020  i               
│ │ │ │  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
│ │ │ │ -00001890: 6564 2030 2e30 3237 3339 3239 7320 2863  ed 0.0273929s (c
│ │ │ │ -000018a0: 7075 293b 2030 2e30 3237 3339 3439 7320  pu); 0.0273949s 
│ │ │ │ +00001890: 6564 2030 2e30 3337 3235 3034 7320 2863  ed 0.0372504s (c
│ │ │ │ +000018a0: 7075 293b 2030 2e30 3337 3235 3839 7320  pu); 0.0372589s 
│ │ │ │  000018b0: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │  000018c0: 2920 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                  
│ │ │ │  00001900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -412,17 +412,17 @@
│ │ │ │  000019b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000019c0: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2074  -------+.|i5 : t
│ │ │ │  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
│ │ │ │ -00001a20: 6564 2030 2e36 3630 3333 3673 2028 6370  ed 0.660336s (cp
│ │ │ │ -00001a30: 7529 3b20 302e 3436 3735 3937 7320 2874  u); 0.467597s (t
│ │ │ │ -00001a40: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │ +00001a20: 6564 2030 2e36 3535 3030 3573 2028 6370  ed 0.655005s (cp
│ │ │ │ +00001a30: 7529 3b20 302e 3438 3836 7320 2874 6872  u); 0.4886s (thr
│ │ │ │ +00001a40: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 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                  
│ │ │ │  00001ab0: 2020 2020 2020 207c 0a7c 6f35 203d 207b         |.|o5 = {
│ │ │ │ @@ -447,18 +447,18 @@
│ │ │ │  00001be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00001bf0: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 2074  -------+.|i6 : t
│ │ │ │  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
│ │ │ │ -00001c50: 6564 2030 2e30 3631 3532 3573 2028 6370  ed 0.061525s (cp
│ │ │ │ -00001c60: 7529 3b20 302e 3036 3135 3330 3873 2028  u); 0.0615308s (
│ │ │ │ -00001c70: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 0s (gc)
│ │ │ │ -00001c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00001c50: 6564 2030 2e30 3731 3339 3632 7320 2863  ed 0.0713962s (c
│ │ │ │ +00001c60: 7075 293b 2030 2e30 3731 3430 3333 7320  pu); 0.0714033s 
│ │ │ │ +00001c70: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │ +00001c80: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  00001c90: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00001ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001ce0: 2020 2020 2020 207c 0a7c 6f36 203d 207b         |.|o6 = {
│ │ │ │  00001cf0: 357d 2020 2020 2020 2020 2020 2020 2020  5}              
│ │ │ │ @@ -482,15 +482,15 @@
│ │ │ │  00001e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00001e20: 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a 2074  -------+.|i7 : t
│ │ │ │  00001e30: 696d 6520 7068 6920 3d20 746f 4d61 7028  ime phi = toMap(
│ │ │ │  00001e40: 7068 6920 2020 2020 2020 2020 2020 2020  phi             
│ │ │ │  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 3032 3234 3937 3973 2028  ed 0.00224979s (
│ │ │ │ +00001e80: 6564 2030 2e30 3032 3631 3739 3173 2028  ed 0.00261791s (
│ │ │ │  00001e90: 6370 7520 2020 2020 2020 2020 2020 2020  cpu             
│ │ │ │  00001ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001ec0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00001ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -567,16 +567,16 @@
│ │ │ │  00002360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002370: 2d2d 2d2d 2d2d 2d7c 0a7c 2c44 6f6d 696e  -------|.|,Domin
│ │ │ │  00002380: 616e 743d 3e4a 2920 2020 2020 2020 2020  ant=>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 3235 3133 3773 2028 7468 7265 6164  0225137s (thread
│ │ │ │ -000023e0: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │ +000023d0: 3032 3632 3334 7320 2874 6872 6561 6429  026234s (thread)
│ │ │ │ +000023e0: 3b20 3073 2028 6763 2920 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                  
│ │ │ │  00002450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -832,16 +832,16 @@
│ │ │ │  000033f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00003400: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 2074  -------+.|i8 : t
│ │ │ │  00003410: 696d 6520 7073 6920 3d20 696e 7665 7273  ime psi = invers
│ │ │ │  00003420: 654d 6170 2070 6869 2020 2020 2020 2020  eMap phi        
│ │ │ │  00003430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003450: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -00003460: 6564 2030 2e34 3634 3631 3873 2028 6370  ed 0.464618s (cp
│ │ │ │ -00003470: 7529 3b20 302e 3338 3539 3731 7320 2874  u); 0.385971s (t
│ │ │ │ +00003460: 6564 2030 2e34 3235 3034 3773 2028 6370  ed 0.425047s (cp
│ │ │ │ +00003470: 7529 3b20 302e 3432 3438 3439 7320 2874  u); 0.424849s (t
│ │ │ │  00003480: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │  00003490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000034a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  000034b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000034c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000034d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000034e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -1117,18 +1117,18 @@
│ │ │ │  000045c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000045d0: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2074  -------+.|i9 : t
│ │ │ │  000045e0: 696d 6520 6973 496e 7665 7273 654d 6170  ime isInverseMap
│ │ │ │  000045f0: 2870 6869 2c70 7369 2920 2020 2020 2020  (phi,psi)       
│ │ │ │  00004600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004620: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -00004630: 6564 2030 2e30 3039 3537 3736 3873 2028  ed 0.00957768s (
│ │ │ │ -00004640: 6370 7529 3b20 302e 3030 3935 3832 3337  cpu); 0.00958237
│ │ │ │ -00004650: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ -00004660: 6763 2920 2020 2020 2020 2020 2020 2020  gc)             
│ │ │ │ +00004630: 6564 2030 2e30 3130 3931 3538 7320 2863  ed 0.0109158s (c
│ │ │ │ +00004640: 7075 293b 2030 2e30 3130 3931 3732 7320  pu); 0.0109172s 
│ │ │ │ +00004650: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │ +00004660: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  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             
│ │ │ │ @@ -1142,17 +1142,17 @@
│ │ │ │  00004750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00004760: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20  -------+.|i10 : 
│ │ │ │  00004770: 7469 6d65 2064 6567 7265 654d 6170 2070  time degreeMap p
│ │ │ │  00004780: 7369 2020 2020 2020 2020 2020 2020 2020  si              
│ │ │ │  00004790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000047a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000047b0: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -000047c0: 6564 2030 2e33 3636 3038 7320 2863 7075  ed 0.36608s (cpu
│ │ │ │ -000047d0: 293b 2030 2e32 3530 3637 3473 2028 7468  ); 0.250674s (th
│ │ │ │ -000047e0: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │ +000047c0: 6564 2030 2e35 3331 3137 3973 2028 6370  ed 0.531179s (cp
│ │ │ │ +000047d0: 7529 3b20 302e 3237 3239 3432 7320 2874  u); 0.272942s (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                  
│ │ │ │  00004850: 2020 2020 2020 207c 0a7c 6f31 3020 3d20         |.|o10 = 
│ │ │ │ @@ -1167,17 +1167,17 @@
│ │ │ │  000048e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000048f0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20  -------+.|i11 : 
│ │ │ │  00004900: 7469 6d65 2070 726f 6a65 6374 6976 6544  time projectiveD
│ │ │ │  00004910: 6567 7265 6573 2070 7369 2020 2020 2020  egrees psi      
│ │ │ │  00004920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004940: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -00004950: 6564 2035 2e31 3232 3933 7320 2863 7075  ed 5.12293s (cpu
│ │ │ │ -00004960: 293b 2034 2e33 3839 3734 7320 2874 6872  ); 4.38974s (thr
│ │ │ │ -00004970: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │ +00004950: 6564 2035 2e36 3836 3237 7320 2863 7075  ed 5.68627s (cpu
│ │ │ │ +00004960: 293b 2035 2e32 3631 3873 2028 7468 7265  ); 5.2618s (thre
│ │ │ │ +00004970: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │  00004980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004990: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  000049a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000049b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000049c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000049d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000049e0: 2020 2020 2020 207c 0a7c 6f31 3120 3d20         |.|o11 = 
│ │ │ │ @@ -1214,16 +1214,16 @@
│ │ │ │  00004bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  00004be0: 0a7c 6931 3220 3a20 7469 6d65 2070 6869  .|i12 : time phi
│ │ │ │  00004bf0: 203d 2072 6174 696f 6e61 6c4d 6170 206d   = rationalMap m
│ │ │ │  00004c00: 696e 6f72 7328 332c 6d61 7472 6978 7b7b  inors(3,matrix{{
│ │ │ │  00004c10: 745f 302e 2e74 5f34 7d2c 7b74 5f31 2e2e  t_0..t_4},{t_1..
│ │ │ │  00004c20: 745f 357d 2c7b 745f 322e 2e74 5f36 207c  t_5},{t_2..t_6 |
│ │ │ │  00004c30: 0a7c 202d 2d20 7573 6564 2030 2e30 3032  .| -- used 0.002
│ │ │ │ -00004c40: 3134 3835 3373 2028 6370 7529 3b20 302e  14853s (cpu); 0.
│ │ │ │ -00004c50: 3030 3231 3439 3039 7320 2874 6872 6561  00214909s (threa
│ │ │ │ +00004c40: 3538 3037 3673 2028 6370 7529 3b20 302e  58076s (cpu); 0.
│ │ │ │ +00004c50: 3030 3235 3833 3138 7320 2874 6872 6561  00258318s (threa
│ │ │ │  00004c60: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │  00004c70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00004c80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00004c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004cc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ @@ -1493,17 +1493,17 @@
│ │ │ │  00005d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00005d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  00005d60: 0a7c 6931 3320 3a20 7469 6d65 2070 6869  .|i13 : time phi
│ │ │ │  00005d70: 203d 2072 6174 696f 6e61 6c4d 6170 2870   = rationalMap(p
│ │ │ │  00005d80: 6869 2c44 6f6d 696e 616e 743d 3e32 2920  hi,Dominant=>2) 
│ │ │ │  00005d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005da0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00005db0: 0a7c 202d 2d20 7573 6564 2030 2e31 3539  .| -- used 0.159
│ │ │ │ -00005dc0: 3836 3973 2028 6370 7529 3b20 302e 3038  869s (cpu); 0.08
│ │ │ │ -00005dd0: 3139 3935 3973 2028 7468 7265 6164 293b  19959s (thread);
│ │ │ │ +00005db0: 0a7c 202d 2d20 7573 6564 2030 2e31 3739  .| -- used 0.179
│ │ │ │ +00005dc0: 3531 3973 2028 6370 7529 3b20 302e 3039  519s (cpu); 0.09
│ │ │ │ +00005dd0: 3730 3836 3373 2028 7468 7265 6164 293b  70863s (thread);
│ │ │ │  00005de0: 2030 7320 2867 6329 2020 2020 2020 2020   0s (gc)        
│ │ │ │  00005df0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00005e00: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00005e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005e40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ @@ -2153,17 +2153,17 @@
│ │ │ │  00008680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00008690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  000086a0: 0a7c 6931 3420 3a20 7469 6d65 2070 6869  .|i14 : time phi
│ │ │ │  000086b0: 5e28 2d31 2920 2020 2020 2020 2020 2020  ^(-1)           
│ │ │ │  000086c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000086d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000086e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000086f0: 0a7c 202d 2d20 7573 6564 2030 2e34 3934  .| -- used 0.494
│ │ │ │ -00008700: 3834 3773 2028 6370 7529 3b20 302e 3431  847s (cpu); 0.41
│ │ │ │ -00008710: 3433 3531 7320 2874 6872 6561 6429 3b20  4351s (thread); 
│ │ │ │ +000086f0: 0a7c 202d 2d20 7573 6564 2030 2e34 3634  .| -- used 0.464
│ │ │ │ +00008700: 3233 3473 2028 6370 7529 3b20 302e 3436  234s (cpu); 0.46
│ │ │ │ +00008710: 3339 3935 7320 2874 6872 6561 6429 3b20  3995s (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 3433  .| -- used 0.343
│ │ │ │ -0000a9b0: 3631 3473 2028 6370 7529 3b20 302e 3236  614s (cpu); 0.26
│ │ │ │ -0000a9c0: 3934 3936 7320 2874 6872 6561 6429 3b20  9496s (thread); 
│ │ │ │ +0000a9a0: 0a7c 202d 2d20 7573 6564 2030 2e34 3536  .| -- used 0.456
│ │ │ │ +0000a9b0: 3630 3873 2028 6370 7529 3b20 302e 3334  608s (cpu); 0.34
│ │ │ │ +0000a9c0: 3536 3738 7320 2874 6872 6561 6429 3b20  5678s (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 2e30 3137  .| -- used 0.017
│ │ │ │ -0000abe0: 3036 3673 2028 6370 7529 3b20 302e 3031  066s (cpu); 0.01
│ │ │ │ -0000abf0: 3637 3831 3373 2028 7468 7265 6164 293b  67813s (thread);
│ │ │ │ -0000ac00: 2030 7320 2867 6329 2020 2020 2020 2020   0s (gc)        
│ │ │ │ +0000abd0: 0a7c 202d 2d20 7573 6564 2030 2e30 3433  .| -- used 0.043
│ │ │ │ +0000abe0: 3835 3432 7320 2863 7075 293b 2030 2e30  8542s (cpu); 0.0
│ │ │ │ +0000abf0: 3233 3836 3836 7320 2874 6872 6561 6429  238686s (thread)
│ │ │ │ +0000ac00: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │  0000ac10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000ac20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000ac30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ac40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ac50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ac60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000ac70: 0a7c 6f31 3620 3d20 7b31 2c20 332c 2039  .|o16 = {1, 3, 9
│ │ │ │ @@ -2778,17 +2778,17 @@
│ │ │ │  0000ad90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000ada0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  0000adb0: 0a7c 6931 3720 3a20 7469 6d65 2064 6573  .|i17 : time des
│ │ │ │  0000adc0: 6372 6962 6520 7068 6920 2020 2020 2020  cribe phi       
│ │ │ │  0000add0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ade0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000adf0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000ae00: 0a7c 202d 2d20 7573 6564 2030 2e30 3032  .| -- used 0.002
│ │ │ │ -0000ae10: 3837 3335 3173 2028 6370 7529 3b20 302e  87351s (cpu); 0.
│ │ │ │ -0000ae20: 3030 3238 3734 3132 7320 2874 6872 6561  00287412s (threa
│ │ │ │ +0000ae00: 0a7c 202d 2d20 7573 6564 2030 2e30 3034  .| -- used 0.004
│ │ │ │ +0000ae10: 3038 3933 3473 2028 6370 7529 3b20 302e  08934s (cpu); 0.
│ │ │ │ +0000ae20: 3030 3430 3935 3932 7320 2874 6872 6561  00409592s (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 3039  .| -- used 0.009
│ │ │ │ -0000b220: 3439 3336 3773 2028 6370 7529 3b20 302e  49367s (cpu); 0.
│ │ │ │ -0000b230: 3030 3934 3934 3634 7320 2874 6872 6561  00949464s (threa
│ │ │ │ -0000b240: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │ +0000b210: 0a7c 202d 2d20 7573 6564 2030 2e30 3131  .| -- used 0.011
│ │ │ │ +0000b220: 3433 3835 7320 2863 7075 293b 2030 2e30  4385s (cpu); 0.0
│ │ │ │ +0000b230: 3131 3434 3631 7320 2874 6872 6561 6429  114461s (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
│ │ │ │ @@ -2923,18 +2923,18 @@
│ │ │ │  0000b6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000b6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  0000b6c0: 0a7c 6931 3920 3a20 7469 6d65 2028 662c  .|i19 : time (f,
│ │ │ │  0000b6d0: 6729 203d 2067 7261 7068 2070 6869 5e2d  g) = graph phi^-
│ │ │ │  0000b6e0: 313b 2066 3b20 2020 2020 2020 2020 2020  1; f;           
│ │ │ │  0000b6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b700: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000b710: 0a7c 202d 2d20 7573 6564 2030 2e30 3039  .| -- used 0.009
│ │ │ │ -0000b720: 3130 3138 3273 2028 6370 7529 3b20 302e  10182s (cpu); 0.
│ │ │ │ -0000b730: 3030 3931 3032 3635 7320 2874 6872 6561  00910265s (threa
│ │ │ │ -0000b740: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │ +0000b710: 0a7c 202d 2d20 7573 6564 2030 2e30 3131  .| -- used 0.011
│ │ │ │ +0000b720: 3639 3838 7320 2863 7075 293b 2030 2e30  6988s (cpu); 0.0
│ │ │ │ +0000b730: 3131 3730 3737 7320 2874 6872 6561 6429  117077s (thread)
│ │ │ │ +0000b740: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │  0000b750: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000b760: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000b770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b7a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000b7b0: 0a7c 6f32 3020 3a20 4d75 6c74 6968 6f6d  .|o20 : Multihom
│ │ │ │ @@ -2958,18 +2958,18 @@
│ │ │ │  0000b8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000b8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  0000b8f0: 0a7c 6932 3120 3a20 7469 6d65 2064 6567  .|i21 : time deg
│ │ │ │  0000b900: 7265 6573 2066 2020 2020 2020 2020 2020  rees f          
│ │ │ │  0000b910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b930: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000b940: 0a7c 202d 2d20 7573 6564 2031 2e32 3831  .| -- used 1.281
│ │ │ │ -0000b950: 3173 2028 6370 7529 3b20 302e 3936 3238  1s (cpu); 0.9628
│ │ │ │ -0000b960: 3739 7320 2874 6872 6561 6429 3b20 3073  79s (thread); 0s
│ │ │ │ -0000b970: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │ +0000b940: 0a7c 202d 2d20 7573 6564 2031 2e32 3039  .| -- used 1.209
│ │ │ │ +0000b950: 3573 2028 6370 7529 3b20 312e 3030 3834  5s (cpu); 1.0084
│ │ │ │ +0000b960: 3273 2028 7468 7265 6164 293b 2030 7320  2s (thread); 0s 
│ │ │ │ +0000b970: 2867 6329 2020 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
│ │ │ │ @@ -2993,18 +2993,18 @@
│ │ │ │  0000bb00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  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 2031 2e35 3838  .| -- used 1.588
│ │ │ │ -0000bb80: 652d 3035 7320 2863 7075 293b 2031 2e35  e-05s (cpu); 1.5
│ │ │ │ -0000bb90: 3530 3965 2d30 3573 2028 7468 7265 6164  509e-05s (thread
│ │ │ │ -0000bba0: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │ +0000bb70: 0a7c 202d 2d20 7573 6564 2031 2e35 3333  .| -- used 1.533
│ │ │ │ +0000bb80: 3865 2d30 3573 2028 6370 7529 3b20 312e  8e-05s (cpu); 1.
│ │ │ │ +0000bb90: 3439 3032 652d 3035 7320 2874 6872 6561  4902e-05s (threa
│ │ │ │ +0000bba0: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 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                 |
│ │ │ │  0000bc10: 0a7c 6f32 3220 3d20 3120 2020 2020 2020  .|o22 = 1       
│ │ │ │ @@ -3019,16 +3019,16 @@
│ │ │ │  0000bca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  0000bcb0: 0a7c 6932 3320 3a20 7469 6d65 2064 6573  .|i23 : time des
│ │ │ │  0000bcc0: 6372 6962 6520 6620 2020 2020 2020 2020  cribe f         
│ │ │ │  0000bcd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bcf0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000bd00: 0a7c 202d 2d20 7573 6564 2030 2e30 3031  .| -- used 0.001
│ │ │ │ -0000bd10: 3736 3234 3673 2028 6370 7529 3b20 302e  76246s (cpu); 0.
│ │ │ │ -0000bd20: 3030 3137 3633 3335 7320 2874 6872 6561  00176335s (threa
│ │ │ │ +0000bd10: 3639 3331 3373 2028 6370 7529 3b20 302e  69313s (cpu); 0.
│ │ │ │ +0000bd20: 3030 3136 3938 3334 7320 2874 6872 6561  00169834s (threa
│ │ │ │  0000bd30: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 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                 |
│ │ │ │ @@ -4676,16 +4676,16 @@
│ │ │ │  00012430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00012440: 2b0a 7c69 3420 3a20 7469 6d65 2070 7369  +.|i4 : time psi
│ │ │ │  00012450: 203d 2061 6273 7472 6163 7452 6174 696f   = abstractRatio
│ │ │ │  00012460: 6e61 6c4d 6170 2850 342c 5035 2c66 2920  nalMap(P4,P5,f) 
│ │ │ │  00012470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012490: 7c0a 7c20 2d2d 2075 7365 6420 302e 3030  |.| -- used 0.00
│ │ │ │ -000124a0: 3034 3136 3534 3273 2028 6370 7529 3b20  0416542s (cpu); 
│ │ │ │ -000124b0: 302e 3030 3034 3038 3835 3773 2028 7468  0.000408857s (th
│ │ │ │ +000124a0: 3034 3039 3438 3473 2028 6370 7529 3b20  0409484s (cpu); 
│ │ │ │ +000124b0: 302e 3030 3034 3032 3239 3573 2028 7468  0.000402295s (th
│ │ │ │  000124c0: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │  000124d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000124e0: 7c0a 7c20 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 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -4746,18 +4746,18 @@
│ │ │ │  00012890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000128a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000128b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │  000128c0: 6935 203a 2074 696d 6520 7072 6f6a 6563  i5 : time projec
│ │ │ │  000128d0: 7469 7665 4465 6772 6565 7328 7073 692c  tiveDegrees(psi,
│ │ │ │  000128e0: 3329 2020 2020 2020 2020 2020 2020 2020  3)              
│ │ │ │  000128f0: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ -00012900: 2d20 7573 6564 2030 2e33 3238 3735 7320  - used 0.32875s 
│ │ │ │ -00012910: 2863 7075 293b 2030 2e31 3835 3037 3773  (cpu); 0.185077s
│ │ │ │ -00012920: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │ -00012930: 6329 2020 2020 2020 207c 0a7c 2020 2020  c)       |.|    
│ │ │ │ +00012900: 2d20 7573 6564 2030 2e33 3834 3634 3673  - used 0.384646s
│ │ │ │ +00012910: 2028 6370 7529 3b20 302e 3231 3831 3437   (cpu); 0.218147
│ │ │ │ +00012920: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ +00012930: 6763 2920 2020 2020 207c 0a7c 2020 2020  gc)      |.|    
│ │ │ │  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 2020 207c 0a7c 6f35 203d 2032         |.|o5 = 2
│ │ │ │  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 2020                  
│ │ │ │ @@ -4766,16 +4766,16 @@
│ │ │ │  000129d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000129e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000129f0: 2d2d 2d2b 0a7c 6936 203a 2074 696d 6520  ---+.|i6 : time 
│ │ │ │  00012a00: 7261 7469 6f6e 616c 4d61 7020 7073 6920  rationalMap psi 
│ │ │ │  00012a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012a30: 207c 0a7c 202d 2d20 7573 6564 2030 2e35   |.| -- used 0.5
│ │ │ │ -00012a40: 3130 3234 3873 2028 6370 7529 3b20 302e  10248s (cpu); 0.
│ │ │ │ -00012a50: 3337 3330 3132 7320 2874 6872 6561 6429  373012s (thread)
│ │ │ │ +00012a40: 3037 3732 3873 2028 6370 7529 3b20 302e  07728s (cpu); 0.
│ │ │ │ +00012a50: 3431 3530 3635 7320 2874 6872 6561 6429  415065s (thread)
│ │ │ │  00012a60: 3b20 3073 2028 6763 2920 2020 2020 207c  ; 0s (gc)      |
│ │ │ │  00012a70: 0a7c 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 207c 0a7c               |.|
│ │ │ │  00012ab0: 6f36 203d 202d 2d20 7261 7469 6f6e 616c  o6 = -- rational
│ │ │ │  00012ac0: 206d 6170 202d 2d20 2020 2020 2020 2020   map --         
│ │ │ │ @@ -5189,16 +5189,16 @@
│ │ │ │  00014440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014450: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3134 203a  --------+.|i14 :
│ │ │ │  00014460: 2074 696d 6520 5420 3d20 6162 7374 7261   time T = abstra
│ │ │ │  00014470: 6374 5261 7469 6f6e 616c 4d61 7028 492c  ctRationalMap(I,
│ │ │ │  00014480: 224f 4144 5022 2920 2020 2020 2020 2020  "OADP")         
│ │ │ │  00014490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000144a0: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365        |.| -- use
│ │ │ │ -000144b0: 6420 302e 3135 3130 3834 7320 2863 7075  d 0.151084s (cpu
│ │ │ │ -000144c0: 293b 2030 2e30 3733 3333 3637 7320 2874  ); 0.0733367s (t
│ │ │ │ +000144b0: 6420 302e 3137 3939 3739 7320 2863 7075  d 0.179979s (cpu
│ │ │ │ +000144c0: 293b 2030 2e30 3939 3232 3837 7320 2874  ); 0.0992287s (t
│ │ │ │  000144d0: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │  000144e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000144f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00014500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -5265,16 +5265,16 @@
│ │ │ │  00014900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014930: 2d2d 2b0a 7c69 3135 203a 2074 696d 6520  --+.|i15 : time 
│ │ │ │  00014940: 7072 6f6a 6563 7469 7665 4465 6772 6565  projectiveDegree
│ │ │ │  00014950: 7328 542c 3229 2020 2020 2020 2020 2020  s(T,2)          
│ │ │ │  00014960: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075          |.| -- u
│ │ │ │ -00014970: 7365 6420 332e 3738 3438 3773 2028 6370  sed 3.78487s (cp
│ │ │ │ -00014980: 7529 3b20 312e 3935 3132 3673 2028 7468  u); 1.95126s (th
│ │ │ │ +00014970: 7365 6420 342e 3337 3334 3573 2028 6370  sed 4.37345s (cp
│ │ │ │ +00014980: 7529 3b20 322e 3239 3437 3373 2028 7468  u); 2.29473s (th
│ │ │ │  00014990: 7265 6164 293b 2030 7320 2867 6329 7c0a  read); 0s (gc)|.
│ │ │ │  000149a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000149b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000149c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000149d0: 2020 2020 7c0a 7c6f 3135 203d 2033 2020      |.|o15 = 3  
│ │ │ │  000149e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000149f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -5291,16 +5291,16 @@
│ │ │ │  00014aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │  00014ad0: 3620 3a20 7469 6d65 2054 3220 3d20 5420  6 : time T2 = T 
│ │ │ │  00014ae0: 2a20 5420 2020 2020 2020 2020 2020 2020  * T             
│ │ │ │  00014af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014b00: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075          |.| -- u
│ │ │ │ -00014b10: 7365 6420 322e 3831 3132 652d 3035 7320  sed 2.8112e-05s 
│ │ │ │ -00014b20: 2863 7075 293b 2032 2e37 3834 3265 2d30  (cpu); 2.7842e-0
│ │ │ │ +00014b10: 7365 6420 332e 3634 3336 652d 3035 7320  sed 3.6436e-05s 
│ │ │ │ +00014b20: 2863 7075 293b 2033 2e33 3434 3965 2d30  (cpu); 3.3449e-0
│ │ │ │  00014b30: 3573 2028 7468 7265 6164 293b 2030 7320  5s (thread); 0s 
│ │ │ │  00014b40: 2867 6329 207c 0a7c 2020 2020 2020 2020  (gc) |.|        
│ │ │ │  00014b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014b80: 2020 7c0a 7c6f 3136 203d 202d 2d20 7261    |.|o16 = -- ra
│ │ │ │  00014b90: 7469 6f6e 616c 206d 6170 202d 2d20 2020  tional map --   
│ │ │ │ @@ -5344,17 +5344,17 @@
│ │ │ │  00014df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014e20: 2d2b 0a7c 6931 3720 3a20 7469 6d65 2070  -+.|i17 : time p
│ │ │ │  00014e30: 726f 6a65 6374 6976 6544 6567 7265 6573  rojectiveDegrees
│ │ │ │  00014e40: 2854 322c 3229 2020 2020 2020 2020 2020  (T2,2)          
│ │ │ │  00014e50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00014e60: 7c20 2d2d 2075 7365 6420 362e 3434 3236  | -- used 6.4426
│ │ │ │ -00014e70: 3373 2028 6370 7529 3b20 332e 3337 3936  3s (cpu); 3.3796
│ │ │ │ -00014e80: 3973 2028 7468 7265 6164 293b 2030 7320  9s (thread); 0s 
│ │ │ │ +00014e60: 7c20 2d2d 2075 7365 6420 372e 3431 3232  | -- used 7.4122
│ │ │ │ +00014e70: 3173 2028 6370 7529 3b20 332e 3636 3434  1s (cpu); 3.6644
│ │ │ │ +00014e80: 3673 2028 7468 7265 6164 293b 2030 7320  6s (thread); 0s 
│ │ │ │  00014e90: 2867 6329 2020 2020 2020 207c 0a7c 2020  (gc)       |.|  
│ │ │ │  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 2020 2020                  
│ │ │ │  00014ed0: 2020 2020 2020 2020 7c0a 7c6f 3137 203d          |.|o17 =
│ │ │ │  00014ee0: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │  00014ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -5430,17 +5430,17 @@
│ │ │ │  00015350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │  00015380: 3120 3a20 7469 6d65 2066 203d 2072 6174  1 : time f = rat
│ │ │ │  00015390: 696f 6e61 6c4d 6170 2054 2020 2020 2020  ionalMap T      
│ │ │ │  000153a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000153b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000153c0: 0a7c 202d 2d20 7573 6564 2035 2e33 3239  .| -- used 5.329
│ │ │ │ -000153d0: 3332 7320 2863 7075 293b 2032 2e38 3733  32s (cpu); 2.873
│ │ │ │ -000153e0: 3136 7320 2874 6872 6561 6429 3b20 3073  16s (thread); 0s
│ │ │ │ +000153c0: 0a7c 202d 2d20 7573 6564 2036 2e30 3731  .| -- used 6.071
│ │ │ │ +000153d0: 3635 7320 2863 7075 293b 2033 2e30 3731  65s (cpu); 3.071
│ │ │ │ +000153e0: 3332 7320 2874 6872 6561 6429 3b20 3073  32s (thread); 0s
│ │ │ │  000153f0: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │  00015400: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00015410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015440: 2020 2020 2020 207c 0a7c 6f32 3120 3d20         |.|o21 = 
│ │ │ │  00015450: 2d2d 2072 6174 696f 6e61 6c20 6d61 7020  -- rational map 
│ │ │ │ @@ -6678,18 +6678,18 @@
│ │ │ │  0001a150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a160: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │  0001a170: 2d20 6170 7072 6f78 696d 6174 6549 6e76  - approximateInv
│ │ │ │  0001a180: 6572 7365 4d61 703a 2073 7465 7020 3130  erseMap: step 10
│ │ │ │  0001a190: 206f 6620 3130 2020 2020 2020 2020 2020   of 10          
│ │ │ │  0001a1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a1b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001a1c0: 2d2d 2075 7365 6420 302e 3233 3831 3931  -- used 0.238191
│ │ │ │ -0001a1d0: 7320 2863 7075 293b 2030 2e31 3836 3231  s (cpu); 0.18621
│ │ │ │ -0001a1e0: 3273 2028 7468 7265 6164 293b 2030 7320  2s (thread); 0s 
│ │ │ │ -0001a1f0: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │ +0001a1c0: 2d2d 2075 7365 6420 302e 3331 3638 3173  -- used 0.31681s
│ │ │ │ +0001a1d0: 2028 6370 7529 3b20 302e 3233 3537 3131   (cpu); 0.235711
│ │ │ │ +0001a1e0: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ +0001a1f0: 6763 2920 2020 2020 2020 2020 2020 2020  gc)             
│ │ │ │  0001a200: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001a210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a250: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │  0001a260: 3320 3d20 2d2d 2072 6174 696f 6e61 6c20  3 = -- rational 
│ │ │ │ @@ -8043,17 +8043,17 @@
│ │ │ │  0001f6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f6b0: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │  0001f6c0: 2d20 6170 7072 6f78 696d 6174 6549 6e76  - approximateInv
│ │ │ │  0001f6d0: 6572 7365 4d61 703a 2073 7465 7020 3320  erseMap: step 3 
│ │ │ │  0001f6e0: 6f66 2033 2020 2020 2020 2020 2020 2020  of 3            
│ │ │ │  0001f6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f700: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001f710: 2d2d 2075 7365 6420 302e 3231 3836 3531  -- used 0.218651
│ │ │ │ -0001f720: 7320 2863 7075 293b 2030 2e31 3535 3437  s (cpu); 0.15547
│ │ │ │ -0001f730: 3173 2028 7468 7265 6164 293b 2030 7320  1s (thread); 0s 
│ │ │ │ +0001f710: 2d2d 2075 7365 6420 302e 3237 3632 3539  -- used 0.276259
│ │ │ │ +0001f720: 7320 2863 7075 293b 2030 2e31 3937 3630  s (cpu); 0.19760
│ │ │ │ +0001f730: 3373 2028 7468 7265 6164 293b 2030 7320  3s (thread); 0s 
│ │ │ │  0001f740: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │  0001f750: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001f760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f7a0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ @@ -10405,17 +10405,17 @@
│ │ │ │  00028a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028a50: 2020 2020 207c 0a7c 2d2d 2061 7070 726f       |.|-- appro
│ │ │ │  00028a60: 7869 6d61 7465 496e 7665 7273 654d 6170  ximateInverseMap
│ │ │ │  00028a70: 3a20 7374 6570 2033 206f 6620 3320 2020  : step 3 of 3   
│ │ │ │  00028a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028aa0: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ -00028ab0: 2032 2e31 3231 3033 7320 2863 7075 293b   2.12103s (cpu);
│ │ │ │ -00028ac0: 2031 2e36 3737 3273 2028 7468 7265 6164   1.6772s (thread
│ │ │ │ -00028ad0: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │ +00028ab0: 2032 2e31 3633 3637 7320 2863 7075 293b   2.16367s (cpu);
│ │ │ │ +00028ac0: 2031 2e38 3135 3236 7320 2874 6872 6561   1.81526s (threa
│ │ │ │ +00028ad0: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │  00028ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028af0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00028b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028b40: 2020 2020 207c 0a7c 6f38 203d 202d 2d20       |.|o8 = -- 
│ │ │ │ @@ -11710,16 +11710,16 @@
│ │ │ │  0002dbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dbe0: 2020 2020 207c 0a7c 4365 7274 6966 793a       |.|Certify:
│ │ │ │  0002dbf0: 206f 7574 7075 7420 6365 7274 6966 6965   output certifie
│ │ │ │  0002dc00: 6421 2020 2020 2020 2020 2020 2020 2020  d!              
│ │ │ │  0002dc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dc30: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ -0002dc40: 2033 2e32 3238 3933 7320 2863 7075 293b   3.22893s (cpu);
│ │ │ │ -0002dc50: 2032 2e35 3633 3331 7320 2874 6872 6561   2.56331s (threa
│ │ │ │ +0002dc40: 2033 2e30 3730 3731 7320 2863 7075 293b   3.07071s (cpu);
│ │ │ │ +0002dc50: 2032 2e36 3533 3132 7320 2874 6872 6561   2.65312s (threa
│ │ │ │  0002dc60: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │  0002dc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dc80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002dc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dcb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -13366,16 +13366,16 @@
│ │ │ │  00034350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034380: 2b0a 7c69 3320 3a20 7469 6d65 2043 6865  +.|i3 : time Che
│ │ │ │  00034390: 726e 5363 6877 6172 747a 4d61 6350 6865  rnSchwartzMacPhe
│ │ │ │  000343a0: 7273 6f6e 2043 2020 2020 2020 2020 2020  rson C          
│ │ │ │  000343b0: 2020 2020 2020 2020 207c 0a7c 202d 2d20           |.| -- 
│ │ │ │ -000343c0: 7573 6564 2032 2e32 3234 3931 7320 2863  used 2.22491s (c
│ │ │ │ -000343d0: 7075 293b 2031 2e32 3634 3738 7320 2874  pu); 1.26478s (t
│ │ │ │ +000343c0: 7573 6564 2032 2e35 3332 3335 7320 2863  used 2.53235s (c
│ │ │ │ +000343d0: 7075 293b 2031 2e32 3139 3537 7320 2874  pu); 1.21957s (t
│ │ │ │  000343e0: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │  000343f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00034400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034420: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00034430: 2020 2020 2034 2020 2020 2033 2020 2020       4     3    
│ │ │ │  00034440: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ @@ -13409,17 +13409,17 @@
│ │ │ │  00034600: 4368 6572 6e53 6368 7761 7274 7a4d 6163  ChernSchwartzMac
│ │ │ │  00034610: 5068 6572 736f 6e28 432c 4365 7274 6966  Pherson(C,Certif
│ │ │ │  00034620: 793d 3e74 7275 6529 2020 2020 7c0a 7c43  y=>true)    |.|C
│ │ │ │  00034630: 6572 7469 6679 3a20 6f75 7470 7574 2063  ertify: output c
│ │ │ │  00034640: 6572 7469 6669 6564 2120 2020 2020 2020  ertified!       
│ │ │ │  00034650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034660: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ -00034670: 2031 2e33 3337 3833 7320 2863 7075 293b   1.33783s (cpu);
│ │ │ │ -00034680: 2030 2e39 3531 3335 3973 2028 7468 7265   0.951359s (thre
│ │ │ │ -00034690: 6164 293b 2030 7320 2867 6329 2020 7c0a  ad); 0s (gc)  |.
│ │ │ │ +00034670: 2031 2e37 3231 3435 7320 2863 7075 293b   1.72145s (cpu);
│ │ │ │ +00034680: 2031 2e30 3733 3735 7320 2874 6872 6561   1.07375s (threa
│ │ │ │ +00034690: 6429 3b20 3073 2028 6763 2920 2020 7c0a  d); 0s (gc)   |.
│ │ │ │  000346a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000346b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000346c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000346d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  000346e0: 2034 2020 2020 2033 2020 2020 2032 2020   4     3     2  
│ │ │ │  000346f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -13619,18 +13619,18 @@
│ │ │ │  00035320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6939  -----------+.|i9
│ │ │ │  00035340: 203a 2074 696d 6520 4368 6572 6e43 6c61   : time ChernCla
│ │ │ │  00035350: 7373 2047 2020 2020 2020 2020 2020 2020  ss G            
│ │ │ │  00035360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035380: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ -00035390: 2d20 7573 6564 2030 2e33 3230 3038 7320  - used 0.32008s 
│ │ │ │ -000353a0: 2863 7075 293b 2030 2e31 3837 3233 7320  (cpu); 0.18723s 
│ │ │ │ -000353b0: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │ -000353c0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +00035390: 2d20 7573 6564 2030 2e33 3933 3131 3773  - used 0.393117s
│ │ │ │ +000353a0: 2028 6370 7529 3b20 302e 3139 3738 3773   (cpu); 0.19787s
│ │ │ │ +000353b0: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │ +000353c0: 6329 2020 2020 2020 2020 2020 2020 2020  c)              
│ │ │ │  000353d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000353e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000353f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035420: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00035430: 2020 2020 2020 3920 2020 2020 2038 2020        9      8  
│ │ │ │ @@ -13679,17 +13679,17 @@
│ │ │ │  000356e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000356f0: 2020 2020 2020 2020 2020 207c 0a7c 4365             |.|Ce
│ │ │ │  00035700: 7274 6966 793a 206f 7574 7075 7420 6365  rtify: output ce
│ │ │ │  00035710: 7274 6966 6965 6421 2020 2020 2020 2020  rtified!        
│ │ │ │  00035720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035740: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ -00035750: 2d20 7573 6564 2030 2e31 3037 3637 3973  - used 0.107679s
│ │ │ │ -00035760: 2028 6370 7529 3b20 302e 3033 3832 3533   (cpu); 0.038253
│ │ │ │ -00035770: 3173 2028 7468 7265 6164 293b 2030 7320  1s (thread); 0s 
│ │ │ │ +00035750: 2d20 7573 6564 2030 2e31 3932 3330 3273  - used 0.192302s
│ │ │ │ +00035760: 2028 6370 7529 3b20 302e 3034 3638 3235   (cpu); 0.046825
│ │ │ │ +00035770: 3473 2028 7468 7265 6164 293b 2030 7320  4s (thread); 0s 
│ │ │ │  00035780: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │  00035790: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000357a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000357b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000357c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000357d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000357e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ @@ -16336,16 +16336,16 @@
│ │ │ │  0003fcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003fd00: 2d2b 0a7c 6935 203a 2074 696d 6520 6465  -+.|i5 : time de
│ │ │ │  0003fd10: 6772 6565 4d61 7020 7068 6920 2020 2020  greeMap phi     
│ │ │ │  0003fd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003fd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003fd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003fd50: 207c 0a7c 202d 2d20 7573 6564 2030 2e30   |.| -- used 0.0
│ │ │ │ -0003fd60: 3435 3736 3334 7320 2863 7075 293b 2030  457634s (cpu); 0
│ │ │ │ -0003fd70: 2e30 3435 3736 3434 7320 2874 6872 6561  .0457644s (threa
│ │ │ │ +0003fd60: 3533 3830 3132 7320 2863 7075 293b 2030  538012s (cpu); 0
│ │ │ │ +0003fd70: 2e30 3533 3830 3031 7320 2874 6872 6561  .0538001s (threa
│ │ │ │  0003fd80: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │  0003fd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003fda0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003fdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003fdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003fdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003fde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -17510,17 +17510,17 @@
│ │ │ │  00044650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00044660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00044670: 2d2b 0a7c 6937 203a 2074 696d 6520 6465  -+.|i7 : time de
│ │ │ │  00044680: 6772 6565 4d61 7020 7068 6927 2020 2020  greeMap phi'    
│ │ │ │  00044690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000446a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000446b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000446c0: 207c 0a7c 202d 2d20 7573 6564 2031 2e32   |.| -- used 1.2
│ │ │ │ -000446d0: 3530 3538 7320 2863 7075 293b 2030 2e37  5058s (cpu); 0.7
│ │ │ │ -000446e0: 3137 3533 3773 2028 7468 7265 6164 293b  17537s (thread);
│ │ │ │ +000446c0: 207c 0a7c 202d 2d20 7573 6564 2031 2e34   |.| -- used 1.4
│ │ │ │ +000446d0: 3533 3733 7320 2863 7075 293b 2030 2e37  5373s (cpu); 0.7
│ │ │ │ +000446e0: 3737 3135 3473 2028 7468 7265 6164 293b  77154s (thread);
│ │ │ │  000446f0: 2030 7320 2867 6329 2020 2020 2020 2020   0s (gc)        
│ │ │ │  00044700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044710: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00044720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -18325,17 +18325,17 @@
│ │ │ │  00047940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00047950: 2d2d 2b0a 7c69 3220 3a20 7469 6d65 2045  --+.|i2 : time E
│ │ │ │  00047960: 756c 6572 4368 6172 6163 7465 7269 7374  ulerCharacterist
│ │ │ │  00047970: 6963 2049 2020 2020 2020 2020 2020 2020  ic I            
│ │ │ │  00047980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000479a0: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e    |.| -- used 0.
│ │ │ │ -000479b0: 3332 3336 3373 2028 6370 7529 3b20 302e  32363s (cpu); 0.
│ │ │ │ -000479c0: 3139 3231 3132 7320 2874 6872 6561 6429  192112s (thread)
│ │ │ │ -000479d0: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │ +000479b0: 3336 3538 3333 7320 2863 7075 293b 2030  365833s (cpu); 0
│ │ │ │ +000479c0: 2e32 3133 3537 3473 2028 7468 7265 6164  .213574s (thread
│ │ │ │ +000479d0: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │  000479e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000479f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00047a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047a40: 2020 7c0a 7c6f 3220 3d20 3130 2020 2020    |.|o2 = 10    
│ │ │ │ @@ -18355,16 +18355,16 @@
│ │ │ │  00047b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047b30: 2020 7c0a 7c43 6572 7469 6679 3a20 6f75    |.|Certify: ou
│ │ │ │  00047b40: 7470 7574 2063 6572 7469 6669 6564 2120  tput certified! 
│ │ │ │  00047b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047b80: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e    |.| -- used 0.
│ │ │ │ -00047b90: 3031 3132 3339 3273 2028 6370 7529 3b20  0112392s (cpu); 
│ │ │ │ -00047ba0: 302e 3031 3039 3534 3973 2028 7468 7265  0.0109549s (thre
│ │ │ │ +00047b90: 3032 3433 3836 3773 2028 6370 7529 3b20  0243867s (cpu); 
│ │ │ │ +00047ba0: 302e 3031 3235 3931 3973 2028 7468 7265  0.0125919s (thre
│ │ │ │  00047bb0: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │  00047bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047bd0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00047be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -19033,17 +19033,17 @@
│ │ │ │  0004a580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004a590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │  0004a5a0: 3420 3a20 7469 6d65 2066 6f72 6365 496d  4 : time forceIm
│ │ │ │  0004a5b0: 6167 6528 5068 692c 6964 6561 6c20 305f  age(Phi,ideal 0_
│ │ │ │  0004a5c0: 2874 6172 6765 7420 5068 6929 2920 2020  (target Phi))   
│ │ │ │  0004a5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a5e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0004a5f0: 2d2d 2075 7365 6420 302e 3030 3036 3530  -- used 0.000650
│ │ │ │ -0004a600: 3232 3973 2028 6370 7529 3b20 302e 3030  229s (cpu); 0.00
│ │ │ │ -0004a610: 3036 3433 3534 3673 2028 7468 7265 6164  0643546s (thread
│ │ │ │ +0004a5f0: 2d2d 2075 7365 6420 302e 3030 3038 3139  -- used 0.000819
│ │ │ │ +0004a600: 3432 3973 2028 6370 7529 3b20 302e 3030  429s (cpu); 0.00
│ │ │ │ +0004a610: 3038 3134 3830 3773 2028 7468 7265 6164  0814807s (thread
│ │ │ │  0004a620: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │  0004a630: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │  0004a640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004a650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004a660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004a670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004a680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ @@ -19645,16 +19645,16 @@
│ │ │ │  0004cbc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004cbd0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2074  -------+.|i3 : t
│ │ │ │  0004cbe0: 696d 6520 2870 312c 7032 2920 3d20 6772  ime (p1,p2) = gr
│ │ │ │  0004cbf0: 6170 6820 7068 693b 2020 2020 2020 2020  aph phi;        
│ │ │ │  0004cc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004cc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004cc20: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -0004cc30: 6564 2030 2e30 3136 3332 3739 7320 2863  ed 0.0163279s (c
│ │ │ │ -0004cc40: 7075 293b 2030 2e30 3136 3033 3833 7320  pu); 0.0160383s 
│ │ │ │ +0004cc30: 6564 2030 2e30 3334 3032 3631 7320 2863  ed 0.0340261s (c
│ │ │ │ +0004cc40: 7075 293b 2030 2e30 3230 3035 3534 7320  pu); 0.0200554s 
│ │ │ │  0004cc50: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │  0004cc60: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  0004cc70: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  0004cc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004cc90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004cca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004ccb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ @@ -20942,16 +20942,16 @@
│ │ │ │  00051cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051ce0: 2d2d 2d2d 2b0a 7c69 3920 3a20 7469 6d65  ----+.|i9 : time
│ │ │ │  00051cf0: 2067 203d 2067 7261 7068 2070 323b 2020   g = graph p2;  
│ │ │ │  00051d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051d30: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ -00051d40: 302e 3033 3333 3539 3573 2028 6370 7529  0.0333595s (cpu)
│ │ │ │ -00051d50: 3b20 302e 3033 3330 3733 3373 2028 7468  ; 0.0330733s (th
│ │ │ │ +00051d40: 302e 3034 3932 3636 3173 2028 6370 7529  0.0492661s (cpu)
│ │ │ │ +00051d50: 3b20 302e 3033 3735 3138 3373 2028 7468  ; 0.0375183s (th
│ │ │ │  00051d60: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │  00051d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051d80: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  00051d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ @@ -21662,17 +21662,17 @@
│ │ │ │  000549d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000549e0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │  000549f0: 7469 6d65 2069 6465 616c 2070 6869 2020  time ideal phi  
│ │ │ │  00054a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054a30: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075          |.| -- u
│ │ │ │ -00054a40: 7365 6420 302e 3030 3334 3939 3934 7320  sed 0.00349994s 
│ │ │ │ -00054a50: 2863 7075 293b 2030 2e30 3033 3439 3436  (cpu); 0.0034946
│ │ │ │ -00054a60: 3673 2028 7468 7265 6164 293b 2030 7320  6s (thread); 0s 
│ │ │ │ +00054a40: 7365 6420 302e 3030 3433 3036 3532 7320  sed 0.00430652s 
│ │ │ │ +00054a50: 2863 7075 293b 2030 2e30 3034 3330 3033  (cpu); 0.0043003
│ │ │ │ +00054a60: 3773 2028 7468 7265 6164 293b 2030 7320  7s (thread); 0s 
│ │ │ │  00054a70: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │  00054a80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  00054a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054ad0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ @@ -22297,18 +22297,18 @@
│ │ │ │  00057180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057190: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20  --------+.|i6 : 
│ │ │ │  000571a0: 7469 6d65 2069 6465 616c 2070 6869 2720  time ideal phi' 
│ │ │ │  000571b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000571c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000571d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000571e0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075          |.| -- u
│ │ │ │ -000571f0: 7365 6420 302e 3039 3538 3236 3373 2028  sed 0.0958263s (
│ │ │ │ -00057200: 6370 7529 3b20 302e 3039 3538 3331 3873  cpu); 0.0958318s
│ │ │ │ -00057210: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │ -00057220: 6329 2020 2020 2020 2020 2020 2020 2020  c)              
│ │ │ │ +000571f0: 7365 6420 302e 3131 3139 3535 7320 2863  sed 0.111955s (c
│ │ │ │ +00057200: 7075 293b 2030 2e31 3131 3935 3673 2028  pu); 0.111956s (
│ │ │ │ +00057210: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 0s (gc)
│ │ │ │ +00057220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057230: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  00057240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057280: 2020 2020 2020 2020 7c0a 7c6f 3620 3d20          |.|o6 = 
│ │ │ │  00057290: 6964 6561 6c20 3120 2020 2020 2020 2020  ideal 1         
│ │ │ │ @@ -24856,17 +24856,17 @@
│ │ │ │  00061170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00061180: 2d2d 2d2d 2d2b 0a7c 6933 203a 2074 696d  -----+.|i3 : tim
│ │ │ │  00061190: 6520 696e 7665 7273 6520 7068 6920 2020  e inverse phi   
│ │ │ │  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 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ -000611e0: 2030 2e30 3631 3036 3873 2028 6370 7529   0.061068s (cpu)
│ │ │ │ -000611f0: 3b20 302e 3036 3130 3634 3573 2028 7468  ; 0.0610645s (th
│ │ │ │ -00061200: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │ +000611e0: 2030 2e30 3639 3530 3333 7320 2863 7075   0.0695033s (cpu
│ │ │ │ +000611f0: 293b 2030 2e30 3639 3531 3838 7320 2874  ); 0.0695188s (t
│ │ │ │ +00061200: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │  00061210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061220: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00061230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061270: 2020 2020 207c 0a7c 6f33 203d 202d 2d20       |.|o3 = -- 
│ │ │ │ @@ -27855,16 +27855,16 @@
│ │ │ │  0006cce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006ccf0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a  ---------+.|i2 :
│ │ │ │  0006cd00: 2074 696d 6520 7073 6920 3d20 696e 7665   time psi = inve
│ │ │ │  0006cd10: 7273 654d 6170 2070 6869 2020 2020 2020  rseMap phi      
│ │ │ │  0006cd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cd40: 2020 2020 2020 2020 207c 0a7c 202d 2d20           |.| -- 
│ │ │ │ -0006cd50: 7573 6564 2030 2e31 3630 3030 3173 2028  used 0.160001s (
│ │ │ │ -0006cd60: 6370 7529 3b20 302e 3130 3530 3632 7320  cpu); 0.105062s 
│ │ │ │ +0006cd50: 7573 6564 2030 2e31 3932 3937 3473 2028  used 0.192974s (
│ │ │ │ +0006cd60: 6370 7529 3b20 302e 3131 3132 3539 7320  cpu); 0.111259s 
│ │ │ │  0006cd70: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │  0006cd80: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  0006cd90: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0006cda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -28540,16 +28540,16 @@
│ │ │ │  0006f7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006f7c0: 2d2b 0a7c 6935 203a 2074 696d 6520 7073  -+.|i5 : time ps
│ │ │ │  0006f7d0: 6920 3d20 696e 7665 7273 654d 6170 2070  i = inverseMap p
│ │ │ │  0006f7e0: 6869 2020 2020 2020 2020 2020 2020 2020  hi              
│ │ │ │  0006f7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f810: 207c 0a7c 202d 2d20 7573 6564 2030 2e33   |.| -- used 0.3
│ │ │ │ -0006f820: 3532 3139 3173 2028 6370 7529 3b20 302e  52191s (cpu); 0.
│ │ │ │ -0006f830: 3230 3739 3235 7320 2874 6872 6561 6429  207925s (thread)
│ │ │ │ +0006f820: 3736 3932 3273 2028 6370 7529 3b20 302e  76922s (cpu); 0.
│ │ │ │ +0006f830: 3231 3232 3534 7320 2874 6872 6561 6429  212254s (thread)
│ │ │ │  0006f840: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │  0006f850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f860: 207c 0a7c 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -29536,18 +29536,18 @@
│ │ │ │  000735f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00073600: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │  00073610: 7469 6d65 2069 7342 6972 6174 696f 6e61  time isBirationa
│ │ │ │  00073620: 6c20 7068 6920 2020 2020 2020 2020 2020  l phi           
│ │ │ │  00073630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00073640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00073650: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075          |.| -- u
│ │ │ │ -00073660: 7365 6420 302e 3031 3739 3232 3873 2028  sed 0.0179228s (
│ │ │ │ -00073670: 6370 7529 3b20 302e 3031 3739 3231 3373  cpu); 0.0179213s
│ │ │ │ -00073680: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │ -00073690: 6329 2020 2020 2020 2020 2020 2020 2020  c)              
│ │ │ │ +00073660: 7365 6420 302e 3032 3139 3933 3473 2028  sed 0.0219934s (
│ │ │ │ +00073670: 6370 7529 3b20 302e 3032 3139 3533 7320  cpu); 0.021953s 
│ │ │ │ +00073680: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │ +00073690: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  000736a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  000736b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000736c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000736d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000736e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000736f0: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20          |.|o3 = 
│ │ │ │  00073700: 7472 7565 2020 2020 2020 2020 2020 2020  true            
│ │ │ │ @@ -29566,16 +29566,16 @@
│ │ │ │  000737d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000737e0: 2020 2020 2020 2020 7c0a 7c43 6572 7469          |.|Certi
│ │ │ │  000737f0: 6679 3a20 6f75 7470 7574 2063 6572 7469  fy: output certi
│ │ │ │  00073800: 6669 6564 2120 2020 2020 2020 2020 2020  fied!           
│ │ │ │  00073810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00073820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00073830: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075          |.| -- u
│ │ │ │ -00073840: 7365 6420 302e 3031 3336 3031 3573 2028  sed 0.0136015s (
│ │ │ │ -00073850: 6370 7529 3b20 302e 3031 3332 3333 3273  cpu); 0.0132332s
│ │ │ │ +00073840: 7365 6420 302e 3032 3735 3730 3573 2028  sed 0.0275705s (
│ │ │ │ +00073850: 6370 7529 3b20 302e 3031 3533 3733 3473  cpu); 0.0153734s
│ │ │ │  00073860: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │  00073870: 6329 2020 2020 2020 2020 2020 2020 2020  c)              
│ │ │ │  00073880: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  00073890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000738a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000738b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000738c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -29739,18 +29739,18 @@
│ │ │ │  000742a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000742b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  000742c0: 0a7c 4365 7274 6966 793a 206f 7574 7075  .|Certify: outpu
│ │ │ │  000742d0: 7420 6365 7274 6966 6965 6421 2020 2020  t certified!    
│ │ │ │  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 202d 2d20 7573 6564 2032 2e35 3030  .| -- used 2.500
│ │ │ │ -00074320: 3339 7320 2863 7075 293b 2031 2e39 3736  39s (cpu); 1.976
│ │ │ │ -00074330: 3436 7320 2874 6872 6561 6429 3b20 3073  46s (thread); 0s
│ │ │ │ -00074340: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │ +00074310: 0a7c 202d 2d20 7573 6564 2032 2e35 3332  .| -- used 2.532
│ │ │ │ +00074320: 3473 2028 6370 7529 3b20 322e 3231 3030  4s (cpu); 2.2100
│ │ │ │ +00074330: 3773 2028 7468 7265 6164 293b 2030 7320  7s (thread); 0s 
│ │ │ │ +00074340: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │  00074350: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00074360: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00074370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00074380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00074390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000743a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  000743b0: 0a7c 6f33 203d 2074 7275 6520 2020 2020  .|o3 = true     
│ │ │ │ @@ -30174,17 +30174,17 @@
│ │ │ │  00075dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00075de0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00075df0: 0a7c 4365 7274 6966 793a 206f 7574 7075  .|Certify: outpu
│ │ │ │  00075e00: 7420 6365 7274 6966 6965 6421 2020 2020  t certified!    
│ │ │ │  00075e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00075e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00075e30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00075e40: 0a7c 202d 2d20 7573 6564 2033 2e38 3530  .| -- used 3.850
│ │ │ │ -00075e50: 3637 7320 2863 7075 293b 2032 2e35 3233  67s (cpu); 2.523
│ │ │ │ -00075e60: 3535 7320 2874 6872 6561 6429 3b20 3073  55s (thread); 0s
│ │ │ │ +00075e40: 0a7c 202d 2d20 7573 6564 2033 2e37 3237  .| -- used 3.727
│ │ │ │ +00075e50: 3838 7320 2863 7075 293b 2032 2e37 3530  88s (cpu); 2.750
│ │ │ │ +00075e60: 3437 7320 2874 6872 6561 6429 3b20 3073  47s (thread); 0s
│ │ │ │  00075e70: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │  00075e80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00075e90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00075ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00075eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00075ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00075ed0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ @@ -31483,18 +31483,18 @@
│ │ │ │  0007afa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007afb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  0007afc0: 7c69 3220 3a20 7469 6d65 206b 6572 6e65  |i2 : time kerne
│ │ │ │  0007afd0: 6c28 7068 692c 3129 2020 2020 2020 2020  l(phi,1)        
│ │ │ │  0007afe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007aff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007b000: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007b010: 7c20 2d2d 2075 7365 6420 302e 3031 3732  | -- used 0.0172
│ │ │ │ -0007b020: 3231 3673 2028 6370 7529 3b20 302e 3031  216s (cpu); 0.01
│ │ │ │ -0007b030: 3732 3138 3673 2028 7468 7265 6164 293b  72186s (thread);
│ │ │ │ -0007b040: 2030 7320 2867 6329 2020 2020 2020 2020   0s (gc)        
│ │ │ │ +0007b010: 7c20 2d2d 2075 7365 6420 302e 3032 3134  | -- used 0.0214
│ │ │ │ +0007b020: 3331 3173 2028 6370 7529 3b20 302e 3032  311s (cpu); 0.02
│ │ │ │ +0007b030: 3134 3331 7320 2874 6872 6561 6429 3b20  1431s (thread); 
│ │ │ │ +0007b040: 3073 2028 6763 2920 2020 2020 2020 2020  0s (gc)         
│ │ │ │  0007b050: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  0007b060: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0007b070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007b080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007b090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007b0a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  0007b0b0: 7c6f 3220 3d20 6964 6561 6c20 2829 2020  |o2 = ideal ()  
│ │ │ │ @@ -31523,18 +31523,18 @@
│ │ │ │  0007b220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007b230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  0007b240: 7c69 3320 3a20 7469 6d65 206b 6572 6e65  |i3 : time kerne
│ │ │ │  0007b250: 6c28 7068 692c 3229 2020 2020 2020 2020  l(phi,2)        
│ │ │ │  0007b260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007b270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007b280: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007b290: 7c20 2d2d 2075 7365 6420 302e 3834 3835  | -- used 0.8485
│ │ │ │ -0007b2a0: 3038 7320 2863 7075 293b 2030 2e34 3433  08s (cpu); 0.443
│ │ │ │ -0007b2b0: 3538 7320 2874 6872 6561 6429 3b20 3073  58s (thread); 0s
│ │ │ │ -0007b2c0: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │ +0007b290: 7c20 2d2d 2075 7365 6420 312e 3131 3238  | -- used 1.1128
│ │ │ │ +0007b2a0: 7320 2863 7075 293b 2030 2e35 3133 3136  s (cpu); 0.51316
│ │ │ │ +0007b2b0: 3473 2028 7468 7265 6164 293b 2030 7320  4s (thread); 0s 
│ │ │ │ +0007b2c0: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │  0007b2d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  0007b2e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0007b2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007b300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007b310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007b320: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  0007b330: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ @@ -32424,18 +32424,18 @@
│ │ │ │  0007ea70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007ea80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  0007ea90: 7c69 3320 3a20 7469 6d65 2070 6172 616d  |i3 : time param
│ │ │ │  0007eaa0: 6574 7269 7a65 204c 2020 2020 2020 2020  etrize L        
│ │ │ │  0007eab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007eac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ead0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007eae0: 7c20 2d2d 2075 7365 6420 302e 3030 3532  | -- used 0.0052
│ │ │ │ -0007eaf0: 3235 3573 2028 6370 7529 3b20 302e 3030  255s (cpu); 0.00
│ │ │ │ -0007eb00: 3532 3230 3632 7320 2874 6872 6561 6429  522062s (thread)
│ │ │ │ -0007eb10: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │ +0007eae0: 7c20 2d2d 2075 7365 6420 302e 3030 3539  | -- used 0.0059
│ │ │ │ +0007eaf0: 3939 3039 7320 2863 7075 293b 2030 2e30  9909s (cpu); 0.0
│ │ │ │ +0007eb00: 3035 3939 3734 3173 2028 7468 7265 6164  0599741s (thread
│ │ │ │ +0007eb10: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │  0007eb20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  0007eb30: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0007eb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007eb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007eb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007eb70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  0007eb80: 7c6f 3320 3d20 2d2d 2072 6174 696f 6e61  |o3 = -- rationa
│ │ │ │ @@ -32934,18 +32934,18 @@
│ │ │ │  00080a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00080a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  00080a70: 7c69 3520 3a20 7469 6d65 2070 6172 616d  |i5 : time param
│ │ │ │  00080a80: 6574 7269 7a65 2051 2020 2020 2020 2020  etrize Q        
│ │ │ │  00080a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080ab0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00080ac0: 7c20 2d2d 2075 7365 6420 302e 3535 3639  | -- used 0.5569
│ │ │ │ -00080ad0: 3837 7320 2863 7075 293b 2030 2e33 3937  87s (cpu); 0.397
│ │ │ │ -00080ae0: 3539 3773 2028 7468 7265 6164 293b 2030  597s (thread); 0
│ │ │ │ -00080af0: 7320 2867 6329 2020 2020 2020 2020 2020  s (gc)          
│ │ │ │ +00080ac0: 7c20 2d2d 2075 7365 6420 302e 3537 3733  | -- used 0.5773
│ │ │ │ +00080ad0: 3473 2028 6370 7529 3b20 302e 3432 3630  4s (cpu); 0.4260
│ │ │ │ +00080ae0: 3138 7320 2874 6872 6561 6429 3b20 3073  18s (thread); 0s
│ │ │ │ +00080af0: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │  00080b00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  00080b10: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00080b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080b50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  00080b60: 7c6f 3520 3d20 2d2d 2072 6174 696f 6e61  |o5 = -- rationa
│ │ │ │ @@ -34395,17 +34395,17 @@
│ │ │ │  000865a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000865b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │  000865c0: 6932 203a 2074 696d 6520 7020 3d20 706f  i2 : time p = po
│ │ │ │  000865d0: 696e 7420 736f 7572 6365 2066 2020 2020  int source f    
│ │ │ │  000865e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000865f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00086600: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00086610: 202d 2d20 7573 6564 2030 2e34 3435 3937   -- used 0.44597
│ │ │ │ -00086620: 7320 2863 7075 293b 2030 2e32 3135 3835  s (cpu); 0.21585
│ │ │ │ -00086630: 3673 2028 7468 7265 6164 293b 2030 7320  6s (thread); 0s 
│ │ │ │ +00086610: 202d 2d20 7573 6564 2030 2e35 3130 3832   -- used 0.51082
│ │ │ │ +00086620: 7320 2863 7075 293b 2030 2e32 3335 3939  s (cpu); 0.23599
│ │ │ │ +00086630: 3473 2028 7468 7265 6164 293b 2030 7320  4s (thread); 0s 
│ │ │ │  00086640: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │  00086650: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00086660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00086670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00086680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00086690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000866a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ @@ -34510,18 +34510,18 @@
│ │ │ │  00086cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00086ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │  00086cf0: 6933 203a 2074 696d 6520 7020 3d3d 2066  i3 : time p == f
│ │ │ │  00086d00: 5e2a 2066 2070 2020 2020 2020 2020 2020  ^* f p          
│ │ │ │  00086d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00086d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00086d30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00086d40: 202d 2d20 7573 6564 2030 2e32 3035 3173   -- used 0.2051s
│ │ │ │ -00086d50: 2028 6370 7529 3b20 302e 3133 3034 3039   (cpu); 0.130409
│ │ │ │ -00086d60: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ -00086d70: 6763 2920 2020 2020 2020 2020 2020 2020  gc)             
│ │ │ │ +00086d40: 202d 2d20 7573 6564 2030 2e32 3234 3836   -- used 0.22486
│ │ │ │ +00086d50: 3673 2028 6370 7529 3b20 302e 3133 3837  6s (cpu); 0.1387
│ │ │ │ +00086d60: 3135 7320 2874 6872 6561 6429 3b20 3073  15s (thread); 0s
│ │ │ │ +00086d70: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │  00086d80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00086d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00086da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00086db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00086dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00086dd0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00086de0: 6f33 203d 2074 7275 6520 2020 2020 2020  o3 = true       
│ │ │ │ @@ -34842,16 +34842,16 @@
│ │ │ │  00088190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000881a0: 2020 207c 0a7c 4365 7274 6966 793a 206f     |.|Certify: o
│ │ │ │  000881b0: 7574 7075 7420 6365 7274 6966 6965 6421  utput certified!
│ │ │ │  000881c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000881d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000881e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000881f0: 2020 207c 0a7c 202d 2d20 7573 6564 2030     |.| -- used 0
│ │ │ │ -00088200: 2e30 3134 3830 3431 7320 2863 7075 293b  .0148041s (cpu);
│ │ │ │ -00088210: 2030 2e30 3134 3438 3933 7320 2874 6872   0.0144893s (thr
│ │ │ │ +00088200: 2e30 3830 3937 3135 7320 2863 7075 293b  .0809715s (cpu);
│ │ │ │ +00088210: 2030 2e30 3234 3430 3639 7320 2874 6872   0.0244069s (thr
│ │ │ │  00088220: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │  00088230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00088240: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00088250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00088260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00088270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00088280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -34972,16 +34972,16 @@
│ │ │ │  000889b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000889c0: 2020 207c 0a7c 4365 7274 6966 793a 206f     |.|Certify: o
│ │ │ │  000889d0: 7574 7075 7420 6365 7274 6966 6965 6421  utput certified!
│ │ │ │  000889e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000889f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00088a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00088a10: 2020 207c 0a7c 202d 2d20 7573 6564 2030     |.| -- used 0
│ │ │ │ -00088a20: 2e30 3131 3234 3931 7320 2863 7075 293b  .0112491s (cpu);
│ │ │ │ -00088a30: 2030 2e30 3130 3936 3031 7320 2874 6872   0.0109601s (thr
│ │ │ │ +00088a20: 2e30 3336 3031 3135 7320 2863 7075 293b  .0360115s (cpu);
│ │ │ │ +00088a30: 2030 2e30 3134 3833 3336 7320 2874 6872   0.0148336s (thr
│ │ │ │  00088a40: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │  00088a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00088a60: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00088a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00088a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00088a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00088aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -35296,17 +35296,17 @@
│ │ │ │  00089df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00089e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00089e10: 2d2d 2d2b 0a7c 6937 203a 2074 696d 6520  ---+.|i7 : time 
│ │ │ │  00089e20: 7072 6f6a 6563 7469 7665 4465 6772 6565  projectiveDegree
│ │ │ │  00089e30: 7320 7068 6920 2020 2020 2020 2020 2020  s phi           
│ │ │ │  00089e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00089e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00089e60: 2020 207c 0a7c 202d 2d20 7573 6564 2035     |.| -- used 5
│ │ │ │ -00089e70: 2e30 3336 3565 2d30 3573 2028 6370 7529  .0365e-05s (cpu)
│ │ │ │ -00089e80: 3b20 342e 3530 3834 652d 3035 7320 2874  ; 4.5084e-05s (t
│ │ │ │ +00089e60: 2020 207c 0a7c 202d 2d20 7573 6564 2036     |.| -- used 6
│ │ │ │ +00089e70: 2e33 3638 3365 2d30 3573 2028 6370 7529  .3683e-05s (cpu)
│ │ │ │ +00089e80: 3b20 352e 3433 3138 652d 3035 7320 2874  ; 5.4318e-05s (t
│ │ │ │  00089e90: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │  00089ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00089eb0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00089ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00089ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00089ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00089ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -35332,16 +35332,16 @@
│ │ │ │  0008a030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008a040: 2d2d 2d2b 0a7c 6938 203a 2074 696d 6520  ---+.|i8 : time 
│ │ │ │  0008a050: 7072 6f6a 6563 7469 7665 4465 6772 6565  projectiveDegree
│ │ │ │  0008a060: 7328 7068 692c 4e75 6d44 6567 7265 6573  s(phi,NumDegrees
│ │ │ │  0008a070: 3d3e 3129 2020 2020 2020 2020 2020 2020  =>1)            
│ │ │ │  0008a080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008a090: 2020 207c 0a7c 202d 2d20 7573 6564 2032     |.| -- used 2
│ │ │ │ -0008a0a0: 2e30 3033 3865 2d30 3573 2028 6370 7529  .0038e-05s (cpu)
│ │ │ │ -0008a0b0: 3b20 312e 3938 3837 652d 3035 7320 2874  ; 1.9887e-05s (t
│ │ │ │ +0008a0a0: 2e39 3139 3565 2d30 3573 2028 6370 7529  .9195e-05s (cpu)
│ │ │ │ +0008a0b0: 3b20 322e 3830 3331 652d 3035 7320 2874  ; 2.8031e-05s (t
│ │ │ │  0008a0c0: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │  0008a0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008a0e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0008a0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008a100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008a110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008a120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -37824,17 +37824,17 @@
│ │ │ │  00093bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00093c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00093c10: 2b0a 7c69 3420 3a20 7469 6d65 2070 6869  +.|i4 : time phi
│ │ │ │  00093c20: 2120 3b20 2020 2020 2020 2020 2020 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: 7c0a 7c20 2d2d 2075 7365 6420 302e 3035  |.| -- used 0.05
│ │ │ │ -00093c70: 3535 3732 3973 2028 6370 7529 3b20 302e  55729s (cpu); 0.
│ │ │ │ -00093c80: 3035 3531 3531 3273 2028 7468 7265 6164  0551512s (thread
│ │ │ │ +00093c60: 7c0a 7c20 2d2d 2075 7365 6420 302e 3039  |.| -- used 0.09
│ │ │ │ +00093c70: 3835 3136 3273 2028 6370 7529 3b20 302e  85162s (cpu); 0.
│ │ │ │ +00093c80: 3036 3732 3734 3573 2028 7468 7265 6164  0672745s (thread
│ │ │ │  00093c90: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │  00093ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093cb0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00093cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -37999,17 +37999,17 @@
│ │ │ │  000946e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000946f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00094700: 2b0a 7c69 3920 3a20 7469 6d65 2070 6869  +.|i9 : time phi
│ │ │ │  00094710: 2120 3b20 2020 2020 2020 2020 2020 2020  ! ;             
│ │ │ │  00094720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00094730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00094740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00094750: 7c0a 7c20 2d2d 2075 7365 6420 302e 3033  |.| -- used 0.03
│ │ │ │ -00094760: 3539 3534 3973 2028 6370 7529 3b20 302e  59549s (cpu); 0.
│ │ │ │ -00094770: 3033 3536 3837 3573 2028 7468 7265 6164  0356875s (thread
│ │ │ │ +00094750: 7c0a 7c20 2d2d 2075 7365 6420 302e 3036  |.| -- used 0.06
│ │ │ │ +00094760: 3235 3839 3873 2028 6370 7529 3b20 302e  25898s (cpu); 0.
│ │ │ │ +00094770: 3034 3630 3537 3473 2028 7468 7265 6164  0460574s (thread
│ │ │ │  00094780: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │  00094790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000947a0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000947b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000947c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000947d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000947e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -40043,18 +40043,18 @@
│ │ │ │  0009c6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009c6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  0009c6c0: 7c69 3620 3a20 7469 6d65 2070 6869 5e2a  |i6 : time phi^*
│ │ │ │  0009c6d0: 2a20 7120 2020 2020 2020 2020 2020 2020  * q             
│ │ │ │  0009c6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009c6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009c700: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009c710: 7c20 2d2d 2075 7365 6420 302e 3135 3537  | -- used 0.1557
│ │ │ │ -0009c720: 3673 2028 6370 7529 3b20 302e 3135 3537  6s (cpu); 0.1557
│ │ │ │ -0009c730: 3631 7320 2874 6872 6561 6429 3b20 3073  61s (thread); 0s
│ │ │ │ -0009c740: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │ +0009c710: 7c20 2d2d 2075 7365 6420 302e 3137 3433  | -- used 0.1743
│ │ │ │ +0009c720: 3032 7320 2863 7075 293b 2030 2e31 3734  02s (cpu); 0.174
│ │ │ │ +0009c730: 3330 3373 2028 7468 7265 6164 293b 2030  303s (thread); 0
│ │ │ │ +0009c740: 7320 2867 6329 2020 2020 2020 2020 2020  s (gc)          
│ │ │ │  0009c750: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  0009c760: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0009c770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009c780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009c790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009c7a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  0009c7b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ @@ -42163,18 +42163,18 @@
│ │ │ │  000a4b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a4b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a4b40: 2d2b 0a7c 6933 203a 2074 696d 6520 7068  -+.|i3 : time ph
│ │ │ │  000a4b50: 6920 3d20 7261 7469 6f6e 616c 4d61 7028  i = rationalMap(
│ │ │ │  000a4b60: 562c 332c 3229 2020 2020 2020 2020 2020  V,3,2)          
│ │ │ │  000a4b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a4b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a4b90: 207c 0a7c 202d 2d20 7573 6564 2030 2e30   |.| -- used 0.0
│ │ │ │ -000a4ba0: 3935 3833 3433 7320 2863 7075 293b 2030  958343s (cpu); 0
│ │ │ │ -000a4bb0: 2e30 3935 3738 3533 7320 2874 6872 6561  .0957853s (threa
│ │ │ │ -000a4bc0: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │ +000a4b90: 207c 0a7c 202d 2d20 7573 6564 2030 2e31   |.| -- used 0.1
│ │ │ │ +000a4ba0: 3131 3973 2028 6370 7529 3b20 302e 3131  119s (cpu); 0.11
│ │ │ │ +000a4bb0: 3138 3938 7320 2874 6872 6561 6429 3b20  1898s (thread); 
│ │ │ │ +000a4bc0: 3073 2028 6763 2920 2020 2020 2020 2020  0s (gc)         
│ │ │ │  000a4bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a4be0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  000a4bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a4c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a4c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a4c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a4c30: 207c 0a7c 6f33 203d 202d 2d20 7261 7469   |.|o3 = -- rati
│ │ │ │ @@ -43696,16 +43696,16 @@
│ │ │ │  000aaaf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000aab00: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20  --------+.|i6 : 
│ │ │ │  000aab10: 7469 6d65 2070 6869 203d 2072 6174 696f  time phi = ratio
│ │ │ │  000aab20: 6e61 6c4d 6170 2044 2020 2020 2020 2020  nalMap D        
│ │ │ │  000aab30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000aab40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000aab50: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075          |.| -- u
│ │ │ │ -000aab60: 7365 6420 302e 3032 3838 3839 3373 2028  sed 0.0288893s (
│ │ │ │ -000aab70: 6370 7529 3b20 302e 3032 3838 3839 3973  cpu); 0.0288899s
│ │ │ │ +000aab60: 7365 6420 302e 3033 3631 3833 3473 2028  sed 0.0361834s (
│ │ │ │ +000aab70: 6370 7529 3b20 302e 3033 3631 3833 3173  cpu); 0.0361831s
│ │ │ │  000aab80: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │  000aab90: 6329 2020 2020 2020 2020 2020 2020 2020  c)              
│ │ │ │  000aaba0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  000aabb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000aabc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000aabd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000aabe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -44706,16 +44706,16 @@
│ │ │ │  000aea10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000aea20: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20  --------+.|i7 : 
│ │ │ │  000aea30: 7469 6d65 203f 2069 6d61 6765 2870 6869  time ? image(phi
│ │ │ │  000aea40: 2c22 4634 2229 2020 2020 2020 2020 2020  ,"F4")          
│ │ │ │  000aea50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000aea60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000aea70: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075          |.| -- u
│ │ │ │ -000aea80: 7365 6420 312e 3238 3336 3573 2028 6370  sed 1.28365s (cp
│ │ │ │ -000aea90: 7529 3b20 302e 3738 3634 3138 7320 2874  u); 0.786418s (t
│ │ │ │ +000aea80: 7365 6420 312e 3739 3138 3573 2028 6370  sed 1.79185s (cp
│ │ │ │ +000aea90: 7529 3b20 302e 3731 3538 3239 7320 2874  u); 0.715829s (t
│ │ │ │  000aeaa0: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │  000aeab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000aeac0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  000aead0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000aeae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000aeaf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000aeb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -46244,17 +46244,17 @@
│ │ │ │  000b4a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000b4a40: 2d2d 2d2d 2b0a 7c69 3420 3a20 7469 6d65  ----+.|i4 : time
│ │ │ │  000b4a50: 2053 6567 7265 436c 6173 7320 5820 2020   SegreClass X   
│ │ │ │  000b4a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b4a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b4a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b4a90: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ -000b4aa0: 302e 3739 3832 3236 7320 2863 7075 293b  0.798226s (cpu);
│ │ │ │ -000b4ab0: 2030 2e35 3039 3630 3273 2028 7468 7265   0.509602s (thre
│ │ │ │ -000b4ac0: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │ +000b4aa0: 302e 3833 3632 3633 7320 2863 7075 293b  0.836263s (cpu);
│ │ │ │ +000b4ab0: 2030 2e35 3530 3134 7320 2874 6872 6561   0.55014s (threa
│ │ │ │ +000b4ac0: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │  000b4ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b4ae0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  000b4af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b4b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b4b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b4b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b4b30: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ @@ -46299,17 +46299,17 @@
│ │ │ │  000b4da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000b4db0: 2d2d 2d2d 2b0a 7c69 3520 3a20 7469 6d65  ----+.|i5 : time
│ │ │ │  000b4dc0: 2053 6567 7265 436c 6173 7320 6c69 6674   SegreClass lift
│ │ │ │  000b4dd0: 2858 2c50 3729 2020 2020 2020 2020 2020  (X,P7)          
│ │ │ │  000b4de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b4df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b4e00: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ -000b4e10: 302e 3534 3931 3331 7320 2863 7075 293b  0.549131s (cpu);
│ │ │ │ -000b4e20: 2030 2e33 3333 3236 7320 2874 6872 6561   0.33326s (threa
│ │ │ │ -000b4e30: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │ +000b4e10: 302e 3633 3536 3734 7320 2863 7075 293b  0.635674s (cpu);
│ │ │ │ +000b4e20: 2030 2e33 3630 3637 3673 2028 7468 7265   0.360676s (thre
│ │ │ │ +000b4e30: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │  000b4e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b4e50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  000b4e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b4e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b4e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b4e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b4ea0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ @@ -46359,16 +46359,16 @@
│ │ │ │  000b5160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b5170: 2020 2020 7c0a 7c43 6572 7469 6679 3a20      |.|Certify: 
│ │ │ │  000b5180: 6f75 7470 7574 2063 6572 7469 6669 6564  output certified
│ │ │ │  000b5190: 2120 2020 2020 2020 2020 2020 2020 2020  !               
│ │ │ │  000b51a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b51b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b51c0: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ -000b51d0: 302e 3032 3132 3632 3473 2028 6370 7529  0.0212624s (cpu)
│ │ │ │ -000b51e0: 3b20 302e 3032 3037 3339 3173 2028 7468  ; 0.0207391s (th
│ │ │ │ +000b51d0: 302e 3034 3232 3530 3773 2028 6370 7529  0.0422507s (cpu)
│ │ │ │ +000b51e0: 3b20 302e 3032 3535 3535 3573 2028 7468  ; 0.0255555s (th
│ │ │ │  000b51f0: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │  000b5200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b5210: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  000b5220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b5230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b5240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b5250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -46419,17 +46419,17 @@
│ │ │ │  000b5520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b5530: 2020 2020 7c0a 7c43 6572 7469 6679 3a20      |.|Certify: 
│ │ │ │  000b5540: 6f75 7470 7574 2063 6572 7469 6669 6564  output certified
│ │ │ │  000b5550: 2120 2020 2020 2020 2020 2020 2020 2020  !               
│ │ │ │  000b5560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b5570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b5580: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ -000b5590: 302e 3039 3734 3937 3573 2028 6370 7529  0.0974975s (cpu)
│ │ │ │ -000b55a0: 3b20 302e 3039 3639 3635 3973 2028 7468  ; 0.0969659s (th
│ │ │ │ -000b55b0: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │ +000b5590: 302e 3133 3538 3836 7320 2863 7075 293b  0.135886s (cpu);
│ │ │ │ +000b55a0: 2030 2e31 3234 3333 3473 2028 7468 7265   0.124334s (thre
│ │ │ │ +000b55b0: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │  000b55c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b55d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  000b55e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b55f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b5600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b5610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b5620: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ @@ -46535,26 +46535,26 @@
│ │ │ │  000b5c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b5c70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  000b5c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b5c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b5ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b5cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b5cc0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000b5cd0: 6f39 2020 2020 205a 5a20 2020 2020 2020  o9     ZZ       
│ │ │ │ +000b5cd0: 2020 2020 2020 205a 5a20 2020 2020 2020         ZZ       
│ │ │ │  000b5ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b5cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b5d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b5d10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000b5d20: 3d20 2d2d 2d2d 2d2d 5b78 202e 2e78 205d  = ------[x ..x ]
│ │ │ │ -000b5d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000b5d20: 6f39 203d 202d 2d2d 2d2d 2d5b 7820 2e2e  o9 = ------[x ..
│ │ │ │ +000b5d30: 7820 5d20 2020 2020 2020 2020 2020 2020  x ]             
│ │ │ │  000b5d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b5d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b5d60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000b5d70: 2020 3130 3030 3033 2020 3020 2020 3620    100003  0   6 
│ │ │ │ -000b5d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000b5d70: 2020 2020 2031 3030 3030 3320 2030 2020       100003  0  
│ │ │ │ +000b5d80: 2036 2020 2020 2020 2020 2020 2020 2020   6              
│ │ │ │  000b5d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b5da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b5db0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  000b5dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b5dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b5de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b5df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -46570,18 +46570,18 @@
│ │ │ │  000b5e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000b5ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │  000b5eb0: 6931 3020 3a20 7469 6d65 2070 6869 203d  i10 : time phi =
│ │ │ │  000b5ec0: 2069 6e76 6572 7365 4d61 7020 746f 4d61   inverseMap toMa
│ │ │ │  000b5ed0: 7028 6d69 6e6f 7273 2832 2c6d 6174 7269  p(minors(2,matri
│ │ │ │  000b5ee0: 787b 7b78 5f30 2c78 5f31 2c78 5f33 2c78  x{{x_0,x_1,x_3,x
│ │ │ │  000b5ef0: 5f34 2c78 5f35 7d2c 7b78 5f31 2c7c 0a7c  _4,x_5},{x_1,|.|
│ │ │ │ -000b5f00: 202d 2d20 7573 6564 2030 2e31 3937 3331   -- used 0.19731
│ │ │ │ -000b5f10: 3573 2028 6370 7529 3b20 302e 3039 3833  5s (cpu); 0.0983
│ │ │ │ -000b5f20: 3935 3273 2028 7468 7265 6164 293b 2030  952s (thread); 0
│ │ │ │ -000b5f30: 7320 2867 6329 2020 2020 2020 2020 2020  s (gc)          
│ │ │ │ +000b5f00: 202d 2d20 7573 6564 2030 2e32 3539 3931   -- used 0.25991
│ │ │ │ +000b5f10: 3773 2028 6370 7529 3b20 302e 3132 3533  7s (cpu); 0.1253
│ │ │ │ +000b5f20: 3638 7320 2874 6872 6561 6429 3b20 3073  68s (thread); 0s
│ │ │ │ +000b5f30: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │  000b5f40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  000b5f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b5f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b5f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b5f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b5f90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  000b5fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -46775,17 +46775,17 @@
│ │ │ │  000b6b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000b6b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │  000b6b80: 6931 3120 3a20 7469 6d65 2053 6567 7265  i11 : time Segre
│ │ │ │  000b6b90: 436c 6173 7320 7068 6920 2020 2020 2020  Class phi       
│ │ │ │  000b6ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b6bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b6bc0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000b6bd0: 202d 2d20 7573 6564 2030 2e31 3639 3530   -- used 0.16950
│ │ │ │ -000b6be0: 3973 2028 6370 7529 3b20 302e 3136 3935  9s (cpu); 0.1695
│ │ │ │ -000b6bf0: 3134 7320 2874 6872 6561 6429 3b20 3073  14s (thread); 0s
│ │ │ │ +000b6bd0: 202d 2d20 7573 6564 2030 2e32 3132 3035   -- used 0.21205
│ │ │ │ +000b6be0: 3873 2028 6370 7529 3b20 302e 3231 3138  8s (cpu); 0.2118
│ │ │ │ +000b6bf0: 3936 7320 2874 6872 6561 6429 3b20 3073  96s (thread); 0s
│ │ │ │  000b6c00: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │  000b6c10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  000b6c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b6c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b6c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b6c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b6c60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ @@ -46935,17 +46935,17 @@
│ │ │ │  000b7560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b7570: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  000b7580: 2020 2020 2020 7469 6d65 2053 6567 7265        time Segre
│ │ │ │  000b7590: 436c 6173 7320 4220 2020 2020 2020 2020  Class B         
│ │ │ │  000b75a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b75b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b75c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000b75d0: 202d 2d20 7573 6564 2030 2e35 3138 3736   -- used 0.51876
│ │ │ │ -000b75e0: 3473 2028 6370 7529 3b20 302e 3331 3537  4s (cpu); 0.3157
│ │ │ │ -000b75f0: 3839 7320 2874 6872 6561 6429 3b20 3073  89s (thread); 0s
│ │ │ │ +000b75d0: 202d 2d20 7573 6564 2030 2e34 3333 3534   -- used 0.43354
│ │ │ │ +000b75e0: 3173 2028 6370 7529 3b20 302e 3239 3138  1s (cpu); 0.2918
│ │ │ │ +000b75f0: 3539 7320 2874 6872 6561 6429 3b20 3073  59s (thread); 0s
│ │ │ │  000b7600: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │  000b7610: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  000b7620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b7630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b7640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b7650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b7660: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ @@ -46995,18 +46995,18 @@
│ │ │ │  000b7920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b7930: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  000b7940: 2020 2020 2020 7469 6d65 2053 6567 7265        time Segre
│ │ │ │  000b7950: 436c 6173 7320 6c69 6674 2842 2c61 6d62  Class lift(B,amb
│ │ │ │  000b7960: 6965 6e74 2072 696e 6720 4229 2020 2020  ient ring B)    
│ │ │ │  000b7970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b7980: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000b7990: 202d 2d20 7573 6564 2031 2e33 3133 3831   -- used 1.31381
│ │ │ │ -000b79a0: 7320 2863 7075 293b 2030 2e38 3630 3831  s (cpu); 0.86081
│ │ │ │ -000b79b0: 3773 2028 7468 7265 6164 293b 2030 7320  7s (thread); 0s 
│ │ │ │ -000b79c0: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │ +000b7990: 202d 2d20 7573 6564 2031 2e38 3434 3434   -- used 1.84444
│ │ │ │ +000b79a0: 7320 2863 7075 293b 2031 2e30 3333 3933  s (cpu); 1.03393
│ │ │ │ +000b79b0: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ +000b79c0: 6763 2920 2020 2020 2020 2020 2020 2020  gc)             
│ │ │ │  000b79d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  000b79e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b79f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b7a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b7a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b7a20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  000b7a30: 2020 2020 2020 2020 2020 2039 2020 2020             9    
│ │ │ │ @@ -47245,17 +47245,17 @@
│ │ │ │  000b88c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000b88d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  000b88e0: 7c69 3120 3a20 7469 6d65 2061 7070 6c79  |i1 : time apply
│ │ │ │  000b88f0: 2831 2e2e 3132 2c69 202d 3e20 6465 7363  (1..12,i -> desc
│ │ │ │  000b8900: 7269 6265 2073 7065 6369 616c 4372 656d  ribe specialCrem
│ │ │ │  000b8910: 6f6e 6154 7261 6e73 666f 726d 6174 696f  onaTransformatio
│ │ │ │  000b8920: 6e28 692c 5a5a 2f33 3333 3129 2920 7c0a  n(i,ZZ/3331)) |.
│ │ │ │ -000b8930: 7c20 2d2d 2075 7365 6420 312e 3437 3538  | -- used 1.4758
│ │ │ │ -000b8940: 3373 2028 6370 7529 3b20 312e 3134 3830  3s (cpu); 1.1480
│ │ │ │ -000b8950: 3273 2028 7468 7265 6164 293b 2030 7320  2s (thread); 0s 
│ │ │ │ +000b8930: 7c20 2d2d 2075 7365 6420 312e 3534 3039  | -- used 1.5409
│ │ │ │ +000b8940: 3573 2028 6370 7529 3b20 312e 3137 3530  5s (cpu); 1.1750
│ │ │ │ +000b8950: 3873 2028 7468 7265 6164 293b 2030 7320  8s (thread); 0s 
│ │ │ │  000b8960: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │  000b8970: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  000b8980: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000b8990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b89a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b89b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b89c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ @@ -48068,16 +48068,16 @@
│ │ │ │  000bbc30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000bbc40: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a  ---------+.|i1 :
│ │ │ │  000bbc50: 2074 696d 6520 7370 6563 6961 6c43 7562   time specialCub
│ │ │ │  000bbc60: 6963 5472 616e 7366 6f72 6d61 7469 6f6e  icTransformation
│ │ │ │  000bbc70: 2039 2020 2020 2020 2020 2020 2020 2020   9              
│ │ │ │  000bbc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bbc90: 2020 2020 2020 2020 207c 0a7c 202d 2d20           |.| -- 
│ │ │ │ -000bbca0: 7573 6564 2030 2e30 3934 3230 3535 7320  used 0.0942055s 
│ │ │ │ -000bbcb0: 2863 7075 293b 2030 2e30 3934 3230 3433  (cpu); 0.0942043
│ │ │ │ +000bbca0: 7573 6564 2030 2e30 3935 3234 3838 7320  used 0.0952488s 
│ │ │ │ +000bbcb0: 2863 7075 293b 2030 2e30 3935 3234 3837  (cpu); 0.0952487
│ │ │ │  000bbcc0: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │  000bbcd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bbce0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000bbcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bbd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bbd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bbd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -50518,16 +50518,16 @@
│ │ │ │  000c5550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000c5560: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a  ---------+.|i2 :
│ │ │ │  000c5570: 2074 696d 6520 6465 7363 7269 6265 206f   time describe o
│ │ │ │  000c5580: 6f20 2020 2020 2020 2020 2020 2020 2020  o               
│ │ │ │  000c5590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000c55a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000c55b0: 2020 2020 2020 2020 207c 0a7c 202d 2d20           |.| -- 
│ │ │ │ -000c55c0: 7573 6564 2030 2e30 3138 3538 3632 7320  used 0.0185862s 
│ │ │ │ -000c55d0: 2863 7075 293b 2030 2e30 3138 3538 3636  (cpu); 0.0185866
│ │ │ │ +000c55c0: 7573 6564 2030 2e30 3230 3332 3534 7320  used 0.0203254s 
│ │ │ │ +000c55d0: 2863 7075 293b 2030 2e30 3230 3332 3532  (cpu); 0.0203252
│ │ │ │  000c55e0: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │  000c55f0: 6763 2920 2020 2020 2020 2020 2020 2020  gc)             
│ │ │ │  000c5600: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000c5610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000c5620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000c5630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000c5640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -50732,17 +50732,17 @@
│ │ │ │  000c62b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000c62c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │  000c62d0: 203a 2074 696d 6520 7370 6563 6961 6c51   : time specialQ
│ │ │ │  000c62e0: 7561 6472 6174 6963 5472 616e 7366 6f72  uadraticTransfor
│ │ │ │  000c62f0: 6d61 7469 6f6e 2034 2020 2020 2020 2020  mation 4        
│ │ │ │  000c6300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000c6310: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ -000c6320: 2d20 7573 6564 2030 2e30 3733 3538 3831  - used 0.0735881
│ │ │ │ -000c6330: 7320 2863 7075 293b 2030 2e30 3733 3537  s (cpu); 0.07357
│ │ │ │ -000c6340: 3032 7320 2874 6872 6561 6429 3b20 3073  02s (thread); 0s
│ │ │ │ +000c6320: 2d20 7573 6564 2030 2e30 3831 3636 3537  - used 0.0816657
│ │ │ │ +000c6330: 7320 2863 7075 293b 2030 2e30 3831 3438  s (cpu); 0.08148
│ │ │ │ +000c6340: 3431 7320 2874 6872 6561 6429 3b20 3073  41s (thread); 0s
│ │ │ │  000c6350: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │  000c6360: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000c6370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000c6380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000c6390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000c63a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000c63b0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ @@ -51287,16 +51287,16 @@
│ │ │ │  000c8560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000c8570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │  000c8580: 203a 2074 696d 6520 6465 7363 7269 6265   : time describe
│ │ │ │  000c8590: 206f 6f20 2020 2020 2020 2020 2020 2020   oo             
│ │ │ │  000c85a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000c85b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000c85c0: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ -000c85d0: 2d20 7573 6564 2030 2e31 3039 3331 3173  - used 0.109311s
│ │ │ │ -000c85e0: 2028 6370 7529 3b20 302e 3032 3938 3438   (cpu); 0.029848
│ │ │ │ +000c85d0: 2d20 7573 6564 2030 2e31 3135 3131 7320  - used 0.11511s 
│ │ │ │ +000c85e0: 2863 7075 293b 2030 2e30 3335 3837 3932  (cpu); 0.0358792
│ │ │ │  000c85f0: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │  000c8600: 6763 2920 2020 2020 2020 2020 2020 2020  gc)             
│ │ │ │  000c8610: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000c8620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000c8630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000c8640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000c8650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -52398,16 +52398,16 @@
│ │ │ │  000ccad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000ccae0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20  --------+.|i4 : 
│ │ │ │  000ccaf0: 7469 6d65 2070 6869 2720 3d20 7661 6c75  time phi' = valu
│ │ │ │  000ccb00: 6520 7374 723b 2020 2020 2020 2020 2020  e str;          
│ │ │ │  000ccb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000ccb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000ccb30: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ -000ccb40: 302e 3032 3232 3432 3473 2028 6370 7529  0.0222424s (cpu)
│ │ │ │ -000ccb50: 3b20 302e 3032 3232 3431 3473 2028 7468  ; 0.0222414s (th
│ │ │ │ +000ccb40: 302e 3034 3533 3537 3173 2028 6370 7529  0.0453571s (cpu)
│ │ │ │ +000ccb50: 3b20 302e 3034 3533 3537 3373 2028 7468  ; 0.0453573s (th
│ │ │ │  000ccb60: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │  000ccb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000ccb80: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000ccb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000ccba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000ccbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000ccbc0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ @@ -52421,18 +52421,18 @@
│ │ │ │  000ccc40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000ccc50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000ccc60: 2d2d 2d2d 2b0a 7c69 3520 3a20 7469 6d65  ----+.|i5 : time
│ │ │ │  000ccc70: 2064 6573 6372 6962 6520 7068 6927 2020   describe phi'  
│ │ │ │  000ccc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000ccc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000ccca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000cccb0: 7c0a 7c20 2d2d 2075 7365 6420 302e 3030  |.| -- used 0.00
│ │ │ │ -000cccc0: 3531 3837 3237 7320 2863 7075 293b 2030  518727s (cpu); 0
│ │ │ │ -000cccd0: 2e30 3035 3138 3736 3273 2028 7468 7265  .00518762s (thre
│ │ │ │ -000ccce0: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │ +000cccb0: 7c0a 7c20 2d2d 2075 7365 6420 302e 3031  |.| -- used 0.01
│ │ │ │ +000cccc0: 3036 3937 3473 2028 6370 7529 3b20 302e  06974s (cpu); 0.
│ │ │ │ +000cccd0: 3031 3037 3031 7320 2874 6872 6561 6429  010701s (thread)
│ │ │ │ +000ccce0: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │  000cccf0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000ccd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000ccd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000ccd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000ccd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000ccd40: 2020 2020 2020 2020 7c0a 7c6f 3520 3d20          |.|o5 = 
│ │ │ │  000ccd50: 7261 7469 6f6e 616c 206d 6170 2064 6566  rational map def
│ │ │ │ @@ -52488,17 +52488,17 @@
│ │ │ │  000cd070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000cd080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │  000cd090: 3620 3a20 7469 6d65 2064 6573 6372 6962  6 : time describ
│ │ │ │  000cd0a0: 6520 696e 7665 7273 6520 7068 6927 2020  e inverse phi'  
│ │ │ │  000cd0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000cd0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000cd0d0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075          |.| -- u
│ │ │ │ -000cd0e0: 7365 6420 302e 3030 3432 3934 3338 7320  sed 0.00429438s 
│ │ │ │ -000cd0f0: 2863 7075 293b 2030 2e30 3034 3239 3530  (cpu); 0.0042950
│ │ │ │ -000cd100: 3473 2028 7468 7265 6164 293b 2030 7320  4s (thread); 0s 
│ │ │ │ +000cd0e0: 7365 6420 302e 3030 3837 3032 3834 7320  sed 0.00870284s 
│ │ │ │ +000cd0f0: 2863 7075 293b 2030 2e30 3038 3730 3933  (cpu); 0.0087093
│ │ │ │ +000cd100: 3673 2028 7468 7265 6164 293b 2030 7320  6s (thread); 0s 
│ │ │ │  000cd110: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │  000cd120: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  000cd130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000cd140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000cd150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000cd160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000cd170: 7c0a 7c6f 3620 3d20 7261 7469 6f6e 616c  |.|o6 = rational
│ │ ├── ./usr/share/info/DGAlgebras.info.gz
│ │ │ ├── DGAlgebras.info
│ │ │ │ @@ -1231,17 +1231,17 @@
│ │ │ │  00004ce0: 3a20 4842 203d 2048 4820 4220 2020 2020  : HB = HH B     
│ │ │ │  00004cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004d20: 2020 2020 2020 2020 2020 7c0a 7c46 696e            |.|Fin
│ │ │ │  00004d30: 6469 6e67 2065 6173 7920 7265 6c61 7469  ding easy relati
│ │ │ │  00004d40: 6f6e 7320 2020 2020 2020 2020 2020 3a20  ons           : 
│ │ │ │ -00004d50: 202d 2d20 7573 6564 2030 2e30 3237 3136   -- used 0.02716
│ │ │ │ -00004d60: 3437 7320 2863 7075 293b 2030 2e30 3235  47s (cpu); 0.025
│ │ │ │ -00004d70: 3731 3539 7320 2020 2020 7c0a 7c20 2020  7159s     |.|   
│ │ │ │ +00004d50: 202d 2d20 7573 6564 2030 2e32 3131 3438   -- used 0.21148
│ │ │ │ +00004d60: 7320 2863 7075 293b 2030 2e30 3435 3831  s (cpu); 0.04581
│ │ │ │ +00004d70: 3438 7320 2020 2020 2020 7c0a 7c20 2020  48s       |.|   
│ │ │ │  00004d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004dc0: 2020 2020 2020 2020 2020 7c0a 7c6f 3520            |.|o5 
│ │ │ │  00004dd0: 3d20 4842 2020 2020 2020 2020 2020 2020  = HB            
│ │ │ │  00004de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -1495,16 +1495,16 @@
│ │ │ │  00005d60: 6779 416c 6765 6272 6128 432c 4765 6e44  gyAlgebra(C,GenD
│ │ │ │  00005d70: 6567 7265 654c 696d 6974 3d3e 342c 5265  egreeLimit=>4,Re
│ │ │ │  00005d80: 6c44 6567 7265 654c 696d 6974 3d3e 3429  lDegreeLimit=>4)
│ │ │ │  00005d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005da0: 207c 0a7c 4669 6e64 696e 6720 6561 7379   |.|Finding easy
│ │ │ │  00005db0: 2072 656c 6174 696f 6e73 2020 2020 2020   relations      
│ │ │ │  00005dc0: 2020 2020 203a 2020 2d2d 2075 7365 6420       :  -- used 
│ │ │ │ -00005dd0: 302e 3031 3839 3338 3173 2028 6370 7529  0.0189381s (cpu)
│ │ │ │ -00005de0: 3b20 302e 3031 3736 3836 3973 2020 2020  ; 0.0176869s    
│ │ │ │ +00005dd0: 302e 3034 3739 3133 3173 2028 6370 7529  0.0479131s (cpu)
│ │ │ │ +00005de0: 3b20 302e 3032 3633 3439 3173 2020 2020  ; 0.0263491s    
│ │ │ │  00005df0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00005e00: 2020 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: 207c 0a7c 2020 2020 2020 205a 5a20 2020   |.|       ZZ   
│ │ │ │  00005e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -2723,17 +2723,17 @@
│ │ │ │  0000aa20: 3720 3a20 484b 5220 3d20 4848 204b 5220  7 : HKR = HH KR 
│ │ │ │  0000aa30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000aa40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000aa50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000aa60: 2020 2020 2020 2020 2020 2020 7c0a 7c46              |.|F
│ │ │ │  0000aa70: 696e 6469 6e67 2065 6173 7920 7265 6c61  inding easy rela
│ │ │ │  0000aa80: 7469 6f6e 7320 2020 2020 2020 2020 2020  tions           
│ │ │ │ -0000aa90: 3a20 202d 2d20 7573 6564 2030 2e31 3235  :  -- used 0.125
│ │ │ │ -0000aaa0: 3530 3473 2028 6370 7529 3b20 302e 3036  504s (cpu); 0.06
│ │ │ │ -0000aab0: 3535 3230 3373 2020 2020 2020 7c0a 7c20  55203s      |.| 
│ │ │ │ +0000aa90: 3a20 202d 2d20 7573 6564 2030 2e32 3231  :  -- used 0.221
│ │ │ │ +0000aaa0: 3234 3573 2028 6370 7529 3b20 302e 3036  245s (cpu); 0.06
│ │ │ │ +0000aab0: 3238 3339 3673 2020 2020 2020 7c0a 7c20  28396s      |.| 
│ │ │ │  0000aac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000aad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000aae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000aaf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ab00: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │  0000ab10: 3720 3d20 484b 5220 2020 2020 2020 2020  7 = HKR         
│ │ │ │  0000ab20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -2838,17 +2838,17 @@
│ │ │ │  0000b150: 3130 203a 2048 4b52 2720 3d20 4848 206b  10 : HKR' = HH k
│ │ │ │  0000b160: 6f73 7a75 6c43 6f6d 706c 6578 4447 4120  oszulComplexDGA 
│ │ │ │  0000b170: 5227 2020 2020 2020 2020 2020 2020 2020  R'              
│ │ │ │  0000b180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b190: 2020 2020 2020 2020 2020 2020 7c0a 7c46              |.|F
│ │ │ │  0000b1a0: 696e 6469 6e67 2065 6173 7920 7265 6c61  inding easy rela
│ │ │ │  0000b1b0: 7469 6f6e 7320 2020 2020 2020 2020 2020  tions           
│ │ │ │ -0000b1c0: 3a20 202d 2d20 7573 6564 2030 2e35 3738  :  -- used 0.578
│ │ │ │ -0000b1d0: 3038 3973 2028 6370 7529 3b20 302e 3439  089s (cpu); 0.49
│ │ │ │ -0000b1e0: 3632 3933 7320 2020 2020 2020 7c0a 7c20  6293s       |.| 
│ │ │ │ +0000b1c0: 3a20 202d 2d20 7573 6564 2030 2e36 3538  :  -- used 0.658
│ │ │ │ +0000b1d0: 3537 3373 2028 6370 7529 3b20 302e 3634  573s (cpu); 0.64
│ │ │ │ +0000b1e0: 3534 3333 7320 2020 2020 2020 7c0a 7c20  5433s       |.| 
│ │ │ │  0000b1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b230: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │  0000b240: 3130 203d 2048 4b52 2720 2020 2020 2020  10 = HKR'       
│ │ │ │  0000b250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -4569,16 +4569,16 @@
│ │ │ │  00011d80: 2048 4820 6720 2020 2020 2020 2020 2020   HH g           
│ │ │ │  00011d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011dc0: 2020 7c0a 7c46 696e 6469 6e67 2065 6173    |.|Finding eas
│ │ │ │  00011dd0: 7920 7265 6c61 7469 6f6e 7320 2020 2020  y relations     
│ │ │ │  00011de0: 2020 2020 2020 3a20 202d 2d20 7573 6564        :  -- used
│ │ │ │ -00011df0: 2030 2e30 3134 3438 3537 7320 2863 7075   0.0144857s (cpu
│ │ │ │ -00011e00: 293b 2030 2e30 3133 3731 3437 7320 2020  ); 0.0137147s   
│ │ │ │ +00011df0: 2030 2e30 3433 3539 3435 7320 2863 7075   0.0435945s (cpu
│ │ │ │ +00011e00: 293b 2030 2e30 3230 3034 3331 7320 2020  ); 0.0200431s   
│ │ │ │  00011e10: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00011e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011e60: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00011e70: 2020 2020 2020 2020 2020 2020 2020 205a                 Z
│ │ │ │ @@ -6296,16 +6296,16 @@
│ │ │ │  00018970: 6c6f 6779 416c 6765 6272 6128 4129 2020  logyAlgebra(A)  
│ │ │ │  00018980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000189a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  000189b0: 0a7c 4669 6e64 696e 6720 6561 7379 2072  .|Finding easy r
│ │ │ │  000189c0: 656c 6174 696f 6e73 2020 2020 2020 2020  elations        
│ │ │ │  000189d0: 2020 203a 2020 2d2d 2075 7365 6420 302e     :  -- used 0.
│ │ │ │ -000189e0: 3032 3434 3732 3273 2028 6370 7529 3b20  0244722s (cpu); 
│ │ │ │ -000189f0: 302e 3032 3038 3431 3973 2020 2020 207c  0.0208419s     |
│ │ │ │ +000189e0: 3033 3437 3738 3873 2028 6370 7529 3b20  0347788s (cpu); 
│ │ │ │ +000189f0: 302e 3032 3135 3334 3473 2020 2020 207c  0.0215344s     |
│ │ │ │  00018a00: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00018a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018a40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00018a50: 0a7c 6f34 203d 2048 4120 2020 2020 2020  .|o4 = HA       
│ │ │ │  00018a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -15306,17 +15306,17 @@
│ │ │ │  0003bc90: 6937 203a 2048 4867 203d 2048 4820 6720  i7 : HHg = HH g 
│ │ │ │  0003bca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bcb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bcd0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0003bce0: 4669 6e64 696e 6720 6561 7379 2072 656c  Finding easy rel
│ │ │ │  0003bcf0: 6174 696f 6e73 2020 2020 2020 2020 2020  ations          
│ │ │ │ -0003bd00: 203a 2020 2d2d 2075 7365 6420 302e 3031   :  -- used 0.01
│ │ │ │ -0003bd10: 3430 3731 7320 2863 7075 293b 2030 2e30  4071s (cpu); 0.0
│ │ │ │ -0003bd20: 3133 3239 3634 7320 2020 2020 207c 0a7c  132964s      |.|
│ │ │ │ +0003bd00: 203a 2020 2d2d 2075 7365 6420 302e 3036   :  -- used 0.06
│ │ │ │ +0003bd10: 3231 3133 3473 2028 6370 7529 3b20 302e  21134s (cpu); 0.
│ │ │ │ +0003bd20: 3032 3238 3936 3673 2020 2020 207c 0a7c  0228966s     |.|
│ │ │ │  0003bd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bd70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0003bd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bd90: 2020 2020 2020 2020 205a 5a20 2020 2020           ZZ     
│ │ │ │ @@ -15749,17 +15749,17 @@
│ │ │ │  0003d840: 3a20 4841 203d 2068 6f6d 6f6c 6f67 7941  : HA = homologyA
│ │ │ │  0003d850: 6c67 6562 7261 2841 2920 2020 2020 2020  lgebra(A)       
│ │ │ │  0003d860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d880: 2020 2020 2020 2020 2020 7c0a 7c46 696e            |.|Fin
│ │ │ │  0003d890: 6469 6e67 2065 6173 7920 7265 6c61 7469  ding easy relati
│ │ │ │  0003d8a0: 6f6e 7320 2020 2020 2020 2020 2020 3a20  ons           : 
│ │ │ │ -0003d8b0: 202d 2d20 7573 6564 2030 2e30 3138 3230   -- used 0.01820
│ │ │ │ -0003d8c0: 3131 7320 2863 7075 293b 2030 2e30 3137  11s (cpu); 0.017
│ │ │ │ -0003d8d0: 3431 3432 7320 2020 2020 7c0a 7c20 2020  4142s     |.|   
│ │ │ │ +0003d8b0: 202d 2d20 7573 6564 2030 2e31 3931 3533   -- used 0.19153
│ │ │ │ +0003d8c0: 3173 2028 6370 7529 3b20 302e 3034 3735  1s (cpu); 0.0475
│ │ │ │ +0003d8d0: 3573 2020 2020 2020 2020 7c0a 7c20 2020  5s        |.|   
│ │ │ │  0003d8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d920: 2020 2020 2020 2020 2020 7c0a 7c6f 3420            |.|o4 
│ │ │ │  0003d930: 3d20 4841 2020 2020 2020 2020 2020 2020  = HA            
│ │ │ │  0003d940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -15912,16 +15912,16 @@
│ │ │ │  0003e270: 6f6c 6f67 7941 6c67 6562 7261 2841 2920  ologyAlgebra(A) 
│ │ │ │  0003e280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e2b0: 7c0a 7c46 696e 6469 6e67 2065 6173 7920  |.|Finding easy 
│ │ │ │  0003e2c0: 7265 6c61 7469 6f6e 7320 2020 2020 2020  relations       
│ │ │ │  0003e2d0: 2020 2020 3a20 202d 2d20 7573 6564 2030      :  -- used 0
│ │ │ │ -0003e2e0: 2e30 3835 3438 3435 7320 2863 7075 293b  .0854845s (cpu);
│ │ │ │ -0003e2f0: 2030 2e30 3832 3431 3573 2020 2020 2020   0.082415s      
│ │ │ │ +0003e2e0: 2e32 3930 3339 3473 2028 6370 7529 3b20  .290394s (cpu); 
│ │ │ │ +0003e2f0: 302e 3132 3139 3239 7320 2020 2020 2020  0.121929s       
│ │ │ │  0003e300: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0003e310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e350: 7c0a 7c6f 3820 3d20 4841 2020 2020 2020  |.|o8 = HA      
│ │ │ │  0003e360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -16277,16 +16277,16 @@
│ │ │ │  0003f940: 6d6f 6c6f 6779 416c 6765 6272 6128 4129  mologyAlgebra(A)
│ │ │ │  0003f950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f980: 7c0a 7c46 696e 6469 6e67 2065 6173 7920  |.|Finding easy 
│ │ │ │  0003f990: 7265 6c61 7469 6f6e 7320 2020 2020 2020  relations       
│ │ │ │  0003f9a0: 2020 2020 3a20 202d 2d20 7573 6564 2030      :  -- used 0
│ │ │ │ -0003f9b0: 2e30 3533 3136 3473 2028 6370 7529 3b20  .053164s (cpu); 
│ │ │ │ -0003f9c0: 302e 3035 3137 3833 3973 2020 2020 2020  0.0517839s      
│ │ │ │ +0003f9b0: 2e31 3237 3937 7320 2863 7075 293b 2030  .12797s (cpu); 0
│ │ │ │ +0003f9c0: 2e30 3635 3838 3832 7320 2020 2020 2020  .0658882s       
│ │ │ │  0003f9d0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0003f9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003fa00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003fa10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003fa20: 7c0a 7c6f 3136 203d 2048 4120 2020 2020  |.|o16 = HA     
│ │ │ │  0003fa30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -16488,17 +16488,17 @@
│ │ │ │  00040670: 3231 203a 2048 4220 3d20 686f 6d6f 6c6f  21 : HB = homolo
│ │ │ │  00040680: 6779 416c 6765 6272 6128 422c 4765 6e44  gyAlgebra(B,GenD
│ │ │ │  00040690: 6567 7265 654c 696d 6974 3d3e 372c 5265  egreeLimit=>7,Re
│ │ │ │  000406a0: 6c44 6567 7265 654c 696d 6974 3d3e 3134  lDegreeLimit=>14
│ │ │ │  000406b0: 2920 2020 2020 2020 2020 2020 7c0a 7c46  )           |.|F
│ │ │ │  000406c0: 696e 6469 6e67 2065 6173 7920 7265 6c61  inding easy rela
│ │ │ │  000406d0: 7469 6f6e 7320 2020 2020 2020 2020 2020  tions           
│ │ │ │ -000406e0: 3a20 202d 2d20 7573 6564 2030 2e30 3138  :  -- used 0.018
│ │ │ │ -000406f0: 3231 3773 2028 6370 7529 3b20 302e 3031  217s (cpu); 0.01
│ │ │ │ -00040700: 3734 3432 3173 2020 2020 2020 7c0a 7c20  74421s      |.| 
│ │ │ │ +000406e0: 3a20 202d 2d20 7573 6564 2030 2e30 3831  :  -- used 0.081
│ │ │ │ +000406f0: 3634 3535 7320 2863 7075 293b 2030 2e30  6455s (cpu); 0.0
│ │ │ │ +00040700: 3238 3637 3337 7320 2020 2020 7c0a 7c20  286737s     |.| 
│ │ │ │  00040710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00040720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00040730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00040740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00040750: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │  00040760: 3231 203d 2048 4220 2020 2020 2020 2020  21 = HB         
│ │ │ │  00040770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -17245,16 +17245,16 @@
│ │ │ │  000435c0: 203d 2048 4828 4b52 2920 2020 2020 2020   = HH(KR)       
│ │ │ │  000435d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000435e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000435f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00043600: 2020 2020 2020 207c 0a7c 4669 6e64 696e         |.|Findin
│ │ │ │  00043610: 6720 6561 7379 2072 656c 6174 696f 6e73  g easy relations
│ │ │ │  00043620: 2020 2020 2020 2020 2020 203a 2020 2d2d             :  --
│ │ │ │ -00043630: 2075 7365 6420 302e 3031 3734 3730 3873   used 0.0174708s
│ │ │ │ -00043640: 2028 6370 7529 3b20 302e 3031 3633 3932   (cpu); 0.016392
│ │ │ │ +00043630: 2075 7365 6420 302e 3033 3637 3837 7320   used 0.036787s 
│ │ │ │ +00043640: 2863 7075 293b 2030 2e30 3139 3636 3233  (cpu); 0.0196623
│ │ │ │  00043650: 7320 2020 2020 207c 0a7c 2020 2020 2020  s      |.|      
│ │ │ │  00043660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00043670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00043680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00043690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000436a0: 2020 2020 2020 207c 0a7c 6f37 203d 2048         |.|o7 = H
│ │ │ │  000436b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -17611,16 +17611,16 @@
│ │ │ │  00044ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00044cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00044cc0: 2b0a 7c69 3620 3a20 484b 5220 3d20 4848  +.|i6 : HKR = HH
│ │ │ │  00044cd0: 284b 5229 2020 2020 2020 2020 2020 2020  (KR)            
│ │ │ │  00044ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044d00: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ -00044d10: 2d20 7573 6564 2030 2e32 3137 3733 3473  - used 0.217734s
│ │ │ │ -00044d20: 2028 6370 7529 3b20 302e 3135 3636 3835   (cpu); 0.156685
│ │ │ │ +00044d10: 2d20 7573 6564 2030 2e32 3530 3631 3773  - used 0.250617s
│ │ │ │ +00044d20: 2028 6370 7529 3b20 302e 3134 3836 3833   (cpu); 0.148683
│ │ │ │  00044d30: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │  00044d40: 6763 2920 2020 2020 2020 2020 2020 2020  gc)             
│ │ │ │  00044d50: 2020 2020 2020 7c0a 7c46 696e 6469 6e67        |.|Finding
│ │ │ │  00044d60: 2065 6173 7920 7265 6c61 7469 6f6e 7320   easy relations 
│ │ │ │  00044d70: 2020 2020 2020 2020 2020 3a20 2020 2020            :     
│ │ │ │  00044d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -22288,16 +22288,16 @@
│ │ │ │  000570f0: 6d61 7373 6579 5472 6970 6c65 5072 6f64  masseyTripleProd
│ │ │ │  00057100: 7563 7428 4b52 2c7a 312c 7a32 2c7a 3329  uct(KR,z1,z2,z3)
│ │ │ │  00057110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057130: 7c0a 7c46 696e 6469 6e67 2065 6173 7920  |.|Finding easy 
│ │ │ │  00057140: 7265 6c61 7469 6f6e 7320 2020 2020 2020  relations       
│ │ │ │  00057150: 2020 2020 3a20 202d 2d20 7573 6564 2030      :  -- used 0
│ │ │ │ -00057160: 2e35 3430 3639 3973 2028 6370 7529 3b20  .540699s (cpu); 
│ │ │ │ -00057170: 302e 3437 3031 3332 7320 2020 2020 2020  0.470132s       
│ │ │ │ +00057160: 2e37 3439 3334 3173 2028 6370 7529 3b20  .749341s (cpu); 
│ │ │ │ +00057170: 302e 3632 3031 3538 7320 2020 2020 2020  0.620158s       
│ │ │ │  00057180: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00057190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000571a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000571b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000571c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000571d0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000571e0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ @@ -22791,17 +22791,17 @@
│ │ │ │  00059060: 7c69 3520 3a20 4820 3d20 4848 284b 5229  |i5 : H = HH(KR)
│ │ │ │  00059070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000590a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  000590b0: 7c46 696e 6469 6e67 2065 6173 7920 7265  |Finding easy re
│ │ │ │  000590c0: 6c61 7469 6f6e 7320 2020 2020 2020 2020  lations         
│ │ │ │ -000590d0: 2020 3a20 202d 2d20 7573 6564 2030 2e32    :  -- used 0.2
│ │ │ │ -000590e0: 3438 3030 3873 2028 6370 7529 3b20 302e  48008s (cpu); 0.
│ │ │ │ -000590f0: 3234 3336 3373 2020 2020 2020 2020 7c0a  24363s        |.
│ │ │ │ +000590d0: 2020 3a20 202d 2d20 7573 6564 2030 2e31    :  -- used 0.1
│ │ │ │ +000590e0: 3732 3735 3573 2028 6370 7529 3b20 302e  72755s (cpu); 0.
│ │ │ │ +000590f0: 3135 3833 3839 7320 2020 2020 2020 7c0a  158389s       |.
│ │ │ │  00059100: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00059110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059140: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  00059150: 7c6f 3520 3d20 4820 2020 2020 2020 2020  |o5 = H         
│ │ │ │  00059160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -26223,18 +26223,18 @@
│ │ │ │  000666e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000666f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00066700: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2048  -------+.|i4 : H
│ │ │ │  00066710: 4220 3d20 746f 7241 6c67 6562 7261 2852  B = torAlgebra(R
│ │ │ │  00066720: 2c53 2c47 656e 4465 6772 6565 4c69 6d69  ,S,GenDegreeLimi
│ │ │ │  00066730: 743d 3e34 2c52 656c 4465 6772 6565 4c69  t=>4,RelDegreeLi
│ │ │ │  00066740: 6d69 743d 3e38 2920 2020 2020 2020 2020  mit=>8)         
│ │ │ │ -00066750: 7c0a 7c20 2d2d 2075 7365 6420 302e 3437  |.| -- used 0.47
│ │ │ │ -00066760: 3232 3773 2028 6370 7529 3b20 302e 3430  227s (cpu); 0.40
│ │ │ │ -00066770: 3737 3935 7320 2874 6872 6561 6429 3b20  7795s (thread); 
│ │ │ │ -00066780: 3073 2028 6763 2920 2020 2020 2020 2020  0s (gc)         
│ │ │ │ +00066750: 7c0a 7c20 2d2d 2075 7365 6420 302e 3731  |.| -- used 0.71
│ │ │ │ +00066760: 3330 3631 7320 2863 7075 293b 2030 2e35  3061s (cpu); 0.5
│ │ │ │ +00066770: 3537 3638 3773 2028 7468 7265 6164 293b  57687s (thread);
│ │ │ │ +00066780: 2030 7320 2867 6329 2020 2020 2020 2020   0s (gc)        
│ │ │ │  00066790: 2020 2020 2020 2020 207c 0a7c 4669 6e64           |.|Find
│ │ │ │  000667a0: 696e 6720 6561 7379 2072 656c 6174 696f  ing easy relatio
│ │ │ │  000667b0: 6e73 2020 2020 2020 2020 2020 203a 2020  ns           :  
│ │ │ │  000667c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000667d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000667e0: 2020 7c0a 7c6f 3420 3d20 4842 2020 2020    |.|o4 = HB    
│ │ │ │  000667f0: 2020 2020 2020 2020 2020 2020 2020 2020
│ │ ├── ./usr/share/info/EdgeIdeals.info.gz
│ │ │ ├── EdgeIdeals.info
│ │ │ │ @@ -7842,16 +7842,16 @@
│ │ │ │  0001ea10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ea20: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0001ea30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ea40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ea50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ea60: 2020 7c0a 7c6f 3420 3d20 4879 7065 7247    |.|o4 = HyperG
│ │ │ │  0001ea70: 7261 7068 7b22 6564 6765 7322 203d 3e20  raph{"edges" => 
│ │ │ │ -0001ea80: 7b7b 632c 2064 7d2c 207b 622c 2063 7d2c  {{c, d}, {b, c},
│ │ │ │ -0001ea90: 207b 612c 2065 7d7d 7d20 2020 2020 2020   {a, e}}}       
│ │ │ │ +0001ea80: 7b7b 622c 2063 7d2c 207b 612c 2065 7d2c  {{b, c}, {a, e},
│ │ │ │ +0001ea90: 207b 632c 2064 7d7d 7d20 2020 2020 2020   {c, d}}}       
│ │ │ │  0001eaa0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0001eab0: 2020 2020 2020 2022 7269 6e67 2220 3d3e         "ring" =>
│ │ │ │  0001eac0: 2053 2020 2020 2020 2020 2020 2020 2020   S              
│ │ │ │  0001ead0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001eae0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001eaf0: 2020 2020 2020 2020 2022 7665 7274 6963           "vertic
│ │ │ │  0001eb00: 6573 2220 3d3e 207b 612c 2062 2c20 632c  es" => {a, b, c,
│ │ │ │ @@ -21417,17 +21417,17 @@
│ │ │ │  00053a80: 2020 207c 0a7c 6f33 203d 2048 7970 6572     |.|o3 = Hyper
│ │ │ │  00053a90: 4772 6170 687b 2265 6467 6573 2220 3d3e  Graph{"edges" =>
│ │ │ │  00053aa0: 207b 7b78 202c 2078 202c 2078 207d 2c20   {{x , x , x }, 
│ │ │ │  00053ab0: 7b78 202c 2078 207d 2c20 7b78 202c 2078  {x , x }, {x , x
│ │ │ │  00053ac0: 202c 2078 202c 2078 207d 7d7d 2020 2020   , x , x }}}    
│ │ │ │  00053ad0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00053ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053af0: 2020 2020 3320 2020 3420 2020 3520 2020      3   4   5   
│ │ │ │ -00053b00: 2020 3120 2020 3420 2020 2020 3120 2020    1   4     1   
│ │ │ │ -00053b10: 3220 2020 3320 2020 3520 2020 2020 2020  2   3   5       
│ │ │ │ +00053af0: 2020 2020 3120 2020 3220 2020 3520 2020      1   2   5   
│ │ │ │ +00053b00: 2020 3220 2020 3320 2020 2020 3120 2020    2   3     1   
│ │ │ │ +00053b10: 3320 2020 3420 2020 3520 2020 2020 2020  3   4   5       
│ │ │ │  00053b20: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00053b30: 2020 2020 2020 2272 696e 6722 203d 3e20        "ring" => 
│ │ │ │  00053b40: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │  00053b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053b70: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00053b80: 2020 2020 2020 2276 6572 7469 6365 7322        "vertices"
│ │ │ │ @@ -21455,2568 +21455,2608 @@
│ │ │ │  00053ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053d00: 2d2d 2d2b 0a7c 6934 203a 2072 616e 646f  ---+.|i4 : rando
│ │ │ │  00053d10: 6d48 7970 6572 4772 6170 6828 522c 7b33  mHyperGraph(R,{3
│ │ │ │  00053d20: 2c32 2c34 7d29 2020 2020 2020 2020 2020  ,2,4})          
│ │ │ │  00053d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053d50: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -00053d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053da0: 2d2d 2d2b 0a7c 6935 203a 2072 616e 646f  ---+.|i5 : rando
│ │ │ │ -00053db0: 6d48 7970 6572 4772 6170 6828 522c 7b34  mHyperGraph(R,{4
│ │ │ │ -00053dc0: 2c34 2c32 2c32 7d29 202d 2d20 696d 706f  ,4,2,2}) -- impo
│ │ │ │ -00053dd0: 7373 6962 6c65 2c20 7265 7475 726e 7320  ssible, returns 
│ │ │ │ -00053de0: 6e75 6c6c 2077 6865 6e20 2020 2020 2020  null when       
│ │ │ │ -00053df0: 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d     |.|----------
│ │ │ │ -00053e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053e40: 2d2d 2d7c 0a7c 7469 6d65 2f62 7261 6e63  ---|.|time/branc
│ │ │ │ -00053e50: 6820 6c69 6d69 7420 7265 6163 6865 6420  h limit reached 
│ │ │ │ -00053e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053d50: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00053d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053da0: 2020 207c 0a7c 6f34 203d 2048 7970 6572     |.|o4 = Hyper
│ │ │ │ +00053db0: 4772 6170 687b 2265 6467 6573 2220 3d3e  Graph{"edges" =>
│ │ │ │ +00053dc0: 207b 7b78 202c 2078 202c 2078 207d 2c20   {{x , x , x }, 
│ │ │ │ +00053dd0: 7b78 202c 2078 207d 2c20 7b78 202c 2078  {x , x }, {x , x
│ │ │ │ +00053de0: 202c 2078 202c 2078 207d 7d7d 2020 2020   , x , x }}}    
│ │ │ │ +00053df0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00053e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053e10: 2020 2020 3120 2020 3420 2020 3520 2020      1   4   5   
│ │ │ │ +00053e20: 2020 3220 2020 3520 2020 2020 3120 2020    2   5     1   
│ │ │ │ +00053e30: 3220 2020 3320 2020 3420 2020 2020 2020  2   3   4       
│ │ │ │ +00053e40: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00053e50: 2020 2020 2020 2272 696e 6722 203d 3e20        "ring" => 
│ │ │ │ +00053e60: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │  00053e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053e90: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -00053ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053ee0: 2d2d 2d2b 0a0a 5468 6520 7261 6e64 6f6d  ---+..The random
│ │ │ │ -00053ef0: 4879 7065 7247 7261 7068 206d 6574 686f  HyperGraph metho
│ │ │ │ -00053f00: 6420 7769 6c6c 2072 6574 7572 6e20 6e75  d will return nu
│ │ │ │ -00053f10: 6c6c 2069 6d6d 6564 6961 7465 6c79 2069  ll immediately i
│ │ │ │ -00053f20: 6620 7468 6520 7369 7a65 7320 6f66 2074  f the sizes of t
│ │ │ │ -00053f30: 6865 0a65 6467 6573 2066 6169 6c20 746f  he.edges fail to
│ │ │ │ -00053f40: 2070 6173 7320 7468 6520 4c59 4d2d 696e   pass the LYM-in
│ │ │ │ -00053f50: 6571 7561 6c69 7479 3a20 2431 2f28 6e20  equality: $1/(n 
│ │ │ │ -00053f60: 6368 6f6f 7365 2044 5f31 2920 2b20 312f  choose D_1) + 1/
│ │ │ │ -00053f70: 286e 2063 686f 6f73 6520 445f 3229 202b  (n choose D_2) +
│ │ │ │ -00053f80: 0a2e 2e2e 202b 2031 2f28 6e20 6368 6f6f  .... + 1/(n choo
│ │ │ │ -00053f90: 7365 2044 5f6d 2920 5c6c 6571 2031 2420  se D_m) \leq 1$ 
│ │ │ │ -00053fa0: 7768 6572 6520 246e 2420 6973 2074 6865  where $n$ is the
│ │ │ │ -00053fb0: 206e 756d 6265 7220 6f66 2076 6172 6961   number of varia
│ │ │ │ -00053fc0: 626c 6573 2069 6e20 5220 616e 640a 246d  bles in R and.$m
│ │ │ │ -00053fd0: 2420 6973 2074 6865 206c 656e 6774 6820  $ is the length 
│ │ │ │ -00053fe0: 6f66 2044 2e20 4e6f 7465 2074 6861 7420  of D. Note that 
│ │ │ │ -00053ff0: 6576 656e 2069 6620 4420 7061 7373 6573  even if D passes
│ │ │ │ -00054000: 2074 6869 7320 696e 6571 7561 6c69 7479   this inequality
│ │ │ │ -00054010: 2c20 6974 2069 7320 6e6f 740a 6e65 6365  , it is not.nece
│ │ │ │ -00054020: 7373 6172 696c 7920 7472 7565 2074 6861  ssarily true tha
│ │ │ │ -00054030: 7420 7468 6572 6520 6973 2073 6f6d 6520  t there is some 
│ │ │ │ -00054040: 6879 7065 7267 7261 7068 2077 6974 6820  hypergraph with 
│ │ │ │ -00054050: 6564 6765 2073 697a 6573 2067 6976 656e  edge sizes given
│ │ │ │ -00054060: 2062 7920 442e 2053 6565 0a44 2e20 4c75   by D. See.D. Lu
│ │ │ │ -00054070: 6265 6c6c 2773 2022 4120 7368 6f72 7420  bell's "A short 
│ │ │ │ -00054080: 7072 6f6f 6620 6f66 2053 7065 726e 6572  proof of Sperner
│ │ │ │ -00054090: 2773 206c 656d 6d61 2c22 204a 2e20 436f  's lemma," J. Co
│ │ │ │ -000540a0: 6d62 696e 2e20 5468 656f 7279 2c20 313a  mbin. Theory, 1:
│ │ │ │ -000540b0: 3239 390a 2831 3936 3629 2e0a 0a53 6565  299.(1966)...See
│ │ │ │ -000540c0: 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a   also.========..
│ │ │ │ -000540d0: 2020 2a20 2a6e 6f74 6520 7261 6e64 6f6d    * *note random
│ │ │ │ -000540e0: 4772 6170 683a 2072 616e 646f 6d47 7261  Graph: randomGra
│ │ │ │ -000540f0: 7068 2c20 2d2d 2072 6574 7572 6e73 2061  ph, -- returns a
│ │ │ │ -00054100: 2072 616e 646f 6d20 6772 6170 680a 2020   random graph.  
│ │ │ │ -00054110: 2a20 2a6e 6f74 6520 7261 6e64 6f6d 556e  * *note randomUn
│ │ │ │ -00054120: 6966 6f72 6d48 7970 6572 4772 6170 683a  iformHyperGraph:
│ │ │ │ -00054130: 2072 616e 646f 6d55 6e69 666f 726d 4879   randomUniformHy
│ │ │ │ -00054140: 7065 7247 7261 7068 2c20 2d2d 2072 6574  perGraph, -- ret
│ │ │ │ -00054150: 7572 6e73 2061 2072 616e 646f 6d0a 2020  urns a random.  
│ │ │ │ -00054160: 2020 756e 6966 6f72 6d20 6879 7065 7267    uniform hyperg
│ │ │ │ -00054170: 7261 7068 0a20 202a 202a 6e6f 7465 2042  raph.  * *note B
│ │ │ │ -00054180: 7261 6e63 684c 696d 6974 3a20 4272 616e  ranchLimit: Bran
│ │ │ │ -00054190: 6368 4c69 6d69 742c 202d 2d20 6f70 7469  chLimit, -- opti
│ │ │ │ -000541a0: 6f6e 616c 2061 7267 756d 656e 7420 666f  onal argument fo
│ │ │ │ -000541b0: 7220 7261 6e64 6f6d 4879 7065 7247 7261  r randomHyperGra
│ │ │ │ -000541c0: 7068 0a20 202a 202a 6e6f 7465 2054 696d  ph.  * *note Tim
│ │ │ │ -000541d0: 654c 696d 6974 3a20 5469 6d65 4c69 6d69  eLimit: TimeLimi
│ │ │ │ -000541e0: 742c 202d 2d20 6f70 7469 6f6e 616c 2061  t, -- optional a
│ │ │ │ -000541f0: 7267 756d 656e 7420 666f 7220 7261 6e64  rgument for rand
│ │ │ │ -00054200: 6f6d 4879 7065 7247 7261 7068 0a20 202a  omHyperGraph.  *
│ │ │ │ -00054210: 202a 6e6f 7465 2043 6f6e 7374 7275 6374   *note Construct
│ │ │ │ -00054220: 6f72 204f 7665 7276 6965 773a 2043 6f6e  or Overview: Con
│ │ │ │ -00054230: 7374 7275 6374 6f72 204f 7665 7276 6965  structor Overvie
│ │ │ │ -00054240: 772c 202d 2d20 6120 7375 6d6d 6172 7920  w, -- a summary 
│ │ │ │ -00054250: 6f66 2074 6865 206d 616e 790a 2020 2020  of the many.    
│ │ │ │ -00054260: 7761 7973 206f 6620 6d61 6b69 6e67 2067  ways of making g
│ │ │ │ -00054270: 7261 7068 7320 616e 6420 6879 7065 7267  raphs and hyperg
│ │ │ │ -00054280: 7261 7068 730a 0a57 6179 7320 746f 2075  raphs..Ways to u
│ │ │ │ -00054290: 7365 2072 616e 646f 6d48 7970 6572 4772  se randomHyperGr
│ │ │ │ -000542a0: 6170 683a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  aph:.===========
│ │ │ │ -000542b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000542c0: 3d3d 0a0a 2020 2a20 2272 616e 646f 6d48  ==..  * "randomH
│ │ │ │ -000542d0: 7970 6572 4772 6170 6828 506f 6c79 6e6f  yperGraph(Polyno
│ │ │ │ -000542e0: 6d69 616c 5269 6e67 2c4c 6973 7429 220a  mialRing,List)".
│ │ │ │ -000542f0: 0a46 6f72 2074 6865 2070 726f 6772 616d  .For the program
│ │ │ │ -00054300: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  mer.============
│ │ │ │ -00054310: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65  ======..The obje
│ │ │ │ -00054320: 6374 202a 6e6f 7465 2072 616e 646f 6d48  ct *note randomH
│ │ │ │ -00054330: 7970 6572 4772 6170 683a 2072 616e 646f  yperGraph: rando
│ │ │ │ -00054340: 6d48 7970 6572 4772 6170 682c 2069 7320  mHyperGraph, is 
│ │ │ │ -00054350: 6120 2a6e 6f74 6520 6d65 7468 6f64 2066  a *note method f
│ │ │ │ -00054360: 756e 6374 696f 6e0a 7769 7468 206f 7074  unction.with opt
│ │ │ │ -00054370: 696f 6e73 3a20 284d 6163 6175 6c61 7932  ions: (Macaulay2
│ │ │ │ -00054380: 446f 6329 4d65 7468 6f64 4675 6e63 7469  Doc)MethodFuncti
│ │ │ │ -00054390: 6f6e 5769 7468 4f70 7469 6f6e 732c 2e0a  onWithOptions,..
│ │ │ │ -000543a0: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .---------------
│ │ │ │ -000543b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000543c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000543d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000543e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000543f0: 0a0a 5468 6520 736f 7572 6365 206f 6620  ..The source of 
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│ │ │ │ -00054510: 6f6d 4879 7065 7247 7261 7068 2c20 5570  omHyperGraph, Up
│ │ │ │ -00054520: 3a20 546f 700a 0a72 616e 646f 6d48 7970  : Top..randomHyp
│ │ │ │ -00054530: 6572 4772 6170 6828 2e2e 2e2c 4272 616e  erGraph(...,Bran
│ │ │ │ -00054540: 6368 4c69 6d69 743d 3e2e 2e2e 2920 2d2d  chLimit=>...) --
│ │ │ │ -00054550: 206c 696d 6974 2072 6563 7572 7369 7665   limit recursive
│ │ │ │ -00054560: 2062 7261 6e63 6869 6e67 2069 6e20 7261   branching in ra
│ │ │ │ -00054570: 6e64 6f6d 4879 7065 7247 7261 7068 0a2a  ndomHyperGraph.*
│ │ │ │ -00054580: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00054590: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000545a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000545b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000545c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000545d0: 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 6167  ******..  * Usag
│ │ │ │ -000545e0: 653a 200a 2020 2020 2020 2020 4720 3d20  e: .        G = 
│ │ │ │ -000545f0: 7261 6e64 6f6d 4879 7065 7247 7261 7068  randomHyperGraph
│ │ │ │ -00054600: 2852 2c44 2c42 7261 6e63 684c 696d 6974  (R,D,BranchLimit
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│ │ │ │ -00054640: 6163 6175 6c61 7932 446f 6329 5a5a 2c2c  acaulay2Doc)ZZ,,
│ │ │ │ -00054650: 206e 756d 6265 7220 6f66 2062 7261 6e63   number of branc
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│ │ │ │ -00054670: 2020 2020 6c65 7665 6c20 696e 2074 6865      level in the
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│ │ │ │ -00054690: 6e64 6f6d 4879 7065 7247 7261 7068 0a0a  ndomHyperGraph..
│ │ │ │ -000546a0: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
│ │ │ │ -000546b0: 3d3d 3d3d 3d3d 3d0a 0a54 6865 202a 6e6f  =======..The *no
│ │ │ │ -000546c0: 7465 2072 616e 646f 6d48 7970 6572 4772  te randomHyperGr
│ │ │ │ -000546d0: 6170 683a 2072 616e 646f 6d48 7970 6572  aph: randomHyper
│ │ │ │ -000546e0: 4772 6170 682c 206d 6574 686f 6420 666f  Graph, method fo
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│ │ │ │ -00054700: 6b69 6e67 0a61 6c67 6f72 6974 686d 2074  king.algorithm t
│ │ │ │ -00054710: 6f20 6765 6e65 7261 7465 2065 6467 6573  o generate edges
│ │ │ │ -00054720: 2077 6974 6820 6e6f 2069 6e63 6c75 7369   with no inclusi
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│ │ │ │ -00054760: 7472 6565 2c20 202a 6e6f 7465 2072 616e  tree,  *note ran
│ │ │ │ -00054770: 646f 6d48 7970 6572 4772 6170 683a 2072  domHyperGraph: r
│ │ │ │ -00054780: 616e 646f 6d48 7970 6572 4772 6170 682c  andomHyperGraph,
│ │ │ │ -00054790: 2077 696c 6c20 6d61 6b65 0a42 7261 6e63   will make.Branc
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│ │ │ │ -000547f0: 7320 6973 2072 6571 7569 7265 642c 202a  s is required, *
│ │ │ │ -00054800: 6e6f 7465 2072 616e 646f 6d48 7970 6572  note randomHyper
│ │ │ │ -00054810: 4772 6170 683a 2072 616e 646f 6d48 7970  Graph: randomHyp
│ │ │ │ -00054820: 6572 4772 6170 682c 206d 6179 2074 616b  erGraph, may tak
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│ │ │ │ -00054840: 7374 6570 7320 6265 666f 7265 2074 6572  steps before ter
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│ │ │ │ -00054860: 6d6f 7265 2070 7265 6369 7365 2c20 7468  more precise, th
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│ │ │ │ -000548d0: 6865 2064 6566 6175 6c74 2076 616c 7565  he default value
│ │ │ │ -000548e0: 2069 7320 332e 0a0a 5365 6520 616c 736f   is 3...See also
│ │ │ │ -000548f0: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a  .========..  * *
│ │ │ │ -00054900: 6e6f 7465 2054 696d 654c 696d 6974 3a20  note TimeLimit: 
│ │ │ │ -00054910: 5469 6d65 4c69 6d69 742c 202d 2d20 6f70  TimeLimit, -- op
│ │ │ │ -00054920: 7469 6f6e 616c 2061 7267 756d 656e 7420  tional argument 
│ │ │ │ -00054930: 666f 7220 7261 6e64 6f6d 4879 7065 7247  for randomHyperG
│ │ │ │ -00054940: 7261 7068 0a20 202a 202a 6e6f 7465 2072  raph.  * *note r
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│ │ │ │ -00054960: 2072 616e 646f 6d48 7970 6572 4772 6170   randomHyperGrap
│ │ │ │ -00054970: 682c 202d 2d20 7265 7475 726e 7320 6120  h, -- returns a 
│ │ │ │ -00054980: 7261 6e64 6f6d 2068 7970 6572 6772 6170  random hypergrap
│ │ │ │ -00054990: 680a 0a46 756e 6374 696f 6e73 2077 6974  h..Functions wit
│ │ │ │ -000549a0: 6820 6f70 7469 6f6e 616c 2061 7267 756d  h optional argum
│ │ │ │ -000549b0: 656e 7420 6e61 6d65 6420 4272 616e 6368  ent named Branch
│ │ │ │ -000549c0: 4c69 6d69 743a 0a3d 3d3d 3d3d 3d3d 3d3d  Limit:.=========
│ │ │ │ -000549d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000549e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000549f0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ -00054a00: 2a6e 6f74 6520 7261 6e64 6f6d 4879 7065  *note randomHype
│ │ │ │ -00054a10: 7247 7261 7068 282e 2e2e 2c42 7261 6e63  rGraph(...,Branc
│ │ │ │ -00054a20: 684c 696d 6974 3d3e 2e2e 2e29 3a0a 2020  hLimit=>...):.  
│ │ │ │ -00054a30: 2020 7261 6e64 6f6d 4879 7065 7247 7261    randomHyperGra
│ │ │ │ -00054a40: 7068 5f6c 705f 7064 5f70 645f 7064 5f63  ph_lp_pd_pd_pd_c
│ │ │ │ -00054a50: 6d42 7261 6e63 684c 696d 6974 3d3e 5f70  mBranchLimit=>_p
│ │ │ │ -00054a60: 645f 7064 5f70 645f 7270 2c20 2d2d 206c  d_pd_pd_rp, -- l
│ │ │ │ -00054a70: 696d 6974 0a20 2020 2072 6563 7572 7369  imit.    recursi
│ │ │ │ -00054a80: 7665 2062 7261 6e63 6869 6e67 2069 6e20  ve branching in 
│ │ │ │ -00054a90: 7261 6e64 6f6d 4879 7065 7247 7261 7068  randomHyperGraph
│ │ │ │ -00054aa0: 0a0a 4675 7274 6865 7220 696e 666f 726d  ..Further inform
│ │ │ │ -00054ab0: 6174 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ation.==========
│ │ │ │ -00054ac0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2044  =========..  * D
│ │ │ │ -00054ad0: 6566 6175 6c74 2076 616c 7565 3a20 330a  efault value: 3.
│ │ │ │ -00054ae0: 2020 2a20 4675 6e63 7469 6f6e 3a20 2a6e    * Function: *n
│ │ │ │ -00054af0: 6f74 6520 7261 6e64 6f6d 4879 7065 7247  ote randomHyperG
│ │ │ │ -00054b00: 7261 7068 3a20 7261 6e64 6f6d 4879 7065  raph: randomHype
│ │ │ │ -00054b10: 7247 7261 7068 2c20 2d2d 2072 6574 7572  rGraph, -- retur
│ │ │ │ -00054b20: 6e73 2061 2072 616e 646f 6d0a 2020 2020  ns a random.    
│ │ │ │ -00054b30: 6879 7065 7267 7261 7068 0a20 202a 204f  hypergraph.  * O
│ │ │ │ -00054b40: 7074 696f 6e20 6b65 793a 202a 6e6f 7465  ption key: *note
│ │ │ │ -00054b50: 2042 7261 6e63 684c 696d 6974 3a20 4272   BranchLimit: Br
│ │ │ │ -00054b60: 616e 6368 4c69 6d69 742c 202d 2d20 6f70  anchLimit, -- op
│ │ │ │ -00054b70: 7469 6f6e 616c 2061 7267 756d 656e 7420  tional argument 
│ │ │ │ -00054b80: 666f 720a 2020 2020 7261 6e64 6f6d 4879  for.    randomHy
│ │ │ │ -00054b90: 7065 7247 7261 7068 0a2d 2d2d 2d2d 2d2d  perGraph.-------
│ │ │ │ -00054ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00054bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00054bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00054bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00054be0: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f  --------..The so
│ │ │ │ -00054bf0: 7572 6365 206f 6620 7468 6973 2064 6f63  urce of this doc
│ │ │ │ -00054c00: 756d 656e 7420 6973 2069 6e0a 2f62 7569  ument is in./bui
│ │ │ │ -00054c10: 6c64 2f72 6570 726f 6475 6369 626c 652d  ld/reproducible-
│ │ │ │ -00054c20: 7061 7468 2f6d 6163 6175 6c61 7932 2d31  path/macaulay2-1
│ │ │ │ -00054c30: 2e32 352e 3131 2b64 732f 4d32 2f4d 6163  .25.11+ds/M2/Mac
│ │ │ │ -00054c40: 6175 6c61 7932 2f70 6163 6b61 6765 732f  aulay2/packages/
│ │ │ │ -00054c50: 4564 6765 4964 6561 6c73 2e0a 6d32 3a34  EdgeIdeals..m2:4
│ │ │ │ -00054c60: 3236 363a 302e 0a1f 0a46 696c 653a 2045  266:0....File: E
│ │ │ │ -00054c70: 6467 6549 6465 616c 732e 696e 666f 2c20  dgeIdeals.info, 
│ │ │ │ -00054c80: 4e6f 6465 3a20 7261 6e64 6f6d 4879 7065  Node: randomHype
│ │ │ │ -00054c90: 7247 7261 7068 5f6c 705f 7064 5f70 645f  rGraph_lp_pd_pd_
│ │ │ │ -00054ca0: 7064 5f63 6d54 696d 654c 696d 6974 3d3e  pd_cmTimeLimit=>
│ │ │ │ -00054cb0: 5f70 645f 7064 5f70 645f 7270 2c20 4e65  _pd_pd_pd_rp, Ne
│ │ │ │ -00054cc0: 7874 3a20 7261 6e64 6f6d 556e 6966 6f72  xt: randomUnifor
│ │ │ │ -00054cd0: 6d48 7970 6572 4772 6170 682c 2050 7265  mHyperGraph, Pre
│ │ │ │ -00054ce0: 763a 2072 616e 646f 6d48 7970 6572 4772  v: randomHyperGr
│ │ │ │ -00054cf0: 6170 685f 6c70 5f70 645f 7064 5f70 645f  aph_lp_pd_pd_pd_
│ │ │ │ -00054d00: 636d 4272 616e 6368 4c69 6d69 743d 3e5f  cmBranchLimit=>_
│ │ │ │ -00054d10: 7064 5f70 645f 7064 5f72 702c 2055 703a  pd_pd_pd_rp, Up:
│ │ │ │ -00054d20: 2054 6f70 0a0a 7261 6e64 6f6d 4879 7065   Top..randomHype
│ │ │ │ -00054d30: 7247 7261 7068 282e 2e2e 2c54 696d 654c  rGraph(...,TimeL
│ │ │ │ -00054d40: 696d 6974 3d3e 2e2e 2e29 202d 2d20 6c69  imit=>...) -- li
│ │ │ │ -00054d50: 6d69 7420 7468 6520 7275 6e6e 696e 6720  mit the running 
│ │ │ │ -00054d60: 7469 6d65 206f 6620 7261 6e64 6f6d 4879  time of randomHy
│ │ │ │ -00054d70: 7065 7247 7261 7068 0a2a 2a2a 2a2a 2a2a  perGraph.*******
│ │ │ │ -00054d80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00054d90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00054da0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00054db0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00054dc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a  ***********..  *
│ │ │ │ -00054dd0: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ -00054de0: 2072 616e 646f 6d48 7970 6572 4772 6170   randomHyperGrap
│ │ │ │ -00054df0: 6828 522c 442c 5469 6d65 4c69 6d69 743d  h(R,D,TimeLimit=
│ │ │ │ -00054e00: 3e73 290a 2020 2a20 496e 7075 7473 3a0a  >s).  * Inputs:.
│ │ │ │ -00054e10: 2020 2020 2020 2a20 732c 2061 6e20 2a6e        * s, an *n
│ │ │ │ -00054e20: 6f74 6520 696e 7465 6765 723a 2028 4d61  ote integer: (Ma
│ │ │ │ -00054e30: 6361 756c 6179 3244 6f63 295a 5a2c 2c20  caulay2Doc)ZZ,, 
│ │ │ │ -00054e40: 7469 6d65 2069 6e20 7365 636f 6e64 7320  time in seconds 
│ │ │ │ -00054e50: 6265 666f 7265 0a20 2020 2020 2020 2072  before.        r
│ │ │ │ -00054e60: 616e 646f 6d48 7970 6572 4772 6170 6820  andomHyperGraph 
│ │ │ │ -00054e70: 7265 7475 726e 7320 6e75 6c6c 0a0a 4465  returns null..De
│ │ │ │ -00054e80: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d  scription.======
│ │ │ │ -00054e90: 3d3d 3d3d 3d0a 0a54 6865 202a 6e6f 7465  =====..The *note
│ │ │ │ -00054ea0: 2072 616e 646f 6d48 7970 6572 4772 6170   randomHyperGrap
│ │ │ │ -00054eb0: 683a 2072 616e 646f 6d48 7970 6572 4772  h: randomHyperGr
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│ │ │ │ -00054f70: 5468 6520 5469 6d65 4c69 6d69 740a 6f70  The TimeLimit.op
│ │ │ │ -00054f80: 7469 6f6e 2069 7320 7468 6520 616d 6f75  tion is the amou
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│ │ │ │ -00054fc0: 6e64 6f6d 4879 7065 7247 7261 7068 3a20  ndomHyperGraph: 
│ │ │ │ -00054fd0: 7261 6e64 6f6d 4879 7065 7247 7261 7068  randomHyperGraph
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│ │ │ │ -00055000: 4120 7661 6c75 6520 6f66 2030 0a69 7320  A value of 0.is 
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│ │ │ │ -00055040: 7365 636f 6e64 7329 2e0a 0a53 6565 2061  seconds)...See a
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│ │ │ │ -00055060: 2a20 2a6e 6f74 6520 4272 616e 6368 4c69  * *note BranchLi
│ │ │ │ -00055070: 6d69 743a 2042 7261 6e63 684c 696d 6974  mit: BranchLimit
│ │ │ │ -00055080: 2c20 2d2d 206f 7074 696f 6e61 6c20 6172  , -- optional ar
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│ │ │ │ -000550a0: 6d48 7970 6572 4772 6170 680a 2020 2a20  mHyperGraph.  * 
│ │ │ │ -000550b0: 2a6e 6f74 6520 7261 6e64 6f6d 4879 7065  *note randomHype
│ │ │ │ -000550c0: 7247 7261 7068 3a20 7261 6e64 6f6d 4879  rGraph: randomHy
│ │ │ │ -000550d0: 7065 7247 7261 7068 2c20 2d2d 2072 6574  perGraph, -- ret
│ │ │ │ -000550e0: 7572 6e73 2061 2072 616e 646f 6d20 6879  urns a random hy
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│ │ │ │ -00055100: 6f6e 7320 7769 7468 206f 7074 696f 6e61  ons with optiona
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│ │ │ │ -00055130: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00055140: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00055150: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20  =============.. 
│ │ │ │ -00055160: 202a 202a 6e6f 7465 2072 616e 646f 6d48   * *note randomH
│ │ │ │ -00055170: 7970 6572 4772 6170 6828 2e2e 2e2c 5469  yperGraph(...,Ti
│ │ │ │ -00055180: 6d65 4c69 6d69 743d 3e2e 2e2e 293a 0a20  meLimit=>...):. 
│ │ │ │ -00055190: 2020 2072 616e 646f 6d48 7970 6572 4772     randomHyperGr
│ │ │ │ -000551a0: 6170 685f 6c70 5f70 645f 7064 5f70 645f  aph_lp_pd_pd_pd_
│ │ │ │ -000551b0: 636d 5469 6d65 4c69 6d69 743d 3e5f 7064  cmTimeLimit=>_pd
│ │ │ │ -000551c0: 5f70 645f 7064 5f72 702c 202d 2d20 6c69  _pd_pd_rp, -- li
│ │ │ │ -000551d0: 6d69 7420 7468 650a 2020 2020 7275 6e6e  mit the.    runn
│ │ │ │ -000551e0: 696e 6720 7469 6d65 206f 6620 7261 6e64  ing time of rand
│ │ │ │ -000551f0: 6f6d 4879 7065 7247 7261 7068 0a0a 4675  omHyperGraph..Fu
│ │ │ │ -00055200: 7274 6865 7220 696e 666f 726d 6174 696f  rther informatio
│ │ │ │ -00055210: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  n.==============
│ │ │ │ -00055220: 3d3d 3d3d 3d0a 0a20 202a 2044 6566 6175  =====..  * Defau
│ │ │ │ -00055230: 6c74 2076 616c 7565 3a20 350a 2020 2a20  lt value: 5.  * 
│ │ │ │ -00055240: 4675 6e63 7469 6f6e 3a20 2a6e 6f74 6520  Function: *note 
│ │ │ │ +00053e90: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00053ea0: 2020 2020 2020 2276 6572 7469 6365 7322        "vertices"
│ │ │ │ +00053eb0: 203d 3e20 7b78 202c 2078 202c 2078 202c   => {x , x , x ,
│ │ │ │ +00053ec0: 2078 202c 2078 207d 2020 2020 2020 2020   x , x }        
│ │ │ │ +00053ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053ee0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00053ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053f00: 2020 2020 2020 3120 2020 3220 2020 3320        1   2   3 
│ │ │ │ +00053f10: 2020 3420 2020 3520 2020 2020 2020 2020    4   5         
│ │ │ │ +00053f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053f30: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00053f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053f80: 2020 207c 0a7c 6f34 203a 2048 7970 6572     |.|o4 : Hyper
│ │ │ │ +00053f90: 4772 6170 6820 2020 2020 2020 2020 2020  Graph           
│ │ │ │ +00053fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053fd0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00053fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00053ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00054000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00054010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00054020: 2d2d 2d2b 0a7c 6935 203a 2072 616e 646f  ---+.|i5 : rando
│ │ │ │ +00054030: 6d48 7970 6572 4772 6170 6828 522c 7b34  mHyperGraph(R,{4
│ │ │ │ +00054040: 2c34 2c32 2c32 7d29 202d 2d20 696d 706f  ,4,2,2}) -- impo
│ │ │ │ +00054050: 7373 6962 6c65 2c20 7265 7475 726e 7320  ssible, returns 
│ │ │ │ +00054060: 6e75 6c6c 2077 6865 6e20 2020 2020 2020  null when       
│ │ │ │ +00054070: 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d     |.|----------
│ │ │ │ +00054080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00054090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000540a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000540b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000540c0: 2d2d 2d7c 0a7c 7469 6d65 2f62 7261 6e63  ---|.|time/branc
│ │ │ │ +000540d0: 6820 6c69 6d69 7420 7265 6163 6865 6420  h limit reached 
│ │ │ │ +000540e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000540f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00054100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00054110: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00054120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00054130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00054140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00054150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00054160: 2d2d 2d2b 0a0a 5468 6520 7261 6e64 6f6d  ---+..The random
│ │ │ │ +00054170: 4879 7065 7247 7261 7068 206d 6574 686f  HyperGraph metho
│ │ │ │ +00054180: 6420 7769 6c6c 2072 6574 7572 6e20 6e75  d will return nu
│ │ │ │ +00054190: 6c6c 2069 6d6d 6564 6961 7465 6c79 2069  ll immediately i
│ │ │ │ +000541a0: 6620 7468 6520 7369 7a65 7320 6f66 2074  f the sizes of t
│ │ │ │ +000541b0: 6865 0a65 6467 6573 2066 6169 6c20 746f  he.edges fail to
│ │ │ │ +000541c0: 2070 6173 7320 7468 6520 4c59 4d2d 696e   pass the LYM-in
│ │ │ │ +000541d0: 6571 7561 6c69 7479 3a20 2431 2f28 6e20  equality: $1/(n 
│ │ │ │ +000541e0: 6368 6f6f 7365 2044 5f31 2920 2b20 312f  choose D_1) + 1/
│ │ │ │ +000541f0: 286e 2063 686f 6f73 6520 445f 3229 202b  (n choose D_2) +
│ │ │ │ +00054200: 0a2e 2e2e 202b 2031 2f28 6e20 6368 6f6f  .... + 1/(n choo
│ │ │ │ +00054210: 7365 2044 5f6d 2920 5c6c 6571 2031 2420  se D_m) \leq 1$ 
│ │ │ │ +00054220: 7768 6572 6520 246e 2420 6973 2074 6865  where $n$ is the
│ │ │ │ +00054230: 206e 756d 6265 7220 6f66 2076 6172 6961   number of varia
│ │ │ │ +00054240: 626c 6573 2069 6e20 5220 616e 640a 246d  bles in R and.$m
│ │ │ │ +00054250: 2420 6973 2074 6865 206c 656e 6774 6820  $ is the length 
│ │ │ │ +00054260: 6f66 2044 2e20 4e6f 7465 2074 6861 7420  of D. Note that 
│ │ │ │ +00054270: 6576 656e 2069 6620 4420 7061 7373 6573  even if D passes
│ │ │ │ +00054280: 2074 6869 7320 696e 6571 7561 6c69 7479   this inequality
│ │ │ │ +00054290: 2c20 6974 2069 7320 6e6f 740a 6e65 6365  , it is not.nece
│ │ │ │ +000542a0: 7373 6172 696c 7920 7472 7565 2074 6861  ssarily true tha
│ │ │ │ +000542b0: 7420 7468 6572 6520 6973 2073 6f6d 6520  t there is some 
│ │ │ │ +000542c0: 6879 7065 7267 7261 7068 2077 6974 6820  hypergraph with 
│ │ │ │ +000542d0: 6564 6765 2073 697a 6573 2067 6976 656e  edge sizes given
│ │ │ │ +000542e0: 2062 7920 442e 2053 6565 0a44 2e20 4c75   by D. See.D. Lu
│ │ │ │ +000542f0: 6265 6c6c 2773 2022 4120 7368 6f72 7420  bell's "A short 
│ │ │ │ +00054300: 7072 6f6f 6620 6f66 2053 7065 726e 6572  proof of Sperner
│ │ │ │ +00054310: 2773 206c 656d 6d61 2c22 204a 2e20 436f  's lemma," J. Co
│ │ │ │ +00054320: 6d62 696e 2e20 5468 656f 7279 2c20 313a  mbin. Theory, 1:
│ │ │ │ +00054330: 3239 390a 2831 3936 3629 2e0a 0a53 6565  299.(1966)...See
│ │ │ │ +00054340: 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a   also.========..
│ │ │ │ +00054350: 2020 2a20 2a6e 6f74 6520 7261 6e64 6f6d    * *note random
│ │ │ │ +00054360: 4772 6170 683a 2072 616e 646f 6d47 7261  Graph: randomGra
│ │ │ │ +00054370: 7068 2c20 2d2d 2072 6574 7572 6e73 2061  ph, -- returns a
│ │ │ │ +00054380: 2072 616e 646f 6d20 6772 6170 680a 2020   random graph.  
│ │ │ │ +00054390: 2a20 2a6e 6f74 6520 7261 6e64 6f6d 556e  * *note randomUn
│ │ │ │ +000543a0: 6966 6f72 6d48 7970 6572 4772 6170 683a  iformHyperGraph:
│ │ │ │ +000543b0: 2072 616e 646f 6d55 6e69 666f 726d 4879   randomUniformHy
│ │ │ │ +000543c0: 7065 7247 7261 7068 2c20 2d2d 2072 6574  perGraph, -- ret
│ │ │ │ +000543d0: 7572 6e73 2061 2072 616e 646f 6d0a 2020  urns a random.  
│ │ │ │ +000543e0: 2020 756e 6966 6f72 6d20 6879 7065 7267    uniform hyperg
│ │ │ │ +000543f0: 7261 7068 0a20 202a 202a 6e6f 7465 2042  raph.  * *note B
│ │ │ │ +00054400: 7261 6e63 684c 696d 6974 3a20 4272 616e  ranchLimit: Bran
│ │ │ │ +00054410: 6368 4c69 6d69 742c 202d 2d20 6f70 7469  chLimit, -- opti
│ │ │ │ +00054420: 6f6e 616c 2061 7267 756d 656e 7420 666f  onal argument fo
│ │ │ │ +00054430: 7220 7261 6e64 6f6d 4879 7065 7247 7261  r randomHyperGra
│ │ │ │ +00054440: 7068 0a20 202a 202a 6e6f 7465 2054 696d  ph.  * *note Tim
│ │ │ │ +00054450: 654c 696d 6974 3a20 5469 6d65 4c69 6d69  eLimit: TimeLimi
│ │ │ │ +00054460: 742c 202d 2d20 6f70 7469 6f6e 616c 2061  t, -- optional a
│ │ │ │ +00054470: 7267 756d 656e 7420 666f 7220 7261 6e64  rgument for rand
│ │ │ │ +00054480: 6f6d 4879 7065 7247 7261 7068 0a20 202a  omHyperGraph.  *
│ │ │ │ +00054490: 202a 6e6f 7465 2043 6f6e 7374 7275 6374   *note Construct
│ │ │ │ +000544a0: 6f72 204f 7665 7276 6965 773a 2043 6f6e  or Overview: Con
│ │ │ │ +000544b0: 7374 7275 6374 6f72 204f 7665 7276 6965  structor Overvie
│ │ │ │ +000544c0: 772c 202d 2d20 6120 7375 6d6d 6172 7920  w, -- a summary 
│ │ │ │ +000544d0: 6f66 2074 6865 206d 616e 790a 2020 2020  of the many.    
│ │ │ │ +000544e0: 7761 7973 206f 6620 6d61 6b69 6e67 2067  ways of making g
│ │ │ │ +000544f0: 7261 7068 7320 616e 6420 6879 7065 7267  raphs and hyperg
│ │ │ │ +00054500: 7261 7068 730a 0a57 6179 7320 746f 2075  raphs..Ways to u
│ │ │ │ +00054510: 7365 2072 616e 646f 6d48 7970 6572 4772  se randomHyperGr
│ │ │ │ +00054520: 6170 683a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  aph:.===========
│ │ │ │ +00054530: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00054540: 3d3d 0a0a 2020 2a20 2272 616e 646f 6d48  ==..  * "randomH
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│ │ │ │ +00054580: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  mer.============
│ │ │ │ +00054590: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65  ======..The obje
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│ │ │ │ +000545b0: 7970 6572 4772 6170 683a 2072 616e 646f  yperGraph: rando
│ │ │ │ +000545c0: 6d48 7970 6572 4772 6170 682c 2069 7320  mHyperGraph, is 
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│ │ │ │ +00054630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00054640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00054650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00054660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00054720: 705f 7064 5f70 645f 7064 5f63 6d42 7261  p_pd_pd_pd_cmBra
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│ │ │ │ +00054740: 5f70 645f 7270 2c20 4e65 7874 3a20 7261  _pd_rp, Next: ra
│ │ │ │ +00054750: 6e64 6f6d 4879 7065 7247 7261 7068 5f6c  ndomHyperGraph_l
│ │ │ │ +00054760: 705f 7064 5f70 645f 7064 5f63 6d54 696d  p_pd_pd_pd_cmTim
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│ │ │ │ +00054790: 6f6d 4879 7065 7247 7261 7068 2c20 5570  omHyperGraph, Up
│ │ │ │ +000547a0: 3a20 546f 700a 0a72 616e 646f 6d48 7970  : Top..randomHyp
│ │ │ │ +000547b0: 6572 4772 6170 6828 2e2e 2e2c 4272 616e  erGraph(...,Bran
│ │ │ │ +000547c0: 6368 4c69 6d69 743d 3e2e 2e2e 2920 2d2d  chLimit=>...) --
│ │ │ │ +000547d0: 206c 696d 6974 2072 6563 7572 7369 7665   limit recursive
│ │ │ │ +000547e0: 2062 7261 6e63 6869 6e67 2069 6e20 7261   branching in ra
│ │ │ │ +000547f0: 6e64 6f6d 4879 7065 7247 7261 7068 0a2a  ndomHyperGraph.*
│ │ │ │ +00054800: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00054810: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00054820: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00054830: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00054840: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00054850: 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 6167  ******..  * Usag
│ │ │ │ +00054860: 653a 200a 2020 2020 2020 2020 4720 3d20  e: .        G = 
│ │ │ │ +00054870: 7261 6e64 6f6d 4879 7065 7247 7261 7068  randomHyperGraph
│ │ │ │ +00054880: 2852 2c44 2c42 7261 6e63 684c 696d 6974  (R,D,BranchLimit
│ │ │ │ +00054890: 3d3e 6e29 0a20 202a 2049 6e70 7574 733a  =>n).  * Inputs:
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│ │ │ │ +000548c0: 6163 6175 6c61 7932 446f 6329 5a5a 2c2c  acaulay2Doc)ZZ,,
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│ │ │ │ +000548f0: 2020 2020 6c65 7665 6c20 696e 2074 6865      level in the
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│ │ │ │ +00054910: 6e64 6f6d 4879 7065 7247 7261 7068 0a0a  ndomHyperGraph..
│ │ │ │ +00054920: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
│ │ │ │ +00054930: 3d3d 3d3d 3d3d 3d0a 0a54 6865 202a 6e6f  =======..The *no
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│ │ │ │ +00054950: 6170 683a 2072 616e 646f 6d48 7970 6572  aph: randomHyper
│ │ │ │ +00054960: 4772 6170 682c 206d 6574 686f 6420 666f  Graph, method fo
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│ │ │ │ +00054980: 6b69 6e67 0a61 6c67 6f72 6974 686d 2074  king.algorithm t
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│ │ │ │ +000549c0: 2e20 4174 2065 6163 6820 7374 6570 2069  . At each step i
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│ │ │ │ +000549e0: 7472 6565 2c20 202a 6e6f 7465 2072 616e  tree,  *note ran
│ │ │ │ +000549f0: 646f 6d48 7970 6572 4772 6170 683a 2072  domHyperGraph: r
│ │ │ │ +00054a00: 616e 646f 6d48 7970 6572 4772 6170 682c  andomHyperGraph,
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│ │ │ │ +00054a30: 746f 2063 6f6d 706c 6574 6520 6974 7320  to complete its 
│ │ │ │ +00054a40: 6c69 7374 206f 6620 6564 6765 732e 2054  list of edges. T
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│ │ │ │ +00054a70: 7320 6973 2072 6571 7569 7265 642c 202a  s is required, *
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│ │ │ │ +00054a90: 4772 6170 683a 2072 616e 646f 6d48 7970  Graph: randomHyp
│ │ │ │ +00054aa0: 6572 4772 6170 682c 206d 6179 2074 616b  erGraph, may tak
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│ │ │ │ +00054ac0: 7374 6570 7320 6265 666f 7265 2074 6572  steps before ter
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│ │ │ │ +00054ae0: 6d6f 7265 2070 7265 6369 7365 2c20 7468  more precise, th
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│ │ │ │ +00054b00: 656d 656e 7465 6420 736f 2074 6861 7420  emented so that 
│ │ │ │ +00054b10: 6974 206d 616b 6573 2042 7261 6e63 684c  it makes BranchL
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│ │ │ │ +00054b40: 7468 650a 7265 6375 7273 696f 6e2e 2054  the.recursion. T
│ │ │ │ +00054b50: 6865 2064 6566 6175 6c74 2076 616c 7565  he default value
│ │ │ │ +00054b60: 2069 7320 332e 0a0a 5365 6520 616c 736f   is 3...See also
│ │ │ │ +00054b70: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a  .========..  * *
│ │ │ │ +00054b80: 6e6f 7465 2054 696d 654c 696d 6974 3a20  note TimeLimit: 
│ │ │ │ +00054b90: 5469 6d65 4c69 6d69 742c 202d 2d20 6f70  TimeLimit, -- op
│ │ │ │ +00054ba0: 7469 6f6e 616c 2061 7267 756d 656e 7420  tional argument 
│ │ │ │ +00054bb0: 666f 7220 7261 6e64 6f6d 4879 7065 7247  for randomHyperG
│ │ │ │ +00054bc0: 7261 7068 0a20 202a 202a 6e6f 7465 2072  raph.  * *note r
│ │ │ │ +00054bd0: 616e 646f 6d48 7970 6572 4772 6170 683a  andomHyperGraph:
│ │ │ │ +00054be0: 2072 616e 646f 6d48 7970 6572 4772 6170   randomHyperGrap
│ │ │ │ +00054bf0: 682c 202d 2d20 7265 7475 726e 7320 6120  h, -- returns a 
│ │ │ │ +00054c00: 7261 6e64 6f6d 2068 7970 6572 6772 6170  random hypergrap
│ │ │ │ +00054c10: 680a 0a46 756e 6374 696f 6e73 2077 6974  h..Functions wit
│ │ │ │ +00054c20: 6820 6f70 7469 6f6e 616c 2061 7267 756d  h optional argum
│ │ │ │ +00054c30: 656e 7420 6e61 6d65 6420 4272 616e 6368  ent named Branch
│ │ │ │ +00054c40: 4c69 6d69 743a 0a3d 3d3d 3d3d 3d3d 3d3d  Limit:.=========
│ │ │ │ +00054c50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00054c60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00054c70: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ +00054c80: 2a6e 6f74 6520 7261 6e64 6f6d 4879 7065  *note randomHype
│ │ │ │ +00054c90: 7247 7261 7068 282e 2e2e 2c42 7261 6e63  rGraph(...,Branc
│ │ │ │ +00054ca0: 684c 696d 6974 3d3e 2e2e 2e29 3a0a 2020  hLimit=>...):.  
│ │ │ │ +00054cb0: 2020 7261 6e64 6f6d 4879 7065 7247 7261    randomHyperGra
│ │ │ │ +00054cc0: 7068 5f6c 705f 7064 5f70 645f 7064 5f63  ph_lp_pd_pd_pd_c
│ │ │ │ +00054cd0: 6d42 7261 6e63 684c 696d 6974 3d3e 5f70  mBranchLimit=>_p
│ │ │ │ +00054ce0: 645f 7064 5f70 645f 7270 2c20 2d2d 206c  d_pd_pd_rp, -- l
│ │ │ │ +00054cf0: 696d 6974 0a20 2020 2072 6563 7572 7369  imit.    recursi
│ │ │ │ +00054d00: 7665 2062 7261 6e63 6869 6e67 2069 6e20  ve branching in 
│ │ │ │ +00054d10: 7261 6e64 6f6d 4879 7065 7247 7261 7068  randomHyperGraph
│ │ │ │ +00054d20: 0a0a 4675 7274 6865 7220 696e 666f 726d  ..Further inform
│ │ │ │ +00054d30: 6174 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ation.==========
│ │ │ │ +00054d40: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2044  =========..  * D
│ │ │ │ +00054d50: 6566 6175 6c74 2076 616c 7565 3a20 330a  efault value: 3.
│ │ │ │ +00054d60: 2020 2a20 4675 6e63 7469 6f6e 3a20 2a6e    * Function: *n
│ │ │ │ +00054d70: 6f74 6520 7261 6e64 6f6d 4879 7065 7247  ote randomHyperG
│ │ │ │ +00054d80: 7261 7068 3a20 7261 6e64 6f6d 4879 7065  raph: randomHype
│ │ │ │ +00054d90: 7247 7261 7068 2c20 2d2d 2072 6574 7572  rGraph, -- retur
│ │ │ │ +00054da0: 6e73 2061 2072 616e 646f 6d0a 2020 2020  ns a random.    
│ │ │ │ +00054db0: 6879 7065 7267 7261 7068 0a20 202a 204f  hypergraph.  * O
│ │ │ │ +00054dc0: 7074 696f 6e20 6b65 793a 202a 6e6f 7465  ption key: *note
│ │ │ │ +00054dd0: 2042 7261 6e63 684c 696d 6974 3a20 4272   BranchLimit: Br
│ │ │ │ +00054de0: 616e 6368 4c69 6d69 742c 202d 2d20 6f70  anchLimit, -- op
│ │ │ │ +00054df0: 7469 6f6e 616c 2061 7267 756d 656e 7420  tional argument 
│ │ │ │ +00054e00: 666f 720a 2020 2020 7261 6e64 6f6d 4879  for.    randomHy
│ │ │ │ +00054e10: 7065 7247 7261 7068 0a2d 2d2d 2d2d 2d2d  perGraph.-------
│ │ │ │ +00054e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00054e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00054e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00054e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00054e60: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f  --------..The so
│ │ │ │ +00054e70: 7572 6365 206f 6620 7468 6973 2064 6f63  urce of this doc
│ │ │ │ +00054e80: 756d 656e 7420 6973 2069 6e0a 2f62 7569  ument is in./bui
│ │ │ │ +00054e90: 6c64 2f72 6570 726f 6475 6369 626c 652d  ld/reproducible-
│ │ │ │ +00054ea0: 7061 7468 2f6d 6163 6175 6c61 7932 2d31  path/macaulay2-1
│ │ │ │ +00054eb0: 2e32 352e 3131 2b64 732f 4d32 2f4d 6163  .25.11+ds/M2/Mac
│ │ │ │ +00054ec0: 6175 6c61 7932 2f70 6163 6b61 6765 732f  aulay2/packages/
│ │ │ │ +00054ed0: 4564 6765 4964 6561 6c73 2e0a 6d32 3a34  EdgeIdeals..m2:4
│ │ │ │ +00054ee0: 3236 363a 302e 0a1f 0a46 696c 653a 2045  266:0....File: E
│ │ │ │ +00054ef0: 6467 6549 6465 616c 732e 696e 666f 2c20  dgeIdeals.info, 
│ │ │ │ +00054f00: 4e6f 6465 3a20 7261 6e64 6f6d 4879 7065  Node: randomHype
│ │ │ │ +00054f10: 7247 7261 7068 5f6c 705f 7064 5f70 645f  rGraph_lp_pd_pd_
│ │ │ │ +00054f20: 7064 5f63 6d54 696d 654c 696d 6974 3d3e  pd_cmTimeLimit=>
│ │ │ │ +00054f30: 5f70 645f 7064 5f70 645f 7270 2c20 4e65  _pd_pd_pd_rp, Ne
│ │ │ │ +00054f40: 7874 3a20 7261 6e64 6f6d 556e 6966 6f72  xt: randomUnifor
│ │ │ │ +00054f50: 6d48 7970 6572 4772 6170 682c 2050 7265  mHyperGraph, Pre
│ │ │ │ +00054f60: 763a 2072 616e 646f 6d48 7970 6572 4772  v: randomHyperGr
│ │ │ │ +00054f70: 6170 685f 6c70 5f70 645f 7064 5f70 645f  aph_lp_pd_pd_pd_
│ │ │ │ +00054f80: 636d 4272 616e 6368 4c69 6d69 743d 3e5f  cmBranchLimit=>_
│ │ │ │ +00054f90: 7064 5f70 645f 7064 5f72 702c 2055 703a  pd_pd_pd_rp, Up:
│ │ │ │ +00054fa0: 2054 6f70 0a0a 7261 6e64 6f6d 4879 7065   Top..randomHype
│ │ │ │ +00054fb0: 7247 7261 7068 282e 2e2e 2c54 696d 654c  rGraph(...,TimeL
│ │ │ │ +00054fc0: 696d 6974 3d3e 2e2e 2e29 202d 2d20 6c69  imit=>...) -- li
│ │ │ │ +00054fd0: 6d69 7420 7468 6520 7275 6e6e 696e 6720  mit the running 
│ │ │ │ +00054fe0: 7469 6d65 206f 6620 7261 6e64 6f6d 4879  time of randomHy
│ │ │ │ +00054ff0: 7065 7247 7261 7068 0a2a 2a2a 2a2a 2a2a  perGraph.*******
│ │ │ │ +00055000: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00055010: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00055020: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00055030: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00055040: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a  ***********..  *
│ │ │ │ +00055050: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ +00055060: 2072 616e 646f 6d48 7970 6572 4772 6170   randomHyperGrap
│ │ │ │ +00055070: 6828 522c 442c 5469 6d65 4c69 6d69 743d  h(R,D,TimeLimit=
│ │ │ │ +00055080: 3e73 290a 2020 2a20 496e 7075 7473 3a0a  >s).  * Inputs:.
│ │ │ │ +00055090: 2020 2020 2020 2a20 732c 2061 6e20 2a6e        * s, an *n
│ │ │ │ +000550a0: 6f74 6520 696e 7465 6765 723a 2028 4d61  ote integer: (Ma
│ │ │ │ +000550b0: 6361 756c 6179 3244 6f63 295a 5a2c 2c20  caulay2Doc)ZZ,, 
│ │ │ │ +000550c0: 7469 6d65 2069 6e20 7365 636f 6e64 7320  time in seconds 
│ │ │ │ +000550d0: 6265 666f 7265 0a20 2020 2020 2020 2072  before.        r
│ │ │ │ +000550e0: 616e 646f 6d48 7970 6572 4772 6170 6820  andomHyperGraph 
│ │ │ │ +000550f0: 7265 7475 726e 7320 6e75 6c6c 0a0a 4465  returns null..De
│ │ │ │ +00055100: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d  scription.======
│ │ │ │ +00055110: 3d3d 3d3d 3d0a 0a54 6865 202a 6e6f 7465  =====..The *note
│ │ │ │ +00055120: 2072 616e 646f 6d48 7970 6572 4772 6170   randomHyperGrap
│ │ │ │ +00055130: 683a 2072 616e 646f 6d48 7970 6572 4772  h: randomHyperGr
│ │ │ │ +00055140: 6170 682c 206d 6574 686f 6420 666f 6c6c  aph, method foll
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│ │ │ │ +00055170: 6765 6e65 7261 7465 2065 6467 6573 2077  generate edges w
│ │ │ │ +00055180: 6974 6820 6e6f 2069 6e63 6c75 7369 6f6e  ith no inclusion
│ │ │ │ +00055190: 7320 6265 7477 6565 6e20 7468 656d 2e20  s between them. 
│ │ │ │ +000551a0: 4173 2069 7420 6d61 7920 7461 6b65 2061  As it may take a
│ │ │ │ +000551b0: 0a6c 6f6e 6720 7469 6d65 2074 6f20 7465  .long time to te
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│ │ │ │ +000551d0: 6c69 6d69 7420 6973 2069 6d70 6f73 6564  limit is imposed
│ │ │ │ +000551e0: 206f 6e20 7468 6520 6d65 7468 6f64 2e20   on the method. 
│ │ │ │ +000551f0: 5468 6520 5469 6d65 4c69 6d69 740a 6f70  The TimeLimit.op
│ │ │ │ +00055200: 7469 6f6e 2069 7320 7468 6520 616d 6f75  tion is the amou
│ │ │ │ +00055210: 6e74 206f 6620 7469 6d65 2c20 696e 2073  nt of time, in s
│ │ │ │ +00055220: 6563 6f6e 6473 2c20 7468 6174 2074 6865  econds, that the
│ │ │ │ +00055230: 206d 6574 686f 6420 2a6e 6f74 650a 7261   method *note.ra
│ │ │ │ +00055240: 6e64 6f6d 4879 7065 7247 7261 7068 3a20  ndomHyperGraph: 
│ │ │ │  00055250: 7261 6e64 6f6d 4879 7065 7247 7261 7068  randomHyperGraph
│ │ │ │ -00055260: 3a20 7261 6e64 6f6d 4879 7065 7247 7261  : randomHyperGra
│ │ │ │ -00055270: 7068 2c20 2d2d 2072 6574 7572 6e73 2061  ph, -- returns a
│ │ │ │ -00055280: 2072 616e 646f 6d0a 2020 2020 6879 7065   random.    hype
│ │ │ │ -00055290: 7267 7261 7068 0a20 202a 204f 7074 696f  rgraph.  * Optio
│ │ │ │ -000552a0: 6e20 6b65 793a 202a 6e6f 7465 2054 696d  n key: *note Tim
│ │ │ │ -000552b0: 654c 696d 6974 3a20 5469 6d65 4c69 6d69  eLimit: TimeLimi
│ │ │ │ -000552c0: 742c 202d 2d20 6f70 7469 6f6e 616c 2061  t, -- optional a
│ │ │ │ -000552d0: 7267 756d 656e 7420 666f 720a 2020 2020  rgument for.    
│ │ │ │ -000552e0: 7261 6e64 6f6d 4879 7065 7247 7261 7068  randomHyperGraph
│ │ │ │ -000552f0: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .---------------
│ │ │ │ -00055300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00055310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00055320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00055330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00055340: 0a0a 5468 6520 736f 7572 6365 206f 6620  ..The source of 
│ │ │ │ -00055350: 7468 6973 2064 6f63 756d 656e 7420 6973  this document is
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│ │ │ │ -00055370: 6475 6369 626c 652d 7061 7468 2f6d 6163  ducible-path/mac
│ │ │ │ -00055380: 6175 6c61 7932 2d31 2e32 352e 3131 2b64  aulay2-1.25.11+d
│ │ │ │ -00055390: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70  s/M2/Macaulay2/p
│ │ │ │ -000553a0: 6163 6b61 6765 732f 4564 6765 4964 6561  ackages/EdgeIdea
│ │ │ │ -000553b0: 6c73 2e0a 6d32 3a34 3339 313a 302e 0a1f  ls..m2:4391:0...
│ │ │ │ -000553c0: 0a46 696c 653a 2045 6467 6549 6465 616c  .File: EdgeIdeal
│ │ │ │ -000553d0: 732e 696e 666f 2c20 4e6f 6465 3a20 7261  s.info, Node: ra
│ │ │ │ -000553e0: 6e64 6f6d 556e 6966 6f72 6d48 7970 6572  ndomUniformHyper
│ │ │ │ -000553f0: 4772 6170 682c 204e 6578 743a 2072 696e  Graph, Next: rin
│ │ │ │ -00055400: 675f 6c70 4879 7065 7247 7261 7068 5f72  g_lpHyperGraph_r
│ │ │ │ -00055410: 702c 2050 7265 763a 2072 616e 646f 6d48  p, Prev: randomH
│ │ │ │ -00055420: 7970 6572 4772 6170 685f 6c70 5f70 645f  yperGraph_lp_pd_
│ │ │ │ -00055430: 7064 5f70 645f 636d 5469 6d65 4c69 6d69  pd_pd_cmTimeLimi
│ │ │ │ -00055440: 743d 3e5f 7064 5f70 645f 7064 5f72 702c  t=>_pd_pd_pd_rp,
│ │ │ │ -00055450: 2055 703a 2054 6f70 0a0a 7261 6e64 6f6d   Up: Top..random
│ │ │ │ -00055460: 556e 6966 6f72 6d48 7970 6572 4772 6170  UniformHyperGrap
│ │ │ │ -00055470: 6820 2d2d 2072 6574 7572 6e73 2061 2072  h -- returns a r
│ │ │ │ -00055480: 616e 646f 6d20 756e 6966 6f72 6d20 6879  andom uniform hy
│ │ │ │ -00055490: 7065 7267 7261 7068 0a2a 2a2a 2a2a 2a2a  pergraph.*******
│ │ │ │ -000554a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000554b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000554c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000554d0: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 7361  *******..  * Usa
│ │ │ │ -000554e0: 6765 3a20 0a20 2020 2020 2020 2048 203d  ge: .        H =
│ │ │ │ -000554f0: 2072 616e 646f 6d55 6e69 666f 726d 4879   randomUniformHy
│ │ │ │ -00055500: 7065 7247 7261 7068 2852 2c63 2c64 290a  perGraph(R,c,d).
│ │ │ │ -00055510: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ -00055520: 2020 2a20 522c 2061 202a 6e6f 7465 2070    * R, a *note p
│ │ │ │ -00055530: 6f6c 796e 6f6d 6961 6c20 7269 6e67 3a20  olynomial ring: 
│ │ │ │ -00055540: 284d 6163 6175 6c61 7932 446f 6329 506f  (Macaulay2Doc)Po
│ │ │ │ -00055550: 6c79 6e6f 6d69 616c 5269 6e67 2c2c 2077  lynomialRing,, w
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│ │ │ │ -00055570: 2020 2074 6865 2076 6572 7465 7820 7365     the vertex se
│ │ │ │ -00055580: 7420 6f66 2048 0a20 2020 2020 202a 2063  t of H.      * c
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│ │ │ │ -000555b0: 6329 5a5a 2c2c 2074 6865 2063 6172 6469  c)ZZ,, the cardi
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│ │ │ │ -000555e0: 2020 2020 2020 2a20 642c 2061 6e20 2a6e        * d, an *n
│ │ │ │ -000555f0: 6f74 6520 696e 7465 6765 723a 2028 4d61  ote integer: (Ma
│ │ │ │ -00055600: 6361 756c 6179 3244 6f63 295a 5a2c 2c20  caulay2Doc)ZZ,, 
│ │ │ │ -00055610: 7468 6520 6e75 6d62 6572 206f 6620 6564  the number of ed
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│ │ │ │ -00055630: 7075 7473 3a0a 2020 2020 2020 2a20 482c  puts:.      * H,
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│ │ │ │ -000556b0: 2052 0a0a 4465 7363 7269 7074 696f 6e0a   R..Description.
│ │ │ │ -000556c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6869  ===========..Thi
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│ │ │ │ -000556f0: 6120 756e 6966 6f72 6d20 6879 7065 7267  a uniform hyperg
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│ │ │ │ -00055770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00055780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00055790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000557a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000557b0: 2d2d 2d2b 0a7c 6931 203a 2052 203d 2051  ---+.|i1 : R = Q
│ │ │ │ -000557c0: 515b 785f 312e 2e78 5f39 5d3b 2020 2020  Q[x_1..x_9];    
│ │ │ │ -000557d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000557e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000557f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055800: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -00055810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00055820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00055830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00055840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00055850: 2d2d 2d2b 0a7c 6932 203a 2072 616e 646f  ---+.|i2 : rando
│ │ │ │ -00055860: 6d55 6e69 666f 726d 4879 7065 7247 7261  mUniformHyperGra
│ │ │ │ -00055870: 7068 2852 2c33 2c34 2920 2020 2020 2020  ph(R,3,4)       
│ │ │ │ -00055880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000558a0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -000558b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000558c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000558d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000558e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000558f0: 2020 207c 0a7c 6f32 203d 2048 7970 6572     |.|o2 = Hyper
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│ │ │ │ -00055920: 7b78 202c 2078 202c 2078 207d 2c20 7b78  {x , x , x }, {x
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│ │ │ │ -00055980: 3420 2020 3720 2020 3820 2020 2020 3320  4   7   8     3 
│ │ │ │ -00055990: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -000559a0: 2020 2020 2020 2272 696e 6722 203d 3e20        "ring" => 
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│ │ │ │ -000559d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000559e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -000559f0: 2020 2020 2020 2276 6572 7469 6365 7322        "vertices"
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│ │ │ │ -00055a20: 2078 202c 2078 207d 2020 2020 2020 2020   x , x }        
│ │ │ │ -00055a30: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00055a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055a50: 2020 2020 2020 3120 2020 3220 2020 3320        1   2   3 
│ │ │ │ -00055a60: 2020 3420 2020 3520 2020 3620 2020 3720    4   5   6   7 
│ │ │ │ -00055a70: 2020 3820 2020 3920 2020 2020 2020 2020    8   9         
│ │ │ │ -00055a80: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00055a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055ad0: 2020 207c 0a7c 6f32 203a 2048 7970 6572     |.|o2 : Hyper
│ │ │ │ -00055ae0: 4772 6170 6820 2020 2020 2020 2020 2020  Graph           
│ │ │ │ -00055af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00055260: 2c20 7769 6c6c 2074 616b 6520 6265 666f  , will take befo
│ │ │ │ +00055270: 7265 2074 6572 6d69 6e61 7469 6e67 2e20  re terminating. 
│ │ │ │ +00055280: 4120 7661 6c75 6520 6f66 2030 0a69 7320  A value of 0.is 
│ │ │ │ +00055290: 696e 7465 7270 7265 7465 6420 6173 206f  interpreted as o
│ │ │ │ +000552a0: 6e65 2064 6179 2e20 5468 6520 6465 6661  ne day. The defa
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│ │ │ │ +000552d0: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020  lso.========..  
│ │ │ │ +000552e0: 2a20 2a6e 6f74 6520 4272 616e 6368 4c69  * *note BranchLi
│ │ │ │ +000552f0: 6d69 743a 2042 7261 6e63 684c 696d 6974  mit: BranchLimit
│ │ │ │ +00055300: 2c20 2d2d 206f 7074 696f 6e61 6c20 6172  , -- optional ar
│ │ │ │ +00055310: 6775 6d65 6e74 2066 6f72 2072 616e 646f  gument for rando
│ │ │ │ +00055320: 6d48 7970 6572 4772 6170 680a 2020 2a20  mHyperGraph.  * 
│ │ │ │ +00055330: 2a6e 6f74 6520 7261 6e64 6f6d 4879 7065  *note randomHype
│ │ │ │ +00055340: 7247 7261 7068 3a20 7261 6e64 6f6d 4879  rGraph: randomHy
│ │ │ │ +00055350: 7065 7247 7261 7068 2c20 2d2d 2072 6574  perGraph, -- ret
│ │ │ │ +00055360: 7572 6e73 2061 2072 616e 646f 6d20 6879  urns a random hy
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│ │ │ │ +00055380: 6f6e 7320 7769 7468 206f 7074 696f 6e61  ons with optiona
│ │ │ │ +00055390: 6c20 6172 6775 6d65 6e74 206e 616d 6564  l argument named
│ │ │ │ +000553a0: 2054 696d 654c 696d 6974 3a0a 3d3d 3d3d   TimeLimit:.====
│ │ │ │ +000553b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +000553c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +000553d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20  =============.. 
│ │ │ │ +000553e0: 202a 202a 6e6f 7465 2072 616e 646f 6d48   * *note randomH
│ │ │ │ +000553f0: 7970 6572 4772 6170 6828 2e2e 2e2c 5469  yperGraph(...,Ti
│ │ │ │ +00055400: 6d65 4c69 6d69 743d 3e2e 2e2e 293a 0a20  meLimit=>...):. 
│ │ │ │ +00055410: 2020 2072 616e 646f 6d48 7970 6572 4772     randomHyperGr
│ │ │ │ +00055420: 6170 685f 6c70 5f70 645f 7064 5f70 645f  aph_lp_pd_pd_pd_
│ │ │ │ +00055430: 636d 5469 6d65 4c69 6d69 743d 3e5f 7064  cmTimeLimit=>_pd
│ │ │ │ +00055440: 5f70 645f 7064 5f72 702c 202d 2d20 6c69  _pd_pd_rp, -- li
│ │ │ │ +00055450: 6d69 7420 7468 650a 2020 2020 7275 6e6e  mit the.    runn
│ │ │ │ +00055460: 696e 6720 7469 6d65 206f 6620 7261 6e64  ing time of rand
│ │ │ │ +00055470: 6f6d 4879 7065 7247 7261 7068 0a0a 4675  omHyperGraph..Fu
│ │ │ │ +00055480: 7274 6865 7220 696e 666f 726d 6174 696f  rther informatio
│ │ │ │ +00055490: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  n.==============
│ │ │ │ +000554a0: 3d3d 3d3d 3d0a 0a20 202a 2044 6566 6175  =====..  * Defau
│ │ │ │ +000554b0: 6c74 2076 616c 7565 3a20 350a 2020 2a20  lt value: 5.  * 
│ │ │ │ +000554c0: 4675 6e63 7469 6f6e 3a20 2a6e 6f74 6520  Function: *note 
│ │ │ │ +000554d0: 7261 6e64 6f6d 4879 7065 7247 7261 7068  randomHyperGraph
│ │ │ │ +000554e0: 3a20 7261 6e64 6f6d 4879 7065 7247 7261  : randomHyperGra
│ │ │ │ +000554f0: 7068 2c20 2d2d 2072 6574 7572 6e73 2061  ph, -- returns a
│ │ │ │ +00055500: 2072 616e 646f 6d0a 2020 2020 6879 7065   random.    hype
│ │ │ │ +00055510: 7267 7261 7068 0a20 202a 204f 7074 696f  rgraph.  * Optio
│ │ │ │ +00055520: 6e20 6b65 793a 202a 6e6f 7465 2054 696d  n key: *note Tim
│ │ │ │ +00055530: 654c 696d 6974 3a20 5469 6d65 4c69 6d69  eLimit: TimeLimi
│ │ │ │ +00055540: 742c 202d 2d20 6f70 7469 6f6e 616c 2061  t, -- optional a
│ │ │ │ +00055550: 7267 756d 656e 7420 666f 720a 2020 2020  rgument for.    
│ │ │ │ +00055560: 7261 6e64 6f6d 4879 7065 7247 7261 7068  randomHyperGraph
│ │ │ │ +00055570: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .---------------
│ │ │ │ +00055580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00055590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000555a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000555b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000555c0: 0a0a 5468 6520 736f 7572 6365 206f 6620  ..The source of 
│ │ │ │ +000555d0: 7468 6973 2064 6f63 756d 656e 7420 6973  this document is
│ │ │ │ +000555e0: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f   in./build/repro
│ │ │ │ +000555f0: 6475 6369 626c 652d 7061 7468 2f6d 6163  ducible-path/mac
│ │ │ │ +00055600: 6175 6c61 7932 2d31 2e32 352e 3131 2b64  aulay2-1.25.11+d
│ │ │ │ +00055610: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70  s/M2/Macaulay2/p
│ │ │ │ +00055620: 6163 6b61 6765 732f 4564 6765 4964 6561  ackages/EdgeIdea
│ │ │ │ +00055630: 6c73 2e0a 6d32 3a34 3339 313a 302e 0a1f  ls..m2:4391:0...
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│ │ │ │ +00055650: 732e 696e 666f 2c20 4e6f 6465 3a20 7261  s.info, Node: ra
│ │ │ │ +00055660: 6e64 6f6d 556e 6966 6f72 6d48 7970 6572  ndomUniformHyper
│ │ │ │ +00055670: 4772 6170 682c 204e 6578 743a 2072 696e  Graph, Next: rin
│ │ │ │ +00055680: 675f 6c70 4879 7065 7247 7261 7068 5f72  g_lpHyperGraph_r
│ │ │ │ +00055690: 702c 2050 7265 763a 2072 616e 646f 6d48  p, Prev: randomH
│ │ │ │ +000556a0: 7970 6572 4772 6170 685f 6c70 5f70 645f  yperGraph_lp_pd_
│ │ │ │ +000556b0: 7064 5f70 645f 636d 5469 6d65 4c69 6d69  pd_pd_cmTimeLimi
│ │ │ │ +000556c0: 743d 3e5f 7064 5f70 645f 7064 5f72 702c  t=>_pd_pd_pd_rp,
│ │ │ │ +000556d0: 2055 703a 2054 6f70 0a0a 7261 6e64 6f6d   Up: Top..random
│ │ │ │ +000556e0: 556e 6966 6f72 6d48 7970 6572 4772 6170  UniformHyperGrap
│ │ │ │ +000556f0: 6820 2d2d 2072 6574 7572 6e73 2061 2072  h -- returns a r
│ │ │ │ +00055700: 616e 646f 6d20 756e 6966 6f72 6d20 6879  andom uniform hy
│ │ │ │ +00055710: 7065 7267 7261 7068 0a2a 2a2a 2a2a 2a2a  pergraph.*******
│ │ │ │ +00055720: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00055730: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00055740: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00055750: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 7361  *******..  * Usa
│ │ │ │ +00055760: 6765 3a20 0a20 2020 2020 2020 2048 203d  ge: .        H =
│ │ │ │ +00055770: 2072 616e 646f 6d55 6e69 666f 726d 4879   randomUniformHy
│ │ │ │ +00055780: 7065 7247 7261 7068 2852 2c63 2c64 290a  perGraph(R,c,d).
│ │ │ │ +00055790: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ +000557a0: 2020 2a20 522c 2061 202a 6e6f 7465 2070    * R, a *note p
│ │ │ │ +000557b0: 6f6c 796e 6f6d 6961 6c20 7269 6e67 3a20  olynomial ring: 
│ │ │ │ +000557c0: 284d 6163 6175 6c61 7932 446f 6329 506f  (Macaulay2Doc)Po
│ │ │ │ +000557d0: 6c79 6e6f 6d69 616c 5269 6e67 2c2c 2077  lynomialRing,, w
│ │ │ │ +000557e0: 6869 6368 2067 6976 6573 0a20 2020 2020  hich gives.     
│ │ │ │ +000557f0: 2020 2074 6865 2076 6572 7465 7820 7365     the vertex se
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│ │ │ │ +00055810: 2c20 616e 202a 6e6f 7465 2069 6e74 6567  , an *note integ
│ │ │ │ +00055820: 6572 3a20 284d 6163 6175 6c61 7932 446f  er: (Macaulay2Do
│ │ │ │ +00055830: 6329 5a5a 2c2c 2074 6865 2063 6172 6469  c)ZZ,, the cardi
│ │ │ │ +00055840: 6e61 6c69 7479 206f 6620 7468 6520 6564  nality of the ed
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│ │ │ │ +00055860: 2020 2020 2020 2a20 642c 2061 6e20 2a6e        * d, an *n
│ │ │ │ +00055870: 6f74 6520 696e 7465 6765 723a 2028 4d61  ote integer: (Ma
│ │ │ │ +00055880: 6361 756c 6179 3244 6f63 295a 5a2c 2c20  caulay2Doc)ZZ,, 
│ │ │ │ +00055890: 7468 6520 6e75 6d62 6572 206f 6620 6564  the number of ed
│ │ │ │ +000558a0: 6765 7320 696e 2048 0a20 202a 204f 7574  ges in H.  * Out
│ │ │ │ +000558b0: 7075 7473 3a0a 2020 2020 2020 2a20 482c  puts:.      * H,
│ │ │ │ +000558c0: 2061 202a 6e6f 7465 2068 7970 6572 6772   a *note hypergr
│ │ │ │ +000558d0: 6170 683a 2048 7970 6572 4772 6170 682c  aph: HyperGraph,
│ │ │ │ +000558e0: 2c20 6120 6879 7065 7267 7261 7068 2077  , a hypergraph w
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│ │ │ │ +00055900: 2020 2020 2020 2063 6172 6469 6e61 6c69         cardinali
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│ │ │ │ +00055930: 2052 0a0a 4465 7363 7269 7074 696f 6e0a   R..Description.
│ │ │ │ +00055940: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6869  ===========..Thi
│ │ │ │ +00055950: 7320 6675 6e63 7469 6f6e 2061 6c6c 6f77  s function allow
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│ │ │ │ +000559a0: 2077 6974 6820 6120 6769 7665 6e20 6e75   with a given nu
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│ │ │ │ +000559c0: 2063 686f 7365 6e20 6564 6765 7320 6f66   chosen edges of
│ │ │ │ +000559d0: 2067 6976 656e 2063 6172 6469 6e61 6c69   given cardinali
│ │ │ │ +000559e0: 7479 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  ty...+----------
│ │ │ │ +000559f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00055a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00055a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00055a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00055a30: 2d2d 2d2b 0a7c 6931 203a 2052 203d 2051  ---+.|i1 : R = Q
│ │ │ │ +00055a40: 515b 785f 312e 2e78 5f39 5d3b 2020 2020  Q[x_1..x_9];    
│ │ │ │ +00055a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00055a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00055a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00055a80: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00055a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00055aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00055ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00055ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00055ad0: 2d2d 2d2b 0a7c 6932 203a 2072 616e 646f  ---+.|i2 : rando
│ │ │ │ +00055ae0: 6d55 6e69 666f 726d 4879 7065 7247 7261  mUniformHyperGra
│ │ │ │ +00055af0: 7068 2852 2c33 2c34 2920 2020 2020 2020  ph(R,3,4)       
│ │ │ │  00055b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055b20: 2020 207c 0a7c 2d2d 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 2d2d  ----------------
│ │ │ │ -00055b70: 2d2d 2d7c 0a7c 7820 2c20 7820 7d7d 7d20  ---|.|x , x }}} 
│ │ │ │ -00055b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055bc0: 2020 207c 0a7c 2035 2020 2037 2020 2020     |.| 5   7    
│ │ │ │ +00055b20: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00055b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00055b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00055b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00055bb0: 202c 2078 202c 2078 207d 2c20 7b78 202c   , x , x }, {x ,
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│ │ │ │ +00055bf0: 2020 3220 2020 3620 2020 3820 2020 2020    2   6   8     
│ │ │ │ +00055c00: 3420 2020 3720 2020 3820 2020 2020 3320  4   7   8     3 
│ │ │ │ +00055c10: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00055c20: 2020 2020 2020 2272 696e 6722 203d 3e20        "ring" => 
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│ │ │ │ -00055d80: 3920 2020 2020 3220 2020 3320 2020 3420  9     2   3   4 
│ │ │ │ -00055d90: 2020 3820 2020 2020 2020 2020 2020 2020    8             
│ │ │ │ -00055da0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
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│ │ │ │ -00055e30: 2078 202c 2078 207d 2020 2020 2020 2020   x , x }        
│ │ │ │ -00055e40: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
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│ │ │ │ +00055cf0: 2020 3820 2020 3920 2020 2020 2020 2020    8   9         
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│ │ │ │ +00055d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00055d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00055d60: 4772 6170 6820 2020 2020 2020 2020 2020  Graph           
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│ │ │ │ +00055d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00055e40: 2020 207c 0a7c 2035 2020 2037 2020 2020     |.| 5   7    
│ │ │ │  00055e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055e60: 2020 2020 2020 3120 2020 3220 2020 3320        1   2   3 
│ │ │ │ -00055e70: 2020 3420 2020 3520 2020 3620 2020 3720    4   5   6   7 
│ │ │ │ -00055e80: 2020 3820 2020 3920 2020 2020 2020 2020    8   9         
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│ │ │ │ -00055ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00055ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00055ef0: 4772 6170 6820 2020 2020 2020 2020 2020  Graph           
│ │ │ │ -00055f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00055e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00055e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00055e90: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00055ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00055eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00055ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00055ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00055ee0: 2d2d 2d2b 0a7c 6933 203a 2072 616e 646f  ---+.|i3 : rando
│ │ │ │ +00055ef0: 6d55 6e69 666f 726d 4879 7065 7247 7261  mUniformHyperGra
│ │ │ │ +00055f00: 7068 2852 2c34 2c32 2920 2020 2020 2020  ph(R,4,2)       
│ │ │ │  00055f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055f30: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -00055f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00055f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00055f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00055f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00055f80: 2d2d 2d2b 0a0a 5365 6520 616c 736f 0a3d  ---+..See also.=
│ │ │ │ -00055f90: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f  =======..  * *no
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│ │ │ │ -00055fc0: 7265 7475 726e 7320 6120 7261 6e64 6f6d  returns a random
│ │ │ │ -00055fd0: 2067 7261 7068 0a20 202a 202a 6e6f 7465   graph.  * *note
│ │ │ │ -00055fe0: 2072 616e 646f 6d48 7970 6572 4772 6170   randomHyperGrap
│ │ │ │ -00055ff0: 683a 2072 616e 646f 6d48 7970 6572 4772  h: randomHyperGr
│ │ │ │ -00056000: 6170 682c 202d 2d20 7265 7475 726e 7320  aph, -- returns 
│ │ │ │ -00056010: 6120 7261 6e64 6f6d 2068 7970 6572 6772  a random hypergr
│ │ │ │ -00056020: 6170 680a 2020 2a20 2a6e 6f74 6520 436f  aph.  * *note Co
│ │ │ │ -00056030: 6e73 7472 7563 746f 7220 4f76 6572 7669  nstructor Overvi
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│ │ │ │ -00056050: 4f76 6572 7669 6577 2c20 2d2d 2061 2073  Overview, -- a s
│ │ │ │ -00056060: 756d 6d61 7279 206f 6620 7468 6520 6d61  ummary of the ma
│ │ │ │ -00056070: 6e79 0a20 2020 2077 6179 7320 6f66 206d  ny.    ways of m
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│ │ │ │ -00056090: 2068 7970 6572 6772 6170 6873 0a0a 5761   hypergraphs..Wa
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│ │ │ │ -000560b0: 556e 6966 6f72 6d48 7970 6572 4772 6170  UniformHyperGrap
│ │ │ │ -000560c0: 683a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  h:.=============
│ │ │ │ -000560d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000560e0: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 7261  =======..  * "ra
│ │ │ │ -000560f0: 6e64 6f6d 556e 6966 6f72 6d48 7970 6572  ndomUniformHyper
│ │ │ │ -00056100: 4772 6170 6828 506f 6c79 6e6f 6d69 616c  Graph(Polynomial
│ │ │ │ -00056110: 5269 6e67 2c5a 5a2c 5a5a 2922 0a0a 466f  Ring,ZZ,ZZ)"..Fo
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│ │ │ │ -00056130: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ -00056140: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420  ===..The object 
│ │ │ │ -00056150: 2a6e 6f74 6520 7261 6e64 6f6d 556e 6966  *note randomUnif
│ │ │ │ -00056160: 6f72 6d48 7970 6572 4772 6170 683a 2072  ormHyperGraph: r
│ │ │ │ -00056170: 616e 646f 6d55 6e69 666f 726d 4879 7065  andomUniformHype
│ │ │ │ -00056180: 7247 7261 7068 2c20 6973 2061 202a 6e6f  rGraph, is a *no
│ │ │ │ -00056190: 7465 0a6d 6574 686f 6420 6675 6e63 7469  te.method functi
│ │ │ │ -000561a0: 6f6e 3a20 284d 6163 6175 6c61 7932 446f  on: (Macaulay2Do
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│ │ │ │ -000561c0: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ,...------------
│ │ │ │ +00055f30: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00055f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00055f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00055f80: 2020 207c 0a7c 6f33 203d 2048 7970 6572     |.|o3 = Hyper
│ │ │ │ +00055f90: 4772 6170 687b 2265 6467 6573 2220 3d3e  Graph{"edges" =>
│ │ │ │ +00055fa0: 207b 7b78 202c 2078 202c 2078 202c 2078   {{x , x , x , x
│ │ │ │ +00055fb0: 207d 2c20 7b78 202c 2078 202c 2078 202c   }, {x , x , x ,
│ │ │ │ +00055fc0: 2078 207d 7d20 2020 7d20 2020 2020 2020   x }}   }       
│ │ │ │ +00055fd0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00055fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00055ff0: 2020 2020 3220 2020 3320 2020 3420 2020      2   3   4   
│ │ │ │ +00056000: 3920 2020 2020 3220 2020 3320 2020 3420  9     2   3   4 
│ │ │ │ +00056010: 2020 3820 2020 2020 2020 2020 2020 2020    8             
│ │ │ │ +00056020: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00056030: 2020 2020 2020 2272 696e 6722 203d 3e20        "ring" => 
│ │ │ │ +00056040: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │ +00056050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056070: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00056080: 2020 2020 2020 2276 6572 7469 6365 7322        "vertices"
│ │ │ │ +00056090: 203d 3e20 7b78 202c 2078 202c 2078 202c   => {x , x , x ,
│ │ │ │ +000560a0: 2078 202c 2078 202c 2078 202c 2078 202c   x , x , x , x ,
│ │ │ │ +000560b0: 2078 202c 2078 207d 2020 2020 2020 2020   x , x }        
│ │ │ │ +000560c0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000560d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000560e0: 2020 2020 2020 3120 2020 3220 2020 3320        1   2   3 
│ │ │ │ +000560f0: 2020 3420 2020 3520 2020 3620 2020 3720    4   5   6   7 
│ │ │ │ +00056100: 2020 3820 2020 3920 2020 2020 2020 2020    8   9         
│ │ │ │ +00056110: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00056120: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00056150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056160: 2020 207c 0a7c 6f33 203a 2048 7970 6572     |.|o3 : Hyper
│ │ │ │ +00056170: 4772 6170 6820 2020 2020 2020 2020 2020  Graph           
│ │ │ │ +00056180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056190: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000561e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000561f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00056240: 7072 6f64 7563 6962 6c65 2d70 6174 682f  producible-path/
│ │ │ │ -00056250: 6d61 6361 756c 6179 322d 312e 3235 2e31  macaulay2-1.25.1
│ │ │ │ -00056260: 312b 6473 2f4d 322f 4d61 6361 756c 6179  1+ds/M2/Macaulay
│ │ │ │ -00056270: 322f 7061 636b 6167 6573 2f45 6467 6549  2/packages/EdgeI
│ │ │ │ -00056280: 6465 616c 732e 0a6d 323a 3339 3234 3a30  deals..m2:3924:0
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│ │ │ │ -000562b0: 2072 696e 675f 6c70 4879 7065 7247 7261   ring_lpHyperGra
│ │ │ │ -000562c0: 7068 5f72 702c 204e 6578 743a 2073 696d  ph_rp, Next: sim
│ │ │ │ -000562d0: 706c 6963 6961 6c43 6f6d 706c 6578 546f  plicialComplexTo
│ │ │ │ -000562e0: 4879 7065 7247 7261 7068 2c20 5072 6576  HyperGraph, Prev
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│ │ │ │ -00056310: 6f70 0a0a 7269 6e67 2848 7970 6572 4772  op..ring(HyperGr
│ │ │ │ -00056320: 6170 6829 202d 2d20 6769 7665 7320 7468  aph) -- gives th
│ │ │ │ -00056330: 6520 7269 6e67 206f 6620 6120 2868 7970  e ring of a (hyp
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│ │ │ │ -00056350: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00056360: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00056370: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20  *************.. 
│ │ │ │ -00056380: 202a 2046 756e 6374 696f 6e3a 202a 6e6f   * Function: *no
│ │ │ │ -00056390: 7465 2072 696e 673a 2028 4d61 6361 756c  te ring: (Macaul
│ │ │ │ -000563a0: 6179 3244 6f63 2972 696e 672c 0a20 202a  ay2Doc)ring,.  *
│ │ │ │ -000563b0: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ -000563c0: 2052 203d 2072 696e 6720 480a 2020 2a20   R = ring H.  * 
│ │ │ │ -000563d0: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ -000563e0: 482c 2061 202a 6e6f 7465 2068 7970 6572  H, a *note hyper
│ │ │ │ -000563f0: 6772 6170 683a 2048 7970 6572 4772 6170  graph: HyperGrap
│ │ │ │ -00056400: 682c 2c20 0a20 202a 204f 7574 7075 7473  h,, .  * Outputs
│ │ │ │ -00056410: 3a0a 2020 2020 2020 2a20 522c 2061 202a  :.      * R, a *
│ │ │ │ -00056420: 6e6f 7465 2072 696e 673a 2028 4d61 6361  note ring: (Maca
│ │ │ │ -00056430: 756c 6179 3244 6f63 2952 696e 672c 2c20  ulay2Doc)Ring,, 
│ │ │ │ -00056440: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d  ..Description.==
│ │ │ │ -00056450: 3d3d 3d3d 3d3d 3d3d 3d0a 0a45 7665 7279  =========..Every
│ │ │ │ -00056460: 2028 6879 7065 7229 6772 6170 6820 6973   (hyper)graph is
│ │ │ │ -00056470: 2064 6566 696e 6564 206f 7665 7220 736f   defined over so
│ │ │ │ -00056480: 6d65 2070 6f6c 796e 6f6d 6961 6c20 7269  me polynomial ri
│ │ │ │ -00056490: 6e67 2e20 5468 6973 206d 6574 686f 6420  ng. This method 
│ │ │ │ -000564a0: 7265 7475 726e 730a 7468 6520 7269 6e67  returns.the ring
│ │ │ │ -000564b0: 206f 6620 6120 6879 7065 7267 7261 7068   of a hypergraph
│ │ │ │ -000564c0: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...+------------
│ │ │ │ -000564d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000564e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000564f0: 2b0a 7c69 3120 3a20 5320 3d20 5151 5b61  +.|i1 : S = QQ[a
│ │ │ │ -00056500: 2e2e 645d 3b20 2020 2020 2020 2020 2020  ..d];           
│ │ │ │ -00056510: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00056520: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00056530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00056550: 7c69 3220 3a20 6720 3d20 6379 636c 6520  |i2 : g = cycle 
│ │ │ │ -00056560: 533b 2020 2020 2020 2020 2020 2020 2020  S;              
│ │ │ │ -00056570: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00056580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000565a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -000565b0: 3320 3a20 6820 3d20 696e 6475 6365 6448  3 : h = inducedH
│ │ │ │ -000565c0: 7970 6572 4772 6170 6828 672c 7b61 2c62  yperGraph(g,{a,b
│ │ │ │ -000565d0: 2c63 7d29 3b20 2020 2020 207c 0a2b 2d2d  ,c});      |.+--
│ │ │ │ -000565e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000565f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056600: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420  ----------+.|i4 
│ │ │ │ -00056610: 3a20 6465 7363 7269 6265 2072 696e 6720  : describe ring 
│ │ │ │ -00056620: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ -00056630: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00056640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056660: 2020 2020 2020 2020 7c0a 7c6f 3420 3d20          |.|o4 = 
│ │ │ │ -00056670: 5151 5b61 2e2e 642c 2044 6567 7265 6573  QQ[a..d, Degrees
│ │ │ │ -00056680: 203d 3e20 7b34 3a31 7d2c 2048 6566 7420   => {4:1}, Heft 
│ │ │ │ -00056690: 3d3e 207b 317d 5d7c 0a2b 2d2d 2d2d 2d2d  => {1}]|.+------
│ │ │ │ -000566a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000566b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000566c0: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 6465  ------+.|i5 : de
│ │ │ │ -000566d0: 7363 7269 6265 2072 696e 6720 6820 2020  scribe ring h   
│ │ │ │ -000566e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000566f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00056700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056720: 2020 2020 7c0a 7c6f 3520 3d20 5151 5b61      |.|o5 = QQ[a
│ │ │ │ -00056730: 2e2e 632c 2044 6567 7265 6573 203d 3e20  ..c, Degrees => 
│ │ │ │ -00056740: 7b33 3a31 7d2c 2048 6566 7420 3d3e 207b  {3:1}, Heft => {
│ │ │ │ -00056750: 317d 5d7c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  1}]|.+----------
│ │ │ │ +00056200: 2d2d 2d2b 0a0a 5365 6520 616c 736f 0a3d  ---+..See also.=
│ │ │ │ +00056210: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f  =======..  * *no
│ │ │ │ +00056220: 7465 2072 616e 646f 6d47 7261 7068 3a20  te randomGraph: 
│ │ │ │ +00056230: 7261 6e64 6f6d 4772 6170 682c 202d 2d20  randomGraph, -- 
│ │ │ │ +00056240: 7265 7475 726e 7320 6120 7261 6e64 6f6d  returns a random
│ │ │ │ +00056250: 2067 7261 7068 0a20 202a 202a 6e6f 7465   graph.  * *note
│ │ │ │ +00056260: 2072 616e 646f 6d48 7970 6572 4772 6170   randomHyperGrap
│ │ │ │ +00056270: 683a 2072 616e 646f 6d48 7970 6572 4772  h: randomHyperGr
│ │ │ │ +00056280: 6170 682c 202d 2d20 7265 7475 726e 7320  aph, -- returns 
│ │ │ │ +00056290: 6120 7261 6e64 6f6d 2068 7970 6572 6772  a random hypergr
│ │ │ │ +000562a0: 6170 680a 2020 2a20 2a6e 6f74 6520 436f  aph.  * *note Co
│ │ │ │ +000562b0: 6e73 7472 7563 746f 7220 4f76 6572 7669  nstructor Overvi
│ │ │ │ +000562c0: 6577 3a20 436f 6e73 7472 7563 746f 7220  ew: Constructor 
│ │ │ │ +000562d0: 4f76 6572 7669 6577 2c20 2d2d 2061 2073  Overview, -- a s
│ │ │ │ +000562e0: 756d 6d61 7279 206f 6620 7468 6520 6d61  ummary of the ma
│ │ │ │ +000562f0: 6e79 0a20 2020 2077 6179 7320 6f66 206d  ny.    ways of m
│ │ │ │ +00056300: 616b 696e 6720 6772 6170 6873 2061 6e64  aking graphs and
│ │ │ │ +00056310: 2068 7970 6572 6772 6170 6873 0a0a 5761   hypergraphs..Wa
│ │ │ │ +00056320: 7973 2074 6f20 7573 6520 7261 6e64 6f6d  ys to use random
│ │ │ │ +00056330: 556e 6966 6f72 6d48 7970 6572 4772 6170  UniformHyperGrap
│ │ │ │ +00056340: 683a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  h:.=============
│ │ │ │ +00056350: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00056360: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 7261  =======..  * "ra
│ │ │ │ +00056370: 6e64 6f6d 556e 6966 6f72 6d48 7970 6572  ndomUniformHyper
│ │ │ │ +00056380: 4772 6170 6828 506f 6c79 6e6f 6d69 616c  Graph(Polynomial
│ │ │ │ +00056390: 5269 6e67 2c5a 5a2c 5a5a 2922 0a0a 466f  Ring,ZZ,ZZ)"..Fo
│ │ │ │ +000563a0: 7220 7468 6520 7072 6f67 7261 6d6d 6572  r the programmer
│ │ │ │ +000563b0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ +000563c0: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420  ===..The object 
│ │ │ │ +000563d0: 2a6e 6f74 6520 7261 6e64 6f6d 556e 6966  *note randomUnif
│ │ │ │ +000563e0: 6f72 6d48 7970 6572 4772 6170 683a 2072  ormHyperGraph: r
│ │ │ │ +000563f0: 616e 646f 6d55 6e69 666f 726d 4879 7065  andomUniformHype
│ │ │ │ +00056400: 7247 7261 7068 2c20 6973 2061 202a 6e6f  rGraph, is a *no
│ │ │ │ +00056410: 7465 0a6d 6574 686f 6420 6675 6e63 7469  te.method functi
│ │ │ │ +00056420: 6f6e 3a20 284d 6163 6175 6c61 7932 446f  on: (Macaulay2Do
│ │ │ │ +00056430: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e  c)MethodFunction
│ │ │ │ +00056440: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ,...------------
│ │ │ │ +00056450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00056460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00056470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00056480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00056490: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520  ---..The source 
│ │ │ │ +000564a0: 6f66 2074 6869 7320 646f 6375 6d65 6e74  of this document
│ │ │ │ +000564b0: 2069 7320 696e 0a2f 6275 696c 642f 7265   is in./build/re
│ │ │ │ +000564c0: 7072 6f64 7563 6962 6c65 2d70 6174 682f  producible-path/
│ │ │ │ +000564d0: 6d61 6361 756c 6179 322d 312e 3235 2e31  macaulay2-1.25.1
│ │ │ │ +000564e0: 312b 6473 2f4d 322f 4d61 6361 756c 6179  1+ds/M2/Macaulay
│ │ │ │ +000564f0: 322f 7061 636b 6167 6573 2f45 6467 6549  2/packages/EdgeI
│ │ │ │ +00056500: 6465 616c 732e 0a6d 323a 3339 3234 3a30  deals..m2:3924:0
│ │ │ │ +00056510: 2e0a 1f0a 4669 6c65 3a20 4564 6765 4964  ....File: EdgeId
│ │ │ │ +00056520: 6561 6c73 2e69 6e66 6f2c 204e 6f64 653a  eals.info, Node:
│ │ │ │ +00056530: 2072 696e 675f 6c70 4879 7065 7247 7261   ring_lpHyperGra
│ │ │ │ +00056540: 7068 5f72 702c 204e 6578 743a 2073 696d  ph_rp, Next: sim
│ │ │ │ +00056550: 706c 6963 6961 6c43 6f6d 706c 6578 546f  plicialComplexTo
│ │ │ │ +00056560: 4879 7065 7247 7261 7068 2c20 5072 6576  HyperGraph, Prev
│ │ │ │ +00056570: 3a20 7261 6e64 6f6d 556e 6966 6f72 6d48  : randomUniformH
│ │ │ │ +00056580: 7970 6572 4772 6170 682c 2055 703a 2054  yperGraph, Up: T
│ │ │ │ +00056590: 6f70 0a0a 7269 6e67 2848 7970 6572 4772  op..ring(HyperGr
│ │ │ │ +000565a0: 6170 6829 202d 2d20 6769 7665 7320 7468  aph) -- gives th
│ │ │ │ +000565b0: 6520 7269 6e67 206f 6620 6120 2868 7970  e ring of a (hyp
│ │ │ │ +000565c0: 6572 2967 7261 7068 0a2a 2a2a 2a2a 2a2a  er)graph.*******
│ │ │ │ +000565d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000565e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000565f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20  *************.. 
│ │ │ │ +00056600: 202a 2046 756e 6374 696f 6e3a 202a 6e6f   * Function: *no
│ │ │ │ +00056610: 7465 2072 696e 673a 2028 4d61 6361 756c  te ring: (Macaul
│ │ │ │ +00056620: 6179 3244 6f63 2972 696e 672c 0a20 202a  ay2Doc)ring,.  *
│ │ │ │ +00056630: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ +00056640: 2052 203d 2072 696e 6720 480a 2020 2a20   R = ring H.  * 
│ │ │ │ +00056650: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ +00056660: 482c 2061 202a 6e6f 7465 2068 7970 6572  H, a *note hyper
│ │ │ │ +00056670: 6772 6170 683a 2048 7970 6572 4772 6170  graph: HyperGrap
│ │ │ │ +00056680: 682c 2c20 0a20 202a 204f 7574 7075 7473  h,, .  * Outputs
│ │ │ │ +00056690: 3a0a 2020 2020 2020 2a20 522c 2061 202a  :.      * R, a *
│ │ │ │ +000566a0: 6e6f 7465 2072 696e 673a 2028 4d61 6361  note ring: (Maca
│ │ │ │ +000566b0: 756c 6179 3244 6f63 2952 696e 672c 2c20  ulay2Doc)Ring,, 
│ │ │ │ +000566c0: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d  ..Description.==
│ │ │ │ +000566d0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a45 7665 7279  =========..Every
│ │ │ │ +000566e0: 2028 6879 7065 7229 6772 6170 6820 6973   (hyper)graph is
│ │ │ │ +000566f0: 2064 6566 696e 6564 206f 7665 7220 736f   defined over so
│ │ │ │ +00056700: 6d65 2070 6f6c 796e 6f6d 6961 6c20 7269  me polynomial ri
│ │ │ │ +00056710: 6e67 2e20 5468 6973 206d 6574 686f 6420  ng. This method 
│ │ │ │ +00056720: 7265 7475 726e 730a 7468 6520 7269 6e67  returns.the ring
│ │ │ │ +00056730: 206f 6620 6120 6879 7065 7267 7261 7068   of a hypergraph
│ │ │ │ +00056740: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...+------------
│ │ │ │ +00056750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056780: 2d2d 2b0a 0a53 6565 2061 6c73 6f0a 3d3d  --+..See also.==
│ │ │ │ -00056790: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74  ======..  * *not
│ │ │ │ -000567a0: 6520 6564 6765 733a 2065 6467 6573 2c20  e edges: edges, 
│ │ │ │ -000567b0: 2d2d 2067 6574 7320 7468 6520 6564 6765  -- gets the edge
│ │ │ │ -000567c0: 7320 6f66 2061 2028 6879 7065 7229 6772  s of a (hyper)gr
│ │ │ │ -000567d0: 6170 680a 2020 2a20 2a6e 6f74 6520 7665  aph.  * *note ve
│ │ │ │ -000567e0: 7274 6963 6573 2848 7970 6572 4772 6170  rtices(HyperGrap
│ │ │ │ -000567f0: 6829 3a20 7665 7274 6963 6573 5f6c 7048  h): vertices_lpH
│ │ │ │ -00056800: 7970 6572 4772 6170 685f 7270 2c20 2d2d  yperGraph_rp, --
│ │ │ │ -00056810: 2067 6574 7320 7468 6520 7665 7274 6963   gets the vertic
│ │ │ │ -00056820: 6573 0a20 2020 206f 6620 6120 2868 7970  es.    of a (hyp
│ │ │ │ -00056830: 6572 2967 7261 7068 0a0a 5761 7973 2074  er)graph..Ways t
│ │ │ │ -00056840: 6f20 7573 6520 7468 6973 206d 6574 686f  o use this metho
│ │ │ │ -00056850: 643a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  d:.=============
│ │ │ │ -00056860: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ -00056870: 202a 6e6f 7465 2072 696e 6728 4879 7065   *note ring(Hype
│ │ │ │ -00056880: 7247 7261 7068 293a 2072 696e 675f 6c70  rGraph): ring_lp
│ │ │ │ -00056890: 4879 7065 7247 7261 7068 5f72 702c 202d  HyperGraph_rp, -
│ │ │ │ -000568a0: 2d20 6769 7665 7320 7468 6520 7269 6e67  - gives the ring
│ │ │ │ -000568b0: 206f 6620 610a 2020 2020 2868 7970 6572   of a.    (hyper
│ │ │ │ -000568c0: 2967 7261 7068 0a2d 2d2d 2d2d 2d2d 2d2d  )graph.---------
│ │ │ │ -000568d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000568e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000568f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056910: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572  ------..The sour
│ │ │ │ -00056920: 6365 206f 6620 7468 6973 2064 6f63 756d  ce of this docum
│ │ │ │ -00056930: 656e 7420 6973 2069 6e0a 2f62 7569 6c64  ent is in./build
│ │ │ │ -00056940: 2f72 6570 726f 6475 6369 626c 652d 7061  /reproducible-pa
│ │ │ │ -00056950: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32  th/macaulay2-1.2
│ │ │ │ -00056960: 352e 3131 2b64 732f 4d32 2f4d 6163 6175  5.11+ds/M2/Macau
│ │ │ │ -00056970: 6c61 7932 2f70 6163 6b61 6765 732f 4564  lay2/packages/Ed
│ │ │ │ -00056980: 6765 4964 6561 6c73 2e0a 6d32 3a33 3935  geIdeals..m2:395
│ │ │ │ -00056990: 353a 302e 0a1f 0a46 696c 653a 2045 6467  5:0....File: Edg
│ │ │ │ -000569a0: 6549 6465 616c 732e 696e 666f 2c20 4e6f  eIdeals.info, No
│ │ │ │ -000569b0: 6465 3a20 7369 6d70 6c69 6369 616c 436f  de: simplicialCo
│ │ │ │ -000569c0: 6d70 6c65 7854 6f48 7970 6572 4772 6170  mplexToHyperGrap
│ │ │ │ -000569d0: 682c 204e 6578 743a 2073 6d61 6c6c 6573  h, Next: smalles
│ │ │ │ -000569e0: 7443 7963 6c65 5369 7a65 2c20 5072 6576  tCycleSize, Prev
│ │ │ │ -000569f0: 3a20 7269 6e67 5f6c 7048 7970 6572 4772  : ring_lpHyperGr
│ │ │ │ -00056a00: 6170 685f 7270 2c20 5570 3a20 546f 700a  aph_rp, Up: Top.
│ │ │ │ -00056a10: 0a73 696d 706c 6963 6961 6c43 6f6d 706c  .simplicialCompl
│ │ │ │ -00056a20: 6578 546f 4879 7065 7247 7261 7068 202d  exToHyperGraph -
│ │ │ │ -00056a30: 2d20 6d61 6b65 7320 6120 2868 7970 6572  - makes a (hyper
│ │ │ │ -00056a40: 2967 7261 7068 2066 726f 6d20 6120 7369  )graph from a si
│ │ │ │ -00056a50: 6d70 6c69 6369 616c 2063 6f6d 706c 6578  mplicial complex
│ │ │ │ -00056a60: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  .***************
│ │ │ │ -00056a70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00056a80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00056a90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00056aa0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00056ab0: 0a0a 2020 2a20 5573 6167 653a 200a 2020  ..  * Usage: .  
│ │ │ │ -00056ac0: 2020 2020 2020 4820 3d20 7369 6d70 6c69        H = simpli
│ │ │ │ -00056ad0: 6369 616c 436f 6d70 6c65 7854 6f48 7970  cialComplexToHyp
│ │ │ │ -00056ae0: 6572 4772 6170 6828 4429 0a20 202a 2049  erGraph(D).  * I
│ │ │ │ -00056af0: 6e70 7574 733a 0a20 2020 2020 202a 2044  nputs:.      * D
│ │ │ │ -00056b00: 2c20 616e 202a 6e6f 7465 2061 6273 7472  , an *note abstr
│ │ │ │ -00056b10: 6163 7420 7369 6d70 6c69 6369 616c 2063  act simplicial c
│ │ │ │ -00056b20: 6f6d 706c 6578 3a0a 2020 2020 2020 2020  omplex:.        
│ │ │ │ -00056b30: 2853 696d 706c 6963 6961 6c43 6f6d 706c  (SimplicialCompl
│ │ │ │ -00056b40: 6578 6573 2953 696d 706c 6963 6961 6c43  exes)SimplicialC
│ │ │ │ -00056b50: 6f6d 706c 6578 2c2c 2074 6865 2069 6e70  omplex,, the inp
│ │ │ │ -00056b60: 7574 0a20 202a 204f 7574 7075 7473 3a0a  ut.  * Outputs:.
│ │ │ │ -00056b70: 2020 2020 2020 2a20 482c 2061 202a 6e6f        * H, a *no
│ │ │ │ -00056b80: 7465 2068 7970 6572 6772 6170 683a 2048  te hypergraph: H
│ │ │ │ -00056b90: 7970 6572 4772 6170 682c 2c20 7768 6f73  yperGraph,, whos
│ │ │ │ -00056ba0: 6520 6564 6765 7320 6172 6520 7468 6520  e edges are the 
│ │ │ │ -00056bb0: 6661 6365 7473 206f 6620 440a 0a44 6573  facets of D..Des
│ │ │ │ -00056bc0: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ -00056bd0: 3d3d 3d3d 0a0a 5468 6973 2066 756e 6374  ====..This funct
│ │ │ │ -00056be0: 696f 6e20 6d61 6b65 7320 6120 2a6e 6f74  ion makes a *not
│ │ │ │ -00056bf0: 6520 4879 7065 7247 7261 7068 3a20 4879  e HyperGraph: Hy
│ │ │ │ -00056c00: 7065 7247 7261 7068 2c20 6672 6f6d 2061  perGraph, from a
│ │ │ │ -00056c10: 202a 6e6f 7465 0a53 696d 706c 6963 6961   *note.Simplicia
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│ │ │ │ -00056c30: 6963 6961 6c43 6f6d 706c 6578 6573 2953  icialComplexes)S
│ │ │ │ -00056c40: 696d 706c 6963 6961 6c43 6f6d 706c 6578  implicialComplex
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│ │ │ │ -00056ca0: 2054 6869 7320 6973 2074 6865 0a69 6e76   This is the.inv
│ │ │ │ -00056cb0: 6572 7365 206f 6620 7468 6520 6675 6e63  erse of the func
│ │ │ │ -00056cc0: 7469 6f6e 202a 6e6f 7465 2068 7970 6572  tion *note hyper
│ │ │ │ -00056cd0: 4772 6170 6854 6f53 696d 706c 6963 6961  GraphToSimplicia
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│ │ │ │ -00056cf0: 7261 7068 546f 5369 6d70 6c69 6369 616c  raphToSimplicial
│ │ │ │ -00056d00: 436f 6d70 6c65 782c 2e0a 0a2b 2d2d 2d2d  Complex,...+----
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│ │ │ │ -00056d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056d50: 2d2d 2d2d 2b0a 7c69 3120 3a20 5320 3d20  ----+.|i1 : S = 
│ │ │ │ -00056d60: 5151 5b61 2e2e 665d 3b20 2020 2020 2020  QQ[a..f];       
│ │ │ │ -00056d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056d90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00056da0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00056db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220  ----------+.|i2 
│ │ │ │ -00056df0: 3a20 4465 6c74 6120 3d20 7369 6d70 6c69  : Delta = simpli
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│ │ │ │ -00056e10: 2a63 2c62 2a63 2a64 2c63 2a64 2a65 2c64  *c,b*c*d,c*d*e,d
│ │ │ │ -00056e20: 2a65 2a66 7d20 2020 2020 2020 2020 2020  *e*f}           
│ │ │ │ -00056e30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00056e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056e80: 7c0a 7c6f 3220 3d20 7369 6d70 6c69 6369  |.|o2 = simplici
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│ │ │ │ -00056eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056ec0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00056ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056f10: 2020 2020 2020 7c0a 7c6f 3220 3a20 5369        |.|o2 : Si
│ │ │ │ -00056f20: 6d70 6c69 6369 616c 436f 6d70 6c65 7820  mplicialComplex 
│ │ │ │ -00056f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00056f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00056780: 2e2e 645d 3b20 2020 2020 2020 2020 2020  ..d];           
│ │ │ │ +00056790: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000567a0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000567b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +000567e0: 533b 2020 2020 2020 2020 2020 2020 2020  S;              
│ │ │ │ +000567f0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00056800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00056810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00056820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00056830: 3320 3a20 6820 3d20 696e 6475 6365 6448  3 : h = inducedH
│ │ │ │ +00056840: 7970 6572 4772 6170 6828 672c 7b61 2c62  yperGraph(g,{a,b
│ │ │ │ +00056850: 2c63 7d29 3b20 2020 2020 207c 0a2b 2d2d  ,c});      |.+--
│ │ │ │ +00056860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00056870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00056880: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420  ----------+.|i4 
│ │ │ │ +00056890: 3a20 6465 7363 7269 6265 2072 696e 6720  : describe ring 
│ │ │ │ +000568a0: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ +000568b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000568c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000568d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000568e0: 2020 2020 2020 2020 7c0a 7c6f 3420 3d20          |.|o4 = 
│ │ │ │ +000568f0: 5151 5b61 2e2e 642c 2044 6567 7265 6573  QQ[a..d, Degrees
│ │ │ │ +00056900: 203d 3e20 7b34 3a31 7d2c 2048 6566 7420   => {4:1}, Heft 
│ │ │ │ +00056910: 3d3e 207b 317d 5d7c 0a2b 2d2d 2d2d 2d2d  => {1}]|.+------
│ │ │ │ +00056920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00056930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00056940: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 6465  ------+.|i5 : de
│ │ │ │ +00056950: 7363 7269 6265 2072 696e 6720 6820 2020  scribe ring h   
│ │ │ │ +00056960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056970: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00056980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000569a0: 2020 2020 7c0a 7c6f 3520 3d20 5151 5b61      |.|o5 = QQ[a
│ │ │ │ +000569b0: 2e2e 632c 2044 6567 7265 6573 203d 3e20  ..c, Degrees => 
│ │ │ │ +000569c0: 7b33 3a31 7d2c 2048 6566 7420 3d3e 207b  {3:1}, Heft => {
│ │ │ │ +000569d0: 317d 5d7c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  1}]|.+----------
│ │ │ │ +000569e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000569f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00056a00: 2d2d 2b0a 0a53 6565 2061 6c73 6f0a 3d3d  --+..See also.==
│ │ │ │ +00056a10: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74  ======..  * *not
│ │ │ │ +00056a20: 6520 6564 6765 733a 2065 6467 6573 2c20  e edges: edges, 
│ │ │ │ +00056a30: 2d2d 2067 6574 7320 7468 6520 6564 6765  -- gets the edge
│ │ │ │ +00056a40: 7320 6f66 2061 2028 6879 7065 7229 6772  s of a (hyper)gr
│ │ │ │ +00056a50: 6170 680a 2020 2a20 2a6e 6f74 6520 7665  aph.  * *note ve
│ │ │ │ +00056a60: 7274 6963 6573 2848 7970 6572 4772 6170  rtices(HyperGrap
│ │ │ │ +00056a70: 6829 3a20 7665 7274 6963 6573 5f6c 7048  h): vertices_lpH
│ │ │ │ +00056a80: 7970 6572 4772 6170 685f 7270 2c20 2d2d  yperGraph_rp, --
│ │ │ │ +00056a90: 2067 6574 7320 7468 6520 7665 7274 6963   gets the vertic
│ │ │ │ +00056aa0: 6573 0a20 2020 206f 6620 6120 2868 7970  es.    of a (hyp
│ │ │ │ +00056ab0: 6572 2967 7261 7068 0a0a 5761 7973 2074  er)graph..Ways t
│ │ │ │ +00056ac0: 6f20 7573 6520 7468 6973 206d 6574 686f  o use this metho
│ │ │ │ +00056ad0: 643a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  d:.=============
│ │ │ │ +00056ae0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ +00056af0: 202a 6e6f 7465 2072 696e 6728 4879 7065   *note ring(Hype
│ │ │ │ +00056b00: 7247 7261 7068 293a 2072 696e 675f 6c70  rGraph): ring_lp
│ │ │ │ +00056b10: 4879 7065 7247 7261 7068 5f72 702c 202d  HyperGraph_rp, -
│ │ │ │ +00056b20: 2d20 6769 7665 7320 7468 6520 7269 6e67  - gives the ring
│ │ │ │ +00056b30: 206f 6620 610a 2020 2020 2868 7970 6572   of a.    (hyper
│ │ │ │ +00056b40: 2967 7261 7068 0a2d 2d2d 2d2d 2d2d 2d2d  )graph.---------
│ │ │ │ +00056b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00056b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00056b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00056b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00056b90: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572  ------..The sour
│ │ │ │ +00056ba0: 6365 206f 6620 7468 6973 2064 6f63 756d  ce of this docum
│ │ │ │ +00056bb0: 656e 7420 6973 2069 6e0a 2f62 7569 6c64  ent is in./build
│ │ │ │ +00056bc0: 2f72 6570 726f 6475 6369 626c 652d 7061  /reproducible-pa
│ │ │ │ +00056bd0: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32  th/macaulay2-1.2
│ │ │ │ +00056be0: 352e 3131 2b64 732f 4d32 2f4d 6163 6175  5.11+ds/M2/Macau
│ │ │ │ +00056bf0: 6c61 7932 2f70 6163 6b61 6765 732f 4564  lay2/packages/Ed
│ │ │ │ +00056c00: 6765 4964 6561 6c73 2e0a 6d32 3a33 3935  geIdeals..m2:395
│ │ │ │ +00056c10: 353a 302e 0a1f 0a46 696c 653a 2045 6467  5:0....File: Edg
│ │ │ │ +00056c20: 6549 6465 616c 732e 696e 666f 2c20 4e6f  eIdeals.info, No
│ │ │ │ +00056c30: 6465 3a20 7369 6d70 6c69 6369 616c 436f  de: simplicialCo
│ │ │ │ +00056c40: 6d70 6c65 7854 6f48 7970 6572 4772 6170  mplexToHyperGrap
│ │ │ │ +00056c50: 682c 204e 6578 743a 2073 6d61 6c6c 6573  h, Next: smalles
│ │ │ │ +00056c60: 7443 7963 6c65 5369 7a65 2c20 5072 6576  tCycleSize, Prev
│ │ │ │ +00056c70: 3a20 7269 6e67 5f6c 7048 7970 6572 4772  : ring_lpHyperGr
│ │ │ │ +00056c80: 6170 685f 7270 2c20 5570 3a20 546f 700a  aph_rp, Up: Top.
│ │ │ │ +00056c90: 0a73 696d 706c 6963 6961 6c43 6f6d 706c  .simplicialCompl
│ │ │ │ +00056ca0: 6578 546f 4879 7065 7247 7261 7068 202d  exToHyperGraph -
│ │ │ │ +00056cb0: 2d20 6d61 6b65 7320 6120 2868 7970 6572  - makes a (hyper
│ │ │ │ +00056cc0: 2967 7261 7068 2066 726f 6d20 6120 7369  )graph from a si
│ │ │ │ +00056cd0: 6d70 6c69 6369 616c 2063 6f6d 706c 6578  mplicial complex
│ │ │ │ +00056ce0: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  .***************
│ │ │ │ +00056cf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00056d00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00056d10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00056d20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00056d30: 0a0a 2020 2a20 5573 6167 653a 200a 2020  ..  * Usage: .  
│ │ │ │ +00056d40: 2020 2020 2020 4820 3d20 7369 6d70 6c69        H = simpli
│ │ │ │ +00056d50: 6369 616c 436f 6d70 6c65 7854 6f48 7970  cialComplexToHyp
│ │ │ │ +00056d60: 6572 4772 6170 6828 4429 0a20 202a 2049  erGraph(D).  * I
│ │ │ │ +00056d70: 6e70 7574 733a 0a20 2020 2020 202a 2044  nputs:.      * D
│ │ │ │ +00056d80: 2c20 616e 202a 6e6f 7465 2061 6273 7472  , an *note abstr
│ │ │ │ +00056d90: 6163 7420 7369 6d70 6c69 6369 616c 2063  act simplicial c
│ │ │ │ +00056da0: 6f6d 706c 6578 3a0a 2020 2020 2020 2020  omplex:.        
│ │ │ │ +00056db0: 2853 696d 706c 6963 6961 6c43 6f6d 706c  (SimplicialCompl
│ │ │ │ +00056dc0: 6578 6573 2953 696d 706c 6963 6961 6c43  exes)SimplicialC
│ │ │ │ +00056dd0: 6f6d 706c 6578 2c2c 2074 6865 2069 6e70  omplex,, the inp
│ │ │ │ +00056de0: 7574 0a20 202a 204f 7574 7075 7473 3a0a  ut.  * Outputs:.
│ │ │ │ +00056df0: 2020 2020 2020 2a20 482c 2061 202a 6e6f        * H, a *no
│ │ │ │ +00056e00: 7465 2068 7970 6572 6772 6170 683a 2048  te hypergraph: H
│ │ │ │ +00056e10: 7970 6572 4772 6170 682c 2c20 7768 6f73  yperGraph,, whos
│ │ │ │ +00056e20: 6520 6564 6765 7320 6172 6520 7468 6520  e edges are the 
│ │ │ │ +00056e30: 6661 6365 7473 206f 6620 440a 0a44 6573  facets of D..Des
│ │ │ │ +00056e40: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ +00056e50: 3d3d 3d3d 0a0a 5468 6973 2066 756e 6374  ====..This funct
│ │ │ │ +00056e60: 696f 6e20 6d61 6b65 7320 6120 2a6e 6f74  ion makes a *not
│ │ │ │ +00056e70: 6520 4879 7065 7247 7261 7068 3a20 4879  e HyperGraph: Hy
│ │ │ │ +00056e80: 7065 7247 7261 7068 2c20 6672 6f6d 2061  perGraph, from a
│ │ │ │ +00056e90: 202a 6e6f 7465 0a53 696d 706c 6963 6961   *note.Simplicia
│ │ │ │ +00056ea0: 6c43 6f6d 706c 6578 3a20 2853 696d 706c  lComplex: (Simpl
│ │ │ │ +00056eb0: 6963 6961 6c43 6f6d 706c 6578 6573 2953  icialComplexes)S
│ │ │ │ +00056ec0: 696d 706c 6963 6961 6c43 6f6d 706c 6578  implicialComplex
│ │ │ │ +00056ed0: 2c2e 2054 6865 2065 6467 6573 206f 6620  ,. The edges of 
│ │ │ │ +00056ee0: 7468 650a 4879 7065 7247 7261 7068 2061  the.HyperGraph a
│ │ │ │ +00056ef0: 7265 2067 6976 656e 2062 7920 7468 6520  re given by the 
│ │ │ │ +00056f00: 6661 6365 7473 206f 6620 7468 6520 5369  facets of the Si
│ │ │ │ +00056f10: 6d70 6c69 6369 616c 436f 6d70 6c65 782e  mplicialComplex.
│ │ │ │ +00056f20: 2054 6869 7320 6973 2074 6865 0a69 6e76   This is the.inv
│ │ │ │ +00056f30: 6572 7365 206f 6620 7468 6520 6675 6e63  erse of the func
│ │ │ │ +00056f40: 7469 6f6e 202a 6e6f 7465 2068 7970 6572  tion *note hyper
│ │ │ │ +00056f50: 4772 6170 6854 6f53 696d 706c 6963 6961  GraphToSimplicia
│ │ │ │ +00056f60: 6c43 6f6d 706c 6578 3a0a 6879 7065 7247  lComplex:.hyperG
│ │ │ │ +00056f70: 7261 7068 546f 5369 6d70 6c69 6369 616c  raphToSimplicial
│ │ │ │ +00056f80: 436f 6d70 6c65 782c 2e0a 0a2b 2d2d 2d2d  Complex,...+----
│ │ │ │  00056f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00056fb0: 3320 3a20 4820 3d20 7369 6d70 6c69 6369  3 : H = simplici
│ │ │ │ -00056fc0: 616c 436f 6d70 6c65 7854 6f48 7970 6572  alComplexToHyper
│ │ │ │ -00056fd0: 4772 6170 6820 4465 6c74 6120 2020 2020  Graph Delta     
│ │ │ │ -00056fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056ff0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00056fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00056fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00056fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00056fd0: 2d2d 2d2d 2b0a 7c69 3120 3a20 5320 3d20  ----+.|i1 : S = 
│ │ │ │ +00056fe0: 5151 5b61 2e2e 665d 3b20 2020 2020 2020  QQ[a..f];       
│ │ │ │ +00056ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057040: 2020 7c0a 7c6f 3320 3d20 4879 7065 7247    |.|o3 = HyperG
│ │ │ │ -00057050: 7261 7068 7b22 6564 6765 7322 203d 3e20  raph{"edges" => 
│ │ │ │ -00057060: 7b7b 612c 2062 2c20 637d 2c20 7b62 2c20  {{a, b, c}, {b, 
│ │ │ │ -00057070: 632c 2064 7d2c 207b 632c 2064 2c20 657d  c, d}, {c, d, e}
│ │ │ │ -00057080: 2c20 7b64 2c20 652c 2066 7d7d 7d7c 0a7c  , {d, e, f}}}|.|
│ │ │ │ -00057090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000570a0: 2272 696e 6722 203d 3e20 5320 2020 2020  "ring" => S     
│ │ │ │ -000570b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057010: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │ -000571e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00057210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00057290: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000572b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00057340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057350: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
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│ │ │ │ +000573c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000573d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00057450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -000575f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00057600: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ -00057970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -000579d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000579e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000579f0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -00057a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00057d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +000576f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00057880: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ +00057900: 6772 6170 683a 2047 7261 7068 2c2c 2074  graph: Graph,, t
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│ │ │ │ +00057c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00057cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00057d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00057dd0: 2031 2020 2032 2020 2020 2032 2020 2033   1   2     2   3
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│ │ │ │ +00057df0: 2020 2035 2020 2020 2035 2020 2036 2020     5     5   6  
│ │ │ │ +00057e00: 2020 2036 207c 0a7c 2020 2020 2020 2020     6 |.|        
│ │ │ │ +00057e10: 2020 2022 7269 6e67 2220 3d3e 2054 2020     "ring" => T  
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│ │ │ │  00057e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057e50: 2020 2020 207c 0a7c 2020 3720 2020 2020       |.|  7     
│ │ │ │ -00057e60: 3720 2020 3820 2020 2020 3120 2020 3920  7   8     1   9 
│ │ │ │ -00057e70: 2020 2020 3820 2020 3920 2020 2020 2020      8   9       
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│ │ │ │ -00057e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00057ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057ef0: 2d2d 2d2d 2d2b 0a7c 6933 203a 2073 6d61  -----+.|i3 : sma
│ │ │ │ -00057f00: 6c6c 6573 7443 7963 6c65 5369 7a65 2067  llestCycleSize g
│ │ │ │ +00057e50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00057e60: 2020 2022 7665 7274 6963 6573 2220 3d3e     "vertices" =>
│ │ │ │ +00057e70: 207b 7820 2c20 7820 2c20 7820 2c20 7820   {x , x , x , x 
│ │ │ │ +00057e80: 2c20 7820 2c20 7820 2c20 7820 2c20 7820  , x , x , x , x 
│ │ │ │ +00057e90: 2c20 7820 7d20 2020 2020 2020 2020 2020  , x }           
│ │ │ │ +00057ea0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00057eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057ec0: 2020 2031 2020 2032 2020 2033 2020 2034     1   2   3   4
│ │ │ │ +00057ed0: 2020 2035 2020 2036 2020 2037 2020 2038     5   6   7   8
│ │ │ │ +00057ee0: 2020 2039 2020 2020 2020 2020 2020 2020     9            
│ │ │ │ +00057ef0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00057f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057f40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00057f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057f40: 2020 2020 207c 0a7c 6f32 203a 2047 7261       |.|o2 : Gra
│ │ │ │ +00057f50: 7068 2020 2020 2020 2020 2020 2020 2020  ph              
│ │ │ │  00057f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057f90: 2020 2020 207c 0a7c 6f33 203d 2039 2020       |.|o3 = 9  
│ │ │ │ -00057fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057fe0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -00057ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058030: 2d2d 2d2d 2d2b 0a7c 6934 203a 2068 203d  -----+.|i4 : h =
│ │ │ │ -00058040: 2067 7261 7068 207b 785f 312a 785f 322c   graph {x_1*x_2,
│ │ │ │ -00058050: 785f 322a 785f 332c 785f 332a 785f 342c  x_2*x_3,x_3*x_4,
│ │ │ │ -00058060: 785f 342a 785f 352c 785f 352a 785f 362c  x_4*x_5,x_5*x_6,
│ │ │ │ -00058070: 785f 362a 785f 372c 785f 372a 785f 382c  x_6*x_7,x_7*x_8,
│ │ │ │ -00058080: 785f 382a 787c 0a7c 2020 2020 2020 2020  x_8*x|.|        
│ │ │ │ -00058090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000580a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057f90: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +00057fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00057fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00057fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00057fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00057fe0: 2d2d 2d2d 2d7c 0a7c 5f39 2c78 5f39 2a78  -----|.|_9,x_9*x
│ │ │ │ +00057ff0: 5f31 7d20 2d2d 2061 2039 2d63 7963 6c65  _1} -- a 9-cycle
│ │ │ │ +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 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00058040: 2020 2020 2020 2020 2020 2020 2020 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 207c 0a7c 2078 207d 2c20 7b78       |.| x }, {x
│ │ │ │ +00058090: 202c 2078 207d 2c20 7b78 202c 2078 207d   , x }, {x , x }
│ │ │ │ +000580a0: 2c20 7b78 202c 2078 207d 7d7d 2020 2020  , {x , x }}}    
│ │ │ │  000580b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000580c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000580d0: 2020 2020 207c 0a7c 6f34 203d 2047 7261       |.|o4 = Gra
│ │ │ │ -000580e0: 7068 7b22 6564 6765 7322 203d 3e20 7b7b  ph{"edges" => {{
│ │ │ │ -000580f0: 7820 2c20 7820 7d2c 207b 7820 2c20 7820  x , x }, {x , x 
│ │ │ │ -00058100: 7d2c 207b 7820 2c20 7820 7d2c 207b 7820  }, {x , x }, {x 
│ │ │ │ -00058110: 2c20 7820 7d2c 207b 7820 2c20 7820 7d2c  , x }, {x , x },
│ │ │ │ -00058120: 207b 7820 2c7c 0a7c 2020 2020 2020 2020   {x ,|.|        
│ │ │ │ -00058130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058140: 2031 2020 2032 2020 2020 2032 2020 2033   1   2     2   3
│ │ │ │ -00058150: 2020 2020 2033 2020 2034 2020 2020 2034       3   4     4
│ │ │ │ -00058160: 2020 2035 2020 2020 2035 2020 2036 2020     5     5   6  
│ │ │ │ -00058170: 2020 2036 207c 0a7c 2020 2020 2020 2020     6 |.|        
│ │ │ │ -00058180: 2020 2022 7269 6e67 2220 3d3e 2054 2020     "ring" => T  
│ │ │ │ +000580d0: 2020 2020 207c 0a7c 2020 3720 2020 2020       |.|  7     
│ │ │ │ +000580e0: 3720 2020 3820 2020 2020 3120 2020 3920  7   8     1   9 
│ │ │ │ +000580f0: 2020 2020 3820 2020 3920 2020 2020 2020      8   9       
│ │ │ │ +00058100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058120: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00058130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00058140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00058150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00058160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00058170: 2d2d 2d2d 2d2b 0a7c 6933 203a 2073 6d61  -----+.|i3 : sma
│ │ │ │ +00058180: 6c6c 6573 7443 7963 6c65 5369 7a65 2067  llestCycleSize g
│ │ │ │  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 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000581d0: 2020 2022 7665 7274 6963 6573 2220 3d3e     "vertices" =>
│ │ │ │ -000581e0: 207b 7820 2c20 7820 2c20 7820 2c20 7820   {x , x , x , x 
│ │ │ │ -000581f0: 2c20 7820 2c20 7820 2c20 7820 2c20 7820  , x , x , x , x 
│ │ │ │ -00058200: 2c20 7820 7d20 2020 2020 2020 2020 2020  , x }           
│ │ │ │ -00058210: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000581d0: 2020 2020 2020 2020 2020 2020 2020 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 207c 0a7c 6f33 203d 2039 2020       |.|o3 = 9  
│ │ │ │  00058220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058230: 2020 2031 2020 2032 2020 2033 2020 2034     1   2   3   4
│ │ │ │ -00058240: 2020 2035 2020 2036 2020 2037 2020 2038     5   6   7   8
│ │ │ │ -00058250: 2020 2039 2020 2020 2020 2020 2020 2020     9            
│ │ │ │ -00058260: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00058270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000582a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000582b0: 2020 2020 207c 0a7c 6f34 203a 2047 7261       |.|o4 : Gra
│ │ │ │ -000582c0: 7068 2020 2020 2020 2020 2020 2020 2020  ph              
│ │ │ │ -000582d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000582e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000582f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058300: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ -00058310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058350: 2d2d 2d2d 2d7c 0a7c 5f39 7d20 2d2d 2061  -----|.|_9} -- a
│ │ │ │ -00058360: 2074 7265 6520 286e 6f20 6379 636c 6573   tree (no cycles
│ │ │ │ -00058370: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -00058380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000583a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00058230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058260: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00058270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00058280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00058290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000582a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000582b0: 2d2d 2d2d 2d2b 0a7c 6934 203a 2068 203d  -----+.|i4 : h =
│ │ │ │ +000582c0: 2067 7261 7068 207b 785f 312a 785f 322c   graph {x_1*x_2,
│ │ │ │ +000582d0: 785f 322a 785f 332c 785f 332a 785f 342c  x_2*x_3,x_3*x_4,
│ │ │ │ +000582e0: 785f 342a 785f 352c 785f 352a 785f 362c  x_4*x_5,x_5*x_6,
│ │ │ │ +000582f0: 785f 362a 785f 372c 785f 372a 785f 382c  x_6*x_7,x_7*x_8,
│ │ │ │ +00058300: 785f 382a 787c 0a7c 2020 2020 2020 2020  x_8*x|.|        
│ │ │ │ +00058310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058350: 2020 2020 207c 0a7c 6f34 203d 2047 7261       |.|o4 = Gra
│ │ │ │ +00058360: 7068 7b22 6564 6765 7322 203d 3e20 7b7b  ph{"edges" => {{
│ │ │ │ +00058370: 7820 2c20 7820 7d2c 207b 7820 2c20 7820  x , x }, {x , x 
│ │ │ │ +00058380: 7d2c 207b 7820 2c20 7820 7d2c 207b 7820  }, {x , x }, {x 
│ │ │ │ +00058390: 2c20 7820 7d2c 207b 7820 2c20 7820 7d2c  , x }, {x , x },
│ │ │ │ +000583a0: 207b 7820 2c7c 0a7c 2020 2020 2020 2020   {x ,|.|        
│ │ │ │  000583b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000583c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000583d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000583e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000583f0: 2020 2020 207c 0a7c 2078 207d 2c20 7b78       |.| x }, {x
│ │ │ │ -00058400: 202c 2078 207d 2c20 7b78 202c 2078 207d   , x }, {x , x }
│ │ │ │ -00058410: 7d7d 2020 2020 2020 2020 2020 2020 2020  }}              
│ │ │ │ +000583c0: 2031 2020 2032 2020 2020 2032 2020 2033   1   2     2   3
│ │ │ │ +000583d0: 2020 2020 2033 2020 2034 2020 2020 2034       3   4     4
│ │ │ │ +000583e0: 2020 2035 2020 2020 2035 2020 2036 2020     5     5   6  
│ │ │ │ +000583f0: 2020 2036 207c 0a7c 2020 2020 2020 2020     6 |.|        
│ │ │ │ +00058400: 2020 2022 7269 6e67 2220 3d3e 2054 2020     "ring" => T  
│ │ │ │ +00058410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00058420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00058430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058440: 2020 2020 207c 0a7c 2020 3720 2020 2020       |.|  7     
│ │ │ │ -00058450: 3720 2020 3820 2020 2020 3820 2020 3920  7   8     8   9 
│ │ │ │ -00058460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058490: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -000584a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000584b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000584c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000584d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000584e0: 2d2d 2d2d 2d2b 0a7c 6935 203a 2073 6d61  -----+.|i5 : sma
│ │ │ │ -000584f0: 6c6c 6573 7443 7963 6c65 5369 7a65 2068  llestCycleSize h
│ │ │ │ +00058440: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00058450: 2020 2022 7665 7274 6963 6573 2220 3d3e     "vertices" =>
│ │ │ │ +00058460: 207b 7820 2c20 7820 2c20 7820 2c20 7820   {x , x , x , x 
│ │ │ │ +00058470: 2c20 7820 2c20 7820 2c20 7820 2c20 7820  , x , x , x , x 
│ │ │ │ +00058480: 2c20 7820 7d20 2020 2020 2020 2020 2020  , x }           
│ │ │ │ +00058490: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000584a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000584b0: 2020 2031 2020 2032 2020 2033 2020 2034     1   2   3   4
│ │ │ │ +000584c0: 2020 2035 2020 2036 2020 2037 2020 2038     5   6   7   8
│ │ │ │ +000584d0: 2020 2039 2020 2020 2020 2020 2020 2020     9            
│ │ │ │ +000584e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000584f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00058500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00058510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00058520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058530: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00058540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058530: 2020 2020 207c 0a7c 6f34 203a 2047 7261       |.|o4 : Gra
│ │ │ │ +00058540: 7068 2020 2020 2020 2020 2020 2020 2020  ph              
│ │ │ │  00058550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00058560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00058570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058580: 2020 2020 207c 0a7c 6f35 203d 2069 6e66       |.|o5 = inf
│ │ │ │ -00058590: 696e 6974 7920 2020 2020 2020 2020 2020  inity           
│ │ │ │ -000585a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000585b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000585c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000585d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000585e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000585f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058580: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +00058590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000585a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000585b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000585c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000585d0: 2d2d 2d2d 2d7c 0a7c 5f39 7d20 2d2d 2061  -----|.|_9} -- a
│ │ │ │ +000585e0: 2074 7265 6520 286e 6f20 6379 636c 6573   tree (no cycles
│ │ │ │ +000585f0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  00058600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00058610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058620: 2020 2020 207c 0a7c 6f35 203a 2049 6e66       |.|o5 : Inf
│ │ │ │ -00058630: 696e 6974 654e 756d 6265 7220 2020 2020  initeNumber     
│ │ │ │ +00058620: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00058630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00058640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00058650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00058660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058670: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -00058680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000586a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000586b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000586c0: 2d2d 2d2d 2d2b 0a7c 6936 203a 206c 203d  -----+.|i6 : l =
│ │ │ │ -000586d0: 2020 6772 6170 6820 7b78 5f31 2a78 5f32    graph {x_1*x_2
│ │ │ │ -000586e0: 2c78 5f32 2a78 5f33 2c78 5f33 2a78 5f34  ,x_2*x_3,x_3*x_4
│ │ │ │ -000586f0: 2c78 5f34 2a78 5f35 2c78 5f35 2a78 5f36  ,x_4*x_5,x_5*x_6
│ │ │ │ -00058700: 2c78 5f36 2a78 5f37 2c78 5f37 2a78 5f38  ,x_6*x_7,x_7*x_8
│ │ │ │ -00058710: 2c78 5f38 2a7c 0a7c 2020 2020 2020 2020  ,x_8*|.|        
│ │ │ │ -00058720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058760: 2020 2020 207c 0a7c 6f36 203d 2047 7261       |.|o6 = Gra
│ │ │ │ -00058770: 7068 7b22 6564 6765 7322 203d 3e20 7b7b  ph{"edges" => {{
│ │ │ │ -00058780: 7820 2c20 7820 7d2c 207b 7820 2c20 7820  x , x }, {x , x 
│ │ │ │ -00058790: 7d2c 207b 7820 2c20 7820 7d2c 207b 7820  }, {x , x }, {x 
│ │ │ │ -000587a0: 2c20 7820 7d2c 207b 7820 2c20 7820 7d2c  , x }, {x , x },
│ │ │ │ -000587b0: 207b 7820 2c7c 0a7c 2020 2020 2020 2020   {x ,|.|        
│ │ │ │ +00058670: 2020 2020 207c 0a7c 2078 207d 2c20 7b78       |.| x }, {x
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│ │ │ │ -00058f70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ -00059660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00059670: 2d2d 2d2b 0a7c 6935 203a 2054 203d 3d20  ---+.|i5 : T == 
│ │ │ │ -00059680: 7370 616e 6e69 6e67 5472 6565 2054 2020  spanningTree T  
│ │ │ │ -00059690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000596a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00059620: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00059630: 2022 7665 7274 6963 6573 2220 3d3e 207b   "vertices" => {
│ │ │ │ +00059640: 7820 2c20 7820 2c20 7820 2c20 7820 2c20  x , x , x , x , 
│ │ │ │ +00059650: 7820 2c20 7820 7d20 2020 2020 2020 2020  x , x }         
│ │ │ │ +00059660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00059670: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00059680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00059690: 2031 2020 2032 2020 2033 2020 2034 2020   1   2   3   4  
│ │ │ │ +000596a0: 2035 2020 2036 2020 2020 2020 2020 2020   5   6          
│ │ │ │  000596b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000596c0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  000596d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000596e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000596f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059710: 2020 207c 0a7c 6f35 203d 2074 7275 6520     |.|o5 = true 
│ │ │ │ +00059710: 2020 207c 0a7c 6f33 203a 2047 7261 7068     |.|o3 : Graph
│ │ │ │  00059720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059760: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  00059770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00059780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00059790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000597a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000597b0: 2d2d 2d2b 0a7c 6936 203a 2047 203d 2067  ---+.|i6 : G = g
│ │ │ │ +000597b0: 2d2d 2d2b 0a7c 6934 203a 2054 203d 2067  ---+.|i4 : T = g
│ │ │ │  000597c0: 7261 7068 207b 785f 312a 785f 322c 785f  raph {x_1*x_2,x_
│ │ │ │ -000597d0: 322a 785f 332c 785f 332a 785f 312c 785f  2*x_3,x_3*x_1,x_
│ │ │ │ -000597e0: 342a 785f 352c 785f 352a 785f 362c 785f  4*x_5,x_5*x_6,x_
│ │ │ │ -000597f0: 362a 785f 347d 3b20 2d2d 2074 776f 2020  6*x_4}; -- two  
│ │ │ │ +000597d0: 322a 785f 332c 2078 5f31 2a78 5f34 2c78  2*x_3, x_1*x_4,x
│ │ │ │ +000597e0: 5f31 2a78 5f35 2c78 5f35 2a78 5f36 7d3b  _1*x_5,x_5*x_6};
│ │ │ │ +000597f0: 202d 2d20 6120 7472 6565 2028 6e6f 2020   -- a tree (no  
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│ │ │ │  00059810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00059820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00059830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00059840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00059850: 2d2d 2d7c 0a7c 7468 7265 6520 6379 636c  ---|.|three cycl
│ │ │ │ -00059860: 6573 2020 2020 2020 2020 2020 2020 2020  es              
│ │ │ │ +00059850: 2d2d 2d7c 0a7c 6379 636c 6573 2920 2020  ---|.|cycles)   
│ │ │ │ +00059860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000598a0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  000598b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000598c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000598d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000598e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000598f0: 2d2d 2d2b 0a7c 6937 203a 2073 7061 6e6e  ---+.|i7 : spann
│ │ │ │ -00059900: 696e 6754 7265 6520 4720 2020 2020 2020  ingTree G       
│ │ │ │ +000598f0: 2d2d 2d2b 0a7c 6935 203a 2054 203d 3d20  ---+.|i5 : T == 
│ │ │ │ +00059900: 7370 616e 6e69 6e67 5472 6565 2054 2020  spanningTree T  
│ │ │ │  00059910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059940: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00059950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059990: 2020 207c 0a7c 6f37 203d 2047 7261 7068     |.|o7 = Graph
│ │ │ │ -000599a0: 7b22 6564 6765 7322 203d 3e20 7b7b 7820  {"edges" => {{x 
│ │ │ │ -000599b0: 2c20 7820 7d2c 207b 7820 2c20 7820 7d2c  , x }, {x , x },
│ │ │ │ -000599c0: 207b 7820 2c20 7820 7d2c 207b 7820 2c20   {x , x }, {x , 
│ │ │ │ -000599d0: 7820 7d7d 7d20 2020 2020 2020 2020 2020  x }}}           
│ │ │ │ -000599e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -000599f0: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ -00059a00: 2020 2032 2020 2020 2031 2020 2033 2020     2     1   3  
│ │ │ │ -00059a10: 2020 2034 2020 2035 2020 2020 2034 2020     4   5     4  
│ │ │ │ -00059a20: 2036 2020 2020 2020 2020 2020 2020 2020   6              
│ │ │ │ -00059a30: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00059a40: 2022 7269 6e67 2220 3d3e 2052 2020 2020   "ring" => R    
│ │ │ │ -00059a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059a80: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00059a90: 2022 7665 7274 6963 6573 2220 3d3e 207b   "vertices" => {
│ │ │ │ -00059aa0: 7820 2c20 7820 2c20 7820 2c20 7820 2c20  x , x , x , x , 
│ │ │ │ -00059ab0: 7820 2c20 7820 7d20 2020 2020 2020 2020  x , x }         
│ │ │ │ -00059ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059ad0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00059ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059af0: 2031 2020 2032 2020 2033 2020 2034 2020   1   2   3   4  
│ │ │ │ -00059b00: 2035 2020 2036 2020 2020 2020 2020 2020   5   6          
│ │ │ │ +00059990: 2020 207c 0a7c 6f35 203d 2074 7275 6520     |.|o5 = true 
│ │ │ │ +000599a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000599b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000599c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000599d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000599e0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +000599f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00059a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00059a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00059a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00059a30: 2d2d 2d2b 0a7c 6936 203a 2047 203d 2067  ---+.|i6 : G = g
│ │ │ │ +00059a40: 7261 7068 207b 785f 312a 785f 322c 785f  raph {x_1*x_2,x_
│ │ │ │ +00059a50: 322a 785f 332c 785f 332a 785f 312c 785f  2*x_3,x_3*x_1,x_
│ │ │ │ +00059a60: 342a 785f 352c 785f 352a 785f 362c 785f  4*x_5,x_5*x_6,x_
│ │ │ │ +00059a70: 362a 785f 347d 3b20 2d2d 2074 776f 2020  6*x_4}; -- two  
│ │ │ │ +00059a80: 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d     |.|----------
│ │ │ │ +00059a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00059aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00059ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00059ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00059ad0: 2d2d 2d7c 0a7c 7468 7265 6520 6379 636c  ---|.|three cycl
│ │ │ │ +00059ae0: 6573 2020 2020 2020 2020 2020 2020 2020  es              
│ │ │ │ +00059af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00059b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059b20: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00059b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059b70: 2020 207c 0a7c 6f37 203a 2047 7261 7068     |.|o7 : Graph
│ │ │ │ -00059b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00059b20: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00059b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00059b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00059b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00059b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00059b70: 2d2d 2d2b 0a7c 6937 203a 2073 7061 6e6e  ---+.|i7 : spann
│ │ │ │ +00059b80: 696e 6754 7265 6520 4720 2020 2020 2020  ingTree G       
│ │ │ │  00059b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059bc0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -00059bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00059be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00059bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00059c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00059c10: 2d2d 2d2b 0a0a 5761 7973 2074 6f20 7573  ---+..Ways to us
│ │ │ │ -00059c20: 6520 7370 616e 6e69 6e67 5472 6565 3a0a  e spanningTree:.
│ │ │ │ -00059c30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00059c40: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022  =========..  * "
│ │ │ │ -00059c50: 7370 616e 6e69 6e67 5472 6565 2847 7261  spanningTree(Gra
│ │ │ │ -00059c60: 7068 2922 0a0a 466f 7220 7468 6520 7072  ph)"..For the pr
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│ │ │ │ -00059d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00059d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00059d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00059d90: 2e31 312b 6473 2f4d 322f 4d61 6361 756c  .11+ds/M2/Macaul
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│ │ │ │ -00059df0: 7874 3a20 7665 7274 6578 436f 7665 724e  xt: vertexCoverN
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│ │ │ │ -00059e10: 6e6e 696e 6754 7265 652c 2055 703a 2054  nningTree, Up: T
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│ │ │ │ -00059e60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00059e70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00059e80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a44 6573  ***********..Des
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│ │ │ │ -00059ea0: 3d3d 3d3d 0a0a 5468 6520 2a6e 6f74 6520  ====..The *note 
│ │ │ │ -00059eb0: 7261 6e64 6f6d 4879 7065 7247 7261 7068  randomHyperGraph
│ │ │ │ -00059ec0: 3a20 7261 6e64 6f6d 4879 7065 7247 7261  : randomHyperGra
│ │ │ │ -00059ed0: 7068 2c20 6d65 7468 6f64 2066 6f6c 6c6f  ph, method follo
│ │ │ │ -00059ee0: 7773 2061 2062 6163 6b74 7261 636b 696e  ws a backtrackin
│ │ │ │ -00059ef0: 670a 616c 676f 7269 7468 6d20 746f 2067  g.algorithm to g
│ │ │ │ -00059f00: 656e 6572 6174 6520 6564 6765 7320 7769  enerate edges wi
│ │ │ │ -00059f10: 7468 206e 6f20 696e 636c 7573 696f 6e73  th no inclusions
│ │ │ │ -00059f20: 2062 6574 7765 656e 2074 6865 6d2e 2041   between them. A
│ │ │ │ -00059f30: 7320 6974 206d 6179 2074 616b 6520 610a  s it may take a.
│ │ │ │ -00059f40: 6c6f 6e67 2074 696d 6520 746f 2074 6572  long time to ter
│ │ │ │ -00059f50: 6d69 6e61 7465 2c20 6120 7469 6d65 206c  minate, a time l
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│ │ │ │ -00059f70: 6f6e 2074 6865 206d 6574 686f 642e 2054  on the method. T
│ │ │ │ -00059f80: 6865 2054 696d 654c 696d 6974 0a6f 7074  he TimeLimit.opt
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│ │ │ │ -00059fb0: 636f 6e64 732c 2074 6861 7420 7468 6520  conds, that the 
│ │ │ │ -00059fc0: 6d65 7468 6f64 202a 6e6f 7465 0a72 616e  method *note.ran
│ │ │ │ -00059fd0: 646f 6d48 7970 6572 4772 6170 683a 2072  domHyperGraph: r
│ │ │ │ -00059fe0: 616e 646f 6d48 7970 6572 4772 6170 682c  andomHyperGraph,
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│ │ │ │ -0005a010: 2076 616c 7565 206f 6620 300a 6973 2069   value of 0.is i
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│ │ │ │ -0005a070: 202a 6e6f 7465 2042 7261 6e63 684c 696d   *note BranchLim
│ │ │ │ -0005a080: 6974 3a20 4272 616e 6368 4c69 6d69 742c  it: BranchLimit,
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│ │ │ │ -0005a0b0: 4879 7065 7247 7261 7068 0a20 202a 202a  HyperGraph.  * *
│ │ │ │ -0005a0c0: 6e6f 7465 2072 616e 646f 6d48 7970 6572  note randomHyper
│ │ │ │ -0005a0d0: 4772 6170 683a 2072 616e 646f 6d48 7970  Graph: randomHyp
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│ │ │ │ +0005a170: 670a 616c 676f 7269 7468 6d20 746f 2067  g.algorithm to g
│ │ │ │ +0005a180: 656e 6572 6174 6520 6564 6765 7320 7769  enerate edges wi
│ │ │ │ +0005a190: 7468 206e 6f20 696e 636c 7573 696f 6e73  th no inclusions
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│ │ │ │ +0005a1b0: 7320 6974 206d 6179 2074 616b 6520 610a  s it may take a.
│ │ │ │ +0005a1c0: 6c6f 6e67 2074 696d 6520 746f 2074 6572  long time to ter
│ │ │ │ +0005a1d0: 6d69 6e61 7465 2c20 6120 7469 6d65 206c  minate, a time l
│ │ │ │ +0005a1e0: 696d 6974 2069 7320 696d 706f 7365 6420  imit is imposed 
│ │ │ │ +0005a1f0: 6f6e 2074 6865 206d 6574 686f 642e 2054  on the method. T
│ │ │ │ +0005a200: 6865 2054 696d 654c 696d 6974 0a6f 7074  he TimeLimit.opt
│ │ │ │ +0005a210: 696f 6e20 6973 2074 6865 2061 6d6f 756e  ion is the amoun
│ │ │ │ +0005a220: 7420 6f66 2074 696d 652c 2069 6e20 7365  t of time, in se
│ │ │ │ +0005a230: 636f 6e64 732c 2074 6861 7420 7468 6520  conds, that the 
│ │ │ │ +0005a240: 6d65 7468 6f64 202a 6e6f 7465 0a72 616e  method *note.ran
│ │ │ │ +0005a250: 646f 6d48 7970 6572 4772 6170 683a 2072  domHyperGraph: r
│ │ │ │ +0005a260: 616e 646f 6d48 7970 6572 4772 6170 682c  andomHyperGraph,
│ │ │ │ +0005a270: 2077 696c 6c20 7461 6b65 2062 6566 6f72   will take befor
│ │ │ │ +0005a280: 6520 7465 726d 696e 6174 696e 672e 2041  e terminating. A
│ │ │ │ +0005a290: 2076 616c 7565 206f 6620 300a 6973 2069   value of 0.is i
│ │ │ │ +0005a2a0: 6e74 6572 7072 6574 6564 2061 7320 6f6e  nterpreted as on
│ │ │ │ +0005a2b0: 6520 6461 792e 2054 6865 2064 6566 6175  e day. The defau
│ │ │ │ +0005a2c0: 6c74 2076 616c 7565 2069 7320 3520 2873  lt value is 5 (s
│ │ │ │ +0005a2d0: 6563 6f6e 6473 292e 0a0a 5365 6520 616c  econds)...See al
│ │ │ │ +0005a2e0: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  so.========..  *
│ │ │ │ +0005a2f0: 202a 6e6f 7465 2042 7261 6e63 684c 696d   *note BranchLim
│ │ │ │ +0005a300: 6974 3a20 4272 616e 6368 4c69 6d69 742c  it: BranchLimit,
│ │ │ │ +0005a310: 202d 2d20 6f70 7469 6f6e 616c 2061 7267   -- optional arg
│ │ │ │ +0005a320: 756d 656e 7420 666f 7220 7261 6e64 6f6d  ument for random
│ │ │ │ +0005a330: 4879 7065 7247 7261 7068 0a20 202a 202a  HyperGraph.  * *
│ │ │ │ +0005a340: 6e6f 7465 2072 616e 646f 6d48 7970 6572  note randomHyper
│ │ │ │ +0005a350: 4772 6170 683a 2072 616e 646f 6d48 7970  Graph: randomHyp
│ │ │ │ +0005a360: 6572 4772 6170 682c 202d 2d20 7265 7475  erGraph, -- retu
│ │ │ │ +0005a370: 726e 7320 6120 7261 6e64 6f6d 2068 7970  rns a random hyp
│ │ │ │ +0005a380: 6572 6772 6170 680a 0a46 756e 6374 696f  ergraph..Functio
│ │ │ │ +0005a390: 6e73 2077 6974 6820 6f70 7469 6f6e 616c  ns with optional
│ │ │ │ +0005a3a0: 2061 7267 756d 656e 7420 6e61 6d65 6420   argument named 
│ │ │ │ +0005a3b0: 5469 6d65 4c69 6d69 743a 0a3d 3d3d 3d3d  TimeLimit:.=====
│ │ │ │ +0005a3c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0005a3d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0005a3e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020  ============..  
│ │ │ │ +0005a3f0: 2a20 2a6e 6f74 6520 7261 6e64 6f6d 4879  * *note randomHy
│ │ │ │ +0005a400: 7065 7247 7261 7068 282e 2e2e 2c54 696d  perGraph(...,Tim
│ │ │ │ +0005a410: 654c 696d 6974 3d3e 2e2e 2e29 3a0a 2020  eLimit=>...):.  
│ │ │ │ +0005a420: 2020 7261 6e64 6f6d 4879 7065 7247 7261    randomHyperGra
│ │ │ │ +0005a430: 7068 5f6c 705f 7064 5f70 645f 7064 5f63  ph_lp_pd_pd_pd_c
│ │ │ │ +0005a440: 6d54 696d 654c 696d 6974 3d3e 5f70 645f  mTimeLimit=>_pd_
│ │ │ │ +0005a450: 7064 5f70 645f 7270 2c20 2d2d 206c 696d  pd_pd_rp, -- lim
│ │ │ │ +0005a460: 6974 2074 6865 0a20 2020 2072 756e 6e69  it the.    runni
│ │ │ │ +0005a470: 6e67 2074 696d 6520 6f66 2072 616e 646f  ng time of rando
│ │ │ │ +0005a480: 6d48 7970 6572 4772 6170 680a 0a46 6f72  mHyperGraph..For
│ │ │ │ +0005a490: 2074 6865 2070 726f 6772 616d 6d65 720a   the programmer.
│ │ │ │ +0005a4a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0005a4b0: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a  ==..The object *
│ │ │ │ +0005a4c0: 6e6f 7465 2054 696d 654c 696d 6974 3a20  note TimeLimit: 
│ │ │ │ +0005a4d0: 5469 6d65 4c69 6d69 742c 2069 7320 6120  TimeLimit, is a 
│ │ │ │ +0005a4e0: 2a6e 6f74 6520 7379 6d62 6f6c 3a0a 284d  *note symbol:.(M
│ │ │ │ +0005a4f0: 6163 6175 6c61 7932 446f 6329 5379 6d62  acaulay2Doc)Symb
│ │ │ │ +0005a500: 6f6c 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d  ol,...----------
│ │ │ │ +0005a510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005a520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005a530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005a540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005a550: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263  -----..The sourc
│ │ │ │ +0005a560: 6520 6f66 2074 6869 7320 646f 6375 6d65  e of this docume
│ │ │ │ +0005a570: 6e74 2069 7320 696e 0a2f 6275 696c 642f  nt is in./build/
│ │ │ │ +0005a580: 7265 7072 6f64 7563 6962 6c65 2d70 6174  reproducible-pat
│ │ │ │ +0005a590: 682f 6d61 6361 756c 6179 322d 312e 3235  h/macaulay2-1.25
│ │ │ │ +0005a5a0: 2e31 312b 6473 2f4d 322f 4d61 6361 756c  .11+ds/M2/Macaul
│ │ │ │ +0005a5b0: 6179 322f 7061 636b 6167 6573 2f45 6467  ay2/packages/Edg
│ │ │ │ +0005a5c0: 6549 6465 616c 732e 0a6d 323a 3433 3638  eIdeals..m2:4368
│ │ │ │ +0005a5d0: 3a30 2e0a 1f0a 4669 6c65 3a20 4564 6765  :0....File: Edge
│ │ │ │ +0005a5e0: 4964 6561 6c73 2e69 6e66 6f2c 204e 6f64  Ideals.info, Nod
│ │ │ │ +0005a5f0: 653a 2076 6572 7465 7843 6f76 6572 4e75  e: vertexCoverNu
│ │ │ │ +0005a600: 6d62 6572 2c20 4e65 7874 3a20 7665 7274  mber, Next: vert
│ │ │ │ +0005a610: 6578 436f 7665 7273 2c20 5072 6576 3a20  exCovers, Prev: 
│ │ │ │ +0005a620: 5469 6d65 4c69 6d69 742c 2055 703a 2054  TimeLimit, Up: T
│ │ │ │ +0005a630: 6f70 0a0a 7665 7274 6578 436f 7665 724e  op..vertexCoverN
│ │ │ │ +0005a640: 756d 6265 7220 2d2d 2066 696e 6420 7468  umber -- find th
│ │ │ │ +0005a650: 6520 7665 7274 6578 2063 6f76 6572 696e  e vertex coverin
│ │ │ │ +0005a660: 6720 6e75 6d62 6572 206f 6620 6120 2868  g number of a (h
│ │ │ │ +0005a670: 7970 6572 2967 7261 7068 0a2a 2a2a 2a2a  yper)graph.*****
│ │ │ │ +0005a680: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0005a690: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0005a6a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0005a6b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0005a6c0: 2a0a 0a20 202a 2055 7361 6765 3a20 0a20  *..  * Usage: . 
│ │ │ │ +0005a6d0: 2020 2020 2020 2063 203d 2076 6572 7465         c = verte
│ │ │ │ +0005a6e0: 7843 6f76 6572 4e75 6d62 6572 2848 290a  xCoverNumber(H).
│ │ │ │ +0005a6f0: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ +0005a700: 2020 2a20 482c 2061 202a 6e6f 7465 2068    * H, a *note h
│ │ │ │ +0005a710: 7970 6572 6772 6170 683a 2048 7970 6572  ypergraph: Hyper
│ │ │ │ +0005a720: 4772 6170 682c 2c20 7468 6520 696e 7075  Graph,, the inpu
│ │ │ │ +0005a730: 740a 2020 2a20 4f75 7470 7574 733a 0a20  t.  * Outputs:. 
│ │ │ │ +0005a740: 2020 2020 202a 2063 2c20 616e 202a 6e6f       * c, an *no
│ │ │ │ +0005a750: 7465 2069 6e74 6567 6572 3a20 284d 6163  te integer: (Mac
│ │ │ │ +0005a760: 6175 6c61 7932 446f 6329 5a5a 2c2c 2074  aulay2Doc)ZZ,, t
│ │ │ │ +0005a770: 6865 2076 6572 7465 7820 636f 7665 7269  he vertex coveri
│ │ │ │ +0005a780: 6e67 206e 756d 6265 720a 0a44 6573 6372  ng number..Descr
│ │ │ │ +0005a790: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d  iption.=========
│ │ │ │ +0005a7a0: 3d3d 0a0a 5468 6973 2066 756e 6374 696f  ==..This functio
│ │ │ │ +0005a7b0: 6e20 7265 7475 726e 7320 7468 6520 7665  n returns the ve
│ │ │ │ +0005a7c0: 7274 6578 2063 6f76 6572 696e 6720 6e75  rtex covering nu
│ │ │ │ +0005a7d0: 6d62 6572 206f 6620 6120 2868 7970 6572  mber of a (hyper
│ │ │ │ +0005a7e0: 2967 7261 7068 2e20 5468 6520 7665 7274  )graph. The vert
│ │ │ │ +0005a7f0: 6578 0a63 6f76 6572 696e 6720 6e75 6d62  ex.covering numb
│ │ │ │ +0005a800: 6572 2069 7320 7468 6520 7369 7a65 206f  er is the size o
│ │ │ │ +0005a810: 6620 736d 616c 6c65 7374 2076 6572 7465  f smallest verte
│ │ │ │ +0005a820: 7820 636f 7665 7220 6f66 2074 6865 2028  x cover of the (
│ │ │ │ +0005a830: 6879 7065 7229 6772 6170 682e 2020 5468  hyper)graph.  Th
│ │ │ │ +0005a840: 6973 0a63 6f72 7265 7370 6f6e 6473 2074  is.corresponds t
│ │ │ │ +0005a850: 6f20 7468 6520 736d 616c 6c65 7374 2064  o the smallest d
│ │ │ │ +0005a860: 6567 7265 6520 6f66 2061 2067 656e 6572  egree of a gener
│ │ │ │ +0005a870: 6174 6f72 206f 6620 7468 6520 636f 7665  ator of the cove
│ │ │ │ +0005a880: 7220 6964 6561 6c20 6f66 2074 6865 0a28  r ideal of the.(
│ │ │ │ +0005a890: 6879 7065 7229 6772 6170 682e 0a0a 2b2d  hyper)graph...+-
│ │ │ │ +0005a8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005a8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005a8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005a8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005a8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0005a8f0: 3120 3a20 5320 3d20 5151 5b61 2e2e 645d  1 : S = QQ[a..d]
│ │ │ │ +0005a900: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │  0005a910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005a920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005a930: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │  0005a940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005a950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005a960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005a970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005a980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0005a990: 3320 3a20 7665 7274 6578 436f 7665 724e  3 : vertexCoverN
│ │ │ │ -0005a9a0: 756d 6265 7220 6720 2020 2020 2020 2020  umber g         
│ │ │ │ -0005a9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a990: 3220 3a20 6720 3d20 6772 6170 6820 7b61  2 : g = graph {a
│ │ │ │ +0005a9a0: 2a62 2c62 2a63 2c63 2a64 2c64 2a61 7d20  *b,b*c,c*d,d*a} 
│ │ │ │ +0005a9b0: 2d2d 2074 6865 2066 6f75 7220 6379 636c  -- the four cycl
│ │ │ │ +0005a9c0: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │  0005a9d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0005a9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005a9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005aa00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005aa10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005aa20: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0005aa30: 3320 3d20 3220 2020 2020 2020 2020 2020  3 = 2           
│ │ │ │ -0005aa40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005aa50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005aa60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005aa70: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -0005aa80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005aa90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005aaa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005aab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005aac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0005aad0: 3420 3a20 5320 3d20 5151 5b61 2e2e 655d  4 : S = QQ[a..e]
│ │ │ │ -0005aae0: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ -0005aaf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005aa30: 3220 3d20 4772 6170 687b 2265 6467 6573  2 = Graph{"edges
│ │ │ │ +0005aa40: 2220 3d3e 207b 7b61 2c20 627d 2c20 7b62  " => {{a, b}, {b
│ │ │ │ +0005aa50: 2c20 637d 2c20 7b61 2c20 647d 2c20 7b63  , c}, {a, d}, {c
│ │ │ │ +0005aa60: 2c20 647d 7d7d 2020 2020 2020 2020 2020  , d}}}          
│ │ │ │ +0005aa70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0005aa80: 2020 2020 2020 2020 2020 2272 696e 6722            "ring"
│ │ │ │ +0005aa90: 203d 3e20 5320 2020 2020 2020 2020 2020   => S           
│ │ │ │ +0005aaa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005aab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005aac0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0005aad0: 2020 2020 2020 2020 2020 2276 6572 7469            "verti
│ │ │ │ +0005aae0: 6365 7322 203d 3e20 7b61 2c20 622c 2063  ces" => {a, b, c
│ │ │ │ +0005aaf0: 2c20 647d 2020 2020 2020 2020 2020 2020  , d}            
│ │ │ │  0005ab00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ab10: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -0005ab20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ab30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ab40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ab50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ab60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0005ab70: 3520 3a20 6720 3d20 6772 6170 6820 7b61  5 : g = graph {a
│ │ │ │ -0005ab80: 2a62 2c61 2a63 2c61 2a64 2c61 2a65 2c62  *b,a*c,a*d,a*e,b
│ │ │ │ -0005ab90: 2a63 2c62 2a64 2c62 2a65 2c63 2a64 2c63  *c,b*d,b*e,c*d,c
│ │ │ │ -0005aba0: 2a65 2c64 2a65 7d20 2d2d 2074 6865 2063  *e,d*e} -- the c
│ │ │ │ -0005abb0: 6f6d 706c 6574 6520 2020 2020 7c0a 7c20  omplete     |.| 
│ │ │ │ -0005abc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005abd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005abe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005abf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ac00: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0005ac10: 3520 3d20 4772 6170 687b 2265 6467 6573  5 = Graph{"edges
│ │ │ │ -0005ac20: 2220 3d3e 207b 7b61 2c20 627d 2c20 7b61  " => {{a, b}, {a
│ │ │ │ -0005ac30: 2c20 637d 2c20 7b62 2c20 637d 2c20 7b61  , c}, {b, c}, {a
│ │ │ │ -0005ac40: 2c20 647d 2c20 7b62 2c20 647d 2c20 7b63  , d}, {b, d}, {c
│ │ │ │ -0005ac50: 2c20 647d 2c20 7b61 2020 2020 7c0a 7c20  , d}, {a    |.| 
│ │ │ │ -0005ac60: 2020 2020 2020 2020 2020 2272 696e 6722            "ring"
│ │ │ │ -0005ac70: 203d 3e20 5320 2020 2020 2020 2020 2020   => S           
│ │ │ │ +0005ab10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0005ab20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ab30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ab40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ab50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ab60: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0005ab70: 3220 3a20 4772 6170 6820 2020 2020 2020  2 : Graph       
│ │ │ │ +0005ab80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ab90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005aba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005abb0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0005abc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005abd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005abe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005abf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005ac00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0005ac10: 3320 3a20 7665 7274 6578 436f 7665 724e  3 : vertexCoverN
│ │ │ │ +0005ac20: 756d 6265 7220 6720 2020 2020 2020 2020  umber g         
│ │ │ │ +0005ac30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ac40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ac50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0005ac60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ac70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ac80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ac90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005aca0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0005acb0: 2020 2020 2020 2020 2020 2276 6572 7469            "verti
│ │ │ │ -0005acc0: 6365 7322 203d 3e20 7b61 2c20 622c 2063  ces" => {a, b, c
│ │ │ │ -0005acd0: 2c20 642c 2065 7d20 2020 2020 2020 2020  , d, e}         
│ │ │ │ +0005aca0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0005acb0: 3320 3d20 3220 2020 2020 2020 2020 2020  3 = 2           
│ │ │ │ +0005acc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005acd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ace0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005acf0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0005ad00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ad10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ad20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ad30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ad40: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0005ad50: 3520 3a20 4772 6170 6820 2020 2020 2020  5 : Graph       
│ │ │ │ -0005ad60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005acf0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0005ad00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005ad10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005ad20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005ad30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005ad40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0005ad50: 3420 3a20 5320 3d20 5151 5b61 2e2e 655d  4 : S = QQ[a..e]
│ │ │ │ +0005ad60: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │  0005ad70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ad80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ad90: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ +0005ad90: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │  0005ada0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005adb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005adc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005add0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ade0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c67  ------------|.|g
│ │ │ │ -0005adf0: 7261 7068 204b 5f35 2020 2020 2020 2020  raph K_5        
│ │ │ │ -0005ae00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ae10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ae20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ae30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0005ade0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0005adf0: 3520 3a20 6720 3d20 6772 6170 6820 7b61  5 : g = graph {a
│ │ │ │ +0005ae00: 2a62 2c61 2a63 2c61 2a64 2c61 2a65 2c62  *b,a*c,a*d,a*e,b
│ │ │ │ +0005ae10: 2a63 2c62 2a64 2c62 2a65 2c63 2a64 2c63  *c,b*d,b*e,c*d,c
│ │ │ │ +0005ae20: 2a65 2c64 2a65 7d20 2d2d 2074 6865 2063  *e,d*e} -- the c
│ │ │ │ +0005ae30: 6f6d 706c 6574 6520 2020 2020 7c0a 7c20  omplete     |.| 
│ │ │ │  0005ae40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ae50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ae60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ae70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ae80: 2020 2020 2020 2020 2020 2020 7c0a 7c2c              |.|,
│ │ │ │ -0005ae90: 2065 7d2c 207b 622c 2065 7d2c 207b 632c   e}, {b, e}, {c,
│ │ │ │ -0005aea0: 2065 7d2c 207b 642c 2065 7d7d 7d20 2020   e}, {d, e}}}   
│ │ │ │ -0005aeb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005aec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005aed0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -0005aee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005aef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005af00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005af10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005af20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0005af30: 3620 3a20 7665 7274 6578 436f 7665 724e  6 : vertexCoverN
│ │ │ │ -0005af40: 756d 6265 7220 6720 2020 2020 2020 2020  umber g         
│ │ │ │ -0005af50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ae80: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0005ae90: 3520 3d20 4772 6170 687b 2265 6467 6573  5 = Graph{"edges
│ │ │ │ +0005aea0: 2220 3d3e 207b 7b61 2c20 627d 2c20 7b61  " => {{a, b}, {a
│ │ │ │ +0005aeb0: 2c20 637d 2c20 7b62 2c20 637d 2c20 7b61  , c}, {b, c}, {a
│ │ │ │ +0005aec0: 2c20 647d 2c20 7b62 2c20 647d 2c20 7b63  , d}, {b, d}, {c
│ │ │ │ +0005aed0: 2c20 647d 2c20 7b61 2020 2020 7c0a 7c20  , d}, {a    |.| 
│ │ │ │ +0005aee0: 2020 2020 2020 2020 2020 2272 696e 6722            "ring"
│ │ │ │ +0005aef0: 203d 3e20 5320 2020 2020 2020 2020 2020   => S           
│ │ │ │ +0005af00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005af10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005af20: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0005af30: 2020 2020 2020 2020 2020 2276 6572 7469            "verti
│ │ │ │ +0005af40: 6365 7322 203d 3e20 7b61 2c20 622c 2063  ces" => {a, b, c
│ │ │ │ +0005af50: 2c20 642c 2065 7d20 2020 2020 2020 2020  , d, e}         
│ │ │ │  0005af60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005af70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0005af80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005af90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005afa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005afb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005afc0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0005afd0: 3620 3d20 3420 2020 2020 2020 2020 2020  6 = 4           
│ │ │ │ +0005afd0: 3520 3a20 4772 6170 6820 2020 2020 2020  5 : Graph       
│ │ │ │  0005afe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005aff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005b000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005b010: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0005b010: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │  0005b020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005b030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005b040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005b050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005b060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0005b070: 3720 3a20 6820 3d20 6879 7065 7247 7261  7 : h = hyperGra
│ │ │ │ -0005b080: 7068 207b 612a 622a 632c 612a 642c 632a  ph {a*b*c,a*d,c*
│ │ │ │ -0005b090: 652c 622a 642a 657d 2020 2020 2020 2020  e,b*d*e}        
│ │ │ │ +0005b060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c67  ------------|.|g
│ │ │ │ +0005b070: 7261 7068 204b 5f35 2020 2020 2020 2020  raph K_5        
│ │ │ │ +0005b080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005b090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005b0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005b0b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0005b0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005b0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005b0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005b0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005b100: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0005b110: 3720 3d20 4879 7065 7247 7261 7068 7b22  7 = HyperGraph{"
│ │ │ │ -0005b120: 6564 6765 7322 203d 3e20 7b7b 612c 2062  edges" => {{a, b
│ │ │ │ -0005b130: 2c20 637d 2c20 7b61 2c20 647d 2c20 7b63  , c}, {a, d}, {c
│ │ │ │ -0005b140: 2c20 657d 2c20 7b62 2c20 642c 2065 7d7d  , e}, {b, d, e}}
│ │ │ │ -0005b150: 7d20 2020 2020 2020 2020 2020 7c0a 7c20  }           |.| 
│ │ │ │ -0005b160: 2020 2020 2020 2020 2020 2020 2020 2022                 "
│ │ │ │ -0005b170: 7269 6e67 2220 3d3e 2053 2020 2020 2020  ring" => S      
│ │ │ │ -0005b180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005b190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005b1a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0005b1b0: 2020 2020 2020 2020 2020 2020 2020 2022                 "
│ │ │ │ -0005b1c0: 7665 7274 6963 6573 2220 3d3e 207b 612c  vertices" => {a,
│ │ │ │ -0005b1d0: 2062 2c20 632c 2064 2c20 657d 2020 2020   b, c, d, e}    
│ │ │ │ +0005b100: 2020 2020 2020 2020 2020 2020 7c0a 7c2c              |.|,
│ │ │ │ +0005b110: 2065 7d2c 207b 622c 2065 7d2c 207b 632c   e}, {b, e}, {c,
│ │ │ │ +0005b120: 2065 7d2c 207b 642c 2065 7d7d 7d20 2020   e}, {d, e}}}   
│ │ │ │ +0005b130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005b140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005b150: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0005b160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005b170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005b180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005b190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005b1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0005b1b0: 3620 3a20 7665 7274 6578 436f 7665 724e  6 : vertexCoverN
│ │ │ │ +0005b1c0: 756d 6265 7220 6720 2020 2020 2020 2020  umber g         
│ │ │ │ +0005b1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005b1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005b1f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0005b200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005b210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005b220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005b230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005b240: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0005b250: 3720 3a20 4879 7065 7247 7261 7068 2020  7 : HyperGraph  
│ │ │ │ +0005b250: 3620 3d20 3420 2020 2020 2020 2020 2020  6 = 4           
│ │ │ │  0005b260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005b270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005b280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005b290: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │  0005b2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005b2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005b2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005b2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005b2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0005b2f0: 3820 3a20 7665 7274 6578 436f 7665 724e  8 : vertexCoverN
│ │ │ │ -0005b300: 756d 6265 7228 6829 2020 2020 2020 2020  umber(h)        
│ │ │ │ -0005b310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005b2f0: 3720 3a20 6820 3d20 6879 7065 7247 7261  7 : h = hyperGra
│ │ │ │ +0005b300: 7068 207b 612a 622a 632c 612a 642c 632a  ph {a*b*c,a*d,c*
│ │ │ │ +0005b310: 652c 622a 642a 657d 2020 2020 2020 2020  e,b*d*e}        
│ │ │ │  0005b320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005b330: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0005b340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005b350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005b360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005b370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005b380: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0005b390: 3820 3d20 3220 2020 2020 2020 2020 2020  8 = 2           
│ │ │ │ -0005b3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005b3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005b3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005b3d0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -0005b3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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 2d2d 2d2d 2d2d 2d2d 2b0a 0a53  ------------+..S
│ │ │ │ -0005b430: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d  ee also.========
│ │ │ │ -0005b440: 0a0a 2020 2a20 2a6e 6f74 6520 636f 7665  ..  * *note cove
│ │ │ │ -0005b450: 7249 6465 616c 3a20 636f 7665 7249 6465  rIdeal: coverIde
│ │ │ │ -0005b460: 616c 2c20 2d2d 2063 7265 6174 6573 2074  al, -- creates t
│ │ │ │ -0005b470: 6865 2063 6f76 6572 2069 6465 616c 206f  he cover ideal o
│ │ │ │ -0005b480: 6620 6120 2868 7970 6572 2967 7261 7068  f a (hyper)graph
│ │ │ │ -0005b490: 0a20 202a 202a 6e6f 7465 2076 6572 7465  .  * *note verte
│ │ │ │ -0005b4a0: 7843 6f76 6572 733a 2076 6572 7465 7843  xCovers: vertexC
│ │ │ │ -0005b4b0: 6f76 6572 732c 202d 2d20 6c69 7374 2074  overs, -- list t
│ │ │ │ -0005b4c0: 6865 206d 696e 696d 616c 2076 6572 7465  he minimal verte
│ │ │ │ -0005b4d0: 7820 636f 7665 7273 206f 6620 610a 2020  x covers of a.  
│ │ │ │ -0005b4e0: 2020 2868 7970 6572 2967 7261 7068 0a0a    (hyper)graph..
│ │ │ │ -0005b4f0: 5761 7973 2074 6f20 7573 6520 7665 7274  Ways to use vert
│ │ │ │ -0005b500: 6578 436f 7665 724e 756d 6265 723a 0a3d  exCoverNumber:.=
│ │ │ │ -0005b510: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0005b520: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20  =============.. 
│ │ │ │ -0005b530: 202a 2022 7665 7274 6578 436f 7665 724e   * "vertexCoverN
│ │ │ │ -0005b540: 756d 6265 7228 4879 7065 7247 7261 7068  umber(HyperGraph
│ │ │ │ -0005b550: 2922 0a0a 466f 7220 7468 6520 7072 6f67  )"..For the prog
│ │ │ │ -0005b560: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d  rammer.=========
│ │ │ │ -0005b570: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f  =========..The o
│ │ │ │ -0005b580: 626a 6563 7420 2a6e 6f74 6520 7665 7274  bject *note vert
│ │ │ │ -0005b590: 6578 436f 7665 724e 756d 6265 723a 2076  exCoverNumber: v
│ │ │ │ -0005b5a0: 6572 7465 7843 6f76 6572 4e75 6d62 6572  ertexCoverNumber
│ │ │ │ -0005b5b0: 2c20 6973 2061 202a 6e6f 7465 206d 6574  , is a *note met
│ │ │ │ -0005b5c0: 686f 640a 6675 6e63 7469 6f6e 3a20 284d  hod.function: (M
│ │ │ │ -0005b5d0: 6163 6175 6c61 7932 446f 6329 4d65 7468  acaulay2Doc)Meth
│ │ │ │ -0005b5e0: 6f64 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d  odFunction,...--
│ │ │ │ -0005b5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005b600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005b610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005b620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005b630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54  -------------..T
│ │ │ │ -0005b640: 6865 2073 6f75 7263 6520 6f66 2074 6869  he source of thi
│ │ │ │ -0005b650: 7320 646f 6375 6d65 6e74 2069 7320 696e  s document is in
│ │ │ │ -0005b660: 0a2f 6275 696c 642f 7265 7072 6f64 7563  ./build/reproduc
│ │ │ │ -0005b670: 6962 6c65 2d70 6174 682f 6d61 6361 756c  ible-path/macaul
│ │ │ │ -0005b680: 6179 322d 312e 3235 2e31 312b 6473 2f4d  ay2-1.25.11+ds/M
│ │ │ │ -0005b690: 322f 4d61 6361 756c 6179 322f 7061 636b  2/Macaulay2/pack
│ │ │ │ -0005b6a0: 6167 6573 2f45 6467 6549 6465 616c 732e  ages/EdgeIdeals.
│ │ │ │ -0005b6b0: 0a6d 323a 3431 3032 3a30 2e0a 1f0a 4669  .m2:4102:0....Fi
│ │ │ │ -0005b6c0: 6c65 3a20 4564 6765 4964 6561 6c73 2e69  le: EdgeIdeals.i
│ │ │ │ -0005b6d0: 6e66 6f2c 204e 6f64 653a 2076 6572 7465  nfo, Node: verte
│ │ │ │ -0005b6e0: 7843 6f76 6572 732c 204e 6578 743a 2076  xCovers, Next: v
│ │ │ │ -0005b6f0: 6572 7469 6365 735f 6c70 4879 7065 7247  ertices_lpHyperG
│ │ │ │ -0005b700: 7261 7068 5f72 702c 2050 7265 763a 2076  raph_rp, Prev: v
│ │ │ │ -0005b710: 6572 7465 7843 6f76 6572 4e75 6d62 6572  ertexCoverNumber
│ │ │ │ -0005b720: 2c20 5570 3a20 546f 700a 0a76 6572 7465  , Up: Top..verte
│ │ │ │ -0005b730: 7843 6f76 6572 7320 2d2d 206c 6973 7420  xCovers -- list 
│ │ │ │ -0005b740: 7468 6520 6d69 6e69 6d61 6c20 7665 7274  the minimal vert
│ │ │ │ -0005b750: 6578 2063 6f76 6572 7320 6f66 2061 2028  ex covers of a (
│ │ │ │ -0005b760: 6879 7065 7229 6772 6170 680a 2a2a 2a2a  hyper)graph.****
│ │ │ │ -0005b770: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0005b780: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0005b790: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0005b7a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020  ************..  
│ │ │ │ -0005b7b0: 2a20 5573 6167 653a 200a 2020 2020 2020  * Usage: .      
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│ │ │ │ -0005ba50: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0005ba10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ +0005bbd0: 6865 2028 6879 7065 7229 6772 6170 6820  he (hyper)graph 
│ │ │ │ +0005bbe0: 6861 730a 6e6f 6e2d 656d 7074 7920 696e  has.non-empty in
│ │ │ │ +0005bbf0: 7465 7273 6563 7469 6f6e 2077 6974 6820  tersection with 
│ │ │ │ +0005bc00: 7468 6973 2073 6574 2e20 2054 6865 206d  this set.  The m
│ │ │ │ +0005bc10: 696e 696d 616c 2076 6572 7465 7820 636f  inimal vertex co
│ │ │ │ +0005bc20: 7665 7273 2061 7265 2067 6976 656e 2062  vers are given b
│ │ │ │ +0005bc30: 790a 7468 6520 6d69 6e69 6d61 6c20 6765  y.the minimal ge
│ │ │ │ +0005bc40: 6e65 7261 746f 7273 206f 6620 7468 6520  nerators of the 
│ │ │ │ +0005bc50: 636f 7665 7220 6964 6561 6c20 6f66 2048  cover ideal of H
│ │ │ │ +0005bc60: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...+------------
│ │ │ │ +0005bc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005bc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005bc90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005bca0: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 5320  ------+.|i1 : S 
│ │ │ │ +0005bcb0: 3d20 5151 5b61 2e2e 645d 3b20 2020 2020  = QQ[a..d];     
│ │ │ │ +0005bcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005bcd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005bce0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0005bcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005bd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005bd10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005bd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005bd30: 2b0a 7c69 3220 3a20 6720 3d20 6772 6170  +.|i2 : g = grap
│ │ │ │ +0005bd40: 6820 7b61 2a62 2c62 2a63 2c63 2a64 2c64  h {a*b,b*c,c*d,d
│ │ │ │ +0005bd50: 2a61 7d20 2d2d 2074 6865 2066 6f75 7220  *a} -- the four 
│ │ │ │ +0005bd60: 6379 636c 6520 2020 2020 2020 2020 2020  cycle           
│ │ │ │ +0005bd70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0005bd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005bd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005bda0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0005bdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005bdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005bdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005bde0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0005bdf0: 3320 3a20 4c69 7374 2020 2020 2020 2020  3 : List        
│ │ │ │ -0005be00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005be10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005bda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005bdb0: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ +0005bdc0: 3d20 4772 6170 687b 2265 6467 6573 2220  = Graph{"edges" 
│ │ │ │ +0005bdd0: 3d3e 207b 7b61 2c20 627d 2c20 7b62 2c20  => {{a, b}, {b, 
│ │ │ │ +0005bde0: 637d 2c20 7b61 2c20 647d 2c20 7b63 2c20  c}, {a, d}, {c, 
│ │ │ │ +0005bdf0: 647d 7d7d 2020 2020 2020 2020 2020 207c  d}}}           |
│ │ │ │ +0005be00: 0a7c 2020 2020 2020 2020 2020 2022 7269  .|           "ri
│ │ │ │ +0005be10: 6e67 2220 3d3e 2053 2020 2020 2020 2020  ng" => S        
│ │ │ │  0005be20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005be30: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ -0005be40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005be50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005be60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005be70: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 636f  ------+.|i4 : co
│ │ │ │ -0005be80: 7665 7249 6465 616c 2067 2020 2020 2020  verIdeal g      
│ │ │ │ +0005be30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005be40: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0005be50: 2020 2276 6572 7469 6365 7322 203d 3e20    "vertices" => 
│ │ │ │ +0005be60: 7b61 2c20 622c 2063 2c20 647d 2020 2020  {a, b, c, d}    
│ │ │ │ +0005be70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005be80: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0005be90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005bea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005beb0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0005bec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005bed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005beb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005bec0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005bed0: 7c6f 3220 3a20 4772 6170 6820 2020 2020  |o2 : Graph     
│ │ │ │  0005bee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005bef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005bf00: 7c0a 7c6f 3420 3d20 6d6f 6e6f 6d69 616c  |.|o4 = monomial
│ │ │ │ -0005bf10: 4964 6561 6c20 2861 2a63 2c20 622a 6429  Ideal (a*c, b*d)
│ │ │ │ -0005bf20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005bf30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005bf40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0005bf50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005bf60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005bf00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005bf10: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0005bf20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005bf30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005bf40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005bf50: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │ +0005bf60: 7665 7274 6578 436f 7665 7273 2067 2020  vertexCovers g  
│ │ │ │  0005bf70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005bf80: 2020 2020 2020 2020 2020 7c0a 7c6f 3420            |.|o4 
│ │ │ │ -0005bf90: 3a20 4d6f 6e6f 6d69 616c 4964 6561 6c20  : MonomialIdeal 
│ │ │ │ -0005bfa0: 6f66 2053 2020 2020 2020 2020 2020 2020  of S            
│ │ │ │ +0005bf80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005bf90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0005bfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005bfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005bfc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0005bfd0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -0005bfe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005bff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c010: 2d2d 2d2d 2b0a 7c69 3520 3a20 666c 6174  ----+.|i5 : flat
│ │ │ │ -0005c020: 7465 6e20 656e 7472 6965 7320 6765 6e73  ten entries gens
│ │ │ │ -0005c030: 2063 6f76 6572 4964 6561 6c20 6720 3d3d   coverIdeal g ==
│ │ │ │ -0005c040: 2076 6572 7465 7843 6f76 6572 7320 6720   vertexCovers g 
│ │ │ │ -0005c050: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0005c060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005bfc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005bfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005bfe0: 2020 7c0a 7c6f 3320 3d20 7b61 2a63 2c20    |.|o3 = {a*c, 
│ │ │ │ +0005bff0: 622a 647d 2020 2020 2020 2020 2020 2020  b*d}            
│ │ │ │ +0005c000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c020: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005c030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c060: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0005c070: 3320 3a20 4c69 7374 2020 2020 2020 2020  3 : List        
│ │ │ │  0005c080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c090: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0005c0a0: 7c6f 3520 3d20 7472 7565 2020 2020 2020  |o5 = true      
│ │ │ │ -0005c0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c0e0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -0005c0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c120: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20  --------+.|i6 : 
│ │ │ │ -0005c130: 5320 3d20 5151 5b61 2e2e 655d 3b20 2020  S = QQ[a..e];   
│ │ │ │ +0005c090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c0b0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0005c0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005c0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005c0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005c0f0: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 636f  ------+.|i4 : co
│ │ │ │ +0005c100: 7665 7249 6465 616c 2067 2020 2020 2020  verIdeal g      
│ │ │ │ +0005c110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c130: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0005c140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c160: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -0005c170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c1b0: 2d2d 2b0a 7c69 3720 3a20 6820 3d20 6879  --+.|i7 : h = hy
│ │ │ │ -0005c1c0: 7065 7247 7261 7068 207b 612a 622a 632c  perGraph {a*b*c,
│ │ │ │ -0005c1d0: 612a 642c 632a 652c 622a 642a 657d 2020  a*d,c*e,b*d*e}  
│ │ │ │ +0005c160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c180: 7c0a 7c6f 3420 3d20 6d6f 6e6f 6d69 616c  |.|o4 = monomial
│ │ │ │ +0005c190: 4964 6561 6c20 2861 2a63 2c20 622a 6429  Ideal (a*c, b*d)
│ │ │ │ +0005c1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c1c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005c1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c1f0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0005c200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c230: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0005c240: 3720 3d20 4879 7065 7247 7261 7068 7b22  7 = HyperGraph{"
│ │ │ │ -0005c250: 6564 6765 7322 203d 3e20 7b7b 612c 2062  edges" => {{a, b
│ │ │ │ -0005c260: 2c20 637d 2c20 7b61 2c20 647d 2c20 7b63  , c}, {a, d}, {c
│ │ │ │ -0005c270: 2c20 657d 2c20 7b62 2c20 642c 2065 7d7d  , e}, {b, d, e}}
│ │ │ │ -0005c280: 7d7c 0a7c 2020 2020 2020 2020 2020 2020  }|.|            
│ │ │ │ -0005c290: 2020 2020 2272 696e 6722 203d 3e20 5320      "ring" => S 
│ │ │ │ -0005c2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c2c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0005c2d0: 2020 2020 2020 2020 2022 7665 7274 6963           "vertic
│ │ │ │ -0005c2e0: 6573 2220 3d3e 207b 612c 2062 2c20 632c  es" => {a, b, c,
│ │ │ │ -0005c2f0: 2064 2c20 657d 2020 2020 2020 2020 2020   d, e}          
│ │ │ │ -0005c300: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0005c310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c200: 2020 2020 2020 2020 2020 7c0a 7c6f 3420            |.|o4 
│ │ │ │ +0005c210: 3a20 4d6f 6e6f 6d69 616c 4964 6561 6c20  : MonomialIdeal 
│ │ │ │ +0005c220: 6f66 2053 2020 2020 2020 2020 2020 2020  of S            
│ │ │ │ +0005c230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c240: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005c250: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0005c260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005c270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005c280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005c290: 2d2d 2d2d 2b0a 7c69 3520 3a20 666c 6174  ----+.|i5 : flat
│ │ │ │ +0005c2a0: 7465 6e20 656e 7472 6965 7320 6765 6e73  ten entries gens
│ │ │ │ +0005c2b0: 2063 6f76 6572 4964 6561 6c20 6720 3d3d   coverIdeal g ==
│ │ │ │ +0005c2c0: 2076 6572 7465 7843 6f76 6572 7320 6720   vertexCovers g 
│ │ │ │ +0005c2d0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +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 7c0a                |.
│ │ │ │ +0005c320: 7c6f 3520 3d20 7472 7565 2020 2020 2020  |o5 = true      
│ │ │ │  0005c330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c350: 7c0a 7c6f 3720 3a20 4879 7065 7247 7261  |.|o7 : HyperGra
│ │ │ │ -0005c360: 7068 2020 2020 2020 2020 2020 2020 2020  ph              
│ │ │ │ -0005c370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c390: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -0005c3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820  ----------+.|i8 
│ │ │ │ -0005c3e0: 3a20 7665 7274 6578 436f 7665 7273 2868  : vertexCovers(h
│ │ │ │ -0005c3f0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -0005c400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c410: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0005c420: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0005c430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c460: 2020 2020 7c0a 7c6f 3820 3d20 7b61 2a62      |.|o8 = {a*b
│ │ │ │ -0005c470: 2a63 2c20 632a 642c 2061 2a65 2c20 622a  *c, c*d, a*e, b*
│ │ │ │ -0005c480: 642a 657d 2020 2020 2020 2020 2020 2020  d*e}            
│ │ │ │ +0005c350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c360: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0005c370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005c380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005c390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005c3a0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20  --------+.|i6 : 
│ │ │ │ +0005c3b0: 5320 3d20 5151 5b61 2e2e 655d 3b20 2020  S = QQ[a..e];   
│ │ │ │ +0005c3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c3e0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0005c3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005c400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005c410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005c420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005c430: 2d2d 2b0a 7c69 3720 3a20 6820 3d20 6879  --+.|i7 : h = hy
│ │ │ │ +0005c440: 7065 7247 7261 7068 207b 612a 622a 632c  perGraph {a*b*c,
│ │ │ │ +0005c450: 612a 642c 632a 652c 622a 642a 657d 2020  a*d,c*e,b*d*e}  
│ │ │ │ +0005c460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c470: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005c480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c4a0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0005c4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c4e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0005c4f0: 7c6f 3820 3a20 4c69 7374 2020 2020 2020  |o8 : List      
│ │ │ │ -0005c500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c4b0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0005c4c0: 3720 3d20 4879 7065 7247 7261 7068 7b22  7 = HyperGraph{"
│ │ │ │ +0005c4d0: 6564 6765 7322 203d 3e20 7b7b 612c 2062  edges" => {{a, b
│ │ │ │ +0005c4e0: 2c20 637d 2c20 7b61 2c20 647d 2c20 7b63  , c}, {a, d}, {c
│ │ │ │ +0005c4f0: 2c20 657d 2c20 7b62 2c20 642c 2065 7d7d  , e}, {b, d, e}}
│ │ │ │ +0005c500: 7d7c 0a7c 2020 2020 2020 2020 2020 2020  }|.|            
│ │ │ │ +0005c510: 2020 2020 2272 696e 6722 203d 3e20 5320      "ring" => S 
│ │ │ │  0005c520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c530: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -0005c540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c570: 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061  --------+..See a
│ │ │ │ -0005c580: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020  lso.========..  
│ │ │ │ -0005c590: 2a20 2a6e 6f74 6520 636f 7665 7249 6465  * *note coverIde
│ │ │ │ -0005c5a0: 616c 3a20 636f 7665 7249 6465 616c 2c20  al: coverIdeal, 
│ │ │ │ -0005c5b0: 2d2d 2063 7265 6174 6573 2074 6865 2063  -- creates the c
│ │ │ │ -0005c5c0: 6f76 6572 2069 6465 616c 206f 6620 6120  over ideal of a 
│ │ │ │ -0005c5d0: 2868 7970 6572 2967 7261 7068 0a20 202a  (hyper)graph.  *
│ │ │ │ -0005c5e0: 202a 6e6f 7465 2076 6572 7465 7843 6f76   *note vertexCov
│ │ │ │ -0005c5f0: 6572 4e75 6d62 6572 3a20 7665 7274 6578  erNumber: vertex
│ │ │ │ -0005c600: 436f 7665 724e 756d 6265 722c 202d 2d20  CoverNumber, -- 
│ │ │ │ -0005c610: 6669 6e64 2074 6865 2076 6572 7465 7820  find the vertex 
│ │ │ │ -0005c620: 636f 7665 7269 6e67 0a20 2020 206e 756d  covering.    num
│ │ │ │ -0005c630: 6265 7220 6f66 2061 2028 6879 7065 7229  ber of a (hyper)
│ │ │ │ -0005c640: 6772 6170 680a 0a57 6179 7320 746f 2075  graph..Ways to u
│ │ │ │ -0005c650: 7365 2076 6572 7465 7843 6f76 6572 733a  se vertexCovers:
│ │ │ │ -0005c660: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ -0005c670: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ -0005c680: 2276 6572 7465 7843 6f76 6572 7328 4879  "vertexCovers(Hy
│ │ │ │ -0005c690: 7065 7247 7261 7068 2922 0a0a 466f 7220  perGraph)"..For 
│ │ │ │ -0005c6a0: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d  the programmer.=
│ │ │ │ -0005c6b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0005c6c0: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e  =..The object *n
│ │ │ │ -0005c6d0: 6f74 6520 7665 7274 6578 436f 7665 7273  ote vertexCovers
│ │ │ │ -0005c6e0: 3a20 7665 7274 6578 436f 7665 7273 2c20  : vertexCovers, 
│ │ │ │ -0005c6f0: 6973 2061 202a 6e6f 7465 206d 6574 686f  is a *note metho
│ │ │ │ -0005c700: 6420 6675 6e63 7469 6f6e 3a0a 284d 6163  d function:.(Mac
│ │ │ │ -0005c710: 6175 6c61 7932 446f 6329 4d65 7468 6f64  aulay2Doc)Method
│ │ │ │ -0005c720: 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d  Function,...----
│ │ │ │ -0005c730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865  -----------..The
│ │ │ │ -0005c780: 2073 6f75 7263 6520 6f66 2074 6869 7320   source of this 
│ │ │ │ -0005c790: 646f 6375 6d65 6e74 2069 7320 696e 0a2f  document is in./
│ │ │ │ -0005c7a0: 6275 696c 642f 7265 7072 6f64 7563 6962  build/reproducib
│ │ │ │ -0005c7b0: 6c65 2d70 6174 682f 6d61 6361 756c 6179  le-path/macaulay
│ │ │ │ -0005c7c0: 322d 312e 3235 2e31 312b 6473 2f4d 322f  2-1.25.11+ds/M2/
│ │ │ │ -0005c7d0: 4d61 6361 756c 6179 322f 7061 636b 6167  Macaulay2/packag
│ │ │ │ -0005c7e0: 6573 2f45 6467 6549 6465 616c 732e 0a6d  es/EdgeIdeals..m
│ │ │ │ -0005c7f0: 323a 3431 3431 3a30 2e0a 1f0a 4669 6c65  2:4141:0....File
│ │ │ │ -0005c800: 3a20 4564 6765 4964 6561 6c73 2e69 6e66  : EdgeIdeals.inf
│ │ │ │ -0005c810: 6f2c 204e 6f64 653a 2076 6572 7469 6365  o, Node: vertice
│ │ │ │ -0005c820: 735f 6c70 4879 7065 7247 7261 7068 5f72  s_lpHyperGraph_r
│ │ │ │ -0005c830: 702c 2050 7265 763a 2076 6572 7465 7843  p, Prev: vertexC
│ │ │ │ -0005c840: 6f76 6572 732c 2055 703a 2054 6f70 0a0a  overs, Up: Top..
│ │ │ │ -0005c850: 7665 7274 6963 6573 2848 7970 6572 4772  vertices(HyperGr
│ │ │ │ -0005c860: 6170 6829 202d 2d20 6765 7473 2074 6865  aph) -- gets the
│ │ │ │ -0005c870: 2076 6572 7469 6365 7320 6f66 2061 2028   vertices of a (
│ │ │ │ -0005c880: 6879 7065 7229 6772 6170 680a 2a2a 2a2a  hyper)graph.****
│ │ │ │ -0005c890: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0005c8a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0005c8b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0005c8c0: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2046 756e  *******..  * Fun
│ │ │ │ -0005c8d0: 6374 696f 6e3a 202a 6e6f 7465 2076 6572  ction: *note ver
│ │ │ │ -0005c8e0: 7469 6365 733a 2028 506f 6c79 6865 6472  tices: (Polyhedr
│ │ │ │ -0005c8f0: 6129 7665 7274 6963 6573 2c0a 2020 2a20  a)vertices,.  * 
│ │ │ │ -0005c900: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ -0005c910: 5620 3d20 7665 7274 6963 6573 2848 290a  V = vertices(H).
│ │ │ │ -0005c920: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ -0005c930: 2020 2a20 482c 2061 202a 6e6f 7465 2068    * H, a *note h
│ │ │ │ -0005c940: 7970 6572 6772 6170 683a 2048 7970 6572  ypergraph: Hyper
│ │ │ │ -0005c950: 4772 6170 682c 2c20 7468 6520 696e 7075  Graph,, the inpu
│ │ │ │ -0005c960: 740a 2020 2a20 4f75 7470 7574 733a 0a20  t.  * Outputs:. 
│ │ │ │ -0005c970: 2020 2020 202a 2056 2c20 6120 2a6e 6f74       * V, a *not
│ │ │ │ -0005c980: 6520 6c69 7374 3a20 284d 6163 6175 6c61  e list: (Macaula
│ │ │ │ -0005c990: 7932 446f 6329 4c69 7374 2c2c 206f 6620  y2Doc)List,, of 
│ │ │ │ -0005c9a0: 7468 6520 7665 7274 6963 6573 206f 6620  the vertices of 
│ │ │ │ -0005c9b0: 480a 0a44 6573 6372 6970 7469 6f6e 0a3d  H..Description.=
│ │ │ │ -0005c9c0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6973  ==========..This
│ │ │ │ -0005c9d0: 2066 756e 6374 696f 6e20 7461 6b65 7320   function takes 
│ │ │ │ -0005c9e0: 6120 6772 6170 6820 6f72 2068 7970 6572  a graph or hyper
│ │ │ │ -0005c9f0: 6772 6170 6820 616e 6420 7265 7475 726e  graph and return
│ │ │ │ -0005ca00: 7320 7468 6520 7665 7274 6578 2073 6574  s the vertex set
│ │ │ │ -0005ca10: 206f 6620 7468 650a 6772 6170 682e 0a0a   of the.graph...
│ │ │ │ -0005ca20: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -0005ca30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ca40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ca50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ca60: 2d2d 2b0a 7c69 3120 3a20 5320 3d20 5151  --+.|i1 : S = QQ
│ │ │ │ -0005ca70: 5b61 2e2e 645d 3b20 2020 2020 2020 2020  [a..d];         
│ │ │ │ -0005ca80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ca90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005caa0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ -0005cab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005cac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005cad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005cae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220  ----------+.|i2 
│ │ │ │ -0005caf0: 3a20 6720 3d20 6772 6170 6820 7b61 2a62  : g = graph {a*b
│ │ │ │ -0005cb00: 2c62 2a63 2c63 2a64 2c64 2a61 7d20 2d2d  ,b*c,c*d,d*a} --
│ │ │ │ -0005cb10: 2074 6865 2066 6f75 7220 6379 636c 6520   the four cycle 
│ │ │ │ -0005cb20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0005cb30: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0005cb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005cb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005cb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005cb70: 2020 7c0a 7c6f 3220 3d20 4772 6170 687b    |.|o2 = Graph{
│ │ │ │ -0005cb80: 2265 6467 6573 2220 3d3e 207b 7b61 2c20  "edges" => {{a, 
│ │ │ │ -0005cb90: 627d 2c20 7b62 2c20 637d 2c20 7b61 2c20  b}, {b, c}, {a, 
│ │ │ │ -0005cba0: 647d 2c20 7b63 2c20 647d 7d7d 2020 2020  d}, {c, d}}}    
│ │ │ │ -0005cbb0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0005cbc0: 2020 2020 2272 696e 6722 203d 3e20 5320      "ring" => S 
│ │ │ │ -0005cbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005cbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005cbf0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0005cc00: 2020 2020 2020 2020 2276 6572 7469 6365          "vertice
│ │ │ │ -0005cc10: 7322 203d 3e20 7b61 2c20 622c 2063 2c20  s" => {a, b, c, 
│ │ │ │ -0005cc20: 647d 2020 2020 2020 2020 2020 2020 2020  d}              
│ │ │ │ -0005cc30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0005cc40: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0005cc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005cc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005cc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005cc80: 2020 7c0a 7c6f 3220 3a20 4772 6170 6820    |.|o2 : Graph 
│ │ │ │ -0005cc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005cca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ccb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ccc0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0005c530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c540: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0005c550: 2020 2020 2020 2020 2022 7665 7274 6963           "vertic
│ │ │ │ +0005c560: 6573 2220 3d3e 207b 612c 2062 2c20 632c  es" => {a, b, c,
│ │ │ │ +0005c570: 2064 2c20 657d 2020 2020 2020 2020 2020   d, e}          
│ │ │ │ +0005c580: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0005c590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c5d0: 7c0a 7c6f 3720 3a20 4879 7065 7247 7261  |.|o7 : HyperGra
│ │ │ │ +0005c5e0: 7068 2020 2020 2020 2020 2020 2020 2020  ph              
│ │ │ │ +0005c5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c610: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0005c620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005c630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005c640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005c650: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820  ----------+.|i8 
│ │ │ │ +0005c660: 3a20 7665 7274 6578 436f 7665 7273 2868  : vertexCovers(h
│ │ │ │ +0005c670: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +0005c680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c690: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005c6a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0005c6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c6e0: 2020 2020 7c0a 7c6f 3820 3d20 7b61 2a62      |.|o8 = {a*b
│ │ │ │ +0005c6f0: 2a63 2c20 632a 642c 2061 2a65 2c20 622a  *c, c*d, a*e, b*
│ │ │ │ +0005c700: 642a 657d 2020 2020 2020 2020 2020 2020  d*e}            
│ │ │ │ +0005c710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c720: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005c730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c760: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005c770: 7c6f 3820 3a20 4c69 7374 2020 2020 2020  |o8 : List      
│ │ │ │ +0005c780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c7b0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0005c7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005c7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005c7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005c7f0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061  --------+..See a
│ │ │ │ +0005c800: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020  lso.========..  
│ │ │ │ +0005c810: 2a20 2a6e 6f74 6520 636f 7665 7249 6465  * *note coverIde
│ │ │ │ +0005c820: 616c 3a20 636f 7665 7249 6465 616c 2c20  al: coverIdeal, 
│ │ │ │ +0005c830: 2d2d 2063 7265 6174 6573 2074 6865 2063  -- creates the c
│ │ │ │ +0005c840: 6f76 6572 2069 6465 616c 206f 6620 6120  over ideal of a 
│ │ │ │ +0005c850: 2868 7970 6572 2967 7261 7068 0a20 202a  (hyper)graph.  *
│ │ │ │ +0005c860: 202a 6e6f 7465 2076 6572 7465 7843 6f76   *note vertexCov
│ │ │ │ +0005c870: 6572 4e75 6d62 6572 3a20 7665 7274 6578  erNumber: vertex
│ │ │ │ +0005c880: 436f 7665 724e 756d 6265 722c 202d 2d20  CoverNumber, -- 
│ │ │ │ +0005c890: 6669 6e64 2074 6865 2076 6572 7465 7820  find the vertex 
│ │ │ │ +0005c8a0: 636f 7665 7269 6e67 0a20 2020 206e 756d  covering.    num
│ │ │ │ +0005c8b0: 6265 7220 6f66 2061 2028 6879 7065 7229  ber of a (hyper)
│ │ │ │ +0005c8c0: 6772 6170 680a 0a57 6179 7320 746f 2075  graph..Ways to u
│ │ │ │ +0005c8d0: 7365 2076 6572 7465 7843 6f76 6572 733a  se vertexCovers:
│ │ │ │ +0005c8e0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ +0005c8f0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ +0005c900: 2276 6572 7465 7843 6f76 6572 7328 4879  "vertexCovers(Hy
│ │ │ │ +0005c910: 7065 7247 7261 7068 2922 0a0a 466f 7220  perGraph)"..For 
│ │ │ │ +0005c920: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d  the programmer.=
│ │ │ │ +0005c930: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0005c940: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e  =..The object *n
│ │ │ │ +0005c950: 6f74 6520 7665 7274 6578 436f 7665 7273  ote vertexCovers
│ │ │ │ +0005c960: 3a20 7665 7274 6578 436f 7665 7273 2c20  : vertexCovers, 
│ │ │ │ +0005c970: 6973 2061 202a 6e6f 7465 206d 6574 686f  is a *note metho
│ │ │ │ +0005c980: 6420 6675 6e63 7469 6f6e 3a0a 284d 6163  d function:.(Mac
│ │ │ │ +0005c990: 6175 6c61 7932 446f 6329 4d65 7468 6f64  aulay2Doc)Method
│ │ │ │ +0005c9a0: 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d  Function,...----
│ │ │ │ +0005c9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005c9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005c9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005c9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005c9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865  -----------..The
│ │ │ │ +0005ca00: 2073 6f75 7263 6520 6f66 2074 6869 7320   source of this 
│ │ │ │ +0005ca10: 646f 6375 6d65 6e74 2069 7320 696e 0a2f  document is in./
│ │ │ │ +0005ca20: 6275 696c 642f 7265 7072 6f64 7563 6962  build/reproducib
│ │ │ │ +0005ca30: 6c65 2d70 6174 682f 6d61 6361 756c 6179  le-path/macaulay
│ │ │ │ +0005ca40: 322d 312e 3235 2e31 312b 6473 2f4d 322f  2-1.25.11+ds/M2/
│ │ │ │ +0005ca50: 4d61 6361 756c 6179 322f 7061 636b 6167  Macaulay2/packag
│ │ │ │ +0005ca60: 6573 2f45 6467 6549 6465 616c 732e 0a6d  es/EdgeIdeals..m
│ │ │ │ +0005ca70: 323a 3431 3431 3a30 2e0a 1f0a 4669 6c65  2:4141:0....File
│ │ │ │ +0005ca80: 3a20 4564 6765 4964 6561 6c73 2e69 6e66  : EdgeIdeals.inf
│ │ │ │ +0005ca90: 6f2c 204e 6f64 653a 2076 6572 7469 6365  o, Node: vertice
│ │ │ │ +0005caa0: 735f 6c70 4879 7065 7247 7261 7068 5f72  s_lpHyperGraph_r
│ │ │ │ +0005cab0: 702c 2050 7265 763a 2076 6572 7465 7843  p, Prev: vertexC
│ │ │ │ +0005cac0: 6f76 6572 732c 2055 703a 2054 6f70 0a0a  overs, Up: Top..
│ │ │ │ +0005cad0: 7665 7274 6963 6573 2848 7970 6572 4772  vertices(HyperGr
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│ │ │ │ -0005cf80: 2022 7269 6e67 2220 3d3e 2053 2020 2020   "ring" => S    
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│ │ │ │ -0005d8a0: 653a 2069 6e64 6570 656e 6465 6e63 654e  e: independenceN
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│ │ │ │ -0005d8c0: 653a 2069 6e64 7563 6564 4772 6170 687f  e: inducedGraph.
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│ │ │ │ -0005da60: 3737 3430 0a4e 6f64 653a 2069 7353 434d  7740.Node: isSCM
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│ │ │ │ -0005dbc0: 616e 6368 4c69 6d69 743d 3e5f 7064 5f70  anchLimit=>_pd_p
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│ │ │ │ -0005dbe0: 6f64 653a 2072 616e 646f 6d48 7970 6572  ode: randomHyper
│ │ │ │ -0005dbf0: 4772 6170 685f 6c70 5f70 645f 7064 5f70  Graph_lp_pd_pd_p
│ │ │ │ -0005dc00: 645f 636d 5469 6d65 4c69 6d69 743d 3e5f  d_cmTimeLimit=>_
│ │ │ │ -0005dc10: 7064 5f70 645f 7064 5f72 707f 3334 3732  pd_pd_pd_rp.3472
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│ │ │ │ -0005dc50: 6e67 5f6c 7048 7970 6572 4772 6170 685f  ng_lpHyperGraph_
│ │ │ │ -0005dc60: 7270 7f33 3532 3931 340a 4e6f 6465 3a20  rp.352914.Node: 
│ │ │ │ -0005dc70: 7369 6d70 6c69 6369 616c 436f 6d70 6c65  simplicialComple
│ │ │ │ -0005dc80: 7854 6f48 7970 6572 4772 6170 687f 3335  xToHyperGraph.35
│ │ │ │ -0005dc90: 3437 3039 0a4e 6f64 653a 2073 6d61 6c6c  4709.Node: small
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│ │ │ │ +0005d400: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
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│ │ │ │ +0005d4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005d4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0005d4f0: 2020 2a20 2a6e 6f74 6520 6564 6765 733a    * *note edges:
│ │ │ │ +0005d500: 2065 6467 6573 2c20 2d2d 2067 6574 7320   edges, -- gets 
│ │ │ │ +0005d510: 7468 6520 6564 6765 7320 6f66 2061 2028  the edges of a (
│ │ │ │ +0005d520: 6879 7065 7229 6772 6170 680a 0a57 6179  hyper)graph..Way
│ │ │ │ +0005d530: 7320 746f 2075 7365 2074 6869 7320 6d65  s to use this me
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│ │ │ │ +0005d550: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ +0005d560: 2020 2a20 2a6e 6f74 6520 7665 7274 6963    * *note vertic
│ │ │ │ +0005d570: 6573 2848 7970 6572 4772 6170 6829 3a20  es(HyperGraph): 
│ │ │ │ +0005d580: 7665 7274 6963 6573 5f6c 7048 7970 6572  vertices_lpHyper
│ │ │ │ +0005d590: 4772 6170 685f 7270 2c20 2d2d 2067 6574  Graph_rp, -- get
│ │ │ │ +0005d5a0: 7320 7468 6520 7665 7274 6963 6573 0a20  s the vertices. 
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│ │ │ │ +0005d5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005d5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005d600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005d610: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365  ----..The source
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│ │ │ │ +0005d640: 6570 726f 6475 6369 626c 652d 7061 7468  eproducible-path
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│ │ │ │ +0005d770: 5f70 645f 7064 5f72 707f 3437 3533 300a  _pd_pd_rp.47530.
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│ │ │ │ +0005d7e0: 7261 7068 7f35 3938 3538 0a4e 6f64 653a  raph.59858.Node:
│ │ │ │ +0005d7f0: 2063 6f6d 706c 6574 6547 7261 7068 7f36   completeGraph.6
│ │ │ │ +0005d800: 3339 3833 0a4e 6f64 653a 2063 6f6d 706c  3983.Node: compl
│ │ │ │ +0005d810: 6574 654d 756c 7469 5061 7274 6974 657f  eteMultiPartite.
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│ │ │ │ +0005d830: 6563 7465 6420 436f 6d70 6f6e 656e 7473  ected Components
│ │ │ │ +0005d840: 2054 7574 6f72 6961 6c7f 3734 3639 390a   Tutorial.74699.
│ │ │ │ +0005d850: 4e6f 6465 3a20 636f 6e6e 6563 7465 6443  Node: connectedC
│ │ │ │ +0005d860: 6f6d 706f 6e65 6e74 737f 3739 3434 350a  omponents.79445.
│ │ │ │ +0005d870: 4e6f 6465 3a20 636f 6e6e 6563 7465 6447  Node: connectedG
│ │ │ │ +0005d880: 7261 7068 436f 6d70 6f6e 656e 7473 7f38  raphComponents.8
│ │ │ │ +0005d890: 3535 3534 0a4e 6f64 653a 2043 6f6e 7374  5554.Node: Const
│ │ │ │ +0005d8a0: 7275 6374 6f72 204f 7665 7276 6965 777f  ructor Overview.
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│ │ │ │ +0005d8c0: 7249 6465 616c 7f31 3039 3934 380a 4e6f  rIdeal.109948.No
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│ │ │ │ +0005d8e0: 0a4e 6f64 653a 2064 6567 7265 6556 6572  .Node: degreeVer
│ │ │ │ +0005d8f0: 7465 787f 3131 3833 3637 0a4e 6f64 653a  tex.118367.Node:
│ │ │ │ +0005d900: 2064 656c 6574 6545 6467 6573 7f31 3233   deleteEdges.123
│ │ │ │ +0005d910: 3632 340a 4e6f 6465 3a20 6564 6765 4964  624.Node: edgeId
│ │ │ │ +0005d920: 6561 6c7f 3132 3738 3234 0a4e 6f64 653a  eal.127824.Node:
│ │ │ │ +0005d930: 2065 6467 6573 7f31 3334 3638 330a 4e6f   edges.134683.No
│ │ │ │ +0005d940: 6465 3a20 4578 7465 6e64 6564 2045 7861  de: Extended Exa
│ │ │ │ +0005d950: 6d70 6c65 7f31 3338 3938 340a 4e6f 6465  mple.138984.Node
│ │ │ │ +0005d960: 3a20 6765 7443 6c69 7175 6573 7f31 3437  : getCliques.147
│ │ │ │ +0005d970: 3534 390a 4e6f 6465 3a20 6765 7445 6467  549.Node: getEdg
│ │ │ │ +0005d980: 657f 3135 3134 3739 0a4e 6f64 653a 2067  e.151479.Node: g
│ │ │ │ +0005d990: 6574 4564 6765 496e 6465 787f 3135 3536  etEdgeIndex.1556
│ │ │ │ +0005d9a0: 3936 0a4e 6f64 653a 2067 6574 476f 6f64  96.Node: getGood
│ │ │ │ +0005d9b0: 4c65 6166 7f31 3630 3538 370a 4e6f 6465  Leaf.160587.Node
│ │ │ │ +0005d9c0: 3a20 6765 7447 6f6f 644c 6561 6649 6e64  : getGoodLeafInd
│ │ │ │ +0005d9d0: 6578 7f31 3632 3738 360a 4e6f 6465 3a20  ex.162786.Node: 
│ │ │ │ +0005d9e0: 6765 744d 6178 436c 6971 7565 737f 3136  getMaxCliques.16
│ │ │ │ +0005d9f0: 3537 3231 0a4e 6f64 653a 2047 696e 737f  5721.Node: Gins.
│ │ │ │ +0005da00: 3136 3931 3637 0a4e 6f64 653a 2047 7261  169167.Node: Gra
│ │ │ │ +0005da10: 7068 7f31 3730 3136 380a 4e6f 6465 3a20  ph.170168.Node: 
│ │ │ │ +0005da20: 6772 6170 687f 3137 3732 3130 0a4e 6f64  graph.177210.Nod
│ │ │ │ +0005da30: 653a 2068 6173 476f 6f64 4c65 6166 7f31  e: hasGoodLeaf.1
│ │ │ │ +0005da40: 3933 3534 320a 4e6f 6465 3a20 6861 734f  93542.Node: hasO
│ │ │ │ +0005da50: 6464 486f 6c65 7f31 3935 3933 330a 4e6f  ddHole.195933.No
│ │ │ │ +0005da60: 6465 3a20 4879 7065 7247 7261 7068 7f32  de: HyperGraph.2
│ │ │ │ +0005da70: 3030 3530 310a 4e6f 6465 3a20 6879 7065  00501.Node: hype
│ │ │ │ +0005da80: 7247 7261 7068 7f32 3039 3430 380a 4e6f  rGraph.209408.No
│ │ │ │ +0005da90: 6465 3a20 4879 7065 7247 7261 7068 203d  de: HyperGraph =
│ │ │ │ +0005daa0: 3d20 4879 7065 7247 7261 7068 7f32 3236  = HyperGraph.226
│ │ │ │ +0005dab0: 3939 350a 4e6f 6465 3a20 6879 7065 7247  995.Node: hyperG
│ │ │ │ +0005dac0: 7261 7068 546f 5369 6d70 6c69 6369 616c  raphToSimplicial
│ │ │ │ +0005dad0: 436f 6d70 6c65 787f 3232 3839 3439 0a4e  Complex.228949.N
│ │ │ │ +0005dae0: 6f64 653a 2069 6e63 6964 656e 6365 4d61  ode: incidenceMa
│ │ │ │ +0005daf0: 7472 6978 7f32 3335 3030 350a 4e6f 6465  trix.235005.Node
│ │ │ │ +0005db00: 3a20 696e 6465 7065 6e64 656e 6365 436f  : independenceCo
│ │ │ │ +0005db10: 6d70 6c65 787f 3234 3034 3334 0a4e 6f64  mplex.240434.Nod
│ │ │ │ +0005db20: 653a 2069 6e64 6570 656e 6465 6e63 654e  e: independenceN
│ │ │ │ +0005db30: 756d 6265 727f 3234 3539 3033 0a4e 6f64  umber.245903.Nod
│ │ │ │ +0005db40: 653a 2069 6e64 7563 6564 4772 6170 687f  e: inducedGraph.
│ │ │ │ +0005db50: 3234 3935 3437 0a4e 6f64 653a 2069 6e64  249547.Node: ind
│ │ │ │ +0005db60: 7563 6564 4772 6170 685f 6c70 5f70 645f  ucedGraph_lp_pd_
│ │ │ │ +0005db70: 7064 5f70 645f 636d 4f72 6967 696e 616c  pd_pd_cmOriginal
│ │ │ │ +0005db80: 5269 6e67 3d3e 5f70 645f 7064 5f70 645f  Ring=>_pd_pd_pd_
│ │ │ │ +0005db90: 7270 7f32 3538 3035 330a 4e6f 6465 3a20  rp.258053.Node: 
│ │ │ │ +0005dba0: 696e 6475 6365 6448 7970 6572 4772 6170  inducedHyperGrap
│ │ │ │ +0005dbb0: 687f 3236 3031 3939 0a4e 6f64 653a 2069  h.260199.Node: i
│ │ │ │ +0005dbc0: 7342 6970 6172 7469 7465 7f32 3639 3235  sBipartite.26925
│ │ │ │ +0005dbd0: 340a 4e6f 6465 3a20 6973 4368 6f72 6461  4.Node: isChorda
│ │ │ │ +0005dbe0: 6c7f 3237 3233 3933 0a4e 6f64 653a 2069  l.272393.Node: i
│ │ │ │ +0005dbf0: 7343 4d7f 3237 3434 3036 0a4e 6f64 653a  sCM.274406.Node:
│ │ │ │ +0005dc00: 2069 7343 6f6e 6e65 6374 6564 7f32 3736   isConnected.276
│ │ │ │ +0005dc10: 3438 330a 4e6f 6465 3a20 6973 436f 6e6e  483.Node: isConn
│ │ │ │ +0005dc20: 6563 7465 6447 7261 7068 7f32 3831 3833  ectedGraph.28183
│ │ │ │ +0005dc30: 360a 4e6f 6465 3a20 6973 4564 6765 7f32  6.Node: isEdge.2
│ │ │ │ +0005dc40: 3838 3037 360a 4e6f 6465 3a20 6973 466f  88076.Node: isFo
│ │ │ │ +0005dc50: 7265 7374 7f32 3931 3231 320a 4e6f 6465  rest.291212.Node
│ │ │ │ +0005dc60: 3a20 6973 476f 6f64 4c65 6166 7f32 3934  : isGoodLeaf.294
│ │ │ │ +0005dc70: 3138 300a 4e6f 6465 3a20 6973 4772 6170  180.Node: isGrap
│ │ │ │ +0005dc80: 687f 3239 3639 3534 0a4e 6f64 653a 2069  h.296954.Node: i
│ │ │ │ +0005dc90: 734c 6561 667f 3239 3833 3933 0a4e 6f64  sLeaf.298393.Nod
│ │ │ │ +0005dca0: 653a 2069 736f 6c61 7465 6456 6572 7469  e: isolatedVerti
│ │ │ │ +0005dcb0: 6365 737f 3330 3230 3030 0a4e 6f64 653a  ces.302000.Node:
│ │ │ │ +0005dcc0: 2069 7350 6572 6665 6374 7f33 3035 3033   isPerfect.30503
│ │ │ │ +0005dcd0: 320a 4e6f 6465 3a20 6973 5343 4d7f 3330  2.Node: isSCM.30
│ │ │ │ +0005dce0: 3737 3430 0a4e 6f64 653a 2069 7353 434d  7740.Node: isSCM
│ │ │ │ +0005dcf0: 5f6c 705f 7064 5f70 645f 7064 5f63 6d47  _lp_pd_pd_pd_cmG
│ │ │ │ +0005dd00: 696e 733d 3e5f 7064 5f70 645f 7064 5f72  ins=>_pd_pd_pd_r
│ │ │ │ +0005dd10: 707f 3331 3138 3539 0a4e 6f64 653a 206c  p.311859.Node: l
│ │ │ │ +0005dd20: 696e 6547 7261 7068 7f33 3133 3332 360a  ineGraph.313326.
│ │ │ │ +0005dd30: 4e6f 6465 3a20 4d61 7869 6d61 6c45 6467  Node: MaximalEdg
│ │ │ │ +0005dd40: 6573 7f33 3135 3933 390a 4e6f 6465 3a20  es.315939.Node: 
│ │ │ │ +0005dd50: 6e65 6967 6862 6f72 737f 3331 3638 3839  neighbors.316889
│ │ │ │ +0005dd60: 0a4e 6f64 653a 206e 756d 436f 6e6e 6563  .Node: numConnec
│ │ │ │ +0005dd70: 7465 6443 6f6d 706f 6e65 6e74 737f 3331  tedComponents.31
│ │ │ │ +0005dd80: 3937 3931 0a4e 6f64 653a 206e 756d 436f  9791.Node: numCo
│ │ │ │ +0005dd90: 6e6e 6563 7465 6447 7261 7068 436f 6d70  nnectedGraphComp
│ │ │ │ +0005dda0: 6f6e 656e 7473 7f33 3236 3434 340a 4e6f  onents.326444.No
│ │ │ │ +0005ddb0: 6465 3a20 6e75 6d54 7269 616e 676c 6573  de: numTriangles
│ │ │ │ +0005ddc0: 7f33 3332 3434 370a 4e6f 6465 3a20 4f72  .332447.Node: Or
│ │ │ │ +0005ddd0: 6967 696e 616c 5269 6e67 7f33 3336 3037  iginalRing.33607
│ │ │ │ +0005dde0: 330a 4e6f 6465 3a20 7261 6e64 6f6d 4772  3.Node: randomGr
│ │ │ │ +0005ddf0: 6170 687f 3333 3736 3135 0a4e 6f64 653a  aph.337615.Node:
│ │ │ │ +0005de00: 2072 616e 646f 6d48 7970 6572 4772 6170   randomHyperGrap
│ │ │ │ +0005de10: 687f 3334 3035 3439 0a4e 6f64 653a 2072  h.340549.Node: r
│ │ │ │ +0005de20: 616e 646f 6d48 7970 6572 4772 6170 685f  andomHyperGraph_
│ │ │ │ +0005de30: 6c70 5f70 645f 7064 5f70 645f 636d 4272  lp_pd_pd_pd_cmBr
│ │ │ │ +0005de40: 616e 6368 4c69 6d69 743d 3e5f 7064 5f70  anchLimit=>_pd_p
│ │ │ │ +0005de50: 645f 7064 5f72 707f 3334 3538 3339 0a4e  d_pd_rp.345839.N
│ │ │ │ +0005de60: 6f64 653a 2072 616e 646f 6d48 7970 6572  ode: randomHyper
│ │ │ │ +0005de70: 4772 6170 685f 6c70 5f70 645f 7064 5f70  Graph_lp_pd_pd_p
│ │ │ │ +0005de80: 645f 636d 5469 6d65 4c69 6d69 743d 3e5f  d_cmTimeLimit=>_
│ │ │ │ +0005de90: 7064 5f70 645f 7064 5f72 707f 3334 3738  pd_pd_pd_rp.3478
│ │ │ │ +0005dea0: 3739 0a4e 6f64 653a 2072 616e 646f 6d55  79.Node: randomU
│ │ │ │ +0005deb0: 6e69 666f 726d 4879 7065 7247 7261 7068  niformHyperGraph
│ │ │ │ +0005dec0: 7f33 3439 3735 390a 4e6f 6465 3a20 7269  .349759.Node: ri
│ │ │ │ +0005ded0: 6e67 5f6c 7048 7970 6572 4772 6170 685f  ng_lpHyperGraph_
│ │ │ │ +0005dee0: 7270 7f33 3533 3535 340a 4e6f 6465 3a20  rp.353554.Node: 
│ │ │ │ +0005def0: 7369 6d70 6c69 6369 616c 436f 6d70 6c65  simplicialComple
│ │ │ │ +0005df00: 7854 6f48 7970 6572 4772 6170 687f 3335  xToHyperGraph.35
│ │ │ │ +0005df10: 3533 3439 0a4e 6f64 653a 2073 6d61 6c6c  5349.Node: small
│ │ │ │ +0005df20: 6573 7443 7963 6c65 5369 7a65 7f33 3538  estCycleSize.358
│ │ │ │ +0005df30: 3330 350a 4e6f 6465 3a20 7370 616e 6e69  305.Node: spanni
│ │ │ │ +0005df40: 6e67 5472 6565 7f33 3634 3837 300a 4e6f  ngTree.364870.No
│ │ │ │ +0005df50: 6465 3a20 5469 6d65 4c69 6d69 747f 3336  de: TimeLimit.36
│ │ │ │ +0005df60: 3837 3038 0a4e 6f64 653a 2076 6572 7465  8708.Node: verte
│ │ │ │ +0005df70: 7843 6f76 6572 4e75 6d62 6572 7f33 3730  xCoverNumber.370
│ │ │ │ +0005df80: 3133 320a 4e6f 6465 3a20 7665 7274 6578  132.Node: vertex
│ │ │ │ +0005df90: 436f 7665 7273 7f33 3735 3130 300a 4e6f  Covers.375100.No
│ │ │ │ +0005dfa0: 6465 3a20 7665 7274 6963 6573 5f6c 7048  de: vertices_lpH
│ │ │ │ +0005dfb0: 7970 6572 4772 6170 685f 7270 7f33 3739  yperGraph_rp.379
│ │ │ │ +0005dfc0: 3531 340a 1f0a 456e 6420 5461 6720 5461  514...End Tag Ta
│ │ │ │ +0005dfd0: 626c 650a                                ble.
│ │ ├── ./usr/share/info/EigenSolver.info.gz
│ │ │ ├── EigenSolver.info
│ │ │ │ @@ -171,15 +171,15 @@
│ │ │ │  00000aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00000ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00000ac0: 2b0a 7c69 3320 3a20 656c 6170 7365 6454  +.|i3 : elapsedT
│ │ │ │  00000ad0: 696d 6520 736f 6c73 203d 207a 6572 6f44  ime sols = zeroD
│ │ │ │  00000ae0: 696d 536f 6c76 6520 493b 2020 2020 2020  imSolve I;      
│ │ │ │  00000af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00000b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00000b10: 7c0a 7c20 2d2d 202e 3232 3138 3234 7320  |.| -- .221824s 
│ │ │ │ +00000b10: 7c0a 7c20 2d2d 202e 3232 3835 3235 7320  |.| -- .228525s 
│ │ │ │  00000b20: 656c 6170 7365 6420 2020 2020 2020 2020  elapsed         
│ │ │ │  00000b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00000b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00000b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00000b60: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00000b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00000b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ ├── ./usr/share/info/Elimination.info.gz
│ │ │ ├── Elimination.info
│ │ │ │ @@ -336,18 +336,18 @@
│ │ │ │  000014f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00001500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00001510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │  00001520: 6934 203a 2074 696d 6520 656c 696d 696e  i4 : time elimin
│ │ │ │  00001530: 6174 6528 782c 6964 6561 6c28 662c 6729  ate(x,ideal(f,g)
│ │ │ │  00001540: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  00001550: 2020 2020 2020 2020 207c 0a7c 202d 2d20           |.| -- 
│ │ │ │ -00001560: 7573 6564 2030 2e30 3032 3733 3032 3573  used 0.00273025s
│ │ │ │ -00001570: 2028 6370 7529 3b20 302e 3030 3237 3236   (cpu); 0.002726
│ │ │ │ -00001580: 3673 2028 7468 7265 6164 293b 2030 7320  6s (thread); 0s 
│ │ │ │ -00001590: 2867 6329 207c 0a7c 2020 2020 2020 2020  (gc) |.|        
│ │ │ │ +00001560: 7573 6564 2030 2e30 3033 3232 3739 3573  used 0.00322795s
│ │ │ │ +00001570: 2028 6370 7529 3b20 302e 3030 3332 3237   (cpu); 0.003227
│ │ │ │ +00001580: 3037 7320 2874 6872 6561 6429 3b20 3073  07s (thread); 0s
│ │ │ │ +00001590: 2028 6763 297c 0a7c 2020 2020 2020 2020   (gc)|.|        
│ │ │ │  000015a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000015b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000015c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000015d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  000015e0: 2020 2020 2020 2020 2020 3220 2020 2032            2    2
│ │ │ │  000015f0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │  00001600: 2020 2020 2020 2020 2032 2020 207c 0a7c           2   |.|
│ │ │ │ @@ -366,17 +366,17 @@
│ │ │ │  000016d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000016e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000016f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │  00001700: 6935 203a 2074 696d 6520 6964 6561 6c20  i5 : time ideal 
│ │ │ │  00001710: 7265 7375 6c74 616e 7428 662c 672c 7829  resultant(f,g,x)
│ │ │ │  00001720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001730: 2020 2020 2020 2020 207c 0a7c 202d 2d20           |.| -- 
│ │ │ │ -00001740: 7573 6564 2030 2e30 3031 3635 3130 3773  used 0.00165107s
│ │ │ │ -00001750: 2028 6370 7529 3b20 302e 3030 3136 3531   (cpu); 0.001651
│ │ │ │ -00001760: 3136 7320 2874 6872 6561 6429 3b20 3073  16s (thread); 0s
│ │ │ │ +00001740: 7573 6564 2030 2e30 3031 3738 3634 3173  used 0.00178641s
│ │ │ │ +00001750: 2028 6370 7529 3b20 302e 3030 3137 3838   (cpu); 0.001788
│ │ │ │ +00001760: 3536 7320 2874 6872 6561 6429 3b20 3073  56s (thread); 0s
│ │ │ │  00001770: 2028 6763 297c 0a7c 2020 2020 2020 2020   (gc)|.|        
│ │ │ │  00001780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000017a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000017b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  000017c0: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │  000017d0: 2032 2020 2020 2020 2020 2020 2020 2032   2             2
│ │ │ │ @@ -620,17 +620,17 @@
│ │ │ │  000026b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000026c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000026d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000026e0: 2b0a 7c69 3420 3a20 7469 6d65 2065 6c69  +.|i4 : time eli
│ │ │ │  000026f0: 6d69 6e61 7465 2878 2c69 6465 616c 2866  minate(x,ideal(f
│ │ │ │  00002700: 2c67 2929 2020 2020 2020 2020 2020 2020  ,g))            
│ │ │ │  00002710: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00002720: 2d2d 2075 7365 6420 302e 3030 3236 3033  -- used 0.002603
│ │ │ │ -00002730: 3634 7320 2863 7075 293b 2030 2e30 3032  64s (cpu); 0.002
│ │ │ │ -00002740: 3630 3037 3373 2028 7468 7265 6164 293b  60073s (thread);
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│ │ │ │ +00002730: 3138 7320 2863 7075 293b 2030 2e30 3033  18s (cpu); 0.003
│ │ │ │ +00002740: 3035 3733 3573 2028 7468 7265 6164 293b  05735s (thread);
│ │ │ │  00002750: 2030 7320 2867 6329 7c0a 7c20 2020 2020   0s (gc)|.|     
│ │ │ │  00002760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002790: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  000027a0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │  000027b0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ @@ -650,18 +650,18 @@
│ │ │ │  00002890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000028a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000028b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000028c0: 2b0a 7c69 3520 3a20 7469 6d65 2069 6465  +.|i5 : time ide
│ │ │ │  000028d0: 616c 2072 6573 756c 7461 6e74 2866 2c67  al resultant(f,g
│ │ │ │  000028e0: 2c78 2920 2020 2020 2020 2020 2020 2020  ,x)             
│ │ │ │  000028f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00002900: 2d2d 2075 7365 6420 302e 3030 3135 3138  -- used 0.001518
│ │ │ │ -00002910: 3731 7320 2863 7075 293b 2030 2e30 3031  71s (cpu); 0.001
│ │ │ │ -00002920: 3531 3937 3373 2028 7468 7265 6164 293b  51973s (thread);
│ │ │ │ -00002930: 2030 7320 2867 6329 7c0a 7c20 2020 2020   0s (gc)|.|     
│ │ │ │ +00002900: 2d2d 2075 7365 6420 302e 3030 3138 3039  -- used 0.001809
│ │ │ │ +00002910: 3934 7320 2863 7075 293b 2030 2e30 3031  94s (cpu); 0.001
│ │ │ │ +00002920: 3831 3038 7320 2874 6872 6561 6429 3b20  8108s (thread); 
│ │ │ │ +00002930: 3073 2028 6763 2920 7c0a 7c20 2020 2020  0s (gc) |.|     
│ │ │ │  00002940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002970: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00002980: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │  00002990: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │  000029a0: 2020 3220 2020 2020 2020 2020 2020 3220    2           2 
│ │ │ │ @@ -995,16 +995,16 @@
│ │ │ │  00003e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00003e30: 2d2d 2b0a 7c69 3420 3a20 7469 6d65 2065  --+.|i4 : time e
│ │ │ │  00003e40: 6c69 6d69 6e61 7465 2869 6465 616c 2866  liminate(ideal(f
│ │ │ │  00003e50: 2c67 292c 7829 2020 2020 2020 2020 2020  ,g),x)          
│ │ │ │  00003e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003e80: 2020 7c0a 7c20 2d2d 2075 7365 6420 312e    |.| -- used 1.
│ │ │ │ -00003e90: 3538 3832 3273 2028 6370 7529 3b20 312e  58822s (cpu); 1.
│ │ │ │ -00003ea0: 3334 3035 3673 2028 7468 7265 6164 293b  34056s (thread);
│ │ │ │ +00003e90: 3439 3934 3473 2028 6370 7529 3b20 312e  49944s (cpu); 1.
│ │ │ │ +00003ea0: 3333 3932 3273 2028 7468 7265 6164 293b  33922s (thread);
│ │ │ │  00003eb0: 2030 7320 2867 6329 2020 2020 2020 2020   0s (gc)        
│ │ │ │  00003ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003ed0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00003ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -1275,17 +1275,17 @@
│ │ │ │  00004fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00004fb0: 2d2d 2b0a 7c69 3520 3a20 7469 6d65 2069  --+.|i5 : time i
│ │ │ │  00004fc0: 6465 616c 2072 6573 756c 7461 6e74 2866  deal resultant(f
│ │ │ │  00004fd0: 2c67 2c78 2920 2020 2020 2020 2020 2020  ,g,x)           
│ │ │ │  00004fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005000: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e    |.| -- used 0.
│ │ │ │ -00005010: 3031 3632 3033 7320 2863 7075 293b 2030  016203s (cpu); 0
│ │ │ │ -00005020: 2e30 3136 3230 3637 7320 2874 6872 6561  .0162067s (threa
│ │ │ │ -00005030: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │ +00005010: 3031 3635 3135 3373 2028 6370 7529 3b20  0165153s (cpu); 
│ │ │ │ +00005020: 302e 3031 3635 3137 3873 2028 7468 7265  0.0165178s (thre
│ │ │ │ +00005030: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │  00005040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005050: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00005060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000050a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ @@ -1917,16 +1917,16 @@
│ │ │ │  000077c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000077d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520  ----------+.|i5 
│ │ │ │  000077e0: 3a20 7469 6d65 2065 6c69 6d69 6e61 7465  : time eliminate
│ │ │ │  000077f0: 2869 6465 616c 2866 2c67 292c 7829 2020  (ideal(f,g),x)  
│ │ │ │  00007800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007820: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d            |.| --
│ │ │ │ -00007830: 2075 7365 6420 312e 3633 3431 3473 2028   used 1.63414s (
│ │ │ │ -00007840: 6370 7529 3b20 312e 3339 3738 3173 2028  cpu); 1.39781s (
│ │ │ │ +00007830: 2075 7365 6420 312e 3634 3638 3873 2028   used 1.64688s (
│ │ │ │ +00007840: 6370 7529 3b20 312e 3439 3434 3173 2028  cpu); 1.49441s (
│ │ │ │  00007850: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 0s (gc)
│ │ │ │  00007860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007870: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00007880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000078a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000078b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -2197,16 +2197,16 @@
│ │ │ │  00008940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00008950: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620  ----------+.|i6 
│ │ │ │  00008960: 3a20 7469 6d65 2069 6465 616c 2072 6573  : time ideal res
│ │ │ │  00008970: 756c 7461 6e74 2866 2c67 2c78 2920 2020  ultant(f,g,x)   
│ │ │ │  00008980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00008990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000089a0: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d            |.| --
│ │ │ │ -000089b0: 2075 7365 6420 302e 3031 3537 3634 3373   used 0.0157643s
│ │ │ │ -000089c0: 2028 6370 7529 3b20 302e 3031 3537 3635   (cpu); 0.015765
│ │ │ │ +000089b0: 2075 7365 6420 302e 3031 3731 3034 3173   used 0.0171041s
│ │ │ │ +000089c0: 2028 6370 7529 3b20 302e 3031 3731 3034   (cpu); 0.017104
│ │ │ │  000089d0: 3873 2028 7468 7265 6164 293b 2030 7320  8s (thread); 0s 
│ │ │ │  000089e0: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │  000089f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00008a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00008a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00008a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00008a30: 2020 2020 2020 2020 2020 2020 2020 2020
│ │ ├── ./usr/share/info/EnumerationCurves.info.gz
│ │ │ ├── EnumerationCurves.info
│ │ │ │ @@ -256,16 +256,16 @@
│ │ │ │  00000ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00001000: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 7469  ------+.|i1 : ti
│ │ │ │  00001010: 6d65 2066 6f72 206e 2066 726f 6d20 3220  me for n from 2 
│ │ │ │  00001020: 746f 2031 3020 6c69 7374 206c 696e 6573  to 10 list lines
│ │ │ │  00001030: 4879 7065 7273 7572 6661 6365 286e 2920  Hypersurface(n) 
│ │ │ │  00001040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001050: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365        |.| -- use
│ │ │ │ -00001060: 6420 302e 3032 3831 3532 3773 2028 6370  d 0.0281527s (cp
│ │ │ │ -00001070: 7529 3b20 302e 3032 3831 3533 3673 2028  u); 0.0281536s (
│ │ │ │ +00001060: 6420 302e 3032 3939 3536 3773 2028 6370  d 0.0299567s (cp
│ │ │ │ +00001070: 7529 3b20 302e 3032 3939 3534 3973 2028  u); 0.0299549s (
│ │ │ │  00001080: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 0s (gc)
│ │ │ │  00001090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000010a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000010b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000010c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000010d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000010e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -649,16 +649,16 @@
│ │ │ │  00002880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000028a0: 2d2d 2b0a 7c69 3720 3a20 7469 6d65 2066  --+.|i7 : time f
│ │ │ │  000028b0: 6f72 2044 2069 6e20 5420 6c69 7374 2072  or D in T list r
│ │ │ │  000028c0: 6174 696f 6e61 6c43 7572 7665 2832 2c44  ationalCurve(2,D
│ │ │ │  000028d0: 2920 2d20 7261 7469 6f6e 616c 4375 7276  ) - rationalCurv
│ │ │ │  000028e0: 6528 312c 4429 2f38 7c0a 7c20 2d2d 2075  e(1,D)/8|.| -- u
│ │ │ │ -000028f0: 7365 6420 302e 3331 3938 3034 7320 2863  sed 0.319804s (c
│ │ │ │ -00002900: 7075 293b 2030 2e32 3731 3831 3173 2028  pu); 0.271811s (
│ │ │ │ +000028f0: 7365 6420 302e 3334 3931 3232 7320 2863  sed 0.349122s (c
│ │ │ │ +00002900: 7075 293b 2030 2e32 3939 3437 3373 2028  pu); 0.299473s (
│ │ │ │  00002910: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 0s (gc)
│ │ │ │  00002920: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  00002930: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00002940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002970: 2020 2020 7c0a 7c6f 3720 3d20 7b36 3039      |.|o7 = {609
│ │ │ │ @@ -685,16 +685,16 @@
│ │ │ │  00002ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │  00002af0: 3820 3a20 7469 6d65 2072 6174 696f 6e61  8 : time rationa
│ │ │ │  00002b00: 6c43 7572 7665 2833 2920 2020 2020 2020  lCurve(3)       
│ │ │ │  00002b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002b20: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -00002b30: 6564 2030 2e32 3133 3936 3573 2028 6370  ed 0.213965s (cp
│ │ │ │ -00002b40: 7529 3b20 302e 3138 3637 3137 7320 2874  u); 0.186717s (t
│ │ │ │ +00002b30: 6564 2030 2e31 3334 3438 3673 2028 6370  ed 0.134486s (cp
│ │ │ │ +00002b40: 7529 3b20 302e 3133 3434 3932 7320 2874  u); 0.134492s (t
│ │ │ │  00002b50: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │  00002b60: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00002b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002b90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00002ba0: 2020 2020 2038 3536 3435 3735 3030 3020       8564575000 
│ │ │ │  00002bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -718,16 +718,16 @@
│ │ │ │  00002cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  00002d00: 0a7c 6939 203a 2074 696d 6520 666f 7220  .|i9 : time for 
│ │ │ │  00002d10: 4420 696e 2054 206c 6973 7420 7261 7469  D in T list rati
│ │ │ │  00002d20: 6f6e 616c 4375 7276 6528 332c 4429 2020  onalCurve(3,D)  
│ │ │ │  00002d30: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d            |.| --
│ │ │ │ -00002d40: 2075 7365 6420 352e 3831 3334 3973 2028   used 5.81349s (
│ │ │ │ -00002d50: 6370 7529 3b20 352e 3039 3031 3773 2028  cpu); 5.09017s (
│ │ │ │ +00002d40: 2075 7365 6420 342e 3936 3535 3973 2028   used 4.96559s (
│ │ │ │ +00002d50: 6370 7529 3b20 342e 3431 3339 3473 2028  cpu); 4.41394s (
│ │ │ │  00002d60: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 0s (gc)
│ │ │ │  00002d70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00002d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002db0: 7c0a 7c20 2020 2020 2038 3536 3435 3735  |.|      8564575
│ │ │ │  00002dc0: 3030 3020 2034 3232 3639 3038 3136 2020  000  422690816  
│ │ │ │ @@ -761,16 +761,16 @@
│ │ │ │  00002f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002fb0: 2d2d 2d2b 0a7c 6931 3020 3a20 7469 6d65  ---+.|i10 : time
│ │ │ │  00002fc0: 2072 6174 696f 6e61 6c43 7572 7665 2833   rationalCurve(3
│ │ │ │  00002fd0: 2920 2d20 7261 7469 6f6e 616c 4375 7276  ) - rationalCurv
│ │ │ │  00002fe0: 6528 3129 2f32 3720 2020 207c 0a7c 202d  e(1)/27    |.| -
│ │ │ │ -00002ff0: 2d20 7573 6564 2030 2e32 3236 3838 3473  - used 0.226884s
│ │ │ │ -00003000: 2028 6370 7529 3b20 302e 3137 3330 3639   (cpu); 0.173069
│ │ │ │ +00002ff0: 2d20 7573 6564 2030 2e31 3336 3634 3773  - used 0.136647s
│ │ │ │ +00003000: 2028 6370 7529 3b20 302e 3133 3636 3532   (cpu); 0.136652
│ │ │ │  00003010: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │  00003020: 6763 297c 0a7c 2020 2020 2020 2020 2020  gc)|.|          
│ │ │ │  00003030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003050: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │  00003060: 3020 3d20 3331 3732 3036 3337 3520 2020  0 = 317206375   
│ │ │ │  00003070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -799,16 +799,16 @@
│ │ │ │  000031e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000031f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00003200: 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20 7469  -----+.|i11 : ti
│ │ │ │  00003210: 6d65 2066 6f72 2044 2069 6e20 5420 6c69  me for D in T li
│ │ │ │  00003220: 7374 2072 6174 696f 6e61 6c43 7572 7665  st rationalCurve
│ │ │ │  00003230: 2833 2c44 2920 2d20 7261 7469 6f6e 616c  (3,D) - rational
│ │ │ │  00003240: 4375 7276 6528 312c 4429 2f32 377c 0a7c  Curve(1,D)/27|.|
│ │ │ │ -00003250: 202d 2d20 7573 6564 2035 2e34 3938 3139   -- used 5.49819
│ │ │ │ -00003260: 7320 2863 7075 293b 2034 2e37 3538 3933  s (cpu); 4.75893
│ │ │ │ +00003250: 202d 2d20 7573 6564 2035 2e33 3035 3732   -- used 5.30572
│ │ │ │ +00003260: 7320 2863 7075 293b 2034 2e36 3532 3839  s (cpu); 4.65289
│ │ │ │  00003270: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │  00003280: 6763 2920 2020 2020 2020 2020 2020 2020  gc)             
│ │ │ │  00003290: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000032a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000032b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000032c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000032d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ @@ -835,17 +835,17 @@
│ │ │ │  00003420: 0a0a 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  ....+-----------
│ │ │ │  00003430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00003440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00003450: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3132 203a  --------+.|i12 :
│ │ │ │  00003460: 2074 696d 6520 7261 7469 6f6e 616c 4375   time rationalCu
│ │ │ │  00003470: 7276 6528 3429 2020 2020 2020 2020 2020  rve(4)          
│ │ │ │  00003480: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00003490: 7c20 2d2d 2075 7365 6420 312e 3631 3939  | -- used 1.6199
│ │ │ │ -000034a0: 3473 2028 6370 7529 3b20 312e 3432 3330  4s (cpu); 1.4230
│ │ │ │ -000034b0: 3173 2028 7468 7265 6164 293b 2030 7320  1s (thread); 0s 
│ │ │ │ +00003490: 7c20 2d2d 2075 7365 6420 312e 3533 3237  | -- used 1.5327
│ │ │ │ +000034a0: 3273 2028 6370 7529 3b20 312e 3430 3532  2s (cpu); 1.4052
│ │ │ │ +000034b0: 3673 2028 7468 7265 6164 293b 2030 7320  6s (thread); 0s 
│ │ │ │  000034c0: 2867 6329 7c0a 7c20 2020 2020 2020 2020  (gc)|.|         
│ │ │ │  000034d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000034e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000034f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00003500: 2020 2031 3535 3137 3932 3637 3936 3837     1551792679687
│ │ │ │  00003510: 3520 2020 2020 2020 2020 2020 2020 2020  5               
│ │ │ │  00003520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -866,17 +866,17 @@
│ │ │ │  00003610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00003620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00003630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  00003640: 7c69 3133 203a 2074 696d 6520 7261 7469  |i13 : time rati
│ │ │ │  00003650: 6f6e 616c 4375 7276 6528 342c 7b34 2c32  onalCurve(4,{4,2
│ │ │ │  00003660: 7d29 2020 2020 2020 2020 2020 2020 2020  })              
│ │ │ │  00003670: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ -00003680: 372e 3431 3539 7320 2863 7075 293b 2035  7.4159s (cpu); 5
│ │ │ │ -00003690: 2e38 3137 3632 7320 2874 6872 6561 6429  .81762s (thread)
│ │ │ │ -000036a0: 3b20 3073 2028 6763 2920 7c0a 7c20 2020  ; 0s (gc) |.|   
│ │ │ │ +00003680: 362e 3930 3336 3773 2028 6370 7529 3b20  6.90367s (cpu); 
│ │ │ │ +00003690: 352e 3735 3736 3773 2028 7468 7265 6164  5.75767s (thread
│ │ │ │ +000036a0: 293b 2030 7320 2867 6329 7c0a 7c20 2020  ); 0s (gc)|.|   
│ │ │ │  000036b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000036c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000036d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000036e0: 7c0a 7c6f 3133 203d 2033 3838 3339 3134  |.|o13 = 3883914
│ │ │ │  000036f0: 3038 3420 2020 2020 2020 2020 2020 2020  084             
│ │ │ │  00003700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003710: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ @@ -899,16 +899,16 @@
│ │ │ │  00003820: 0a0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...+------------
│ │ │ │  00003830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00003840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00003850: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3420 3a20  -------+.|i14 : 
│ │ │ │  00003860: 7469 6d65 2072 6174 696f 6e61 6c43 7572  time rationalCur
│ │ │ │  00003870: 7665 2834 2920 2d20 7261 7469 6f6e 616c  ve(4) - rational
│ │ │ │  00003880: 4375 7276 6528 3229 2f38 2020 207c 0a7c  Curve(2)/8   |.|
│ │ │ │ -00003890: 202d 2d20 7573 6564 2031 2e36 3931 3239   -- used 1.69129
│ │ │ │ -000038a0: 7320 2863 7075 293b 2031 2e34 3638 3637  s (cpu); 1.46867
│ │ │ │ +00003890: 202d 2d20 7573 6564 2031 2e36 3633 3533   -- used 1.66353
│ │ │ │ +000038a0: 7320 2863 7075 293b 2031 2e34 3536 3237  s (cpu); 1.45627
│ │ │ │  000038b0: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │  000038c0: 6763 297c 0a7c 2020 2020 2020 2020 2020  gc)|.|          
│ │ │ │  000038d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000038e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000038f0: 2020 2020 2020 2020 207c 0a7c 6f31 3420           |.|o14 
│ │ │ │  00003900: 3d20 3234 3234 3637 3533 3030 3030 2020  = 242467530000  
│ │ │ │  00003910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -937,16 +937,16 @@
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│ │ │ │  00003a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00003aa0: 2d2d 2d2d 2d2b 0a7c 6931 3520 3a20 7469  -----+.|i15 : ti
│ │ │ │  00003ab0: 6d65 2072 6174 696f 6e61 6c43 7572 7665  me rationalCurve
│ │ │ │  00003ac0: 2834 2c7b 342c 327d 2920 2d20 7261 7469  (4,{4,2}) - rati
│ │ │ │  00003ad0: 6f6e 616c 4375 7276 6528 322c 7b34 2c32  onalCurve(2,{4,2
│ │ │ │  00003ae0: 7d29 2f38 7c0a 7c20 2d2d 2075 7365 6420  })/8|.| -- used 
│ │ │ │ -00003af0: 372e 3439 3630 3873 2028 6370 7529 3b20  7.49608s (cpu); 
│ │ │ │ -00003b00: 352e 3936 3437 3973 2028 7468 7265 6164  5.96479s (thread
│ │ │ │ +00003af0: 362e 3932 3737 3373 2028 6370 7529 3b20  6.92773s (cpu); 
│ │ │ │ +00003b00: 352e 3636 3134 3973 2028 7468 7265 6164  5.66149s (thread
│ │ │ │  00003b10: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │  00003b20: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00003b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003b60: 2020 7c0a 7c6f 3135 203d 2033 3838 3339    |.|o15 = 38839
│ │ │ │  00003b70: 3032 3532 3820 2020 2020 2020 2020 2020  02528           
│ │ │ │ @@ -964,16 +964,16 @@
│ │ │ │  00003c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00003c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00003c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  00003c60: 7c69 3136 203a 2074 696d 6520 7261 7469  |i16 : time rati
│ │ │ │  00003c70: 6f6e 616c 4375 7276 6528 342c 7b33 2c33  onalCurve(4,{3,3
│ │ │ │  00003c80: 7d29 202d 2072 6174 696f 6e61 6c43 7572  }) - rationalCur
│ │ │ │  00003c90: 7665 2832 2c7b 332c 337d 292f 387c 0a7c  ve(2,{3,3})/8|.|
│ │ │ │ -00003ca0: 202d 2d20 7573 6564 2038 2e31 3430 3334   -- used 8.14034
│ │ │ │ -00003cb0: 7320 2863 7075 293b 2036 2e32 3331 3431  s (cpu); 6.23141
│ │ │ │ +00003ca0: 202d 2d20 7573 6564 2037 2e30 3432 3832   -- used 7.04282
│ │ │ │ +00003cb0: 7320 2863 7075 293b 2035 2e38 3732 3132  s (cpu); 5.87212
│ │ │ │  00003cc0: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │  00003cd0: 6763 2920 2020 2020 2020 2020 7c0a 7c20  gc)         |.| 
│ │ │ │  00003ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003d10: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │  00003d20: 3620 3d20 3131 3339 3434 3833 3834 2020  6 = 1139448384
│ │ ├── ./usr/share/info/EquivariantGB.info.gz
│ │ │ ├── EquivariantGB.info
│ │ │ │ @@ -1917,21 +1917,21 @@
│ │ │ │  000077c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000077d0: 2020 2020 2020 2020 2020 7c0a 7c33 2020            |.|3  
│ │ │ │  000077e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000077f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007820: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00007830: 2020 2d2d 2075 7365 6420 2e30 3031 3933    -- used .00193
│ │ │ │ -00007840: 3339 3520 7365 636f 6e64 7320 2020 2020  395 seconds     
│ │ │ │ +00007830: 2020 2d2d 2075 7365 6420 2e30 3031 3939    -- used .00199
│ │ │ │ +00007840: 3731 3220 7365 636f 6e64 7320 2020 2020  712 seconds     
│ │ │ │  00007850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007870: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00007880: 2020 2d2d 2075 7365 6420 2e30 3030 3537    -- used .00057
│ │ │ │ -00007890: 3637 3631 2073 6563 6f6e 6473 2020 2020  6761 seconds    
│ │ │ │ +00007880: 2020 2d2d 2075 7365 6420 2e30 3030 3538    -- used .00058
│ │ │ │ +00007890: 3732 3920 7365 636f 6e64 7320 2020 2020  729 seconds     
│ │ │ │  000078a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000078b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000078c0: 2020 2020 2020 2020 2020 7c0a 7c28 392c            |.|(9,
│ │ │ │  000078d0: 2039 2920 2020 2020 2020 2020 2020 2020   9)             
│ │ │ │  000078e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000078f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -1942,21 +1942,21 @@
│ │ │ │  00007950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007960: 2020 2020 2020 2020 2020 7c0a 7c34 2020            |.|4  
│ │ │ │  00007970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000079a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000079b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -000079c0: 2020 2d2d 2075 7365 6420 2e30 3033 3432    -- used .00342
│ │ │ │ -000079d0: 3637 3820 7365 636f 6e64 7320 2020 2020  678 seconds     
│ │ │ │ +000079c0: 2020 2d2d 2075 7365 6420 2e30 3034 3033    -- used .00403
│ │ │ │ +000079d0: 3938 3920 7365 636f 6e64 7320 2020 2020  989 seconds     
│ │ │ │  000079e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000079f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007a00: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00007a10: 2020 2d2d 2075 7365 6420 2e30 3034 3337    -- used .00437
│ │ │ │ -00007a20: 3531 3720 7365 636f 6e64 7320 2020 2020  517 seconds     
│ │ │ │ +00007a10: 2020 2d2d 2075 7365 6420 2e30 3034 3939    -- used .00499
│ │ │ │ +00007a20: 3031 3920 7365 636f 6e64 7320 2020 2020  019 seconds     
│ │ │ │  00007a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007a50: 2020 2020 2020 2020 2020 7c0a 7c28 3136            |.|(16
│ │ │ │  00007a60: 2c20 3236 2920 2020 2020 2020 2020 2020  , 26)           
│ │ │ │  00007a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -1967,61 +1967,61 @@
│ │ │ │  00007ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007af0: 2020 2020 2020 2020 2020 7c0a 7c35 2020            |.|5  
│ │ │ │  00007b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007b40: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00007b50: 2020 2d2d 2075 7365 6420 2e30 3038 3138    -- used .00818
│ │ │ │ -00007b60: 3235 3720 7365 636f 6e64 7320 2020 2020  257 seconds     
│ │ │ │ +00007b50: 2020 2d2d 2075 7365 6420 2e30 3038 3231    -- used .00821
│ │ │ │ +00007b60: 3630 3620 7365 636f 6e64 7320 2020 2020  606 seconds     
│ │ │ │  00007b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007b90: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00007ba0: 2020 2d2d 2075 7365 6420 2e30 3236 3339    -- used .02639
│ │ │ │ -00007bb0: 3031 2073 6563 6f6e 6473 2020 2020 2020  01 seconds      
│ │ │ │ +00007bb0: 3536 2073 6563 6f6e 6473 2020 2020 2020  56 seconds      
│ │ │ │  00007bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007be0: 2020 2020 2020 2020 2020 7c0a 7c28 3235            |.|(25
│ │ │ │  00007bf0: 2c20 3630 2920 2020 2020 2020 2020 2020  , 60)           
│ │ │ │  00007c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007c30: 2020 2020 2020 2020 2020 7c0a 7c36 2020            |.|6  
│ │ │ │  00007c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007c80: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00007c90: 2020 2d2d 2075 7365 6420 2e30 3137 3835    -- used .01785
│ │ │ │ -00007ca0: 3333 2073 6563 6f6e 6473 2020 2020 2020  33 seconds      
│ │ │ │ +00007c90: 2020 2d2d 2075 7365 6420 2e30 3139 3934    -- used .01994
│ │ │ │ +00007ca0: 3938 2073 6563 6f6e 6473 2020 2020 2020  98 seconds      
│ │ │ │  00007cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007cd0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00007ce0: 2020 2d2d 2075 7365 6420 2e32 3034 3632    -- used .20462
│ │ │ │ -00007cf0: 3920 7365 636f 6e64 7320 2020 2020 2020  9 seconds       
│ │ │ │ +00007ce0: 2020 2d2d 2075 7365 6420 2e32 3235 3130    -- used .22510
│ │ │ │ +00007cf0: 3320 7365 636f 6e64 7320 2020 2020 2020  3 seconds       
│ │ │ │  00007d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007d20: 2020 2020 2020 2020 2020 7c0a 7c28 3336            |.|(36
│ │ │ │  00007d30: 2c20 3132 3029 2020 2020 2020 2020 2020  , 120)          
│ │ │ │  00007d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007d70: 2020 2020 2020 2020 2020 7c0a 7c37 2020            |.|7  
│ │ │ │  00007d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007dc0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00007dd0: 2020 2d2d 2075 7365 6420 2e30 3338 3639    -- used .03869
│ │ │ │ -00007de0: 3338 2073 6563 6f6e 6473 2020 2020 2020  38 seconds      
│ │ │ │ +00007dd0: 2020 2d2d 2075 7365 6420 2e30 3433 3033    -- used .04303
│ │ │ │ +00007de0: 3535 2073 6563 6f6e 6473 2020 2020 2020  55 seconds      
│ │ │ │  00007df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007e10: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00007e20: 2020 2d2d 2075 7365 6420 2e38 3233 3834    -- used .82384
│ │ │ │ -00007e30: 3120 7365 636f 6e64 7320 2020 2020 2020  1 seconds       
│ │ │ │ +00007e20: 2020 2d2d 2075 7365 6420 2e39 3239 3338    -- used .92938
│ │ │ │ +00007e30: 2073 6563 6f6e 6473 2020 2020 2020 2020   seconds        
│ │ │ │  00007e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007e60: 2020 2020 2020 2020 2020 7c0a 7c28 3439            |.|(49
│ │ │ │  00007e70: 2c20 3231 3729 2020 2020 2020 2020 2020  , 217)          
│ │ │ │  00007e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007ea0: 2020 2020 2020 2020 2020 2020 2020 2020
│ │ ├── ./usr/share/info/FastMinors.info.gz
│ │ │ ├── FastMinors.info
│ │ │ │ @@ -4406,16 +4406,16 @@
│ │ │ │  00011350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011360: 2d2d 2d2d 2d2d 2b0a 7c69 3238 203a 2074  ------+.|i28 : t
│ │ │ │  00011370: 696d 6520 6469 6d20 284a 202b 2063 686f  ime dim (J + cho
│ │ │ │  00011380: 6f73 6547 6f6f 644d 696e 6f72 7328 382c  oseGoodMinors(8,
│ │ │ │  00011390: 2036 2c20 4d2c 204a 2c20 5374 7261 7465   6, M, J, Strate
│ │ │ │  000113a0: 6779 3d3e 5261 6e64 6f6d 2929 2020 2020  gy=>Random))    
│ │ │ │  000113b0: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365        |.| -- use
│ │ │ │ -000113c0: 6420 302e 3135 3734 3432 7320 2863 7075  d 0.157442s (cpu
│ │ │ │ -000113d0: 293b 2030 2e31 3230 3639 3873 2028 7468  ); 0.120698s (th
│ │ │ │ +000113c0: 6420 302e 3232 3734 3134 7320 2863 7075  d 0.227414s (cpu
│ │ │ │ +000113d0: 293b 2030 2e31 3533 3130 3473 2028 7468  ); 0.153104s (th
│ │ │ │  000113e0: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │  000113f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011400: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00011410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -4431,16 +4431,16 @@
│ │ │ │  000114e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000114f0: 2d2d 2d2d 2d2d 2b0a 7c69 3239 203a 2074  ------+.|i29 : t
│ │ │ │  00011500: 696d 6520 6469 6d20 284a 202b 2063 686f  ime dim (J + cho
│ │ │ │  00011510: 6f73 6547 6f6f 644d 696e 6f72 7328 382c  oseGoodMinors(8,
│ │ │ │  00011520: 2036 2c20 4d2c 204a 2c20 5374 7261 7465   6, M, J, Strate
│ │ │ │  00011530: 6779 3d3e 4c65 7853 6d61 6c6c 6573 7429  gy=>LexSmallest)
│ │ │ │  00011540: 2920 2020 2020 7c0a 7c20 2d2d 2075 7365  )     |.| -- use
│ │ │ │ -00011550: 6420 302e 3330 3234 3836 7320 2863 7075  d 0.302486s (cpu
│ │ │ │ -00011560: 293b 2030 2e32 3038 3039 3973 2028 7468  ); 0.208099s (th
│ │ │ │ +00011550: 6420 302e 3335 3634 3636 7320 2863 7075  d 0.356466s (cpu
│ │ │ │ +00011560: 293b 2030 2e32 3139 3631 3273 2028 7468  ); 0.219612s (th
│ │ │ │  00011570: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │  00011580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011590: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000115a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000115b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000115c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000115d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -4456,16 +4456,16 @@
│ │ │ │  00011670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011680: 2d2d 2d2d 2d2d 2b0a 7c69 3330 203a 2074  ------+.|i30 : t
│ │ │ │  00011690: 696d 6520 6469 6d20 284a 202b 2063 686f  ime dim (J + cho
│ │ │ │  000116a0: 6f73 6547 6f6f 644d 696e 6f72 7328 382c  oseGoodMinors(8,
│ │ │ │  000116b0: 2036 2c20 4d2c 204a 2c20 5374 7261 7465   6, M, J, Strate
│ │ │ │  000116c0: 6779 3d3e 4c65 7853 6d61 6c6c 6573 7454  gy=>LexSmallestT
│ │ │ │  000116d0: 6572 6d29 2920 7c0a 7c20 2d2d 2075 7365  erm)) |.| -- use
│ │ │ │ -000116e0: 6420 302e 3439 3530 3835 7320 2863 7075  d 0.495085s (cpu
│ │ │ │ -000116f0: 293b 2030 2e33 3139 3930 3473 2028 7468  ); 0.319904s (th
│ │ │ │ +000116e0: 6420 302e 3539 3730 3538 7320 2863 7075  d 0.597058s (cpu
│ │ │ │ +000116f0: 293b 2030 2e33 3834 3337 3173 2028 7468  ); 0.384371s (th
│ │ │ │  00011700: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │  00011710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011720: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00011730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -4481,17 +4481,17 @@
│ │ │ │  00011800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011810: 2d2d 2d2d 2d2d 2b0a 7c69 3331 203a 2074  ------+.|i31 : t
│ │ │ │  00011820: 696d 6520 6469 6d20 284a 202b 2063 686f  ime dim (J + cho
│ │ │ │  00011830: 6f73 6547 6f6f 644d 696e 6f72 7328 382c  oseGoodMinors(8,
│ │ │ │  00011840: 2036 2c20 4d2c 204a 2c20 5374 7261 7465   6, M, J, Strate
│ │ │ │  00011850: 6779 3d3e 4c65 784c 6172 6765 7374 2929  gy=>LexLargest))
│ │ │ │  00011860: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365        |.| -- use
│ │ │ │ -00011870: 6420 302e 3231 3136 3273 2028 6370 7529  d 0.21162s (cpu)
│ │ │ │ -00011880: 3b20 302e 3136 3935 3536 7320 2874 6872  ; 0.169556s (thr
│ │ │ │ -00011890: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │ +00011870: 6420 302e 3237 3638 3935 7320 2863 7075  d 0.276895s (cpu
│ │ │ │ +00011880: 293b 2030 2e32 3034 3334 3573 2028 7468  ); 0.204345s (th
│ │ │ │ +00011890: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │  000118a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000118b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000118c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000118d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000118e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000118f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011900: 2020 2020 2020 7c0a 7c6f 3331 203d 2032        |.|o31 = 2
│ │ │ │ @@ -4506,16 +4506,16 @@
│ │ │ │  00011990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000119a0: 2d2d 2d2d 2d2d 2b0a 7c69 3332 203a 2074  ------+.|i32 : t
│ │ │ │  000119b0: 696d 6520 6469 6d20 284a 202b 2063 686f  ime dim (J + cho
│ │ │ │  000119c0: 6f73 6547 6f6f 644d 696e 6f72 7328 382c  oseGoodMinors(8,
│ │ │ │  000119d0: 2036 2c20 4d2c 204a 2c20 5374 7261 7465   6, M, J, Strate
│ │ │ │  000119e0: 6779 3d3e 4752 6576 4c65 7853 6d61 6c6c  gy=>GRevLexSmall
│ │ │ │  000119f0: 6573 7429 2920 7c0a 7c20 2d2d 2075 7365  est)) |.| -- use
│ │ │ │ -00011a00: 6420 302e 3337 3530 3032 7320 2863 7075  d 0.375002s (cpu
│ │ │ │ -00011a10: 293b 2030 2e32 3035 3234 3773 2028 7468  ); 0.205247s (th
│ │ │ │ +00011a00: 6420 302e 3434 3339 3639 7320 2863 7075  d 0.443969s (cpu
│ │ │ │ +00011a10: 293b 2030 2e32 3232 3334 3973 2028 7468  ); 0.222349s (th
│ │ │ │  00011a20: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │  00011a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011a40: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00011a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -4528,20 +4528,20 @@
│ │ │ │  00011af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011b30: 2d2d 2d2d 2d2d 2b0a 7c69 3333 203a 2074  ------+.|i33 : t
│ │ │ │  00011b40: 696d 6520 6469 6d20 284a 202b 2063 686f  ime dim (J + cho
│ │ │ │  00011b50: 6f73 6547 6f6f 644d 696e 6f72 7328 382c  oseGoodMinors(8,
│ │ │ │ -00011b60: 2036 2c20 4d2c 204a 2c20 5374 7261 7465   6, M, J, Strate
│ │ │ │ -00011b70: 6779 3d3e 2020 2020 2020 2020 2020 2020  gy=>            
│ │ │ │ +00011b60: 2036 2c20 4d2c 204a 2c20 2020 2020 2020   6, M, J,       
│ │ │ │ +00011b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011b80: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365        |.| -- use
│ │ │ │ -00011b90: 6420 302e 3333 3831 3438 7320 2863 7075  d 0.338148s (cpu
│ │ │ │ -00011ba0: 293b 2030 2e32 3333 3633 7320 2874 6872  ); 0.23363s (thr
│ │ │ │ -00011bb0: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │ +00011b90: 6420 302e 3430 3835 3132 7320 2863 7075  d 0.408512s (cpu
│ │ │ │ +00011ba0: 293b 2030 2e32 3732 3435 3273 2028 7468  ); 0.272452s (th
│ │ │ │ +00011bb0: 7265 6164 293b 2030 7320 2020 2020 2020  read); 0s       
│ │ │ │  00011bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011bd0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00011be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011c20: 2020 2020 2020 7c0a 7c6f 3333 203d 2033        |.|o33 = 3
│ │ │ │ @@ -4550,6019 +4550,6019 @@
│ │ │ │  00011c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011c70: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │  00011c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00011cc0: 2d2d 2d2d 2d2d 7c0a 7c47 5265 764c 6578  ------|.|GRevLex
│ │ │ │ -00011cd0: 536d 616c 6c65 7374 5465 726d 2929 2020  SmallestTerm))  
│ │ │ │ -00011ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011cc0: 2d2d 2d2d 2d2d 7c0a 7c53 7472 6174 6567  ------|.|Strateg
│ │ │ │ +00011cd0: 793d 3e47 5265 764c 6578 536d 616c 6c65  y=>GRevLexSmalle
│ │ │ │ +00011ce0: 7374 5465 726d 2929 2020 2020 2020 2020  stTerm))        
│ │ │ │  00011cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011d10: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ -00011d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00011d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00011d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00011d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00011d60: 2d2d 2d2d 2d2d 2b0a 7c69 3334 203a 2074  ------+.|i34 : t
│ │ │ │ -00011d70: 696d 6520 6469 6d20 284a 202b 2063 686f  ime dim (J + cho
│ │ │ │ -00011d80: 6f73 6547 6f6f 644d 696e 6f72 7328 382c  oseGoodMinors(8,
│ │ │ │ -00011d90: 2036 2c20 4d2c 204a 2c20 5374 7261 7465   6, M, J, Strate
│ │ │ │ -00011da0: 6779 3d3e 4752 6576 4c65 784c 6172 6765  gy=>GRevLexLarge
│ │ │ │ -00011db0: 7374 2929 2020 7c0a 7c20 2d2d 2075 7365  st))  |.| -- use
│ │ │ │ -00011dc0: 6420 302e 3239 3836 3234 7320 2863 7075  d 0.298624s (cpu
│ │ │ │ -00011dd0: 293b 2030 2e31 3931 3534 3273 2028 7468  ); 0.191542s (th
│ │ │ │ -00011de0: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │ -00011df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011e00: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00011e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011d10: 2020 2020 2020 7c0a 7c28 6763 2920 2020        |.|(gc)   
│ │ │ │ +00011d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011d60: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00011d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00011d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00011d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00011da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00011db0: 2d2d 2d2d 2d2d 2b0a 7c69 3334 203a 2074  ------+.|i34 : t
│ │ │ │ +00011dc0: 696d 6520 6469 6d20 284a 202b 2063 686f  ime dim (J + cho
│ │ │ │ +00011dd0: 6f73 6547 6f6f 644d 696e 6f72 7328 382c  oseGoodMinors(8,
│ │ │ │ +00011de0: 2036 2c20 4d2c 204a 2c20 5374 7261 7465   6, M, J, Strate
│ │ │ │ +00011df0: 6779 3d3e 4752 6576 4c65 784c 6172 6765  gy=>GRevLexLarge
│ │ │ │ +00011e00: 7374 2929 2020 7c0a 7c20 2d2d 2075 7365  st))  |.| -- use
│ │ │ │ +00011e10: 6420 302e 3337 3839 3833 7320 2863 7075  d 0.378983s (cpu
│ │ │ │ +00011e20: 293b 2030 2e32 3338 3837 3473 2028 7468  ); 0.238874s (th
│ │ │ │ +00011e30: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │  00011e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011e50: 2020 2020 2020 7c0a 7c6f 3334 203d 2033        |.|o34 = 3
│ │ │ │ +00011e50: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00011e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011ea0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ -00011eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00011ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00011ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00011ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00011ef0: 2d2d 2d2d 2d2d 2b0a 7c69 3335 203a 2074  ------+.|i35 : t
│ │ │ │ -00011f00: 696d 6520 6469 6d20 284a 202b 2063 686f  ime dim (J + cho
│ │ │ │ -00011f10: 6f73 6547 6f6f 644d 696e 6f72 7328 382c  oseGoodMinors(8,
│ │ │ │ -00011f20: 2036 2c20 4d2c 204a 2c20 5374 7261 7465   6, M, J, Strate
│ │ │ │ -00011f30: 6779 3d3e 506f 696e 7473 2929 2020 2020  gy=>Points))    
│ │ │ │ -00011f40: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365        |.| -- use
│ │ │ │ -00011f50: 6420 3134 2e39 3935 3573 2028 6370 7529  d 14.9955s (cpu)
│ │ │ │ -00011f60: 3b20 3130 2e31 3634 3773 2028 7468 7265  ; 10.1647s (thre
│ │ │ │ -00011f70: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │ -00011f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011f90: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00011fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011ea0: 2020 2020 2020 7c0a 7c6f 3334 203d 2033        |.|o34 = 3
│ │ │ │ +00011eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011ef0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00011f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00011f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00011f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00011f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00011f40: 2d2d 2d2d 2d2d 2b0a 7c69 3335 203a 2074  ------+.|i35 : t
│ │ │ │ +00011f50: 696d 6520 6469 6d20 284a 202b 2063 686f  ime dim (J + cho
│ │ │ │ +00011f60: 6f73 6547 6f6f 644d 696e 6f72 7328 382c  oseGoodMinors(8,
│ │ │ │ +00011f70: 2036 2c20 4d2c 204a 2c20 5374 7261 7465   6, M, J, Strate
│ │ │ │ +00011f80: 6779 3d3e 506f 696e 7473 2929 2020 2020  gy=>Points))    
│ │ │ │ +00011f90: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365        |.| -- use
│ │ │ │ +00011fa0: 6420 3138 2e37 3631 3473 2028 6370 7529  d 18.7614s (cpu)
│ │ │ │ +00011fb0: 3b20 3131 2e39 3032 3573 2028 7468 7265  ; 11.9025s (thre
│ │ │ │ +00011fc0: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │  00011fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011fe0: 2020 2020 2020 7c0a 7c6f 3335 203d 2031        |.|o35 = 1
│ │ │ │ +00011fe0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00011ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012030: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ -00012040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012080: 2d2d 2d2d 2d2d 2b0a 0a49 6e64 6565 642c  ------+..Indeed,
│ │ │ │ -00012090: 2069 6e20 7468 6973 2065 7861 6d70 6c65   in this example
│ │ │ │ -000120a0: 2c20 6576 656e 2063 6f6d 7075 7469 6e67  , even computing
│ │ │ │ -000120b0: 2064 6574 6572 6d69 6e61 6e74 7320 6f66   determinants of
│ │ │ │ -000120c0: 2031 2c30 3030 2072 616e 646f 6d0a 7375   1,000 random.su
│ │ │ │ -000120d0: 626d 6174 7269 6365 7320 6973 206e 6f74  bmatrices is not
│ │ │ │ -000120e0: 2074 7970 6963 616c 6c79 2065 6e6f 7567   typically enoug
│ │ │ │ -000120f0: 6820 746f 2076 6572 6966 7920 7468 6174  h to verify that
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│ │ │ │ -00012110: 6172 2069 6e0a 636f 6469 6d65 6e73 696f  ar in.codimensio
│ │ │ │ -00012120: 6e20 312e 2020 4f6e 2074 6865 206f 7468  n 1.  On the oth
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│ │ │ │ -00012140: 6973 2061 6c6d 6f73 7420 616c 7761 7973  is almost always
│ │ │ │ -00012150: 2071 7569 7465 2065 6666 6563 7469 7665   quite effective
│ │ │ │ -00012160: 2061 740a 6669 6e64 696e 6720 7661 6c75   at.finding valu
│ │ │ │ -00012170: 6162 6c65 2073 7562 6d61 7472 6963 6573  able submatrices
│ │ │ │ -00012180: 2c20 6275 7420 6361 6e20 6265 2071 7569  , but can be qui
│ │ │ │ -00012190: 7465 2073 6c6f 772e 2020 496e 2074 6869  te slow.  In thi
│ │ │ │ -000121a0: 7320 7061 7274 6963 756c 6172 0a65 7861  s particular.exa
│ │ │ │ -000121b0: 6d70 6c65 2c20 7765 2063 616e 2073 6565  mple, we can see
│ │ │ │ -000121c0: 2074 6861 7420 4c65 7853 6d61 6c6c 6573   that LexSmalles
│ │ │ │ -000121d0: 7454 6572 6d20 616c 736f 2070 6572 666f  tTerm also perfo
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│ │ │ │ -000121f0: 6e64 2064 6f65 7320 6974 0a71 7569 636b  nd does it.quick
│ │ │ │ -00012200: 6c79 292e 2053 696e 6365 2064 6966 6665  ly). Since diffe
│ │ │ │ -00012210: 7265 6e74 2073 7472 6174 6567 6965 7320  rent strategies 
│ │ │ │ -00012220: 776f 726b 2062 6574 7465 7220 6f72 2077  work better or w
│ │ │ │ -00012230: 6f72 7365 206f 6e20 6469 6666 6572 656e  orse on differen
│ │ │ │ -00012240: 740a 6578 616d 706c 6573 2c20 7468 6520  t.examples, the 
│ │ │ │ -00012250: 6465 6661 756c 7420 7374 7261 7465 6779  default strategy
│ │ │ │ -00012260: 2061 6374 7561 6c6c 7920 6d69 7865 7320   actually mixes 
│ │ │ │ -00012270: 616e 6420 6d61 7463 6865 7320 7661 7269  and matches vari
│ │ │ │ -00012280: 6f75 7320 7374 7261 7465 6769 6573 2e0a  ous strategies..
│ │ │ │ -00012290: 5468 6520 6465 6661 756c 7420 7374 7261  The default stra
│ │ │ │ -000122a0: 7465 6779 2c20 7768 6963 6820 7765 206e  tegy, which we n
│ │ │ │ -000122b0: 6f77 2065 6c75 6369 6461 7465 2c0a 0a2b  ow elucidate,..+
│ │ │ │ -000122c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000122d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000122e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -000122f0: 3336 203a 2070 6565 6b20 5374 7261 7465  36 : peek Strate
│ │ │ │ -00012300: 6779 4465 6661 756c 7420 2020 2020 2020  gyDefault       
│ │ │ │ -00012310: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00012320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012340: 2020 2020 2020 2020 2020 7c0a 7c6f 3336            |.|o36
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│ │ │ │ -00012370: 2030 2020 2020 2020 7d7c 0a7c 2020 2020   0      }|.|    
│ │ │ │ -00012380: 2020 2020 2020 2020 2020 2020 2020 4752                GR
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│ │ │ │ -000123b0: 2020 2020 2020 2020 2020 2020 2047 5265               GRe
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│ │ │ │ -000123d0: 203d 3e20 3136 207c 0a7c 2020 2020 2020   => 16 |.|      
│ │ │ │ -000123e0: 2020 2020 2020 2020 2020 2020 4c65 784c              LexL
│ │ │ │ -000123f0: 6172 6765 7374 203d 3e20 3020 2020 2020  argest => 0     
│ │ │ │ -00012400: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00012410: 2020 2020 2020 2020 2020 204c 6578 536d             LexSm
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│ │ │ │ -00012430: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00012440: 2020 2020 2020 2020 2020 4c65 7853 6d61            LexSma
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│ │ │ │ -00012460: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00012470: 2020 2020 2020 2020 2050 6f69 6e74 7320           Points 
│ │ │ │ -00012480: 3d3e 2030 2020 2020 2020 2020 2020 2020  => 0            
│ │ │ │ -00012490: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -000124a0: 2020 2020 2020 2020 5261 6e64 6f6d 203d          Random =
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│ │ │ │ -000124c0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -000124d0: 2020 2020 2020 2052 616e 646f 6d4e 6f6e         RandomNon
│ │ │ │ -000124e0: 7a65 726f 203d 3e20 3136 2020 2020 2020  zero => 16      
│ │ │ │ -000124f0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ -00012500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -000125f0: 646f 6d20 7375 626d 6174 7269 6365 7320  dom submatrices 
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│ │ │ │ -00012670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000126a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000126b0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3337 203a  --------+.|i37 :
│ │ │ │ -000126c0: 2074 696d 6520 6368 6f6f 7365 476f 6f64   time chooseGood
│ │ │ │ -000126d0: 4d69 6e6f 7273 2832 302c 2036 2c20 4d2c  Minors(20, 6, M,
│ │ │ │ -000126e0: 204a 2c20 5374 7261 7465 6779 3d3e 5374   J, Strategy=>St
│ │ │ │ -000126f0: 7261 7465 6779 4465 6661 756c 742c 2020  rategyDefault,  
│ │ │ │ -00012700: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075          |.| -- u
│ │ │ │ -00012710: 7365 6420 302e 3438 3633 3273 2028 6370  sed 0.48632s (cp
│ │ │ │ -00012720: 7529 3b20 302e 3434 3436 3936 7320 2874  u); 0.444696s (t
│ │ │ │ -00012730: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │ -00012740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012750: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ -00012760: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ -00012770: 4368 6f6f 7369 6e67 2052 616e 646f 6d20  Choosing Random 
│ │ │ │ -00012780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012030: 2020 2020 2020 7c0a 7c6f 3335 203d 2031        |.|o35 = 1
│ │ │ │ +00012040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012080: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00012090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000120a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000120b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000120c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000120d0: 2d2d 2d2d 2d2d 2b0a 0a49 6e64 6565 642c  ------+..Indeed,
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│ │ │ │ +000120f0: 2c20 6576 656e 2063 6f6d 7075 7469 6e67  , even computing
│ │ │ │ +00012100: 2064 6574 6572 6d69 6e61 6e74 7320 6f66   determinants of
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│ │ │ │ +00012270: 776f 726b 2062 6574 7465 7220 6f72 2077  work better or w
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│ │ │ │ +000122b0: 2061 6374 7561 6c6c 7920 6d69 7865 7320   actually mixes 
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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 2b0a 7c69  ------------+.|i
│ │ │ │ +00012340: 3336 203a 2070 6565 6b20 5374 7261 7465  36 : peek Strate
│ │ │ │ +00012350: 6779 4465 6661 756c 7420 2020 2020 2020  gyDefault       
│ │ │ │ +00012360: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00012370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012390: 2020 2020 2020 2020 2020 7c0a 7c6f 3336            |.|o36
│ │ │ │ +000123a0: 203d 204f 7074 696f 6e54 6162 6c65 7b47   = OptionTable{G
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│ │ │ │ +000123c0: 2030 2020 2020 2020 7d7c 0a7c 2020 2020   0      }|.|    
│ │ │ │ +000123d0: 2020 2020 2020 2020 2020 2020 2020 4752                GR
│ │ │ │ +000123e0: 6576 4c65 7853 6d61 6c6c 6573 7420 3d3e  evLexSmallest =>
│ │ │ │ +000123f0: 2031 3620 2020 2020 7c0a 7c20 2020 2020   16     |.|     
│ │ │ │ +00012400: 2020 2020 2020 2020 2020 2020 2047 5265               GRe
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│ │ │ │ +00012420: 203d 3e20 3136 207c 0a7c 2020 2020 2020   => 16 |.|      
│ │ │ │ +00012430: 2020 2020 2020 2020 2020 2020 4c65 784c              LexL
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│ │ │ │ +00012450: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00012460: 2020 2020 2020 2020 2020 204c 6578 536d             LexSm
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│ │ │ │ +00012480: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00012490: 2020 2020 2020 2020 2020 4c65 7853 6d61            LexSma
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│ │ │ │ +000124b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000124c0: 2020 2020 2020 2020 2050 6f69 6e74 7320           Points 
│ │ │ │ +000124d0: 3d3e 2030 2020 2020 2020 2020 2020 2020  => 0            
│ │ │ │ +000124e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000124f0: 2020 2020 2020 2020 5261 6e64 6f6d 203d          Random =
│ │ │ │ +00012500: 3e20 3136 2020 2020 2020 2020 2020 2020  > 16            
│ │ │ │ +00012510: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00012520: 2020 2020 2020 2052 616e 646f 6d4e 6f6e         RandomNon
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│ │ │ │ +00012540: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00012550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012570: 2b0a 0a73 6179 7320 7468 6174 2077 6520  +..says that we 
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│ │ │ │ +000125f0: 7561 6c20 7072 6f62 6162 696c 6974 7920  ual probability 
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│ │ │ │ +00012640: 646f 6d20 7375 626d 6174 7269 6365 7320  dom submatrices 
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│ │ │ │ +000126a0: 466f 720a 696e 7374 616e 6365 2c20 6966  For.instance, if
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│ │ │ │ +000126c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000126d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000126e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000126f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012700: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3337 203a  --------+.|i37 :
│ │ │ │ +00012710: 2074 696d 6520 6368 6f6f 7365 476f 6f64   time chooseGood
│ │ │ │ +00012720: 4d69 6e6f 7273 2832 302c 2036 2c20 4d2c  Minors(20, 6, M,
│ │ │ │ +00012730: 204a 2c20 5374 7261 7465 6779 3d3e 5374   J, Strategy=>St
│ │ │ │ +00012740: 7261 7465 6779 4465 6661 756c 742c 2020  rategyDefault,  
│ │ │ │ +00012750: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075          |.| -- u
│ │ │ │ +00012760: 7365 6420 302e 3433 3934 3131 7320 2863  sed 0.439411s (c
│ │ │ │ +00012770: 7075 293b 2030 2e33 3636 3733 3673 2028  pu); 0.366736s (
│ │ │ │ +00012780: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 0s (gc)
│ │ │ │  00012790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000127a0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │  000127b0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ -000127c0: 4368 6f6f 7369 6e67 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ -000127d0: 6c65 7374 2020 2020 2020 2020 2020 2020  lest            
│ │ │ │ +000127c0: 4368 6f6f 7369 6e67 2052 616e 646f 6d20  Choosing Random 
│ │ │ │ +000127d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000127e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000127f0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │  00012800: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ -00012810: 4368 6f6f 7369 6e67 2052 616e 646f 6d20  Choosing Random 
│ │ │ │ -00012820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012810: 4368 6f6f 7369 6e67 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ +00012820: 6c65 7374 2020 2020 2020 2020 2020 2020  lest            
│ │ │ │  00012830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012840: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │  00012850: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ -00012860: 4368 6f6f 7369 6e67 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ -00012870: 536d 616c 6c65 7374 5465 726d 2020 2020  SmallestTerm    
│ │ │ │ +00012860: 4368 6f6f 7369 6e67 2052 616e 646f 6d20  Choosing Random 
│ │ │ │ +00012870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012890: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │  000128a0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ -000128b0: 4368 6f6f 7369 6e67 2052 616e 646f 6d4e  Choosing RandomN
│ │ │ │ -000128c0: 6f6e 5a65 726f 2020 2020 2020 2020 2020  onZero          
│ │ │ │ +000128b0: 4368 6f6f 7369 6e67 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ +000128c0: 536d 616c 6c65 7374 5465 726d 2020 2020  SmallestTerm    
│ │ │ │  000128d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000128e0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │  000128f0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │  00012900: 4368 6f6f 7369 6e67 2052 616e 646f 6d4e  Choosing RandomN
│ │ │ │  00012910: 6f6e 5a65 726f 2020 2020 2020 2020 2020  onZero          
│ │ │ │  00012920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012930: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │  00012940: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ -00012950: 4368 6f6f 7369 6e67 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ -00012960: 6c65 7374 2020 2020 2020 2020 2020 2020  lest            
│ │ │ │ +00012950: 4368 6f6f 7369 6e67 2052 616e 646f 6d4e  Choosing RandomN
│ │ │ │ +00012960: 6f6e 5a65 726f 2020 2020 2020 2020 2020  onZero          
│ │ │ │  00012970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012980: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │  00012990: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ -000129a0: 4368 6f6f 7369 6e67 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ -000129b0: 536d 616c 6c65 7374 5465 726d 2020 2020  SmallestTerm    
│ │ │ │ +000129a0: 4368 6f6f 7369 6e67 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ +000129b0: 6c65 7374 2020 2020 2020 2020 2020 2020  lest            
│ │ │ │  000129c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000129d0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │  000129e0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ -000129f0: 4368 6f6f 7369 6e67 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ -00012a00: 6c65 7374 5465 726d 2020 2020 2020 2020  lestTerm        
│ │ │ │ +000129f0: 4368 6f6f 7369 6e67 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ +00012a00: 536d 616c 6c65 7374 5465 726d 2020 2020  SmallestTerm    
│ │ │ │  00012a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012a20: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │  00012a30: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ -00012a40: 4368 6f6f 7369 6e67 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ -00012a50: 536d 616c 6c65 7374 5465 726d 2020 2020  SmallestTerm    
│ │ │ │ +00012a40: 4368 6f6f 7369 6e67 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ +00012a50: 6c65 7374 5465 726d 2020 2020 2020 2020  lestTerm        
│ │ │ │  00012a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012a70: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │  00012a80: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ -00012a90: 4368 6f6f 7369 6e67 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ -00012aa0: 6c65 7374 5465 726d 2020 2020 2020 2020  lestTerm        
│ │ │ │ +00012a90: 4368 6f6f 7369 6e67 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ +00012aa0: 536d 616c 6c65 7374 5465 726d 2020 2020  SmallestTerm    
│ │ │ │  00012ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012ac0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │  00012ad0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ -00012ae0: 4368 6f6f 7369 6e67 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ -00012af0: 536d 616c 6c65 7374 2020 2020 2020 2020  Smallest        
│ │ │ │ +00012ae0: 4368 6f6f 7369 6e67 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ +00012af0: 6c65 7374 5465 726d 2020 2020 2020 2020  lestTerm        
│ │ │ │  00012b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012b10: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │  00012b20: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │  00012b30: 4368 6f6f 7369 6e67 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │  00012b40: 536d 616c 6c65 7374 2020 2020 2020 2020  Smallest        
│ │ │ │  00012b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012b60: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │  00012b70: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ -00012b80: 4368 6f6f 7369 6e67 2052 616e 646f 6d20  Choosing Random 
│ │ │ │ -00012b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012b80: 4368 6f6f 7369 6e67 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ +00012b90: 536d 616c 6c65 7374 2020 2020 2020 2020  Smallest        
│ │ │ │  00012ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012bb0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │  00012bc0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ -00012bd0: 4368 6f6f 7369 6e67 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ -00012be0: 6c65 7374 5465 726d 2020 2020 2020 2020  lestTerm        
│ │ │ │ +00012bd0: 4368 6f6f 7369 6e67 2052 616e 646f 6d20  Choosing Random 
│ │ │ │ +00012be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012c00: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │  00012c10: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │  00012c20: 4368 6f6f 7369 6e67 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │  00012c30: 6c65 7374 5465 726d 2020 2020 2020 2020  lestTerm        
│ │ │ │  00012c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012c50: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │  00012c60: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ -00012c70: 4368 6f6f 7369 6e67 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ -00012c80: 536d 616c 6c65 7374 5465 726d 2020 2020  SmallestTerm    
│ │ │ │ +00012c70: 4368 6f6f 7369 6e67 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ +00012c80: 6c65 7374 5465 726d 2020 2020 2020 2020  lestTerm        
│ │ │ │  00012c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012ca0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │  00012cb0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ -00012cc0: 4368 6f6f 7369 6e67 2052 616e 646f 6d4e  Choosing RandomN
│ │ │ │ -00012cd0: 6f6e 5a65 726f 2020 2020 2020 2020 2020  onZero          
│ │ │ │ +00012cc0: 4368 6f6f 7369 6e67 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ +00012cd0: 536d 616c 6c65 7374 5465 726d 2020 2020  SmallestTerm    
│ │ │ │  00012ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00012d00: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ -00012d10: 4368 6f6f 7369 6e67 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ -00012d20: 536d 616c 6c65 7374 2020 2020 2020 2020  Smallest        
│ │ │ │ +00012d10: 4368 6f6f 7369 6e67 2052 616e 646f 6d4e  Choosing RandomN
│ │ │ │ +00012d20: 6f6e 5a65 726f 2020 2020 2020 2020 2020  onZero          
│ │ │ │  00012d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012d40: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │  00012d50: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ -00012d60: 4368 6f6f 7369 6e67 2052 616e 646f 6d20  Choosing Random 
│ │ │ │ -00012d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012d60: 4368 6f6f 7369 6e67 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ +00012d70: 536d 616c 6c65 7374 2020 2020 2020 2020  Smallest        
│ │ │ │  00012d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012d90: 2020 2020 2020 2020 7c0a 7c63 686f 6f73          |.|choos
│ │ │ │ -00012da0: 6547 6f6f 644d 696e 6f72 733a 2066 6f75  eGoodMinors: fou
│ │ │ │ -00012db0: 6e64 203d 3230 2c20 6174 7465 6d70 7465  nd =20, attempte
│ │ │ │ -00012dc0: 6420 3d20 3230 2020 2020 2020 2020 2020  d = 20          
│ │ │ │ +00012d90: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +00012da0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +00012db0: 4368 6f6f 7369 6e67 2052 616e 646f 6d20  Choosing Random 
│ │ │ │ +00012dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012de0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00012df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012de0: 2020 2020 2020 2020 7c0a 7c63 686f 6f73          |.|choos
│ │ │ │ +00012df0: 6547 6f6f 644d 696e 6f72 733a 2066 6f75  eGoodMinors: fou
│ │ │ │ +00012e00: 6e64 203d 3230 2c20 6174 7465 6d70 7465  nd =20, attempte
│ │ │ │ +00012e10: 6420 3d20 3230 2020 2020 2020 2020 2020  d = 20          
│ │ │ │  00012e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012e30: 2020 2020 2020 2020 7c0a 7c6f 3337 203a          |.|o37 :
│ │ │ │ -00012e40: 2049 6465 616c 206f 6620 5320 2020 2020   Ideal of S     
│ │ │ │ +00012e30: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00012e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00012ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012ed0: 2d2d 2d2d 2d2d 2d2d 7c0a 7c56 6572 626f  --------|.|Verbo
│ │ │ │ -00012ee0: 7365 3d3e 7472 7565 293b 2020 2020 2020  se=>true);      
│ │ │ │ -00012ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00012f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012f70: 2d2d 2d2d 2d2d 2d2d 2b0a 0a77 6520 6361  --------+..we ca
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│ │ │ │ -00012fd0: 2063 686f 6f73 6547 6f6f 644d 696e 6f72   chooseGoodMinor
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│ │ │ │ -00013000: 616e 7920 7469 6d65 2061 0a50 6f69 6e74  any time a.Point
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│ │ │ │ -00013030: 7420 6973 2066 6f75 6e64 206f 6e20 244a  t is found on $J
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│ │ │ │ -00013070: 2e0a 0a4c 6574 2075 7320 7461 6b65 2061  ...Let us take a
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│ │ │ │ -00013090: 6865 7220 6275 696c 742d 696e 2073 7472  her built-in str
│ │ │ │ -000130a0: 6174 6567 6965 732e 0a0a 2b2d 2d2d 2d2d  ategies...+-----
│ │ │ │ -000130b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000130c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000130d0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 3820 3a20  -------+.|i38 : 
│ │ │ │ -000130e0: 7065 656b 2053 7472 6174 6567 7944 6566  peek StrategyDef
│ │ │ │ -000130f0: 6175 6c74 4e6f 6e52 616e 646f 6d20 2020  aultNonRandom   
│ │ │ │ -00013100: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00013110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013130: 2020 2020 207c 0a7c 6f33 3820 3d20 4f70       |.|o38 = Op
│ │ │ │ -00013140: 7469 6f6e 5461 626c 657b 4752 6576 4c65  tionTable{GRevLe
│ │ │ │ -00013150: 784c 6172 6765 7374 203d 3e20 3020 2020  xLargest => 0   
│ │ │ │ -00013160: 2020 207d 7c0a 7c20 2020 2020 2020 2020     }|.|         
│ │ │ │ -00013170: 2020 2020 2020 2020 2047 5265 764c 6578           GRevLex
│ │ │ │ -00013180: 536d 616c 6c65 7374 203d 3e20 3235 2020  Smallest => 25  
│ │ │ │ -00013190: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -000131a0: 2020 2020 2020 2020 4752 6576 4c65 7853          GRevLexS
│ │ │ │ -000131b0: 6d61 6c6c 6573 7454 6572 6d20 3d3e 2032  mallestTerm => 2
│ │ │ │ -000131c0: 3520 7c0a 7c20 2020 2020 2020 2020 2020  5 |.|           
│ │ │ │ -000131d0: 2020 2020 2020 204c 6578 4c61 7267 6573         LexLarges
│ │ │ │ -000131e0: 7420 3d3e 2030 2020 2020 2020 2020 2020  t => 0          
│ │ │ │ -000131f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00013200: 2020 2020 2020 4c65 7853 6d61 6c6c 6573        LexSmalles
│ │ │ │ -00013210: 7420 3d3e 2032 3520 2020 2020 2020 2020  t => 25         
│ │ │ │ -00013220: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00013230: 2020 2020 204c 6578 536d 616c 6c65 7374       LexSmallest
│ │ │ │ -00013240: 5465 726d 203d 3e20 3235 2020 2020 207c  Term => 25     |
│ │ │ │ -00013250: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00013260: 2020 2020 506f 696e 7473 203d 3e20 3020      Points => 0 
│ │ │ │ -00013270: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00013280: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00013290: 2020 2052 616e 646f 6d20 3d3e 2030 2020     Random => 0  
│ │ │ │ -000132a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000132b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000132c0: 2020 5261 6e64 6f6d 4e6f 6e7a 6572 6f20    RandomNonzero 
│ │ │ │ -000132d0: 3d3e 2030 2020 2020 2020 2020 7c0a 2b2d  => 0        |.+-
│ │ │ │ -000132e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000132f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933  -----------+.|i3
│ │ │ │ -00013310: 3920 3a20 7065 656b 2053 7472 6174 6567  9 : peek Strateg
│ │ │ │ -00013320: 7944 6566 6175 6c74 5769 7468 506f 696e  yDefaultWithPoin
│ │ │ │ -00013330: 7473 2020 2020 2020 2020 7c0a 7c20 2020  ts        |.|   
│ │ │ │ -00013340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013360: 2020 2020 2020 2020 207c 0a7c 6f33 3920           |.|o39 
│ │ │ │ -00013370: 3d20 4f70 7469 6f6e 5461 626c 657b 4752  = OptionTable{GR
│ │ │ │ -00013380: 6576 4c65 784c 6172 6765 7374 203d 3e20  evLexLargest => 
│ │ │ │ -00013390: 3020 2020 2020 207d 7c0a 7c20 2020 2020  0      }|.|     
│ │ │ │ -000133a0: 2020 2020 2020 2020 2020 2020 2047 5265               GRe
│ │ │ │ -000133b0: 764c 6578 536d 616c 6c65 7374 203d 3e20  vLexSmallest => 
│ │ │ │ -000133c0: 3136 2020 2020 207c 0a7c 2020 2020 2020  16     |.|      
│ │ │ │ -000133d0: 2020 2020 2020 2020 2020 2020 4752 6576              GRev
│ │ │ │ -000133e0: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │ -000133f0: 3d3e 2031 3620 7c0a 7c20 2020 2020 2020  => 16 |.|       
│ │ │ │ -00013400: 2020 2020 2020 2020 2020 204c 6578 4c61             LexLa
│ │ │ │ -00013410: 7267 6573 7420 3d3e 2030 2020 2020 2020  rgest => 0      
│ │ │ │ -00013420: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00013430: 2020 2020 2020 2020 2020 4c65 7853 6d61            LexSma
│ │ │ │ -00013440: 6c6c 6573 7420 3d3e 2031 3620 2020 2020  llest => 16     
│ │ │ │ -00013450: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00013460: 2020 2020 2020 2020 204c 6578 536d 616c           LexSmal
│ │ │ │ -00013470: 6c65 7374 5465 726d 203d 3e20 3136 2020  lestTerm => 16  
│ │ │ │ -00013480: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00013490: 2020 2020 2020 2020 506f 696e 7473 203d          Points =
│ │ │ │ -000134a0: 3e20 3332 2020 2020 2020 2020 2020 2020  > 32            
│ │ │ │ -000134b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -000134c0: 2020 2020 2020 2052 616e 646f 6d20 3d3e         Random =>
│ │ │ │ -000134d0: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ -000134e0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000134f0: 2020 2020 2020 5261 6e64 6f6d 4e6f 6e7a        RandomNonz
│ │ │ │ -00013500: 6572 6f20 3d3e 2030 2020 2020 2020 2020  ero => 0        
│ │ │ │ -00013510: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ -00013520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00013540: 0a7c 6934 3020 3a20 7065 656b 2053 7472  .|i40 : peek Str
│ │ │ │ -00013550: 6174 6567 7950 6f69 6e74 7320 2020 2020  ategyPoints     
│ │ │ │ -00013560: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00013570: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00013580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013590: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000135a0: 6f34 3020 3d20 4f70 7469 6f6e 5461 626c  o40 = OptionTabl
│ │ │ │ -000135b0: 657b 4752 6576 4c65 784c 6172 6765 7374  e{GRevLexLargest
│ │ │ │ -000135c0: 203d 3e20 3020 2020 2020 7d20 7c0a 7c20   => 0     } |.| 
│ │ │ │ +00012e80: 2020 2020 2020 2020 7c0a 7c6f 3337 203a          |.|o37 :
│ │ │ │ +00012e90: 2049 6465 616c 206f 6620 5320 2020 2020   Ideal of S     
│ │ │ │ +00012ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012ed0: 2020 2020 2020 2020 7c0a 7c2d 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 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012f20: 2d2d 2d2d 2d2d 2d2d 7c0a 7c56 6572 626f  --------|.|Verbo
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│ │ │ │ +00012f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012fc0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a77 6520 6361  --------+..we ca
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│ │ │ │ +00012fe0: 6d69 6e6f 7273 2062 6569 6e67 2063 686f  minors being cho
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│ │ │ │ +000130b0: 6f6d 7075 7465 6420 7468 7573 2066 6172  omputed thus far
│ │ │ │ +000130c0: 2e0a 0a4c 6574 2075 7320 7461 6b65 2061  ...Let us take a
│ │ │ │ +000130d0: 206c 6f6f 6b20 6174 2073 6f6d 6520 6f74   look at some ot
│ │ │ │ +000130e0: 6865 7220 6275 696c 742d 696e 2073 7472  her built-in str
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│ │ │ │ +000131e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000131f0: 2020 2020 2020 2020 4752 6576 4c65 7853          GRevLexS
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│ │ │ │ +00013270: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
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│ │ │ │ +00013290: 5465 726d 203d 3e20 3235 2020 2020 207c  Term => 25     |
│ │ │ │ +000132a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000132b0: 2020 2020 506f 696e 7473 203d 3e20 3020      Points => 0 
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│ │ │ │ +000132d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
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│ │ │ │ +000133f0: 2020 2020 2020 2020 2020 2020 2047 5265               GRe
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│ │ │ │ +00013450: 2020 2020 2020 2020 2020 204c 6578 4c61             LexLa
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│ │ │ │ +00013480: 2020 2020 2020 2020 2020 4c65 7853 6d61            LexSma
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│ │ │ │ +00013570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
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│ │ │ │ -00013660: 2020 2020 2020 2020 2020 2020 2020 4c65                Le
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│ │ │ │ -000136c0: 2020 2020 2020 2020 2020 2020 506f 696e              Poin
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│ │ │ │ -000136f0: 2020 2020 2020 2020 2020 2052 616e 646f             Rando
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│ │ │ │ -00013760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -000137e0: 2062 656e 6566 6963 6961 6c20 696e 2073   beneficial in s
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│ │ │ │ -00013880: 2061 6e64 204c 6578 536d 616c 6c65 7374   and LexSmallest
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│ │ │ │ -000139e0: 7468 6520 756e 6465 726c 7969 6e67 206d  the underlying m
│ │ │ │ -000139f0: 6174 7269 7820 746f 2061 766f 6964 2074  atrix to avoid t
│ │ │ │ -00013a00: 6869 7320 736f 7274 206f 6620 6c6f 6f70  his sort of loop
│ │ │ │ -00013a10: 2e0a 0a50 6f69 6e74 733a 204e 6f74 6963  ...Points: Notic
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│ │ │ │ -00013a30: 3d3e 2053 7472 6174 6567 7950 6f69 6e74  => StrategyPoint
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│ │ │ │ -00013a60: 7361 6d65 2074 6869 6e67 2e20 5765 2062  same thing. We b
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│ │ │ │ -00013ac0: 7468 6520 6964 6561 6c20 6f66 206d 696e  the ideal of min
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│ │ │ │ -00013ae0: 6661 7220 2870 6c75 7320 244a 2429 2c20  far (plus $J$), 
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│ │ │ │ -00013b20: 616e 2062 6520 736c 6f77 292c 2065 7661  an be slow), eva
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│ │ │ │ -00013b40: 7820 244d 240a 6174 2074 6861 7420 706f  x $M$.at that po
│ │ │ │ -00013b50: 696e 742c 2061 6e64 2074 6865 6e20 6669  int, and then fi
│ │ │ │ -00013b60: 6e61 6c6c 7920 636f 6d70 7574 6573 2074  nally computes t
│ │ │ │ -00013b70: 6865 2063 6f72 7265 7370 6f6e 6469 6e67  he corresponding
│ │ │ │ -00013b80: 2064 6574 6572 6d69 6e61 6e74 206f 6620   determinant of 
│ │ │ │ -00013b90: 7468 650a 7375 626d 6174 7269 782e 2020  the.submatrix.  
│ │ │ │ -00013ba0: 5468 6973 2073 7562 6d61 7472 6978 2077  This submatrix w
│ │ │ │ -00013bb0: 696c 6c20 616c 7761 7973 2070 726f 6475  ill always produ
│ │ │ │ -00013bc0: 6365 2061 206d 696e 6f72 2077 6869 6368  ce a minor which
│ │ │ │ -00013bd0: 2073 6872 696e 6b73 206f 7572 0a76 616e   shrinks our.van
│ │ │ │ -00013be0: 6973 6869 6e67 206c 6f63 7573 2e0a 0a42  ishing locus...B
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│ │ │ │ -00013c30: 2057 6869 6368 2063 616e 2062 650a 736f   Which can be.so
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│ │ │ │ -00013c50: 6275 7420 7768 6963 6820 6172 6520 616c  but which are al
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│ │ │ │ -00013c70: 6578 6973 742c 2061 6e64 2061 7265 206c  exist, and are l
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│ │ │ │ -00013cb0: 696e 6469 6e67 2061 2070 6f69 6e74 292e  inding a point).
│ │ │ │ -00013cc0: 2020 466f 7220 696e 7374 616e 6365 2c20    For instance, 
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│ │ │ │ -00013cf0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00013d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00013d40: 0a7c 6934 3120 3a20 7074 7353 7472 6174  .|i41 : ptsStrat
│ │ │ │ -00013d50: 4765 6f6d 6574 7269 6320 3d20 6e65 7720  Geometric = new 
│ │ │ │ -00013d60: 4f70 7469 6f6e 5461 626c 6520 6672 6f6d  OptionTable from
│ │ │ │ -00013d70: 2028 6f70 7469 6f6e 7320 2020 2020 2020   (options       
│ │ │ │ -00013d80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00013d90: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 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 2d7c  ---------------|
│ │ │ │ -00013de0: 0a7c 6368 6f6f 7365 476f 6f64 4d69 6e6f  .|chooseGoodMino
│ │ │ │ -00013df0: 7273 2923 506f 696e 744f 7074 696f 6e73  rs)#PointOptions
│ │ │ │ -00013e00: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ -00013e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013e20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00013e30: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00013e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00013e80: 0a7c 6934 3220 3a20 7074 7353 7472 6174  .|i42 : ptsStrat
│ │ │ │ -00013e90: 4765 6f6d 6574 7269 6323 4578 7465 6e64  Geometric#Extend
│ │ │ │ -00013ea0: 4669 656c 6420 2d2d 6c6f 6f6b 2061 7420  Field --look at 
│ │ │ │ -00013eb0: 7468 6520 6465 6661 756c 7420 7661 6c75  the default valu
│ │ │ │ -00013ec0: 6520 2020 2020 2020 2020 2020 2020 207c  e              |
│ │ │ │ -00013ed0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00013ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013f10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00013f20: 0a7c 6f34 3220 3d20 7472 7565 2020 2020  .|o42 = true    
│ │ │ │ +000135e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000135f0: 6f34 3020 3d20 4f70 7469 6f6e 5461 626c  o40 = OptionTabl
│ │ │ │ +00013600: 657b 4752 6576 4c65 784c 6172 6765 7374  e{GRevLexLargest
│ │ │ │ +00013610: 203d 3e20 3020 2020 2020 7d20 7c0a 7c20   => 0     } |.| 
│ │ │ │ +00013620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013630: 2047 5265 764c 6578 536d 616c 6c65 7374   GRevLexSmallest
│ │ │ │ +00013640: 203d 3e20 3020 2020 2020 207c 0a7c 2020   => 0      |.|  
│ │ │ │ +00013650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013660: 4752 6576 4c65 7853 6d61 6c6c 6573 7454  GRevLexSmallestT
│ │ │ │ +00013670: 6572 6d20 3d3e 2030 2020 7c0a 7c20 2020  erm => 0  |.|   
│ │ │ │ +00013680: 2020 2020 2020 2020 2020 2020 2020 204c                 L
│ │ │ │ +00013690: 6578 4c61 7267 6573 7420 3d3e 2030 2020  exLargest => 0  
│ │ │ │ +000136a0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000136b0: 2020 2020 2020 2020 2020 2020 2020 4c65                Le
│ │ │ │ +000136c0: 7853 6d61 6c6c 6573 7420 3d3e 2030 2020  xSmallest => 0  
│ │ │ │ +000136d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000136e0: 2020 2020 2020 2020 2020 2020 204c 6578               Lex
│ │ │ │ +000136f0: 536d 616c 6c65 7374 5465 726d 203d 3e20  SmallestTerm => 
│ │ │ │ +00013700: 3020 2020 2020 207c 0a7c 2020 2020 2020  0      |.|      
│ │ │ │ +00013710: 2020 2020 2020 2020 2020 2020 506f 696e              Poin
│ │ │ │ +00013720: 7473 203d 3e20 3130 3020 2020 2020 2020  ts => 100       
│ │ │ │ +00013730: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00013740: 2020 2020 2020 2020 2020 2052 616e 646f             Rando
│ │ │ │ +00013750: 6d20 3d3e 2030 2020 2020 2020 2020 2020  m => 0          
│ │ │ │ +00013760: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00013770: 2020 2020 2020 2020 2020 5261 6e64 6f6d            Random
│ │ │ │ +00013780: 4e6f 6e7a 6572 6f20 3d3e 2030 2020 2020  Nonzero => 0    
│ │ │ │ +00013790: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +000137a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000137b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000137c0: 2d2d 2d2b 0a0a 5374 7261 7465 6779 4465  ---+..StrategyDe
│ │ │ │ +000137d0: 6661 756c 744e 6f6e 5261 6e64 6f6d 2069  faultNonRandom i
│ │ │ │ +000137e0: 7320 6c69 6b65 2053 7472 6174 6567 7944  s like StrategyD
│ │ │ │ +000137f0: 6566 6175 6c74 2062 7574 2072 656d 6f76  efault but remov
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│ │ │ │ +00013820: 2062 6520 7375 7270 7269 7369 6e67 6c79   be surprisingly
│ │ │ │ +00013830: 2062 656e 6566 6963 6961 6c20 696e 2073   beneficial in s
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│ │ │ │ +00013850: 7465 6779 4465 6661 756c 7457 6974 6850  tegyDefaultWithP
│ │ │ │ +00013860: 6f69 6e74 7320 7265 6d6f 7665 7320 7261  oints removes ra
│ │ │ │ +00013870: 6e64 6f6d 6e65 7373 2062 7574 2061 6464  ndomness but add
│ │ │ │ +00013880: 7320 696e 2070 6f69 6e74 7320 696e 7374  s in points inst
│ │ │ │ +00013890: 6561 642e 0a0a 4120 7761 726e 696e 6720  ead...A warning 
│ │ │ │ +000138a0: 6f6e 2063 686f 6f73 6547 6f6f 644d 696e  on chooseGoodMin
│ │ │ │ +000138b0: 6f72 733a 2020 5468 6520 7374 7261 7465  ors:  The strate
│ │ │ │ +000138c0: 6769 6573 204c 6578 536d 616c 6c65 7374  gies LexSmallest
│ │ │ │ +000138d0: 2061 6e64 204c 6578 536d 616c 6c65 7374   and LexSmallest
│ │ │ │ +000138e0: 5465 726d 0a77 696c 6c20 7665 7279 2066  Term.will very f
│ │ │ │ +000138f0: 7265 7175 656e 746c 7920 7265 7065 6174  requently repeat
│ │ │ │ +00013900: 6564 6c79 2063 686f 6f73 6520 7468 6520  edly choose the 
│ │ │ │ +00013910: 7361 6d65 2073 7562 6d61 7472 6978 206f  same submatrix o
│ │ │ │ +00013920: 6620 7468 6520 6769 7665 6e20 6d61 7472  f the given matr
│ │ │ │ +00013930: 6978 2e0a 4865 6e63 6520 6966 206f 6e65  ix..Hence if one
│ │ │ │ +00013940: 2074 7269 6573 2074 6f20 7275 6e20 6368   tries to run ch
│ │ │ │ +00013950: 6f6f 7365 476f 6f64 4d69 6e6f 7273 2061  ooseGoodMinors a
│ │ │ │ +00013960: 6e64 2063 686f 6f73 6520 746f 6f20 6d61  nd choose too ma
│ │ │ │ +00013970: 6e79 206d 696e 6f72 7320 7769 7468 2073  ny minors with s
│ │ │ │ +00013980: 7563 680a 6120 7374 7261 7465 6779 2c20  uch.a strategy, 
│ │ │ │ +00013990: 6f6e 6520 6361 6e20 6765 7420 696e 746f  one can get into
│ │ │ │ +000139a0: 2061 206c 6f6e 6720 6c6f 6f70 2028 7468   a long loop (th
│ │ │ │ +000139b0: 6520 6675 6e63 7469 6f6e 2067 6976 6520  e function give 
│ │ │ │ +000139c0: 7570 2065 7665 6e74 7561 6c6c 792c 2062  up eventually, b
│ │ │ │ +000139d0: 7574 0a6f 6e6c 7920 6166 7465 7220 646f  ut.only after do
│ │ │ │ +000139e0: 696e 6720 7761 7920 746f 6f20 6d75 6368  ing way too much
│ │ │ │ +000139f0: 2077 6f72 6b29 2e20 2054 6865 2047 5265   work).  The GRe
│ │ │ │ +00013a00: 764c 6578 2073 7472 6174 6567 6965 7320  vLex strategies 
│ │ │ │ +00013a10: 7065 7269 6f64 6963 616c 6c79 0a74 656d  periodically.tem
│ │ │ │ +00013a20: 706f 7261 7269 6c79 2063 6861 6e67 6520  porarily change 
│ │ │ │ +00013a30: 7468 6520 756e 6465 726c 7969 6e67 206d  the underlying m
│ │ │ │ +00013a40: 6174 7269 7820 746f 2061 766f 6964 2074  atrix to avoid t
│ │ │ │ +00013a50: 6869 7320 736f 7274 206f 6620 6c6f 6f70  his sort of loop
│ │ │ │ +00013a60: 2e0a 0a50 6f69 6e74 733a 204e 6f74 6963  ...Points: Notic
│ │ │ │ +00013a70: 6520 7468 6174 2053 7472 6174 6567 7920  e that Strategy 
│ │ │ │ +00013a80: 3d3e 2053 7472 6174 6567 7950 6f69 6e74  => StrategyPoint
│ │ │ │ +00013a90: 7320 616e 6420 5374 7261 7465 6779 203d  s and Strategy =
│ │ │ │ +00013aa0: 3e20 506f 696e 7473 2064 6f20 7468 650a  > Points do the.
│ │ │ │ +00013ab0: 7361 6d65 2074 6869 6e67 2e20 5765 2062  same thing. We b
│ │ │ │ +00013ac0: 7269 6566 6c79 2064 6573 6372 6962 6520  riefly describe 
│ │ │ │ +00013ad0: 686f 7720 6368 6f6f 7365 476f 6f64 4d69  how chooseGoodMi
│ │ │ │ +00013ae0: 6e6f 7273 2069 6e74 6572 6163 7473 2077  nors interacts w
│ │ │ │ +00013af0: 6974 6820 506f 696e 7473 2e0a 496e 6465  ith Points..Inde
│ │ │ │ +00013b00: 6564 2050 6f69 6e74 7320 666f 726d 7320  ed Points forms 
│ │ │ │ +00013b10: 7468 6520 6964 6561 6c20 6f66 206d 696e  the ideal of min
│ │ │ │ +00013b20: 6f72 7320 636f 6d70 7574 6564 2073 6f20  ors computed so 
│ │ │ │ +00013b30: 6661 7220 2870 6c75 7320 244a 2429 2c20  far (plus $J$), 
│ │ │ │ +00013b40: 6669 6e64 7320 610a 706f 696e 7420 7768  finds a.point wh
│ │ │ │ +00013b50: 6572 6520 7468 6174 2069 6465 616c 2076  ere that ideal v
│ │ │ │ +00013b60: 616e 6973 6865 7320 2877 6869 6368 2063  anishes (which c
│ │ │ │ +00013b70: 616e 2062 6520 736c 6f77 292c 2065 7661  an be slow), eva
│ │ │ │ +00013b80: 6c75 6174 6573 2074 6865 206d 6174 7269  luates the matri
│ │ │ │ +00013b90: 7820 244d 240a 6174 2074 6861 7420 706f  x $M$.at that po
│ │ │ │ +00013ba0: 696e 742c 2061 6e64 2074 6865 6e20 6669  int, and then fi
│ │ │ │ +00013bb0: 6e61 6c6c 7920 636f 6d70 7574 6573 2074  nally computes t
│ │ │ │ +00013bc0: 6865 2063 6f72 7265 7370 6f6e 6469 6e67  he corresponding
│ │ │ │ +00013bd0: 2064 6574 6572 6d69 6e61 6e74 206f 6620   determinant of 
│ │ │ │ +00013be0: 7468 650a 7375 626d 6174 7269 782e 2020  the.submatrix.  
│ │ │ │ +00013bf0: 5468 6973 2073 7562 6d61 7472 6978 2077  This submatrix w
│ │ │ │ +00013c00: 696c 6c20 616c 7761 7973 2070 726f 6475  ill always produ
│ │ │ │ +00013c10: 6365 2061 206d 696e 6f72 2077 6869 6368  ce a minor which
│ │ │ │ +00013c20: 2073 6872 696e 6b73 206f 7572 0a76 616e   shrinks our.van
│ │ │ │ +00013c30: 6973 6869 6e67 206c 6f63 7573 2e0a 0a42  ishing locus...B
│ │ │ │ +00013c40: 7920 6465 6661 756c 742c 2074 6865 2050  y default, the P
│ │ │ │ +00013c50: 6f69 6e74 7320 7374 7261 7465 6779 2061  oints strategy a
│ │ │ │ +00013c60: 6374 7561 6c6c 7920 6669 6e64 7320 6765  ctually finds ge
│ │ │ │ +00013c70: 6f6d 6574 7269 6320 706f 696e 7473 2e20  ometric points. 
│ │ │ │ +00013c80: 2057 6869 6368 2063 616e 2062 650a 736f   Which can be.so
│ │ │ │ +00013c90: 6d65 7469 6d65 7320 736c 6f77 6572 2028  metimes slower (
│ │ │ │ +00013ca0: 6275 7420 7768 6963 6820 6172 6520 616c  but which are al
│ │ │ │ +00013cb0: 6d6f 7374 2063 6572 7461 696e 2074 6f20  most certain to 
│ │ │ │ +00013cc0: 6578 6973 742c 2061 6e64 2061 7265 206c  exist, and are l
│ │ │ │ +00013cd0: 6573 7320 6c69 6b65 6c79 2074 6f0a 6861  ess likely to.ha
│ │ │ │ +00013ce0: 6e67 2069 6620 7468 6520 6675 6e63 7469  ng if the functi
│ │ │ │ +00013cf0: 6f6e 2068 6173 2074 726f 7562 6c65 2066  on has trouble f
│ │ │ │ +00013d00: 696e 6469 6e67 2061 2070 6f69 6e74 292e  inding a point).
│ │ │ │ +00013d10: 2020 466f 7220 696e 7374 616e 6365 2c20    For instance, 
│ │ │ │ +00013d20: 7765 2063 616e 0a63 6f6e 7472 6f6c 2074  we can.control t
│ │ │ │ +00013d30: 6861 7420 6173 2066 6f6c 6c6f 7773 2e0a  hat as follows..
│ │ │ │ +00013d40: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00013d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00013d90: 0a7c 6934 3120 3a20 7074 7353 7472 6174  .|i41 : ptsStrat
│ │ │ │ +00013da0: 4765 6f6d 6574 7269 6320 3d20 6e65 7720  Geometric = new 
│ │ │ │ +00013db0: 4f70 7469 6f6e 5461 626c 6520 6672 6f6d  OptionTable from
│ │ │ │ +00013dc0: 2028 6f70 7469 6f6e 7320 2020 2020 2020   (options       
│ │ │ │ +00013dd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00013de0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00013df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00013e30: 0a7c 6368 6f6f 7365 476f 6f64 4d69 6e6f  .|chooseGoodMino
│ │ │ │ +00013e40: 7273 2923 506f 696e 744f 7074 696f 6e73  rs)#PointOptions
│ │ │ │ +00013e50: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ +00013e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013e70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00013e80: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00013e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00013ed0: 0a7c 6934 3220 3a20 7074 7353 7472 6174  .|i42 : ptsStrat
│ │ │ │ +00013ee0: 4765 6f6d 6574 7269 6323 4578 7465 6e64  Geometric#Extend
│ │ │ │ +00013ef0: 4669 656c 6420 2d2d 6c6f 6f6b 2061 7420  Field --look at 
│ │ │ │ +00013f00: 7468 6520 6465 6661 756c 7420 7661 6c75  the default valu
│ │ │ │ +00013f10: 6520 2020 2020 2020 2020 2020 2020 207c  e              |
│ │ │ │ +00013f20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  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 2020 2020                  
│ │ │ │  00013f60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00013f70: 0a2b 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 2d2d 2d2d  ----------------
│ │ │ │ -00013fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00013fc0: 0a7c 6934 3320 3a20 7469 6d65 2064 696d  .|i43 : time dim
│ │ │ │ -00013fd0: 2028 4a20 2b20 6368 6f6f 7365 476f 6f64   (J + chooseGood
│ │ │ │ -00013fe0: 4d69 6e6f 7273 2831 2c20 362c 204d 2c20  Minors(1, 6, M, 
│ │ │ │ -00013ff0: 4a2c 2053 7472 6174 6567 793d 3e50 6f69  J, Strategy=>Poi
│ │ │ │ -00014000: 6e74 732c 2020 2020 2020 2020 2020 207c  nts,           |
│ │ │ │ -00014010: 0a7c 202d 2d20 7573 6564 2030 2e34 3935  .| -- used 0.495
│ │ │ │ -00014020: 3035 3773 2028 6370 7529 3b20 302e 3436  057s (cpu); 0.46
│ │ │ │ -00014030: 3132 3637 7320 2874 6872 6561 6429 3b20  1267s (thread); 
│ │ │ │ -00014040: 3073 2028 6763 2920 2020 2020 2020 2020  0s (gc)         
│ │ │ │ -00014050: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014060: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00014070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013f70: 0a7c 6f34 3220 3d20 7472 7565 2020 2020  .|o42 = true    
│ │ │ │ +00013f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013fb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00013fc0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00013fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00014010: 0a7c 6934 3320 3a20 7469 6d65 2064 696d  .|i43 : time dim
│ │ │ │ +00014020: 2028 4a20 2b20 6368 6f6f 7365 476f 6f64   (J + chooseGood
│ │ │ │ +00014030: 4d69 6e6f 7273 2831 2c20 362c 204d 2c20  Minors(1, 6, M, 
│ │ │ │ +00014040: 4a2c 2053 7472 6174 6567 793d 3e50 6f69  J, Strategy=>Poi
│ │ │ │ +00014050: 6e74 732c 2020 2020 2020 2020 2020 207c  nts,           |
│ │ │ │ +00014060: 0a7c 202d 2d20 7573 6564 2030 2e37 3034  .| -- used 0.704
│ │ │ │ +00014070: 3530 3273 2028 6370 7529 3b20 302e 3632  502s (cpu); 0.62
│ │ │ │ +00014080: 3030 3335 7320 2874 6872 6561 6429 3b20  0035s (thread); 
│ │ │ │ +00014090: 3073 2028 6763 2920 2020 2020 2020 2020  0s (gc)         
│ │ │ │  000140a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000140b0: 0a7c 6f34 3320 3d20 3220 2020 2020 2020  .|o43 = 2       
│ │ │ │ +000140b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000140c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000140d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000140e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000140f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014100: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ -00014110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00014150: 0a7c 506f 696e 744f 7074 696f 6e73 3d3e  .|PointOptions=>
│ │ │ │ -00014160: 7074 7353 7472 6174 4765 6f6d 6574 7269  ptsStratGeometri
│ │ │ │ -00014170: 6329 2920 2020 2020 2020 2020 2020 2020  c))             
│ │ │ │ -00014180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014190: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000141a0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -000141b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000141c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000141d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000141e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000141f0: 0a7c 6934 3420 3a20 7074 7353 7472 6174  .|i44 : ptsStrat
│ │ │ │ -00014200: 5261 7469 6f6e 616c 203d 2070 7473 5374  Rational = ptsSt
│ │ │ │ -00014210: 7261 7447 656f 6d65 7472 6963 2b2b 7b45  ratGeometric++{E
│ │ │ │ -00014220: 7874 656e 6446 6965 6c64 3d3e 6661 6c73  xtendField=>fals
│ │ │ │ -00014230: 657d 202d 2d63 6861 6e67 6520 2020 207c  e} --change    |
│ │ │ │ -00014240: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00014250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014280: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014290: 0a7c 6f34 3420 3d20 4f70 7469 6f6e 5461  .|o44 = OptionTa
│ │ │ │ -000142a0: 626c 657b 4465 636f 6d70 6f73 6974 696f  ble{Decompositio
│ │ │ │ -000142b0: 6e53 7472 6174 6567 7920 3d3e 2044 6563  nStrategy => Dec
│ │ │ │ -000142c0: 6f6d 706f 7365 7d20 2020 2020 2020 2020  ompose}         
│ │ │ │ +00014100: 0a7c 6f34 3320 3d20 3220 2020 2020 2020  .|o43 = 2       
│ │ │ │ +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 207c                 |
│ │ │ │ +00014150: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00014160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +000141a0: 0a7c 506f 696e 744f 7074 696f 6e73 3d3e  .|PointOptions=>
│ │ │ │ +000141b0: 7074 7353 7472 6174 4765 6f6d 6574 7269  ptsStratGeometri
│ │ │ │ +000141c0: 6329 2920 2020 2020 2020 2020 2020 2020  c))             
│ │ │ │ +000141d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000141e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000141f0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00014200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00014240: 0a7c 6934 3420 3a20 7074 7353 7472 6174  .|i44 : ptsStrat
│ │ │ │ +00014250: 5261 7469 6f6e 616c 203d 2070 7473 5374  Rational = ptsSt
│ │ │ │ +00014260: 7261 7447 656f 6d65 7472 6963 2b2b 7b45  ratGeometric++{E
│ │ │ │ +00014270: 7874 656e 6446 6965 6c64 3d3e 6661 6c73  xtendField=>fals
│ │ │ │ +00014280: 657d 202d 2d63 6861 6e67 6520 2020 207c  e} --change    |
│ │ │ │ +00014290: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000142a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000142b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000142c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000142d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000142e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000142f0: 2020 2020 4469 6d65 6e73 696f 6e46 756e      DimensionFun
│ │ │ │ -00014300: 6374 696f 6e20 3d3e 2064 696d 2020 2020  ction => dim    
│ │ │ │ -00014310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000142e0: 0a7c 6f34 3420 3d20 4f70 7469 6f6e 5461  .|o44 = OptionTa
│ │ │ │ +000142f0: 626c 657b 4465 636f 6d70 6f73 6974 696f  ble{Decompositio
│ │ │ │ +00014300: 6e53 7472 6174 6567 7920 3d3e 2044 6563  nStrategy => Dec
│ │ │ │ +00014310: 6f6d 706f 7365 7d20 2020 2020 2020 2020  ompose}         
│ │ │ │  00014320: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00014330: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00014340: 2020 2020 4578 7465 6e64 4669 656c 6420      ExtendField 
│ │ │ │ -00014350: 3d3e 2066 616c 7365 2020 2020 2020 2020  => false        
│ │ │ │ +00014340: 2020 2020 4469 6d65 6e73 696f 6e46 756e      DimensionFun
│ │ │ │ +00014350: 6374 696f 6e20 3d3e 2064 696d 2020 2020  ction => dim    
│ │ │ │  00014360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014370: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00014380: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00014390: 2020 2020 486f 6d6f 6765 6e65 6f75 7320      Homogeneous 
│ │ │ │ +00014390: 2020 2020 4578 7465 6e64 4669 656c 6420      ExtendField 
│ │ │ │  000143a0: 3d3e 2066 616c 7365 2020 2020 2020 2020  => false        
│ │ │ │  000143b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000143c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  000143d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000143e0: 2020 2020 4e75 6d54 6872 6561 6473 546f      NumThreadsTo
│ │ │ │ -000143f0: 5573 6520 3d3e 2031 2020 2020 2020 2020  Use => 1        
│ │ │ │ +000143e0: 2020 2020 486f 6d6f 6765 6e65 6f75 7320      Homogeneous 
│ │ │ │ +000143f0: 3d3e 2066 616c 7365 2020 2020 2020 2020  => false        
│ │ │ │  00014400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014410: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00014420: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00014430: 2020 2020 506f 696e 7443 6865 636b 4174      PointCheckAt
│ │ │ │ -00014440: 7465 6d70 7473 203d 3e20 3020 2020 2020  tempts => 0     
│ │ │ │ +00014430: 2020 2020 4e75 6d54 6872 6561 6473 546f      NumThreadsTo
│ │ │ │ +00014440: 5573 6520 3d3e 2031 2020 2020 2020 2020  Use => 1        
│ │ │ │  00014450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014460: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00014470: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00014480: 2020 2020 5265 706c 6163 656d 656e 7420      Replacement 
│ │ │ │ -00014490: 3d3e 2042 696e 6f6d 6961 6c20 2020 2020  => Binomial     
│ │ │ │ +00014480: 2020 2020 506f 696e 7443 6865 636b 4174      PointCheckAt
│ │ │ │ +00014490: 7465 6d70 7473 203d 3e20 3020 2020 2020  tempts => 0     
│ │ │ │  000144a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000144b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  000144c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000144d0: 2020 2020 5374 7261 7465 6779 203d 3e20      Strategy => 
│ │ │ │ -000144e0: 4465 6661 756c 7420 2020 2020 2020 2020  Default         
│ │ │ │ +000144d0: 2020 2020 5265 706c 6163 656d 656e 7420      Replacement 
│ │ │ │ +000144e0: 3d3e 2042 696e 6f6d 6961 6c20 2020 2020  => Binomial     
│ │ │ │  000144f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014500: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00014510: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00014520: 2020 2020 5665 7262 6f73 6520 3d3e 2066      Verbose => f
│ │ │ │ -00014530: 616c 7365 2020 2020 2020 2020 2020 2020  alse            
│ │ │ │ +00014520: 2020 2020 5374 7261 7465 6779 203d 3e20      Strategy => 
│ │ │ │ +00014530: 4465 6661 756c 7420 2020 2020 2020 2020  Default         
│ │ │ │  00014540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014550: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00014560: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00014570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014570: 2020 2020 5665 7262 6f73 6520 3d3e 2066      Verbose => f
│ │ │ │ +00014580: 616c 7365 2020 2020 2020 2020 2020 2020  alse            
│ │ │ │  00014590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000145a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000145b0: 0a7c 6f34 3420 3a20 4f70 7469 6f6e 5461  .|o44 : OptionTa
│ │ │ │ -000145c0: 626c 6520 2020 2020 2020 2020 2020 2020  ble             
│ │ │ │ +000145b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000145c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000145d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000145e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000145f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014600: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ -00014610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00014650: 0a7c 7468 6174 2076 616c 7565 2020 2020  .|that value    
│ │ │ │ -00014660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014690: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000146a0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -000146b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000146c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000146d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000146e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000146f0: 0a7c 6934 3520 3a20 7074 7353 7472 6174  .|i45 : ptsStrat
│ │ │ │ -00014700: 5261 7469 6f6e 616c 2e45 7874 656e 6446  Rational.ExtendF
│ │ │ │ -00014710: 6965 6c64 202d 2d6c 6f6f 6b20 6174 206f  ield --look at o
│ │ │ │ -00014720: 7572 2063 6861 6e67 6564 2076 616c 7565  ur changed value
│ │ │ │ -00014730: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014740: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00014750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014600: 0a7c 6f34 3420 3a20 4f70 7469 6f6e 5461  .|o44 : OptionTa
│ │ │ │ +00014610: 626c 6520 2020 2020 2020 2020 2020 2020  ble             
│ │ │ │ +00014620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014640: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00014650: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00014660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +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 2d7c  ---------------|
│ │ │ │ +000146a0: 0a7c 7468 6174 2076 616c 7565 2020 2020  .|that value    
│ │ │ │ +000146b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000146c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000146d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000146e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000146f0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00014700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00014740: 0a7c 6934 3520 3a20 7074 7353 7472 6174  .|i45 : ptsStrat
│ │ │ │ +00014750: 5261 7469 6f6e 616c 2e45 7874 656e 6446  Rational.ExtendF
│ │ │ │ +00014760: 6965 6c64 202d 2d6c 6f6f 6b20 6174 206f  ield --look at o
│ │ │ │ +00014770: 7572 2063 6861 6e67 6564 2076 616c 7565  ur changed value
│ │ │ │  00014780: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014790: 0a7c 6f34 3520 3d20 6661 6c73 6520 2020  .|o45 = false   
│ │ │ │ +00014790: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000147a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000147b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000147c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000147d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000147e0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -000147f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00014830: 0a7c 6934 3620 3a20 7469 6d65 2064 696d  .|i46 : time dim
│ │ │ │ -00014840: 2028 4a20 2b20 6368 6f6f 7365 476f 6f64   (J + chooseGood
│ │ │ │ -00014850: 4d69 6e6f 7273 2831 2c20 362c 204d 2c20  Minors(1, 6, M, 
│ │ │ │ -00014860: 4a2c 2053 7472 6174 6567 793d 3e50 6f69  J, Strategy=>Poi
│ │ │ │ -00014870: 6e74 732c 2020 2020 2020 2020 2020 207c  nts,           |
│ │ │ │ -00014880: 0a7c 202d 2d20 7573 6564 2030 2e35 3132  .| -- used 0.512
│ │ │ │ -00014890: 3132 3773 2028 6370 7529 3b20 302e 3339  127s (cpu); 0.39
│ │ │ │ -000148a0: 3038 3936 7320 2874 6872 6561 6429 3b20  0896s (thread); 
│ │ │ │ -000148b0: 3073 2028 6763 2920 2020 2020 2020 2020  0s (gc)         
│ │ │ │ -000148c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000148d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000148e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000148f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000147e0: 0a7c 6f34 3520 3d20 6661 6c73 6520 2020  .|o45 = false   
│ │ │ │ +000147f0: 2020 2020 2020 2020 2020 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 2020 207c                 |
│ │ │ │ +00014830: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00014840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00014880: 0a7c 6934 3620 3a20 7469 6d65 2064 696d  .|i46 : time dim
│ │ │ │ +00014890: 2028 4a20 2b20 6368 6f6f 7365 476f 6f64   (J + chooseGood
│ │ │ │ +000148a0: 4d69 6e6f 7273 2831 2c20 362c 204d 2c20  Minors(1, 6, M, 
│ │ │ │ +000148b0: 4a2c 2053 7472 6174 6567 793d 3e50 6f69  J, Strategy=>Poi
│ │ │ │ +000148c0: 6e74 732c 2020 2020 2020 2020 2020 207c  nts,           |
│ │ │ │ +000148d0: 0a7c 202d 2d20 7573 6564 2030 2e35 3938  .| -- used 0.598
│ │ │ │ +000148e0: 3932 3773 2028 6370 7529 3b20 302e 3434  927s (cpu); 0.44
│ │ │ │ +000148f0: 3635 3673 2028 7468 7265 6164 293b 2030  656s (thread); 0
│ │ │ │ +00014900: 7320 2867 6329 2020 2020 2020 2020 2020  s (gc)          
│ │ │ │  00014910: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014920: 0a7c 6f34 3620 3d20 3220 2020 2020 2020  .|o46 = 2       
│ │ │ │ +00014920: 0a7c 2020 2020 2020 2020 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 2020                  
│ │ │ │  00014960: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014970: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ -00014980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000149a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000149b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -000149c0: 0a7c 506f 696e 744f 7074 696f 6e73 3d3e  .|PointOptions=>
│ │ │ │ -000149d0: 7074 7353 7472 6174 5261 7469 6f6e 616c  ptsStratRational
│ │ │ │ -000149e0: 2929 2020 2020 2020 2020 2020 2020 2020  ))              
│ │ │ │ -000149f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014a00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014a10: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00014a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00014a60: 0a0a 4f74 6865 7220 6f70 7469 6f6e 7320  ..Other options 
│ │ │ │ -00014a70: 6d61 7920 616c 736f 2062 6520 7061 7373  may also be pass
│ │ │ │ -00014a80: 6564 2074 6f20 7468 6520 2a6e 6f74 6520  ed to the *note 
│ │ │ │ -00014a90: 5261 6e64 6f6d 506f 696e 7473 3a20 2852  RandomPoints: (R
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│ │ │ │ -00014ab0: 0a70 6163 6b61 6765 2076 6961 2074 6865  .package via the
│ │ │ │ -00014ac0: 202a 6e6f 7465 2050 6f69 6e74 4f70 7469   *note PointOpti
│ │ │ │ -00014ad0: 6f6e 733a 2050 6f69 6e74 4f70 7469 6f6e  ons: PointOption
│ │ │ │ -00014ae0: 732c 206f 7074 696f 6e2e 0a0a 7265 6775  s, option...regu
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│ │ │ │ -00014b00: 3a20 2049 7420 6973 2072 6561 736f 6e61  :  It is reasona
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│ │ │ │ -00014b20: 7420 796f 7520 7368 6f75 6c64 2066 696e  t you should fin
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│ │ │ │ -00014b40: 7769 7468 206f 6e65 2073 7472 6174 6567  with one strateg
│ │ │ │ -00014b50: 7920 6f72 2061 6e6f 7468 6572 292c 2061  y or another), a
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│ │ │ │ -00014b70: 7320 7468 6520 6d69 6e6f 7273 2079 6f75  s the minors you
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│ │ │ │ -00014ba0: 2074 6f20 7665 7269 6679 206f 7572 2072   to verify our r
│ │ │ │ -00014bb0: 696e 6720 6973 2072 6567 756c 6172 2069  ing is regular i
│ │ │ │ -00014bc0: 6e20 636f 6469 6d65 6e73 696f 6e20 312e  n codimension 1.
│ │ │ │ -00014bd0: 0a54 6869 7320 6973 2065 7861 6374 6c79  .This is exactly
│ │ │ │ -00014be0: 2077 6861 7420 7265 6775 6c61 7249 6e43   what regularInC
│ │ │ │ -00014bf0: 6f64 696d 656e 7369 6f6e 2064 6f65 732e  odimension does.
│ │ │ │ -00014c00: 2020 4f6e 6520 6361 6e20 636f 6e74 726f    One can contro
│ │ │ │ -00014c10: 6c20 6174 2061 2066 696e 650a 6c65 7665  l at a fine.leve
│ │ │ │ -00014c20: 6c20 686f 7720 6672 6571 7565 6e74 6c79  l how frequently
│ │ │ │ -00014c30: 206e 6577 206d 696e 6f72 7320 6172 6520   new minors are 
│ │ │ │ -00014c40: 636f 6d70 7574 6564 2c20 616e 6420 686f  computed, and ho
│ │ │ │ -00014c50: 7720 6672 6571 7565 6e74 6c79 2074 6865  w frequently the
│ │ │ │ -00014c60: 2064 696d 656e 7369 6f6e 0a6f 6620 7768   dimension.of wh
│ │ │ │ -00014c70: 6174 2077 6520 6861 7665 2063 6f6d 7075  at we have compu
│ │ │ │ -00014c80: 7465 6420 736f 2066 6172 2069 7320 6368  ted so far is ch
│ │ │ │ -00014c90: 6563 6b65 642c 2062 7920 7468 6520 6f70  ecked, by the op
│ │ │ │ -00014ca0: 7469 6f6e 2063 6f64 696d 4368 6563 6b46  tion codimCheckF
│ │ │ │ -00014cb0: 756e 6374 696f 6e2e 0a46 6f72 206d 6f72  unction..For mor
│ │ │ │ -00014cc0: 6520 6f6e 2074 6861 742c 2073 6565 202a  e on that, see *
│ │ │ │ -00014cd0: 6e6f 7465 2052 6567 756c 6172 496e 436f  note RegularInCo
│ │ │ │ -00014ce0: 6469 6d65 6e73 696f 6e54 7574 6f72 6961  dimensionTutoria
│ │ │ │ -00014cf0: 6c3a 0a52 6567 756c 6172 496e 436f 6469  l:.RegularInCodi
│ │ │ │ -00014d00: 6d65 6e73 696f 6e54 7574 6f72 6961 6c2c  mensionTutorial,
│ │ │ │ -00014d10: 2061 6e64 202a 6e6f 7465 2072 6567 756c   and *note regul
│ │ │ │ -00014d20: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ -00014d30: 0a72 6567 756c 6172 496e 436f 6469 6d65  .regularInCodime
│ │ │ │ -00014d40: 6e73 696f 6e2c 2e20 204c 6574 2075 7320  nsion,.  Let us 
│ │ │ │ -00014d50: 6669 6e69 7368 2072 756e 6e69 6e67 2072  finish running r
│ │ │ │ -00014d60: 6567 756c 6172 496e 436f 6469 6d65 6e73  egularInCodimens
│ │ │ │ -00014d70: 696f 6e20 6f6e 206f 7572 0a65 7861 6d70  ion on our.examp
│ │ │ │ -00014d80: 6c65 2077 6974 6820 7365 7665 7261 6c20  le with several 
│ │ │ │ -00014d90: 6469 6666 6572 656e 7420 7374 7261 7465  different strate
│ │ │ │ -00014da0: 6769 6573 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  gies...+--------
│ │ │ │ -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 2d2d 2d2d  ----------------
│ │ │ │ -00014df0: 2d2d 2d2d 2d2b 0a7c 6934 3720 3a20 7469  -----+.|i47 : ti
│ │ │ │ -00014e00: 6d65 2072 6567 756c 6172 496e 436f 6469  me regularInCodi
│ │ │ │ -00014e10: 6d65 6e73 696f 6e28 312c 2053 2f4a 2c20  mension(1, S/J, 
│ │ │ │ -00014e20: 4d61 784d 696e 6f72 7320 3d3e 2031 3030  MaxMinors => 100
│ │ │ │ -00014e30: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ -00014e40: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ -00014e50: 2033 2e33 3536 3438 7320 2863 7075 293b   3.35648s (cpu);
│ │ │ │ -00014e60: 2033 2e30 3333 3873 2028 7468 7265 6164   3.0338s (thread
│ │ │ │ -00014e70: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │ -00014e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014e90: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ -00014ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014ee0: 2d2d 2d2d 2d7c 0a7c 5374 7261 7465 6779  -----|.|Strategy
│ │ │ │ -00014ef0: 3d3e 5374 7261 7465 6779 4465 6661 756c  =>StrategyDefaul
│ │ │ │ -00014f00: 7429 2020 2020 2020 2020 2020 2020 2020  t)              
│ │ │ │ -00014f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014f30: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -00014f40: 2d2d 2d2d 2d2d 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 2d2b 0a7c 6934 3820 3a20 7469  -----+.|i48 : ti
│ │ │ │ -00014f90: 6d65 2072 6567 756c 6172 496e 436f 6469  me regularInCodi
│ │ │ │ -00014fa0: 6d65 6e73 696f 6e28 312c 2053 2f4a 2c20  mension(1, S/J, 
│ │ │ │ -00014fb0: 4d61 784d 696e 6f72 7320 3d3e 2031 3030  MaxMinors => 100
│ │ │ │ -00014fc0: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ -00014fd0: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ -00014fe0: 2030 2e37 3238 3039 3973 2028 6370 7529   0.728099s (cpu)
│ │ │ │ -00014ff0: 3b20 302e 3633 3631 3438 7320 2874 6872  ; 0.636148s (thr
│ │ │ │ -00015000: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │ -00015010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015020: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00015030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014970: 0a7c 6f34 3620 3d20 3220 2020 2020 2020  .|o46 = 2       
│ │ │ │ +00014980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000149a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000149b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000149c0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +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 2d2d  ----------------
│ │ │ │ +00014a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00014a10: 0a7c 506f 696e 744f 7074 696f 6e73 3d3e  .|PointOptions=>
│ │ │ │ +00014a20: 7074 7353 7472 6174 5261 7469 6f6e 616c  ptsStratRational
│ │ │ │ +00014a30: 2929 2020 2020 2020 2020 2020 2020 2020  ))              
│ │ │ │ +00014a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014a50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00014a60: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00014a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +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 2d2b  ---------------+
│ │ │ │ +00014ab0: 0a0a 4f74 6865 7220 6f70 7469 6f6e 7320  ..Other options 
│ │ │ │ +00014ac0: 6d61 7920 616c 736f 2062 6520 7061 7373  may also be pass
│ │ │ │ +00014ad0: 6564 2074 6f20 7468 6520 2a6e 6f74 6520  ed to the *note 
│ │ │ │ +00014ae0: 5261 6e64 6f6d 506f 696e 7473 3a20 2852  RandomPoints: (R
│ │ │ │ +00014af0: 616e 646f 6d50 6f69 6e74 7329 546f 702c  andomPoints)Top,
│ │ │ │ +00014b00: 0a70 6163 6b61 6765 2076 6961 2074 6865  .package via the
│ │ │ │ +00014b10: 202a 6e6f 7465 2050 6f69 6e74 4f70 7469   *note PointOpti
│ │ │ │ +00014b20: 6f6e 733a 2050 6f69 6e74 4f70 7469 6f6e  ons: PointOption
│ │ │ │ +00014b30: 732c 206f 7074 696f 6e2e 0a0a 7265 6775  s, option...regu
│ │ │ │ +00014b40: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ +00014b50: 3a20 2049 7420 6973 2072 6561 736f 6e61  :  It is reasona
│ │ │ │ +00014b60: 626c 6520 746f 2074 6869 6e6b 2074 6861  ble to think tha
│ │ │ │ +00014b70: 7420 796f 7520 7368 6f75 6c64 2066 696e  t you should fin
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│ │ │ │ +00014b90: 7769 7468 206f 6e65 2073 7472 6174 6567  with one strateg
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│ │ │ │ +00014bc0: 7320 7468 6520 6d69 6e6f 7273 2079 6f75  s the minors you
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│ │ │ │ +00014c00: 696e 6720 6973 2072 6567 756c 6172 2069  ing is regular i
│ │ │ │ +00014c10: 6e20 636f 6469 6d65 6e73 696f 6e20 312e  n codimension 1.
│ │ │ │ +00014c20: 0a54 6869 7320 6973 2065 7861 6374 6c79  .This is exactly
│ │ │ │ +00014c30: 2077 6861 7420 7265 6775 6c61 7249 6e43   what regularInC
│ │ │ │ +00014c40: 6f64 696d 656e 7369 6f6e 2064 6f65 732e  odimension does.
│ │ │ │ +00014c50: 2020 4f6e 6520 6361 6e20 636f 6e74 726f    One can contro
│ │ │ │ +00014c60: 6c20 6174 2061 2066 696e 650a 6c65 7665  l at a fine.leve
│ │ │ │ +00014c70: 6c20 686f 7720 6672 6571 7565 6e74 6c79  l how frequently
│ │ │ │ +00014c80: 206e 6577 206d 696e 6f72 7320 6172 6520   new minors are 
│ │ │ │ +00014c90: 636f 6d70 7574 6564 2c20 616e 6420 686f  computed, and ho
│ │ │ │ +00014ca0: 7720 6672 6571 7565 6e74 6c79 2074 6865  w frequently the
│ │ │ │ +00014cb0: 2064 696d 656e 7369 6f6e 0a6f 6620 7768   dimension.of wh
│ │ │ │ +00014cc0: 6174 2077 6520 6861 7665 2063 6f6d 7075  at we have compu
│ │ │ │ +00014cd0: 7465 6420 736f 2066 6172 2069 7320 6368  ted so far is ch
│ │ │ │ +00014ce0: 6563 6b65 642c 2062 7920 7468 6520 6f70  ecked, by the op
│ │ │ │ +00014cf0: 7469 6f6e 2063 6f64 696d 4368 6563 6b46  tion codimCheckF
│ │ │ │ +00014d00: 756e 6374 696f 6e2e 0a46 6f72 206d 6f72  unction..For mor
│ │ │ │ +00014d10: 6520 6f6e 2074 6861 742c 2073 6565 202a  e on that, see *
│ │ │ │ +00014d20: 6e6f 7465 2052 6567 756c 6172 496e 436f  note RegularInCo
│ │ │ │ +00014d30: 6469 6d65 6e73 696f 6e54 7574 6f72 6961  dimensionTutoria
│ │ │ │ +00014d40: 6c3a 0a52 6567 756c 6172 496e 436f 6469  l:.RegularInCodi
│ │ │ │ +00014d50: 6d65 6e73 696f 6e54 7574 6f72 6961 6c2c  mensionTutorial,
│ │ │ │ +00014d60: 2061 6e64 202a 6e6f 7465 2072 6567 756c   and *note regul
│ │ │ │ +00014d70: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +00014d80: 0a72 6567 756c 6172 496e 436f 6469 6d65  .regularInCodime
│ │ │ │ +00014d90: 6e73 696f 6e2c 2e20 204c 6574 2075 7320  nsion,.  Let us 
│ │ │ │ +00014da0: 6669 6e69 7368 2072 756e 6e69 6e67 2072  finish running r
│ │ │ │ +00014db0: 6567 756c 6172 496e 436f 6469 6d65 6e73  egularInCodimens
│ │ │ │ +00014dc0: 696f 6e20 6f6e 206f 7572 0a65 7861 6d70  ion on our.examp
│ │ │ │ +00014dd0: 6c65 2077 6974 6820 7365 7665 7261 6c20  le with several 
│ │ │ │ +00014de0: 6469 6666 6572 656e 7420 7374 7261 7465  different strate
│ │ │ │ +00014df0: 6769 6573 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  gies...+--------
│ │ │ │ +00014e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014e40: 2d2d 2d2d 2d2b 0a7c 6934 3720 3a20 7469  -----+.|i47 : ti
│ │ │ │ +00014e50: 6d65 2072 6567 756c 6172 496e 436f 6469  me regularInCodi
│ │ │ │ +00014e60: 6d65 6e73 696f 6e28 312c 2053 2f4a 2c20  mension(1, S/J, 
│ │ │ │ +00014e70: 4d61 784d 696e 6f72 7320 3d3e 2031 3030  MaxMinors => 100
│ │ │ │ +00014e80: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +00014e90: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ +00014ea0: 2034 2e31 3038 3173 2028 6370 7529 3b20   4.1081s (cpu); 
│ │ │ │ +00014eb0: 332e 3631 3832 3473 2028 7468 7265 6164  3.61824s (thread
│ │ │ │ +00014ec0: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │ +00014ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014ee0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +00014ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014f30: 2d2d 2d2d 2d7c 0a7c 5374 7261 7465 6779  -----|.|Strategy
│ │ │ │ +00014f40: 3d3e 5374 7261 7465 6779 4465 6661 756c  =>StrategyDefaul
│ │ │ │ +00014f50: 7429 2020 2020 2020 2020 2020 2020 2020  t)              
│ │ │ │ +00014f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014f80: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00014f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014fd0: 2d2d 2d2d 2d2b 0a7c 6934 3820 3a20 7469  -----+.|i48 : ti
│ │ │ │ +00014fe0: 6d65 2072 6567 756c 6172 496e 436f 6469  me regularInCodi
│ │ │ │ +00014ff0: 6d65 6e73 696f 6e28 312c 2053 2f4a 2c20  mension(1, S/J, 
│ │ │ │ +00015000: 4d61 784d 696e 6f72 7320 3d3e 2031 3030  MaxMinors => 100
│ │ │ │ +00015010: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +00015020: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ +00015030: 2030 2e39 3135 3837 3273 2028 6370 7529   0.915872s (cpu)
│ │ │ │ +00015040: 3b20 302e 3737 3137 3431 7320 2874 6872  ; 0.771741s (thr
│ │ │ │ +00015050: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │  00015060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015070: 2020 2020 207c 0a7c 6f34 3820 3d20 7472       |.|o48 = tr
│ │ │ │ -00015080: 7565 2020 2020 2020 2020 2020 2020 2020  ue              
│ │ │ │ +00015070: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00015080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ -000150d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000150e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000150f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015110: 2d2d 2d2d 2d7c 0a7c 5374 7261 7465 6779  -----|.|Strategy
│ │ │ │ -00015120: 3d3e 5374 7261 7465 6779 4465 6661 756c  =>StrategyDefaul
│ │ │ │ -00015130: 744e 6f6e 5261 6e64 6f6d 2920 2020 2020  tNonRandom)     
│ │ │ │ -00015140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015160: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -00015170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000151a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000151b0: 2d2d 2d2d 2d2b 0a7c 6934 3920 3a20 7469  -----+.|i49 : ti
│ │ │ │ -000151c0: 6d65 2072 6567 756c 6172 496e 436f 6469  me regularInCodi
│ │ │ │ -000151d0: 6d65 6e73 696f 6e28 312c 2053 2f4a 2c20  mension(1, S/J, 
│ │ │ │ -000151e0: 4d61 784d 696e 6f72 7320 3d3e 2031 3030  MaxMinors => 100
│ │ │ │ -000151f0: 2c20 5374 7261 7465 6779 3d3e 5261 6e64  , Strategy=>Rand
│ │ │ │ -00015200: 6f6d 2920 207c 0a7c 202d 2d20 7573 6564  om)  |.| -- used
│ │ │ │ -00015210: 2032 2e37 3138 3638 7320 2863 7075 293b   2.71868s (cpu);
│ │ │ │ -00015220: 2032 2e35 3237 3135 7320 2874 6872 6561   2.52715s (threa
│ │ │ │ -00015230: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │ -00015240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015250: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -00015260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000152a0: 2d2d 2d2d 2d2b 0a7c 6935 3020 3a20 7469  -----+.|i50 : ti
│ │ │ │ -000152b0: 6d65 2072 6567 756c 6172 496e 436f 6469  me regularInCodi
│ │ │ │ -000152c0: 6d65 6e73 696f 6e28 312c 2053 2f4a 2c20  mension(1, S/J, 
│ │ │ │ -000152d0: 4d61 784d 696e 6f72 7320 3d3e 2031 3030  MaxMinors => 100
│ │ │ │ -000152e0: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ -000152f0: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ -00015300: 2032 2e33 3537 3273 2028 6370 7529 3b20   2.3572s (cpu); 
│ │ │ │ -00015310: 322e 3030 3434 3673 2028 7468 7265 6164  2.00446s (thread
│ │ │ │ -00015320: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │ -00015330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015340: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ -00015350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015390: 2d2d 2d2d 2d7c 0a7c 5374 7261 7465 6779  -----|.|Strategy
│ │ │ │ -000153a0: 3d3e 4c65 7853 6d61 6c6c 6573 7429 2020  =>LexSmallest)  
│ │ │ │ -000153b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000153c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000153d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000153e0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -000153f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015430: 2d2d 2d2d 2d2b 0a7c 6935 3120 3a20 7469  -----+.|i51 : ti
│ │ │ │ -00015440: 6d65 2072 6567 756c 6172 496e 436f 6469  me regularInCodi
│ │ │ │ -00015450: 6d65 6e73 696f 6e28 312c 2053 2f4a 2c20  mension(1, S/J, 
│ │ │ │ -00015460: 4d61 784d 696e 6f72 7320 3d3e 2031 3030  MaxMinors => 100
│ │ │ │ -00015470: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ -00015480: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ -00015490: 2030 2e38 3734 3735 3173 2028 6370 7529   0.874751s (cpu)
│ │ │ │ -000154a0: 3b20 302e 3737 3734 3833 7320 2874 6872  ; 0.777483s (thr
│ │ │ │ -000154b0: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │ -000154c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000154d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000154e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000154f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000150c0: 2020 2020 207c 0a7c 6f34 3820 3d20 7472       |.|o48 = tr
│ │ │ │ +000150d0: 7565 2020 2020 2020 2020 2020 2020 2020  ue              
│ │ │ │ +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 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +00015120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015160: 2d2d 2d2d 2d7c 0a7c 5374 7261 7465 6779  -----|.|Strategy
│ │ │ │ +00015170: 3d3e 5374 7261 7465 6779 4465 6661 756c  =>StrategyDefaul
│ │ │ │ +00015180: 744e 6f6e 5261 6e64 6f6d 2920 2020 2020  tNonRandom)     
│ │ │ │ +00015190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000151a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000151b0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +000151c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000151d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000151e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000151f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015200: 2d2d 2d2d 2d2b 0a7c 6934 3920 3a20 7469  -----+.|i49 : ti
│ │ │ │ +00015210: 6d65 2072 6567 756c 6172 496e 436f 6469  me regularInCodi
│ │ │ │ +00015220: 6d65 6e73 696f 6e28 312c 2053 2f4a 2c20  mension(1, S/J, 
│ │ │ │ +00015230: 4d61 784d 696e 6f72 7320 3d3e 2031 3030  MaxMinors => 100
│ │ │ │ +00015240: 2c20 5374 7261 7465 6779 3d3e 5261 6e64  , Strategy=>Rand
│ │ │ │ +00015250: 6f6d 2920 207c 0a7c 202d 2d20 7573 6564  om)  |.| -- used
│ │ │ │ +00015260: 2033 2e35 3030 3631 7320 2863 7075 293b   3.50061s (cpu);
│ │ │ │ +00015270: 2033 2e32 3739 3436 7320 2874 6872 6561   3.27946s (threa
│ │ │ │ +00015280: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │ +00015290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000152a0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +000152b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000152c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000152d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000152e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000152f0: 2d2d 2d2d 2d2b 0a7c 6935 3020 3a20 7469  -----+.|i50 : ti
│ │ │ │ +00015300: 6d65 2072 6567 756c 6172 496e 436f 6469  me regularInCodi
│ │ │ │ +00015310: 6d65 6e73 696f 6e28 312c 2053 2f4a 2c20  mension(1, S/J, 
│ │ │ │ +00015320: 4d61 784d 696e 6f72 7320 3d3e 2031 3030  MaxMinors => 100
│ │ │ │ +00015330: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +00015340: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ +00015350: 2032 2e37 3438 3934 7320 2863 7075 293b   2.74894s (cpu);
│ │ │ │ +00015360: 2032 2e32 3131 3139 7320 2874 6872 6561   2.21119s (threa
│ │ │ │ +00015370: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │ +00015380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015390: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +000153a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000153b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000153c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000153d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000153e0: 2d2d 2d2d 2d7c 0a7c 5374 7261 7465 6779  -----|.|Strategy
│ │ │ │ +000153f0: 3d3e 4c65 7853 6d61 6c6c 6573 7429 2020  =>LexSmallest)  
│ │ │ │ +00015400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015430: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00015440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015480: 2d2d 2d2d 2d2b 0a7c 6935 3120 3a20 7469  -----+.|i51 : ti
│ │ │ │ +00015490: 6d65 2072 6567 756c 6172 496e 436f 6469  me regularInCodi
│ │ │ │ +000154a0: 6d65 6e73 696f 6e28 312c 2053 2f4a 2c20  mension(1, S/J, 
│ │ │ │ +000154b0: 4d61 784d 696e 6f72 7320 3d3e 2031 3030  MaxMinors => 100
│ │ │ │ +000154c0: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +000154d0: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ +000154e0: 2030 2e39 3033 3334 7320 2863 7075 293b   0.90334s (cpu);
│ │ │ │ +000154f0: 2030 2e38 3331 3531 3373 2028 7468 7265   0.831513s (thre
│ │ │ │ +00015500: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │  00015510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015520: 2020 2020 207c 0a7c 6f35 3120 3d20 7472       |.|o51 = tr
│ │ │ │ -00015530: 7565 2020 2020 2020 2020 2020 2020 2020  ue              
│ │ │ │ +00015520: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00015530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015570: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ -00015580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000155a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000155b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000155c0: 2d2d 2d2d 2d7c 0a7c 5374 7261 7465 6779  -----|.|Strategy
│ │ │ │ -000155d0: 3d3e 4c65 7853 6d61 6c6c 6573 7454 6572  =>LexSmallestTer
│ │ │ │ -000155e0: 6d29 2020 2020 2020 2020 2020 2020 2020  m)              
│ │ │ │ -000155f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015610: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -00015620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015660: 2d2d 2d2d 2d2b 0a7c 6935 3220 3a20 7469  -----+.|i52 : ti
│ │ │ │ -00015670: 6d65 2072 6567 756c 6172 496e 436f 6469  me regularInCodi
│ │ │ │ -00015680: 6d65 6e73 696f 6e28 312c 2053 2f4a 2c20  mension(1, S/J, 
│ │ │ │ -00015690: 4d61 784d 696e 6f72 7320 3d3e 2031 3030  MaxMinors => 100
│ │ │ │ -000156a0: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ -000156b0: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ -000156c0: 2032 2e37 3036 3535 7320 2863 7075 293b   2.70655s (cpu);
│ │ │ │ -000156d0: 2032 2e32 3839 3035 7320 2874 6872 6561   2.28905s (threa
│ │ │ │ -000156e0: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │ -000156f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015700: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ -00015710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015750: 2d2d 2d2d 2d7c 0a7c 5374 7261 7465 6779  -----|.|Strategy
│ │ │ │ -00015760: 3d3e 4752 6576 4c65 7853 6d61 6c6c 6573  =>GRevLexSmalles
│ │ │ │ -00015770: 7429 2020 2020 2020 2020 2020 2020 2020  t)              
│ │ │ │ -00015780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000157a0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -000157b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000157c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000157d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000157e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000157f0: 2d2d 2d2d 2d2b 0a7c 6935 3320 3a20 7469  -----+.|i53 : ti
│ │ │ │ -00015800: 6d65 2072 6567 756c 6172 496e 436f 6469  me regularInCodi
│ │ │ │ -00015810: 6d65 6e73 696f 6e28 312c 2053 2f4a 2c20  mension(1, S/J, 
│ │ │ │ -00015820: 4d61 784d 696e 6f72 7320 3d3e 2031 3030  MaxMinors => 100
│ │ │ │ -00015830: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ -00015840: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ -00015850: 2033 2e31 3037 3673 2028 6370 7529 3b20   3.1076s (cpu); 
│ │ │ │ -00015860: 322e 3638 3735 3873 2028 7468 7265 6164  2.68758s (thread
│ │ │ │ -00015870: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │ -00015880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015890: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ -000158a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000158b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000158c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000158d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000158e0: 2d2d 2d2d 2d7c 0a7c 5374 7261 7465 6779  -----|.|Strategy
│ │ │ │ -000158f0: 3d3e 4752 6576 4c65 7853 6d61 6c6c 6573  =>GRevLexSmalles
│ │ │ │ -00015900: 7454 6572 6d29 2020 2020 2020 2020 2020  tTerm)          
│ │ │ │ -00015910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015930: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -00015940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015980: 2d2d 2d2d 2d2b 0a7c 6935 3420 3a20 7469  -----+.|i54 : ti
│ │ │ │ -00015990: 6d65 2072 6567 756c 6172 496e 436f 6469  me regularInCodi
│ │ │ │ -000159a0: 6d65 6e73 696f 6e28 312c 2053 2f4a 2c20  mension(1, S/J, 
│ │ │ │ -000159b0: 4d61 784d 696e 6f72 7320 3d3e 2031 3030  MaxMinors => 100
│ │ │ │ -000159c0: 2c20 5374 7261 7465 6779 3d3e 506f 696e  , Strategy=>Poin
│ │ │ │ -000159d0: 7473 2920 207c 0a7c 202d 2d20 7573 6564  ts)  |.| -- used
│ │ │ │ -000159e0: 2039 2e32 3634 3432 7320 2863 7075 293b   9.26442s (cpu);
│ │ │ │ -000159f0: 2037 2e37 3133 3138 7320 2874 6872 6561   7.71318s (threa
│ │ │ │ -00015a00: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │ -00015a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015a20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00015a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015570: 2020 2020 207c 0a7c 6f35 3120 3d20 7472       |.|o51 = tr
│ │ │ │ +00015580: 7565 2020 2020 2020 2020 2020 2020 2020  ue              
│ │ │ │ +00015590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000155a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000155b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000155c0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +000155d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000155e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000155f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015610: 2d2d 2d2d 2d7c 0a7c 5374 7261 7465 6779  -----|.|Strategy
│ │ │ │ +00015620: 3d3e 4c65 7853 6d61 6c6c 6573 7454 6572  =>LexSmallestTer
│ │ │ │ +00015630: 6d29 2020 2020 2020 2020 2020 2020 2020  m)              
│ │ │ │ +00015640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015660: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00015670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000156a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000156b0: 2d2d 2d2d 2d2b 0a7c 6935 3220 3a20 7469  -----+.|i52 : ti
│ │ │ │ +000156c0: 6d65 2072 6567 756c 6172 496e 436f 6469  me regularInCodi
│ │ │ │ +000156d0: 6d65 6e73 696f 6e28 312c 2053 2f4a 2c20  mension(1, S/J, 
│ │ │ │ +000156e0: 4d61 784d 696e 6f72 7320 3d3e 2031 3030  MaxMinors => 100
│ │ │ │ +000156f0: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +00015700: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ +00015710: 2033 2e31 3232 3336 7320 2863 7075 293b   3.12236s (cpu);
│ │ │ │ +00015720: 2032 2e35 3236 3937 7320 2874 6872 6561   2.52697s (threa
│ │ │ │ +00015730: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │ +00015740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015750: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +00015760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000157a0: 2d2d 2d2d 2d7c 0a7c 5374 7261 7465 6779  -----|.|Strategy
│ │ │ │ +000157b0: 3d3e 4752 6576 4c65 7853 6d61 6c6c 6573  =>GRevLexSmalles
│ │ │ │ +000157c0: 7429 2020 2020 2020 2020 2020 2020 2020  t)              
│ │ │ │ +000157d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000157e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000157f0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00015800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015840: 2d2d 2d2d 2d2b 0a7c 6935 3320 3a20 7469  -----+.|i53 : ti
│ │ │ │ +00015850: 6d65 2072 6567 756c 6172 496e 436f 6469  me regularInCodi
│ │ │ │ +00015860: 6d65 6e73 696f 6e28 312c 2053 2f4a 2c20  mension(1, S/J, 
│ │ │ │ +00015870: 4d61 784d 696e 6f72 7320 3d3e 2031 3030  MaxMinors => 100
│ │ │ │ +00015880: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +00015890: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ +000158a0: 2033 2e39 3333 3937 7320 2863 7075 293b   3.93397s (cpu);
│ │ │ │ +000158b0: 2033 2e33 3034 3039 7320 2874 6872 6561   3.30409s (threa
│ │ │ │ +000158c0: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │ +000158d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000158e0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +000158f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015930: 2d2d 2d2d 2d7c 0a7c 5374 7261 7465 6779  -----|.|Strategy
│ │ │ │ +00015940: 3d3e 4752 6576 4c65 7853 6d61 6c6c 6573  =>GRevLexSmalles
│ │ │ │ +00015950: 7454 6572 6d29 2020 2020 2020 2020 2020  tTerm)          
│ │ │ │ +00015960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015980: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00015990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000159a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000159b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000159c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000159d0: 2d2d 2d2d 2d2b 0a7c 6935 3420 3a20 7469  -----+.|i54 : ti
│ │ │ │ +000159e0: 6d65 2072 6567 756c 6172 496e 436f 6469  me regularInCodi
│ │ │ │ +000159f0: 6d65 6e73 696f 6e28 312c 2053 2f4a 2c20  mension(1, S/J, 
│ │ │ │ +00015a00: 4d61 784d 696e 6f72 7320 3d3e 2031 3030  MaxMinors => 100
│ │ │ │ +00015a10: 2c20 5374 7261 7465 6779 3d3e 506f 696e  , Strategy=>Poin
│ │ │ │ +00015a20: 7473 2920 207c 0a7c 202d 2d20 7573 6564  ts)  |.| -- used
│ │ │ │ +00015a30: 2031 312e 3537 3731 7320 2863 7075 293b   11.5771s (cpu);
│ │ │ │ +00015a40: 2039 2e34 3135 3739 7320 2874 6872 6561   9.41579s (threa
│ │ │ │ +00015a50: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │  00015a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015a70: 2020 2020 207c 0a7c 6f35 3420 3d20 7472       |.|o54 = tr
│ │ │ │ -00015a80: 7565 2020 2020 2020 2020 2020 2020 2020  ue              
│ │ │ │ +00015a70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00015a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015ac0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -00015ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015b10: 2d2d 2d2d 2d2b 0a7c 6935 3520 3a20 7469  -----+.|i55 : ti
│ │ │ │ -00015b20: 6d65 2072 6567 756c 6172 496e 436f 6469  me regularInCodi
│ │ │ │ -00015b30: 6d65 6e73 696f 6e28 312c 2053 2f4a 2c20  mension(1, S/J, 
│ │ │ │ -00015b40: 4d61 784d 696e 6f72 7320 3d3e 2031 3030  MaxMinors => 100
│ │ │ │ -00015b50: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ -00015b60: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ -00015b70: 2037 2e33 3031 3839 7320 2863 7075 293b   7.30189s (cpu);
│ │ │ │ -00015b80: 2036 2e30 3639 3473 2028 7468 7265 6164   6.0694s (thread
│ │ │ │ -00015b90: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │ -00015ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015bb0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00015bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015ac0: 2020 2020 207c 0a7c 6f35 3420 3d20 7472       |.|o54 = tr
│ │ │ │ +00015ad0: 7565 2020 2020 2020 2020 2020 2020 2020  ue              
│ │ │ │ +00015ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015b10: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00015b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015b60: 2d2d 2d2d 2d2b 0a7c 6935 3520 3a20 7469  -----+.|i55 : ti
│ │ │ │ +00015b70: 6d65 2072 6567 756c 6172 496e 436f 6469  me regularInCodi
│ │ │ │ +00015b80: 6d65 6e73 696f 6e28 312c 2053 2f4a 2c20  mension(1, S/J, 
│ │ │ │ +00015b90: 4d61 784d 696e 6f72 7320 3d3e 2031 3030  MaxMinors => 100
│ │ │ │ +00015ba0: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +00015bb0: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ +00015bc0: 2039 2e32 3338 3337 7320 2863 7075 293b   9.23837s (cpu);
│ │ │ │ +00015bd0: 2037 2e32 3236 3434 7320 2874 6872 6561   7.22644s (threa
│ │ │ │ +00015be0: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │  00015bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015c00: 2020 2020 207c 0a7c 6f35 3520 3d20 7472       |.|o55 = tr
│ │ │ │ -00015c10: 7565 2020 2020 2020 2020 2020 2020 2020  ue              
│ │ │ │ +00015c00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +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 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ -00015c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015ca0: 2d2d 2d2d 2d7c 0a7c 5374 7261 7465 6779  -----|.|Strategy
│ │ │ │ -00015cb0: 3d3e 5374 7261 7465 6779 4465 6661 756c  =>StrategyDefaul
│ │ │ │ -00015cc0: 7457 6974 6850 6f69 6e74 7329 2020 2020  tWithPoints)    
│ │ │ │ -00015cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015cf0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -00015d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015d40: 2d2d 2d2d 2d2b 0a0a 4966 2072 6567 756c  -----+..If regul
│ │ │ │ -00015d50: 6172 496e 436f 6469 6d65 6e73 696f 6e20  arInCodimension 
│ │ │ │ -00015d60: 6f75 7470 7574 7320 6e6f 7468 696e 672c  outputs nothing,
│ │ │ │ -00015d70: 2074 6865 6e20 6974 2063 6f75 6c64 6e27   then it couldn'
│ │ │ │ -00015d80: 7420 7665 7269 6679 2074 6861 7420 7468  t verify that th
│ │ │ │ -00015d90: 6520 7269 6e67 0a77 6173 2072 6567 756c  e ring.was regul
│ │ │ │ -00015da0: 6172 2069 6e20 7468 6174 2063 6f64 696d  ar in that codim
│ │ │ │ -00015db0: 656e 7369 6f6e 2e20 2057 6520 7365 7420  ension.  We set 
│ │ │ │ -00015dc0: 4d61 784d 696e 6f72 7320 3d3e 2031 3030  MaxMinors => 100
│ │ │ │ -00015dd0: 2074 6f20 6b65 6570 2069 7420 6672 6f6d   to keep it from
│ │ │ │ -00015de0: 0a72 756e 6e69 6e67 2074 6f6f 206c 6f6e  .running too lon
│ │ │ │ -00015df0: 6720 7769 7468 2061 6e20 696e 6566 6665  g with an ineffe
│ │ │ │ -00015e00: 6374 6976 6520 7374 7261 7465 6779 2e20  ctive strategy. 
│ │ │ │ -00015e10: 2041 6761 696e 2c20 6576 656e 2074 686f   Again, even tho
│ │ │ │ -00015e20: 7567 680a 4752 6576 4c65 7853 6d61 6c6c  ugh.GRevLexSmall
│ │ │ │ -00015e30: 6573 7420 616e 6420 4752 6576 4c65 7853  est and GRevLexS
│ │ │ │ -00015e40: 6d61 6c6c 6573 7454 6572 6d20 6172 6520  mallestTerm are 
│ │ │ │ -00015e50: 6e6f 7420 6566 6665 6374 6976 6520 696e  not effective in
│ │ │ │ -00015e60: 2074 6869 7320 7061 7274 6963 756c 6172   this particular
│ │ │ │ -00015e70: 0a65 7861 6d70 6c65 2c20 696e 206f 7468  .example, in oth
│ │ │ │ -00015e80: 6572 7320 7468 6579 2070 6572 666f 726d  ers they perform
│ │ │ │ -00015e90: 2062 6574 7465 7220 7468 616e 206f 7468   better than oth
│ │ │ │ -00015ea0: 6572 2073 7472 6174 6567 6965 732e 2020  er strategies.  
│ │ │ │ -00015eb0: 4e6f 7465 2073 696d 696c 6172 0a63 6f6e  Note similar.con
│ │ │ │ -00015ec0: 7369 6465 7261 7469 6f6e 7320 616c 736f  siderations also
│ │ │ │ -00015ed0: 2061 7070 6c79 2074 6f20 2a6e 6f74 6520   apply to *note 
│ │ │ │ -00015ee0: 7072 6f6a 4469 6d3a 2070 726f 6a44 696d  projDim: projDim
│ │ │ │ -00015ef0: 2c2e 0a0a 5365 6520 616c 736f 0a3d 3d3d  ,...See also.===
│ │ │ │ -00015f00: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
│ │ │ │ -00015f10: 2063 686f 6f73 6547 6f6f 644d 696e 6f72   chooseGoodMinor
│ │ │ │ -00015f20: 7328 2e2e 2e2c 5374 7261 7465 6779 3d3e  s(...,Strategy=>
│ │ │ │ -00015f30: 2e2e 2e29 3a20 5374 7261 7465 6779 4465  ...): StrategyDe
│ │ │ │ -00015f40: 6661 756c 742c 202d 2d20 7374 7261 7465  fault, -- strate
│ │ │ │ -00015f50: 6769 6573 0a20 2020 2066 6f72 2063 686f  gies.    for cho
│ │ │ │ -00015f60: 6f73 696e 6720 7375 626d 6174 7269 6365  osing submatrice
│ │ │ │ -00015f70: 730a 2020 2a20 2a6e 6f74 6520 5265 6775  s.  * *note Regu
│ │ │ │ -00015f80: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ -00015f90: 5475 746f 7269 616c 3a20 5265 6775 6c61  Tutorial: Regula
│ │ │ │ -00015fa0: 7249 6e43 6f64 696d 656e 7369 6f6e 5475  rInCodimensionTu
│ │ │ │ -00015fb0: 746f 7269 616c 2c20 2d2d 2041 0a20 2020  torial, -- A.   
│ │ │ │ -00015fc0: 2074 7574 6f72 6961 6c20 666f 7220 686f   tutorial for ho
│ │ │ │ -00015fd0: 7720 746f 2075 7365 2074 6865 2061 6476  w to use the adv
│ │ │ │ -00015fe0: 616e 6365 6420 6f70 7469 6f6e 7320 6f66  anced options of
│ │ │ │ -00015ff0: 2074 6865 2072 6567 756c 6172 496e 436f   the regularInCo
│ │ │ │ -00016000: 6469 6d65 6e73 696f 6e0a 2020 2020 6675  dimension.    fu
│ │ │ │ -00016010: 6e63 7469 6f6e 0a0a 466f 7220 7468 6520  nction..For the 
│ │ │ │ -00016020: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d  programmer.=====
│ │ │ │ -00016030: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  =============..T
│ │ │ │ -00016040: 6865 206f 626a 6563 7420 2a6e 6f74 6520  he object *note 
│ │ │ │ -00016050: 4661 7374 4d69 6e6f 7273 5374 7261 7465  FastMinorsStrate
│ │ │ │ -00016060: 6779 5475 746f 7269 616c 3a20 4661 7374  gyTutorial: Fast
│ │ │ │ -00016070: 4d69 6e6f 7273 5374 7261 7465 6779 5475  MinorsStrategyTu
│ │ │ │ -00016080: 746f 7269 616c 2c20 6973 2061 0a2a 6e6f  torial, is a.*no
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│ │ │ │ -000160c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -000160e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000160f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016100: 2d2d 0a0a 5468 6520 736f 7572 6365 206f  --..The source o
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│ │ │ │ -00016120: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570  is in./build/rep
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│ │ │ │ -00016140: 6163 6175 6c61 7932 2d31 2e32 352e 3131  acaulay2-1.25.11
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│ │ │ │ -000161a0: 6765 7453 7562 6d61 7472 6978 4f66 5261  getSubmatrixOfRa
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│ │ │ │ -000161c0: 6d41 744c 6561 7374 2c20 5072 6576 3a20  mAtLeast, Prev: 
│ │ │ │ -000161d0: 4661 7374 4d69 6e6f 7273 5374 7261 7465  FastMinorsStrate
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│ │ │ │ -000161f0: 546f 700a 0a67 6574 5375 626d 6174 7269  Top..getSubmatri
│ │ │ │ -00016200: 784f 6652 616e 6b20 2d2d 2074 7269 6573  xOfRank -- tries
│ │ │ │ -00016210: 2074 6f20 6669 6e64 2061 2073 7562 6d61   to find a subma
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│ │ │ │ -00016230: 6e20 7261 6e6b 0a2a 2a2a 2a2a 2a2a 2a2a  n rank.*********
│ │ │ │ -00016240: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00016250: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00016260: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00016270: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573  ********..  * Us
│ │ │ │ -00016280: 6167 653a 200a 2020 2020 2020 2020 6765  age: .        ge
│ │ │ │ -00016290: 7453 7562 6d61 7472 6978 4f66 5261 6e6b  tSubmatrixOfRank
│ │ │ │ -000162a0: 286e 312c 204d 3129 0a20 202a 2049 6e70  (n1, M1).  * Inp
│ │ │ │ -000162b0: 7574 733a 0a20 2020 2020 202a 206e 312c  uts:.      * n1,
│ │ │ │ -000162c0: 2061 6e20 2a6e 6f74 6520 696e 7465 6765   an *note intege
│ │ │ │ -000162d0: 723a 2028 4d61 6361 756c 6179 3244 6f63  r: (Macaulay2Doc
│ │ │ │ -000162e0: 295a 5a2c 2c20 0a20 2020 2020 202a 204d  )ZZ,, .      * M
│ │ │ │ -000162f0: 312c 2061 202a 6e6f 7465 206d 6174 7269  1, a *note matri
│ │ │ │ -00016300: 783a 2028 4d61 6361 756c 6179 3244 6f63  x: (Macaulay2Doc
│ │ │ │ -00016310: 294d 6174 7269 782c 2c20 0a20 202a 202a  )Matrix,, .  * *
│ │ │ │ -00016320: 6e6f 7465 204f 7074 696f 6e61 6c20 696e  note Optional in
│ │ │ │ -00016330: 7075 7473 3a20 284d 6163 6175 6c61 7932  puts: (Macaulay2
│ │ │ │ -00016340: 446f 6329 7573 696e 6720 6675 6e63 7469  Doc)using functi
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│ │ │ │ -00016360: 6c20 696e 7075 7473 2c3a 0a20 2020 2020  l inputs,:.     
│ │ │ │ -00016370: 202a 202a 6e6f 7465 2044 6574 5374 7261   * *note DetStra
│ │ │ │ -00016380: 7465 6779 3a20 4465 7453 7472 6174 6567  tegy: DetStrateg
│ │ │ │ -00016390: 792c 203d 3e20 2e2e 2e2c 2064 6566 6175  y, => ..., defau
│ │ │ │ -000163a0: 6c74 2076 616c 7565 2052 616e 6b2c 2044  lt value Rank, D
│ │ │ │ -000163b0: 6574 5374 7261 7465 6779 0a20 2020 2020  etStrategy.     
│ │ │ │ -000163c0: 2020 2069 7320 6120 7374 7261 7465 6779     is a strategy
│ │ │ │ -000163d0: 2066 6f72 2061 6c6c 6f77 696e 6720 7468   for allowing th
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│ │ │ │ -000163f0: 2068 6f77 2064 6574 6572 6d69 6e61 6e74   how determinant
│ │ │ │ -00016400: 7320 286f 720a 2020 2020 2020 2020 7261  s (or.        ra
│ │ │ │ -00016410: 6e6b 292c 2069 7320 636f 6d70 7574 6564  nk), is computed
│ │ │ │ -00016420: 0a20 2020 2020 202a 202a 6e6f 7465 204d  .      * *note M
│ │ │ │ -00016430: 6178 4d69 6e6f 7273 3a20 4d61 784d 696e  axMinors: MaxMin
│ │ │ │ -00016440: 6f72 732c 203d 3e20 2e2e 2e2c 2064 6566  ors, => ..., def
│ │ │ │ -00016450: 6175 6c74 2076 616c 7565 206e 756c 6c2c  ault value null,
│ │ │ │ -00016460: 2061 6e20 6f70 7469 6f6e 2074 6f0a 2020   an option to.  
│ │ │ │ -00016470: 2020 2020 2020 636f 6e74 726f 6c20 6465        control de
│ │ │ │ -00016480: 7074 6820 6f66 2073 6561 7263 680a 2020  pth of search.  
│ │ │ │ -00016490: 2020 2020 2a20 2a6e 6f74 6520 506f 696e      * *note Poin
│ │ │ │ -000164a0: 744f 7074 696f 6e73 3a20 506f 696e 744f  tOptions: PointO
│ │ │ │ -000164b0: 7074 696f 6e73 2c20 3d3e 202e 2e2e 2c20  ptions, => ..., 
│ │ │ │ -000164c0: 6465 6661 756c 7420 7661 6c75 6520 7b53  default value {S
│ │ │ │ -000164d0: 7472 6174 6567 7920 3d3e 0a20 2020 2020  trategy =>.     
│ │ │ │ -000164e0: 2020 2044 6566 6175 6c74 2c20 486f 6d6f     Default, Homo
│ │ │ │ -000164f0: 6765 6e65 6f75 7320 3d3e 2066 616c 7365  geneous => false
│ │ │ │ -00016500: 2c20 5265 706c 6163 656d 656e 7420 3d3e  , Replacement =>
│ │ │ │ -00016510: 2042 696e 6f6d 6961 6c2c 2045 7874 656e   Binomial, Exten
│ │ │ │ -00016520: 6446 6965 6c64 203d 3e0a 2020 2020 2020  dField =>.      
│ │ │ │ -00016530: 2020 7472 7565 2c20 506f 696e 7443 6865    true, PointChe
│ │ │ │ -00016540: 636b 4174 7465 6d70 7473 203d 3e20 302c  ckAttempts => 0,
│ │ │ │ -00016550: 2044 6563 6f6d 706f 7369 7469 6f6e 5374   DecompositionSt
│ │ │ │ -00016560: 7261 7465 6779 203d 3e20 4465 636f 6d70  rategy => Decomp
│ │ │ │ -00016570: 6f73 652c 0a20 2020 2020 2020 204e 756d  ose,.        Num
│ │ │ │ -00016580: 5468 7265 6164 7354 6f55 7365 203d 3e20  ThreadsToUse => 
│ │ │ │ -00016590: 312c 2044 696d 656e 7369 6f6e 4675 6e63  1, DimensionFunc
│ │ │ │ -000165a0: 7469 6f6e 203d 3e20 6469 6d2c 2056 6572  tion => dim, Ver
│ │ │ │ -000165b0: 626f 7365 203d 3e20 6661 6c73 657d 2c0a  bose => false},.
│ │ │ │ -000165c0: 2020 2020 2020 2020 6f70 7469 6f6e 7320          options 
│ │ │ │ -000165d0: 746f 2070 6173 7320 746f 2066 756e 6374  to pass to funct
│ │ │ │ -000165e0: 696f 6e73 2069 6e20 7468 6520 7061 636b  ions in the pack
│ │ │ │ -000165f0: 6167 6520 5261 6e64 6f6d 506f 696e 7473  age RandomPoints
│ │ │ │ -00016600: 0a20 2020 2020 202a 202a 6e6f 7465 2053  .      * *note S
│ │ │ │ -00016610: 7472 6174 6567 793a 2053 7472 6174 6567  trategy: Strateg
│ │ │ │ -00016620: 7944 6566 6175 6c74 2c20 3d3e 202e 2e2e  yDefault, => ...
│ │ │ │ -00016630: 2c20 6465 6661 756c 7420 7661 6c75 6520  , default value 
│ │ │ │ -00016640: 6e65 7720 4f70 7469 6f6e 5461 626c 650a  new OptionTable.
│ │ │ │ -00016650: 2020 2020 2020 2020 6672 6f6d 207b 506f          from {Po
│ │ │ │ -00016660: 696e 7473 203d 3e20 302c 2052 616e 646f  ints => 0, Rando
│ │ │ │ -00016670: 6d20 3d3e 2030 2c20 4752 6576 4c65 784c  m => 0, GRevLexL
│ │ │ │ -00016680: 6172 6765 7374 203d 3e20 302c 204c 6578  argest => 0, Lex
│ │ │ │ -00016690: 536d 616c 6c65 7374 5465 726d 203d 3e0a  SmallestTerm =>.
│ │ │ │ -000166a0: 2020 2020 2020 2020 3235 2c20 4c65 784c          25, LexL
│ │ │ │ -000166b0: 6172 6765 7374 203d 3e20 302c 204c 6578  argest => 0, Lex
│ │ │ │ -000166c0: 536d 616c 6c65 7374 203d 3e20 3235 2c20  Smallest => 25, 
│ │ │ │ -000166d0: 4752 6576 4c65 7853 6d61 6c6c 6573 7454  GRevLexSmallestT
│ │ │ │ -000166e0: 6572 6d20 3d3e 2032 352c 0a20 2020 2020  erm => 25,.     
│ │ │ │ -000166f0: 2020 2052 616e 646f 6d4e 6f6e 7a65 726f     RandomNonzero
│ │ │ │ -00016700: 203d 3e20 302c 2047 5265 764c 6578 536d   => 0, GRevLexSm
│ │ │ │ -00016710: 616c 6c65 7374 203d 3e20 3235 7d2c 2073  allest => 25}, s
│ │ │ │ -00016720: 7472 6174 6567 6965 7320 666f 7220 6368  trategies for ch
│ │ │ │ -00016730: 6f6f 7369 6e67 0a20 2020 2020 2020 2073  oosing.        s
│ │ │ │ -00016740: 7562 6d61 7472 6963 6573 0a20 2020 2020  ubmatrices.     
│ │ │ │ -00016750: 202a 202a 6e6f 7465 2054 6872 6561 6473   * *note Threads
│ │ │ │ -00016760: 3a20 6973 5261 6e6b 4174 4c65 6173 745f  : isRankAtLeast_
│ │ │ │ -00016770: 6c70 5f70 645f 7064 5f70 645f 636d 5468  lp_pd_pd_pd_cmTh
│ │ │ │ -00016780: 7265 6164 733d 3e5f 7064 5f70 645f 7064  reads=>_pd_pd_pd
│ │ │ │ -00016790: 5f72 702c 203d 3e0a 2020 2020 2020 2020  _rp, =>.        
│ │ │ │ -000167a0: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c  ..., default val
│ │ │ │ -000167b0: 7565 2031 2c20 616e 206f 7074 696f 6e20  ue 1, an option 
│ │ │ │ -000167c0: 666f 7220 7661 7269 6f75 7320 6675 6e63  for various func
│ │ │ │ -000167d0: 7469 6f6e 730a 2020 2020 2020 2a20 5665  tions.      * Ve
│ │ │ │ -000167e0: 7262 6f73 6520 3d3e 202e 2e2e 2c20 6465  rbose => ..., de
│ │ │ │ -000167f0: 6661 756c 7420 7661 6c75 6520 6661 6c73  fault value fals
│ │ │ │ -00016800: 650a 2020 2a20 4f75 7470 7574 733a 0a20  e.  * Outputs:. 
│ │ │ │ -00016810: 2020 2020 202a 2061 202a 6e6f 7465 206c       * a *note l
│ │ │ │ -00016820: 6973 743a 2028 4d61 6361 756c 6179 3244  ist: (Macaulay2D
│ │ │ │ -00016830: 6f63 294c 6973 742c 2c20 7468 6520 6669  oc)List,, the fi
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│ │ │ │ -00016850: 6973 7420 6f66 2072 6f77 0a20 2020 2020  ist of row.     
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│ │ │ │ -00016870: 7365 636f 6e64 2069 7320 6120 6c69 7374  second is a list
│ │ │ │ -00016880: 206f 6620 636f 6c75 6d6e 2069 6e64 6963   of column indic
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│ │ │ │ -000168a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6869  ===========..Thi
│ │ │ │ -000168b0: 7320 6675 6e63 7469 6f6e 206c 6f6f 6b73  s function looks
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│ │ │ │ -000168d0: 6f66 2074 6865 2067 6976 656e 206d 6174  of the given mat
│ │ │ │ -000168e0: 7269 782c 2061 6e64 2074 7269 6573 2074  rix, and tries t
│ │ │ │ -000168f0: 6f20 6669 6e64 206f 6e65 0a6f 6620 7468  o find one.of th
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│ │ │ │ -00016910: 2e20 2049 6620 6974 2073 7563 6365 6564  .  If it succeed
│ │ │ │ -00016920: 732c 2069 7420 7265 7475 726e 7320 6120  s, it returns a 
│ │ │ │ -00016930: 6c69 7374 206f 6620 7477 6f20 6c69 7374  list of two list
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│ │ │ │ -00016950: 7468 6520 6c69 7374 206f 6620 726f 7720  the list of row 
│ │ │ │ -00016960: 696e 6469 6365 732c 2074 6865 2073 6563  indices, the sec
│ │ │ │ -00016970: 6f6e 6420 6973 2074 6865 206c 6973 7420  ond is the list 
│ │ │ │ -00016980: 6f66 2063 6f6c 756d 6e73 2c20 6f66 2074  of columns, of t
│ │ │ │ -00016990: 6865 0a64 6573 6972 6564 2072 616e 6b20  he.desired rank 
│ │ │ │ -000169a0: 7375 626d 6174 7269 782e 2049 6620 6974  submatrix. If it
│ │ │ │ -000169b0: 2066 6169 6c73 2074 6f20 6669 6e64 2073   fails to find s
│ │ │ │ -000169c0: 7563 6820 6120 6d61 7472 6978 2c20 7468  uch a matrix, th
│ │ │ │ -000169d0: 6520 6675 6e63 7469 6f6e 2072 6574 7572  e function retur
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│ │ │ │ -000169f0: 7469 6f6e 204d 6178 4d69 6e6f 7273 2069  tion MaxMinors i
│ │ │ │ -00016a00: 7320 7573 6564 2074 6f20 636f 6e74 726f  s used to contro
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│ │ │ │ -00016a20: 7320 746f 2063 6f6e 7369 6465 722e 2020  s to consider.  
│ │ │ │ -00016a30: 4966 0a6c 6566 7420 6e75 6c6c 2c20 7468  If.left null, th
│ │ │ │ -00016a40: 6520 6e75 6d62 6572 2063 6f6e 7369 6465  e number conside
│ │ │ │ -00016a50: 7265 6420 6973 2062 6173 6564 206f 6e20  red is based on 
│ │ │ │ -00016a60: 7468 6520 7369 7a65 206f 6620 7468 6520  the size of the 
│ │ │ │ -00016a70: 6d61 7472 6978 2e0a 0a2b 2d2d 2d2d 2d2d  matrix...+------
│ │ │ │ -00016a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016ab0: 2d2d 2d2d 2d2b 0a7c 6931 203a 2052 203d  -----+.|i1 : R =
│ │ │ │ -00016ac0: 2051 515b 782c 795d 3b20 2020 2020 2020   QQ[x,y];       
│ │ │ │ -00016ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016af0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -00016b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016b30: 2d2b 0a7c 6932 203a 204d 203d 206d 6174  -+.|i2 : M = mat
│ │ │ │ -00016b40: 7269 787b 7b78 2c79 2c32 2c30 2c32 2a78  rix{{x,y,2,0,2*x
│ │ │ │ -00016b50: 2b79 7d2c 207b 302c 302c 312c 302c 787d  +y}, {0,0,1,0,x}
│ │ │ │ -00016b60: 2c20 7b78 2c79 2c30 2c30 2c79 7d7d 3b7c  , {x,y,0,0,y}};|
│ │ │ │ -00016b70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00016b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ba0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00016bb0: 2020 2020 2020 2020 2020 2020 2033 2020               3  
│ │ │ │ -00016bc0: 2020 2020 3520 2020 2020 2020 2020 2020      5           
│ │ │ │ +00015c50: 2020 2020 207c 0a7c 6f35 3520 3d20 7472       |.|o55 = tr
│ │ │ │ +00015c60: 7565 2020 2020 2020 2020 2020 2020 2020  ue              
│ │ │ │ +00015c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015ca0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +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 2d7c 0a7c 5374 7261 7465 6779  -----|.|Strategy
│ │ │ │ +00015d00: 3d3e 5374 7261 7465 6779 4465 6661 756c  =>StrategyDefaul
│ │ │ │ +00015d10: 7457 6974 6850 6f69 6e74 7329 2020 2020  tWithPoints)    
│ │ │ │ +00015d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015d40: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00015d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015d90: 2d2d 2d2d 2d2b 0a0a 4966 2072 6567 756c  -----+..If regul
│ │ │ │ +00015da0: 6172 496e 436f 6469 6d65 6e73 696f 6e20  arInCodimension 
│ │ │ │ +00015db0: 6f75 7470 7574 7320 6e6f 7468 696e 672c  outputs nothing,
│ │ │ │ +00015dc0: 2074 6865 6e20 6974 2063 6f75 6c64 6e27   then it couldn'
│ │ │ │ +00015dd0: 7420 7665 7269 6679 2074 6861 7420 7468  t verify that th
│ │ │ │ +00015de0: 6520 7269 6e67 0a77 6173 2072 6567 756c  e ring.was regul
│ │ │ │ +00015df0: 6172 2069 6e20 7468 6174 2063 6f64 696d  ar in that codim
│ │ │ │ +00015e00: 656e 7369 6f6e 2e20 2057 6520 7365 7420  ension.  We set 
│ │ │ │ +00015e10: 4d61 784d 696e 6f72 7320 3d3e 2031 3030  MaxMinors => 100
│ │ │ │ +00015e20: 2074 6f20 6b65 6570 2069 7420 6672 6f6d   to keep it from
│ │ │ │ +00015e30: 0a72 756e 6e69 6e67 2074 6f6f 206c 6f6e  .running too lon
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│ │ │ │ +00015f60: 2063 686f 6f73 6547 6f6f 644d 696e 6f72   chooseGoodMinor
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│ │ │ │ +00015f80: 2e2e 2e29 3a20 5374 7261 7465 6779 4465  ...): StrategyDe
│ │ │ │ +00015f90: 6661 756c 742c 202d 2d20 7374 7261 7465  fault, -- strate
│ │ │ │ +00015fa0: 6769 6573 0a20 2020 2066 6f72 2063 686f  gies.    for cho
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│ │ │ │ +00015fc0: 730a 2020 2a20 2a6e 6f74 6520 5265 6775  s.  * *note Regu
│ │ │ │ +00015fd0: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ +00015fe0: 5475 746f 7269 616c 3a20 5265 6775 6c61  Tutorial: Regula
│ │ │ │ +00015ff0: 7249 6e43 6f64 696d 656e 7369 6f6e 5475  rInCodimensionTu
│ │ │ │ +00016000: 746f 7269 616c 2c20 2d2d 2041 0a20 2020  torial, -- A.   
│ │ │ │ +00016010: 2074 7574 6f72 6961 6c20 666f 7220 686f   tutorial for ho
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│ │ │ │ +00016030: 616e 6365 6420 6f70 7469 6f6e 7320 6f66  anced options of
│ │ │ │ +00016040: 2074 6865 2072 6567 756c 6172 496e 436f   the regularInCo
│ │ │ │ +00016050: 6469 6d65 6e73 696f 6e0a 2020 2020 6675  dimension.    fu
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│ │ │ │ +00016070: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d  programmer.=====
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│ │ │ │ +000160b0: 6779 5475 746f 7269 616c 3a20 4661 7374  gyTutorial: Fast
│ │ │ │ +000160c0: 4d69 6e6f 7273 5374 7261 7465 6779 5475  MinorsStrategyTu
│ │ │ │ +000160d0: 746f 7269 616c 2c20 6973 2061 0a2a 6e6f  torial, is a.*no
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│ │ │ │ +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  ----------------
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│ │ │ │ +000161f0: 6765 7453 7562 6d61 7472 6978 4f66 5261  getSubmatrixOfRa
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│ │ │ │ +00016210: 6d41 744c 6561 7374 2c20 5072 6576 3a20  mAtLeast, Prev: 
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│ │ │ │ +00016250: 784f 6652 616e 6b20 2d2d 2074 7269 6573  xOfRank -- tries
│ │ │ │ +00016260: 2074 6f20 6669 6e64 2061 2073 7562 6d61   to find a subma
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│ │ │ │ +00016280: 6e20 7261 6e6b 0a2a 2a2a 2a2a 2a2a 2a2a  n rank.*********
│ │ │ │ +00016290: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000162a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000162b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000162c0: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573  ********..  * Us
│ │ │ │ +000162d0: 6167 653a 200a 2020 2020 2020 2020 6765  age: .        ge
│ │ │ │ +000162e0: 7453 7562 6d61 7472 6978 4f66 5261 6e6b  tSubmatrixOfRank
│ │ │ │ +000162f0: 286e 312c 204d 3129 0a20 202a 2049 6e70  (n1, M1).  * Inp
│ │ │ │ +00016300: 7574 733a 0a20 2020 2020 202a 206e 312c  uts:.      * n1,
│ │ │ │ +00016310: 2061 6e20 2a6e 6f74 6520 696e 7465 6765   an *note intege
│ │ │ │ +00016320: 723a 2028 4d61 6361 756c 6179 3244 6f63  r: (Macaulay2Doc
│ │ │ │ +00016330: 295a 5a2c 2c20 0a20 2020 2020 202a 204d  )ZZ,, .      * M
│ │ │ │ +00016340: 312c 2061 202a 6e6f 7465 206d 6174 7269  1, a *note matri
│ │ │ │ +00016350: 783a 2028 4d61 6361 756c 6179 3244 6f63  x: (Macaulay2Doc
│ │ │ │ +00016360: 294d 6174 7269 782c 2c20 0a20 202a 202a  )Matrix,, .  * *
│ │ │ │ +00016370: 6e6f 7465 204f 7074 696f 6e61 6c20 696e  note Optional in
│ │ │ │ +00016380: 7075 7473 3a20 284d 6163 6175 6c61 7932  puts: (Macaulay2
│ │ │ │ +00016390: 446f 6329 7573 696e 6720 6675 6e63 7469  Doc)using functi
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│ │ │ │ +000163c0: 202a 202a 6e6f 7465 2044 6574 5374 7261   * *note DetStra
│ │ │ │ +000163d0: 7465 6779 3a20 4465 7453 7472 6174 6567  tegy: DetStrateg
│ │ │ │ +000163e0: 792c 203d 3e20 2e2e 2e2c 2064 6566 6175  y, => ..., defau
│ │ │ │ +000163f0: 6c74 2076 616c 7565 2052 616e 6b2c 2044  lt value Rank, D
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│ │ │ │ +00016460: 6e6b 292c 2069 7320 636f 6d70 7574 6564  nk), is computed
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│ │ │ │ +000164e0: 2020 2020 2a20 2a6e 6f74 6520 506f 696e      * *note Poin
│ │ │ │ +000164f0: 744f 7074 696f 6e73 3a20 506f 696e 744f  tOptions: PointO
│ │ │ │ +00016500: 7074 696f 6e73 2c20 3d3e 202e 2e2e 2c20  ptions, => ..., 
│ │ │ │ +00016510: 6465 6661 756c 7420 7661 6c75 6520 7b53  default value {S
│ │ │ │ +00016520: 7472 6174 6567 7920 3d3e 0a20 2020 2020  trategy =>.     
│ │ │ │ +00016530: 2020 2044 6566 6175 6c74 2c20 486f 6d6f     Default, Homo
│ │ │ │ +00016540: 6765 6e65 6f75 7320 3d3e 2066 616c 7365  geneous => false
│ │ │ │ +00016550: 2c20 5265 706c 6163 656d 656e 7420 3d3e  , Replacement =>
│ │ │ │ +00016560: 2042 696e 6f6d 6961 6c2c 2045 7874 656e   Binomial, Exten
│ │ │ │ +00016570: 6446 6965 6c64 203d 3e0a 2020 2020 2020  dField =>.      
│ │ │ │ +00016580: 2020 7472 7565 2c20 506f 696e 7443 6865    true, PointChe
│ │ │ │ +00016590: 636b 4174 7465 6d70 7473 203d 3e20 302c  ckAttempts => 0,
│ │ │ │ +000165a0: 2044 6563 6f6d 706f 7369 7469 6f6e 5374   DecompositionSt
│ │ │ │ +000165b0: 7261 7465 6779 203d 3e20 4465 636f 6d70  rategy => Decomp
│ │ │ │ +000165c0: 6f73 652c 0a20 2020 2020 2020 204e 756d  ose,.        Num
│ │ │ │ +000165d0: 5468 7265 6164 7354 6f55 7365 203d 3e20  ThreadsToUse => 
│ │ │ │ +000165e0: 312c 2044 696d 656e 7369 6f6e 4675 6e63  1, DimensionFunc
│ │ │ │ +000165f0: 7469 6f6e 203d 3e20 6469 6d2c 2056 6572  tion => dim, Ver
│ │ │ │ +00016600: 626f 7365 203d 3e20 6661 6c73 657d 2c0a  bose => false},.
│ │ │ │ +00016610: 2020 2020 2020 2020 6f70 7469 6f6e 7320          options 
│ │ │ │ +00016620: 746f 2070 6173 7320 746f 2066 756e 6374  to pass to funct
│ │ │ │ +00016630: 696f 6e73 2069 6e20 7468 6520 7061 636b  ions in the pack
│ │ │ │ +00016640: 6167 6520 5261 6e64 6f6d 506f 696e 7473  age RandomPoints
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│ │ │ │ +00016660: 7472 6174 6567 793a 2053 7472 6174 6567  trategy: Strateg
│ │ │ │ +00016670: 7944 6566 6175 6c74 2c20 3d3e 202e 2e2e  yDefault, => ...
│ │ │ │ +00016680: 2c20 6465 6661 756c 7420 7661 6c75 6520  , default value 
│ │ │ │ +00016690: 6e65 7720 4f70 7469 6f6e 5461 626c 650a  new OptionTable.
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│ │ │ │ +000166c0: 6d20 3d3e 2030 2c20 4752 6576 4c65 784c  m => 0, GRevLexL
│ │ │ │ +000166d0: 6172 6765 7374 203d 3e20 302c 204c 6578  argest => 0, Lex
│ │ │ │ +000166e0: 536d 616c 6c65 7374 5465 726d 203d 3e0a  SmallestTerm =>.
│ │ │ │ +000166f0: 2020 2020 2020 2020 3235 2c20 4c65 784c          25, LexL
│ │ │ │ +00016700: 6172 6765 7374 203d 3e20 302c 204c 6578  argest => 0, Lex
│ │ │ │ +00016710: 536d 616c 6c65 7374 203d 3e20 3235 2c20  Smallest => 25, 
│ │ │ │ +00016720: 4752 6576 4c65 7853 6d61 6c6c 6573 7454  GRevLexSmallestT
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│ │ │ │ +00016740: 2020 2052 616e 646f 6d4e 6f6e 7a65 726f     RandomNonzero
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│ │ │ │ +00016b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016b80: 2d2b 0a7c 6932 203a 204d 203d 206d 6174  -+.|i2 : M = mat
│ │ │ │ +00016b90: 7269 787b 7b78 2c79 2c32 2c30 2c32 2a78  rix{{x,y,2,0,2*x
│ │ │ │ +00016ba0: 2b79 7d2c 207b 302c 302c 312c 302c 787d  +y}, {0,0,1,0,x}
│ │ │ │ +00016bb0: 2c20 7b78 2c79 2c30 2c30 2c79 7d7d 3b7c  , {x,y,0,0,y}};|
│ │ │ │ +00016bc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00016bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016be0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ -00016bf0: 203a 204d 6174 7269 7820 5220 203c 2d2d   : Matrix R  <--
│ │ │ │ -00016c00: 2052 2020 2020 2020 2020 2020 2020 2020   R              
│ │ │ │ -00016c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016c20: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -00016c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016c60: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 206c  -------+.|i3 : l
│ │ │ │ -00016c70: 203d 2067 6574 5375 626d 6174 7269 784f   = getSubmatrixO
│ │ │ │ -00016c80: 6652 616e 6b28 322c 204d 2920 2020 2020  fRank(2, M)     
│ │ │ │ -00016c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ca0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00016cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ce0: 2020 207c 0a7c 6f33 203d 207b 7b31 2c20     |.|o3 = {{1, 
│ │ │ │ -00016cf0: 327d 2c20 7b32 2c20 317d 7d20 2020 2020  2}, {2, 1}}     
│ │ │ │ +00016be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016bf0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00016c00: 2020 2020 2020 2020 2020 2020 2033 2020               3  
│ │ │ │ +00016c10: 2020 2020 3520 2020 2020 2020 2020 2020      5           
│ │ │ │ +00016c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016c30: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +00016c40: 203a 204d 6174 7269 7820 5220 203c 2d2d   : Matrix R  <--
│ │ │ │ +00016c50: 2052 2020 2020 2020 2020 2020 2020 2020   R              
│ │ │ │ +00016c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016c70: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00016c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016cb0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 206c  -------+.|i3 : l
│ │ │ │ +00016cc0: 203d 2067 6574 5375 626d 6174 7269 784f   = getSubmatrixO
│ │ │ │ +00016cd0: 6652 616e 6b28 322c 204d 2920 2020 2020  fRank(2, M)     
│ │ │ │ +00016ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016cf0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00016d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016d20: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00016d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016d50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016d60: 0a7c 6f33 203a 204c 6973 7420 2020 2020  .|o3 : List     
│ │ │ │ -00016d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016d30: 2020 207c 0a7c 6f33 203d 207b 7b31 2c20     |.|o3 = {{1, 
│ │ │ │ +00016d40: 327d 2c20 7b32 2c20 317d 7d20 2020 2020  2}, {2, 1}}     
│ │ │ │ +00016d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016d70: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00016d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016d90: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00016da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934  -----------+.|i4
│ │ │ │ -00016de0: 203a 2028 4d5e 286c 2330 2929 5f28 6c23   : (M^(l#0))_(l#
│ │ │ │ -00016df0: 3129 2020 2020 2020 2020 2020 2020 2020  1)              
│ │ │ │ -00016e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016e10: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00016e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016e50: 2020 2020 2020 207c 0a7c 6f34 203d 207c         |.|o4 = |
│ │ │ │ -00016e60: 2031 2030 207c 2020 2020 2020 2020 2020   1 0 |          
│ │ │ │ +00016d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016da0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00016db0: 0a7c 6f33 203a 204c 6973 7420 2020 2020  .|o3 : List     
│ │ │ │ +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 207c 0a2b               |.+
│ │ │ │ +00016df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934  -----------+.|i4
│ │ │ │ +00016e30: 203a 2028 4d5e 286c 2330 2929 5f28 6c23   : (M^(l#0))_(l#
│ │ │ │ +00016e40: 3129 2020 2020 2020 2020 2020 2020 2020  1)              
│ │ │ │ +00016e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016e60: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00016e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016e90: 2020 2020 207c 0a7c 2020 2020 207c 2030       |.|     | 0
│ │ │ │ -00016ea0: 2079 207c 2020 2020 2020 2020 2020 2020   y |            
│ │ │ │ -00016eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016ea0: 2020 2020 2020 207c 0a7c 6f34 203d 207c         |.|o4 = |
│ │ │ │ +00016eb0: 2031 2030 207c 2020 2020 2020 2020 2020   1 0 |          
│ │ │ │  00016ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ed0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00016ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016ee0: 2020 2020 207c 0a7c 2020 2020 207c 2030       |.|     | 0
│ │ │ │ +00016ef0: 2079 207c 2020 2020 2020 2020 2020 2020   y |            
│ │ │ │  00016f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016f10: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00016f20: 2032 2020 2020 2020 3220 2020 2020 2020   2      2       
│ │ │ │ +00016f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016f20: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00016f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016f40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016f50: 0a7c 6f34 203a 204d 6174 7269 7820 5220  .|o4 : Matrix R 
│ │ │ │ -00016f60: 203c 2d2d 2052 2020 2020 2020 2020 2020   <-- R          
│ │ │ │ -00016f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016f80: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00016f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935  -----------+.|i5
│ │ │ │ -00016fd0: 203a 206c 203d 2067 6574 5375 626d 6174   : l = getSubmat
│ │ │ │ -00016fe0: 7269 784f 6652 616e 6b28 322c 204d 2920  rixOfRank(2, M) 
│ │ │ │ -00016ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017000: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00017010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017040: 2020 2020 2020 207c 0a7c 6f35 203d 207b         |.|o5 = {
│ │ │ │ -00017050: 7b31 2c20 307d 2c20 7b32 2c20 317d 7d20  {1, 0}, {2, 1}} 
│ │ │ │ +00016f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016f60: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00016f70: 2032 2020 2020 2020 3220 2020 2020 2020   2      2       
│ │ │ │ +00016f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016f90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00016fa0: 0a7c 6f34 203a 204d 6174 7269 7820 5220  .|o4 : Matrix R 
│ │ │ │ +00016fb0: 203c 2d2d 2052 2020 2020 2020 2020 2020   <-- R          
│ │ │ │ +00016fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016fd0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00016fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935  -----------+.|i5
│ │ │ │ +00017020: 203a 206c 203d 2067 6574 5375 626d 6174   : l = getSubmat
│ │ │ │ +00017030: 7269 784f 6652 616e 6b28 322c 204d 2920  rixOfRank(2, M) 
│ │ │ │ +00017040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017050: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00017060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017080: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00017090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000170a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017090: 2020 2020 2020 207c 0a7c 6f35 203d 207b         |.|o5 = {
│ │ │ │ +000170a0: 7b31 2c20 307d 2c20 7b32 2c20 317d 7d20  {1, 0}, {2, 1}} 
│ │ │ │  000170b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000170c0: 2020 207c 0a7c 6f35 203a 204c 6973 7420     |.|o5 : List 
│ │ │ │ -000170d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000170c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000170d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000170e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000170f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017100: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ -00017110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00017140: 0a7c 6936 203a 2028 4d5e 286c 2330 2929  .|i6 : (M^(l#0))
│ │ │ │ -00017150: 5f28 6c23 3129 2020 2020 2020 2020 2020  _(l#1)          
│ │ │ │ -00017160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017170: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00017180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000171a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000171b0: 2020 2020 2020 2020 2020 207c 0a7c 6f36             |.|o6
│ │ │ │ -000171c0: 203d 207c 2031 2030 207c 2020 2020 2020   = | 1 0 |      
│ │ │ │ +00017100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017110: 2020 207c 0a7c 6f35 203a 204c 6973 7420     |.|o5 : List 
│ │ │ │ +00017120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017150: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00017160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00017190: 0a7c 6936 203a 2028 4d5e 286c 2330 2929  .|i6 : (M^(l#0))
│ │ │ │ +000171a0: 5f28 6c23 3129 2020 2020 2020 2020 2020  _(l#1)          
│ │ │ │ +000171b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000171c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  000171d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000171e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000171f0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00017200: 207c 2032 2079 207c 2020 2020 2020 2020   | 2 y |        
│ │ │ │ -00017210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000171f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017200: 2020 2020 2020 2020 2020 207c 0a7c 6f36             |.|o6
│ │ │ │ +00017210: 203d 207c 2031 2030 207c 2020 2020 2020   = | 1 0 |      
│ │ │ │  00017220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017230: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00017240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017240: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00017250: 207c 2032 2079 207c 2020 2020 2020 2020   | 2 y |        
│ │ │ │  00017260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017270: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00017280: 2020 2020 2032 2020 2020 2020 3220 2020       2      2   
│ │ │ │ +00017270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017280: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00017290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000172a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000172b0: 2020 207c 0a7c 6f36 203a 204d 6174 7269     |.|o6 : Matri
│ │ │ │ -000172c0: 7820 5220 203c 2d2d 2052 2020 2020 2020  x R  <-- R      
│ │ │ │ -000172d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000172b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000172c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000172d0: 2020 2020 2032 2020 2020 2020 3220 2020       2      2   
│ │ │ │  000172e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000172f0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ -00017300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00017330: 0a7c 6937 203a 2067 6574 5375 626d 6174  .|i7 : getSubmat
│ │ │ │ -00017340: 7269 784f 6652 616e 6b28 332c 204d 2920  rixOfRank(3, M) 
│ │ │ │ -00017350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017360: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00017370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000173a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5468  -----------+..Th
│ │ │ │ -000173b0: 6520 6f70 7469 6f6e 2053 7472 6174 6567  e option Strateg
│ │ │ │ -000173c0: 7920 6973 2075 7365 6420 746f 2075 7365  y is used to use
│ │ │ │ -000173d0: 6420 746f 2063 6f6e 7472 6f6c 2068 6f77  d to control how
│ │ │ │ -000173e0: 2074 6865 2066 756e 6374 696f 6e20 636f   the function co
│ │ │ │ -000173f0: 6d70 7574 6573 2074 6865 0a72 616e 6b20  mputes the.rank 
│ │ │ │ -00017400: 6f66 2074 6865 2073 7562 6d61 7472 6963  of the submatric
│ │ │ │ -00017410: 6573 2063 6f6e 7369 6465 7265 642e 2020  es considered.  
│ │ │ │ -00017420: 5365 6520 2a6e 6f74 650a 6765 7453 7562  See *note.getSub
│ │ │ │ -00017430: 6d61 7472 6978 4f66 5261 6e6b 282e 2e2e  matrixOfRank(...
│ │ │ │ -00017440: 2c53 7472 6174 6567 793d 3e2e 2e2e 293a  ,Strategy=>...):
│ │ │ │ -00017450: 2053 7472 6174 6567 7944 6566 6175 6c74   StrategyDefault
│ │ │ │ -00017460: 2c2e 2049 6e20 7468 6520 6675 7475 7265  ,. In the future
│ │ │ │ -00017470: 2c20 7765 2068 6f70 650a 746f 2073 7065  , we hope.to spe
│ │ │ │ -00017480: 6564 2075 7020 7468 6520 6675 6e63 7469  ed up the functi
│ │ │ │ -00017490: 6f6e 2074 6f20 7573 6520 6d75 6c74 6970  on to use multip
│ │ │ │ -000174a0: 6c65 2074 6872 6561 6473 206f 6620 6578  le threads of ex
│ │ │ │ -000174b0: 6563 7574 696f 6e2c 2069 6e20 7768 6963  ecution, in whic
│ │ │ │ -000174c0: 6820 6361 7365 0a74 6865 2074 6872 6561  h case.the threa
│ │ │ │ -000174d0: 6469 6e67 2077 6f75 6c64 2062 6520 636f  ding would be co
│ │ │ │ -000174e0: 6e74 726f 6c6c 6564 2062 7920 7468 6520  ntrolled by the 
│ │ │ │ -000174f0: 6f70 7469 6f6e 2054 6872 6561 6473 2e0a  option Threads..
│ │ │ │ -00017500: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d  .See also.======
│ │ │ │ -00017510: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6973  ==..  * *note is
│ │ │ │ -00017520: 5261 6e6b 4174 4c65 6173 743a 2069 7352  RankAtLeast: isR
│ │ │ │ -00017530: 616e 6b41 744c 6561 7374 2c20 2d2d 2064  ankAtLeast, -- d
│ │ │ │ -00017540: 6574 6572 6d69 6e65 7320 6966 2074 6865  etermines if the
│ │ │ │ -00017550: 206d 6174 7269 7820 6861 7320 7261 6e6b   matrix has rank
│ │ │ │ -00017560: 2061 740a 2020 2020 6c65 6173 7420 6120   at.    least a 
│ │ │ │ -00017570: 6e75 6d62 6572 0a20 202a 202a 6e6f 7465  number.  * *note
│ │ │ │ -00017580: 2067 6574 5375 626d 6174 7269 784f 6652   getSubmatrixOfR
│ │ │ │ -00017590: 616e 6b28 2e2e 2e2c 5374 7261 7465 6779  ank(...,Strategy
│ │ │ │ -000175a0: 3d3e 2e2e 2e29 3a20 5374 7261 7465 6779  =>...): Strategy
│ │ │ │ -000175b0: 4465 6661 756c 742c 202d 2d20 7374 7261  Default, -- stra
│ │ │ │ -000175c0: 7465 6769 6573 0a20 2020 2066 6f72 2063  tegies.    for c
│ │ │ │ -000175d0: 686f 6f73 696e 6720 7375 626d 6174 7269  hoosing submatri
│ │ │ │ -000175e0: 6365 730a 0a57 6179 7320 746f 2075 7365  ces..Ways to use
│ │ │ │ -000175f0: 2067 6574 5375 626d 6174 7269 784f 6652   getSubmatrixOfR
│ │ │ │ -00017600: 616e 6b3a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  ank:.===========
│ │ │ │ -00017610: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00017620: 3d3d 3d3d 0a0a 2020 2a20 2267 6574 5375  ====..  * "getSu
│ │ │ │ -00017630: 626d 6174 7269 784f 6652 616e 6b28 5a5a  bmatrixOfRank(ZZ
│ │ │ │ -00017640: 2c4d 6174 7269 7829 220a 0a46 6f72 2074  ,Matrix)"..For t
│ │ │ │ -00017650: 6865 2070 726f 6772 616d 6d65 720a 3d3d  he programmer.==
│ │ │ │ +000172f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017300: 2020 207c 0a7c 6f36 203a 204d 6174 7269     |.|o6 : Matri
│ │ │ │ +00017310: 7820 5220 203c 2d2d 2052 2020 2020 2020  x R  <-- R      
│ │ │ │ +00017320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017340: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00017350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00017380: 0a7c 6937 203a 2067 6574 5375 626d 6174  .|i7 : getSubmat
│ │ │ │ +00017390: 7269 784f 6652 616e 6b28 332c 204d 2920  rixOfRank(3, M) 
│ │ │ │ +000173a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000173b0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +000173c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000173d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000173e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000173f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5468  -----------+..Th
│ │ │ │ +00017400: 6520 6f70 7469 6f6e 2053 7472 6174 6567  e option Strateg
│ │ │ │ +00017410: 7920 6973 2075 7365 6420 746f 2075 7365  y is used to use
│ │ │ │ +00017420: 6420 746f 2063 6f6e 7472 6f6c 2068 6f77  d to control how
│ │ │ │ +00017430: 2074 6865 2066 756e 6374 696f 6e20 636f   the function co
│ │ │ │ +00017440: 6d70 7574 6573 2074 6865 0a72 616e 6b20  mputes the.rank 
│ │ │ │ +00017450: 6f66 2074 6865 2073 7562 6d61 7472 6963  of the submatric
│ │ │ │ +00017460: 6573 2063 6f6e 7369 6465 7265 642e 2020  es considered.  
│ │ │ │ +00017470: 5365 6520 2a6e 6f74 650a 6765 7453 7562  See *note.getSub
│ │ │ │ +00017480: 6d61 7472 6978 4f66 5261 6e6b 282e 2e2e  matrixOfRank(...
│ │ │ │ +00017490: 2c53 7472 6174 6567 793d 3e2e 2e2e 293a  ,Strategy=>...):
│ │ │ │ +000174a0: 2053 7472 6174 6567 7944 6566 6175 6c74   StrategyDefault
│ │ │ │ +000174b0: 2c2e 2049 6e20 7468 6520 6675 7475 7265  ,. In the future
│ │ │ │ +000174c0: 2c20 7765 2068 6f70 650a 746f 2073 7065  , we hope.to spe
│ │ │ │ +000174d0: 6564 2075 7020 7468 6520 6675 6e63 7469  ed up the functi
│ │ │ │ +000174e0: 6f6e 2074 6f20 7573 6520 6d75 6c74 6970  on to use multip
│ │ │ │ +000174f0: 6c65 2074 6872 6561 6473 206f 6620 6578  le threads of ex
│ │ │ │ +00017500: 6563 7574 696f 6e2c 2069 6e20 7768 6963  ecution, in whic
│ │ │ │ +00017510: 6820 6361 7365 0a74 6865 2074 6872 6561  h case.the threa
│ │ │ │ +00017520: 6469 6e67 2077 6f75 6c64 2062 6520 636f  ding would be co
│ │ │ │ +00017530: 6e74 726f 6c6c 6564 2062 7920 7468 6520  ntrolled by the 
│ │ │ │ +00017540: 6f70 7469 6f6e 2054 6872 6561 6473 2e0a  option Threads..
│ │ │ │ +00017550: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d  .See also.======
│ │ │ │ +00017560: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6973  ==..  * *note is
│ │ │ │ +00017570: 5261 6e6b 4174 4c65 6173 743a 2069 7352  RankAtLeast: isR
│ │ │ │ +00017580: 616e 6b41 744c 6561 7374 2c20 2d2d 2064  ankAtLeast, -- d
│ │ │ │ +00017590: 6574 6572 6d69 6e65 7320 6966 2074 6865  etermines if the
│ │ │ │ +000175a0: 206d 6174 7269 7820 6861 7320 7261 6e6b   matrix has rank
│ │ │ │ +000175b0: 2061 740a 2020 2020 6c65 6173 7420 6120   at.    least a 
│ │ │ │ +000175c0: 6e75 6d62 6572 0a20 202a 202a 6e6f 7465  number.  * *note
│ │ │ │ +000175d0: 2067 6574 5375 626d 6174 7269 784f 6652   getSubmatrixOfR
│ │ │ │ +000175e0: 616e 6b28 2e2e 2e2c 5374 7261 7465 6779  ank(...,Strategy
│ │ │ │ +000175f0: 3d3e 2e2e 2e29 3a20 5374 7261 7465 6779  =>...): Strategy
│ │ │ │ +00017600: 4465 6661 756c 742c 202d 2d20 7374 7261  Default, -- stra
│ │ │ │ +00017610: 7465 6769 6573 0a20 2020 2066 6f72 2063  tegies.    for c
│ │ │ │ +00017620: 686f 6f73 696e 6720 7375 626d 6174 7269  hoosing submatri
│ │ │ │ +00017630: 6365 730a 0a57 6179 7320 746f 2075 7365  ces..Ways to use
│ │ │ │ +00017640: 2067 6574 5375 626d 6174 7269 784f 6652   getSubmatrixOfR
│ │ │ │ +00017650: 616e 6b3a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  ank:.===========
│ │ │ │  00017660: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00017670: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f  ..The object *no
│ │ │ │ -00017680: 7465 2067 6574 5375 626d 6174 7269 784f  te getSubmatrixO
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│ │ │ │ -000176a0: 7269 784f 6652 616e 6b2c 2069 7320 6120  rixOfRank, is a 
│ │ │ │ -000176b0: 2a6e 6f74 6520 6d65 7468 6f64 0a66 756e  *note method.fun
│ │ │ │ -000176c0: 6374 696f 6e20 7769 7468 206f 7074 696f  ction with optio
│ │ │ │ -000176d0: 6e73 3a20 284d 6163 6175 6c61 7932 446f  ns: (Macaulay2Do
│ │ │ │ -000176e0: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e  c)MethodFunction
│ │ │ │ -000176f0: 5769 7468 4f70 7469 6f6e 732c 2e0a 0a2d  WithOptions,...-
│ │ │ │ -00017700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a  --------------..
│ │ │ │ -00017750: 5468 6520 736f 7572 6365 206f 6620 7468  The source of th
│ │ │ │ -00017760: 6973 2064 6f63 756d 656e 7420 6973 2069  is document is i
│ │ │ │ -00017770: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475  n./build/reprodu
│ │ │ │ -00017780: 6369 626c 652d 7061 7468 2f6d 6163 6175  cible-path/macau
│ │ │ │ -00017790: 6c61 7932 2d31 2e32 352e 3131 2b64 732f  lay2-1.25.11+ds/
│ │ │ │ -000177a0: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163  M2/Macaulay2/pac
│ │ │ │ -000177b0: 6b61 6765 732f 4661 7374 4d69 6e6f 7273  kages/FastMinors
│ │ │ │ -000177c0: 2e0a 6d32 3a31 3737 323a 302e 0a1f 0a46  ..m2:1772:0....F
│ │ │ │ -000177d0: 696c 653a 2046 6173 744d 696e 6f72 732e  ile: FastMinors.
│ │ │ │ -000177e0: 696e 666f 2c20 4e6f 6465 3a20 6973 436f  info, Node: isCo
│ │ │ │ -000177f0: 6469 6d41 744c 6561 7374 2c20 4e65 7874  dimAtLeast, Next
│ │ │ │ -00017800: 3a20 6973 4469 6d41 744d 6f73 742c 2050  : isDimAtMost, P
│ │ │ │ -00017810: 7265 763a 2067 6574 5375 626d 6174 7269  rev: getSubmatri
│ │ │ │ -00017820: 784f 6652 616e 6b2c 2055 703a 2054 6f70  xOfRank, Up: Top
│ │ │ │ -00017830: 0a0a 6973 436f 6469 6d41 744c 6561 7374  ..isCodimAtLeast
│ │ │ │ -00017840: 202d 2d20 7265 7475 726e 7320 7472 7565   -- returns true
│ │ │ │ -00017850: 2069 6620 7765 2063 616e 2071 7569 636b   if we can quick
│ │ │ │ -00017860: 6c79 2073 6565 2077 6865 7468 6572 2074  ly see whether t
│ │ │ │ -00017870: 6865 2063 6f64 696d 2069 7320 6174 206c  he codim is at l
│ │ │ │ -00017880: 6561 7374 2061 2067 6976 656e 206e 756d  east a given num
│ │ │ │ -00017890: 6265 720a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ber.************
│ │ │ │ -000178a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000178b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000178c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000178d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000178e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000178f0: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765  *****..  * Usage
│ │ │ │ -00017900: 3a20 0a20 2020 2020 2020 2069 7343 6f64  : .        isCod
│ │ │ │ -00017910: 696d 4174 4c65 6173 7428 6e2c 2049 290a  imAtLeast(n, I).
│ │ │ │ -00017920: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ -00017930: 2020 2a20 6e2c 2061 6e20 2a6e 6f74 6520    * n, an *note 
│ │ │ │ -00017940: 696e 7465 6765 723a 2028 4d61 6361 756c  integer: (Macaul
│ │ │ │ -00017950: 6179 3244 6f63 295a 5a2c 2c20 616e 2069  ay2Doc)ZZ,, an i
│ │ │ │ -00017960: 6e74 6567 6572 0a20 2020 2020 202a 2049  nteger.      * I
│ │ │ │ -00017970: 2c20 616e 202a 6e6f 7465 2069 6465 616c  , an *note ideal
│ │ │ │ -00017980: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ -00017990: 4964 6561 6c2c 2c20 616e 2069 6465 616c  Ideal,, an ideal
│ │ │ │ -000179a0: 2069 6e20 6120 706f 6c79 6e6f 6d69 616c   in a polynomial
│ │ │ │ -000179b0: 2072 696e 670a 2020 2020 2020 2020 6f76   ring.        ov
│ │ │ │ -000179c0: 6572 2061 2066 6965 6c64 2c20 6f72 2061  er a field, or a
│ │ │ │ -000179d0: 2071 756f 7469 656e 7420 7269 6e67 0a20   quotient ring. 
│ │ │ │ -000179e0: 202a 202a 6e6f 7465 204f 7074 696f 6e61   * *note Optiona
│ │ │ │ -000179f0: 6c20 696e 7075 7473 3a20 284d 6163 6175  l inputs: (Macau
│ │ │ │ -00017a00: 6c61 7932 446f 6329 7573 696e 6720 6675  lay2Doc)using fu
│ │ │ │ -00017a10: 6e63 7469 6f6e 7320 7769 7468 206f 7074  nctions with opt
│ │ │ │ -00017a20: 696f 6e61 6c20 696e 7075 7473 2c3a 0a20  ional inputs,:. 
│ │ │ │ -00017a30: 2020 2020 202a 2053 5061 6972 7346 756e       * SPairsFun
│ │ │ │ -00017a40: 6374 696f 6e20 3d3e 2061 202a 6e6f 7465  ction => a *note
│ │ │ │ -00017a50: 2066 756e 6374 696f 6e3a 2028 4d61 6361   function: (Maca
│ │ │ │ -00017a60: 756c 6179 3244 6f63 2946 756e 6374 696f  ulay2Doc)Functio
│ │ │ │ -00017a70: 6e2c 2c20 6465 6661 756c 740a 2020 2020  n,, default.    
│ │ │ │ -00017a80: 2020 2020 7661 6c75 6520 4675 6e63 7469      value Functi
│ │ │ │ -00017a90: 6f6e 436c 6f73 7572 655b 2e2e 2f46 6173  onClosure[../Fas
│ │ │ │ -00017aa0: 744d 696e 6f72 732e 6d32 3a32 3131 3a32  tMinors.m2:211:2
│ │ │ │ -00017ab0: 332d 3231 313a 3432 5d2c 2061 2066 756e  3-211:42], a fun
│ │ │ │ -00017ac0: 6374 696f 6e20 746f 0a20 2020 2020 2020  ction to.       
│ │ │ │ -00017ad0: 2063 6f6e 7472 6f6c 2068 6f77 2077 6865   control how whe
│ │ │ │ -00017ae0: 6e20 7468 6520 636f 6469 6d65 6e73 696f  n the codimensio
│ │ │ │ -00017af0: 6e20 6f66 206d 696e 6f72 7320 6973 2063  n of minors is c
│ │ │ │ -00017b00: 6f6d 7075 7465 642c 2064 6566 6175 6c74  omputed, default
│ │ │ │ -00017b10: 2069 730a 2020 2020 2020 2020 692d 3e63   is.        i->c
│ │ │ │ -00017b20: 6569 6c69 6e67 2831 2e35 5e69 290a 2020  eiling(1.5^i).  
│ │ │ │ -00017b30: 2020 2020 2a20 5061 6972 4c69 6d69 7420      * PairLimit 
│ │ │ │ -00017b40: 3d3e 2061 202a 6e6f 7465 206e 756d 6265  => a *note numbe
│ │ │ │ -00017b50: 723a 2028 4d61 6361 756c 6179 3244 6f63  r: (Macaulay2Doc
│ │ │ │ -00017b60: 294e 756d 6265 722c 2c20 6465 6661 756c  )Number,, defaul
│ │ │ │ -00017b70: 7420 7661 6c75 6520 3130 302c 0a20 2020  t value 100,.   
│ │ │ │ -00017b80: 2020 2020 2074 6865 206d 6178 2076 616c       the max val
│ │ │ │ -00017b90: 7565 2074 6f20 6265 2070 6c75 6767 6564  ue to be plugged
│ │ │ │ -00017ba0: 2069 6e74 6f20 5350 6169 7273 4675 6e63   into SPairsFunc
│ │ │ │ -00017bb0: 7469 6f6e 0a20 2020 2020 202a 2056 6572  tion.      * Ver
│ │ │ │ -00017bc0: 626f 7365 203d 3e20 2e2e 2e2c 2064 6566  bose => ..., def
│ │ │ │ -00017bd0: 6175 6c74 2076 616c 7565 2066 616c 7365  ault value false
│ │ │ │ -00017be0: 0a20 202a 204f 7574 7075 7473 3a0a 2020  .  * Outputs:.  
│ │ │ │ -00017bf0: 2020 2020 2a20 7472 7565 2069 6620 7468      * true if th
│ │ │ │ -00017c00: 6520 636f 6469 6d65 6e73 696f 6e20 6f66  e codimension of
│ │ │ │ -00017c10: 2049 2069 7320 6174 206c 6561 7374 206e   I is at least n
│ │ │ │ -00017c20: 206f 7220 6e75 6c6c 2069 6620 7468 6520   or null if the 
│ │ │ │ -00017c30: 6675 6e63 7469 6f6e 0a20 2020 2020 2020  function.       
│ │ │ │ -00017c40: 2063 616e 6e6f 7420 7465 6c6c 2077 6865   cannot tell whe
│ │ │ │ -00017c50: 7468 6572 2074 6865 2063 6f64 696d 656e  ther the codimen
│ │ │ │ -00017c60: 7369 6f6e 2069 7320 6174 206c 6561 7374  sion is at least
│ │ │ │ -00017c70: 206e 0a0a 4465 7363 7269 7074 696f 6e0a   n..Description.
│ │ │ │ -00017c80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6869  ===========..Thi
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│ │ │ │ -00017cb0: 7369 732c 2074 616b 6573 2074 6865 2069  sis, takes the i
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│ │ │ │ -00017cf0: 696e 6974 6961 6c20 6964 6561 6c20 6861  initial ideal ha
│ │ │ │ -00017d00: 7320 636f 6469 6d65 6e73 696f 6e20 6174  s codimension at
│ │ │ │ -00017d10: 206c 6561 7374 206e 2e20 436f 6e73 6964   least n. Consid
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│ │ │ │ -00017d30: 2065 7861 6d70 6c65 2e20 2057 6520 6372   example.  We cr
│ │ │ │ -00017d40: 6561 7465 2061 6e20 6964 6561 6c20 6f66  eate an ideal of
│ │ │ │ -00017d50: 2031 3520 6d69 6e6f 7273 206f 6620 7468   15 minors of th
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│ │ │ │ -00017d70: 2861 0a6d 6174 7269 7820 636f 6e73 7472  (a.matrix constr
│ │ │ │ -00017d80: 7563 7465 6420 696e 2061 2077 6179 2074  ucted in a way t
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│ │ │ │ -00017e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00017eb0: 6931 203a 2052 203d 205a 5a2f 3132 375b  i1 : R = ZZ/127[
│ │ │ │ -00017ec0: 785f 3120 2e2e 2078 5f28 3132 295d 3b20  x_1 .. x_(12)]; 
│ │ │ │ -00017ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017ef0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00017f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00017f50: 6932 203a 2050 203d 206d 696e 6f72 7328  i2 : P = minors(
│ │ │ │ -00017f60: 332c 6765 6e65 7269 634d 6174 7269 7828  3,genericMatrix(
│ │ │ │ -00017f70: 522c 785f 312c 332c 3429 293b 2020 2020  R,x_1,3,4));    
│ │ │ │ -00017f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017f90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00017fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017670: 3d3d 3d3d 0a0a 2020 2a20 2267 6574 5375  ====..  * "getSu
│ │ │ │ +00017680: 626d 6174 7269 784f 6652 616e 6b28 5a5a  bmatrixOfRank(ZZ
│ │ │ │ +00017690: 2c4d 6174 7269 7829 220a 0a46 6f72 2074  ,Matrix)"..For t
│ │ │ │ +000176a0: 6865 2070 726f 6772 616d 6d65 720a 3d3d  he programmer.==
│ │ │ │ +000176b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +000176c0: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f  ..The object *no
│ │ │ │ +000176d0: 7465 2067 6574 5375 626d 6174 7269 784f  te getSubmatrixO
│ │ │ │ +000176e0: 6652 616e 6b3a 2067 6574 5375 626d 6174  fRank: getSubmat
│ │ │ │ +000176f0: 7269 784f 6652 616e 6b2c 2069 7320 6120  rixOfRank, is a 
│ │ │ │ +00017700: 2a6e 6f74 6520 6d65 7468 6f64 0a66 756e  *note method.fun
│ │ │ │ +00017710: 6374 696f 6e20 7769 7468 206f 7074 696f  ction with optio
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│ │ │ │ +00017730: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e  c)MethodFunction
│ │ │ │ +00017740: 5769 7468 4f70 7469 6f6e 732c 2e0a 0a2d  WithOptions,...-
│ │ │ │ +00017750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a  --------------..
│ │ │ │ +000177a0: 5468 6520 736f 7572 6365 206f 6620 7468  The source of th
│ │ │ │ +000177b0: 6973 2064 6f63 756d 656e 7420 6973 2069  is document is i
│ │ │ │ +000177c0: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475  n./build/reprodu
│ │ │ │ +000177d0: 6369 626c 652d 7061 7468 2f6d 6163 6175  cible-path/macau
│ │ │ │ +000177e0: 6c61 7932 2d31 2e32 352e 3131 2b64 732f  lay2-1.25.11+ds/
│ │ │ │ +000177f0: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163  M2/Macaulay2/pac
│ │ │ │ +00017800: 6b61 6765 732f 4661 7374 4d69 6e6f 7273  kages/FastMinors
│ │ │ │ +00017810: 2e0a 6d32 3a31 3737 323a 302e 0a1f 0a46  ..m2:1772:0....F
│ │ │ │ +00017820: 696c 653a 2046 6173 744d 696e 6f72 732e  ile: FastMinors.
│ │ │ │ +00017830: 696e 666f 2c20 4e6f 6465 3a20 6973 436f  info, Node: isCo
│ │ │ │ +00017840: 6469 6d41 744c 6561 7374 2c20 4e65 7874  dimAtLeast, Next
│ │ │ │ +00017850: 3a20 6973 4469 6d41 744d 6f73 742c 2050  : isDimAtMost, P
│ │ │ │ +00017860: 7265 763a 2067 6574 5375 626d 6174 7269  rev: getSubmatri
│ │ │ │ +00017870: 784f 6652 616e 6b2c 2055 703a 2054 6f70  xOfRank, Up: Top
│ │ │ │ +00017880: 0a0a 6973 436f 6469 6d41 744c 6561 7374  ..isCodimAtLeast
│ │ │ │ +00017890: 202d 2d20 7265 7475 726e 7320 7472 7565   -- returns true
│ │ │ │ +000178a0: 2069 6620 7765 2063 616e 2071 7569 636b   if we can quick
│ │ │ │ +000178b0: 6c79 2073 6565 2077 6865 7468 6572 2074  ly see whether t
│ │ │ │ +000178c0: 6865 2063 6f64 696d 2069 7320 6174 206c  he codim is at l
│ │ │ │ +000178d0: 6561 7374 2061 2067 6976 656e 206e 756d  east a given num
│ │ │ │ +000178e0: 6265 720a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ber.************
│ │ │ │ +000178f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00017900: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00017910: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00017920: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00017930: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00017940: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765  *****..  * Usage
│ │ │ │ +00017950: 3a20 0a20 2020 2020 2020 2069 7343 6f64  : .        isCod
│ │ │ │ +00017960: 696d 4174 4c65 6173 7428 6e2c 2049 290a  imAtLeast(n, I).
│ │ │ │ +00017970: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ +00017980: 2020 2a20 6e2c 2061 6e20 2a6e 6f74 6520    * n, an *note 
│ │ │ │ +00017990: 696e 7465 6765 723a 2028 4d61 6361 756c  integer: (Macaul
│ │ │ │ +000179a0: 6179 3244 6f63 295a 5a2c 2c20 616e 2069  ay2Doc)ZZ,, an i
│ │ │ │ +000179b0: 6e74 6567 6572 0a20 2020 2020 202a 2049  nteger.      * I
│ │ │ │ +000179c0: 2c20 616e 202a 6e6f 7465 2069 6465 616c  , an *note ideal
│ │ │ │ +000179d0: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +000179e0: 4964 6561 6c2c 2c20 616e 2069 6465 616c  Ideal,, an ideal
│ │ │ │ +000179f0: 2069 6e20 6120 706f 6c79 6e6f 6d69 616c   in a polynomial
│ │ │ │ +00017a00: 2072 696e 670a 2020 2020 2020 2020 6f76   ring.        ov
│ │ │ │ +00017a10: 6572 2061 2066 6965 6c64 2c20 6f72 2061  er a field, or a
│ │ │ │ +00017a20: 2071 756f 7469 656e 7420 7269 6e67 0a20   quotient ring. 
│ │ │ │ +00017a30: 202a 202a 6e6f 7465 204f 7074 696f 6e61   * *note Optiona
│ │ │ │ +00017a40: 6c20 696e 7075 7473 3a20 284d 6163 6175  l inputs: (Macau
│ │ │ │ +00017a50: 6c61 7932 446f 6329 7573 696e 6720 6675  lay2Doc)using fu
│ │ │ │ +00017a60: 6e63 7469 6f6e 7320 7769 7468 206f 7074  nctions with opt
│ │ │ │ +00017a70: 696f 6e61 6c20 696e 7075 7473 2c3a 0a20  ional inputs,:. 
│ │ │ │ +00017a80: 2020 2020 202a 2053 5061 6972 7346 756e       * SPairsFun
│ │ │ │ +00017a90: 6374 696f 6e20 3d3e 2061 202a 6e6f 7465  ction => a *note
│ │ │ │ +00017aa0: 2066 756e 6374 696f 6e3a 2028 4d61 6361   function: (Maca
│ │ │ │ +00017ab0: 756c 6179 3244 6f63 2946 756e 6374 696f  ulay2Doc)Functio
│ │ │ │ +00017ac0: 6e2c 2c20 6465 6661 756c 740a 2020 2020  n,, default.    
│ │ │ │ +00017ad0: 2020 2020 7661 6c75 6520 4675 6e63 7469      value Functi
│ │ │ │ +00017ae0: 6f6e 436c 6f73 7572 655b 2e2e 2f46 6173  onClosure[../Fas
│ │ │ │ +00017af0: 744d 696e 6f72 732e 6d32 3a32 3131 3a32  tMinors.m2:211:2
│ │ │ │ +00017b00: 332d 3231 313a 3432 5d2c 2061 2066 756e  3-211:42], a fun
│ │ │ │ +00017b10: 6374 696f 6e20 746f 0a20 2020 2020 2020  ction to.       
│ │ │ │ +00017b20: 2063 6f6e 7472 6f6c 2068 6f77 2077 6865   control how whe
│ │ │ │ +00017b30: 6e20 7468 6520 636f 6469 6d65 6e73 696f  n the codimensio
│ │ │ │ +00017b40: 6e20 6f66 206d 696e 6f72 7320 6973 2063  n of minors is c
│ │ │ │ +00017b50: 6f6d 7075 7465 642c 2064 6566 6175 6c74  omputed, default
│ │ │ │ +00017b60: 2069 730a 2020 2020 2020 2020 692d 3e63   is.        i->c
│ │ │ │ +00017b70: 6569 6c69 6e67 2831 2e35 5e69 290a 2020  eiling(1.5^i).  
│ │ │ │ +00017b80: 2020 2020 2a20 5061 6972 4c69 6d69 7420      * PairLimit 
│ │ │ │ +00017b90: 3d3e 2061 202a 6e6f 7465 206e 756d 6265  => a *note numbe
│ │ │ │ +00017ba0: 723a 2028 4d61 6361 756c 6179 3244 6f63  r: (Macaulay2Doc
│ │ │ │ +00017bb0: 294e 756d 6265 722c 2c20 6465 6661 756c  )Number,, defaul
│ │ │ │ +00017bc0: 7420 7661 6c75 6520 3130 302c 0a20 2020  t value 100,.   
│ │ │ │ +00017bd0: 2020 2020 2074 6865 206d 6178 2076 616c       the max val
│ │ │ │ +00017be0: 7565 2074 6f20 6265 2070 6c75 6767 6564  ue to be plugged
│ │ │ │ +00017bf0: 2069 6e74 6f20 5350 6169 7273 4675 6e63   into SPairsFunc
│ │ │ │ +00017c00: 7469 6f6e 0a20 2020 2020 202a 2056 6572  tion.      * Ver
│ │ │ │ +00017c10: 626f 7365 203d 3e20 2e2e 2e2c 2064 6566  bose => ..., def
│ │ │ │ +00017c20: 6175 6c74 2076 616c 7565 2066 616c 7365  ault value false
│ │ │ │ +00017c30: 0a20 202a 204f 7574 7075 7473 3a0a 2020  .  * Outputs:.  
│ │ │ │ +00017c40: 2020 2020 2a20 7472 7565 2069 6620 7468      * true if th
│ │ │ │ +00017c50: 6520 636f 6469 6d65 6e73 696f 6e20 6f66  e codimension of
│ │ │ │ +00017c60: 2049 2069 7320 6174 206c 6561 7374 206e   I is at least n
│ │ │ │ +00017c70: 206f 7220 6e75 6c6c 2069 6620 7468 6520   or null if the 
│ │ │ │ +00017c80: 6675 6e63 7469 6f6e 0a20 2020 2020 2020  function.       
│ │ │ │ +00017c90: 2063 616e 6e6f 7420 7465 6c6c 2077 6865   cannot tell whe
│ │ │ │ +00017ca0: 7468 6572 2074 6865 2063 6f64 696d 656e  ther the codimen
│ │ │ │ +00017cb0: 7369 6f6e 2069 7320 6174 206c 6561 7374  sion is at least
│ │ │ │ +00017cc0: 206e 0a0a 4465 7363 7269 7074 696f 6e0a   n..Description.
│ │ │ │ +00017cd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6869  ===========..Thi
│ │ │ │ +00017ce0: 7320 636f 6d70 7574 6573 2061 2070 6172  s computes a par
│ │ │ │ +00017cf0: 7469 616c 2047 726f 6562 6e65 7220 6261  tial Groebner ba
│ │ │ │ +00017d00: 7369 732c 2074 616b 6573 2074 6865 2069  sis, takes the i
│ │ │ │ +00017d10: 6e69 7469 616c 2074 6572 6d73 2c20 616e  nitial terms, an
│ │ │ │ +00017d20: 6420 6368 6563 6b73 0a77 6865 7468 6572  d checks.whether
│ │ │ │ +00017d30: 2074 6861 7420 2870 6172 7469 616c 2920   that (partial) 
│ │ │ │ +00017d40: 696e 6974 6961 6c20 6964 6561 6c20 6861  initial ideal ha
│ │ │ │ +00017d50: 7320 636f 6469 6d65 6e73 696f 6e20 6174  s codimension at
│ │ │ │ +00017d60: 206c 6561 7374 206e 2e20 436f 6e73 6964   least n. Consid
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│ │ │ │ +00017d80: 2065 7861 6d70 6c65 2e20 2057 6520 6372   example.  We cr
│ │ │ │ +00017d90: 6561 7465 2061 6e20 6964 6561 6c20 6f66  eate an ideal of
│ │ │ │ +00017da0: 2031 3520 6d69 6e6f 7273 206f 6620 7468   15 minors of th
│ │ │ │ +00017db0: 6520 6d61 7472 6978 206d 7944 6966 6620  e matrix myDiff 
│ │ │ │ +00017dc0: 2861 0a6d 6174 7269 7820 636f 6e73 7472  (a.matrix constr
│ │ │ │ +00017dd0: 7563 7465 6420 696e 2061 2077 6179 2074  ucted in a way t
│ │ │ │ +00017de0: 7970 6963 616c 206f 6620 6170 706c 6963  ypical of applic
│ │ │ │ +00017df0: 6174 696f 6e73 292e 2020 5765 2077 6f75  ations).  We wou
│ │ │ │ +00017e00: 6c64 206c 696b 6520 746f 2076 6572 6966  ld like to verif
│ │ │ │ +00017e10: 790a 7468 6174 2074 6865 2063 6f64 696d  y.that the codim
│ │ │ │ +00017e20: 656e 7369 6f6e 206f 6620 7468 6973 2069  ension of this i
│ │ │ │ +00017e30: 6465 616c 2069 7320 6174 206c 6561 7374  deal is at least
│ │ │ │ +00017e40: 2033 2e20 2054 6865 2062 7569 6c74 2d69   3.  The built-i
│ │ │ │ +00017e50: 6e20 636f 6469 6d20 6675 6e63 7469 6f6e  n codim function
│ │ │ │ +00017e60: 0a74 7970 6963 616c 6c79 2064 6f65 7320  .typically does 
│ │ │ │ +00017e70: 6e6f 7420 7465 726d 696e 6174 652e 2048  not terminate. H
│ │ │ │ +00017e80: 6f77 6576 6572 2c20 6973 436f 6469 6d41  owever, isCodimA
│ │ │ │ +00017e90: 744c 6561 7374 2069 7320 6e6f 726d 616c  tLeast is normal
│ │ │ │ +00017ea0: 6c79 2076 6572 7920 6661 7374 2e0a 0a2b  ly very fast...+
│ │ │ │ +00017eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00017f00: 6931 203a 2052 203d 205a 5a2f 3132 375b  i1 : R = ZZ/127[
│ │ │ │ +00017f10: 785f 3120 2e2e 2078 5f28 3132 295d 3b20  x_1 .. x_(12)]; 
│ │ │ │ +00017f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017f40: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00017f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00017fa0: 6932 203a 2050 203d 206d 696e 6f72 7328  i2 : P = minors(
│ │ │ │ +00017fb0: 332c 6765 6e65 7269 634d 6174 7269 7828  3,genericMatrix(
│ │ │ │ +00017fc0: 522c 785f 312c 332c 3429 293b 2020 2020  R,x_1,3,4));    
│ │ │ │  00017fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017fe0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00017ff0: 6f32 203a 2049 6465 616c 206f 6620 5220  o2 : Ideal of R 
│ │ │ │ +00017ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018030: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00018040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00018090: 6933 203a 2043 203d 2072 6573 2028 525e  i3 : C = res (R^
│ │ │ │ -000180a0: 312f 2850 5e33 2929 3b20 2020 2020 2020  1/(P^3));       
│ │ │ │ -000180b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000180c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000180d0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -000180e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000180f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00018130: 6934 203a 206d 7944 6966 6620 3d20 432e  i4 : myDiff = C.
│ │ │ │ -00018140: 6464 5f33 3b20 2020 2020 2020 2020 2020  dd_3;           
│ │ │ │ -00018150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018170: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00018180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018030: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00018040: 6f32 203a 2049 6465 616c 206f 6620 5220  o2 : Ideal of R 
│ │ │ │ +00018050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018080: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00018090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000180a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000180b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000180c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000180d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +000180e0: 6933 203a 2043 203d 2072 6573 2028 525e  i3 : C = res (R^
│ │ │ │ +000180f0: 312f 2850 5e33 2929 3b20 2020 2020 2020  1/(P^3));       
│ │ │ │ +00018100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018120: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00018130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00018180: 6934 203a 206d 7944 6966 6620 3d20 432e  i4 : myDiff = C.
│ │ │ │ +00018190: 6464 5f33 3b20 2020 2020 2020 2020 2020  dd_3;           
│ │ │ │  000181a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000181b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000181c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000181d0: 2020 2020 2020 2020 2020 2020 2033 3020               30 
│ │ │ │ -000181e0: 2020 2020 2031 3220 2020 2020 2020 2020       12         
│ │ │ │ +000181d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000181e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000181f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018210: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00018220: 6f34 203a 204d 6174 7269 7820 5220 2020  o4 : Matrix R   
│ │ │ │ -00018230: 3c2d 2d20 5220 2020 2020 2020 2020 2020  <-- R           
│ │ │ │ +00018220: 2020 2020 2020 2020 2020 2020 2033 3020               30 
│ │ │ │ +00018230: 2020 2020 2031 3220 2020 2020 2020 2020       12         
│ │ │ │  00018240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018260: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00018270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000182a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000182b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -000182c0: 6935 203a 2072 203d 2072 616e 6b20 6d79  i5 : r = rank my
│ │ │ │ -000182d0: 4469 6666 3b20 2020 2020 2020 2020 2020  Diff;           
│ │ │ │ -000182e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000182f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018300: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00018310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00018360: 6936 203a 204a 203d 2063 686f 6f73 6547  i6 : J = chooseG
│ │ │ │ -00018370: 6f6f 644d 696e 6f72 7328 3135 2c20 722c  oodMinors(15, r,
│ │ │ │ -00018380: 206d 7944 6966 662c 2053 7472 6174 6567   myDiff, Strateg
│ │ │ │ -00018390: 793d 3e53 7472 6174 6567 7944 6566 6175  y=>StrategyDefau
│ │ │ │ -000183a0: 6c74 4e6f 6e52 616e 646f 6d29 3b7c 0a7c  ltNonRandom);|.|
│ │ │ │ -000183b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000183c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000183d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000183e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000183f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00018400: 6f36 203a 2049 6465 616c 206f 6620 5220  o6 : Ideal of R 
│ │ │ │ +00018260: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00018270: 6f34 203a 204d 6174 7269 7820 5220 2020  o4 : Matrix R   
│ │ │ │ +00018280: 3c2d 2d20 5220 2020 2020 2020 2020 2020  <-- R           
│ │ │ │ +00018290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000182a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000182b0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +000182c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000182d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000182e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000182f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00018310: 6935 203a 2072 203d 2072 616e 6b20 6d79  i5 : r = rank my
│ │ │ │ +00018320: 4469 6666 3b20 2020 2020 2020 2020 2020  Diff;           
│ │ │ │ +00018330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018350: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00018360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000183a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +000183b0: 6936 203a 204a 203d 2063 686f 6f73 6547  i6 : J = chooseG
│ │ │ │ +000183c0: 6f6f 644d 696e 6f72 7328 3135 2c20 722c  oodMinors(15, r,
│ │ │ │ +000183d0: 206d 7944 6966 662c 2053 7472 6174 6567   myDiff, Strateg
│ │ │ │ +000183e0: 793d 3e53 7472 6174 6567 7944 6566 6175  y=>StrategyDefau
│ │ │ │ +000183f0: 6c74 4e6f 6e52 616e 646f 6d29 3b7c 0a7c  ltNonRandom);|.|
│ │ │ │ +00018400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018440: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00018450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -000184a0: 6937 203a 2074 696d 6520 6973 436f 6469  i7 : time isCodi
│ │ │ │ -000184b0: 6d41 744c 6561 7374 2833 2c20 4a29 2020  mAtLeast(3, J)  
│ │ │ │ -000184c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000184d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000184e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000184f0: 202d 2d20 7573 6564 2030 2e30 3034 3030   -- used 0.00400
│ │ │ │ -00018500: 3037 3273 2028 6370 7529 3b20 302e 3030  072s (cpu); 0.00
│ │ │ │ -00018510: 3236 3430 3439 7320 2874 6872 6561 6429  264049s (thread)
│ │ │ │ -00018520: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │ +00018440: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00018450: 6f36 203a 2049 6465 616c 206f 6620 5220  o6 : Ideal of R 
│ │ │ │ +00018460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018490: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +000184a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000184b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000184c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000184d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000184e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +000184f0: 6937 203a 2074 696d 6520 6973 436f 6469  i7 : time isCodi
│ │ │ │ +00018500: 6d41 744c 6561 7374 2833 2c20 4a29 2020  mAtLeast(3, J)  
│ │ │ │ +00018510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018530: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00018540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018540: 202d 2d20 7573 6564 2030 2e30 3033 3932   -- used 0.00392
│ │ │ │ +00018550: 3733 3373 2028 6370 7529 3b20 302e 3030  733s (cpu); 0.00
│ │ │ │ +00018560: 3332 3033 3538 7320 2874 6872 6561 6429  320358s (thread)
│ │ │ │ +00018570: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │  00018580: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00018590: 6f37 203d 2074 7275 6520 2020 2020 2020  o7 = true       
│ │ │ │ +00018590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000185a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000185b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000185c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000185d0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -000185e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000185f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ -00018630: 5468 6520 6675 6e63 7469 6f6e 2077 6f72  The function wor
│ │ │ │ -00018640: 6b73 2062 7920 636f 6d70 7574 696e 6720  ks by computing 
│ │ │ │ -00018650: 6762 2849 2c20 5061 6972 4c69 6d69 743d  gb(I, PairLimit=
│ │ │ │ -00018660: 3e66 2869 2929 2066 6f72 2073 7563 6365  >f(i)) for succe
│ │ │ │ -00018670: 7373 6976 6520 7661 6c75 6573 206f 660a  ssive values of.
│ │ │ │ -00018680: 692e 2020 4865 7265 2066 2869 2920 6973  i.  Here f(i) is
│ │ │ │ -00018690: 2061 2066 756e 6374 696f 6e20 7468 6174   a function that
│ │ │ │ -000186a0: 2074 616b 6573 2074 2c20 736f 6d65 2061   takes t, some a
│ │ │ │ -000186b0: 7070 726f 7869 6d61 7469 6f6e 206f 6620  pproximation of 
│ │ │ │ -000186c0: 7468 6520 6261 7365 2064 6567 7265 650a  the base degree.
│ │ │ │ -000186d0: 7661 6c75 6520 6f66 2074 6865 2070 6f6c  value of the pol
│ │ │ │ -000186e0: 796e 6f6d 6961 6c20 7269 6e67 2028 666f  ynomial ring (fo
│ │ │ │ -000186f0: 7220 6578 616d 706c 652c 2069 6e20 6120  r example, in a 
│ │ │ │ -00018700: 7374 616e 6461 7264 2067 7261 6465 6420  standard graded 
│ │ │ │ -00018710: 706f 6c79 6e6f 6d69 616c 0a72 696e 672c  polynomial.ring,
│ │ │ │ -00018720: 2074 6869 7320 6973 2070 726f 6261 626c   this is probabl
│ │ │ │ -00018730: 7920 6578 7065 6374 6564 2074 6f20 6265  y expected to be
│ │ │ │ -00018740: 205c 7b31 5c7d 292e 2020 416e 6420 6920   \{1\}).  And i 
│ │ │ │ -00018750: 6973 2061 2063 6f75 6e74 696e 6720 7661  is a counting va
│ │ │ │ -00018760: 7269 6162 6c65 2e0a 596f 7520 6361 6e20  riable..You can 
│ │ │ │ -00018770: 7072 6f76 6964 6520 796f 7572 206f 776e  provide your own
│ │ │ │ -00018780: 2066 756e 6374 696f 6e20 6279 2063 616c   function by cal
│ │ │ │ -00018790: 6c69 6e67 2069 7343 6f64 696d 4174 4c65  ling isCodimAtLe
│ │ │ │ -000187a0: 6173 7428 6e2c 2049 2c0a 5350 6169 7273  ast(n, I,.SPairs
│ │ │ │ -000187b0: 4675 6e63 7469 6f6e 3d3e 2820 2869 2920  Function=>( (i) 
│ │ │ │ -000187c0: 2d3e 2066 2869 2920 292c 2074 6865 2064  -> f(i) ), the d
│ │ │ │ -000187d0: 6566 6175 6c74 2066 756e 6374 696f 6e20  efault function 
│ │ │ │ -000187e0: 6973 0a53 5061 6972 7346 756e 6374 696f  is.SPairsFunctio
│ │ │ │ -000187f0: 6e3d 3e69 2d3e 6365 696c 696e 6728 312e  n=>i->ceiling(1.
│ │ │ │ -00018800: 355e 6929 2020 2050 6572 6861 7073 206d  5^i)   Perhaps m
│ │ │ │ -00018810: 6f72 6520 636f 6d6d 6f6e 6c79 2068 6f77  ore commonly how
│ │ │ │ -00018820: 6576 6572 2c20 7468 6520 7573 6572 206d  ever, the user m
│ │ │ │ -00018830: 6179 0a77 616e 7420 746f 2069 6e73 7465  ay.want to inste
│ │ │ │ -00018840: 6164 2074 656c 6c20 7468 6520 6675 6e63  ad tell the func
│ │ │ │ -00018850: 7469 6f6e 2074 6f20 636f 6d70 7574 6520  tion to compute 
│ │ │ │ -00018860: 666f 7220 6c61 7267 6572 2076 616c 7565  for larger value
│ │ │ │ -00018870: 7320 6f66 2069 2e20 2054 6869 7320 6973  s of i.  This is
│ │ │ │ -00018880: 0a64 6f6e 6520 7669 6120 7468 6520 6f70  .done via the op
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│ │ │ │ -000188a0: 2054 6869 7320 6973 2074 6865 206d 6178   This is the max
│ │ │ │ -000188b0: 2076 616c 7565 206f 6620 6920 746f 2062   value of i to b
│ │ │ │ -000188c0: 6520 706c 7567 6765 6420 696e 746f 0a53  e plugged into.S
│ │ │ │ -000188d0: 5061 6972 7346 756e 6374 696f 6e20 6265  PairsFunction be
│ │ │ │ -000188e0: 666f 7265 2074 6865 2066 756e 6374 696f  fore the functio
│ │ │ │ -000188f0: 6e20 6769 7665 7320 7570 2e20 2049 6e20  n gives up.  In 
│ │ │ │ -00018900: 6f74 6865 7220 776f 7264 732c 2050 6169  other words, Pai
│ │ │ │ -00018910: 724c 696d 6974 3d3e 3520 7769 6c6c 0a74  rLimit=>5 will.t
│ │ │ │ -00018920: 656c 6c20 7468 6520 6675 6e63 7469 6f6e  ell the function
│ │ │ │ -00018930: 2074 6f20 6368 6563 6b20 636f 6469 6d65   to check codime
│ │ │ │ -00018940: 6e73 696f 6e20 3520 7469 6d65 732e 0a0a  nsion 5 times...
│ │ │ │ -00018950: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -00018960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -000189a0: 7c69 3820 3a20 4920 3d20 6964 6561 6c28  |i8 : I = ideal(
│ │ │ │ -000189b0: 785f 325e 382a 785f 3130 5e33 2d33 2a78  x_2^8*x_10^3-3*x
│ │ │ │ -000189c0: 5f31 2a78 5f32 5e37 2a78 5f31 305e 322a  _1*x_2^7*x_10^2*
│ │ │ │ -000189d0: 785f 3131 2b33 2a78 5f31 5e32 2a78 5f32  x_11+3*x_1^2*x_2
│ │ │ │ -000189e0: 5e36 2a78 5f31 302a 785f 3131 5e32 7c0a  ^6*x_10*x_11^2|.
│ │ │ │ -000189f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00018a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018a30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000185d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000185e0: 6f37 203d 2074 7275 6520 2020 2020 2020  o7 = true       
│ │ │ │ +000185f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018620: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00018630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ +00018680: 5468 6520 6675 6e63 7469 6f6e 2077 6f72  The function wor
│ │ │ │ +00018690: 6b73 2062 7920 636f 6d70 7574 696e 6720  ks by computing 
│ │ │ │ +000186a0: 6762 2849 2c20 5061 6972 4c69 6d69 743d  gb(I, PairLimit=
│ │ │ │ +000186b0: 3e66 2869 2929 2066 6f72 2073 7563 6365  >f(i)) for succe
│ │ │ │ +000186c0: 7373 6976 6520 7661 6c75 6573 206f 660a  ssive values of.
│ │ │ │ +000186d0: 692e 2020 4865 7265 2066 2869 2920 6973  i.  Here f(i) is
│ │ │ │ +000186e0: 2061 2066 756e 6374 696f 6e20 7468 6174   a function that
│ │ │ │ +000186f0: 2074 616b 6573 2074 2c20 736f 6d65 2061   takes t, some a
│ │ │ │ +00018700: 7070 726f 7869 6d61 7469 6f6e 206f 6620  pproximation of 
│ │ │ │ +00018710: 7468 6520 6261 7365 2064 6567 7265 650a  the base degree.
│ │ │ │ +00018720: 7661 6c75 6520 6f66 2074 6865 2070 6f6c  value of the pol
│ │ │ │ +00018730: 796e 6f6d 6961 6c20 7269 6e67 2028 666f  ynomial ring (fo
│ │ │ │ +00018740: 7220 6578 616d 706c 652c 2069 6e20 6120  r example, in a 
│ │ │ │ +00018750: 7374 616e 6461 7264 2067 7261 6465 6420  standard graded 
│ │ │ │ +00018760: 706f 6c79 6e6f 6d69 616c 0a72 696e 672c  polynomial.ring,
│ │ │ │ +00018770: 2074 6869 7320 6973 2070 726f 6261 626c   this is probabl
│ │ │ │ +00018780: 7920 6578 7065 6374 6564 2074 6f20 6265  y expected to be
│ │ │ │ +00018790: 205c 7b31 5c7d 292e 2020 416e 6420 6920   \{1\}).  And i 
│ │ │ │ +000187a0: 6973 2061 2063 6f75 6e74 696e 6720 7661  is a counting va
│ │ │ │ +000187b0: 7269 6162 6c65 2e0a 596f 7520 6361 6e20  riable..You can 
│ │ │ │ +000187c0: 7072 6f76 6964 6520 796f 7572 206f 776e  provide your own
│ │ │ │ +000187d0: 2066 756e 6374 696f 6e20 6279 2063 616c   function by cal
│ │ │ │ +000187e0: 6c69 6e67 2069 7343 6f64 696d 4174 4c65  ling isCodimAtLe
│ │ │ │ +000187f0: 6173 7428 6e2c 2049 2c0a 5350 6169 7273  ast(n, I,.SPairs
│ │ │ │ +00018800: 4675 6e63 7469 6f6e 3d3e 2820 2869 2920  Function=>( (i) 
│ │ │ │ +00018810: 2d3e 2066 2869 2920 292c 2074 6865 2064  -> f(i) ), the d
│ │ │ │ +00018820: 6566 6175 6c74 2066 756e 6374 696f 6e20  efault function 
│ │ │ │ +00018830: 6973 0a53 5061 6972 7346 756e 6374 696f  is.SPairsFunctio
│ │ │ │ +00018840: 6e3d 3e69 2d3e 6365 696c 696e 6728 312e  n=>i->ceiling(1.
│ │ │ │ +00018850: 355e 6929 2020 2050 6572 6861 7073 206d  5^i)   Perhaps m
│ │ │ │ +00018860: 6f72 6520 636f 6d6d 6f6e 6c79 2068 6f77  ore commonly how
│ │ │ │ +00018870: 6576 6572 2c20 7468 6520 7573 6572 206d  ever, the user m
│ │ │ │ +00018880: 6179 0a77 616e 7420 746f 2069 6e73 7465  ay.want to inste
│ │ │ │ +00018890: 6164 2074 656c 6c20 7468 6520 6675 6e63  ad tell the func
│ │ │ │ +000188a0: 7469 6f6e 2074 6f20 636f 6d70 7574 6520  tion to compute 
│ │ │ │ +000188b0: 666f 7220 6c61 7267 6572 2076 616c 7565  for larger value
│ │ │ │ +000188c0: 7320 6f66 2069 2e20 2054 6869 7320 6973  s of i.  This is
│ │ │ │ +000188d0: 0a64 6f6e 6520 7669 6120 7468 6520 6f70  .done via the op
│ │ │ │ +000188e0: 7469 6f6e 2050 6169 724c 696d 6974 2e20  tion PairLimit. 
│ │ │ │ +000188f0: 2054 6869 7320 6973 2074 6865 206d 6178   This is the max
│ │ │ │ +00018900: 2076 616c 7565 206f 6620 6920 746f 2062   value of i to b
│ │ │ │ +00018910: 6520 706c 7567 6765 6420 696e 746f 0a53  e plugged into.S
│ │ │ │ +00018920: 5061 6972 7346 756e 6374 696f 6e20 6265  PairsFunction be
│ │ │ │ +00018930: 666f 7265 2074 6865 2066 756e 6374 696f  fore the functio
│ │ │ │ +00018940: 6e20 6769 7665 7320 7570 2e20 2049 6e20  n gives up.  In 
│ │ │ │ +00018950: 6f74 6865 7220 776f 7264 732c 2050 6169  other words, Pai
│ │ │ │ +00018960: 724c 696d 6974 3d3e 3520 7769 6c6c 0a74  rLimit=>5 will.t
│ │ │ │ +00018970: 656c 6c20 7468 6520 6675 6e63 7469 6f6e  ell the function
│ │ │ │ +00018980: 2074 6f20 6368 6563 6b20 636f 6469 6d65   to check codime
│ │ │ │ +00018990: 6e73 696f 6e20 3520 7469 6d65 732e 0a0a  nsion 5 times...
│ │ │ │ +000189a0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +000189b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000189c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000189d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000189e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000189f0: 7c69 3820 3a20 4920 3d20 6964 6561 6c28  |i8 : I = ideal(
│ │ │ │ +00018a00: 785f 325e 382a 785f 3130 5e33 2d33 2a78  x_2^8*x_10^3-3*x
│ │ │ │ +00018a10: 5f31 2a78 5f32 5e37 2a78 5f31 305e 322a  _1*x_2^7*x_10^2*
│ │ │ │ +00018a20: 785f 3131 2b33 2a78 5f31 5e32 2a78 5f32  x_11+3*x_1^2*x_2
│ │ │ │ +00018a30: 5e36 2a78 5f31 302a 785f 3131 5e32 7c0a  ^6*x_10*x_11^2|.
│ │ │ │  00018a40: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00018a50: 5a5a 2020 2020 2020 2020 2020 2020 2020  ZZ              
│ │ │ │ +00018a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018a80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00018a90: 7c6f 3820 3a20 4964 6561 6c20 6f66 202d  |o8 : Ideal of -
│ │ │ │ -00018aa0: 2d2d 5b78 2020 2c20 7820 2c20 7820 2c20  --[x  , x , x , 
│ │ │ │ -00018ab0: 7820 2c20 7820 202c 2078 202c 2078 202c  x , x  , x , x ,
│ │ │ │ -00018ac0: 2078 2020 2c20 7820 2c20 7820 2c20 7820   x  , x , x , x 
│ │ │ │ -00018ad0: 2c20 7820 5d20 2020 2020 2020 2020 7c0a  , x ]         |.
│ │ │ │ -00018ae0: 7c20 2020 2020 2020 2020 2020 2020 2031  |              1
│ │ │ │ -00018af0: 3237 2020 3131 2020 2038 2020 2031 2020  27  11   8   1  
│ │ │ │ -00018b00: 2039 2020 2031 3220 2020 3620 2020 3520   9   12   6   5 
│ │ │ │ -00018b10: 2020 3130 2020 2032 2020 2034 2020 2033    10   2   4   3
│ │ │ │ -00018b20: 2020 2037 2020 2020 2020 2020 2020 7c0a     7          |.
│ │ │ │ -00018b30: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ -00018b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -00018b80: 7c2d 785f 315e 332a 785f 325e 352a 785f  |-x_1^3*x_2^5*x_
│ │ │ │ -00018b90: 3131 5e33 2c78 5f35 5e35 2a78 5f36 5e33  11^3,x_5^5*x_6^3
│ │ │ │ -00018ba0: 2a78 5f31 315e 332d 332a 785f 355e 362a  *x_11^3-3*x_5^6*
│ │ │ │ -00018bb0: 785f 365e 322a 785f 3131 5e32 2a78 5f31  x_6^2*x_11^2*x_1
│ │ │ │ -00018bc0: 322b 332a 785f 355e 372a 785f 362a 7c0a  2+3*x_5^7*x_6*|.
│ │ │ │ -00018bd0: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ -00018be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -00018c20: 7c78 5f31 312a 785f 3132 5e32 2d78 5f35  |x_11*x_12^2-x_5
│ │ │ │ -00018c30: 5e38 2a78 5f31 325e 332c 785f 315e 352a  ^8*x_12^3,x_1^5*
│ │ │ │ -00018c40: 785f 325e 332a 785f 345e 332d 332a 785f  x_2^3*x_4^3-3*x_
│ │ │ │ -00018c50: 315e 362a 785f 325e 322a 785f 345e 322a  1^6*x_2^2*x_4^2*
│ │ │ │ -00018c60: 785f 352b 332a 785f 315e 372a 785f 7c0a  x_5+3*x_1^7*x_|.
│ │ │ │ -00018c70: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ -00018c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -00018cc0: 7c32 2a78 5f34 2a78 5f35 5e32 2d78 5f31  |2*x_4*x_5^2-x_1
│ │ │ │ -00018cd0: 5e38 2a78 5f35 5e33 2c78 5f36 5e38 2a78  ^8*x_5^3,x_6^8*x
│ │ │ │ -00018ce0: 5f31 315e 332d 332a 785f 352a 785f 365e  _11^3-3*x_5*x_6^
│ │ │ │ -00018cf0: 372a 785f 3131 5e32 2a78 5f31 322b 332a  7*x_11^2*x_12+3*
│ │ │ │ -00018d00: 785f 355e 322a 785f 365e 362a 785f 7c0a  x_5^2*x_6^6*x_|.
│ │ │ │ -00018d10: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ -00018d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -00018d60: 7c31 312a 785f 3132 5e32 2d78 5f35 5e33  |11*x_12^2-x_5^3
│ │ │ │ -00018d70: 2a78 5f36 5e35 2a78 5f31 325e 332c 785f  *x_6^5*x_12^3,x_
│ │ │ │ -00018d80: 385e 332a 785f 3130 5e38 2d33 2a78 5f37  8^3*x_10^8-3*x_7
│ │ │ │ -00018d90: 2a78 5f38 5e32 2a78 5f31 305e 372a 785f  *x_8^2*x_10^7*x_
│ │ │ │ -00018da0: 3131 2b33 2a78 5f37 5e32 2a78 5f38 7c0a  11+3*x_7^2*x_8|.
│ │ │ │ -00018db0: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ -00018dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -00018e00: 7c2a 785f 3130 5e36 2a78 5f31 315e 322d  |*x_10^6*x_11^2-
│ │ │ │ -00018e10: 785f 375e 332a 785f 3130 5e35 2a78 5f31  x_7^3*x_10^5*x_1
│ │ │ │ -00018e20: 315e 332c 785f 325e 382a 785f 345e 332d  1^3,x_2^8*x_4^3-
│ │ │ │ -00018e30: 332a 785f 312a 785f 325e 372a 785f 345e  3*x_1*x_2^7*x_4^
│ │ │ │ -00018e40: 322a 785f 352b 332a 785f 315e 322a 7c0a  2*x_5+3*x_1^2*|.
│ │ │ │ -00018e50: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ -00018e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -00018ea0: 7c78 5f32 5e36 2a78 5f34 2a78 5f35 5e32  |x_2^6*x_4*x_5^2
│ │ │ │ -00018eb0: 2d78 5f31 5e33 2a78 5f32 5e35 2a78 5f35  -x_1^3*x_2^5*x_5
│ │ │ │ -00018ec0: 5e33 2c2d 785f 365e 332a 785f 3131 5e38  ^3,-x_6^3*x_11^8
│ │ │ │ -00018ed0: 2b33 2a78 5f35 2a78 5f36 5e32 2a78 5f31  +3*x_5*x_6^2*x_1
│ │ │ │ -00018ee0: 315e 372a 785f 3132 2d33 2a78 5f35 7c0a  1^7*x_12-3*x_5|.
│ │ │ │ -00018ef0: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ -00018f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -00018f40: 7c5e 322a 785f 362a 785f 3131 5e36 2a78  |^2*x_6*x_11^6*x
│ │ │ │ -00018f50: 5f31 325e 322b 785f 355e 332a 785f 3131  _12^2+x_5^3*x_11
│ │ │ │ -00018f60: 5e35 2a78 5f31 325e 332c 2d78 5f36 5e33  ^5*x_12^3,-x_6^3
│ │ │ │ -00018f70: 2a78 5f37 5e33 2a78 5f39 5e35 2b33 2a78  *x_7^3*x_9^5+3*x
│ │ │ │ -00018f80: 5f34 2a78 5f36 5e32 2a78 5f37 5e32 7c0a  _4*x_6^2*x_7^2|.
│ │ │ │ -00018f90: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ -00018fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -00018fe0: 7c2a 785f 395e 362d 332a 785f 345e 322a  |*x_9^6-3*x_4^2*
│ │ │ │ -00018ff0: 785f 362a 785f 372a 785f 395e 372b 785f  x_6*x_7*x_9^7+x_
│ │ │ │ -00019000: 345e 332a 785f 395e 382c 785f 385e 382a  4^3*x_9^8,x_8^8*
│ │ │ │ -00019010: 785f 3130 5e33 2d33 2a78 5f37 2a78 5f38  x_10^3-3*x_7*x_8
│ │ │ │ -00019020: 5e37 2a78 5f31 305e 322a 785f 3131 7c0a  ^7*x_10^2*x_11|.
│ │ │ │ -00019030: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ -00019040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -00019080: 7c2b 332a 785f 375e 322a 785f 385e 362a  |+3*x_7^2*x_8^6*
│ │ │ │ -00019090: 785f 3130 2a78 5f31 315e 322d 785f 375e  x_10*x_11^2-x_7^
│ │ │ │ -000190a0: 332a 785f 385e 352a 785f 3131 5e33 2c78  3*x_8^5*x_11^3,x
│ │ │ │ -000190b0: 5f32 5e35 2a78 5f33 5e33 2a78 5f31 315e  _2^5*x_3^3*x_11^
│ │ │ │ -000190c0: 332d 332a 785f 325e 362a 785f 335e 7c0a  3-3*x_2^6*x_3^|.
│ │ │ │ -000190d0: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ -000190e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000190f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -00019120: 7c32 2a78 5f31 315e 322a 785f 3132 2b33  |2*x_11^2*x_12+3
│ │ │ │ -00019130: 2a78 5f32 5e37 2a78 5f33 2a78 5f31 312a  *x_2^7*x_3*x_11*
│ │ │ │ -00019140: 785f 3132 5e32 2d78 5f32 5e38 2a78 5f31  x_12^2-x_2^8*x_1
│ │ │ │ -00019150: 325e 3329 3b20 2020 2020 2020 2020 2020  2^3);           
│ │ │ │ -00019160: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00019170: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -00019180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000191a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000191b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -000191c0: 7c69 3920 3a20 7469 6d65 2069 7343 6f64  |i9 : time isCod
│ │ │ │ -000191d0: 696d 4174 4c65 6173 7428 352c 2049 2c20  imAtLeast(5, I, 
│ │ │ │ -000191e0: 5061 6972 4c69 6d69 7420 3d3e 2035 2c20  PairLimit => 5, 
│ │ │ │ -000191f0: 5665 7262 6f73 653d 3e74 7275 6529 2020  Verbose=>true)  
│ │ │ │ -00019200: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00019210: 7c20 2d2d 2075 7365 6420 302e 3030 3234  | -- used 0.0024
│ │ │ │ -00019220: 3531 3235 7320 2863 7075 293b 2030 2e30  5125s (cpu); 0.0
│ │ │ │ -00019230: 3032 3438 3231 3573 2028 7468 7265 6164  0248215s (thread
│ │ │ │ -00019240: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │ +00018a90: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00018aa0: 5a5a 2020 2020 2020 2020 2020 2020 2020  ZZ              
│ │ │ │ +00018ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018ad0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00018ae0: 7c6f 3820 3a20 4964 6561 6c20 6f66 202d  |o8 : Ideal of -
│ │ │ │ +00018af0: 2d2d 5b78 2020 2c20 7820 2c20 7820 2c20  --[x  , x , x , 
│ │ │ │ +00018b00: 7820 2c20 7820 202c 2078 202c 2078 202c  x , x  , x , x ,
│ │ │ │ +00018b10: 2078 2020 2c20 7820 2c20 7820 2c20 7820   x  , x , x , x 
│ │ │ │ +00018b20: 2c20 7820 5d20 2020 2020 2020 2020 7c0a  , x ]         |.
│ │ │ │ +00018b30: 7c20 2020 2020 2020 2020 2020 2020 2031  |              1
│ │ │ │ +00018b40: 3237 2020 3131 2020 2038 2020 2031 2020  27  11   8   1  
│ │ │ │ +00018b50: 2039 2020 2031 3220 2020 3620 2020 3520   9   12   6   5 
│ │ │ │ +00018b60: 2020 3130 2020 2032 2020 2034 2020 2033    10   2   4   3
│ │ │ │ +00018b70: 2020 2037 2020 2020 2020 2020 2020 7c0a     7          |.
│ │ │ │ +00018b80: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +00018b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +00018bd0: 7c2d 785f 315e 332a 785f 325e 352a 785f  |-x_1^3*x_2^5*x_
│ │ │ │ +00018be0: 3131 5e33 2c78 5f35 5e35 2a78 5f36 5e33  11^3,x_5^5*x_6^3
│ │ │ │ +00018bf0: 2a78 5f31 315e 332d 332a 785f 355e 362a  *x_11^3-3*x_5^6*
│ │ │ │ +00018c00: 785f 365e 322a 785f 3131 5e32 2a78 5f31  x_6^2*x_11^2*x_1
│ │ │ │ +00018c10: 322b 332a 785f 355e 372a 785f 362a 7c0a  2+3*x_5^7*x_6*|.
│ │ │ │ +00018c20: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +00018c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +00018c70: 7c78 5f31 312a 785f 3132 5e32 2d78 5f35  |x_11*x_12^2-x_5
│ │ │ │ +00018c80: 5e38 2a78 5f31 325e 332c 785f 315e 352a  ^8*x_12^3,x_1^5*
│ │ │ │ +00018c90: 785f 325e 332a 785f 345e 332d 332a 785f  x_2^3*x_4^3-3*x_
│ │ │ │ +00018ca0: 315e 362a 785f 325e 322a 785f 345e 322a  1^6*x_2^2*x_4^2*
│ │ │ │ +00018cb0: 785f 352b 332a 785f 315e 372a 785f 7c0a  x_5+3*x_1^7*x_|.
│ │ │ │ +00018cc0: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +00018cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +00018d10: 7c32 2a78 5f34 2a78 5f35 5e32 2d78 5f31  |2*x_4*x_5^2-x_1
│ │ │ │ +00018d20: 5e38 2a78 5f35 5e33 2c78 5f36 5e38 2a78  ^8*x_5^3,x_6^8*x
│ │ │ │ +00018d30: 5f31 315e 332d 332a 785f 352a 785f 365e  _11^3-3*x_5*x_6^
│ │ │ │ +00018d40: 372a 785f 3131 5e32 2a78 5f31 322b 332a  7*x_11^2*x_12+3*
│ │ │ │ +00018d50: 785f 355e 322a 785f 365e 362a 785f 7c0a  x_5^2*x_6^6*x_|.
│ │ │ │ +00018d60: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +00018d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +00018db0: 7c31 312a 785f 3132 5e32 2d78 5f35 5e33  |11*x_12^2-x_5^3
│ │ │ │ +00018dc0: 2a78 5f36 5e35 2a78 5f31 325e 332c 785f  *x_6^5*x_12^3,x_
│ │ │ │ +00018dd0: 385e 332a 785f 3130 5e38 2d33 2a78 5f37  8^3*x_10^8-3*x_7
│ │ │ │ +00018de0: 2a78 5f38 5e32 2a78 5f31 305e 372a 785f  *x_8^2*x_10^7*x_
│ │ │ │ +00018df0: 3131 2b33 2a78 5f37 5e32 2a78 5f38 7c0a  11+3*x_7^2*x_8|.
│ │ │ │ +00018e00: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +00018e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +00018e50: 7c2a 785f 3130 5e36 2a78 5f31 315e 322d  |*x_10^6*x_11^2-
│ │ │ │ +00018e60: 785f 375e 332a 785f 3130 5e35 2a78 5f31  x_7^3*x_10^5*x_1
│ │ │ │ +00018e70: 315e 332c 785f 325e 382a 785f 345e 332d  1^3,x_2^8*x_4^3-
│ │ │ │ +00018e80: 332a 785f 312a 785f 325e 372a 785f 345e  3*x_1*x_2^7*x_4^
│ │ │ │ +00018e90: 322a 785f 352b 332a 785f 315e 322a 7c0a  2*x_5+3*x_1^2*|.
│ │ │ │ +00018ea0: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +00018eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +00018ef0: 7c78 5f32 5e36 2a78 5f34 2a78 5f35 5e32  |x_2^6*x_4*x_5^2
│ │ │ │ +00018f00: 2d78 5f31 5e33 2a78 5f32 5e35 2a78 5f35  -x_1^3*x_2^5*x_5
│ │ │ │ +00018f10: 5e33 2c2d 785f 365e 332a 785f 3131 5e38  ^3,-x_6^3*x_11^8
│ │ │ │ +00018f20: 2b33 2a78 5f35 2a78 5f36 5e32 2a78 5f31  +3*x_5*x_6^2*x_1
│ │ │ │ +00018f30: 315e 372a 785f 3132 2d33 2a78 5f35 7c0a  1^7*x_12-3*x_5|.
│ │ │ │ +00018f40: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +00018f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +00018f90: 7c5e 322a 785f 362a 785f 3131 5e36 2a78  |^2*x_6*x_11^6*x
│ │ │ │ +00018fa0: 5f31 325e 322b 785f 355e 332a 785f 3131  _12^2+x_5^3*x_11
│ │ │ │ +00018fb0: 5e35 2a78 5f31 325e 332c 2d78 5f36 5e33  ^5*x_12^3,-x_6^3
│ │ │ │ +00018fc0: 2a78 5f37 5e33 2a78 5f39 5e35 2b33 2a78  *x_7^3*x_9^5+3*x
│ │ │ │ +00018fd0: 5f34 2a78 5f36 5e32 2a78 5f37 5e32 7c0a  _4*x_6^2*x_7^2|.
│ │ │ │ +00018fe0: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +00018ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +00019030: 7c2a 785f 395e 362d 332a 785f 345e 322a  |*x_9^6-3*x_4^2*
│ │ │ │ +00019040: 785f 362a 785f 372a 785f 395e 372b 785f  x_6*x_7*x_9^7+x_
│ │ │ │ +00019050: 345e 332a 785f 395e 382c 785f 385e 382a  4^3*x_9^8,x_8^8*
│ │ │ │ +00019060: 785f 3130 5e33 2d33 2a78 5f37 2a78 5f38  x_10^3-3*x_7*x_8
│ │ │ │ +00019070: 5e37 2a78 5f31 305e 322a 785f 3131 7c0a  ^7*x_10^2*x_11|.
│ │ │ │ +00019080: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +00019090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000190a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000190b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000190c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +000190d0: 7c2b 332a 785f 375e 322a 785f 385e 362a  |+3*x_7^2*x_8^6*
│ │ │ │ +000190e0: 785f 3130 2a78 5f31 315e 322d 785f 375e  x_10*x_11^2-x_7^
│ │ │ │ +000190f0: 332a 785f 385e 352a 785f 3131 5e33 2c78  3*x_8^5*x_11^3,x
│ │ │ │ +00019100: 5f32 5e35 2a78 5f33 5e33 2a78 5f31 315e  _2^5*x_3^3*x_11^
│ │ │ │ +00019110: 332d 332a 785f 325e 362a 785f 335e 7c0a  3-3*x_2^6*x_3^|.
│ │ │ │ +00019120: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +00019130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +00019170: 7c32 2a78 5f31 315e 322a 785f 3132 2b33  |2*x_11^2*x_12+3
│ │ │ │ +00019180: 2a78 5f32 5e37 2a78 5f33 2a78 5f31 312a  *x_2^7*x_3*x_11*
│ │ │ │ +00019190: 785f 3132 5e32 2d78 5f32 5e38 2a78 5f31  x_12^2-x_2^8*x_1
│ │ │ │ +000191a0: 325e 3329 3b20 2020 2020 2020 2020 2020  2^3);           
│ │ │ │ +000191b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000191c0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +000191d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000191e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000191f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00019210: 7c69 3920 3a20 7469 6d65 2069 7343 6f64  |i9 : time isCod
│ │ │ │ +00019220: 696d 4174 4c65 6173 7428 352c 2049 2c20  imAtLeast(5, I, 
│ │ │ │ +00019230: 5061 6972 4c69 6d69 7420 3d3e 2035 2c20  PairLimit => 5, 
│ │ │ │ +00019240: 5665 7262 6f73 653d 3e74 7275 6529 2020  Verbose=>true)  
│ │ │ │  00019250: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00019260: 7c69 7343 6f64 696d 4174 4c65 6173 743a  |isCodimAtLeast:
│ │ │ │ -00019270: 2043 6f6d 7075 7469 6e67 2063 6f64 696d   Computing codim
│ │ │ │ -00019280: 206f 6620 6d6f 6e6f 6d69 616c 7320 6261   of monomials ba
│ │ │ │ -00019290: 7365 6420 6f6e 2069 6465 616c 2067 656e  sed on ideal gen
│ │ │ │ -000192a0: 6572 6174 6f72 732e 2020 2020 2020 7c0a  erators.      |.
│ │ │ │ -000192b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -000192c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000192d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000192e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000192f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00019300: 7c6f 3920 3d20 7472 7565 2020 2020 2020  |o9 = true      
│ │ │ │ +00019260: 7c20 2d2d 2075 7365 6420 302e 3030 3234  | -- used 0.0024
│ │ │ │ +00019270: 3833 3537 7320 2863 7075 293b 2030 2e30  8357s (cpu); 0.0
│ │ │ │ +00019280: 3032 3939 3638 3973 2028 7468 7265 6164  0299689s (thread
│ │ │ │ +00019290: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │ +000192a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000192b0: 7c69 7343 6f64 696d 4174 4c65 6173 743a  |isCodimAtLeast:
│ │ │ │ +000192c0: 2043 6f6d 7075 7469 6e67 2063 6f64 696d   Computing codim
│ │ │ │ +000192d0: 206f 6620 6d6f 6e6f 6d69 616c 7320 6261   of monomials ba
│ │ │ │ +000192e0: 7365 6420 6f6e 2069 6465 616c 2067 656e  sed on ideal gen
│ │ │ │ +000192f0: 6572 6174 6f72 732e 2020 2020 2020 7c0a  erators.      |.
│ │ │ │ +00019300: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00019310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019340: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00019350: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -00019360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -000193a0: 7c69 3130 203a 2074 696d 6520 6973 436f  |i10 : time isCo
│ │ │ │ -000193b0: 6469 6d41 744c 6561 7374 2835 2c20 492c  dimAtLeast(5, I,
│ │ │ │ -000193c0: 2050 6169 724c 696d 6974 203d 3e20 3230   PairLimit => 20
│ │ │ │ -000193d0: 302c 2056 6572 626f 7365 3d3e 6661 6c73  0, Verbose=>fals
│ │ │ │ -000193e0: 6529 2020 2020 2020 2020 2020 2020 7c0a  e)            |.
│ │ │ │ -000193f0: 7c20 2d2d 2075 7365 6420 302e 3030 3233  | -- used 0.0023
│ │ │ │ -00019400: 3131 3937 7320 2863 7075 293b 2030 2e30  1197s (cpu); 0.0
│ │ │ │ -00019410: 3032 3331 3031 7320 2874 6872 6561 6429  023101s (thread)
│ │ │ │ -00019420: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │ -00019430: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00019440: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00019450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019350: 7c6f 3920 3d20 7472 7565 2020 2020 2020  |o9 = true      
│ │ │ │ +00019360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019390: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000193a0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +000193b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000193c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000193d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000193e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000193f0: 7c69 3130 203a 2074 696d 6520 6973 436f  |i10 : time isCo
│ │ │ │ +00019400: 6469 6d41 744c 6561 7374 2835 2c20 492c  dimAtLeast(5, I,
│ │ │ │ +00019410: 2050 6169 724c 696d 6974 203d 3e20 3230   PairLimit => 20
│ │ │ │ +00019420: 302c 2056 6572 626f 7365 3d3e 6661 6c73  0, Verbose=>fals
│ │ │ │ +00019430: 6529 2020 2020 2020 2020 2020 2020 7c0a  e)            |.
│ │ │ │ +00019440: 7c20 2d2d 2075 7365 6420 302e 3030 3135  | -- used 0.0015
│ │ │ │ +00019450: 3938 3836 7320 2863 7075 293b 2030 2e30  9886s (cpu); 0.0
│ │ │ │ +00019460: 3032 3835 3538 3973 2028 7468 7265 6164  0285589s (thread
│ │ │ │ +00019470: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │  00019480: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00019490: 7c6f 3130 203d 2074 7275 6520 2020 2020  |o10 = true     
│ │ │ │ +00019490: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000194a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000194b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000194c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000194d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000194e0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -000194f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00019530: 0a4e 6f74 6963 6520 696e 2074 6865 2066  .Notice in the f
│ │ │ │ -00019540: 6972 7374 2063 6173 6520 7468 6520 6675  irst case the fu
│ │ │ │ -00019550: 6e63 7469 6f6e 2072 6574 7572 6e65 6420  nction returned 
│ │ │ │ -00019560: 6e75 6c6c 2c20 6265 6361 7573 6520 7468  null, because th
│ │ │ │ -00019570: 6520 6465 7074 6820 6f66 0a73 6561 7263  e depth of.searc
│ │ │ │ -00019580: 6820 7761 7320 6e6f 7420 6869 6768 2065  h was not high e
│ │ │ │ -00019590: 6e6f 7567 682e 2020 4974 206f 6e6c 7920  nough.  It only 
│ │ │ │ -000195a0: 636f 6d70 7574 6564 2063 6f64 696d 2035  computed codim 5
│ │ │ │ -000195b0: 2074 696d 6573 2e20 2054 6865 2073 6563   times.  The sec
│ │ │ │ -000195c0: 6f6e 640a 7265 7475 726e 6564 2074 7275  ond.returned tru
│ │ │ │ -000195d0: 652c 2062 7574 2069 7420 6469 6420 736f  e, but it did so
│ │ │ │ -000195e0: 2061 7320 736f 6f6e 2061 7320 7468 6520   as soon as the 
│ │ │ │ -000195f0: 616e 7377 6572 2077 6173 2066 6f75 6e64  answer was found
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│ │ │ │ -00019650: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ -00019670: 4174 4c65 6173 7428 5a5a 2c49 6465 616c  AtLeast(ZZ,Ideal
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│ │ │ │ -00019730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -000197c0: 756c 6179 322d 312e 3235 2e31 312b 6473  ulay2-1.25.11+ds
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│ │ │ │ -000197e0: 636b 6167 6573 2f46 6173 744d 696e 6f72  ckages/FastMinor
│ │ │ │ -000197f0: 732e 0a6d 323a 3232 3436 3a30 2e0a 1f0a  s..m2:2246:0....
│ │ │ │ -00019800: 4669 6c65 3a20 4661 7374 4d69 6e6f 7273  File: FastMinors
│ │ │ │ -00019810: 2e69 6e66 6f2c 204e 6f64 653a 2069 7344  .info, Node: isD
│ │ │ │ -00019820: 696d 4174 4d6f 7374 2c20 4e65 7874 3a20  imAtMost, Next: 
│ │ │ │ -00019830: 6973 5261 6e6b 4174 4c65 6173 742c 2050  isRankAtLeast, P
│ │ │ │ -00019840: 7265 763a 2069 7343 6f64 696d 4174 4c65  rev: isCodimAtLe
│ │ │ │ -00019850: 6173 742c 2055 703a 2054 6f70 0a0a 6973  ast, Up: Top..is
│ │ │ │ -00019860: 4469 6d41 744d 6f73 7420 2d2d 2072 6574  DimAtMost -- ret
│ │ │ │ -00019870: 7572 6e73 2074 7275 6520 6966 2077 6520  urns true if we 
│ │ │ │ -00019880: 6361 6e20 7175 6963 6b6c 7920 7365 6520  can quickly see 
│ │ │ │ -00019890: 7768 6574 6865 7220 7468 6520 6469 6d20  whether the dim 
│ │ │ │ -000198a0: 6973 2061 7420 6d6f 7374 2061 2067 6976  is at most a giv
│ │ │ │ -000198b0: 656e 206e 756d 6265 720a 2a2a 2a2a 2a2a  en number.******
│ │ │ │ -000198c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000198d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000198e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000198f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00019900: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00019910: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765  *****..  * Usage
│ │ │ │ -00019920: 3a20 0a20 2020 2020 2020 2069 7344 696d  : .        isDim
│ │ │ │ -00019930: 4174 4d6f 7374 286e 2c20 4929 0a20 202a  AtMost(n, I).  *
│ │ │ │ -00019940: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ -00019950: 206e 2c20 616e 202a 6e6f 7465 2069 6e74   n, an *note int
│ │ │ │ -00019960: 6567 6572 3a20 284d 6163 6175 6c61 7932  eger: (Macaulay2
│ │ │ │ -00019970: 446f 6329 5a5a 2c2c 2061 6e20 696e 7465  Doc)ZZ,, an inte
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│ │ │ │ -00019990: 6e20 2a6e 6f74 6520 6964 6561 6c3a 2028  n *note ideal: (
│ │ │ │ -000199a0: 4d61 6361 756c 6179 3244 6f63 2949 6465  Macaulay2Doc)Ide
│ │ │ │ -000199b0: 616c 2c2c 2061 6e20 6964 6561 6c20 696e  al,, an ideal in
│ │ │ │ -000199c0: 2061 2070 6f6c 796e 6f6d 6961 6c20 7269   a polynomial ri
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│ │ │ │ -000199e0: 6120 6669 656c 642c 206f 7220 6120 7175  a field, or a qu
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│ │ │ │ -00019a20: 4d61 6361 756c 6179 3244 6f63 2975 7369  Macaulay2Doc)usi
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│ │ │ │ -00019a60: 4c69 6d69 7420 3d3e 202e 2e2e 2c20 6465  Limit => ..., de
│ │ │ │ -00019a70: 6661 756c 7420 7661 6c75 6520 3130 300a  fault value 100.
│ │ │ │ -00019a80: 2020 2020 2020 2a20 5350 6169 7273 4675        * SPairsFu
│ │ │ │ -00019a90: 6e63 7469 6f6e 203d 3e20 2e2e 2e2c 2064  nction => ..., d
│ │ │ │ -00019aa0: 6566 6175 6c74 2076 616c 7565 0a20 2020  efault value.   
│ │ │ │ -00019ab0: 2020 2020 2046 756e 6374 696f 6e43 6c6f       FunctionClo
│ │ │ │ -00019ac0: 7375 7265 5b2e 2e2f 4661 7374 4d69 6e6f  sure[../FastMino
│ │ │ │ -00019ad0: 7273 2e6d 323a 3231 313a 3233 2d32 3131  rs.m2:211:23-211
│ │ │ │ -00019ae0: 3a34 325d 0a20 2020 2020 202a 2056 6572  :42].      * Ver
│ │ │ │ -00019af0: 626f 7365 203d 3e20 2e2e 2e2c 2064 6566  bose => ..., def
│ │ │ │ -00019b00: 6175 6c74 2076 616c 7565 2066 616c 7365  ault value false
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│ │ │ │ -00019b20: 2020 2020 2a20 7472 7565 2069 6620 7468      * true if th
│ │ │ │ -00019b30: 6520 6469 6d65 6e73 696f 6e20 6f66 2049  e dimension of I
│ │ │ │ -00019b40: 2069 7320 6174 206d 6f73 7420 6e20 6f72   is at most n or
│ │ │ │ -00019b50: 206e 756c 6c20 6966 2074 6865 2066 756e   null if the fun
│ │ │ │ -00019b60: 6374 696f 6e20 6361 6e6e 6f74 0a20 2020  ction cannot.   
│ │ │ │ -00019b70: 2020 2020 2074 656c 6c20 7768 6574 6865       tell whethe
│ │ │ │ -00019b80: 7220 7468 6520 6469 6d65 6e73 696f 6e20  r the dimension 
│ │ │ │ -00019b90: 6973 2061 7420 6d6f 7374 206e 0a0a 4465  is at most n..De
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│ │ │ │ -00019bb0: 3d3d 3d3d 3d0a 0a54 6869 7320 7369 6d70  =====..This simp
│ │ │ │ -00019bc0: 6c79 2063 616c 6c73 2069 7343 6f64 696d  ly calls isCodim
│ │ │ │ -00019bd0: 4174 4c65 6173 742c 2070 6173 7369 6e67  AtLeast, passing
│ │ │ │ -00019be0: 206f 7074 696f 6e73 2061 7320 6465 7363   options as desc
│ │ │ │ -00019bf0: 7269 6265 6420 7468 6572 652e 0a0a 5365  ribed there...Se
│ │ │ │ -00019c00: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a  e also.========.
│ │ │ │ -00019c10: 0a20 202a 202a 6e6f 7465 2069 7343 6f64  .  * *note isCod
│ │ │ │ -00019c20: 696d 4174 4c65 6173 743a 2069 7343 6f64  imAtLeast: isCod
│ │ │ │ -00019c30: 696d 4174 4c65 6173 742c 202d 2d20 7265  imAtLeast, -- re
│ │ │ │ -00019c40: 7475 726e 7320 7472 7565 2069 6620 7765  turns true if we
│ │ │ │ -00019c50: 2063 616e 2071 7569 636b 6c79 2073 6565   can quickly see
│ │ │ │ -00019c60: 0a20 2020 2077 6865 7468 6572 2074 6865  .    whether the
│ │ │ │ -00019c70: 2063 6f64 696d 2069 7320 6174 206c 6561   codim is at lea
│ │ │ │ -00019c80: 7374 2061 2067 6976 656e 206e 756d 6265  st a given numbe
│ │ │ │ -00019c90: 720a 0a57 6179 7320 746f 2075 7365 2069  r..Ways to use i
│ │ │ │ -00019ca0: 7344 696d 4174 4d6f 7374 3a0a 3d3d 3d3d  sDimAtMost:.====
│ │ │ │ -00019cb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00019cc0: 3d3d 3d3d 0a0a 2020 2a20 2269 7344 696d  ====..  * "isDim
│ │ │ │ -00019cd0: 4174 4d6f 7374 285a 5a2c 4964 6561 6c29  AtMost(ZZ,Ideal)
│ │ │ │ -00019ce0: 220a 0a46 6f72 2074 6865 2070 726f 6772  "..For the progr
│ │ │ │ -00019cf0: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d  ammer.==========
│ │ │ │ -00019d00: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62  ========..The ob
│ │ │ │ -00019d10: 6a65 6374 202a 6e6f 7465 2069 7344 696d  ject *note isDim
│ │ │ │ -00019d20: 4174 4d6f 7374 3a20 6973 4469 6d41 744d  AtMost: isDimAtM
│ │ │ │ -00019d30: 6f73 742c 2069 7320 6120 2a6e 6f74 6520  ost, is a *note 
│ │ │ │ -00019d40: 6d65 7468 6f64 2066 756e 6374 696f 6e20  method function 
│ │ │ │ -00019d50: 7769 7468 0a6f 7074 696f 6e73 3a20 284d  with.options: (M
│ │ │ │ -00019d60: 6163 6175 6c61 7932 446f 6329 4d65 7468  acaulay2Doc)Meth
│ │ │ │ -00019d70: 6f64 4675 6e63 7469 6f6e 5769 7468 4f70  odFunctionWithOp
│ │ │ │ -00019d80: 7469 6f6e 732c 2e0a 0a2d 2d2d 2d2d 2d2d  tions,...-------
│ │ │ │ -00019d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019dd0: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f  --------..The so
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│ │ │ │ -00019e30: 6175 6c61 7932 2f70 6163 6b61 6765 732f  aulay2/packages/
│ │ │ │ -00019e40: 4661 7374 4d69 6e6f 7273 2e0a 6d32 3a32  FastMinors..m2:2
│ │ │ │ -00019e50: 3237 323a 302e 0a1f 0a46 696c 653a 2046  272:0....File: F
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│ │ │ │ -00019e70: 4e6f 6465 3a20 6973 5261 6e6b 4174 4c65  Node: isRankAtLe
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│ │ │ │ -00019eb0: 5f70 645f 7064 5f70 645f 7270 2c20 5072  _pd_pd_pd_rp, Pr
│ │ │ │ -00019ec0: 6576 3a20 6973 4469 6d41 744d 6f73 742c  ev: isDimAtMost,
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│ │ │ │ -00019ee0: 4174 4c65 6173 7420 2d2d 2064 6574 6572  AtLeast -- deter
│ │ │ │ -00019ef0: 6d69 6e65 7320 6966 2074 6865 206d 6174  mines if the mat
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│ │ │ │ -00019f10: 6c65 6173 7420 6120 6e75 6d62 6572 0a2a  least a number.*
│ │ │ │ -00019f20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00019f30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00019f40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00019f50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00019f60: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20  ***..  * Usage: 
│ │ │ │ -00019f70: 0a20 2020 2020 2020 2069 7352 616e 6b41  .        isRankA
│ │ │ │ -00019f80: 744c 6561 7374 286e 312c 204d 3129 0a20  tLeast(n1, M1). 
│ │ │ │ -00019f90: 202a 2049 6e70 7574 733a 0a20 2020 2020   * Inputs:.     
│ │ │ │ -00019fa0: 202a 206e 312c 2061 6e20 2a6e 6f74 6520   * n1, an *note 
│ │ │ │ -00019fb0: 696e 7465 6765 723a 2028 4d61 6361 756c  integer: (Macaul
│ │ │ │ -00019fc0: 6179 3244 6f63 295a 5a2c 2c20 0a20 2020  ay2Doc)ZZ,, .   
│ │ │ │ -00019fd0: 2020 202a 204d 312c 2061 202a 6e6f 7465     * M1, a *note
│ │ │ │ -00019fe0: 206d 6174 7269 783a 2028 4d61 6361 756c   matrix: (Macaul
│ │ │ │ -00019ff0: 6179 3244 6f63 294d 6174 7269 782c 2c20  ay2Doc)Matrix,, 
│ │ │ │ -0001a000: 0a20 202a 202a 6e6f 7465 204f 7074 696f  .  * *note Optio
│ │ │ │ -0001a010: 6e61 6c20 696e 7075 7473 3a20 284d 6163  nal inputs: (Mac
│ │ │ │ -0001a020: 6175 6c61 7932 446f 6329 7573 696e 6720  aulay2Doc)using 
│ │ │ │ -0001a030: 6675 6e63 7469 6f6e 7320 7769 7468 206f  functions with o
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│ │ │ │ -0001a060: 6574 5374 7261 7465 6779 3a20 4465 7453  etStrategy: DetS
│ │ │ │ -0001a070: 7472 6174 6567 792c 203d 3e20 2e2e 2e2c  trategy, => ...,
│ │ │ │ -0001a080: 2064 6566 6175 6c74 2076 616c 7565 2052   default value R
│ │ │ │ -0001a090: 616e 6b2c 2044 6574 5374 7261 7465 6779  ank, DetStrategy
│ │ │ │ -0001a0a0: 0a20 2020 2020 2020 2069 7320 6120 7374  .        is a st
│ │ │ │ -0001a0b0: 7261 7465 6779 2066 6f72 2061 6c6c 6f77  rategy for allow
│ │ │ │ -0001a0c0: 696e 6720 7468 6520 7573 6572 2074 6f20  ing the user to 
│ │ │ │ -0001a0d0: 6368 6f6f 7365 2068 6f77 2064 6574 6572  choose how deter
│ │ │ │ -0001a0e0: 6d69 6e61 6e74 7320 286f 720a 2020 2020  minants (or.    
│ │ │ │ -0001a0f0: 2020 2020 7261 6e6b 292c 2069 7320 636f      rank), is co
│ │ │ │ -0001a100: 6d70 7574 6564 0a20 2020 2020 202a 202a  mputed.      * *
│ │ │ │ -0001a110: 6e6f 7465 204d 6178 4d69 6e6f 7273 3a20  note MaxMinors: 
│ │ │ │ -0001a120: 4d61 784d 696e 6f72 732c 203d 3e20 2e2e  MaxMinors, => ..
│ │ │ │ -0001a130: 2e2c 2064 6566 6175 6c74 2076 616c 7565  ., default value
│ │ │ │ -0001a140: 206e 756c 6c2c 2061 6e20 6f70 7469 6f6e   null, an option
│ │ │ │ -0001a150: 2074 6f0a 2020 2020 2020 2020 636f 6e74   to.        cont
│ │ │ │ -0001a160: 726f 6c20 6465 7074 6820 6f66 2073 6561  rol depth of sea
│ │ │ │ -0001a170: 7263 680a 2020 2020 2020 2a20 2a6e 6f74  rch.      * *not
│ │ │ │ -0001a180: 6520 506f 696e 744f 7074 696f 6e73 3a20  e PointOptions: 
│ │ │ │ -0001a190: 506f 696e 744f 7074 696f 6e73 2c20 3d3e  PointOptions, =>
│ │ │ │ -0001a1a0: 202e 2e2e 2c20 6465 6661 756c 7420 7661   ..., default va
│ │ │ │ -0001a1b0: 6c75 6520 7b53 7472 6174 6567 7920 3d3e  lue {Strategy =>
│ │ │ │ -0001a1c0: 0a20 2020 2020 2020 2044 6566 6175 6c74  .        Default
│ │ │ │ -0001a1d0: 2c20 486f 6d6f 6765 6e65 6f75 7320 3d3e  , Homogeneous =>
│ │ │ │ -0001a1e0: 2066 616c 7365 2c20 5265 706c 6163 656d   false, Replacem
│ │ │ │ -0001a1f0: 656e 7420 3d3e 2042 696e 6f6d 6961 6c2c  ent => Binomial,
│ │ │ │ -0001a200: 2045 7874 656e 6446 6965 6c64 203d 3e0a   ExtendField =>.
│ │ │ │ -0001a210: 2020 2020 2020 2020 7472 7565 2c20 506f          true, Po
│ │ │ │ -0001a220: 696e 7443 6865 636b 4174 7465 6d70 7473  intCheckAttempts
│ │ │ │ -0001a230: 203d 3e20 302c 2044 6563 6f6d 706f 7369   => 0, Decomposi
│ │ │ │ -0001a240: 7469 6f6e 5374 7261 7465 6779 203d 3e20  tionStrategy => 
│ │ │ │ -0001a250: 4465 636f 6d70 6f73 652c 0a20 2020 2020  Decompose,.     
│ │ │ │ -0001a260: 2020 204e 756d 5468 7265 6164 7354 6f55     NumThreadsToU
│ │ │ │ -0001a270: 7365 203d 3e20 312c 2044 696d 656e 7369  se => 1, Dimensi
│ │ │ │ -0001a280: 6f6e 4675 6e63 7469 6f6e 203d 3e20 6469  onFunction => di
│ │ │ │ -0001a290: 6d2c 2056 6572 626f 7365 203d 3e20 6661  m, Verbose => fa
│ │ │ │ -0001a2a0: 6c73 657d 2c0a 2020 2020 2020 2020 6f70  lse},.        op
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│ │ │ │ -0001a340: 6f6d 207b 506f 696e 7473 203d 3e20 302c  om {Points => 0,
│ │ │ │ -0001a350: 2052 616e 646f 6d20 3d3e 2030 2c20 4752   Random => 0, GR
│ │ │ │ -0001a360: 6576 4c65 784c 6172 6765 7374 203d 3e20  evLexLargest => 
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│ │ │ │ -0001a3a0: 302c 204c 6578 536d 616c 6c65 7374 203d  0, LexSmallest =
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│ │ │ │ -0001a3f0: 764c 6578 536d 616c 6c65 7374 203d 3e20  vLexSmallest => 
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│ │ │ │ -0001a5c0: 6b3a 2067 6574 5375 626d 6174 7269 784f  k: getSubmatrixO
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│ │ │ │ -0001a6e0: 6520 7468 6520 6f70 7469 6f6e 204d 6178  e the option Max
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│ │ │ │ -0001a700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a730: 2d2d 2d2d 2d2b 0a7c 6931 203a 2052 203d  -----+.|i1 : R =
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│ │ │ │ -0001a760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a770: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -0001a780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a7b0: 2d2b 0a7c 6932 203a 204d 203d 206d 6174  -+.|i2 : M = mat
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│ │ │ │ -0001a7d0: 2b79 7d2c 207b 302c 302c 312c 302c 787d  +y}, {0,0,1,0,x}
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│ │ │ │ -0001a800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a820: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │ -0001a840: 2020 2020 3520 2020 2020 2020 2020 2020      5           
│ │ │ │ +000194e0: 7c6f 3130 203d 2074 7275 6520 2020 2020  |o10 = true     
│ │ │ │ +000194f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019520: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00019530: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00019540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
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│ │ │ │ +00019690: 4c65 6173 743a 0a3d 3d3d 3d3d 3d3d 3d3d  Least:.=========
│ │ │ │ +000196a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ +00019720: 6469 6d41 744c 6561 7374 2c20 6973 2061  dimAtLeast, is a
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│ │ │ │ +00019790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000197a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000197b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000197c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a  ---------------.
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│ │ │ │ +00019880: 6973 5261 6e6b 4174 4c65 6173 742c 2050  isRankAtLeast, P
│ │ │ │ +00019890: 7265 763a 2069 7343 6f64 696d 4174 4c65  rev: isCodimAtLe
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│ │ │ │ +000198b0: 4469 6d41 744d 6f73 7420 2d2d 2072 6574  DimAtMost -- ret
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│ │ │ │ +000198d0: 6361 6e20 7175 6963 6b6c 7920 7365 6520  can quickly see 
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│ │ │ │ +000198f0: 6973 2061 7420 6d6f 7374 2061 2067 6976  is at most a giv
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│ │ │ │ +00019910: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00019920: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00019930: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00019940: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00019950: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00019960: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765  *****..  * Usage
│ │ │ │ +00019970: 3a20 0a20 2020 2020 2020 2069 7344 696d  : .        isDim
│ │ │ │ +00019980: 4174 4d6f 7374 286e 2c20 4929 0a20 202a  AtMost(n, I).  *
│ │ │ │ +00019990: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ +000199a0: 206e 2c20 616e 202a 6e6f 7465 2069 6e74   n, an *note int
│ │ │ │ +000199b0: 6567 6572 3a20 284d 6163 6175 6c61 7932  eger: (Macaulay2
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│ │ │ │ +000199e0: 6e20 2a6e 6f74 6520 6964 6561 6c3a 2028  n *note ideal: (
│ │ │ │ +000199f0: 4d61 6361 756c 6179 3244 6f63 2949 6465  Macaulay2Doc)Ide
│ │ │ │ +00019a00: 616c 2c2c 2061 6e20 6964 6561 6c20 696e  al,, an ideal in
│ │ │ │ +00019a10: 2061 2070 6f6c 796e 6f6d 6961 6c20 7269   a polynomial ri
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│ │ │ │ +00019a30: 6120 6669 656c 642c 206f 7220 6120 7175  a field, or a qu
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│ │ │ │ +00019b00: 2020 2020 2046 756e 6374 696f 6e43 6c6f       FunctionClo
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│ │ │ │ +00019b50: 6175 6c74 2076 616c 7565 2066 616c 7365  ault value false
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│ │ │ │ +00019b70: 2020 2020 2a20 7472 7565 2069 6620 7468      * true if th
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│ │ │ │ +00019d00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ +00019d50: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62  ========..The ob
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│ │ │ │ +00019de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00019e60: 7061 7468 2f6d 6163 6175 6c61 7932 2d31  path/macaulay2-1
│ │ │ │ +00019e70: 2e32 352e 3131 2b64 732f 4d32 2f4d 6163  .25.11+ds/M2/Mac
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│ │ │ │ +00019ee0: 6b41 744c 6561 7374 5f6c 705f 7064 5f70  kAtLeast_lp_pd_p
│ │ │ │ +00019ef0: 645f 7064 5f63 6d54 6872 6561 6473 3d3e  d_pd_cmThreads=>
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│ │ │ │ +00019f10: 6576 3a20 6973 4469 6d41 744d 6f73 742c  ev: isDimAtMost,
│ │ │ │ +00019f20: 2055 703a 2054 6f70 0a0a 6973 5261 6e6b   Up: Top..isRank
│ │ │ │ +00019f30: 4174 4c65 6173 7420 2d2d 2064 6574 6572  AtLeast -- deter
│ │ │ │ +00019f40: 6d69 6e65 7320 6966 2074 6865 206d 6174  mines if the mat
│ │ │ │ +00019f50: 7269 7820 6861 7320 7261 6e6b 2061 7420  rix has rank at 
│ │ │ │ +00019f60: 6c65 6173 7420 6120 6e75 6d62 6572 0a2a  least a number.*
│ │ │ │ +00019f70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00019f80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00019f90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00019fa0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00019fb0: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20  ***..  * Usage: 
│ │ │ │ +00019fc0: 0a20 2020 2020 2020 2069 7352 616e 6b41  .        isRankA
│ │ │ │ +00019fd0: 744c 6561 7374 286e 312c 204d 3129 0a20  tLeast(n1, M1). 
│ │ │ │ +00019fe0: 202a 2049 6e70 7574 733a 0a20 2020 2020   * Inputs:.     
│ │ │ │ +00019ff0: 202a 206e 312c 2061 6e20 2a6e 6f74 6520   * n1, an *note 
│ │ │ │ +0001a000: 696e 7465 6765 723a 2028 4d61 6361 756c  integer: (Macaul
│ │ │ │ +0001a010: 6179 3244 6f63 295a 5a2c 2c20 0a20 2020  ay2Doc)ZZ,, .   
│ │ │ │ +0001a020: 2020 202a 204d 312c 2061 202a 6e6f 7465     * M1, a *note
│ │ │ │ +0001a030: 206d 6174 7269 783a 2028 4d61 6361 756c   matrix: (Macaul
│ │ │ │ +0001a040: 6179 3244 6f63 294d 6174 7269 782c 2c20  ay2Doc)Matrix,, 
│ │ │ │ +0001a050: 0a20 202a 202a 6e6f 7465 204f 7074 696f  .  * *note Optio
│ │ │ │ +0001a060: 6e61 6c20 696e 7075 7473 3a20 284d 6163  nal inputs: (Mac
│ │ │ │ +0001a070: 6175 6c61 7932 446f 6329 7573 696e 6720  aulay2Doc)using 
│ │ │ │ +0001a080: 6675 6e63 7469 6f6e 7320 7769 7468 206f  functions with o
│ │ │ │ +0001a090: 7074 696f 6e61 6c20 696e 7075 7473 2c3a  ptional inputs,:
│ │ │ │ +0001a0a0: 0a20 2020 2020 202a 202a 6e6f 7465 2044  .      * *note D
│ │ │ │ +0001a0b0: 6574 5374 7261 7465 6779 3a20 4465 7453  etStrategy: DetS
│ │ │ │ +0001a0c0: 7472 6174 6567 792c 203d 3e20 2e2e 2e2c  trategy, => ...,
│ │ │ │ +0001a0d0: 2064 6566 6175 6c74 2076 616c 7565 2052   default value R
│ │ │ │ +0001a0e0: 616e 6b2c 2044 6574 5374 7261 7465 6779  ank, DetStrategy
│ │ │ │ +0001a0f0: 0a20 2020 2020 2020 2069 7320 6120 7374  .        is a st
│ │ │ │ +0001a100: 7261 7465 6779 2066 6f72 2061 6c6c 6f77  rategy for allow
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│ │ │ │ +0001a120: 6368 6f6f 7365 2068 6f77 2064 6574 6572  choose how deter
│ │ │ │ +0001a130: 6d69 6e61 6e74 7320 286f 720a 2020 2020  minants (or.    
│ │ │ │ +0001a140: 2020 2020 7261 6e6b 292c 2069 7320 636f      rank), is co
│ │ │ │ +0001a150: 6d70 7574 6564 0a20 2020 2020 202a 202a  mputed.      * *
│ │ │ │ +0001a160: 6e6f 7465 204d 6178 4d69 6e6f 7273 3a20  note MaxMinors: 
│ │ │ │ +0001a170: 4d61 784d 696e 6f72 732c 203d 3e20 2e2e  MaxMinors, => ..
│ │ │ │ +0001a180: 2e2c 2064 6566 6175 6c74 2076 616c 7565  ., default value
│ │ │ │ +0001a190: 206e 756c 6c2c 2061 6e20 6f70 7469 6f6e   null, an option
│ │ │ │ +0001a1a0: 2074 6f0a 2020 2020 2020 2020 636f 6e74   to.        cont
│ │ │ │ +0001a1b0: 726f 6c20 6465 7074 6820 6f66 2073 6561  rol depth of sea
│ │ │ │ +0001a1c0: 7263 680a 2020 2020 2020 2a20 2a6e 6f74  rch.      * *not
│ │ │ │ +0001a1d0: 6520 506f 696e 744f 7074 696f 6e73 3a20  e PointOptions: 
│ │ │ │ +0001a1e0: 506f 696e 744f 7074 696f 6e73 2c20 3d3e  PointOptions, =>
│ │ │ │ +0001a1f0: 202e 2e2e 2c20 6465 6661 756c 7420 7661   ..., default va
│ │ │ │ +0001a200: 6c75 6520 7b53 7472 6174 6567 7920 3d3e  lue {Strategy =>
│ │ │ │ +0001a210: 0a20 2020 2020 2020 2044 6566 6175 6c74  .        Default
│ │ │ │ +0001a220: 2c20 486f 6d6f 6765 6e65 6f75 7320 3d3e  , Homogeneous =>
│ │ │ │ +0001a230: 2066 616c 7365 2c20 5265 706c 6163 656d   false, Replacem
│ │ │ │ +0001a240: 656e 7420 3d3e 2042 696e 6f6d 6961 6c2c  ent => Binomial,
│ │ │ │ +0001a250: 2045 7874 656e 6446 6965 6c64 203d 3e0a   ExtendField =>.
│ │ │ │ +0001a260: 2020 2020 2020 2020 7472 7565 2c20 506f          true, Po
│ │ │ │ +0001a270: 696e 7443 6865 636b 4174 7465 6d70 7473  intCheckAttempts
│ │ │ │ +0001a280: 203d 3e20 302c 2044 6563 6f6d 706f 7369   => 0, Decomposi
│ │ │ │ +0001a290: 7469 6f6e 5374 7261 7465 6779 203d 3e20  tionStrategy => 
│ │ │ │ +0001a2a0: 4465 636f 6d70 6f73 652c 0a20 2020 2020  Decompose,.     
│ │ │ │ +0001a2b0: 2020 204e 756d 5468 7265 6164 7354 6f55     NumThreadsToU
│ │ │ │ +0001a2c0: 7365 203d 3e20 312c 2044 696d 656e 7369  se => 1, Dimensi
│ │ │ │ +0001a2d0: 6f6e 4675 6e63 7469 6f6e 203d 3e20 6469  onFunction => di
│ │ │ │ +0001a2e0: 6d2c 2056 6572 626f 7365 203d 3e20 6661  m, Verbose => fa
│ │ │ │ +0001a2f0: 6c73 657d 2c0a 2020 2020 2020 2020 6f70  lse},.        op
│ │ │ │ +0001a300: 7469 6f6e 7320 746f 2070 6173 7320 746f  tions to pass to
│ │ │ │ +0001a310: 2066 756e 6374 696f 6e73 2069 6e20 7468   functions in th
│ │ │ │ +0001a320: 6520 7061 636b 6167 6520 5261 6e64 6f6d  e package Random
│ │ │ │ +0001a330: 506f 696e 7473 0a20 2020 2020 202a 202a  Points.      * *
│ │ │ │ +0001a340: 6e6f 7465 2053 7472 6174 6567 793a 2053  note Strategy: S
│ │ │ │ +0001a350: 7472 6174 6567 7944 6566 6175 6c74 2c20  trategyDefault, 
│ │ │ │ +0001a360: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420  => ..., default 
│ │ │ │ +0001a370: 7661 6c75 6520 6e65 7720 4f70 7469 6f6e  value new Option
│ │ │ │ +0001a380: 5461 626c 650a 2020 2020 2020 2020 6672  Table.        fr
│ │ │ │ +0001a390: 6f6d 207b 506f 696e 7473 203d 3e20 302c  om {Points => 0,
│ │ │ │ +0001a3a0: 2052 616e 646f 6d20 3d3e 2030 2c20 4752   Random => 0, GR
│ │ │ │ +0001a3b0: 6576 4c65 784c 6172 6765 7374 203d 3e20  evLexLargest => 
│ │ │ │ +0001a3c0: 302c 204c 6578 536d 616c 6c65 7374 5465  0, LexSmallestTe
│ │ │ │ +0001a3d0: 726d 203d 3e0a 2020 2020 2020 2020 3235  rm =>.        25
│ │ │ │ +0001a3e0: 2c20 4c65 784c 6172 6765 7374 203d 3e20  , LexLargest => 
│ │ │ │ +0001a3f0: 302c 204c 6578 536d 616c 6c65 7374 203d  0, LexSmallest =
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│ │ │ │ +0001a410: 6c6c 6573 7454 6572 6d20 3d3e 2032 352c  llestTerm => 25,
│ │ │ │ +0001a420: 0a20 2020 2020 2020 2052 616e 646f 6d4e  .        RandomN
│ │ │ │ +0001a430: 6f6e 7a65 726f 203d 3e20 302c 2047 5265  onzero => 0, GRe
│ │ │ │ +0001a440: 764c 6578 536d 616c 6c65 7374 203d 3e20  vLexSmallest => 
│ │ │ │ +0001a450: 3235 7d2c 2073 7472 6174 6567 6965 7320  25}, strategies 
│ │ │ │ +0001a460: 666f 7220 6368 6f6f 7369 6e67 0a20 2020  for choosing.   
│ │ │ │ +0001a470: 2020 2020 2073 7562 6d61 7472 6963 6573       submatrices
│ │ │ │ +0001a480: 0a20 2020 2020 202a 202a 6e6f 7465 2054  .      * *note T
│ │ │ │ +0001a490: 6872 6561 6473 3a20 6973 5261 6e6b 4174  hreads: isRankAt
│ │ │ │ +0001a4a0: 4c65 6173 745f 6c70 5f70 645f 7064 5f70  Least_lp_pd_pd_p
│ │ │ │ +0001a4b0: 645f 636d 5468 7265 6164 733d 3e5f 7064  d_cmThreads=>_pd
│ │ │ │ +0001a4c0: 5f70 645f 7064 5f72 702c 203d 3e0a 2020  _pd_pd_rp, =>.  
│ │ │ │ +0001a4d0: 2020 2020 2020 2e2e 2e2c 2064 6566 6175        ..., defau
│ │ │ │ +0001a4e0: 6c74 2076 616c 7565 2031 2c20 616e 206f  lt value 1, an o
│ │ │ │ +0001a4f0: 7074 696f 6e20 666f 7220 7661 7269 6f75  ption for variou
│ │ │ │ +0001a500: 7320 6675 6e63 7469 6f6e 730a 2020 2020  s functions.    
│ │ │ │ +0001a510: 2020 2a20 5665 7262 6f73 6520 3d3e 202e    * Verbose => .
│ │ │ │ +0001a520: 2e2e 2c20 6465 6661 756c 7420 7661 6c75  .., default valu
│ │ │ │ +0001a530: 6520 6661 6c73 650a 2020 2a20 4f75 7470  e false.  * Outp
│ │ │ │ +0001a540: 7574 733a 0a20 2020 2020 202a 2061 202a  uts:.      * a *
│ │ │ │ +0001a550: 6e6f 7465 2042 6f6f 6c65 616e 2076 616c  note Boolean val
│ │ │ │ +0001a560: 7565 3a20 284d 6163 6175 6c61 7932 446f  ue: (Macaulay2Do
│ │ │ │ +0001a570: 6329 426f 6f6c 6561 6e2c 2c20 0a0a 4465  c)Boolean,, ..De
│ │ │ │ +0001a580: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d  scription.======
│ │ │ │ +0001a590: 3d3d 3d3d 3d0a 0a54 6869 7320 6675 6e63  =====..This func
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│ │ │ │ +0001a5c0: 7768 6574 6865 7220 7468 6520 6d61 7472  whether the matr
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│ │ │ │ +0001a5e0: 616e 6b2e 0a69 7352 616e 6b41 744c 6561  ank..isRankAtLea
│ │ │ │ +0001a5f0: 7374 2063 616c 6c73 202a 6e6f 7465 2067  st calls *note g
│ │ │ │ +0001a600: 6574 5375 626d 6174 7269 784f 6652 616e  etSubmatrixOfRan
│ │ │ │ +0001a610: 6b3a 2067 6574 5375 626d 6174 7269 784f  k: getSubmatrixO
│ │ │ │ +0001a620: 6652 616e 6b2c 2e20 2049 6620 7468 6174  fRank,.  If that
│ │ │ │ +0001a630: 0a66 756e 6374 696f 6e20 6669 6e64 7320  .function finds 
│ │ │ │ +0001a640: 6120 7375 626d 6174 7269 7820 6f66 2061  a submatrix of a
│ │ │ │ +0001a650: 2063 6572 7461 696e 2072 616e 6b2c 2074   certain rank, t
│ │ │ │ +0001a660: 6869 7320 7265 7475 726e 7320 7472 7565  his returns true
│ │ │ │ +0001a670: 2e20 2049 6620 7468 6174 0a66 756e 6374  .  If that.funct
│ │ │ │ +0001a680: 696f 6e20 6661 696c 7320 746f 2066 696e  ion fails to fin
│ │ │ │ +0001a690: 6420 6120 7375 626d 6174 7269 7820 6f66  d a submatrix of
│ │ │ │ +0001a6a0: 2061 2063 6572 7461 696e 2072 616e 6b2c   a certain rank,
│ │ │ │ +0001a6b0: 2074 6869 7320 7369 6d70 6c79 2063 616c   this simply cal
│ │ │ │ +0001a6c0: 6c73 202a 6e6f 7465 0a72 616e 6b3a 2028  ls *note.rank: (
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│ │ │ │ +0001a6e0: 6b2c 2e20 2054 6f20 636f 6e74 726f 6c20  k,.  To control 
│ │ │ │ +0001a6f0: 7468 6520 6e75 6d62 6572 206f 6620 7469  the number of ti
│ │ │ │ +0001a700: 6d65 7320 6765 7453 7562 6d61 7472 6978  mes getSubmatrix
│ │ │ │ +0001a710: 4f66 5261 6e6b 0a63 6f6e 7369 6465 7273  OfRank.considers
│ │ │ │ +0001a720: 2073 7562 6d61 7472 6963 6573 2c20 7573   submatrices, us
│ │ │ │ +0001a730: 6520 7468 6520 6f70 7469 6f6e 204d 6178  e the option Max
│ │ │ │ +0001a740: 4d69 6e6f 7273 2e0a 0a2b 2d2d 2d2d 2d2d  Minors...+------
│ │ │ │ +0001a750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a780: 2d2d 2d2d 2d2b 0a7c 6931 203a 2052 203d  -----+.|i1 : R =
│ │ │ │ +0001a790: 2051 515b 782c 795d 3b20 2020 2020 2020   QQ[x,y];       
│ │ │ │ +0001a7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a7c0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0001a7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a800: 2d2b 0a7c 6932 203a 204d 203d 206d 6174  -+.|i2 : M = mat
│ │ │ │ +0001a810: 7269 787b 7b78 2c79 2c32 2c30 2c32 2a78  rix{{x,y,2,0,2*x
│ │ │ │ +0001a820: 2b79 7d2c 207b 302c 302c 312c 302c 787d  +y}, {0,0,1,0,x}
│ │ │ │ +0001a830: 2c20 7b78 2c79 2c30 2c30 2c79 7d7d 3b7c  , {x,y,0,0,y}};|
│ │ │ │ +0001a840: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0001a850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a860: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ -0001a870: 203a 204d 6174 7269 7820 5220 203c 2d2d   : Matrix R  <--
│ │ │ │ -0001a880: 2052 2020 2020 2020 2020 2020 2020 2020   R              
│ │ │ │ -0001a890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a8a0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -0001a8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a8e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2072  -------+.|i3 : r
│ │ │ │ -0001a8f0: 616e 6b20 4d20 2020 2020 2020 2020 2020  ank M           
│ │ │ │ -0001a900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a920: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0001a930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a870: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001a880: 2020 2020 2020 2020 2020 2020 2033 2020               3  
│ │ │ │ +0001a890: 2020 2020 3520 2020 2020 2020 2020 2020      5           
│ │ │ │ +0001a8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a8b0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +0001a8c0: 203a 204d 6174 7269 7820 5220 203c 2d2d   : Matrix R  <--
│ │ │ │ +0001a8d0: 2052 2020 2020 2020 2020 2020 2020 2020   R              
│ │ │ │ +0001a8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a8f0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0001a900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a930: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2072  -------+.|i3 : r
│ │ │ │ +0001a940: 616e 6b20 4d20 2020 2020 2020 2020 2020  ank M           
│ │ │ │  0001a950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a960: 2020 207c 0a7c 6f33 203d 2032 2020 2020     |.|o3 = 2    
│ │ │ │ -0001a970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a970: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0001a980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a9a0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ -0001a9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0001a9e0: 0a7c 6934 203a 2069 7352 616e 6b41 744c  .|i4 : isRankAtL
│ │ │ │ -0001a9f0: 6561 7374 2832 2c20 4d29 2020 2020 2020  east(2, M)      
│ │ │ │ -0001aa00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aa10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0001aa20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aa30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aa40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aa50: 2020 2020 2020 2020 2020 207c 0a7c 6f34             |.|o4
│ │ │ │ -0001aa60: 203d 2074 7275 6520 2020 2020 2020 2020   = true         
│ │ │ │ +0001a9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a9b0: 2020 207c 0a7c 6f33 203d 2032 2020 2020     |.|o3 = 2    
│ │ │ │ +0001a9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a9f0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0001aa00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001aa10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001aa20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0001aa30: 0a7c 6934 203a 2069 7352 616e 6b41 744c  .|i4 : isRankAtL
│ │ │ │ +0001aa40: 6561 7374 2832 2c20 4d29 2020 2020 2020  east(2, M)      
│ │ │ │ +0001aa50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001aa60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0001aa70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001aa80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aa90: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -0001aaa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001aab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001aac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001aad0: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2069  -------+.|i5 : i
│ │ │ │ -0001aae0: 7352 616e 6b41 744c 6561 7374 2833 2c20  sRankAtLeast(3, 
│ │ │ │ -0001aaf0: 4d29 2020 2020 2020 2020 2020 2020 2020  M)              
│ │ │ │ -0001ab00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ab10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0001ab20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ab30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ab40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ab50: 2020 207c 0a7c 6f35 203d 2066 616c 7365     |.|o5 = false
│ │ │ │ -0001ab60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001aa90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001aaa0: 2020 2020 2020 2020 2020 207c 0a7c 6f34             |.|o4
│ │ │ │ +0001aab0: 203d 2074 7275 6520 2020 2020 2020 2020   = true         
│ │ │ │ +0001aac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001aad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001aae0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0001aaf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ab00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ab10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ab20: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2069  -------+.|i5 : i
│ │ │ │ +0001ab30: 7352 616e 6b41 744c 6561 7374 2833 2c20  sRankAtLeast(3, 
│ │ │ │ +0001ab40: 4d29 2020 2020 2020 2020 2020 2020 2020  M)              
│ │ │ │ +0001ab50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ab60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0001ab70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ab80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ab90: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ -0001aba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001abb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001abc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0001abd0: 0a0a 5468 6520 6f70 7469 6f6e 2054 6872  ..The option Thr
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│ │ │ │ -0001abf0: 2061 6c6c 6f77 2074 6865 2066 756e 6374   allow the funct
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│ │ │ │ -0001ac60: 312c 2074 6865 6e0a 7468 6973 2066 756e  1, then.this fun
│ │ │ │ -0001ac70: 6374 696f 6e20 7769 6c6c 2074 7279 2074  ction will try t
│ │ │ │ -0001ac80: 6f20 7369 6d75 6c74 616e 656f 7573 6c79  o simultaneously
│ │ │ │ -0001ac90: 2063 6f6d 7075 7465 2074 6865 2072 616e   compute the ran
│ │ │ │ -0001aca0: 6b20 6f66 2074 6865 206d 6174 7269 7820  k of the matrix 
│ │ │ │ -0001acb0: 7768 696c 650a 6c6f 6f6b 696e 6720 666f  while.looking fo
│ │ │ │ -0001acc0: 7220 6120 7375 626d 6174 7269 7820 6f66  r a submatrix of
│ │ │ │ -0001acd0: 2061 2063 6572 7461 696e 2072 616e 6b2e   a certain rank.
│ │ │ │ -0001ace0: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d  ..See also.=====
│ │ │ │ -0001acf0: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2067  ===..  * *note g
│ │ │ │ -0001ad00: 6574 5375 626d 6174 7269 784f 6652 616e  etSubmatrixOfRan
│ │ │ │ -0001ad10: 6b3a 2067 6574 5375 626d 6174 7269 784f  k: getSubmatrixO
│ │ │ │ -0001ad20: 6652 616e 6b2c 202d 2d20 7472 6965 7320  fRank, -- tries 
│ │ │ │ -0001ad30: 746f 2066 696e 6420 6120 7375 626d 6174  to find a submat
│ │ │ │ -0001ad40: 7269 780a 2020 2020 6f66 2074 6865 2067  rix.    of the g
│ │ │ │ -0001ad50: 6976 656e 2072 616e 6b0a 2020 2a20 2a6e  iven rank.  * *n
│ │ │ │ -0001ad60: 6f74 6520 6973 5261 6e6b 4174 4c65 6173  ote isRankAtLeas
│ │ │ │ -0001ad70: 7428 2e2e 2e2c 5374 7261 7465 6779 3d3e  t(...,Strategy=>
│ │ │ │ -0001ad80: 2e2e 2e29 3a20 5374 7261 7465 6779 4465  ...): StrategyDe
│ │ │ │ -0001ad90: 6661 756c 742c 202d 2d20 7374 7261 7465  fault, -- strate
│ │ │ │ -0001ada0: 6769 6573 2066 6f72 0a20 2020 2063 686f  gies for.    cho
│ │ │ │ -0001adb0: 6f73 696e 6720 7375 626d 6174 7269 6365  osing submatrice
│ │ │ │ -0001adc0: 730a 0a57 6179 7320 746f 2075 7365 2069  s..Ways to use i
│ │ │ │ -0001add0: 7352 616e 6b41 744c 6561 7374 3a0a 3d3d  sRankAtLeast:.==
│ │ │ │ -0001ade0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001adf0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2269  ========..  * "i
│ │ │ │ -0001ae00: 7352 616e 6b41 744c 6561 7374 285a 5a2c  sRankAtLeast(ZZ,
│ │ │ │ -0001ae10: 4d61 7472 6978 2922 0a0a 466f 7220 7468  Matrix)"..For th
│ │ │ │ -0001ae20: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d  e programmer.===
│ │ │ │ -0001ae30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ -0001ae40: 0a54 6865 206f 626a 6563 7420 2a6e 6f74  .The object *not
│ │ │ │ -0001ae50: 6520 6973 5261 6e6b 4174 4c65 6173 743a  e isRankAtLeast:
│ │ │ │ -0001ae60: 2069 7352 616e 6b41 744c 6561 7374 2c20   isRankAtLeast, 
│ │ │ │ -0001ae70: 6973 2061 202a 6e6f 7465 206d 6574 686f  is a *note metho
│ │ │ │ -0001ae80: 6420 6675 6e63 7469 6f6e 2077 6974 680a  d function with.
│ │ │ │ -0001ae90: 6f70 7469 6f6e 733a 2028 4d61 6361 756c  options: (Macaul
│ │ │ │ -0001aea0: 6179 3244 6f63 294d 6574 686f 6446 756e  ay2Doc)MethodFun
│ │ │ │ -0001aeb0: 6374 696f 6e57 6974 684f 7074 696f 6e73  ctionWithOptions
│ │ │ │ -0001aec0: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ,...------------
│ │ │ │ -0001aed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001aee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001aef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001af00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001af10: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520  ---..The source 
│ │ │ │ -0001af20: 6f66 2074 6869 7320 646f 6375 6d65 6e74  of this document
│ │ │ │ -0001af30: 2069 7320 696e 0a2f 6275 696c 642f 7265   is in./build/re
│ │ │ │ -0001af40: 7072 6f64 7563 6962 6c65 2d70 6174 682f  producible-path/
│ │ │ │ -0001af50: 6d61 6361 756c 6179 322d 312e 3235 2e31  macaulay2-1.25.1
│ │ │ │ -0001af60: 312b 6473 2f4d 322f 4d61 6361 756c 6179  1+ds/M2/Macaulay
│ │ │ │ -0001af70: 322f 7061 636b 6167 6573 2f46 6173 744d  2/packages/FastM
│ │ │ │ -0001af80: 696e 6f72 732e 0a6d 323a 3137 3331 3a30  inors..m2:1731:0
│ │ │ │ -0001af90: 2e0a 1f0a 4669 6c65 3a20 4661 7374 4d69  ....File: FastMi
│ │ │ │ -0001afa0: 6e6f 7273 2e69 6e66 6f2c 204e 6f64 653a  nors.info, Node:
│ │ │ │ -0001afb0: 2069 7352 616e 6b41 744c 6561 7374 5f6c   isRankAtLeast_l
│ │ │ │ -0001afc0: 705f 7064 5f70 645f 7064 5f63 6d54 6872  p_pd_pd_pd_cmThr
│ │ │ │ -0001afd0: 6561 6473 3d3e 5f70 645f 7064 5f70 645f  eads=>_pd_pd_pd_
│ │ │ │ -0001afe0: 7270 2c20 4e65 7874 3a20 4d61 784d 696e  rp, Next: MaxMin
│ │ │ │ -0001aff0: 6f72 732c 2050 7265 763a 2069 7352 616e  ors, Prev: isRan
│ │ │ │ -0001b000: 6b41 744c 6561 7374 2c20 5570 3a20 546f  kAtLeast, Up: To
│ │ │ │ -0001b010: 700a 0a69 7352 616e 6b41 744c 6561 7374  p..isRankAtLeast
│ │ │ │ -0001b020: 282e 2e2e 2c54 6872 6561 6473 3d3e 2e2e  (...,Threads=>..
│ │ │ │ -0001b030: 2e29 202d 2d20 616e 206f 7074 696f 6e20  .) -- an option 
│ │ │ │ -0001b040: 666f 7220 7661 7269 6f75 7320 6675 6e63  for various func
│ │ │ │ -0001b050: 7469 6f6e 730a 2a2a 2a2a 2a2a 2a2a 2a2a  tions.**********
│ │ │ │ -0001b060: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001b070: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001b080: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001b090: 2a2a 2a2a 2a2a 2a2a 0a0a 4465 7363 7269  ********..Descri
│ │ │ │ -0001b0a0: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ -0001b0b0: 3d0a 0a49 6e63 7265 6173 696e 6720 7468  =..Increasing th
│ │ │ │ -0001b0c0: 6973 206f 7074 696f 6e20 6d61 7920 7465  is option may te
│ │ │ │ -0001b0d0: 6c6c 2076 6172 696f 7573 2066 756e 6374  ll various funct
│ │ │ │ -0001b0e0: 696f 6e73 2074 6f20 6d75 6c74 6974 6872  ions to multithr
│ │ │ │ -0001b0f0: 6561 6420 7468 6569 720a 6f70 6572 6174  ead their.operat
│ │ │ │ -0001b100: 696f 6e73 2e20 2059 6f75 206d 6179 2061  ions.  You may a
│ │ │ │ -0001b110: 6c73 6f20 7761 6e74 2074 6f20 696e 6372  lso want to incr
│ │ │ │ -0001b120: 6561 7365 2061 6c6c 6f77 6162 6c65 5468  ease allowableTh
│ │ │ │ -0001b130: 7265 6164 732e 0a0a 5365 6520 616c 736f  reads...See also
│ │ │ │ -0001b140: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a  .========..  * *
│ │ │ │ -0001b150: 6e6f 7465 2069 7352 616e 6b41 744c 6561  note isRankAtLea
│ │ │ │ -0001b160: 7374 3a20 6973 5261 6e6b 4174 4c65 6173  st: isRankAtLeas
│ │ │ │ -0001b170: 742c 202d 2d20 6465 7465 726d 696e 6573  t, -- determines
│ │ │ │ -0001b180: 2069 6620 7468 6520 6d61 7472 6978 2068   if the matrix h
│ │ │ │ -0001b190: 6173 2072 616e 6b20 6174 0a20 2020 206c  as rank at.    l
│ │ │ │ -0001b1a0: 6561 7374 2061 206e 756d 6265 720a 2020  east a number.  
│ │ │ │ -0001b1b0: 2a20 2a6e 6f74 6520 6765 7453 7562 6d61  * *note getSubma
│ │ │ │ -0001b1c0: 7472 6978 4f66 5261 6e6b 3a20 6765 7453  trixOfRank: getS
│ │ │ │ -0001b1d0: 7562 6d61 7472 6978 4f66 5261 6e6b 2c20  ubmatrixOfRank, 
│ │ │ │ -0001b1e0: 2d2d 2074 7269 6573 2074 6f20 6669 6e64  -- tries to find
│ │ │ │ -0001b1f0: 2061 2073 7562 6d61 7472 6978 0a20 2020   a submatrix.   
│ │ │ │ -0001b200: 206f 6620 7468 6520 6769 7665 6e20 7261   of the given ra
│ │ │ │ -0001b210: 6e6b 0a20 202a 202a 6e6f 7465 2072 6563  nk.  * *note rec
│ │ │ │ -0001b220: 7572 7369 7665 4d69 6e6f 7273 3a20 7265  ursiveMinors: re
│ │ │ │ -0001b230: 6375 7273 6976 654d 696e 6f72 732c 202d  cursiveMinors, -
│ │ │ │ -0001b240: 2d20 7573 6573 2061 2072 6563 7572 7369  - uses a recursi
│ │ │ │ -0001b250: 7665 2063 6f66 6163 746f 720a 2020 2020  ve cofactor.    
│ │ │ │ -0001b260: 616c 676f 7269 7468 6d20 746f 2063 6f6d  algorithm to com
│ │ │ │ -0001b270: 7075 7465 2074 6865 2069 6465 616c 206f  pute the ideal o
│ │ │ │ -0001b280: 6620 6d69 6e6f 7273 206f 6620 6120 6d61  f minors of a ma
│ │ │ │ -0001b290: 7472 6978 0a0a 4675 6e63 7469 6f6e 7320  trix..Functions 
│ │ │ │ -0001b2a0: 7769 7468 206f 7074 696f 6e61 6c20 6172  with optional ar
│ │ │ │ -0001b2b0: 6775 6d65 6e74 206e 616d 6564 2054 6872  gument named Thr
│ │ │ │ -0001b2c0: 6561 6473 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d  eads:.==========
│ │ │ │ -0001b2d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001b2e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001b2f0: 3d3d 3d3d 3d0a 0a20 202a 2022 6765 7453  =====..  * "getS
│ │ │ │ -0001b300: 7562 6d61 7472 6978 4f66 5261 6e6b 282e  ubmatrixOfRank(.
│ │ │ │ -0001b310: 2e2e 2c54 6872 6561 6473 3d3e 2e2e 2e29  ..,Threads=>...)
│ │ │ │ -0001b320: 220a 2020 2a20 2a6e 6f74 6520 6973 5261  ".  * *note isRa
│ │ │ │ -0001b330: 6e6b 4174 4c65 6173 7428 2e2e 2e2c 5468  nkAtLeast(...,Th
│ │ │ │ -0001b340: 7265 6164 733d 3e2e 2e2e 293a 0a20 2020  reads=>...):.   
│ │ │ │ -0001b350: 2069 7352 616e 6b41 744c 6561 7374 5f6c   isRankAtLeast_l
│ │ │ │ -0001b360: 705f 7064 5f70 645f 7064 5f63 6d54 6872  p_pd_pd_pd_cmThr
│ │ │ │ -0001b370: 6561 6473 3d3e 5f70 645f 7064 5f70 645f  eads=>_pd_pd_pd_
│ │ │ │ -0001b380: 7270 2c20 2d2d 2061 6e20 6f70 7469 6f6e  rp, -- an option
│ │ │ │ -0001b390: 2066 6f72 2076 6172 696f 7573 0a20 2020   for various.   
│ │ │ │ -0001b3a0: 2066 756e 6374 696f 6e73 0a20 202a 2022   functions.  * "
│ │ │ │ -0001b3b0: 7265 6375 7273 6976 654d 696e 6f72 7328  recursiveMinors(
│ │ │ │ -0001b3c0: 2e2e 2e2c 5468 7265 6164 733d 3e2e 2e2e  ...,Threads=>...
│ │ │ │ -0001b3d0: 2922 0a0a 4675 7274 6865 7220 696e 666f  )"..Further info
│ │ │ │ -0001b3e0: 726d 6174 696f 6e0a 3d3d 3d3d 3d3d 3d3d  rmation.========
│ │ │ │ -0001b3f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ -0001b400: 2044 6566 6175 6c74 2076 616c 7565 3a20   Default value: 
│ │ │ │ -0001b410: 310a 2020 2a20 4675 6e63 7469 6f6e 3a20  1.  * Function: 
│ │ │ │ -0001b420: 2a6e 6f74 6520 6973 5261 6e6b 4174 4c65  *note isRankAtLe
│ │ │ │ -0001b430: 6173 743a 2069 7352 616e 6b41 744c 6561  ast: isRankAtLea
│ │ │ │ -0001b440: 7374 2c20 2d2d 2064 6574 6572 6d69 6e65  st, -- determine
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│ │ │ │ -0001b4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468  ------------..Th
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│ │ │ │ -0001b610: 2c20 5570 3a20 546f 700a 0a4d 6178 4d69  , Up: Top..MaxMi
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│ │ │ │ -0001b650: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001b660: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001b670: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a  **************..
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│ │ │ │ -0001b7c0: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d  .See also.======
│ │ │ │ -0001b7d0: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 7265  ==..  * *note re
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│ │ │ │ -0001b830: 6973 2072 6567 756c 6172 2069 6e20 636f  is regular in co
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│ │ │ │ -0001b860: 726f 6a44 696d 2c20 2d2d 2066 696e 6473  rojDim, -- finds
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│ │ │ │ -0001b880: 666f 7220 7468 6520 7072 6f6a 6563 7469  for the projecti
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│ │ │ │ -0001b8c0: 6561 7374 3a20 6973 5261 6e6b 4174 4c65  east: isRankAtLe
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│ │ │ │ -0001b8e0: 6573 2069 6620 7468 6520 6d61 7472 6978  es if the matrix
│ │ │ │ -0001b8f0: 2068 6173 2072 616e 6b20 6174 0a20 2020   has rank at.   
│ │ │ │ -0001b900: 206c 6561 7374 2061 206e 756d 6265 720a   least a number.
│ │ │ │ -0001b910: 2020 2a20 2a6e 6f74 6520 6765 7453 7562    * *note getSub
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│ │ │ │ -0001b960: 2020 206f 6620 7468 6520 6769 7665 6e20     of the given 
│ │ │ │ -0001b970: 7261 6e6b 0a0a 4675 6e63 7469 6f6e 7320  rank..Functions 
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│ │ │ │ -0001b9c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001b9d0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022  =========..  * "
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│ │ │ │ -0001ba30: 2020 2a20 2270 726f 6a44 696d 282e 2e2e    * "projDim(...
│ │ │ │ -0001ba40: 2c4d 6178 4d69 6e6f 7273 3d3e 2e2e 2e29  ,MaxMinors=>...)
│ │ │ │ -0001ba50: 220a 2020 2a20 2272 6567 756c 6172 496e  ".  * "regularIn
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│ │ │ │ -0001ba80: 0a0a 466f 7220 7468 6520 7072 6f67 7261  ..For the progra
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│ │ │ │ -0001bb10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bb20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bb30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bb40: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520  ----------..The 
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│ │ │ │ -0001bba0: 6163 6175 6c61 7932 2f70 6163 6b61 6765  acaulay2/package
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│ │ │ │ -0001bbf0: 7369 6f6e 2c20 4e65 7874 3a20 4d6f 6475  sion, Next: Modu
│ │ │ │ -0001bc00: 6c75 732c 2050 7265 763a 204d 6178 4d69  lus, Prev: MaxMi
│ │ │ │ -0001bc10: 6e6f 7273 2c20 5570 3a20 546f 700a 0a4d  nors, Up: Top..M
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│ │ │ │ -0001bc50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001bc60: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 4465 7363  **********..Desc
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│ │ │ │ -0001bcc0: 6b20 666f 7220 7072 6f6a 6563 7469 7665  k for projective
│ │ │ │ -0001bcd0: 0a64 696d 656e 7369 6f6e 2062 656c 6f77  .dimension below
│ │ │ │ -0001bce0: 2074 6865 206f 7074 696f 6e20 7661 6c75   the option valu
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│ │ │ │ -0001bd00: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
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│ │ │ │ -0001bd30: 7070 6572 2062 6f75 6e64 2066 6f72 2074  pper bound for t
│ │ │ │ -0001bd40: 6865 2070 726f 6a65 6374 6976 650a 2020  he projective.  
│ │ │ │ -0001bd50: 2020 6469 6d65 6e73 696f 6e20 6f66 2061    dimension of a
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│ │ │ │ -0001bd90: 4d69 6e44 696d 656e 7369 6f6e 3a0a 3d3d  MinDimension:.==
│ │ │ │ -0001bda0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001bdb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001bdc0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001bdd0: 3d3d 0a0a 2020 2a20 2270 726f 6a44 696d  ==..  * "projDim
│ │ │ │ -0001bde0: 282e 2e2e 2c4d 696e 4469 6d65 6e73 696f  (...,MinDimensio
│ │ │ │ -0001bdf0: 6e3d 3e2e 2e2e 2922 202d 2d20 7365 6520  n=>...)" -- see 
│ │ │ │ -0001be00: 2a6e 6f74 6520 7072 6f6a 4469 6d3a 2070  *note projDim: p
│ │ │ │ -0001be10: 726f 6a44 696d 2c20 2d2d 2066 696e 6473  rojDim, -- finds
│ │ │ │ -0001be20: 2061 6e0a 2020 2020 7570 7065 7220 626f   an.    upper bo
│ │ │ │ -0001be30: 756e 6420 666f 7220 7468 6520 7072 6f6a  und for the proj
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│ │ │ │ -0001be60: 7220 7468 6520 7072 6f67 7261 6d6d 6572  r the programmer
│ │ │ │ -0001be70: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ -0001be80: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420  ===..The object 
│ │ │ │ -0001be90: 2a6e 6f74 6520 4d69 6e44 696d 656e 7369  *note MinDimensi
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│ │ │ │ -0001beb0: 2c20 6973 2061 202a 6e6f 7465 2073 796d  , is a *note sym
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│ │ │ │ -0001bee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bf00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bf10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bf20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468  ------------..Th
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│ │ │ │ -0001bf40: 2064 6f63 756d 656e 7420 6973 2069 6e0a   document is in.
│ │ │ │ -0001bf50: 2f62 7569 6c64 2f72 6570 726f 6475 6369  /build/reproduci
│ │ │ │ -0001bf60: 626c 652d 7061 7468 2f6d 6163 6175 6c61  ble-path/macaula
│ │ │ │ -0001bf70: 7932 2d31 2e32 352e 3131 2b64 732f 4d32  y2-1.25.11+ds/M2
│ │ │ │ -0001bf80: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61  /Macaulay2/packa
│ │ │ │ -0001bf90: 6765 732f 4661 7374 4d69 6e6f 7273 2e0a  ges/FastMinors..
│ │ │ │ -0001bfa0: 6d32 3a32 3039 323a 302e 0a1f 0a46 696c  m2:2092:0....Fil
│ │ │ │ -0001bfb0: 653a 2046 6173 744d 696e 6f72 732e 696e  e: FastMinors.in
│ │ │ │ -0001bfc0: 666f 2c20 4e6f 6465 3a20 4d6f 6475 6c75  fo, Node: Modulu
│ │ │ │ -0001bfd0: 732c 204e 6578 743a 2050 6f69 6e74 4f70  s, Next: PointOp
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│ │ │ │ -0001c030: 6e0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  n.**************
│ │ │ │ -0001c040: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001c050: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a  ***************.
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│ │ │ │ +0001b2a0: 7665 2063 6f66 6163 746f 720a 2020 2020  ve cofactor.    
│ │ │ │ +0001b2b0: 616c 676f 7269 7468 6d20 746f 2063 6f6d  algorithm to com
│ │ │ │ +0001b2c0: 7075 7465 2074 6865 2069 6465 616c 206f  pute the ideal o
│ │ │ │ +0001b2d0: 6620 6d69 6e6f 7273 206f 6620 6120 6d61  f minors of a ma
│ │ │ │ +0001b2e0: 7472 6978 0a0a 4675 6e63 7469 6f6e 7320  trix..Functions 
│ │ │ │ +0001b2f0: 7769 7468 206f 7074 696f 6e61 6c20 6172  with optional ar
│ │ │ │ +0001b300: 6775 6d65 6e74 206e 616d 6564 2054 6872  gument named Thr
│ │ │ │ +0001b310: 6561 6473 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d  eads:.==========
│ │ │ │ +0001b320: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0001b330: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0001b340: 3d3d 3d3d 3d0a 0a20 202a 2022 6765 7453  =====..  * "getS
│ │ │ │ +0001b350: 7562 6d61 7472 6978 4f66 5261 6e6b 282e  ubmatrixOfRank(.
│ │ │ │ +0001b360: 2e2e 2c54 6872 6561 6473 3d3e 2e2e 2e29  ..,Threads=>...)
│ │ │ │ +0001b370: 220a 2020 2a20 2a6e 6f74 6520 6973 5261  ".  * *note isRa
│ │ │ │ +0001b380: 6e6b 4174 4c65 6173 7428 2e2e 2e2c 5468  nkAtLeast(...,Th
│ │ │ │ +0001b390: 7265 6164 733d 3e2e 2e2e 293a 0a20 2020  reads=>...):.   
│ │ │ │ +0001b3a0: 2069 7352 616e 6b41 744c 6561 7374 5f6c   isRankAtLeast_l
│ │ │ │ +0001b3b0: 705f 7064 5f70 645f 7064 5f63 6d54 6872  p_pd_pd_pd_cmThr
│ │ │ │ +0001b3c0: 6561 6473 3d3e 5f70 645f 7064 5f70 645f  eads=>_pd_pd_pd_
│ │ │ │ +0001b3d0: 7270 2c20 2d2d 2061 6e20 6f70 7469 6f6e  rp, -- an option
│ │ │ │ +0001b3e0: 2066 6f72 2076 6172 696f 7573 0a20 2020   for various.   
│ │ │ │ +0001b3f0: 2066 756e 6374 696f 6e73 0a20 202a 2022   functions.  * "
│ │ │ │ +0001b400: 7265 6375 7273 6976 654d 696e 6f72 7328  recursiveMinors(
│ │ │ │ +0001b410: 2e2e 2e2c 5468 7265 6164 733d 3e2e 2e2e  ...,Threads=>...
│ │ │ │ +0001b420: 2922 0a0a 4675 7274 6865 7220 696e 666f  )"..Further info
│ │ │ │ +0001b430: 726d 6174 696f 6e0a 3d3d 3d3d 3d3d 3d3d  rmation.========
│ │ │ │ +0001b440: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ +0001b450: 2044 6566 6175 6c74 2076 616c 7565 3a20   Default value: 
│ │ │ │ +0001b460: 310a 2020 2a20 4675 6e63 7469 6f6e 3a20  1.  * Function: 
│ │ │ │ +0001b470: 2a6e 6f74 6520 6973 5261 6e6b 4174 4c65  *note isRankAtLe
│ │ │ │ +0001b480: 6173 743a 2069 7352 616e 6b41 744c 6561  ast: isRankAtLea
│ │ │ │ +0001b490: 7374 2c20 2d2d 2064 6574 6572 6d69 6e65  st, -- determine
│ │ │ │ +0001b4a0: 7320 6966 2074 6865 206d 6174 7269 780a  s if the matrix.
│ │ │ │ +0001b4b0: 2020 2020 6861 7320 7261 6e6b 2061 7420      has rank at 
│ │ │ │ +0001b4c0: 6c65 6173 7420 6120 6e75 6d62 6572 0a20  least a number. 
│ │ │ │ +0001b4d0: 202a 204f 7074 696f 6e20 6b65 793a 202a   * Option key: *
│ │ │ │ +0001b4e0: 6e6f 7465 2054 6872 6561 6473 3a20 284d  note Threads: (M
│ │ │ │ +0001b4f0: 6163 6175 6c61 7932 446f 6329 5468 7265  acaulay2Doc)Thre
│ │ │ │ +0001b500: 6164 732c 202d 2d20 616e 206f 7074 696f  ads, -- an optio
│ │ │ │ +0001b510: 6e61 6c20 6172 6775 6d65 6e74 0a2d 2d2d  nal argument.---
│ │ │ │ +0001b520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468  ------------..Th
│ │ │ │ +0001b570: 6520 736f 7572 6365 206f 6620 7468 6973  e source of this
│ │ │ │ +0001b580: 2064 6f63 756d 656e 7420 6973 2069 6e0a   document is in.
│ │ │ │ +0001b590: 2f62 7569 6c64 2f72 6570 726f 6475 6369  /build/reproduci
│ │ │ │ +0001b5a0: 626c 652d 7061 7468 2f6d 6163 6175 6c61  ble-path/macaula
│ │ │ │ +0001b5b0: 7932 2d31 2e32 352e 3131 2b64 732f 4d32  y2-1.25.11+ds/M2
│ │ │ │ +0001b5c0: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61  /Macaulay2/packa
│ │ │ │ +0001b5d0: 6765 732f 4661 7374 4d69 6e6f 7273 2e0a  ges/FastMinors..
│ │ │ │ +0001b5e0: 6d32 3a32 3132 303a 302e 0a1f 0a46 696c  m2:2120:0....Fil
│ │ │ │ +0001b5f0: 653a 2046 6173 744d 696e 6f72 732e 696e  e: FastMinors.in
│ │ │ │ +0001b600: 666f 2c20 4e6f 6465 3a20 4d61 784d 696e  fo, Node: MaxMin
│ │ │ │ +0001b610: 6f72 732c 204e 6578 743a 204d 696e 4469  ors, Next: MinDi
│ │ │ │ +0001b620: 6d65 6e73 696f 6e2c 2050 7265 763a 2069  mension, Prev: i
│ │ │ │ +0001b630: 7352 616e 6b41 744c 6561 7374 5f6c 705f  sRankAtLeast_lp_
│ │ │ │ +0001b640: 7064 5f70 645f 7064 5f63 6d54 6872 6561  pd_pd_pd_cmThrea
│ │ │ │ +0001b650: 6473 3d3e 5f70 645f 7064 5f70 645f 7270  ds=>_pd_pd_pd_rp
│ │ │ │ +0001b660: 2c20 5570 3a20 546f 700a 0a4d 6178 4d69  , Up: Top..MaxMi
│ │ │ │ +0001b670: 6e6f 7273 202d 2d20 616e 206f 7074 696f  nors -- an optio
│ │ │ │ +0001b680: 6e20 746f 2063 6f6e 7472 6f6c 2064 6570  n to control dep
│ │ │ │ +0001b690: 7468 206f 6620 7365 6172 6368 0a2a 2a2a  th of search.***
│ │ │ │ +0001b6a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0001b6b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0001b6c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a  **************..
│ │ │ │ +0001b6d0: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
│ │ │ │ +0001b6e0: 3d3d 3d3d 3d3d 3d0a 0a54 6869 7320 6f70  =======..This op
│ │ │ │ +0001b6f0: 7469 6f6e 2063 6f6e 7472 6f6c 7320 686f  tion controls ho
│ │ │ │ +0001b700: 7720 6d61 6e79 206d 696e 6f72 7320 7661  w many minors va
│ │ │ │ +0001b710: 7269 6f75 7320 6675 6e63 7469 6f6e 7320  rious functions 
│ │ │ │ +0001b720: 636f 6e73 6964 6572 2e20 2049 6e63 7265  consider.  Incre
│ │ │ │ +0001b730: 6173 696e 6720 6974 0a77 696c 6c20 6d61  asing it.will ma
│ │ │ │ +0001b740: 6b65 2063 6572 7461 696e 2066 756e 6374  ke certain funct
│ │ │ │ +0001b750: 696f 6e73 2073 6561 7263 6820 6c6f 6e67  ions search long
│ │ │ │ +0001b760: 6572 2c20 6275 7420 6d61 7920 6d61 6b65  er, but may make
│ │ │ │ +0001b770: 2074 6865 6d20 6769 7665 206d 6f72 6520   them give more 
│ │ │ │ +0001b780: 7573 6566 756c 0a6f 7574 7075 7473 2e20  useful.outputs. 
│ │ │ │ +0001b790: 2054 6865 2066 756e 6374 696f 6e73 2070   The functions p
│ │ │ │ +0001b7a0: 726f 6a44 696d 2061 6e64 2072 6567 756c  rojDim and regul
│ │ │ │ +0001b7b0: 6172 496e 436f 6469 6d65 6e73 696f 6e20  arInCodimension 
│ │ │ │ +0001b7c0: 6361 6e20 616c 736f 2074 616b 6520 696e  can also take in
│ │ │ │ +0001b7d0: 206d 6f72 650a 636f 6d70 6c69 6361 7465   more.complicate
│ │ │ │ +0001b7e0: 6420 696e 7075 7473 2e20 2053 6565 2074  d inputs.  See t
│ │ │ │ +0001b7f0: 6865 6972 2064 6f63 756d 656e 7461 7469  heir documentati
│ │ │ │ +0001b800: 6f6e 2066 6f72 2064 6574 6169 6c73 2e0a  on for details..
│ │ │ │ +0001b810: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d  .See also.======
│ │ │ │ +0001b820: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 7265  ==..  * *note re
│ │ │ │ +0001b830: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
│ │ │ │ +0001b840: 6f6e 3a20 7265 6775 6c61 7249 6e43 6f64  on: regularInCod
│ │ │ │ +0001b850: 696d 656e 7369 6f6e 2c20 2d2d 2061 7474  imension, -- att
│ │ │ │ +0001b860: 656d 7074 7320 746f 2073 686f 7720 7468  empts to show th
│ │ │ │ +0001b870: 6174 0a20 2020 2074 6865 2072 696e 6720  at.    the ring 
│ │ │ │ +0001b880: 6973 2072 6567 756c 6172 2069 6e20 636f  is regular in co
│ │ │ │ +0001b890: 6469 6d65 6e73 696f 6e20 6e0a 2020 2a20  dimension n.  * 
│ │ │ │ +0001b8a0: 2a6e 6f74 6520 7072 6f6a 4469 6d3a 2070  *note projDim: p
│ │ │ │ +0001b8b0: 726f 6a44 696d 2c20 2d2d 2066 696e 6473  rojDim, -- finds
│ │ │ │ +0001b8c0: 2061 6e20 7570 7065 7220 626f 756e 6420   an upper bound 
│ │ │ │ +0001b8d0: 666f 7220 7468 6520 7072 6f6a 6563 7469  for the projecti
│ │ │ │ +0001b8e0: 7665 0a20 2020 2064 696d 656e 7369 6f6e  ve.    dimension
│ │ │ │ +0001b8f0: 206f 6620 6120 6d6f 6475 6c65 0a20 202a   of a module.  *
│ │ │ │ +0001b900: 202a 6e6f 7465 2069 7352 616e 6b41 744c   *note isRankAtL
│ │ │ │ +0001b910: 6561 7374 3a20 6973 5261 6e6b 4174 4c65  east: isRankAtLe
│ │ │ │ +0001b920: 6173 742c 202d 2d20 6465 7465 726d 696e  ast, -- determin
│ │ │ │ +0001b930: 6573 2069 6620 7468 6520 6d61 7472 6978  es if the matrix
│ │ │ │ +0001b940: 2068 6173 2072 616e 6b20 6174 0a20 2020   has rank at.   
│ │ │ │ +0001b950: 206c 6561 7374 2061 206e 756d 6265 720a   least a number.
│ │ │ │ +0001b960: 2020 2a20 2a6e 6f74 6520 6765 7453 7562    * *note getSub
│ │ │ │ +0001b970: 6d61 7472 6978 4f66 5261 6e6b 3a20 6765  matrixOfRank: ge
│ │ │ │ +0001b980: 7453 7562 6d61 7472 6978 4f66 5261 6e6b  tSubmatrixOfRank
│ │ │ │ +0001b990: 2c20 2d2d 2074 7269 6573 2074 6f20 6669  , -- tries to fi
│ │ │ │ +0001b9a0: 6e64 2061 2073 7562 6d61 7472 6978 0a20  nd a submatrix. 
│ │ │ │ +0001b9b0: 2020 206f 6620 7468 6520 6769 7665 6e20     of the given 
│ │ │ │ +0001b9c0: 7261 6e6b 0a0a 4675 6e63 7469 6f6e 7320  rank..Functions 
│ │ │ │ +0001b9d0: 7769 7468 206f 7074 696f 6e61 6c20 6172  with optional ar
│ │ │ │ +0001b9e0: 6775 6d65 6e74 206e 616d 6564 204d 6178  gument named Max
│ │ │ │ +0001b9f0: 4d69 6e6f 7273 3a0a 3d3d 3d3d 3d3d 3d3d  Minors:.========
│ │ │ │ +0001ba00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0001ba10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0001ba20: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022  =========..  * "
│ │ │ │ +0001ba30: 6765 7453 7562 6d61 7472 6978 4f66 5261  getSubmatrixOfRa
│ │ │ │ +0001ba40: 6e6b 282e 2e2e 2c4d 6178 4d69 6e6f 7273  nk(...,MaxMinors
│ │ │ │ +0001ba50: 3d3e 2e2e 2e29 220a 2020 2a20 2269 7352  =>...)".  * "isR
│ │ │ │ +0001ba60: 616e 6b41 744c 6561 7374 282e 2e2e 2c4d  ankAtLeast(...,M
│ │ │ │ +0001ba70: 6178 4d69 6e6f 7273 3d3e 2e2e 2e29 220a  axMinors=>...)".
│ │ │ │ +0001ba80: 2020 2a20 2270 726f 6a44 696d 282e 2e2e    * "projDim(...
│ │ │ │ +0001ba90: 2c4d 6178 4d69 6e6f 7273 3d3e 2e2e 2e29  ,MaxMinors=>...)
│ │ │ │ +0001baa0: 220a 2020 2a20 2272 6567 756c 6172 496e  ".  * "regularIn
│ │ │ │ +0001bab0: 436f 6469 6d65 6e73 696f 6e28 2e2e 2e2c  Codimension(...,
│ │ │ │ +0001bac0: 4d61 784d 696e 6f72 733d 3e2e 2e2e 2922  MaxMinors=>...)"
│ │ │ │ +0001bad0: 0a0a 466f 7220 7468 6520 7072 6f67 7261  ..For the progra
│ │ │ │ +0001bae0: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  mmer.===========
│ │ │ │ +0001baf0: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a  =======..The obj
│ │ │ │ +0001bb00: 6563 7420 2a6e 6f74 6520 4d61 784d 696e  ect *note MaxMin
│ │ │ │ +0001bb10: 6f72 733a 204d 6178 4d69 6e6f 7273 2c20  ors: MaxMinors, 
│ │ │ │ +0001bb20: 6973 2061 202a 6e6f 7465 2073 796d 626f  is a *note symbo
│ │ │ │ +0001bb30: 6c3a 0a28 4d61 6361 756c 6179 3244 6f63  l:.(Macaulay2Doc
│ │ │ │ +0001bb40: 2953 796d 626f 6c2c 2e0a 0a2d 2d2d 2d2d  )Symbol,...-----
│ │ │ │ +0001bb50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bb60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bb80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bb90: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520  ----------..The 
│ │ │ │ +0001bba0: 736f 7572 6365 206f 6620 7468 6973 2064  source of this d
│ │ │ │ +0001bbb0: 6f63 756d 656e 7420 6973 2069 6e0a 2f62  ocument is in./b
│ │ │ │ +0001bbc0: 7569 6c64 2f72 6570 726f 6475 6369 626c  uild/reproducibl
│ │ │ │ +0001bbd0: 652d 7061 7468 2f6d 6163 6175 6c61 7932  e-path/macaulay2
│ │ │ │ +0001bbe0: 2d31 2e32 352e 3131 2b64 732f 4d32 2f4d  -1.25.11+ds/M2/M
│ │ │ │ +0001bbf0: 6163 6175 6c61 7932 2f70 6163 6b61 6765  acaulay2/package
│ │ │ │ +0001bc00: 732f 4661 7374 4d69 6e6f 7273 2e0a 6d32  s/FastMinors..m2
│ │ │ │ +0001bc10: 3a32 3133 393a 302e 0a1f 0a46 696c 653a  :2139:0....File:
│ │ │ │ +0001bc20: 2046 6173 744d 696e 6f72 732e 696e 666f   FastMinors.info
│ │ │ │ +0001bc30: 2c20 4e6f 6465 3a20 4d69 6e44 696d 656e  , Node: MinDimen
│ │ │ │ +0001bc40: 7369 6f6e 2c20 4e65 7874 3a20 4d6f 6475  sion, Next: Modu
│ │ │ │ +0001bc50: 6c75 732c 2050 7265 763a 204d 6178 4d69  lus, Prev: MaxMi
│ │ │ │ +0001bc60: 6e6f 7273 2c20 5570 3a20 546f 700a 0a4d  nors, Up: Top..M
│ │ │ │ +0001bc70: 696e 4469 6d65 6e73 696f 6e20 2d2d 2061  inDimension -- a
│ │ │ │ +0001bc80: 6e20 6f70 7469 6f6e 2066 6f72 2070 726f  n option for pro
│ │ │ │ +0001bc90: 6a44 696d 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a  jDim.***********
│ │ │ │ +0001bca0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0001bcb0: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 4465 7363  **********..Desc
│ │ │ │ +0001bcc0: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
│ │ │ │ +0001bcd0: 3d3d 3d0a 0a54 6869 7320 6f70 7469 6f6e  ===..This option
│ │ │ │ +0001bce0: 2069 7320 7573 6564 2074 6f20 7465 6c6c   is used to tell
│ │ │ │ +0001bcf0: 2074 6865 2066 756e 6374 696f 6e20 7072   the function pr
│ │ │ │ +0001bd00: 6f6a 4469 6d20 6e6f 7420 746f 206c 6f6f  ojDim not to loo
│ │ │ │ +0001bd10: 6b20 666f 7220 7072 6f6a 6563 7469 7665  k for projective
│ │ │ │ +0001bd20: 0a64 696d 656e 7369 6f6e 2062 656c 6f77  .dimension below
│ │ │ │ +0001bd30: 2074 6865 206f 7074 696f 6e20 7661 6c75   the option valu
│ │ │ │ +0001bd40: 652e 0a0a 5365 6520 616c 736f 0a3d 3d3d  e...See also.===
│ │ │ │ +0001bd50: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
│ │ │ │ +0001bd60: 2070 726f 6a44 696d 3a20 7072 6f6a 4469   projDim: projDi
│ │ │ │ +0001bd70: 6d2c 202d 2d20 6669 6e64 7320 616e 2075  m, -- finds an u
│ │ │ │ +0001bd80: 7070 6572 2062 6f75 6e64 2066 6f72 2074  pper bound for t
│ │ │ │ +0001bd90: 6865 2070 726f 6a65 6374 6976 650a 2020  he projective.  
│ │ │ │ +0001bda0: 2020 6469 6d65 6e73 696f 6e20 6f66 2061    dimension of a
│ │ │ │ +0001bdb0: 206d 6f64 756c 650a 0a46 756e 6374 696f   module..Functio
│ │ │ │ +0001bdc0: 6e73 2077 6974 6820 6f70 7469 6f6e 616c  ns with optional
│ │ │ │ +0001bdd0: 2061 7267 756d 656e 7420 6e61 6d65 6420   argument named 
│ │ │ │ +0001bde0: 4d69 6e44 696d 656e 7369 6f6e 3a0a 3d3d  MinDimension:.==
│ │ │ │ +0001bdf0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0001be00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0001be10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0001be20: 3d3d 0a0a 2020 2a20 2270 726f 6a44 696d  ==..  * "projDim
│ │ │ │ +0001be30: 282e 2e2e 2c4d 696e 4469 6d65 6e73 696f  (...,MinDimensio
│ │ │ │ +0001be40: 6e3d 3e2e 2e2e 2922 202d 2d20 7365 6520  n=>...)" -- see 
│ │ │ │ +0001be50: 2a6e 6f74 6520 7072 6f6a 4469 6d3a 2070  *note projDim: p
│ │ │ │ +0001be60: 726f 6a44 696d 2c20 2d2d 2066 696e 6473  rojDim, -- finds
│ │ │ │ +0001be70: 2061 6e0a 2020 2020 7570 7065 7220 626f   an.    upper bo
│ │ │ │ +0001be80: 756e 6420 666f 7220 7468 6520 7072 6f6a  und for the proj
│ │ │ │ +0001be90: 6563 7469 7665 2064 696d 656e 7369 6f6e  ective dimension
│ │ │ │ +0001bea0: 206f 6620 6120 6d6f 6475 6c65 0a0a 466f   of a module..Fo
│ │ │ │ +0001beb0: 7220 7468 6520 7072 6f67 7261 6d6d 6572  r the programmer
│ │ │ │ +0001bec0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ +0001bed0: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420  ===..The object 
│ │ │ │ +0001bee0: 2a6e 6f74 6520 4d69 6e44 696d 656e 7369  *note MinDimensi
│ │ │ │ +0001bef0: 6f6e 3a20 4d69 6e44 696d 656e 7369 6f6e  on: MinDimension
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│ │ │ │ -0001c960: 2d2d 2d2d 2b0a 7c69 3220 3a20 6f70 7469  ----+.|i2 : opti
│ │ │ │ -0001c970: 6f6e 7320 6669 6e64 414e 6f6e 5a65 726f  ons findANonZero
│ │ │ │ -0001c980: 4d69 6e6f 7220 2020 2020 2020 2020 2020  Minor           
│ │ │ │ -0001c990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c9b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001c9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c910: 2020 2020 7c0a 7c6f 3120 3a20 4c69 7374      |.|o1 : List
│ │ │ │ +0001c920: 2020 2020 2020 2020 2020 2020 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 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0001c970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c9b0: 2d2d 2d2d 2b0a 7c69 3220 3a20 6f70 7469  ----+.|i2 : opti
│ │ │ │ +0001c9c0: 6f6e 7320 6669 6e64 414e 6f6e 5a65 726f  ons findANonZero
│ │ │ │ +0001c9d0: 4d69 6e6f 7220 2020 2020 2020 2020 2020  Minor           
│ │ │ │  0001c9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ca00: 2020 2020 7c0a 7c6f 3220 3d20 4f70 7469      |.|o2 = Opti
│ │ │ │ -0001ca10: 6f6e 5461 626c 657b 4465 636f 6d70 6f73  onTable{Decompos
│ │ │ │ -0001ca20: 6974 696f 6e53 7472 6174 6567 7920 3d3e  itionStrategy =>
│ │ │ │ -0001ca30: 206e 756c 6c7d 2020 2020 2020 2020 2020   null}          
│ │ │ │ +0001ca00: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001ca10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ca20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ca30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ca40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ca50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001ca60: 2020 2020 2020 2020 4469 6d65 6e73 696f          Dimensio
│ │ │ │ -0001ca70: 6e46 756e 6374 696f 6e20 3d3e 2064 696d  nFunction => dim
│ │ │ │ -0001ca80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ca50: 2020 2020 7c0a 7c6f 3220 3d20 4f70 7469      |.|o2 = Opti
│ │ │ │ +0001ca60: 6f6e 5461 626c 657b 4465 636f 6d70 6f73  onTable{Decompos
│ │ │ │ +0001ca70: 6974 696f 6e53 7472 6174 6567 7920 3d3e  itionStrategy =>
│ │ │ │ +0001ca80: 206e 756c 6c7d 2020 2020 2020 2020 2020   null}          
│ │ │ │  0001ca90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001caa0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001cab0: 2020 2020 2020 2020 4578 7465 6e64 4669          ExtendFi
│ │ │ │ -0001cac0: 656c 6420 3d3e 2074 7275 6520 2020 2020  eld => true     
│ │ │ │ +0001cab0: 2020 2020 2020 2020 4469 6d65 6e73 696f          Dimensio
│ │ │ │ +0001cac0: 6e46 756e 6374 696f 6e20 3d3e 2064 696d  nFunction => dim
│ │ │ │  0001cad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001caf0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001cb00: 2020 2020 2020 2020 486f 6d6f 6765 6e65          Homogene
│ │ │ │ -0001cb10: 6f75 7320 3d3e 2066 616c 7365 2020 2020  ous => false    
│ │ │ │ +0001cb00: 2020 2020 2020 2020 4578 7465 6e64 4669          ExtendFi
│ │ │ │ +0001cb10: 656c 6420 3d3e 2074 7275 6520 2020 2020  eld => true     
│ │ │ │  0001cb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cb40: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001cb50: 2020 2020 2020 2020 4d69 6e6f 7250 6f69          MinorPoi
│ │ │ │ -0001cb60: 6e74 4174 7465 6d70 7473 203d 3e20 3520  ntAttempts => 5 
│ │ │ │ +0001cb50: 2020 2020 2020 2020 486f 6d6f 6765 6e65          Homogene
│ │ │ │ +0001cb60: 6f75 7320 3d3e 2066 616c 7365 2020 2020  ous => false    
│ │ │ │  0001cb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cb90: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001cba0: 2020 2020 2020 2020 4e75 6d54 6872 6561          NumThrea
│ │ │ │ -0001cbb0: 6473 546f 5573 6520 3d3e 2031 2020 2020  dsToUse => 1    
│ │ │ │ +0001cba0: 2020 2020 2020 2020 4d69 6e6f 7250 6f69          MinorPoi
│ │ │ │ +0001cbb0: 6e74 4174 7465 6d70 7473 203d 3e20 3520  ntAttempts => 5 
│ │ │ │  0001cbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cbe0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001cbf0: 2020 2020 2020 2020 506f 696e 7443 6865          PointChe
│ │ │ │ -0001cc00: 636b 4174 7465 6d70 7473 203d 3e20 3020  ckAttempts => 0 
│ │ │ │ +0001cbf0: 2020 2020 2020 2020 4e75 6d54 6872 6561          NumThrea
│ │ │ │ +0001cc00: 6473 546f 5573 6520 3d3e 2031 2020 2020  dsToUse => 1    
│ │ │ │  0001cc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cc30: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001cc40: 2020 2020 2020 2020 5265 706c 6163 656d          Replacem
│ │ │ │ -0001cc50: 656e 7420 3d3e 2042 696e 6f6d 6961 6c20  ent => Binomial 
│ │ │ │ +0001cc40: 2020 2020 2020 2020 506f 696e 7443 6865          PointChe
│ │ │ │ +0001cc50: 636b 4174 7465 6d70 7473 203d 3e20 3020  ckAttempts => 0 
│ │ │ │  0001cc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cc80: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001cc90: 2020 2020 2020 2020 5374 7261 7465 6779          Strategy
│ │ │ │ -0001cca0: 203d 3e20 4465 6661 756c 7420 2020 2020   => Default     
│ │ │ │ +0001cc90: 2020 2020 2020 2020 5265 706c 6163 656d          Replacem
│ │ │ │ +0001cca0: 656e 7420 3d3e 2042 696e 6f6d 6961 6c20  ent => Binomial 
│ │ │ │  0001ccb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ccc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ccd0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001cce0: 2020 2020 2020 2020 5665 7262 6f73 6520          Verbose 
│ │ │ │ -0001ccf0: 3d3e 2066 616c 7365 2020 2020 2020 2020  => false        
│ │ │ │ +0001cce0: 2020 2020 2020 2020 5374 7261 7465 6779          Strategy
│ │ │ │ +0001ccf0: 203d 3e20 4465 6661 756c 7420 2020 2020   => Default     
│ │ │ │  0001cd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cd20: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001cd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cd30: 2020 2020 2020 2020 5665 7262 6f73 6520          Verbose 
│ │ │ │ +0001cd40: 3d3e 2066 616c 7365 2020 2020 2020 2020  => false        
│ │ │ │  0001cd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cd70: 2020 2020 7c0a 7c6f 3220 3a20 4f70 7469      |.|o2 : Opti
│ │ │ │ -0001cd80: 6f6e 5461 626c 6520 2020 2020 2020 2020  onTable         
│ │ │ │ +0001cd70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001cd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cdc0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -0001cdd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cdf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ce00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ce10: 2d2d 2d2d 2b0a 0a54 6865 2064 6566 6175  ----+..The defau
│ │ │ │ -0001ce20: 6c74 2073 6574 7469 6e67 2045 7874 656e  lt setting Exten
│ │ │ │ -0001ce30: 6446 6965 6c64 203d 3e20 7472 7565 206d  dField => true m
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│ │ │ │ -0001ce50: 2077 686f 7365 2072 6573 6964 7565 2066   whose residue f
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│ │ │ │ -0001ce70: 6578 7465 6e73 696f 6e73 206f 6620 7468  extensions of th
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│ │ │ │ -0001ced0: 2073 6574 2048 6f6d 6f67 656e 656f 7573   set Homogeneous
│ │ │ │ -0001cee0: 3d3e 6661 6c73 6520 6279 2064 6566 6175  =>false by defau
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│ │ │ │ -0001cf20: 706f 696e 742e 2020 5365 7474 696e 6720  point.  Setting 
│ │ │ │ -0001cf30: 4578 7465 6e64 4669 656c 643d 3e66 616c  ExtendField=>fal
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│ │ │ │ -0001cf50: 730a 7370 6565 6420 7570 2063 6f6d 7075  s.speed up compu
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│ │ │ │ -0001cf70: 616c 736f 206d 6973 7320 736f 6d65 2069  also miss some i
│ │ │ │ -0001cf80: 6d70 6f72 7461 6e74 2073 7562 6d61 7472  mportant submatr
│ │ │ │ -0001cf90: 6963 6573 2069 6620 7468 6174 0a64 6574  ices if that.det
│ │ │ │ -0001cfa0: 6572 6d69 6e61 6e74 2028 706c 7573 2077  erminant (plus w
│ │ │ │ -0001cfb0: 6861 7420 6861 7320 616c 7265 6164 7920  hat has already 
│ │ │ │ -0001cfc0: 6265 656e 2063 6f6d 7075 7465 6429 2064  been computed) d
│ │ │ │ -0001cfd0: 6566 696e 6573 2061 2073 6368 656d 6520  efines a scheme 
│ │ │ │ -0001cfe0: 7769 7468 206e 6f20 6f72 0a72 656c 6174  with no or.relat
│ │ │ │ -0001cff0: 6976 656c 7920 6665 7720 7261 7469 6f6e  ively few ration
│ │ │ │ -0001d000: 616c 2070 6f69 6e74 732e 2020 496e 2073  al points.  In s
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│ │ │ │ -0001d020: 6e64 4669 656c 6420 3d3e 2066 616c 7365  ndField => false
│ │ │ │ -0001d030: 2077 696c 6c0a 7479 7069 6361 6c6c 7920   will.typically 
│ │ │ │ -0001d040: 7375 6273 7461 6e74 6961 6c6c 7920 736c  substantially sl
│ │ │ │ -0001d050: 6f77 2064 6f77 6e20 636f 6d70 7574 6174  ow down computat
│ │ │ │ -0001d060: 696f 6e73 2e0a 0a53 6565 2061 6c73 6f0a  ions...See also.
│ │ │ │ -0001d070: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ -0001d080: 6f74 6520 6669 6e64 414e 6f6e 5a65 726f  ote findANonZero
│ │ │ │ -0001d090: 4d69 6e6f 723a 2028 5261 6e64 6f6d 506f  Minor: (RandomPo
│ │ │ │ -0001d0a0: 696e 7473 2966 696e 6441 4e6f 6e5a 6572  ints)findANonZer
│ │ │ │ -0001d0b0: 6f4d 696e 6f72 2c20 2d2d 2066 696e 6473  oMinor, -- finds
│ │ │ │ -0001d0c0: 2061 0a20 2020 206e 6f6e 2d76 616e 6973   a.    non-vanis
│ │ │ │ -0001d0d0: 6869 6e67 206d 696e 6f72 2061 7420 736f  hing minor at so
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│ │ │ │ -0001d0f0: 656e 2070 6f69 6e74 0a0a 4675 6e63 7469  en point..Functi
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│ │ │ │ -0001d120: 2050 6f69 6e74 4f70 7469 6f6e 733a 0a3d   PointOptions:.=
│ │ │ │ -0001d130: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001d140: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001d150: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001d160: 3d3d 3d0a 0a20 202a 2022 6368 6f6f 7365  ===..  * "choose
│ │ │ │ -0001d170: 476f 6f64 4d69 6e6f 7273 282e 2e2e 2c50  GoodMinors(...,P
│ │ │ │ -0001d180: 6f69 6e74 4f70 7469 6f6e 733d 3e2e 2e2e  ointOptions=>...
│ │ │ │ -0001d190: 2922 0a20 202a 2022 6765 7453 7562 6d61  )".  * "getSubma
│ │ │ │ -0001d1a0: 7472 6978 4f66 5261 6e6b 282e 2e2e 2c50  trixOfRank(...,P
│ │ │ │ -0001d1b0: 6f69 6e74 4f70 7469 6f6e 733d 3e2e 2e2e  ointOptions=>...
│ │ │ │ -0001d1c0: 2922 0a20 202a 2022 6973 5261 6e6b 4174  )".  * "isRankAt
│ │ │ │ -0001d1d0: 4c65 6173 7428 2e2e 2e2c 506f 696e 744f  Least(...,PointO
│ │ │ │ -0001d1e0: 7074 696f 6e73 3d3e 2e2e 2e29 220a 2020  ptions=>...)".  
│ │ │ │ -0001d1f0: 2a20 2270 726f 6a44 696d 282e 2e2e 2c50  * "projDim(...,P
│ │ │ │ +0001cdc0: 2020 2020 7c0a 7c6f 3220 3a20 4f70 7469      |.|o2 : Opti
│ │ │ │ +0001cdd0: 6f6e 5461 626c 6520 2020 2020 2020 2020  onTable         
│ │ │ │ +0001cde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ce00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ce10: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0001ce20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ce30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ce40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ce50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ce60: 2d2d 2d2d 2b0a 0a54 6865 2064 6566 6175  ----+..The defau
│ │ │ │ +0001ce70: 6c74 2073 6574 7469 6e67 2045 7874 656e  lt setting Exten
│ │ │ │ +0001ce80: 6446 6965 6c64 203d 3e20 7472 7565 206d  dField => true m
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│ │ │ │ +0001cea0: 2077 686f 7365 2072 6573 6964 7565 2066   whose residue f
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│ │ │ │ +0001cec0: 6578 7465 6e73 696f 6e73 206f 6620 7468  extensions of th
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│ │ │ │ +0001cee0: 6520 7661 6c69 642c 2061 6e64 2061 7265  e valid, and are
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│ │ │ │ +0001cf10: 6865 726d 6f72 652c 2077 6520 6861 7665  hermore, we have
│ │ │ │ +0001cf20: 2073 6574 2048 6f6d 6f67 656e 656f 7573   set Homogeneous
│ │ │ │ +0001cf30: 3d3e 6661 6c73 6520 6279 2064 6566 6175  =>false by defau
│ │ │ │ +0001cf40: 6c74 2077 6869 6368 206d 6561 6e73 2074  lt which means t
│ │ │ │ +0001cf50: 6865 0a6f 7269 6769 6e20 6973 2074 7265  he.origin is tre
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│ │ │ │ +0001cf70: 706f 696e 742e 2020 5365 7474 696e 6720  point.  Setting 
│ │ │ │ +0001cf80: 4578 7465 6e64 4669 656c 643d 3e66 616c  ExtendField=>fal
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│ │ │ │ +0001cfa0: 730a 7370 6565 6420 7570 2063 6f6d 7075  s.speed up compu
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│ │ │ │ +0001cfc0: 616c 736f 206d 6973 7320 736f 6d65 2069  also miss some i
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│ │ │ │ +0001cfe0: 6963 6573 2069 6620 7468 6174 0a64 6574  ices if that.det
│ │ │ │ +0001cff0: 6572 6d69 6e61 6e74 2028 706c 7573 2077  erminant (plus w
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│ │ │ │ +0001d030: 7769 7468 206e 6f20 6f72 0a72 656c 6174  with no or.relat
│ │ │ │ +0001d040: 6976 656c 7920 6665 7720 7261 7469 6f6e  ively few ration
│ │ │ │ +0001d050: 616c 2070 6f69 6e74 732e 2020 496e 2073  al points.  In s
│ │ │ │ +0001d060: 7563 6820 6120 6361 7365 2c20 4578 7465  uch a case, Exte
│ │ │ │ +0001d070: 6e64 4669 656c 6420 3d3e 2066 616c 7365  ndField => false
│ │ │ │ +0001d080: 2077 696c 6c0a 7479 7069 6361 6c6c 7920   will.typically 
│ │ │ │ +0001d090: 7375 6273 7461 6e74 6961 6c6c 7920 736c  substantially sl
│ │ │ │ +0001d0a0: 6f77 2064 6f77 6e20 636f 6d70 7574 6174  ow down computat
│ │ │ │ +0001d0b0: 696f 6e73 2e0a 0a53 6565 2061 6c73 6f0a  ions...See also.
│ │ │ │ +0001d0c0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ +0001d0d0: 6f74 6520 6669 6e64 414e 6f6e 5a65 726f  ote findANonZero
│ │ │ │ +0001d0e0: 4d69 6e6f 723a 2028 5261 6e64 6f6d 506f  Minor: (RandomPo
│ │ │ │ +0001d0f0: 696e 7473 2966 696e 6441 4e6f 6e5a 6572  ints)findANonZer
│ │ │ │ +0001d100: 6f4d 696e 6f72 2c20 2d2d 2066 696e 6473  oMinor, -- finds
│ │ │ │ +0001d110: 2061 0a20 2020 206e 6f6e 2d76 616e 6973   a.    non-vanis
│ │ │ │ +0001d120: 6869 6e67 206d 696e 6f72 2061 7420 736f  hing minor at so
│ │ │ │ +0001d130: 6d65 2072 616e 646f 6d6c 7920 6368 6f73  me randomly chos
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│ │ │ │ +0001d160: 6c20 6172 6775 6d65 6e74 206e 616d 6564  l argument named
│ │ │ │ +0001d170: 2050 6f69 6e74 4f70 7469 6f6e 733a 0a3d   PointOptions:.=
│ │ │ │ +0001d180: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0001d190: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0001d1a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0001d1b0: 3d3d 3d0a 0a20 202a 2022 6368 6f6f 7365  ===..  * "choose
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│ │ │ │ +0001d1e0: 2922 0a20 202a 2022 6765 7453 7562 6d61  )".  * "getSubma
│ │ │ │ +0001d1f0: 7472 6978 4f66 5261 6e6b 282e 2e2e 2c50  trixOfRank(...,P
│ │ │ │  0001d200: 6f69 6e74 4f70 7469 6f6e 733d 3e2e 2e2e  ointOptions=>...
│ │ │ │ -0001d210: 2922 0a20 202a 2022 7265 6775 6c61 7249  )".  * "regularI
│ │ │ │ -0001d220: 6e43 6f64 696d 656e 7369 6f6e 282e 2e2e  nCodimension(...
│ │ │ │ -0001d230: 2c50 6f69 6e74 4f70 7469 6f6e 733d 3e2e  ,PointOptions=>.
│ │ │ │ -0001d240: 2e2e 2922 0a0a 466f 7220 7468 6520 7072  ..)"..For the pr
│ │ │ │ -0001d250: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d  ogrammer.=======
│ │ │ │ -0001d260: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865  ===========..The
│ │ │ │ -0001d270: 206f 626a 6563 7420 2a6e 6f74 6520 506f   object *note Po
│ │ │ │ -0001d280: 696e 744f 7074 696f 6e73 3a20 506f 696e  intOptions: Poin
│ │ │ │ -0001d290: 744f 7074 696f 6e73 2c20 6973 2061 202a  tOptions, is a *
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│ │ │ │ -0001d2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d310: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365  ----..The source
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│ │ │ │ -0001d340: 6570 726f 6475 6369 626c 652d 7061 7468  eproducible-path
│ │ │ │ -0001d350: 2f6d 6163 6175 6c61 7932 2d31 2e32 352e  /macaulay2-1.25.
│ │ │ │ -0001d360: 3131 2b64 732f 4d32 2f4d 6163 6175 6c61  11+ds/M2/Macaula
│ │ │ │ -0001d370: 7932 2f70 6163 6b61 6765 732f 4661 7374  y2/packages/Fast
│ │ │ │ -0001d380: 4d69 6e6f 7273 2e0a 6d32 3a31 3739 343a  Minors..m2:1794:
│ │ │ │ -0001d390: 302e 0a1f 0a46 696c 653a 2046 6173 744d  0....File: FastM
│ │ │ │ -0001d3a0: 696e 6f72 732e 696e 666f 2c20 4e6f 6465  inors.info, Node
│ │ │ │ -0001d3b0: 3a20 7072 6f6a 4469 6d2c 204e 6578 743a  : projDim, Next:
│ │ │ │ -0001d3c0: 2072 6563 7572 7369 7665 4d69 6e6f 7273   recursiveMinors
│ │ │ │ -0001d3d0: 2c20 5072 6576 3a20 506f 696e 744f 7074  , Prev: PointOpt
│ │ │ │ -0001d3e0: 696f 6e73 2c20 5570 3a20 546f 700a 0a70  ions, Up: Top..p
│ │ │ │ -0001d3f0: 726f 6a44 696d 202d 2d20 6669 6e64 7320  rojDim -- finds 
│ │ │ │ -0001d400: 616e 2075 7070 6572 2062 6f75 6e64 2066  an upper bound f
│ │ │ │ -0001d410: 6f72 2074 6865 2070 726f 6a65 6374 6976  or the projectiv
│ │ │ │ -0001d420: 6520 6469 6d65 6e73 696f 6e20 6f66 2061  e dimension of a
│ │ │ │ -0001d430: 206d 6f64 756c 650a 2a2a 2a2a 2a2a 2a2a   module.********
│ │ │ │ -0001d440: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001d450: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001d460: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001d470: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001d480: 0a0a 2020 2a20 5573 6167 653a 200a 2020  ..  * Usage: .  
│ │ │ │ -0001d490: 2020 2020 2020 6e20 3d20 7072 6f6a 4469        n = projDi
│ │ │ │ -0001d4a0: 6d28 4e2c 204d 696e 4469 6d65 6e73 696f  m(N, MinDimensio
│ │ │ │ -0001d4b0: 6e3d 3e64 290a 2020 2a20 496e 7075 7473  n=>d).  * Inputs
│ │ │ │ -0001d4c0: 3a0a 2020 2020 2020 2a20 4e2c 2061 202a  :.      * N, a *
│ │ │ │ -0001d4d0: 6e6f 7465 206d 6f64 756c 653a 2028 4d61  note module: (Ma
│ │ │ │ -0001d4e0: 6361 756c 6179 3244 6f63 294d 6f64 756c  caulay2Doc)Modul
│ │ │ │ -0001d4f0: 652c 2c20 6120 6d6f 6475 6c65 206f 7665  e,, a module ove
│ │ │ │ -0001d500: 7220 6120 706f 6c79 6e6f 6d69 616c 0a20  r a polynomial. 
│ │ │ │ -0001d510: 2020 2020 2020 2072 696e 670a 2020 2020         ring.    
│ │ │ │ -0001d520: 2020 2a20 642c 2061 6e20 2a6e 6f74 6520    * d, an *note 
│ │ │ │ -0001d530: 696e 7465 6765 723a 2028 4d61 6361 756c  integer: (Macaul
│ │ │ │ -0001d540: 6179 3244 6f63 295a 5a2c 2c20 7468 6520  ay2Doc)ZZ,, the 
│ │ │ │ -0001d550: 6d69 6e69 6d75 6d20 7072 6f6a 6563 7469  minimum projecti
│ │ │ │ -0001d560: 7665 0a20 2020 2020 2020 2064 696d 656e  ve.        dimen
│ │ │ │ -0001d570: 7369 6f6e 206f 6620 7468 6520 6d6f 6475  sion of the modu
│ │ │ │ -0001d580: 6c65 0a20 202a 202a 6e6f 7465 204f 7074  le.  * *note Opt
│ │ │ │ -0001d590: 696f 6e61 6c20 696e 7075 7473 3a20 284d  ional inputs: (M
│ │ │ │ -0001d5a0: 6163 6175 6c61 7932 446f 6329 7573 696e  acaulay2Doc)usin
│ │ │ │ -0001d5b0: 6720 6675 6e63 7469 6f6e 7320 7769 7468  g functions with
│ │ │ │ -0001d5c0: 206f 7074 696f 6e61 6c20 696e 7075 7473   optional inputs
│ │ │ │ -0001d5d0: 2c3a 0a20 2020 2020 202a 204d 696e 4469  ,:.      * MinDi
│ │ │ │ -0001d5e0: 6d65 6e73 696f 6e20 3d3e 2061 202a 6e6f  mension => a *no
│ │ │ │ -0001d5f0: 7465 206e 756d 6265 723a 2028 4d61 6361  te number: (Maca
│ │ │ │ -0001d600: 756c 6179 3244 6f63 294e 756d 6265 722c  ulay2Doc)Number,
│ │ │ │ -0001d610: 2c20 6465 6661 756c 7420 7661 6c75 6520  , default value 
│ │ │ │ -0001d620: 302c 0a20 2020 2020 2020 2073 746f 7020  0,.        stop 
│ │ │ │ -0001d630: 6166 7465 7220 7665 7269 6679 696e 6720  after verifying 
│ │ │ │ -0001d640: 7468 6520 6d6f 6475 6c65 2068 6173 2061  the module has a
│ │ │ │ -0001d650: 7420 6d6f 7374 2061 2063 6572 7461 696e  t most a certain
│ │ │ │ -0001d660: 2070 726f 6a65 6374 6976 650a 2020 2020   projective.    
│ │ │ │ -0001d670: 2020 2020 6469 6d65 6e73 696f 6e0a 2020      dimension.  
│ │ │ │ -0001d680: 2020 2020 2a20 2a6e 6f74 6520 506f 696e      * *note Poin
│ │ │ │ -0001d690: 744f 7074 696f 6e73 3a20 506f 696e 744f  tOptions: PointO
│ │ │ │ -0001d6a0: 7074 696f 6e73 2c20 3d3e 2061 202a 6e6f  ptions, => a *no
│ │ │ │ -0001d6b0: 7465 206c 6973 743a 2028 4d61 6361 756c  te list: (Macaul
│ │ │ │ -0001d6c0: 6179 3244 6f63 294c 6973 742c 2c0a 2020  ay2Doc)List,,.  
│ │ │ │ -0001d6d0: 2020 2020 2020 6465 6661 756c 7420 7661        default va
│ │ │ │ -0001d6e0: 6c75 6520 7b53 7472 6174 6567 7920 3d3e  lue {Strategy =>
│ │ │ │ -0001d6f0: 2044 6566 6175 6c74 2c20 486f 6d6f 6765   Default, Homoge
│ │ │ │ -0001d700: 6e65 6f75 7320 3d3e 2066 616c 7365 2c20  neous => false, 
│ │ │ │ -0001d710: 5265 706c 6163 656d 656e 740a 2020 2020  Replacement.    
│ │ │ │ -0001d720: 2020 2020 3d3e 2042 696e 6f6d 6961 6c2c      => Binomial,
│ │ │ │ -0001d730: 2045 7874 656e 6446 6965 6c64 203d 3e20   ExtendField => 
│ │ │ │ -0001d740: 7472 7565 2c20 506f 696e 7443 6865 636b  true, PointCheck
│ │ │ │ -0001d750: 4174 7465 6d70 7473 203d 3e20 302c 0a20  Attempts => 0,. 
│ │ │ │ -0001d760: 2020 2020 2020 2044 6563 6f6d 706f 7369         Decomposi
│ │ │ │ -0001d770: 7469 6f6e 5374 7261 7465 6779 203d 3e20  tionStrategy => 
│ │ │ │ -0001d780: 4465 636f 6d70 6f73 652c 204e 756d 5468  Decompose, NumTh
│ │ │ │ -0001d790: 7265 6164 7354 6f55 7365 203d 3e20 312c  readsToUse => 1,
│ │ │ │ -0001d7a0: 0a20 2020 2020 2020 2044 696d 656e 7369  .        Dimensi
│ │ │ │ -0001d7b0: 6f6e 4675 6e63 7469 6f6e 203d 3e20 6469  onFunction => di
│ │ │ │ -0001d7c0: 6d2c 2056 6572 626f 7365 203d 3e20 6661  m, Verbose => fa
│ │ │ │ -0001d7d0: 6c73 657d 2c20 6f70 7469 6f6e 7320 746f  lse}, options to
│ │ │ │ -0001d7e0: 2062 6520 7061 7373 6564 2074 6f0a 2020   be passed to.  
│ │ │ │ -0001d7f0: 2020 2020 2020 7468 6520 5261 6e64 6f6d        the Random
│ │ │ │ -0001d800: 506f 696e 7473 2070 6163 6b61 6765 0a20  Points package. 
│ │ │ │ -0001d810: 2020 2020 202a 202a 6e6f 7465 204d 6178       * *note Max
│ │ │ │ -0001d820: 4d69 6e6f 7273 3a20 4d61 784d 696e 6f72  Minors: MaxMinor
│ │ │ │ -0001d830: 732c 203d 3e20 2e2e 2e2c 2064 6566 6175  s, => ..., defau
│ │ │ │ -0001d840: 6c74 2076 616c 7565 0a20 2020 2020 2020  lt value.       
│ │ │ │ -0001d850: 2046 756e 6374 696f 6e43 6c6f 7375 7265   FunctionClosure
│ │ │ │ -0001d860: 5b2e 2e2f 4661 7374 4d69 6e6f 7273 2e6d  [../FastMinors.m
│ │ │ │ -0001d870: 323a 3138 363a 3138 2d31 3836 3a34 355d  2:186:18-186:45]
│ │ │ │ -0001d880: 2c20 7573 6564 2074 6f20 636f 6e74 726f  , used to contro
│ │ │ │ -0001d890: 6c20 686f 770a 2020 2020 2020 2020 6d61  l how.        ma
│ │ │ │ -0001d8a0: 6e79 206d 696e 6f72 7320 6172 6520 636f  ny minors are co
│ │ │ │ -0001d8b0: 6d70 7574 6564 206f 6620 7468 6520 6d61  mputed of the ma
│ │ │ │ -0001d8c0: 7472 6963 6573 2069 6e20 6120 7072 6f6a  trices in a proj
│ │ │ │ -0001d8d0: 6563 7469 7665 2072 6573 6f6c 7574 696f  ective resolutio
│ │ │ │ -0001d8e0: 6e0a 2020 2020 2020 2a20 2a6e 6f74 6520  n.      * *note 
│ │ │ │ -0001d8f0: 4465 7453 7472 6174 6567 793a 2044 6574  DetStrategy: Det
│ │ │ │ -0001d900: 5374 7261 7465 6779 2c20 3d3e 202e 2e2e  Strategy, => ...
│ │ │ │ -0001d910: 2c20 6465 6661 756c 7420 7661 6c75 6520  , default value 
│ │ │ │ -0001d920: 436f 6661 6374 6f72 2c0a 2020 2020 2020  Cofactor,.      
│ │ │ │ -0001d930: 2020 4465 7453 7472 6174 6567 7920 6973    DetStrategy is
│ │ │ │ -0001d940: 2061 2073 7472 6174 6567 7920 666f 7220   a strategy for 
│ │ │ │ -0001d950: 616c 6c6f 7769 6e67 2074 6865 2075 7365  allowing the use
│ │ │ │ -0001d960: 7220 746f 2063 686f 6f73 6520 686f 770a  r to choose how.
│ │ │ │ -0001d970: 2020 2020 2020 2020 6465 7465 726d 696e          determin
│ │ │ │ -0001d980: 616e 7473 2028 6f72 2072 616e 6b29 2c20  ants (or rank), 
│ │ │ │ -0001d990: 6973 2063 6f6d 7075 7465 640a 2020 2020  is computed.    
│ │ │ │ -0001d9a0: 2020 2a20 2a6e 6f74 6520 5374 7261 7465    * *note Strate
│ │ │ │ -0001d9b0: 6779 3a20 5374 7261 7465 6779 4465 6661  gy: StrategyDefa
│ │ │ │ -0001d9c0: 756c 742c 203d 3e20 2e2e 2e2c 2064 6566  ult, => ..., def
│ │ │ │ -0001d9d0: 6175 6c74 2076 616c 7565 206e 6577 204f  ault value new O
│ │ │ │ -0001d9e0: 7074 696f 6e54 6162 6c65 0a20 2020 2020  ptionTable.     
│ │ │ │ -0001d9f0: 2020 2066 726f 6d20 7b50 6f69 6e74 7320     from {Points 
│ │ │ │ -0001da00: 3d3e 2030 2c20 5261 6e64 6f6d 203d 3e20  => 0, Random => 
│ │ │ │ -0001da10: 3136 2c20 4752 6576 4c65 784c 6172 6765  16, GRevLexLarge
│ │ │ │ -0001da20: 7374 203d 3e20 302c 204c 6578 536d 616c  st => 0, LexSmal
│ │ │ │ -0001da30: 6c65 7374 5465 726d 0a20 2020 2020 2020  lestTerm.       
│ │ │ │ -0001da40: 203d 3e20 3136 2c20 4c65 784c 6172 6765   => 16, LexLarge
│ │ │ │ -0001da50: 7374 203d 3e20 302c 204c 6578 536d 616c  st => 0, LexSmal
│ │ │ │ -0001da60: 6c65 7374 203d 3e20 3136 2c20 4752 6576  lest => 16, GRev
│ │ │ │ -0001da70: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │ -0001da80: 3d3e 2031 362c 0a20 2020 2020 2020 2052  => 16,.        R
│ │ │ │ -0001da90: 616e 646f 6d4e 6f6e 7a65 726f 203d 3e20  andomNonzero => 
│ │ │ │ -0001daa0: 3136 2c20 4752 6576 4c65 7853 6d61 6c6c  16, GRevLexSmall
│ │ │ │ -0001dab0: 6573 7420 3d3e 2031 367d 2c20 7374 7261  est => 16}, stra
│ │ │ │ -0001dac0: 7465 6769 6573 2066 6f72 2063 686f 6f73  tegies for choos
│ │ │ │ -0001dad0: 696e 670a 2020 2020 2020 2020 7375 626d  ing.        subm
│ │ │ │ -0001dae0: 6174 7269 6365 730a 2020 2020 2020 2a20  atrices.      * 
│ │ │ │ -0001daf0: 5665 7262 6f73 6520 3d3e 202e 2e2e 2c20  Verbose => ..., 
│ │ │ │ -0001db00: 6465 6661 756c 7420 7661 6c75 6520 6661  default value fa
│ │ │ │ -0001db10: 6c73 650a 2020 2a20 4f75 7470 7574 733a  lse.  * Outputs:
│ │ │ │ -0001db20: 0a20 2020 2020 202a 206e 2c20 616e 202a  .      * n, an *
│ │ │ │ -0001db30: 6e6f 7465 2069 6e74 6567 6572 3a20 284d  note integer: (M
│ │ │ │ -0001db40: 6163 6175 6c61 7932 446f 6329 5a5a 2c2c  acaulay2Doc)ZZ,,
│ │ │ │ -0001db50: 2061 6e20 7570 7065 7220 626f 756e 6420   an upper bound 
│ │ │ │ -0001db60: 666f 7220 7468 650a 2020 2020 2020 2020  for the.        
│ │ │ │ -0001db70: 7072 6f6a 6563 7469 7665 2064 696d 656e  projective dimen
│ │ │ │ -0001db80: 7369 6f6e 206f 6620 4e0a 0a44 6573 6372  sion of N..Descr
│ │ │ │ -0001db90: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d  iption.=========
│ │ │ │ -0001dba0: 3d3d 0a0a 5468 6520 6675 6e63 7469 6f6e  ==..The function
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│ │ │ │ -0001dbc0: 6520 6c65 6e67 7468 206f 6620 6120 7072  e length of a pr
│ │ │ │ -0001dbd0: 6f6a 6563 7469 7665 2072 6573 6f6c 7574  ojective resolut
│ │ │ │ -0001dbe0: 696f 6e2e 2049 6620 7468 6520 6d6f 6475  ion. If the modu
│ │ │ │ -0001dbf0: 6c65 0a70 6173 7365 6420 6973 206e 6f74  le.passed is not
│ │ │ │ -0001dc00: 2068 6f6d 6f67 656e 656f 7573 2c20 7468   homogeneous, th
│ │ │ │ -0001dc10: 656e 2074 6865 2070 726f 6a65 6374 6976  en the projectiv
│ │ │ │ -0001dc20: 6520 7265 736f 6c75 7469 6f6e 206d 6179  e resolution may
│ │ │ │ -0001dc30: 206e 6f74 2062 6520 6d69 6e69 6d61 6c0a   not be minimal.
│ │ │ │ -0001dc40: 616e 6420 736f 2070 6469 6d20 6361 6e20  and so pdim can 
│ │ │ │ -0001dc50: 6769 7665 2074 6865 2077 726f 6e67 2061  give the wrong a
│ │ │ │ -0001dc60: 6e73 7765 722e 2020 5468 6973 2066 756e  nswer.  This fun
│ │ │ │ -0001dc70: 6374 696f 6e20 7072 6f6a 4469 6d20 7472  ction projDim tr
│ │ │ │ -0001dc80: 6965 7320 746f 2069 6d70 726f 7665 0a74  ies to improve.t
│ │ │ │ -0001dc90: 6869 7320 626f 756e 6420 6279 2063 6f6e  his bound by con
│ │ │ │ -0001dca0: 7369 6465 7269 6e67 2069 6465 616c 7320  sidering ideals 
│ │ │ │ -0001dcb0: 6f66 2061 7070 726f 7072 6961 7465 6c79  of appropriately
│ │ │ │ -0001dcc0: 2073 697a 6564 206d 696e 6f72 7320 6f66   sized minors of
│ │ │ │ -0001dcd0: 2074 6865 0a72 6573 6f6c 7574 696f 6e20   the.resolution 
│ │ │ │ -0001dce0: 2873 7461 7274 696e 6720 6672 6f6d 2074  (starting from t
│ │ │ │ -0001dcf0: 6865 2065 6e64 206f 6620 7468 6520 7265  he end of the re
│ │ │ │ -0001dd00: 736f 6c75 7469 6f6e 2061 6e64 2077 6f72  solution and wor
│ │ │ │ -0001dd10: 6b69 6e67 2062 6163 6b77 6172 6473 292e  king backwards).
│ │ │ │ -0001dd20: 0a55 7369 6e67 2074 6865 206f 7074 696f  .Using the optio
│ │ │ │ -0001dd30: 6e20 4d69 6e44 696d 656e 7369 6f6e 2028  n MinDimension (
│ │ │ │ -0001dd40: 6465 6661 756c 7420 7661 6c75 6520 3029  default value 0)
│ │ │ │ -0001dd50: 2067 6976 6573 2061 206c 6f77 6572 2062   gives a lower b
│ │ │ │ -0001dd60: 6f75 6e64 206f 6e20 7468 650a 7072 6f6a  ound on the.proj
│ │ │ │ -0001dd70: 6563 7469 7665 2064 696d 656e 7369 6f6e  ective dimension
│ │ │ │ -0001dd80: 2c20 696e 6372 6561 7369 6e67 2069 7420  , increasing it 
│ │ │ │ -0001dd90: 6361 6e20 7468 7573 2069 6d70 726f 7665  can thus improve
│ │ │ │ -0001dda0: 2074 6865 2073 7065 6564 206f 6620 636f   the speed of co
│ │ │ │ -0001ddb0: 6d70 7574 6174 696f 6e2e 0a0a 2b2d 2d2d  mputation...+---
│ │ │ │ -0001ddc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ddd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001dde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ddf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001de00: 2d2d 2d2d 2d2b 0a7c 6931 203a 2052 203d  -----+.|i1 : R =
│ │ │ │ -0001de10: 2051 515b 782c 795d 3b20 2020 2020 2020   QQ[x,y];       
│ │ │ │ -0001de20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001de30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001de40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0001d290: 2e2e 2922 0a0a 466f 7220 7468 6520 7072  ..)"..For the pr
│ │ │ │ +0001d2a0: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d  ogrammer.=======
│ │ │ │ +0001d2b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865  ===========..The
│ │ │ │ +0001d2c0: 206f 626a 6563 7420 2a6e 6f74 6520 506f   object *note Po
│ │ │ │ +0001d2d0: 696e 744f 7074 696f 6e73 3a20 506f 696e  intOptions: Poin
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│ │ │ │ +0001d360: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365  ----..The source
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│ │ │ │ +0001d380: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72  t is in./build/r
│ │ │ │ +0001d390: 6570 726f 6475 6369 626c 652d 7061 7468  eproducible-path
│ │ │ │ +0001d3a0: 2f6d 6163 6175 6c61 7932 2d31 2e32 352e  /macaulay2-1.25.
│ │ │ │ +0001d3b0: 3131 2b64 732f 4d32 2f4d 6163 6175 6c61  11+ds/M2/Macaula
│ │ │ │ +0001d3c0: 7932 2f70 6163 6b61 6765 732f 4661 7374  y2/packages/Fast
│ │ │ │ +0001d3d0: 4d69 6e6f 7273 2e0a 6d32 3a31 3739 343a  Minors..m2:1794:
│ │ │ │ +0001d3e0: 302e 0a1f 0a46 696c 653a 2046 6173 744d  0....File: FastM
│ │ │ │ +0001d3f0: 696e 6f72 732e 696e 666f 2c20 4e6f 6465  inors.info, Node
│ │ │ │ +0001d400: 3a20 7072 6f6a 4469 6d2c 204e 6578 743a  : projDim, Next:
│ │ │ │ +0001d410: 2072 6563 7572 7369 7665 4d69 6e6f 7273   recursiveMinors
│ │ │ │ +0001d420: 2c20 5072 6576 3a20 506f 696e 744f 7074  , Prev: PointOpt
│ │ │ │ +0001d430: 696f 6e73 2c20 5570 3a20 546f 700a 0a70  ions, Up: Top..p
│ │ │ │ +0001d440: 726f 6a44 696d 202d 2d20 6669 6e64 7320  rojDim -- finds 
│ │ │ │ +0001d450: 616e 2075 7070 6572 2062 6f75 6e64 2066  an upper bound f
│ │ │ │ +0001d460: 6f72 2074 6865 2070 726f 6a65 6374 6976  or the projectiv
│ │ │ │ +0001d470: 6520 6469 6d65 6e73 696f 6e20 6f66 2061  e dimension of a
│ │ │ │ +0001d480: 206d 6f64 756c 650a 2a2a 2a2a 2a2a 2a2a   module.********
│ │ │ │ +0001d490: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0001d4a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0001d4b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0001d4c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0001d4d0: 0a0a 2020 2a20 5573 6167 653a 200a 2020  ..  * Usage: .  
│ │ │ │ +0001d4e0: 2020 2020 2020 6e20 3d20 7072 6f6a 4469        n = projDi
│ │ │ │ +0001d4f0: 6d28 4e2c 204d 696e 4469 6d65 6e73 696f  m(N, MinDimensio
│ │ │ │ +0001d500: 6e3d 3e64 290a 2020 2a20 496e 7075 7473  n=>d).  * Inputs
│ │ │ │ +0001d510: 3a0a 2020 2020 2020 2a20 4e2c 2061 202a  :.      * N, a *
│ │ │ │ +0001d520: 6e6f 7465 206d 6f64 756c 653a 2028 4d61  note module: (Ma
│ │ │ │ +0001d530: 6361 756c 6179 3244 6f63 294d 6f64 756c  caulay2Doc)Modul
│ │ │ │ +0001d540: 652c 2c20 6120 6d6f 6475 6c65 206f 7665  e,, a module ove
│ │ │ │ +0001d550: 7220 6120 706f 6c79 6e6f 6d69 616c 0a20  r a polynomial. 
│ │ │ │ +0001d560: 2020 2020 2020 2072 696e 670a 2020 2020         ring.    
│ │ │ │ +0001d570: 2020 2a20 642c 2061 6e20 2a6e 6f74 6520    * d, an *note 
│ │ │ │ +0001d580: 696e 7465 6765 723a 2028 4d61 6361 756c  integer: (Macaul
│ │ │ │ +0001d590: 6179 3244 6f63 295a 5a2c 2c20 7468 6520  ay2Doc)ZZ,, the 
│ │ │ │ +0001d5a0: 6d69 6e69 6d75 6d20 7072 6f6a 6563 7469  minimum projecti
│ │ │ │ +0001d5b0: 7665 0a20 2020 2020 2020 2064 696d 656e  ve.        dimen
│ │ │ │ +0001d5c0: 7369 6f6e 206f 6620 7468 6520 6d6f 6475  sion of the modu
│ │ │ │ +0001d5d0: 6c65 0a20 202a 202a 6e6f 7465 204f 7074  le.  * *note Opt
│ │ │ │ +0001d5e0: 696f 6e61 6c20 696e 7075 7473 3a20 284d  ional inputs: (M
│ │ │ │ +0001d5f0: 6163 6175 6c61 7932 446f 6329 7573 696e  acaulay2Doc)usin
│ │ │ │ +0001d600: 6720 6675 6e63 7469 6f6e 7320 7769 7468  g functions with
│ │ │ │ +0001d610: 206f 7074 696f 6e61 6c20 696e 7075 7473   optional inputs
│ │ │ │ +0001d620: 2c3a 0a20 2020 2020 202a 204d 696e 4469  ,:.      * MinDi
│ │ │ │ +0001d630: 6d65 6e73 696f 6e20 3d3e 2061 202a 6e6f  mension => a *no
│ │ │ │ +0001d640: 7465 206e 756d 6265 723a 2028 4d61 6361  te number: (Maca
│ │ │ │ +0001d650: 756c 6179 3244 6f63 294e 756d 6265 722c  ulay2Doc)Number,
│ │ │ │ +0001d660: 2c20 6465 6661 756c 7420 7661 6c75 6520  , default value 
│ │ │ │ +0001d670: 302c 0a20 2020 2020 2020 2073 746f 7020  0,.        stop 
│ │ │ │ +0001d680: 6166 7465 7220 7665 7269 6679 696e 6720  after verifying 
│ │ │ │ +0001d690: 7468 6520 6d6f 6475 6c65 2068 6173 2061  the module has a
│ │ │ │ +0001d6a0: 7420 6d6f 7374 2061 2063 6572 7461 696e  t most a certain
│ │ │ │ +0001d6b0: 2070 726f 6a65 6374 6976 650a 2020 2020   projective.    
│ │ │ │ +0001d6c0: 2020 2020 6469 6d65 6e73 696f 6e0a 2020      dimension.  
│ │ │ │ +0001d6d0: 2020 2020 2a20 2a6e 6f74 6520 506f 696e      * *note Poin
│ │ │ │ +0001d6e0: 744f 7074 696f 6e73 3a20 506f 696e 744f  tOptions: PointO
│ │ │ │ +0001d6f0: 7074 696f 6e73 2c20 3d3e 2061 202a 6e6f  ptions, => a *no
│ │ │ │ +0001d700: 7465 206c 6973 743a 2028 4d61 6361 756c  te list: (Macaul
│ │ │ │ +0001d710: 6179 3244 6f63 294c 6973 742c 2c0a 2020  ay2Doc)List,,.  
│ │ │ │ +0001d720: 2020 2020 2020 6465 6661 756c 7420 7661        default va
│ │ │ │ +0001d730: 6c75 6520 7b53 7472 6174 6567 7920 3d3e  lue {Strategy =>
│ │ │ │ +0001d740: 2044 6566 6175 6c74 2c20 486f 6d6f 6765   Default, Homoge
│ │ │ │ +0001d750: 6e65 6f75 7320 3d3e 2066 616c 7365 2c20  neous => false, 
│ │ │ │ +0001d760: 5265 706c 6163 656d 656e 740a 2020 2020  Replacement.    
│ │ │ │ +0001d770: 2020 2020 3d3e 2042 696e 6f6d 6961 6c2c      => Binomial,
│ │ │ │ +0001d780: 2045 7874 656e 6446 6965 6c64 203d 3e20   ExtendField => 
│ │ │ │ +0001d790: 7472 7565 2c20 506f 696e 7443 6865 636b  true, PointCheck
│ │ │ │ +0001d7a0: 4174 7465 6d70 7473 203d 3e20 302c 0a20  Attempts => 0,. 
│ │ │ │ +0001d7b0: 2020 2020 2020 2044 6563 6f6d 706f 7369         Decomposi
│ │ │ │ +0001d7c0: 7469 6f6e 5374 7261 7465 6779 203d 3e20  tionStrategy => 
│ │ │ │ +0001d7d0: 4465 636f 6d70 6f73 652c 204e 756d 5468  Decompose, NumTh
│ │ │ │ +0001d7e0: 7265 6164 7354 6f55 7365 203d 3e20 312c  readsToUse => 1,
│ │ │ │ +0001d7f0: 0a20 2020 2020 2020 2044 696d 656e 7369  .        Dimensi
│ │ │ │ +0001d800: 6f6e 4675 6e63 7469 6f6e 203d 3e20 6469  onFunction => di
│ │ │ │ +0001d810: 6d2c 2056 6572 626f 7365 203d 3e20 6661  m, Verbose => fa
│ │ │ │ +0001d820: 6c73 657d 2c20 6f70 7469 6f6e 7320 746f  lse}, options to
│ │ │ │ +0001d830: 2062 6520 7061 7373 6564 2074 6f0a 2020   be passed to.  
│ │ │ │ +0001d840: 2020 2020 2020 7468 6520 5261 6e64 6f6d        the Random
│ │ │ │ +0001d850: 506f 696e 7473 2070 6163 6b61 6765 0a20  Points package. 
│ │ │ │ +0001d860: 2020 2020 202a 202a 6e6f 7465 204d 6178       * *note Max
│ │ │ │ +0001d870: 4d69 6e6f 7273 3a20 4d61 784d 696e 6f72  Minors: MaxMinor
│ │ │ │ +0001d880: 732c 203d 3e20 2e2e 2e2c 2064 6566 6175  s, => ..., defau
│ │ │ │ +0001d890: 6c74 2076 616c 7565 0a20 2020 2020 2020  lt value.       
│ │ │ │ +0001d8a0: 2046 756e 6374 696f 6e43 6c6f 7375 7265   FunctionClosure
│ │ │ │ +0001d8b0: 5b2e 2e2f 4661 7374 4d69 6e6f 7273 2e6d  [../FastMinors.m
│ │ │ │ +0001d8c0: 323a 3138 363a 3138 2d31 3836 3a34 355d  2:186:18-186:45]
│ │ │ │ +0001d8d0: 2c20 7573 6564 2074 6f20 636f 6e74 726f  , used to contro
│ │ │ │ +0001d8e0: 6c20 686f 770a 2020 2020 2020 2020 6d61  l how.        ma
│ │ │ │ +0001d8f0: 6e79 206d 696e 6f72 7320 6172 6520 636f  ny minors are co
│ │ │ │ +0001d900: 6d70 7574 6564 206f 6620 7468 6520 6d61  mputed of the ma
│ │ │ │ +0001d910: 7472 6963 6573 2069 6e20 6120 7072 6f6a  trices in a proj
│ │ │ │ +0001d920: 6563 7469 7665 2072 6573 6f6c 7574 696f  ective resolutio
│ │ │ │ +0001d930: 6e0a 2020 2020 2020 2a20 2a6e 6f74 6520  n.      * *note 
│ │ │ │ +0001d940: 4465 7453 7472 6174 6567 793a 2044 6574  DetStrategy: Det
│ │ │ │ +0001d950: 5374 7261 7465 6779 2c20 3d3e 202e 2e2e  Strategy, => ...
│ │ │ │ +0001d960: 2c20 6465 6661 756c 7420 7661 6c75 6520  , default value 
│ │ │ │ +0001d970: 436f 6661 6374 6f72 2c0a 2020 2020 2020  Cofactor,.      
│ │ │ │ +0001d980: 2020 4465 7453 7472 6174 6567 7920 6973    DetStrategy is
│ │ │ │ +0001d990: 2061 2073 7472 6174 6567 7920 666f 7220   a strategy for 
│ │ │ │ +0001d9a0: 616c 6c6f 7769 6e67 2074 6865 2075 7365  allowing the use
│ │ │ │ +0001d9b0: 7220 746f 2063 686f 6f73 6520 686f 770a  r to choose how.
│ │ │ │ +0001d9c0: 2020 2020 2020 2020 6465 7465 726d 696e          determin
│ │ │ │ +0001d9d0: 616e 7473 2028 6f72 2072 616e 6b29 2c20  ants (or rank), 
│ │ │ │ +0001d9e0: 6973 2063 6f6d 7075 7465 640a 2020 2020  is computed.    
│ │ │ │ +0001d9f0: 2020 2a20 2a6e 6f74 6520 5374 7261 7465    * *note Strate
│ │ │ │ +0001da00: 6779 3a20 5374 7261 7465 6779 4465 6661  gy: StrategyDefa
│ │ │ │ +0001da10: 756c 742c 203d 3e20 2e2e 2e2c 2064 6566  ult, => ..., def
│ │ │ │ +0001da20: 6175 6c74 2076 616c 7565 206e 6577 204f  ault value new O
│ │ │ │ +0001da30: 7074 696f 6e54 6162 6c65 0a20 2020 2020  ptionTable.     
│ │ │ │ +0001da40: 2020 2066 726f 6d20 7b50 6f69 6e74 7320     from {Points 
│ │ │ │ +0001da50: 3d3e 2030 2c20 5261 6e64 6f6d 203d 3e20  => 0, Random => 
│ │ │ │ +0001da60: 3136 2c20 4752 6576 4c65 784c 6172 6765  16, GRevLexLarge
│ │ │ │ +0001da70: 7374 203d 3e20 302c 204c 6578 536d 616c  st => 0, LexSmal
│ │ │ │ +0001da80: 6c65 7374 5465 726d 0a20 2020 2020 2020  lestTerm.       
│ │ │ │ +0001da90: 203d 3e20 3136 2c20 4c65 784c 6172 6765   => 16, LexLarge
│ │ │ │ +0001daa0: 7374 203d 3e20 302c 204c 6578 536d 616c  st => 0, LexSmal
│ │ │ │ +0001dab0: 6c65 7374 203d 3e20 3136 2c20 4752 6576  lest => 16, GRev
│ │ │ │ +0001dac0: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │ +0001dad0: 3d3e 2031 362c 0a20 2020 2020 2020 2052  => 16,.        R
│ │ │ │ +0001dae0: 616e 646f 6d4e 6f6e 7a65 726f 203d 3e20  andomNonzero => 
│ │ │ │ +0001daf0: 3136 2c20 4752 6576 4c65 7853 6d61 6c6c  16, GRevLexSmall
│ │ │ │ +0001db00: 6573 7420 3d3e 2031 367d 2c20 7374 7261  est => 16}, stra
│ │ │ │ +0001db10: 7465 6769 6573 2066 6f72 2063 686f 6f73  tegies for choos
│ │ │ │ +0001db20: 696e 670a 2020 2020 2020 2020 7375 626d  ing.        subm
│ │ │ │ +0001db30: 6174 7269 6365 730a 2020 2020 2020 2a20  atrices.      * 
│ │ │ │ +0001db40: 5665 7262 6f73 6520 3d3e 202e 2e2e 2c20  Verbose => ..., 
│ │ │ │ +0001db50: 6465 6661 756c 7420 7661 6c75 6520 6661  default value fa
│ │ │ │ +0001db60: 6c73 650a 2020 2a20 4f75 7470 7574 733a  lse.  * Outputs:
│ │ │ │ +0001db70: 0a20 2020 2020 202a 206e 2c20 616e 202a  .      * n, an *
│ │ │ │ +0001db80: 6e6f 7465 2069 6e74 6567 6572 3a20 284d  note integer: (M
│ │ │ │ +0001db90: 6163 6175 6c61 7932 446f 6329 5a5a 2c2c  acaulay2Doc)ZZ,,
│ │ │ │ +0001dba0: 2061 6e20 7570 7065 7220 626f 756e 6420   an upper bound 
│ │ │ │ +0001dbb0: 666f 7220 7468 650a 2020 2020 2020 2020  for the.        
│ │ │ │ +0001dbc0: 7072 6f6a 6563 7469 7665 2064 696d 656e  projective dimen
│ │ │ │ +0001dbd0: 7369 6f6e 206f 6620 4e0a 0a44 6573 6372  sion of N..Descr
│ │ │ │ +0001dbe0: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d  iption.=========
│ │ │ │ +0001dbf0: 3d3d 0a0a 5468 6520 6675 6e63 7469 6f6e  ==..The function
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│ │ │ │ +0001dc10: 6520 6c65 6e67 7468 206f 6620 6120 7072  e length of a pr
│ │ │ │ +0001dc20: 6f6a 6563 7469 7665 2072 6573 6f6c 7574  ojective resolut
│ │ │ │ +0001dc30: 696f 6e2e 2049 6620 7468 6520 6d6f 6475  ion. If the modu
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│ │ │ │ +0001dc50: 2068 6f6d 6f67 656e 656f 7573 2c20 7468   homogeneous, th
│ │ │ │ +0001dc60: 656e 2074 6865 2070 726f 6a65 6374 6976  en the projectiv
│ │ │ │ +0001dc70: 6520 7265 736f 6c75 7469 6f6e 206d 6179  e resolution may
│ │ │ │ +0001dc80: 206e 6f74 2062 6520 6d69 6e69 6d61 6c0a   not be minimal.
│ │ │ │ +0001dc90: 616e 6420 736f 2070 6469 6d20 6361 6e20  and so pdim can 
│ │ │ │ +0001dca0: 6769 7665 2074 6865 2077 726f 6e67 2061  give the wrong a
│ │ │ │ +0001dcb0: 6e73 7765 722e 2020 5468 6973 2066 756e  nswer.  This fun
│ │ │ │ +0001dcc0: 6374 696f 6e20 7072 6f6a 4469 6d20 7472  ction projDim tr
│ │ │ │ +0001dcd0: 6965 7320 746f 2069 6d70 726f 7665 0a74  ies to improve.t
│ │ │ │ +0001dce0: 6869 7320 626f 756e 6420 6279 2063 6f6e  his bound by con
│ │ │ │ +0001dcf0: 7369 6465 7269 6e67 2069 6465 616c 7320  sidering ideals 
│ │ │ │ +0001dd00: 6f66 2061 7070 726f 7072 6961 7465 6c79  of appropriately
│ │ │ │ +0001dd10: 2073 697a 6564 206d 696e 6f72 7320 6f66   sized minors of
│ │ │ │ +0001dd20: 2074 6865 0a72 6573 6f6c 7574 696f 6e20   the.resolution 
│ │ │ │ +0001dd30: 2873 7461 7274 696e 6720 6672 6f6d 2074  (starting from t
│ │ │ │ +0001dd40: 6865 2065 6e64 206f 6620 7468 6520 7265  he end of the re
│ │ │ │ +0001dd50: 736f 6c75 7469 6f6e 2061 6e64 2077 6f72  solution and wor
│ │ │ │ +0001dd60: 6b69 6e67 2062 6163 6b77 6172 6473 292e  king backwards).
│ │ │ │ +0001dd70: 0a55 7369 6e67 2074 6865 206f 7074 696f  .Using the optio
│ │ │ │ +0001dd80: 6e20 4d69 6e44 696d 656e 7369 6f6e 2028  n MinDimension (
│ │ │ │ +0001dd90: 6465 6661 756c 7420 7661 6c75 6520 3029  default value 0)
│ │ │ │ +0001dda0: 2067 6976 6573 2061 206c 6f77 6572 2062   gives a lower b
│ │ │ │ +0001ddb0: 6f75 6e64 206f 6e20 7468 650a 7072 6f6a  ound on the.proj
│ │ │ │ +0001ddc0: 6563 7469 7665 2064 696d 656e 7369 6f6e  ective dimension
│ │ │ │ +0001ddd0: 2c20 696e 6372 6561 7369 6e67 2069 7420  , increasing it 
│ │ │ │ +0001dde0: 6361 6e20 7468 7573 2069 6d70 726f 7665  can thus improve
│ │ │ │ +0001ddf0: 2074 6865 2073 7065 6564 206f 6620 636f   the speed of co
│ │ │ │ +0001de00: 6d70 7574 6174 696f 6e2e 0a0a 2b2d 2d2d  mputation...+---
│ │ │ │ +0001de10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001de20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001de30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001de40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001de50: 2d2d 2d2d 2d2b 0a7c 6931 203a 2052 203d  -----+.|i1 : R =
│ │ │ │ +0001de60: 2051 515b 782c 795d 3b20 2020 2020 2020   QQ[x,y];       
│ │ │ │ +0001de70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001de80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001de90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001dea0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0001deb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001dec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ded0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001dee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +0001def0: 203a 2049 203d 2069 6465 616c 2828 785e   : I = ideal((x^
│ │ │ │ +0001df00: 332b 7929 5e32 2c20 2878 5e32 2b79 5e32  3+y)^2, (x^2+y^2
│ │ │ │ +0001df10: 295e 322c 2028 782b 795e 3329 5e32 2c20  )^2, (x+y^3)^2, 
│ │ │ │ +0001df20: 2878 2a79 295e 3229 3b20 2020 2020 2020  (x*y)^2);       
│ │ │ │ +0001df30: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001df40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001df50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001df60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001df70: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -0001df80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001df90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001dfa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001dfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001dfc0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2070  -------+.|i3 : p
│ │ │ │ -0001dfd0: 6469 6d28 6d6f 6475 6c65 2049 2920 2020  dim(module I)   
│ │ │ │ -0001dfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e010: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0001e020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001df70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001df80: 207c 0a7c 6f32 203a 2049 6465 616c 206f   |.|o2 : Ideal o
│ │ │ │ +0001df90: 6620 5220 2020 2020 2020 2020 2020 2020  f R             
│ │ │ │ +0001dfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001dfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001dfc0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0001dfd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001dfe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001dff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e010: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2070  -------+.|i3 : p
│ │ │ │ +0001e020: 6469 6d28 6d6f 6475 6c65 2049 2920 2020  dim(module I)   
│ │ │ │  0001e030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e050: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0001e060: 6f33 203d 2032 2020 2020 2020 2020 2020  o3 = 2          
│ │ │ │ +0001e050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e060: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0001e070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e0a0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -0001e0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e0f0: 2d2d 2d2b 0a7c 6934 203a 2074 696d 6520  ---+.|i4 : time 
│ │ │ │ -0001e100: 7072 6f6a 4469 6d28 6d6f 6475 6c65 2049  projDim(module I
│ │ │ │ -0001e110: 2c20 5374 7261 7465 6779 3d3e 5374 7261  , Strategy=>Stra
│ │ │ │ -0001e120: 7465 6779 5261 6e64 6f6d 2920 2020 2020  tegyRandom)     
│ │ │ │ -0001e130: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001e140: 7c20 2d2d 2075 7365 6420 302e 3233 3735  | -- used 0.2375
│ │ │ │ -0001e150: 3739 7320 2863 7075 293b 2030 2e31 3432  79s (cpu); 0.142
│ │ │ │ -0001e160: 3935 3673 2028 7468 7265 6164 293b 2030  956s (thread); 0
│ │ │ │ -0001e170: 7320 2867 6329 2020 2020 2020 2020 2020  s (gc)          
│ │ │ │ -0001e180: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0001e190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e1d0: 2020 2020 7c0a 7c6f 3420 3d20 3120 2020      |.|o4 = 1   
│ │ │ │ +0001e0a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001e0b0: 6f33 203d 2032 2020 2020 2020 2020 2020  o3 = 2          
│ │ │ │ +0001e0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e0f0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0001e100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e140: 2d2d 2d2b 0a7c 6934 203a 2074 696d 6520  ---+.|i4 : time 
│ │ │ │ +0001e150: 7072 6f6a 4469 6d28 6d6f 6475 6c65 2049  projDim(module I
│ │ │ │ +0001e160: 2c20 5374 7261 7465 6779 3d3e 5374 7261  , Strategy=>Stra
│ │ │ │ +0001e170: 7465 6779 5261 6e64 6f6d 2920 2020 2020  tegyRandom)     
│ │ │ │ +0001e180: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001e190: 7c20 2d2d 2075 7365 6420 302e 3330 3934  | -- used 0.3094
│ │ │ │ +0001e1a0: 3173 2028 6370 7529 3b20 302e 3137 3034  1s (cpu); 0.1704
│ │ │ │ +0001e1b0: 3937 7320 2874 6872 6561 6429 3b20 3073  97s (thread); 0s
│ │ │ │ +0001e1c0: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │ +0001e1d0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0001e1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e210: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001e220: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -0001e230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e260: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520  ----------+.|i5 
│ │ │ │ -0001e270: 3a20 7469 6d65 2070 726f 6a44 696d 286d  : time projDim(m
│ │ │ │ -0001e280: 6f64 756c 6520 492c 2053 7472 6174 6567  odule I, Strateg
│ │ │ │ -0001e290: 793d 3e53 7472 6174 6567 7952 616e 646f  y=>StrategyRando
│ │ │ │ -0001e2a0: 6d2c 204d 696e 4469 6d65 6e73 696f 6e20  m, MinDimension 
│ │ │ │ -0001e2b0: 3d3e 2031 297c 0a7c 202d 2d20 7573 6564  => 1)|.| -- used
│ │ │ │ -0001e2c0: 2030 2e30 3130 3939 3436 7320 2863 7075   0.0109946s (cpu
│ │ │ │ -0001e2d0: 293b 2030 2e30 3132 3830 3634 7320 2874  ); 0.0128064s (t
│ │ │ │ -0001e2e0: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │ -0001e2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e300: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0001e310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e340: 2020 2020 2020 2020 2020 207c 0a7c 6f35             |.|o5
│ │ │ │ -0001e350: 203d 2031 2020 2020 2020 2020 2020 2020   = 1            
│ │ │ │ +0001e210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e220: 2020 2020 7c0a 7c6f 3420 3d20 3120 2020      |.|o4 = 1   
│ │ │ │ +0001e230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e260: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001e270: 0a2b 2d2d 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 2b0a 7c69 3520  ----------+.|i5 
│ │ │ │ +0001e2c0: 3a20 7469 6d65 2070 726f 6a44 696d 286d  : time projDim(m
│ │ │ │ +0001e2d0: 6f64 756c 6520 492c 2053 7472 6174 6567  odule I, Strateg
│ │ │ │ +0001e2e0: 793d 3e53 7472 6174 6567 7952 616e 646f  y=>StrategyRando
│ │ │ │ +0001e2f0: 6d2c 204d 696e 4469 6d65 6e73 696f 6e20  m, MinDimension 
│ │ │ │ +0001e300: 3d3e 2031 297c 0a7c 202d 2d20 7573 6564  => 1)|.| -- used
│ │ │ │ +0001e310: 2030 2e30 3134 3031 3231 7320 2863 7075   0.0140121s (cpu
│ │ │ │ +0001e320: 293b 2030 2e30 3135 3231 3835 7320 2874  ); 0.0152185s (t
│ │ │ │ +0001e330: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │ +0001e340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e350: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0001e360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e390: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ -0001e3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e3b0: 2d2d 2d2d 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: 2d2b 0a0a 5468 6520 6f70 7469 6f6e 204d  -+..The option M
│ │ │ │ -0001e3f0: 6178 4d69 6e6f 7273 2063 616e 2062 6520  axMinors can be 
│ │ │ │ -0001e400: 7573 6564 2074 6f20 636f 6e74 726f 6c20  used to control 
│ │ │ │ -0001e410: 686f 7720 6d61 6e79 206d 696e 6f72 7320  how many minors 
│ │ │ │ -0001e420: 6172 6520 636f 6d70 7574 6564 2061 740a  are computed at.
│ │ │ │ -0001e430: 6561 6368 2073 7465 702e 2049 6620 7468  each step. If th
│ │ │ │ -0001e440: 6973 2069 7320 6e6f 7420 7370 6563 6966  is is not specif
│ │ │ │ -0001e450: 6965 642c 2074 6865 206e 756d 6265 7220  ied, the number 
│ │ │ │ -0001e460: 6f66 206d 696e 6f72 7320 6973 2061 2066  of minors is a f
│ │ │ │ -0001e470: 756e 6374 696f 6e20 6f66 2074 6865 0a64  unction of the.d
│ │ │ │ -0001e480: 696d 656e 7369 6f6e 2024 6424 206f 6620  imension $d$ of 
│ │ │ │ -0001e490: 7468 6520 706f 6c79 6e6f 6d69 616c 2072  the polynomial r
│ │ │ │ -0001e4a0: 696e 6720 616e 6420 7468 6520 706f 7373  ing and the poss
│ │ │ │ -0001e4b0: 6962 6c65 206d 696e 6f72 7320 2463 242e  ible minors $c$.
│ │ │ │ -0001e4c0: 2053 7065 6369 6669 6361 6c6c 790a 6974   Specifically.it
│ │ │ │ -0001e4d0: 2069 7320 3130 202a 2064 202b 2032 202a   is 10 * d + 2 *
│ │ │ │ -0001e4e0: 206c 6f67 5f31 2e33 2863 292e 204f 7468   log_1.3(c). Oth
│ │ │ │ -0001e4f0: 6572 7769 7365 2074 6865 2075 7365 7220  erwise the user 
│ │ │ │ -0001e500: 6361 6e20 7365 7420 7468 6520 6f70 7469  can set the opti
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│ │ │ │ -0001e520: 5a5a 2074 6f20 7370 6563 6966 7920 7468  ZZ to specify th
│ │ │ │ -0001e530: 6174 2061 2066 6978 6564 2069 6e74 6567  at a fixed integ
│ │ │ │ -0001e540: 6572 2069 7320 7573 6564 2066 6f72 2065  er is used for e
│ │ │ │ -0001e550: 6163 6820 7374 6570 2e20 2041 6c74 6572  ach step.  Alter
│ │ │ │ -0001e560: 6e61 7469 7665 6c79 2c0a 7468 6520 7573  natively,.the us
│ │ │ │ -0001e570: 6572 2063 616e 2063 6f6e 7472 6f6c 2074  er can control t
│ │ │ │ -0001e580: 6865 206e 756d 6265 7220 6f66 206d 696e  he number of min
│ │ │ │ -0001e590: 6f72 7320 636f 6d70 7574 6564 2061 7420  ors computed at 
│ │ │ │ -0001e5a0: 6561 6368 2073 7465 7020 6279 2073 6574  each step by set
│ │ │ │ -0001e5b0: 7469 6e67 2074 6865 0a6f 7074 696f 6e20  ting the.option 
│ │ │ │ -0001e5c0: 4d61 784d 696e 6f72 7320 3d3e 204c 6973  MaxMinors => Lis
│ │ │ │ -0001e5d0: 742e 2020 496e 2074 6869 7320 6361 7365  t.  In this case
│ │ │ │ -0001e5e0: 2c20 7468 6520 6c69 7374 2073 7065 6369  , the list speci
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│ │ │ │ -0001e600: 6e6f 7273 2074 6f0a 6265 2063 6f6d 7075  nors to.be compu
│ │ │ │ -0001e610: 7465 6420 6174 2065 6163 6820 7374 6570  ted at each step
│ │ │ │ -0001e620: 2c20 2877 6f72 6b69 6e67 2062 6163 6b77  , (working backw
│ │ │ │ -0001e630: 6172 6473 292e 2046 696e 616c 6c79 2c20  ards). Finally, 
│ │ │ │ -0001e640: 796f 7520 6361 6e20 616c 736f 2073 6574  you can also set
│ │ │ │ -0001e650: 0a4d 6178 4d69 6e6f 7273 2074 6f20 6265  .MaxMinors to be
│ │ │ │ -0001e660: 2061 2063 7573 746f 6d20 6675 6e63 7469   a custom functi
│ │ │ │ -0001e670: 6f6e 206f 6620 7468 6520 6469 6d65 6e73  on of the dimens
│ │ │ │ -0001e680: 696f 6e20 2464 2420 6f66 2074 6865 2070  ion $d$ of the p
│ │ │ │ -0001e690: 6f6c 796e 6f6d 6961 6c20 7269 6e67 0a61  olynomial ring.a
│ │ │ │ -0001e6a0: 6e64 2074 6865 206d 6178 696d 756d 206e  nd the maximum n
│ │ │ │ -0001e6b0: 756d 6265 7220 6f66 206d 696e 6f72 732e  umber of minors.
│ │ │ │ -0001e6c0: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d  ..See also.=====
│ │ │ │ -0001e6d0: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2070  ===..  * *note p
│ │ │ │ -0001e6e0: 6469 6d3a 2028 4d61 6361 756c 6179 3244  dim: (Macaulay2D
│ │ │ │ -0001e6f0: 6f63 2970 6469 6d2c 202d 2d20 636f 6d70  oc)pdim, -- comp
│ │ │ │ -0001e700: 7574 6520 7468 6520 7072 6f6a 6563 7469  ute the projecti
│ │ │ │ -0001e710: 7665 2064 696d 656e 7369 6f6e 0a0a 5761  ve dimension..Wa
│ │ │ │ -0001e720: 7973 2074 6f20 7573 6520 7072 6f6a 4469  ys to use projDi
│ │ │ │ -0001e730: 6d3a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  m:.=============
│ │ │ │ -0001e740: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 7072  =======..  * "pr
│ │ │ │ -0001e750: 6f6a 4469 6d28 4d6f 6475 6c65 2922 0a0a  ojDim(Module)"..
│ │ │ │ -0001e760: 466f 7220 7468 6520 7072 6f67 7261 6d6d  For the programm
│ │ │ │ -0001e770: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  er.=============
│ │ │ │ -0001e780: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563  =====..The objec
│ │ │ │ -0001e790: 7420 2a6e 6f74 6520 7072 6f6a 4469 6d3a  t *note projDim:
│ │ │ │ -0001e7a0: 2070 726f 6a44 696d 2c20 6973 2061 202a   projDim, is a *
│ │ │ │ -0001e7b0: 6e6f 7465 206d 6574 686f 6420 6675 6e63  note method func
│ │ │ │ -0001e7c0: 7469 6f6e 2077 6974 6820 6f70 7469 6f6e  tion with option
│ │ │ │ -0001e7d0: 733a 0a28 4d61 6361 756c 6179 3244 6f63  s:.(Macaulay2Doc
│ │ │ │ -0001e7e0: 294d 6574 686f 6446 756e 6374 696f 6e57  )MethodFunctionW
│ │ │ │ -0001e7f0: 6974 684f 7074 696f 6e73 2c2e 0a0a 2d2d  ithOptions,...--
│ │ │ │ -0001e800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54  -------------..T
│ │ │ │ -0001e850: 6865 2073 6f75 7263 6520 6f66 2074 6869  he source of thi
│ │ │ │ -0001e860: 7320 646f 6375 6d65 6e74 2069 7320 696e  s document is in
│ │ │ │ -0001e870: 0a2f 6275 696c 642f 7265 7072 6f64 7563  ./build/reproduc
│ │ │ │ -0001e880: 6962 6c65 2d70 6174 682f 6d61 6361 756c  ible-path/macaul
│ │ │ │ -0001e890: 6179 322d 312e 3235 2e31 312b 6473 2f4d  ay2-1.25.11+ds/M
│ │ │ │ -0001e8a0: 322f 4d61 6361 756c 6179 322f 7061 636b  2/Macaulay2/pack
│ │ │ │ -0001e8b0: 6167 6573 2f46 6173 744d 696e 6f72 732e  ages/FastMinors.
│ │ │ │ -0001e8c0: 0a6d 323a 3230 3739 3a30 2e0a 1f0a 4669  .m2:2079:0....Fi
│ │ │ │ -0001e8d0: 6c65 3a20 4661 7374 4d69 6e6f 7273 2e69  le: FastMinors.i
│ │ │ │ -0001e8e0: 6e66 6f2c 204e 6f64 653a 2072 6563 7572  nfo, Node: recur
│ │ │ │ -0001e8f0: 7369 7665 4d69 6e6f 7273 2c20 4e65 7874  siveMinors, Next
│ │ │ │ -0001e900: 3a20 7265 6775 6c61 7249 6e43 6f64 696d  : regularInCodim
│ │ │ │ -0001e910: 656e 7369 6f6e 2c20 5072 6576 3a20 7072  ension, Prev: pr
│ │ │ │ -0001e920: 6f6a 4469 6d2c 2055 703a 2054 6f70 0a0a  ojDim, Up: Top..
│ │ │ │ -0001e930: 7265 6375 7273 6976 654d 696e 6f72 7320  recursiveMinors 
│ │ │ │ -0001e940: 2d2d 2075 7365 7320 6120 7265 6375 7273  -- uses a recurs
│ │ │ │ -0001e950: 6976 6520 636f 6661 6374 6f72 2061 6c67  ive cofactor alg
│ │ │ │ -0001e960: 6f72 6974 686d 2074 6f20 636f 6d70 7574  orithm to comput
│ │ │ │ -0001e970: 6520 7468 6520 6964 6561 6c20 6f66 206d  e the ideal of m
│ │ │ │ -0001e980: 696e 6f72 7320 6f66 2061 206d 6174 7269  inors of a matri
│ │ │ │ -0001e990: 780a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  x.**************
│ │ │ │ -0001e9a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001e9b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001e9c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001e9d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001e9e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001e9f0: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20  ***..  * Usage: 
│ │ │ │ -0001ea00: 0a20 2020 2020 2020 2049 203d 2072 6563  .        I = rec
│ │ │ │ -0001ea10: 7572 7369 7665 4d69 6e6f 7273 286e 2c20  ursiveMinors(n, 
│ │ │ │ -0001ea20: 4d2c 2054 6872 6561 6473 3d3e 742c 204d  M, Threads=>t, M
│ │ │ │ -0001ea30: 696e 6f72 7343 6163 6865 3d3e 6229 0a20  inorsCache=>b). 
│ │ │ │ -0001ea40: 202a 2049 6e70 7574 733a 0a20 2020 2020   * Inputs:.     
│ │ │ │ -0001ea50: 202a 206e 2c20 616e 202a 6e6f 7465 2069   * n, an *note i
│ │ │ │ -0001ea60: 6e74 6567 6572 3a20 284d 6163 6175 6c61  nteger: (Macaula
│ │ │ │ -0001ea70: 7932 446f 6329 5a5a 2c2c 2074 6865 2073  y2Doc)ZZ,, the s
│ │ │ │ -0001ea80: 697a 6520 6f66 206d 696e 6f72 7320 746f  ize of minors to
│ │ │ │ -0001ea90: 2063 6f6d 7075 7465 0a20 2020 2020 202a   compute.      *
│ │ │ │ -0001eaa0: 204d 2c20 6120 2a6e 6f74 6520 6d61 7472   M, a *note matr
│ │ │ │ -0001eab0: 6978 3a20 284d 6163 6175 6c61 7932 446f  ix: (Macaulay2Do
│ │ │ │ -0001eac0: 6329 4d61 7472 6978 2c2c 200a 2020 2020  c)Matrix,, .    
│ │ │ │ -0001ead0: 2020 2a20 742c 2061 6e20 2a6e 6f74 6520    * t, an *note 
│ │ │ │ -0001eae0: 696e 7465 6765 723a 2028 4d61 6361 756c  integer: (Macaul
│ │ │ │ -0001eaf0: 6179 3244 6f63 295a 5a2c 2c20 616e 206f  ay2Doc)ZZ,, an o
│ │ │ │ -0001eb00: 7074 696f 6e61 6c20 696e 7075 742c 2077  ptional input, w
│ │ │ │ -0001eb10: 6869 6368 0a20 2020 2020 2020 2064 6573  hich.        des
│ │ │ │ -0001eb20: 6372 6962 6573 2074 6865 206e 756d 6265  cribes the numbe
│ │ │ │ -0001eb30: 7220 6f66 2074 6872 6561 6473 2074 6f20  r of threads to 
│ │ │ │ -0001eb40: 7573 6573 0a20 2020 2020 202a 2062 2c20  uses.      * b, 
│ │ │ │ -0001eb50: 6120 2a6e 6f74 6520 426f 6f6c 6561 6e20  a *note Boolean 
│ │ │ │ -0001eb60: 7661 6c75 653a 2028 4d61 6361 756c 6179  value: (Macaulay
│ │ │ │ -0001eb70: 3244 6f63 2942 6f6f 6c65 616e 2c2c 2061  2Doc)Boolean,, a
│ │ │ │ -0001eb80: 6e20 6f70 7469 6f6e 616c 2069 6e70 7574  n optional input
│ │ │ │ -0001eb90: 2c0a 2020 2020 2020 2020 7768 6963 6820  ,.        which 
│ │ │ │ -0001eba0: 7361 7973 2077 6865 7468 6572 2074 6f20  says whether to 
│ │ │ │ -0001ebb0: 6361 6368 6520 696e 2069 6e70 7574 0a20  cache in input. 
│ │ │ │ -0001ebc0: 202a 202a 6e6f 7465 204f 7074 696f 6e61   * *note Optiona
│ │ │ │ -0001ebd0: 6c20 696e 7075 7473 3a20 284d 6163 6175  l inputs: (Macau
│ │ │ │ -0001ebe0: 6c61 7932 446f 6329 7573 696e 6720 6675  lay2Doc)using fu
│ │ │ │ -0001ebf0: 6e63 7469 6f6e 7320 7769 7468 206f 7074  nctions with opt
│ │ │ │ -0001ec00: 696f 6e61 6c20 696e 7075 7473 2c3a 0a20  ional inputs,:. 
│ │ │ │ -0001ec10: 2020 2020 202a 204d 696e 6f72 7343 6163       * MinorsCac
│ │ │ │ -0001ec20: 6865 203d 3e20 2e2e 2e2c 2064 6566 6175  he => ..., defau
│ │ │ │ -0001ec30: 6c74 2076 616c 7565 2074 7275 650a 2020  lt value true.  
│ │ │ │ -0001ec40: 2020 2020 2a20 2a6e 6f74 6520 5468 7265      * *note Thre
│ │ │ │ -0001ec50: 6164 733a 2069 7352 616e 6b41 744c 6561  ads: isRankAtLea
│ │ │ │ -0001ec60: 7374 5f6c 705f 7064 5f70 645f 7064 5f63  st_lp_pd_pd_pd_c
│ │ │ │ -0001ec70: 6d54 6872 6561 6473 3d3e 5f70 645f 7064  mThreads=>_pd_pd
│ │ │ │ -0001ec80: 5f70 645f 7270 2c20 3d3e 0a20 2020 2020  _pd_rp, =>.     
│ │ │ │ -0001ec90: 2020 202e 2e2e 2c20 6465 6661 756c 7420     ..., default 
│ │ │ │ -0001eca0: 7661 6c75 6520 302c 2061 6e20 6f70 7469  value 0, an opti
│ │ │ │ -0001ecb0: 6f6e 2066 6f72 2076 6172 696f 7573 2066  on for various f
│ │ │ │ -0001ecc0: 756e 6374 696f 6e73 0a20 2020 2020 202a  unctions.      *
│ │ │ │ -0001ecd0: 2056 6572 626f 7365 203d 3e20 2e2e 2e2c   Verbose => ...,
│ │ │ │ -0001ece0: 2064 6566 6175 6c74 2076 616c 7565 2066   default value f
│ │ │ │ -0001ecf0: 616c 7365 0a20 202a 204f 7574 7075 7473  alse.  * Outputs
│ │ │ │ -0001ed00: 3a0a 2020 2020 2020 2a20 492c 2061 6e20  :.      * I, an 
│ │ │ │ -0001ed10: 2a6e 6f74 6520 6964 6561 6c3a 2028 4d61  *note ideal: (Ma
│ │ │ │ -0001ed20: 6361 756c 6179 3244 6f63 2949 6465 616c  caulay2Doc)Ideal
│ │ │ │ -0001ed30: 2c2c 2074 6865 2069 6465 616c 206f 6620  ,, the ideal of 
│ │ │ │ -0001ed40: 6d69 6e6f 7273 206f 6620 4d0a 0a44 6573  minors of M..Des
│ │ │ │ -0001ed50: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ -0001ed60: 3d3d 3d3d 0a0a 4769 7665 6e20 6120 6d61  ====..Given a ma
│ │ │ │ -0001ed70: 7472 6978 2024 4d24 2c20 7468 6973 2063  trix $M$, this c
│ │ │ │ -0001ed80: 6f6d 7075 7465 7320 7468 6520 6964 6561  omputes the idea
│ │ │ │ -0001ed90: 6c20 6f66 2064 6574 6572 6d69 6e61 6e74  l of determinant
│ │ │ │ -0001eda0: 7320 6f66 2073 697a 6520 246e 205c 7469  s of size $n \ti
│ │ │ │ -0001edb0: 6d65 730a 6e24 2073 7562 6d61 7472 6963  mes.n$ submatric
│ │ │ │ -0001edc0: 6573 2e20 5468 6520 7265 6375 7273 6976  es. The recursiv
│ │ │ │ -0001edd0: 654d 696e 6f72 7320 6675 6e63 7469 6f6e  eMinors function
│ │ │ │ -0001ede0: 2075 7365 7320 6120 7265 6375 7273 6976   uses a recursiv
│ │ │ │ -0001edf0: 6520 7374 7261 7465 6779 2c20 6b65 6570  e strategy, keep
│ │ │ │ -0001ee00: 696e 670a 7472 6163 6b20 6f66 2074 6865  ing.track of the
│ │ │ │ -0001ee10: 2073 6d61 6c6c 6572 206d 696e 6f72 7320   smaller minors 
│ │ │ │ -0001ee20: 636f 6d70 7574 6564 2073 6f20 6661 722c  computed so far,
│ │ │ │ -0001ee30: 2075 6e6c 696b 6520 7468 6520 6275 696c   unlike the buil
│ │ │ │ -0001ee40: 742d 696e 2043 6f66 6163 746f 720a 7374  t-in Cofactor.st
│ │ │ │ -0001ee50: 7261 7465 6779 2066 6f72 206d 696e 6f72  rategy for minor
│ │ │ │ -0001ee60: 730a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  s..+------------
│ │ │ │ -0001ee70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ee80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ee90: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a  ---------+.|i1 :
│ │ │ │ -0001eea0: 2052 203d 2051 515b 782c 795d 3b20 2020   R = QQ[x,y];   
│ │ │ │ -0001eeb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001eec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001eed0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ -0001eee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001eef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ef00: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a  ---------+.|i2 :
│ │ │ │ -0001ef10: 204d 203d 2072 616e 646f 6d28 525e 7b35   M = random(R^{5
│ │ │ │ -0001ef20: 2c35 2c35 2c35 2c35 2c35 7d2c 2052 5e37  ,5,5,5,5,5}, R^7
│ │ │ │ -0001ef30: 293b 2020 2020 2020 2020 2020 2020 2020  );              
│ │ │ │ -0001ef40: 207c 0a7c 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 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0001ef80: 2020 2020 2020 2020 2036 2020 2020 2020           6      
│ │ │ │ -0001ef90: 3720 2020 2020 2020 2020 2020 2020 2020  7               
│ │ │ │ +0001e390: 2020 2020 2020 2020 2020 207c 0a7c 6f35             |.|o5
│ │ │ │ +0001e3a0: 203d 2031 2020 2020 2020 2020 2020 2020   = 1            
│ │ │ │ +0001e3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e3e0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0001e3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e430: 2d2b 0a0a 5468 6520 6f70 7469 6f6e 204d  -+..The option M
│ │ │ │ +0001e440: 6178 4d69 6e6f 7273 2063 616e 2062 6520  axMinors can be 
│ │ │ │ +0001e450: 7573 6564 2074 6f20 636f 6e74 726f 6c20  used to control 
│ │ │ │ +0001e460: 686f 7720 6d61 6e79 206d 696e 6f72 7320  how many minors 
│ │ │ │ +0001e470: 6172 6520 636f 6d70 7574 6564 2061 740a  are computed at.
│ │ │ │ +0001e480: 6561 6368 2073 7465 702e 2049 6620 7468  each step. If th
│ │ │ │ +0001e490: 6973 2069 7320 6e6f 7420 7370 6563 6966  is is not specif
│ │ │ │ +0001e4a0: 6965 642c 2074 6865 206e 756d 6265 7220  ied, the number 
│ │ │ │ +0001e4b0: 6f66 206d 696e 6f72 7320 6973 2061 2066  of minors is a f
│ │ │ │ +0001e4c0: 756e 6374 696f 6e20 6f66 2074 6865 0a64  unction of the.d
│ │ │ │ +0001e4d0: 696d 656e 7369 6f6e 2024 6424 206f 6620  imension $d$ of 
│ │ │ │ +0001e4e0: 7468 6520 706f 6c79 6e6f 6d69 616c 2072  the polynomial r
│ │ │ │ +0001e4f0: 696e 6720 616e 6420 7468 6520 706f 7373  ing and the poss
│ │ │ │ +0001e500: 6962 6c65 206d 696e 6f72 7320 2463 242e  ible minors $c$.
│ │ │ │ +0001e510: 2053 7065 6369 6669 6361 6c6c 790a 6974   Specifically.it
│ │ │ │ +0001e520: 2069 7320 3130 202a 2064 202b 2032 202a   is 10 * d + 2 *
│ │ │ │ +0001e530: 206c 6f67 5f31 2e33 2863 292e 204f 7468   log_1.3(c). Oth
│ │ │ │ +0001e540: 6572 7769 7365 2074 6865 2075 7365 7220  erwise the user 
│ │ │ │ +0001e550: 6361 6e20 7365 7420 7468 6520 6f70 7469  can set the opti
│ │ │ │ +0001e560: 6f6e 204d 6178 4d69 6e6f 7273 0a3d 3e20  on MaxMinors.=> 
│ │ │ │ +0001e570: 5a5a 2074 6f20 7370 6563 6966 7920 7468  ZZ to specify th
│ │ │ │ +0001e580: 6174 2061 2066 6978 6564 2069 6e74 6567  at a fixed integ
│ │ │ │ +0001e590: 6572 2069 7320 7573 6564 2066 6f72 2065  er is used for e
│ │ │ │ +0001e5a0: 6163 6820 7374 6570 2e20 2041 6c74 6572  ach step.  Alter
│ │ │ │ +0001e5b0: 6e61 7469 7665 6c79 2c0a 7468 6520 7573  natively,.the us
│ │ │ │ +0001e5c0: 6572 2063 616e 2063 6f6e 7472 6f6c 2074  er can control t
│ │ │ │ +0001e5d0: 6865 206e 756d 6265 7220 6f66 206d 696e  he number of min
│ │ │ │ +0001e5e0: 6f72 7320 636f 6d70 7574 6564 2061 7420  ors computed at 
│ │ │ │ +0001e5f0: 6561 6368 2073 7465 7020 6279 2073 6574  each step by set
│ │ │ │ +0001e600: 7469 6e67 2074 6865 0a6f 7074 696f 6e20  ting the.option 
│ │ │ │ +0001e610: 4d61 784d 696e 6f72 7320 3d3e 204c 6973  MaxMinors => Lis
│ │ │ │ +0001e620: 742e 2020 496e 2074 6869 7320 6361 7365  t.  In this case
│ │ │ │ +0001e630: 2c20 7468 6520 6c69 7374 2073 7065 6369  , the list speci
│ │ │ │ +0001e640: 6669 6573 2068 6f77 206d 616e 7920 6d69  fies how many mi
│ │ │ │ +0001e650: 6e6f 7273 2074 6f0a 6265 2063 6f6d 7075  nors to.be compu
│ │ │ │ +0001e660: 7465 6420 6174 2065 6163 6820 7374 6570  ted at each step
│ │ │ │ +0001e670: 2c20 2877 6f72 6b69 6e67 2062 6163 6b77  , (working backw
│ │ │ │ +0001e680: 6172 6473 292e 2046 696e 616c 6c79 2c20  ards). Finally, 
│ │ │ │ +0001e690: 796f 7520 6361 6e20 616c 736f 2073 6574  you can also set
│ │ │ │ +0001e6a0: 0a4d 6178 4d69 6e6f 7273 2074 6f20 6265  .MaxMinors to be
│ │ │ │ +0001e6b0: 2061 2063 7573 746f 6d20 6675 6e63 7469   a custom functi
│ │ │ │ +0001e6c0: 6f6e 206f 6620 7468 6520 6469 6d65 6e73  on of the dimens
│ │ │ │ +0001e6d0: 696f 6e20 2464 2420 6f66 2074 6865 2070  ion $d$ of the p
│ │ │ │ +0001e6e0: 6f6c 796e 6f6d 6961 6c20 7269 6e67 0a61  olynomial ring.a
│ │ │ │ +0001e6f0: 6e64 2074 6865 206d 6178 696d 756d 206e  nd the maximum n
│ │ │ │ +0001e700: 756d 6265 7220 6f66 206d 696e 6f72 732e  umber of minors.
│ │ │ │ +0001e710: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d  ..See also.=====
│ │ │ │ +0001e720: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2070  ===..  * *note p
│ │ │ │ +0001e730: 6469 6d3a 2028 4d61 6361 756c 6179 3244  dim: (Macaulay2D
│ │ │ │ +0001e740: 6f63 2970 6469 6d2c 202d 2d20 636f 6d70  oc)pdim, -- comp
│ │ │ │ +0001e750: 7574 6520 7468 6520 7072 6f6a 6563 7469  ute the projecti
│ │ │ │ +0001e760: 7665 2064 696d 656e 7369 6f6e 0a0a 5761  ve dimension..Wa
│ │ │ │ +0001e770: 7973 2074 6f20 7573 6520 7072 6f6a 4469  ys to use projDi
│ │ │ │ +0001e780: 6d3a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  m:.=============
│ │ │ │ +0001e790: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 7072  =======..  * "pr
│ │ │ │ +0001e7a0: 6f6a 4469 6d28 4d6f 6475 6c65 2922 0a0a  ojDim(Module)"..
│ │ │ │ +0001e7b0: 466f 7220 7468 6520 7072 6f67 7261 6d6d  For the programm
│ │ │ │ +0001e7c0: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  er.=============
│ │ │ │ +0001e7d0: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563  =====..The objec
│ │ │ │ +0001e7e0: 7420 2a6e 6f74 6520 7072 6f6a 4469 6d3a  t *note projDim:
│ │ │ │ +0001e7f0: 2070 726f 6a44 696d 2c20 6973 2061 202a   projDim, is a *
│ │ │ │ +0001e800: 6e6f 7465 206d 6574 686f 6420 6675 6e63  note method func
│ │ │ │ +0001e810: 7469 6f6e 2077 6974 6820 6f70 7469 6f6e  tion with option
│ │ │ │ +0001e820: 733a 0a28 4d61 6361 756c 6179 3244 6f63  s:.(Macaulay2Doc
│ │ │ │ +0001e830: 294d 6574 686f 6446 756e 6374 696f 6e57  )MethodFunctionW
│ │ │ │ +0001e840: 6974 684f 7074 696f 6e73 2c2e 0a0a 2d2d  ithOptions,...--
│ │ │ │ +0001e850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54  -------------..T
│ │ │ │ +0001e8a0: 6865 2073 6f75 7263 6520 6f66 2074 6869  he source of thi
│ │ │ │ +0001e8b0: 7320 646f 6375 6d65 6e74 2069 7320 696e  s document is in
│ │ │ │ +0001e8c0: 0a2f 6275 696c 642f 7265 7072 6f64 7563  ./build/reproduc
│ │ │ │ +0001e8d0: 6962 6c65 2d70 6174 682f 6d61 6361 756c  ible-path/macaul
│ │ │ │ +0001e8e0: 6179 322d 312e 3235 2e31 312b 6473 2f4d  ay2-1.25.11+ds/M
│ │ │ │ +0001e8f0: 322f 4d61 6361 756c 6179 322f 7061 636b  2/Macaulay2/pack
│ │ │ │ +0001e900: 6167 6573 2f46 6173 744d 696e 6f72 732e  ages/FastMinors.
│ │ │ │ +0001e910: 0a6d 323a 3230 3739 3a30 2e0a 1f0a 4669  .m2:2079:0....Fi
│ │ │ │ +0001e920: 6c65 3a20 4661 7374 4d69 6e6f 7273 2e69  le: FastMinors.i
│ │ │ │ +0001e930: 6e66 6f2c 204e 6f64 653a 2072 6563 7572  nfo, Node: recur
│ │ │ │ +0001e940: 7369 7665 4d69 6e6f 7273 2c20 4e65 7874  siveMinors, Next
│ │ │ │ +0001e950: 3a20 7265 6775 6c61 7249 6e43 6f64 696d  : regularInCodim
│ │ │ │ +0001e960: 656e 7369 6f6e 2c20 5072 6576 3a20 7072  ension, Prev: pr
│ │ │ │ +0001e970: 6f6a 4469 6d2c 2055 703a 2054 6f70 0a0a  ojDim, Up: Top..
│ │ │ │ +0001e980: 7265 6375 7273 6976 654d 696e 6f72 7320  recursiveMinors 
│ │ │ │ +0001e990: 2d2d 2075 7365 7320 6120 7265 6375 7273  -- uses a recurs
│ │ │ │ +0001e9a0: 6976 6520 636f 6661 6374 6f72 2061 6c67  ive cofactor alg
│ │ │ │ +0001e9b0: 6f72 6974 686d 2074 6f20 636f 6d70 7574  orithm to comput
│ │ │ │ +0001e9c0: 6520 7468 6520 6964 6561 6c20 6f66 206d  e the ideal of m
│ │ │ │ +0001e9d0: 696e 6f72 7320 6f66 2061 206d 6174 7269  inors of a matri
│ │ │ │ +0001e9e0: 780a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  x.**************
│ │ │ │ +0001e9f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0001ea00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0001ea10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0001ea20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0001ea30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0001ea40: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20  ***..  * Usage: 
│ │ │ │ +0001ea50: 0a20 2020 2020 2020 2049 203d 2072 6563  .        I = rec
│ │ │ │ +0001ea60: 7572 7369 7665 4d69 6e6f 7273 286e 2c20  ursiveMinors(n, 
│ │ │ │ +0001ea70: 4d2c 2054 6872 6561 6473 3d3e 742c 204d  M, Threads=>t, M
│ │ │ │ +0001ea80: 696e 6f72 7343 6163 6865 3d3e 6229 0a20  inorsCache=>b). 
│ │ │ │ +0001ea90: 202a 2049 6e70 7574 733a 0a20 2020 2020   * Inputs:.     
│ │ │ │ +0001eaa0: 202a 206e 2c20 616e 202a 6e6f 7465 2069   * n, an *note i
│ │ │ │ +0001eab0: 6e74 6567 6572 3a20 284d 6163 6175 6c61  nteger: (Macaula
│ │ │ │ +0001eac0: 7932 446f 6329 5a5a 2c2c 2074 6865 2073  y2Doc)ZZ,, the s
│ │ │ │ +0001ead0: 697a 6520 6f66 206d 696e 6f72 7320 746f  ize of minors to
│ │ │ │ +0001eae0: 2063 6f6d 7075 7465 0a20 2020 2020 202a   compute.      *
│ │ │ │ +0001eaf0: 204d 2c20 6120 2a6e 6f74 6520 6d61 7472   M, a *note matr
│ │ │ │ +0001eb00: 6978 3a20 284d 6163 6175 6c61 7932 446f  ix: (Macaulay2Do
│ │ │ │ +0001eb10: 6329 4d61 7472 6978 2c2c 200a 2020 2020  c)Matrix,, .    
│ │ │ │ +0001eb20: 2020 2a20 742c 2061 6e20 2a6e 6f74 6520    * t, an *note 
│ │ │ │ +0001eb30: 696e 7465 6765 723a 2028 4d61 6361 756c  integer: (Macaul
│ │ │ │ +0001eb40: 6179 3244 6f63 295a 5a2c 2c20 616e 206f  ay2Doc)ZZ,, an o
│ │ │ │ +0001eb50: 7074 696f 6e61 6c20 696e 7075 742c 2077  ptional input, w
│ │ │ │ +0001eb60: 6869 6368 0a20 2020 2020 2020 2064 6573  hich.        des
│ │ │ │ +0001eb70: 6372 6962 6573 2074 6865 206e 756d 6265  cribes the numbe
│ │ │ │ +0001eb80: 7220 6f66 2074 6872 6561 6473 2074 6f20  r of threads to 
│ │ │ │ +0001eb90: 7573 6573 0a20 2020 2020 202a 2062 2c20  uses.      * b, 
│ │ │ │ +0001eba0: 6120 2a6e 6f74 6520 426f 6f6c 6561 6e20  a *note Boolean 
│ │ │ │ +0001ebb0: 7661 6c75 653a 2028 4d61 6361 756c 6179  value: (Macaulay
│ │ │ │ +0001ebc0: 3244 6f63 2942 6f6f 6c65 616e 2c2c 2061  2Doc)Boolean,, a
│ │ │ │ +0001ebd0: 6e20 6f70 7469 6f6e 616c 2069 6e70 7574  n optional input
│ │ │ │ +0001ebe0: 2c0a 2020 2020 2020 2020 7768 6963 6820  ,.        which 
│ │ │ │ +0001ebf0: 7361 7973 2077 6865 7468 6572 2074 6f20  says whether to 
│ │ │ │ +0001ec00: 6361 6368 6520 696e 2069 6e70 7574 0a20  cache in input. 
│ │ │ │ +0001ec10: 202a 202a 6e6f 7465 204f 7074 696f 6e61   * *note Optiona
│ │ │ │ +0001ec20: 6c20 696e 7075 7473 3a20 284d 6163 6175  l inputs: (Macau
│ │ │ │ +0001ec30: 6c61 7932 446f 6329 7573 696e 6720 6675  lay2Doc)using fu
│ │ │ │ +0001ec40: 6e63 7469 6f6e 7320 7769 7468 206f 7074  nctions with opt
│ │ │ │ +0001ec50: 696f 6e61 6c20 696e 7075 7473 2c3a 0a20  ional inputs,:. 
│ │ │ │ +0001ec60: 2020 2020 202a 204d 696e 6f72 7343 6163       * MinorsCac
│ │ │ │ +0001ec70: 6865 203d 3e20 2e2e 2e2c 2064 6566 6175  he => ..., defau
│ │ │ │ +0001ec80: 6c74 2076 616c 7565 2074 7275 650a 2020  lt value true.  
│ │ │ │ +0001ec90: 2020 2020 2a20 2a6e 6f74 6520 5468 7265      * *note Thre
│ │ │ │ +0001eca0: 6164 733a 2069 7352 616e 6b41 744c 6561  ads: isRankAtLea
│ │ │ │ +0001ecb0: 7374 5f6c 705f 7064 5f70 645f 7064 5f63  st_lp_pd_pd_pd_c
│ │ │ │ +0001ecc0: 6d54 6872 6561 6473 3d3e 5f70 645f 7064  mThreads=>_pd_pd
│ │ │ │ +0001ecd0: 5f70 645f 7270 2c20 3d3e 0a20 2020 2020  _pd_rp, =>.     
│ │ │ │ +0001ece0: 2020 202e 2e2e 2c20 6465 6661 756c 7420     ..., default 
│ │ │ │ +0001ecf0: 7661 6c75 6520 302c 2061 6e20 6f70 7469  value 0, an opti
│ │ │ │ +0001ed00: 6f6e 2066 6f72 2076 6172 696f 7573 2066  on for various f
│ │ │ │ +0001ed10: 756e 6374 696f 6e73 0a20 2020 2020 202a  unctions.      *
│ │ │ │ +0001ed20: 2056 6572 626f 7365 203d 3e20 2e2e 2e2c   Verbose => ...,
│ │ │ │ +0001ed30: 2064 6566 6175 6c74 2076 616c 7565 2066   default value f
│ │ │ │ +0001ed40: 616c 7365 0a20 202a 204f 7574 7075 7473  alse.  * Outputs
│ │ │ │ +0001ed50: 3a0a 2020 2020 2020 2a20 492c 2061 6e20  :.      * I, an 
│ │ │ │ +0001ed60: 2a6e 6f74 6520 6964 6561 6c3a 2028 4d61  *note ideal: (Ma
│ │ │ │ +0001ed70: 6361 756c 6179 3244 6f63 2949 6465 616c  caulay2Doc)Ideal
│ │ │ │ +0001ed80: 2c2c 2074 6865 2069 6465 616c 206f 6620  ,, the ideal of 
│ │ │ │ +0001ed90: 6d69 6e6f 7273 206f 6620 4d0a 0a44 6573  minors of M..Des
│ │ │ │ +0001eda0: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ +0001edb0: 3d3d 3d3d 0a0a 4769 7665 6e20 6120 6d61  ====..Given a ma
│ │ │ │ +0001edc0: 7472 6978 2024 4d24 2c20 7468 6973 2063  trix $M$, this c
│ │ │ │ +0001edd0: 6f6d 7075 7465 7320 7468 6520 6964 6561  omputes the idea
│ │ │ │ +0001ede0: 6c20 6f66 2064 6574 6572 6d69 6e61 6e74  l of determinant
│ │ │ │ +0001edf0: 7320 6f66 2073 697a 6520 246e 205c 7469  s of size $n \ti
│ │ │ │ +0001ee00: 6d65 730a 6e24 2073 7562 6d61 7472 6963  mes.n$ submatric
│ │ │ │ +0001ee10: 6573 2e20 5468 6520 7265 6375 7273 6976  es. The recursiv
│ │ │ │ +0001ee20: 654d 696e 6f72 7320 6675 6e63 7469 6f6e  eMinors function
│ │ │ │ +0001ee30: 2075 7365 7320 6120 7265 6375 7273 6976   uses a recursiv
│ │ │ │ +0001ee40: 6520 7374 7261 7465 6779 2c20 6b65 6570  e strategy, keep
│ │ │ │ +0001ee50: 696e 670a 7472 6163 6b20 6f66 2074 6865  ing.track of the
│ │ │ │ +0001ee60: 2073 6d61 6c6c 6572 206d 696e 6f72 7320   smaller minors 
│ │ │ │ +0001ee70: 636f 6d70 7574 6564 2073 6f20 6661 722c  computed so far,
│ │ │ │ +0001ee80: 2075 6e6c 696b 6520 7468 6520 6275 696c   unlike the buil
│ │ │ │ +0001ee90: 742d 696e 2043 6f66 6163 746f 720a 7374  t-in Cofactor.st
│ │ │ │ +0001eea0: 7261 7465 6779 2066 6f72 206d 696e 6f72  rategy for minor
│ │ │ │ +0001eeb0: 730a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  s..+------------
│ │ │ │ +0001eec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001eed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001eee0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a  ---------+.|i1 :
│ │ │ │ +0001eef0: 2052 203d 2051 515b 782c 795d 3b20 2020   R = QQ[x,y];   
│ │ │ │ +0001ef00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ef10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ef20: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0001ef30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ef40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ef50: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a  ---------+.|i2 :
│ │ │ │ +0001ef60: 204d 203d 2072 616e 646f 6d28 525e 7b35   M = random(R^{5
│ │ │ │ +0001ef70: 2c35 2c35 2c35 2c35 2c35 7d2c 2052 5e37  ,5,5,5,5,5}, R^7
│ │ │ │ +0001ef80: 293b 2020 2020 2020 2020 2020 2020 2020  );              
│ │ │ │ +0001ef90: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001efa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001efb0: 207c 0a7c 6f32 203a 204d 6174 7269 7820   |.|o2 : Matrix 
│ │ │ │ -0001efc0: 5220 203c 2d2d 2052 2020 2020 2020 2020  R  <-- R        
│ │ │ │ -0001efd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001efe0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -0001eff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f020: 2d2b 0a7c 6933 203a 2074 696d 6520 4932  -+.|i3 : time I2
│ │ │ │ -0001f030: 203d 2072 6563 7572 7369 7665 4d69 6e6f   = recursiveMino
│ │ │ │ -0001f040: 7273 2834 2c20 4d2c 2054 6872 6561 6473  rs(4, M, Threads
│ │ │ │ -0001f050: 3d3e 3029 3b20 2020 207c 0a7c 202d 2d20  =>0);    |.| -- 
│ │ │ │ -0001f060: 7573 6564 2030 2e34 3933 3631 3373 2028  used 0.493613s (
│ │ │ │ -0001f070: 6370 7529 3b20 302e 3434 3934 3333 7320  cpu); 0.449433s 
│ │ │ │ -0001f080: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │ -0001f090: 297c 0a7c 2020 2020 2020 2020 2020 2020  )|.|            
│ │ │ │ -0001f0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f0c0: 2020 2020 2020 2020 207c 0a7c 6f33 203a           |.|o3 :
│ │ │ │ -0001f0d0: 2049 6465 616c 206f 6620 5220 2020 2020   Ideal of R     
│ │ │ │ -0001f0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001efb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001efc0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001efd0: 2020 2020 2020 2020 2036 2020 2020 2020           6      
│ │ │ │ +0001efe0: 3720 2020 2020 2020 2020 2020 2020 2020  7               
│ │ │ │ +0001eff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f000: 207c 0a7c 6f32 203a 204d 6174 7269 7820   |.|o2 : Matrix 
│ │ │ │ +0001f010: 5220 203c 2d2d 2052 2020 2020 2020 2020  R  <-- R        
│ │ │ │ +0001f020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f030: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0001f040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f070: 2d2b 0a7c 6933 203a 2074 696d 6520 4932  -+.|i3 : time I2
│ │ │ │ +0001f080: 203d 2072 6563 7572 7369 7665 4d69 6e6f   = recursiveMino
│ │ │ │ +0001f090: 7273 2834 2c20 4d2c 2054 6872 6561 6473  rs(4, M, Threads
│ │ │ │ +0001f0a0: 3d3e 3029 3b20 2020 207c 0a7c 202d 2d20  =>0);    |.| -- 
│ │ │ │ +0001f0b0: 7573 6564 2030 2e35 3730 3636 3273 2028  used 0.570662s (
│ │ │ │ +0001f0c0: 6370 7529 3b20 302e 3531 3330 3037 7320  cpu); 0.513007s 
│ │ │ │ +0001f0d0: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │ +0001f0e0: 297c 0a7c 2020 2020 2020 2020 2020 2020  )|.|            
│ │ │ │  0001f0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f100: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ -0001f110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f130: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a  ---------+.|i4 :
│ │ │ │ -0001f140: 2074 696d 6520 4931 203d 206d 696e 6f72   time I1 = minor
│ │ │ │ -0001f150: 7328 342c 204d 2c20 5374 7261 7465 6779  s(4, M, Strategy
│ │ │ │ -0001f160: 3d3e 436f 6661 6374 6f72 293b 2020 2020  =>Cofactor);    
│ │ │ │ -0001f170: 207c 0a7c 202d 2d20 7573 6564 2031 2e35   |.| -- used 1.5
│ │ │ │ -0001f180: 3135 3233 7320 2863 7075 293b 2031 2e33  1523s (cpu); 1.3
│ │ │ │ -0001f190: 3130 3336 7320 2874 6872 6561 6429 3b20  1036s (thread); 
│ │ │ │ -0001f1a0: 3073 2028 6763 2920 207c 0a7c 2020 2020  0s (gc)  |.|    
│ │ │ │ -0001f1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f1e0: 207c 0a7c 6f34 203a 2049 6465 616c 206f   |.|o4 : Ideal o
│ │ │ │ -0001f1f0: 6620 5220 2020 2020 2020 2020 2020 2020  f R             
│ │ │ │ +0001f100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f110: 2020 2020 2020 2020 207c 0a7c 6f33 203a           |.|o3 :
│ │ │ │ +0001f120: 2049 6465 616c 206f 6620 5220 2020 2020   Ideal of R     
│ │ │ │ +0001f130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f150: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0001f160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f180: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a  ---------+.|i4 :
│ │ │ │ +0001f190: 2074 696d 6520 4931 203d 206d 696e 6f72   time I1 = minor
│ │ │ │ +0001f1a0: 7328 342c 204d 2c20 5374 7261 7465 6779  s(4, M, Strategy
│ │ │ │ +0001f1b0: 3d3e 436f 6661 6374 6f72 293b 2020 2020  =>Cofactor);    
│ │ │ │ +0001f1c0: 207c 0a7c 202d 2d20 7573 6564 2031 2e35   |.| -- used 1.5
│ │ │ │ +0001f1d0: 3339 3735 7320 2863 7075 293b 2031 2e33  3975s (cpu); 1.3
│ │ │ │ +0001f1e0: 3937 3973 2028 7468 7265 6164 293b 2030  979s (thread); 0
│ │ │ │ +0001f1f0: 7320 2867 6329 2020 207c 0a7c 2020 2020  s (gc)   |.|    
│ │ │ │  0001f200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f210: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -0001f220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f250: 2d2b 0a7c 6935 203a 2049 3120 3d3d 2049  -+.|i5 : I1 == I
│ │ │ │ -0001f260: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -0001f270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f280: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0001f290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f2c0: 207c 0a7c 6f35 203d 2074 7275 6520 2020   |.|o5 = true   
│ │ │ │ -0001f2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f230: 207c 0a7c 6f34 203a 2049 6465 616c 206f   |.|o4 : Ideal o
│ │ │ │ +0001f240: 6620 5220 2020 2020 2020 2020 2020 2020  f R             
│ │ │ │ +0001f250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f260: 2020 2020 2020 2020 207c 0a2b 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: 2d2b 0a7c 6935 203a 2049 3120 3d3d 2049  -+.|i5 : I1 == I
│ │ │ │ +0001f2b0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +0001f2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f2d0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0001f2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f2f0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -0001f300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f330: 2d2b 0a0a 5365 6520 616c 736f 0a3d 3d3d  -+..See also.===
│ │ │ │ -0001f340: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
│ │ │ │ -0001f350: 206d 696e 6f72 733a 2028 4d61 6361 756c   minors: (Macaul
│ │ │ │ -0001f360: 6179 3244 6f63 296d 696e 6f72 735f 6c70  ay2Doc)minors_lp
│ │ │ │ -0001f370: 5a5a 5f63 6d4d 6174 7269 785f 7270 2c20  ZZ_cmMatrix_rp, 
│ │ │ │ -0001f380: 2d2d 2069 6465 616c 2067 656e 6572 6174  -- ideal generat
│ │ │ │ -0001f390: 6564 2062 790a 2020 2020 6d69 6e6f 7273  ed by.    minors
│ │ │ │ -0001f3a0: 0a0a 5761 7973 2074 6f20 7573 6520 7265  ..Ways to use re
│ │ │ │ -0001f3b0: 6375 7273 6976 654d 696e 6f72 733a 0a3d  cursiveMinors:.=
│ │ │ │ -0001f3c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001f3d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ -0001f3e0: 2022 7265 6375 7273 6976 654d 696e 6f72   "recursiveMinor
│ │ │ │ -0001f3f0: 7328 5a5a 2c4d 6174 7269 7829 220a 0a46  s(ZZ,Matrix)"..F
│ │ │ │ -0001f400: 6f72 2074 6865 2070 726f 6772 616d 6d65  or the programme
│ │ │ │ -0001f410: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  r.==============
│ │ │ │ -0001f420: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374  ====..The object
│ │ │ │ -0001f430: 202a 6e6f 7465 2072 6563 7572 7369 7665   *note recursive
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│ │ │ │ -0001f650: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001f660: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001f670: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001f680: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ -0001f9e0: 2020 2a20 4d69 6e4d 696e 6f72 7346 756e    * MinMinorsFun
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│ │ │ │ -0001fac0: 4d69 6e6f 7273 3a20 4d61 784d 696e 6f72  Minors: MaxMinor
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│ │ │ │ -0001fbd0: 2020 2020 2020 6465 6661 756c 7420 7661        default va
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│ │ │ │ -0001fc40: 696e 2062 6574 7765 656e 2063 616c 6c73  in between calls
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│ │ │ │ -0001fc70: 6567 793a 2044 6574 5374 7261 7465 6779  egy: DetStrategy
│ │ │ │ -0001fc80: 2c20 3d3e 202e 2e2e 2c20 6465 6661 756c  , => ..., defaul
│ │ │ │ -0001fc90: 7420 7661 6c75 6520 436f 6661 6374 6f72  t value Cofactor
│ │ │ │ -0001fca0: 2c0a 2020 2020 2020 2020 4465 7453 7472  ,.        DetStr
│ │ │ │ -0001fcb0: 6174 6567 7920 6973 2061 2073 7472 6174  ategy is a strat
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│ │ │ │ -0001fcf0: 6465 7465 726d 696e 616e 7473 2028 6f72  determinants (or
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│ │ │ │ -0001fda0: 2045 7874 656e 6446 6965 6c64 203d 3e0a   ExtendField =>.
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│ │ │ │ -0001fdc0: 696e 7443 6865 636b 4174 7465 6d70 7473  intCheckAttempts
│ │ │ │ -0001fdd0: 203d 3e20 302c 2044 6563 6f6d 706f 7369   => 0, Decomposi
│ │ │ │ -0001fde0: 7469 6f6e 5374 7261 7465 6779 203d 3e20  tionStrategy => 
│ │ │ │ -0001fdf0: 4465 636f 6d70 6f73 652c 0a20 2020 2020  Decompose,.     
│ │ │ │ -0001fe00: 2020 204e 756d 5468 7265 6164 7354 6f55     NumThreadsToU
│ │ │ │ -0001fe10: 7365 203d 3e20 312c 2044 696d 656e 7369  se => 1, Dimensi
│ │ │ │ -0001fe20: 6f6e 4675 6e63 7469 6f6e 203d 3e20 6469  onFunction => di
│ │ │ │ -0001fe30: 6d2c 2056 6572 626f 7365 203d 3e20 6661  m, Verbose => fa
│ │ │ │ -0001fe40: 6c73 657d 2c0a 2020 2020 2020 2020 6f70  lse},.        op
│ │ │ │ -0001fe50: 7469 6f6e 7320 746f 2070 6173 7320 746f  tions to pass to
│ │ │ │ -0001fe60: 2066 756e 6374 696f 6e73 2069 6e20 7468   functions in th
│ │ │ │ -0001fe70: 6520 7061 636b 6167 6520 5261 6e64 6f6d  e package Random
│ │ │ │ -0001fe80: 506f 696e 7473 0a20 2020 2020 202a 202a  Points.      * *
│ │ │ │ -0001fe90: 6e6f 7465 2053 7472 6174 6567 793a 2053  note Strategy: S
│ │ │ │ -0001fea0: 7472 6174 6567 7944 6566 6175 6c74 2c20  trategyDefault, 
│ │ │ │ -0001feb0: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420  => ..., default 
│ │ │ │ -0001fec0: 7661 6c75 6520 6e65 7720 4f70 7469 6f6e  value new Option
│ │ │ │ -0001fed0: 5461 626c 650a 2020 2020 2020 2020 6672  Table.        fr
│ │ │ │ -0001fee0: 6f6d 207b 506f 696e 7473 203d 3e20 302c  om {Points => 0,
│ │ │ │ -0001fef0: 2052 616e 646f 6d20 3d3e 2031 362c 2047   Random => 16, G
│ │ │ │ -0001ff00: 5265 764c 6578 4c61 7267 6573 7420 3d3e  RevLexLargest =>
│ │ │ │ -0001ff10: 2030 2c20 4c65 7853 6d61 6c6c 6573 7454   0, LexSmallestT
│ │ │ │ -0001ff20: 6572 6d0a 2020 2020 2020 2020 3d3e 2031  erm.        => 1
│ │ │ │ -0001ff30: 362c 204c 6578 4c61 7267 6573 7420 3d3e  6, LexLargest =>
│ │ │ │ -0001ff40: 2030 2c20 4c65 7853 6d61 6c6c 6573 7420   0, LexSmallest 
│ │ │ │ -0001ff50: 3d3e 2031 362c 2047 5265 764c 6578 536d  => 16, GRevLexSm
│ │ │ │ -0001ff60: 616c 6c65 7374 5465 726d 203d 3e20 3136  allestTerm => 16
│ │ │ │ -0001ff70: 2c0a 2020 2020 2020 2020 5261 6e64 6f6d  ,.        Random
│ │ │ │ -0001ff80: 4e6f 6e7a 6572 6f20 3d3e 2031 362c 2047  Nonzero => 16, G
│ │ │ │ -0001ff90: 5265 764c 6578 536d 616c 6c65 7374 203d  RevLexSmallest =
│ │ │ │ -0001ffa0: 3e20 3136 7d2c 2073 7472 6174 6567 6965  > 16}, strategie
│ │ │ │ -0001ffb0: 7320 666f 7220 6368 6f6f 7369 6e67 0a20  s for choosing. 
│ │ │ │ -0001ffc0: 2020 2020 2020 2073 7562 6d61 7472 6963         submatric
│ │ │ │ -0001ffd0: 6573 0a20 2020 2020 202a 2056 6572 626f  es.      * Verbo
│ │ │ │ -0001ffe0: 7365 203d 3e20 2e2e 2e2c 2064 6566 6175  se => ..., defau
│ │ │ │ -0001fff0: 6c74 2076 616c 7565 2066 616c 7365 0a20  lt value false. 
│ │ │ │ -00020000: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ -00020010: 2020 2a20 7472 7565 2c20 6966 2074 6865    * true, if the
│ │ │ │ -00020020: 2072 696e 6720 6973 2072 6567 756c 6172   ring is regular
│ │ │ │ -00020030: 2069 6e20 636f 6469 6d65 6e73 696f 6e20   in codimension 
│ │ │ │ -00020040: 6e2c 2066 616c 7365 2069 6620 6974 2064  n, false if it d
│ │ │ │ -00020050: 6574 6572 6d69 6e65 730a 2020 2020 2020  etermines.      
│ │ │ │ -00020060: 2020 6974 2069 7320 6e6f 742c 2061 6e64    it is not, and
│ │ │ │ -00020070: 206e 756c 6c20 6966 206e 6f20 6465 7465   null if no dete
│ │ │ │ -00020080: 726d 696e 6174 696f 6e20 6973 206d 6164  rmination is mad
│ │ │ │ -00020090: 650a 0a44 6573 6372 6970 7469 6f6e 0a3d  e..Description.=
│ │ │ │ -000200a0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6973  ==========..This
│ │ │ │ -000200b0: 2066 756e 6374 696f 6e20 7265 7475 726e   function return
│ │ │ │ -000200c0: 7320 7472 7565 2069 6620 5220 6973 2072  s true if R is r
│ │ │ │ -000200d0: 6567 756c 6172 2069 6e20 636f 6469 6d65  egular in codime
│ │ │ │ -000200e0: 6e73 696f 6e20 6e2c 2066 616c 7365 2069  nsion n, false i
│ │ │ │ -000200f0: 6620 6974 2069 730a 6e6f 742c 2061 6e64  f it is.not, and
│ │ │ │ -00020100: 206e 756c 6c20 6966 2069 7420 6469 6420   null if it did 
│ │ │ │ -00020110: 6e6f 7420 6d61 6b65 2061 2064 6574 6572  not make a deter
│ │ │ │ -00020120: 6d69 6e61 7469 6f6e 2e20 4974 2063 6f6e  mination. It con
│ │ │ │ -00020130: 7369 6465 7273 2069 6e74 6572 6573 7469  siders interesti
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│ │ │ │ +0001fe80: 6d2c 2056 6572 626f 7365 203d 3e20 6661  m, Verbose => fa
│ │ │ │ +0001fe90: 6c73 657d 2c0a 2020 2020 2020 2020 6f70  lse},.        op
│ │ │ │ +0001fea0: 7469 6f6e 7320 746f 2070 6173 7320 746f  tions to pass to
│ │ │ │ +0001feb0: 2066 756e 6374 696f 6e73 2069 6e20 7468   functions in th
│ │ │ │ +0001fec0: 6520 7061 636b 6167 6520 5261 6e64 6f6d  e package Random
│ │ │ │ +0001fed0: 506f 696e 7473 0a20 2020 2020 202a 202a  Points.      * *
│ │ │ │ +0001fee0: 6e6f 7465 2053 7472 6174 6567 793a 2053  note Strategy: S
│ │ │ │ +0001fef0: 7472 6174 6567 7944 6566 6175 6c74 2c20  trategyDefault, 
│ │ │ │ +0001ff00: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420  => ..., default 
│ │ │ │ +0001ff10: 7661 6c75 6520 6e65 7720 4f70 7469 6f6e  value new Option
│ │ │ │ +0001ff20: 5461 626c 650a 2020 2020 2020 2020 6672  Table.        fr
│ │ │ │ +0001ff30: 6f6d 207b 506f 696e 7473 203d 3e20 302c  om {Points => 0,
│ │ │ │ +0001ff40: 2052 616e 646f 6d20 3d3e 2031 362c 2047   Random => 16, G
│ │ │ │ +0001ff50: 5265 764c 6578 4c61 7267 6573 7420 3d3e  RevLexLargest =>
│ │ │ │ +0001ff60: 2030 2c20 4c65 7853 6d61 6c6c 6573 7454   0, LexSmallestT
│ │ │ │ +0001ff70: 6572 6d0a 2020 2020 2020 2020 3d3e 2031  erm.        => 1
│ │ │ │ +0001ff80: 362c 204c 6578 4c61 7267 6573 7420 3d3e  6, LexLargest =>
│ │ │ │ +0001ff90: 2030 2c20 4c65 7853 6d61 6c6c 6573 7420   0, LexSmallest 
│ │ │ │ +0001ffa0: 3d3e 2031 362c 2047 5265 764c 6578 536d  => 16, GRevLexSm
│ │ │ │ +0001ffb0: 616c 6c65 7374 5465 726d 203d 3e20 3136  allestTerm => 16
│ │ │ │ +0001ffc0: 2c0a 2020 2020 2020 2020 5261 6e64 6f6d  ,.        Random
│ │ │ │ +0001ffd0: 4e6f 6e7a 6572 6f20 3d3e 2031 362c 2047  Nonzero => 16, G
│ │ │ │ +0001ffe0: 5265 764c 6578 536d 616c 6c65 7374 203d  RevLexSmallest =
│ │ │ │ +0001fff0: 3e20 3136 7d2c 2073 7472 6174 6567 6965  > 16}, strategie
│ │ │ │ +00020000: 7320 666f 7220 6368 6f6f 7369 6e67 0a20  s for choosing. 
│ │ │ │ +00020010: 2020 2020 2020 2073 7562 6d61 7472 6963         submatric
│ │ │ │ +00020020: 6573 0a20 2020 2020 202a 2056 6572 626f  es.      * Verbo
│ │ │ │ +00020030: 7365 203d 3e20 2e2e 2e2c 2064 6566 6175  se => ..., defau
│ │ │ │ +00020040: 6c74 2076 616c 7565 2066 616c 7365 0a20  lt value false. 
│ │ │ │ +00020050: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ +00020060: 2020 2a20 7472 7565 2c20 6966 2074 6865    * true, if the
│ │ │ │ +00020070: 2072 696e 6720 6973 2072 6567 756c 6172   ring is regular
│ │ │ │ +00020080: 2069 6e20 636f 6469 6d65 6e73 696f 6e20   in codimension 
│ │ │ │ +00020090: 6e2c 2066 616c 7365 2069 6620 6974 2064  n, false if it d
│ │ │ │ +000200a0: 6574 6572 6d69 6e65 730a 2020 2020 2020  etermines.      
│ │ │ │ +000200b0: 2020 6974 2069 7320 6e6f 742c 2061 6e64    it is not, and
│ │ │ │ +000200c0: 206e 756c 6c20 6966 206e 6f20 6465 7465   null if no dete
│ │ │ │ +000200d0: 726d 696e 6174 696f 6e20 6973 206d 6164  rmination is mad
│ │ │ │ +000200e0: 650a 0a44 6573 6372 6970 7469 6f6e 0a3d  e..Description.=
│ │ │ │ +000200f0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6973  ==========..This
│ │ │ │ +00020100: 2066 756e 6374 696f 6e20 7265 7475 726e   function return
│ │ │ │ +00020110: 7320 7472 7565 2069 6620 5220 6973 2072  s true if R is r
│ │ │ │ +00020120: 6567 756c 6172 2069 6e20 636f 6469 6d65  egular in codime
│ │ │ │ +00020130: 6e73 696f 6e20 6e2c 2066 616c 7365 2069  nsion n, false i
│ │ │ │ +00020140: 6620 6974 2069 730a 6e6f 742c 2061 6e64  f it is.not, and
│ │ │ │ +00020150: 206e 756c 6c20 6966 2069 7420 6469 6420   null if it did 
│ │ │ │ +00020160: 6e6f 7420 6d61 6b65 2061 2064 6574 6572  not make a deter
│ │ │ │ +00020170: 6d69 6e61 7469 6f6e 2e20 4974 2063 6f6e  mination. It con
│ │ │ │ +00020180: 7369 6465 7273 2069 6e74 6572 6573 7469  siders interesti
│ │ │ │ +00020190: 6e67 0a6d 696e 6f72 7320 6f66 2074 6865  ng.minors of the
│ │ │ │ +000201a0: 206a 6163 6f62 6961 6e20 6d61 7472 6978   jacobian matrix
│ │ │ │ +000201b0: 2074 6f20 7472 7920 746f 2076 6572 6966   to try to verif
│ │ │ │ +000201c0: 7920 7468 6174 2074 6865 2072 696e 6720  y that the ring 
│ │ │ │ +000201d0: 6973 2072 6567 756c 6172 2069 6e0a 636f  is regular in.co
│ │ │ │ +000201e0: 6469 6d65 6e73 696f 6e20 6e2e 2049 7420  dimension n. It 
│ │ │ │ +000201f0: 6973 2066 7265 7175 656e 746c 7920 6d75  is frequently mu
│ │ │ │ +00020200: 6368 2066 6173 7465 7220 6174 2067 6976  ch faster at giv
│ │ │ │ +00020210: 696e 6720 616e 2061 6666 6972 6d61 7469  ing an affirmati
│ │ │ │ +00020220: 7665 2061 6e73 7765 720a 7468 616e 2063  ve answer.than c
│ │ │ │ +00020230: 6f6d 7075 7469 6e67 2074 6865 2064 696d  omputing the dim
│ │ │ │ +00020240: 656e 7369 6f6e 206f 6620 7468 6520 6964  ension of the id
│ │ │ │ +00020250: 6561 6c20 6f66 2061 6c6c 206d 696e 6f72  eal of all minor
│ │ │ │ +00020260: 7320 6f66 2074 6865 204a 6163 6f62 6961  s of the Jacobia
│ │ │ │ +00020270: 6e2e 2057 650a 6265 6769 6e20 7769 7468  n. We.begin with
│ │ │ │ +00020280: 2061 2073 696d 706c 6520 6578 616d 706c   a simple exampl
│ │ │ │ +00020290: 6520 7768 6963 6820 6973 2052 312c 2062  e which is R1, b
│ │ │ │ +000202a0: 7574 206e 6f74 2052 322e 0a0a 2b2d 2d2d  ut not R2...+---
│ │ │ │  000202b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000202c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000202d0: 0a7c 6932 203a 2072 6567 756c 6172 496e  .|i2 : regularIn
│ │ │ │ -000202e0: 436f 6469 6d65 6e73 696f 6e28 312c 2052  Codimension(1, R
│ │ │ │ -000202f0: 2920 2020 2020 7c0a 7c20 2020 2020 2020  )     |.|       
│ │ │ │ -00020300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020310: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00020320: 6f32 203d 2074 7275 6520 2020 2020 2020  o2 = true       
│ │ │ │ -00020330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020340: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -00020350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933  -----------+.|i3
│ │ │ │ -00020370: 203a 2072 6567 756c 6172 496e 436f 6469   : regularInCodi
│ │ │ │ -00020380: 6d65 6e73 696f 6e28 322c 2052 2920 2020  mension(2, R)   
│ │ │ │ -00020390: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +000202c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000202d0: 2d2b 0a7c 6931 203a 2052 203d 2051 515b  -+.|i1 : R = QQ[
│ │ │ │ +000202e0: 782c 2079 2c20 7a5d 2f69 6465 616c 2878  x, y, z]/ideal(x
│ │ │ │ +000202f0: 2a79 2d7a 5e32 293b 7c0a 2b2d 2d2d 2d2d  *y-z^2);|.+-----
│ │ │ │ +00020300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00020320: 0a7c 6932 203a 2072 6567 756c 6172 496e  .|i2 : regularIn
│ │ │ │ +00020330: 436f 6469 6d65 6e73 696f 6e28 312c 2052  Codimension(1, R
│ │ │ │ +00020340: 2920 2020 2020 7c0a 7c20 2020 2020 2020  )     |.|       
│ │ │ │ +00020350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020360: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00020370: 6f32 203d 2074 7275 6520 2020 2020 2020  o2 = true       
│ │ │ │ +00020380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020390: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  000203a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000203b0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4e65 7874  ---------+..Next
│ │ │ │ -000203c0: 2077 6520 636f 6e73 6964 6572 2061 206d   we consider a m
│ │ │ │ -000203d0: 6f72 6520 696e 7465 7265 7374 696e 6720  ore interesting 
│ │ │ │ -000203e0: 6578 616d 706c 6520 7468 6174 2069 7320  example that is 
│ │ │ │ -000203f0: 5231 2062 7574 206e 6f74 2052 322c 2061  R1 but not R2, a
│ │ │ │ -00020400: 6e64 0a68 6967 686c 6967 6874 2074 6865  nd.highlight the
│ │ │ │ -00020410: 2073 7065 6564 2064 6966 6665 7265 6e63   speed differenc
│ │ │ │ -00020420: 6573 2e20 204e 6f74 6520 7468 6174 2072  es.  Note that r
│ │ │ │ -00020430: 6567 756c 6172 496e 436f 6469 6d65 6e73  egularInCodimens
│ │ │ │ -00020440: 696f 6e28 322c 2052 2920 7265 7475 726e  ion(2, R) return
│ │ │ │ -00020450: 730a 6e6f 7468 696e 672c 2061 7320 7468  s.nothing, as th
│ │ │ │ -00020460: 6520 6675 6e63 7469 6f6e 2064 6964 206e  e function did n
│ │ │ │ -00020470: 6f74 2064 6574 6572 6d69 6e65 2077 6865  ot determine whe
│ │ │ │ -00020480: 7468 6572 2074 6865 2072 696e 6720 7761  ther the ring wa
│ │ │ │ -00020490: 7320 7265 6775 6c61 7220 696e 0a63 6f64  s regular in.cod
│ │ │ │ -000204a0: 696d 656e 7369 6f6e 206e 2e0a 0a2b 2d2d  imension n...+--
│ │ │ │ -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 2d2d  ----------------
│ │ │ │ -000204e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000204f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934  -----------+.|i4
│ │ │ │ -00020500: 203a 2054 203d 205a 5a2f 3130 315b 7831   : T = ZZ/101[x1
│ │ │ │ -00020510: 2c78 322c 7833 2c78 342c 7835 2c78 362c  ,x2,x3,x4,x5,x6,
│ │ │ │ -00020520: 7837 5d3b 2020 2020 2020 2020 2020 2020  x7];            
│ │ │ │ -00020530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020540: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -00020550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935  -----------+.|i5
│ │ │ │ -000205a0: 203a 2049 203d 2020 6964 6561 6c28 7835   : I =  ideal(x5
│ │ │ │ -000205b0: 2a78 362d 7834 2a78 372c 7831 2a78 362d  *x6-x4*x7,x1*x6-
│ │ │ │ -000205c0: 7832 2a78 372c 7835 5e32 2d78 312a 7837  x2*x7,x5^2-x1*x7
│ │ │ │ -000205d0: 2c78 342a 7835 2d78 322a 7837 2c78 345e  ,x4*x5-x2*x7,x4^
│ │ │ │ -000205e0: 322d 7832 2a78 362c 7831 2a7c 0a7c 2020  2-x2*x6,x1*|.|  
│ │ │ │ -000205f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020630: 2020 2020 2020 2020 2020 207c 0a7c 6f35             |.|o5
│ │ │ │ -00020640: 203a 2049 6465 616c 206f 6620 5420 2020   : Ideal of T   
│ │ │ │ +000203b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933  -----------+.|i3
│ │ │ │ +000203c0: 203a 2072 6567 756c 6172 496e 436f 6469   : regularInCodi
│ │ │ │ +000203d0: 6d65 6e73 696f 6e28 322c 2052 2920 2020  mension(2, R)   
│ │ │ │ +000203e0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +000203f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020400: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4e65 7874  ---------+..Next
│ │ │ │ +00020410: 2077 6520 636f 6e73 6964 6572 2061 206d   we consider a m
│ │ │ │ +00020420: 6f72 6520 696e 7465 7265 7374 696e 6720  ore interesting 
│ │ │ │ +00020430: 6578 616d 706c 6520 7468 6174 2069 7320  example that is 
│ │ │ │ +00020440: 5231 2062 7574 206e 6f74 2052 322c 2061  R1 but not R2, a
│ │ │ │ +00020450: 6e64 0a68 6967 686c 6967 6874 2074 6865  nd.highlight the
│ │ │ │ +00020460: 2073 7065 6564 2064 6966 6665 7265 6e63   speed differenc
│ │ │ │ +00020470: 6573 2e20 204e 6f74 6520 7468 6174 2072  es.  Note that r
│ │ │ │ +00020480: 6567 756c 6172 496e 436f 6469 6d65 6e73  egularInCodimens
│ │ │ │ +00020490: 696f 6e28 322c 2052 2920 7265 7475 726e  ion(2, R) return
│ │ │ │ +000204a0: 730a 6e6f 7468 696e 672c 2061 7320 7468  s.nothing, as th
│ │ │ │ +000204b0: 6520 6675 6e63 7469 6f6e 2064 6964 206e  e function did n
│ │ │ │ +000204c0: 6f74 2064 6574 6572 6d69 6e65 2077 6865  ot determine whe
│ │ │ │ +000204d0: 7468 6572 2074 6865 2072 696e 6720 7761  ther the ring wa
│ │ │ │ +000204e0: 7320 7265 6775 6c61 7220 696e 0a63 6f64  s regular in.cod
│ │ │ │ +000204f0: 696d 656e 7369 6f6e 206e 2e0a 0a2b 2d2d  imension n...+--
│ │ │ │ +00020500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934  -----------+.|i4
│ │ │ │ +00020550: 203a 2054 203d 205a 5a2f 3130 315b 7831   : T = ZZ/101[x1
│ │ │ │ +00020560: 2c78 322c 7833 2c78 342c 7835 2c78 362c  ,x2,x3,x4,x5,x6,
│ │ │ │ +00020570: 7837 5d3b 2020 2020 2020 2020 2020 2020  x7];            
│ │ │ │ +00020580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020590: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +000205a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000205b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000205c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000205d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000205e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935  -----------+.|i5
│ │ │ │ +000205f0: 203a 2049 203d 2020 6964 6561 6c28 7835   : I =  ideal(x5
│ │ │ │ +00020600: 2a78 362d 7834 2a78 372c 7831 2a78 362d  *x6-x4*x7,x1*x6-
│ │ │ │ +00020610: 7832 2a78 372c 7835 5e32 2d78 312a 7837  x2*x7,x5^2-x1*x7
│ │ │ │ +00020620: 2c78 342a 7835 2d78 322a 7837 2c78 345e  ,x4*x5-x2*x7,x4^
│ │ │ │ +00020630: 322d 7832 2a78 362c 7831 2a7c 0a7c 2020  2-x2*x6,x1*|.|  
│ │ │ │ +00020640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020680: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │ -00020690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000206a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000206b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000206c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000206d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 7834  -----------|.|x4
│ │ │ │ -000206e0: 2d78 322a 7835 2c78 322a 7833 5e33 2a78  -x2*x5,x2*x3^3*x
│ │ │ │ -000206f0: 352b 332a 7832 2a78 335e 322a 7837 2b38  5+3*x2*x3^2*x7+8
│ │ │ │ -00020700: 2a78 325e 322a 7835 2b33 2a78 332a 7834  *x2^2*x5+3*x3*x4
│ │ │ │ -00020710: 2a78 372d 382a 7834 2a78 372b 7836 2a78  *x7-8*x4*x7+x6*x
│ │ │ │ -00020720: 372c 7831 2a78 335e 332a 207c 0a7c 2d2d  7,x1*x3^3* |.|--
│ │ │ │ -00020730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 7835  -----------|.|x5
│ │ │ │ -00020780: 2b33 2a78 312a 7833 5e32 2a78 372b 382a  +3*x1*x3^2*x7+8*
│ │ │ │ -00020790: 7831 2a78 322a 7835 2b33 2a78 332a 7835  x1*x2*x5+3*x3*x5
│ │ │ │ -000207a0: 2a78 372d 382a 7835 2a78 372b 7837 5e32  *x7-8*x5*x7+x7^2
│ │ │ │ -000207b0: 2c78 322a 7833 5e33 2a78 342b 332a 7832  ,x2*x3^3*x4+3*x2
│ │ │ │ -000207c0: 2a78 335e 322a 7836 2b38 2a7c 0a7c 2d2d  *x3^2*x6+8*|.|--
│ │ │ │ -000207d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000207e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000207f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00020830: 2d38 2a78 342a 7836 2b78 365e 322c 7832  -8*x4*x6+x6^2,x2
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│ │ │ │ -00020850: 322a 7834 2b38 2a78 325e 332b 332a 7832  2*x4+8*x2^3+3*x2
│ │ │ │ -00020860: 2a78 332a 7836 2d38 2a78 327c 0a7c 2d2d  *x3*x6-8*x2|.|--
│ │ │ │ -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 2d2d  ----------------
│ │ │ │ -000208a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -000208c0: 362b 7834 2a78 362c 7831 2a78 322a 7833  6+x4*x6,x1*x2*x3
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│ │ │ │ -00020900: 7837 2c78 315e 322a 7833 5e7c 0a7c 2d2d  x7,x1^2*x3^|.|--
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│ │ │ │ -00020920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 332b  -----------|.|3+
│ │ │ │ -00020960: 332a 7831 2a78 335e 322a 7835 2b38 2a78  3*x1*x3^2*x5+8*x
│ │ │ │ -00020970: 315e 322a 7832 2b33 2a78 312a 7833 2a78  1^2*x2+3*x1*x3*x
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│ │ │ │ -00020990: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ -000209a0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -000209b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -000209d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000209e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000209f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936  -----------+.|i6
│ │ │ │ -00020a00: 203a 2053 203d 2054 2f49 3b20 2020 2020   : S = T/I;     
│ │ │ │ -00020a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020a40: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -00020a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937  -----------+.|i7
│ │ │ │ -00020aa0: 203a 2064 696d 2053 2020 2020 2020 2020   : dim S        
│ │ │ │ -00020ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020ae0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00020af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020680: 2020 2020 2020 2020 2020 207c 0a7c 6f35             |.|o5
│ │ │ │ +00020690: 203a 2049 6465 616c 206f 6620 5420 2020   : Ideal of T   
│ │ │ │ +000206a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000206b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000206c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000206d0: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │ +000206e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000206f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 7834  -----------|.|x4
│ │ │ │ +00020730: 2d78 322a 7835 2c78 322a 7833 5e33 2a78  -x2*x5,x2*x3^3*x
│ │ │ │ +00020740: 352b 332a 7832 2a78 335e 322a 7837 2b38  5+3*x2*x3^2*x7+8
│ │ │ │ +00020750: 2a78 325e 322a 7835 2b33 2a78 332a 7834  *x2^2*x5+3*x3*x4
│ │ │ │ +00020760: 2a78 372d 382a 7834 2a78 372b 7836 2a78  *x7-8*x4*x7+x6*x
│ │ │ │ +00020770: 372c 7831 2a78 335e 332a 207c 0a7c 2d2d  7,x1*x3^3* |.|--
│ │ │ │ +00020780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000207a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000207b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000207c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 7835  -----------|.|x5
│ │ │ │ +000207d0: 2b33 2a78 312a 7833 5e32 2a78 372b 382a  +3*x1*x3^2*x7+8*
│ │ │ │ +000207e0: 7831 2a78 322a 7835 2b33 2a78 332a 7835  x1*x2*x5+3*x3*x5
│ │ │ │ +000207f0: 2a78 372d 382a 7835 2a78 372b 7837 5e32  *x7-8*x5*x7+x7^2
│ │ │ │ +00020800: 2c78 322a 7833 5e33 2a78 342b 332a 7832  ,x2*x3^3*x4+3*x2
│ │ │ │ +00020810: 2a78 335e 322a 7836 2b38 2a7c 0a7c 2d2d  *x3^2*x6+8*|.|--
│ │ │ │ +00020820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 7832  -----------|.|x2
│ │ │ │ +00020870: 5e32 2a78 342b 332a 7833 2a78 342a 7836  ^2*x4+3*x3*x4*x6
│ │ │ │ +00020880: 2d38 2a78 342a 7836 2b78 365e 322c 7832  -8*x4*x6+x6^2,x2
│ │ │ │ +00020890: 5e32 2a78 335e 332b 332a 7832 2a78 335e  ^2*x3^3+3*x2*x3^
│ │ │ │ +000208a0: 322a 7834 2b38 2a78 325e 332b 332a 7832  2*x4+8*x2^3+3*x2
│ │ │ │ +000208b0: 2a78 332a 7836 2d38 2a78 327c 0a7c 2d2d  *x3*x6-8*x2|.|--
│ │ │ │ +000208c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000208d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000208e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000208f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2a78  -----------|.|*x
│ │ │ │ +00020910: 362b 7834 2a78 362c 7831 2a78 322a 7833  6+x4*x6,x1*x2*x3
│ │ │ │ +00020920: 5e33 2b33 2a78 322a 7833 5e32 2a78 352b  ^3+3*x2*x3^2*x5+
│ │ │ │ +00020930: 382a 7831 2a78 325e 322b 332a 7832 2a78  8*x1*x2^2+3*x2*x
│ │ │ │ +00020940: 332a 7837 2d38 2a78 322a 7837 2b78 342a  3*x7-8*x2*x7+x4*
│ │ │ │ +00020950: 7837 2c78 315e 322a 7833 5e7c 0a7c 2d2d  x7,x1^2*x3^|.|--
│ │ │ │ +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 2d2d  ----------------
│ │ │ │ +00020990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000209a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 332b  -----------|.|3+
│ │ │ │ +000209b0: 332a 7831 2a78 335e 322a 7835 2b38 2a78  3*x1*x3^2*x5+8*x
│ │ │ │ +000209c0: 315e 322a 7832 2b33 2a78 312a 7833 2a78  1^2*x2+3*x1*x3*x
│ │ │ │ +000209d0: 372d 382a 7831 2a78 372b 7835 2a78 3729  7-8*x1*x7+x5*x7)
│ │ │ │ +000209e0: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ +000209f0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00020a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936  -----------+.|i6
│ │ │ │ +00020a50: 203a 2053 203d 2054 2f49 3b20 2020 2020   : S = T/I;     
│ │ │ │ +00020a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020a90: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00020aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937  -----------+.|i7
│ │ │ │ +00020af0: 203a 2064 696d 2053 2020 2020 2020 2020   : dim S        
│ │ │ │  00020b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020b30: 2020 2020 2020 2020 2020 207c 0a7c 6f37             |.|o7
│ │ │ │ -00020b40: 203d 2033 2020 2020 2020 2020 2020 2020   = 3            
│ │ │ │ +00020b30: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00020b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020b80: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -00020b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938  -----------+.|i8
│ │ │ │ -00020be0: 203a 2074 696d 6520 7265 6775 6c61 7249   : time regularI
│ │ │ │ -00020bf0: 6e43 6f64 696d 656e 7369 6f6e 2831 2c20  nCodimension(1, 
│ │ │ │ -00020c00: 5329 2020 2020 2020 2020 2020 2020 2020  S)              
│ │ │ │ -00020c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020c20: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ -00020c30: 2d20 7573 6564 2030 2e36 3731 3938 7320  - used 0.67198s 
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│ │ │ │ -00020c50: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │ -00020c60: 6329 2020 2020 2020 2020 2020 2020 2020  c)              
│ │ │ │ -00020c70: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00020c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020cc0: 2020 2020 2020 2020 2020 207c 0a7c 6f38             |.|o8
│ │ │ │ -00020cd0: 203d 2074 7275 6520 2020 2020 2020 2020   = true         
│ │ │ │ +00020b80: 2020 2020 2020 2020 2020 207c 0a7c 6f37             |.|o7
│ │ │ │ +00020b90: 203d 2033 2020 2020 2020 2020 2020 2020   = 3            
│ │ │ │ +00020ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00020be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938  -----------+.|i8
│ │ │ │ +00020c30: 203a 2074 696d 6520 7265 6775 6c61 7249   : time regularI
│ │ │ │ +00020c40: 6e43 6f64 696d 656e 7369 6f6e 2831 2c20  nCodimension(1, 
│ │ │ │ +00020c50: 5329 2020 2020 2020 2020 2020 2020 2020  S)              
│ │ │ │ +00020c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020c70: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
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│ │ │ │ +00020cc0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │  00020ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00020d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020d10: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -00020d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6939  -----------+.|i9
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│ │ │ │ -00020d80: 6e43 6f64 696d 656e 7369 6f6e 2832 2c20  nCodimension(2, 
│ │ │ │ -00020d90: 5329 2020 2020 2020 2020 2020 2020 2020  S)              
│ │ │ │ -00020da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020db0: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ -00020dc0: 2d20 7573 6564 2036 2e34 3832 3536 7320  - used 6.48256s 
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│ │ │ │ -00020de0: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │ -00020df0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -00020e00: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -00020e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5468  -----------+..Th
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│ │ │ │ -00020e80: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ -00020e90: 7369 6f6e 2069 7320 7365 7665 7261 6c20  sion is several 
│ │ │ │ -00020ea0: 6f72 6465 7273 206f 660a 6d61 676e 6974  orders of.magnit
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│ │ │ │ -00020ec0: 6361 6c6c 7320 6f66 2064 696d 2073 696e  calls of dim sin
│ │ │ │ -00020ed0: 6775 6c61 724c 6f63 7573 2e0a 0a54 6865  gularLocus...The
│ │ │ │ -00020ee0: 2066 6f6c 6c6f 7769 6e67 2069 7320 6120   following is a 
│ │ │ │ -00020ef0: 2870 7275 6e65 6429 2061 6666 696e 6520  (pruned) affine 
│ │ │ │ -00020f00: 6368 6172 7420 6f6e 2061 6e20 4162 656c  chart on an Abel
│ │ │ │ -00020f10: 6961 6e20 7375 7266 6163 6520 6f62 7461  ian surface obta
│ │ │ │ -00020f20: 696e 6564 2061 7320 610a 7072 6f64 7563  ined as a.produc
│ │ │ │ -00020f30: 7420 6f66 2074 776f 2065 6c6c 6970 7469  t of two ellipti
│ │ │ │ -00020f40: 6320 6375 7276 6573 2e20 2049 7420 6973  c curves.  It is
│ │ │ │ -00020f50: 206e 6f6e 7369 6e67 756c 6172 2c20 6173   nonsingular, as
│ │ │ │ -00020f60: 206f 7572 2066 756e 6374 696f 6e20 7665   our function ve
│ │ │ │ -00020f70: 7269 6669 6573 2e0a 4966 206f 6e65 2064  rifies..If one d
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│ │ │ │ -00020f90: 2c20 7468 656e 2074 6865 2064 696d 2073  , then the dim s
│ │ │ │ -00020fa0: 696e 6775 6c61 724c 6f63 7573 2063 616c  ingularLocus cal
│ │ │ │ -00020fb0: 6c20 7461 6b65 7320 616e 2065 6e6f 726d  l takes an enorm
│ │ │ │ -00020fc0: 6f75 730a 616d 6f75 6e74 206f 6620 7469  ous.amount of ti
│ │ │ │ -00020fd0: 6d65 2c20 6f74 6865 7277 6973 6520 7468  me, otherwise th
│ │ │ │ -00020fe0: 6520 7275 6e6e 696e 6720 7469 6d65 7320  e running times 
│ │ │ │ -00020ff0: 6f66 2064 696d 2073 696e 6775 6c61 724c  of dim singularL
│ │ │ │ -00021000: 6f63 7573 2061 6e64 206f 7572 0a66 756e  ocus and our.fun
│ │ │ │ -00021010: 6374 696f 6e20 6172 6520 6672 6571 7565  ction are freque
│ │ │ │ -00021020: 6e74 6c79 2061 626f 7574 2074 6865 2073  ntly about the s
│ │ │ │ -00021030: 616d 652e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d  ame...+---------
│ │ │ │ -00021040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021080: 2d2d 2d2d 2b0a 7c69 3130 203a 2052 203d  ----+.|i10 : R =
│ │ │ │ -00021090: 2051 515b 632c 2066 2c20 672c 2068 5d2f   QQ[c, f, g, h]/
│ │ │ │ -000210a0: 6964 6561 6c28 675e 332b 685e 332b 312c  ideal(g^3+h^3+1,
│ │ │ │ -000210b0: 662a 675e 332b 662a 685e 332b 662c 632a  f*g^3+f*h^3+f,c*
│ │ │ │ -000210c0: 675e 332b 632a 685e 332b 632c 665e 322a  g^3+c*h^3+c,f^2*
│ │ │ │ -000210d0: 675e 332b 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  g^3+|.|---------
│ │ │ │ -000210e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000210f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021120: 2d2d 2d2d 7c0a 7c66 5e32 2a68 5e33 2b66  ----|.|f^2*h^3+f
│ │ │ │ -00021130: 5e32 2c63 2a66 2a67 5e33 2b63 2a66 2a68  ^2,c*f*g^3+c*f*h
│ │ │ │ -00021140: 5e33 2b63 2a66 2c63 5e32 2a67 5e33 2b63  ^3+c*f,c^2*g^3+c
│ │ │ │ -00021150: 5e32 2a68 5e33 2b63 5e32 2c66 5e33 2a67  ^2*h^3+c^2,f^3*g
│ │ │ │ -00021160: 5e33 2b66 5e33 2a68 5e33 2b66 5e33 2c63  ^3+f^3*h^3+f^3,c
│ │ │ │ -00021170: 2a66 5e32 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  *f^2|.|---------
│ │ │ │ -00021180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000211a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000211b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000211c0: 2d2d 2d2d 7c0a 7c2a 675e 332b 632a 665e  ----|.|*g^3+c*f^
│ │ │ │ -000211d0: 322a 685e 332b 632a 665e 322c 635e 322a  2*h^3+c*f^2,c^2*
│ │ │ │ -000211e0: 662a 675e 332b 635e 322a 662a 685e 332b  f*g^3+c^2*f*h^3+
│ │ │ │ -000211f0: 635e 322a 662c 635e 332d 665e 322d 632c  c^2*f,c^3-f^2-c,
│ │ │ │ -00021200: 635e 332a 682d 665e 322a 682d 632a 682c  c^3*h-f^2*h-c*h,
│ │ │ │ -00021210: 635e 332a 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  c^3*|.|---------
│ │ │ │ -00021220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021260: 2d2d 2d2d 7c0a 7c67 2d66 5e32 2a67 2d63  ----|.|g-f^2*g-c
│ │ │ │ -00021270: 2a67 2c63 5e33 2a68 5e32 2d66 5e32 2a68  *g,c^3*h^2-f^2*h
│ │ │ │ -00021280: 5e32 2d63 2a68 5e32 2c63 5e33 2a67 2a68  ^2-c*h^2,c^3*g*h
│ │ │ │ -00021290: 2d66 5e32 2a67 2a68 2d63 2a67 2a68 2c63  -f^2*g*h-c*g*h,c
│ │ │ │ -000212a0: 5e33 2a67 5e32 2d66 5e32 2a67 5e32 2d63  ^3*g^2-f^2*g^2-c
│ │ │ │ -000212b0: 2a67 5e32 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  *g^2|.|---------
│ │ │ │ -000212c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000212d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000212e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000212f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021300: 2d2d 2d2d 7c0a 7c2c 635e 332a 685e 332d  ----|.|,c^3*h^3-
│ │ │ │ -00021310: 665e 322a 685e 332d 632a 685e 332c 635e  f^2*h^3-c*h^3,c^
│ │ │ │ -00021320: 332a 672a 685e 322d 665e 322a 672a 685e  3*g*h^2-f^2*g*h^
│ │ │ │ -00021330: 322d 632a 672a 685e 322c 635e 332a 675e  2-c*g*h^2,c^3*g^
│ │ │ │ -00021340: 322a 682d 665e 322a 675e 322a 682d 632a  2*h-f^2*g^2*h-c*
│ │ │ │ -00021350: 675e 322a 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  g^2*|.|---------
│ │ │ │ -00021360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000213a0: 2d2d 2d2d 7c0a 7c68 2c63 5e33 2a67 5e33  ----|.|h,c^3*g^3
│ │ │ │ -000213b0: 2b66 5e32 2a68 5e33 2b63 2a68 5e33 2b66  +f^2*h^3+c*h^3+f
│ │ │ │ -000213c0: 5e32 2b63 293b 2020 2020 2020 2020 2020  ^2+c);          
│ │ │ │ -000213d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000213e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000213f0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -00021400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021440: 2d2d 2d2d 2b0a 7c69 3131 203a 2064 696d  ----+.|i11 : dim
│ │ │ │ -00021450: 2852 2920 2020 2020 2020 2020 2020 2020  (R)             
│ │ │ │ -00021460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021490: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000214a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020d10: 2020 2020 2020 2020 2020 207c 0a7c 6f38             |.|o8
│ │ │ │ +00020d20: 203d 2074 7275 6520 2020 2020 2020 2020   = true         
│ │ │ │ +00020d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020d60: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00020d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6939  -----------+.|i9
│ │ │ │ +00020dc0: 203a 2074 696d 6520 7265 6775 6c61 7249   : time regularI
│ │ │ │ +00020dd0: 6e43 6f64 696d 656e 7369 6f6e 2832 2c20  nCodimension(2, 
│ │ │ │ +00020de0: 5329 2020 2020 2020 2020 2020 2020 2020  S)              
│ │ │ │ +00020df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020e00: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ +00020e10: 2d20 7573 6564 2037 2e35 3330 3937 7320  - used 7.53097s 
│ │ │ │ +00020e20: 2863 7075 293b 2035 2e34 3733 3835 7320  (cpu); 5.47385s 
│ │ │ │ +00020e30: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │ +00020e40: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +00020e50: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00020e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5468  -----------+..Th
│ │ │ │ +00020eb0: 6572 6520 6172 6520 6e75 6d65 726f 7573  ere are numerous
│ │ │ │ +00020ec0: 2065 7861 6d70 6c65 7320 7768 6572 6520   examples where 
│ │ │ │ +00020ed0: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ +00020ee0: 7369 6f6e 2069 7320 7365 7665 7261 6c20  sion is several 
│ │ │ │ +00020ef0: 6f72 6465 7273 206f 660a 6d61 676e 6974  orders of.magnit
│ │ │ │ +00020f00: 7564 6520 6661 7374 6572 2074 6861 7420  ude faster that 
│ │ │ │ +00020f10: 6361 6c6c 7320 6f66 2064 696d 2073 696e  calls of dim sin
│ │ │ │ +00020f20: 6775 6c61 724c 6f63 7573 2e0a 0a54 6865  gularLocus...The
│ │ │ │ +00020f30: 2066 6f6c 6c6f 7769 6e67 2069 7320 6120   following is a 
│ │ │ │ +00020f40: 2870 7275 6e65 6429 2061 6666 696e 6520  (pruned) affine 
│ │ │ │ +00020f50: 6368 6172 7420 6f6e 2061 6e20 4162 656c  chart on an Abel
│ │ │ │ +00020f60: 6961 6e20 7375 7266 6163 6520 6f62 7461  ian surface obta
│ │ │ │ +00020f70: 696e 6564 2061 7320 610a 7072 6f64 7563  ined as a.produc
│ │ │ │ +00020f80: 7420 6f66 2074 776f 2065 6c6c 6970 7469  t of two ellipti
│ │ │ │ +00020f90: 6320 6375 7276 6573 2e20 2049 7420 6973  c curves.  It is
│ │ │ │ +00020fa0: 206e 6f6e 7369 6e67 756c 6172 2c20 6173   nonsingular, as
│ │ │ │ +00020fb0: 206f 7572 2066 756e 6374 696f 6e20 7665   our function ve
│ │ │ │ +00020fc0: 7269 6669 6573 2e0a 4966 206f 6e65 2064  rifies..If one d
│ │ │ │ +00020fd0: 6f65 7320 6e6f 7420 7072 756e 6520 6974  oes not prune it
│ │ │ │ +00020fe0: 2c20 7468 656e 2074 6865 2064 696d 2073  , then the dim s
│ │ │ │ +00020ff0: 696e 6775 6c61 724c 6f63 7573 2063 616c  ingularLocus cal
│ │ │ │ +00021000: 6c20 7461 6b65 7320 616e 2065 6e6f 726d  l takes an enorm
│ │ │ │ +00021010: 6f75 730a 616d 6f75 6e74 206f 6620 7469  ous.amount of ti
│ │ │ │ +00021020: 6d65 2c20 6f74 6865 7277 6973 6520 7468  me, otherwise th
│ │ │ │ +00021030: 6520 7275 6e6e 696e 6720 7469 6d65 7320  e running times 
│ │ │ │ +00021040: 6f66 2064 696d 2073 696e 6775 6c61 724c  of dim singularL
│ │ │ │ +00021050: 6f63 7573 2061 6e64 206f 7572 0a66 756e  ocus and our.fun
│ │ │ │ +00021060: 6374 696f 6e20 6172 6520 6672 6571 7565  ction are freque
│ │ │ │ +00021070: 6e74 6c79 2061 626f 7574 2074 6865 2073  ntly about the s
│ │ │ │ +00021080: 616d 652e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d  ame...+---------
│ │ │ │ +00021090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000210a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000210b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000210c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000210d0: 2d2d 2d2d 2b0a 7c69 3130 203a 2052 203d  ----+.|i10 : R =
│ │ │ │ +000210e0: 2051 515b 632c 2066 2c20 672c 2068 5d2f   QQ[c, f, g, h]/
│ │ │ │ +000210f0: 6964 6561 6c28 675e 332b 685e 332b 312c  ideal(g^3+h^3+1,
│ │ │ │ +00021100: 662a 675e 332b 662a 685e 332b 662c 632a  f*g^3+f*h^3+f,c*
│ │ │ │ +00021110: 675e 332b 632a 685e 332b 632c 665e 322a  g^3+c*h^3+c,f^2*
│ │ │ │ +00021120: 675e 332b 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  g^3+|.|---------
│ │ │ │ +00021130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021170: 2d2d 2d2d 7c0a 7c66 5e32 2a68 5e33 2b66  ----|.|f^2*h^3+f
│ │ │ │ +00021180: 5e32 2c63 2a66 2a67 5e33 2b63 2a66 2a68  ^2,c*f*g^3+c*f*h
│ │ │ │ +00021190: 5e33 2b63 2a66 2c63 5e32 2a67 5e33 2b63  ^3+c*f,c^2*g^3+c
│ │ │ │ +000211a0: 5e32 2a68 5e33 2b63 5e32 2c66 5e33 2a67  ^2*h^3+c^2,f^3*g
│ │ │ │ +000211b0: 5e33 2b66 5e33 2a68 5e33 2b66 5e33 2c63  ^3+f^3*h^3+f^3,c
│ │ │ │ +000211c0: 2a66 5e32 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  *f^2|.|---------
│ │ │ │ +000211d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000211e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000211f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021210: 2d2d 2d2d 7c0a 7c2a 675e 332b 632a 665e  ----|.|*g^3+c*f^
│ │ │ │ +00021220: 322a 685e 332b 632a 665e 322c 635e 322a  2*h^3+c*f^2,c^2*
│ │ │ │ +00021230: 662a 675e 332b 635e 322a 662a 685e 332b  f*g^3+c^2*f*h^3+
│ │ │ │ +00021240: 635e 322a 662c 635e 332d 665e 322d 632c  c^2*f,c^3-f^2-c,
│ │ │ │ +00021250: 635e 332a 682d 665e 322a 682d 632a 682c  c^3*h-f^2*h-c*h,
│ │ │ │ +00021260: 635e 332a 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  c^3*|.|---------
│ │ │ │ +00021270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000212a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000212b0: 2d2d 2d2d 7c0a 7c67 2d66 5e32 2a67 2d63  ----|.|g-f^2*g-c
│ │ │ │ +000212c0: 2a67 2c63 5e33 2a68 5e32 2d66 5e32 2a68  *g,c^3*h^2-f^2*h
│ │ │ │ +000212d0: 5e32 2d63 2a68 5e32 2c63 5e33 2a67 2a68  ^2-c*h^2,c^3*g*h
│ │ │ │ +000212e0: 2d66 5e32 2a67 2a68 2d63 2a67 2a68 2c63  -f^2*g*h-c*g*h,c
│ │ │ │ +000212f0: 5e33 2a67 5e32 2d66 5e32 2a67 5e32 2d63  ^3*g^2-f^2*g^2-c
│ │ │ │ +00021300: 2a67 5e32 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  *g^2|.|---------
│ │ │ │ +00021310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021350: 2d2d 2d2d 7c0a 7c2c 635e 332a 685e 332d  ----|.|,c^3*h^3-
│ │ │ │ +00021360: 665e 322a 685e 332d 632a 685e 332c 635e  f^2*h^3-c*h^3,c^
│ │ │ │ +00021370: 332a 672a 685e 322d 665e 322a 672a 685e  3*g*h^2-f^2*g*h^
│ │ │ │ +00021380: 322d 632a 672a 685e 322c 635e 332a 675e  2-c*g*h^2,c^3*g^
│ │ │ │ +00021390: 322a 682d 665e 322a 675e 322a 682d 632a  2*h-f^2*g^2*h-c*
│ │ │ │ +000213a0: 675e 322a 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  g^2*|.|---------
│ │ │ │ +000213b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000213c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000213d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000213e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000213f0: 2d2d 2d2d 7c0a 7c68 2c63 5e33 2a67 5e33  ----|.|h,c^3*g^3
│ │ │ │ +00021400: 2b66 5e32 2a68 5e33 2b63 2a68 5e33 2b66  +f^2*h^3+c*h^3+f
│ │ │ │ +00021410: 5e32 2b63 293b 2020 2020 2020 2020 2020  ^2+c);          
│ │ │ │ +00021420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021440: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00021450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021490: 2d2d 2d2d 2b0a 7c69 3131 203a 2064 696d  ----+.|i11 : dim
│ │ │ │ +000214a0: 2852 2920 2020 2020 2020 2020 2020 2020  (R)             
│ │ │ │  000214b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000214c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000214d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000214e0: 2020 2020 7c0a 7c6f 3131 203d 2032 2020      |.|o11 = 2  
│ │ │ │ +000214e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  000214f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021530: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -00021540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021580: 2d2d 2d2d 2b0a 7c69 3132 203a 2074 696d  ----+.|i12 : tim
│ │ │ │ -00021590: 6520 2864 696d 2073 696e 6775 6c61 724c  e (dim singularL
│ │ │ │ -000215a0: 6f63 7573 2028 5229 2920 2020 2020 2020  ocus (R))       
│ │ │ │ -000215b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000215c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000215d0: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ -000215e0: 302e 3032 3030 3237 7320 2863 7075 293b  0.020027s (cpu);
│ │ │ │ -000215f0: 2030 2e30 3139 3831 3534 7320 2874 6872   0.0198154s (thr
│ │ │ │ -00021600: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │ +00021530: 2020 2020 7c0a 7c6f 3131 203d 2032 2020      |.|o11 = 2  
│ │ │ │ +00021540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021580: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00021590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000215a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000215b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000215c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000215d0: 2d2d 2d2d 2b0a 7c69 3132 203a 2074 696d  ----+.|i12 : tim
│ │ │ │ +000215e0: 6520 2864 696d 2073 696e 6775 6c61 724c  e (dim singularL
│ │ │ │ +000215f0: 6f63 7573 2028 5229 2920 2020 2020 2020  ocus (R))       
│ │ │ │ +00021600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021620: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00021630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021620: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ +00021630: 302e 3032 3431 3030 3173 2028 6370 7529  0.0241001s (cpu)
│ │ │ │ +00021640: 3b20 302e 3032 3131 3832 7320 2874 6872  ; 0.021182s (thr
│ │ │ │ +00021650: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │  00021660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021670: 2020 2020 7c0a 7c6f 3132 203d 202d 3120      |.|o12 = -1 
│ │ │ │ +00021670: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00021680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000216a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000216b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000216c0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -000216d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000216e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000216f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021710: 2d2d 2d2d 2b0a 7c69 3133 203a 2074 696d  ----+.|i13 : tim
│ │ │ │ -00021720: 6520 7265 6775 6c61 7249 6e43 6f64 696d  e regularInCodim
│ │ │ │ -00021730: 656e 7369 6f6e 2832 2c20 5229 2020 2020  ension(2, R)    
│ │ │ │ -00021740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021760: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ -00021770: 302e 3139 3230 3136 7320 2863 7075 293b  0.192016s (cpu);
│ │ │ │ -00021780: 2030 2e31 3630 3834 3973 2028 7468 7265   0.160849s (thre
│ │ │ │ -00021790: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │ +000216c0: 2020 2020 7c0a 7c6f 3132 203d 202d 3120      |.|o12 = -1 
│ │ │ │ +000216d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000216e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000216f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021710: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00021720: 2d2d 2d2d 2d2d 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 2b0a 7c69 3133 203a 2074 696d  ----+.|i13 : tim
│ │ │ │ +00021770: 6520 7265 6775 6c61 7249 6e43 6f64 696d  e regularInCodim
│ │ │ │ +00021780: 656e 7369 6f6e 2832 2c20 5229 2020 2020  ension(2, R)    
│ │ │ │ +00021790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000217a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000217b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000217c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000217d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000217e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000217b0: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ +000217c0: 302e 3231 3531 3831 7320 2863 7075 293b  0.215181s (cpu);
│ │ │ │ +000217d0: 2030 2e31 3533 3036 3573 2028 7468 7265   0.153065s (thre
│ │ │ │ +000217e0: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │  000217f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021800: 2020 2020 7c0a 7c6f 3133 203d 2074 7275      |.|o13 = tru
│ │ │ │ -00021810: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │ +00021800: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00021810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021850: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -00021860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000218a0: 2d2d 2d2d 2b0a 7c69 3134 203a 2074 696d  ----+.|i14 : tim
│ │ │ │ -000218b0: 6520 7265 6775 6c61 7249 6e43 6f64 696d  e regularInCodim
│ │ │ │ -000218c0: 656e 7369 6f6e 2832 2c20 5229 2020 2020  ension(2, R)    
│ │ │ │ -000218d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000218e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000218f0: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ -00021900: 302e 3930 3935 3338 7320 2863 7075 293b  0.909538s (cpu);
│ │ │ │ -00021910: 2030 2e36 3030 3332 3473 2028 7468 7265   0.600324s (thre
│ │ │ │ -00021920: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │ +00021850: 2020 2020 7c0a 7c6f 3133 203d 2074 7275      |.|o13 = tru
│ │ │ │ +00021860: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │ +00021870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000218a0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +000218b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000218c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000218d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000218e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000218f0: 2d2d 2d2d 2b0a 7c69 3134 203a 2074 696d  ----+.|i14 : tim
│ │ │ │ +00021900: 6520 7265 6775 6c61 7249 6e43 6f64 696d  e regularInCodim
│ │ │ │ +00021910: 656e 7369 6f6e 2832 2c20 5229 2020 2020  ension(2, R)    
│ │ │ │ +00021920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021940: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00021950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021940: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ +00021950: 312e 3036 3235 3573 2028 6370 7529 3b20  1.06255s (cpu); 
│ │ │ │ +00021960: 302e 3635 3534 3637 7320 2874 6872 6561  0.655467s (threa
│ │ │ │ +00021970: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │  00021980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021990: 2020 2020 7c0a 7c6f 3134 203d 2074 7275      |.|o14 = tru
│ │ │ │ -000219a0: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │ +00021990: 2020 2020 7c0a 7c20 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000219e0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -000219f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021a30: 2d2d 2d2d 2b0a 7c69 3135 203a 2074 696d  ----+.|i15 : tim
│ │ │ │ -00021a40: 6520 7265 6775 6c61 7249 6e43 6f64 696d  e regularInCodim
│ │ │ │ -00021a50: 656e 7369 6f6e 2832 2c20 5229 2020 2020  ension(2, R)    
│ │ │ │ -00021a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021a80: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ -00021a90: 312e 3332 3839 7320 2863 7075 293b 2030  1.3289s (cpu); 0
│ │ │ │ -00021aa0: 2e38 3939 3039 3473 2028 7468 7265 6164  .899094s (thread
│ │ │ │ -00021ab0: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │ +000219e0: 2020 2020 7c0a 7c6f 3134 203d 2074 7275      |.|o14 = tru
│ │ │ │ +000219f0: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │ +00021a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021a30: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00021a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021a80: 2d2d 2d2d 2b0a 7c69 3135 203a 2074 696d  ----+.|i15 : tim
│ │ │ │ +00021a90: 6520 7265 6775 6c61 7249 6e43 6f64 696d  e regularInCodim
│ │ │ │ +00021aa0: 656e 7369 6f6e 2832 2c20 5229 2020 2020  ension(2, R)    
│ │ │ │ +00021ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021ad0: 2020 2020 7c0a 7c20 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                  
│ │ │ │ +00021ad0: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ +00021ae0: 312e 3534 3438 3673 2028 6370 7529 3b20  1.54486s (cpu); 
│ │ │ │ +00021af0: 312e 3030 3331 3873 2028 7468 7265 6164  1.00318s (thread
│ │ │ │ +00021b00: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │  00021b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021b20: 2020 2020 7c0a 7c6f 3135 203d 2074 7275      |.|o15 = tru
│ │ │ │ -00021b30: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │ +00021b20: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00021b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021b70: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -00021b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021bc0: 2d2d 2d2d 2b0a 0a54 6865 2066 756e 6374  ----+..The funct
│ │ │ │ -00021bd0: 696f 6e20 776f 726b 7320 6279 2063 686f  ion works by cho
│ │ │ │ -00021be0: 6f73 696e 6720 696e 7465 7265 7374 696e  osing interestin
│ │ │ │ -00021bf0: 6720 6c6f 6f6b 696e 6720 7375 626d 6174  g looking submat
│ │ │ │ -00021c00: 7269 6365 732c 2063 6f6d 7075 7469 6e67  rices, computing
│ │ │ │ -00021c10: 2074 6865 6972 0a64 6574 6572 6d69 6e61   their.determina
│ │ │ │ -00021c20: 6e74 732c 2061 6e64 2070 6572 696f 6469  nts, and periodi
│ │ │ │ -00021c30: 6361 6c6c 7920 2862 6173 6564 206f 6e20  cally (based on 
│ │ │ │ -00021c40: 6120 6c6f 6761 7269 7468 6d69 6320 6772  a logarithmic gr
│ │ │ │ -00021c50: 6f77 7468 2073 6574 7469 6e67 292c 0a63  owth setting),.c
│ │ │ │ -00021c60: 6f6d 7075 7469 6e67 2074 6865 2064 696d  omputing the dim
│ │ │ │ -00021c70: 656e 7369 6f6e 206f 6620 6120 7375 6269  ension of a subi
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│ │ │ │  00022440: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -00022460: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +00022460: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │  00022480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022490: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
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│ │ │ │ -000224c0: 6162 6f75 7420 746f 2063 6f6d 7075 7465  about to compute
│ │ │ │ -000224d0: 2064 696d 656e 7369 6f6e 2e20 2053 7562   dimension.  Sub
│ │ │ │ -000224e0: 6d61 7472 6963 6573 2063 6f7c 0a7c 7265  matrices co|.|re
│ │ │ │ +00022490: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ +000224d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000224e0: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
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│ │ │ │ -00022510: 6173 7420 6661 696c 6564 2c20 636f 6d70  ast failed, comp
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│ │ │ │ -00022530: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
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│ │ │ │ +00022520: 2064 696d 656e 7369 6f6e 2e20 2053 7562   dimension.  Sub
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│ │ │ │  00022540: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
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│ │ │ │ -000225a0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ -000225d0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ +00022580: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
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│ │ │ │ -000225f0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │ +000225f0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00022600: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │  00022610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022620: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │  00022640: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │ +00022650: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │  00022660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022670: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -00022690: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +00022690: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │  000226b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000226e0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ +000226e0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │  00022700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022710: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -00022760: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
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│ │ │ │ -000227a0: 2064 696d 656e 7369 6f6e 2e20 2053 7562   dimension.  Sub
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│ │ │ │ +00022760: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ +000227f0: 2064 696d 656e 7369 6f6e 2e20 2053 7562   dimension.  Sub
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│ │ │ │ -000228c0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +000228c0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │  00022910: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ -00022960: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │ +00022960: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │  00022980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022990: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -000229b0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +000229b0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │  00022a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022a30: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ +00022a50: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │  00022a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022a80: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
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│ │ │ │ -00022ac0: 2064 696d 656e 7369 6f6e 2e20 2053 7562   dimension.  Sub
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│ │ │ │ +00022a80: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ +00022aa0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │ +00022ad0: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
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│ │ │ │ -00022b00: 6173 7420 6661 696c 6564 2c20 636f 6d70  ast failed, comp
│ │ │ │ -00022b10: 7574 696e 6720 636f 6469 6d2e 2020 2020  uting codim.    
│ │ │ │ -00022b20: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
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│ │ │ │ +00022b00: 6162 6f75 7420 746f 2063 6f6d 7075 7465  about to compute
│ │ │ │ +00022b10: 2064 696d 656e 7369 6f6e 2e20 2053 7562   dimension.  Sub
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│ │ │ │  00022b30: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
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│ │ │ │ -00022b90: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │ -00022bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022bc0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ +00022b50: 6173 7420 6661 696c 6564 2c20 636f 6d70  ast failed, comp
│ │ │ │ +00022b60: 7574 696e 6720 636f 6469 6d2e 2020 2020  uting codim.    
│ │ │ │ +00022b70: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
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│ │ │ │ +00022bc0: 3d20 3120 2020 2020 2020 207c 0a7c 696e  = 1        |.|in
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│ │ │ │ -00022be0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -00022bf0: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │ +00022be0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +00022bf0: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │  00022c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022c10: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00022c20: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  00022c30: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │  00022c40: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │  00022c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022c60: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00022c70: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00022c80: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -00022c90: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │ +00022c80: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +00022c90: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │  00022ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022cb0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00022cc0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00022cd0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00022ce0: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │ +00022cd0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00022ce0: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │  00022cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022d00: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00022d10: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00022d20: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -00022d30: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │ +00022d20: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │ -00022e80: 2064 696d 656e 7369 6f6e 2e20 2053 7562   dimension.  Sub
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│ │ │ │ -000234f0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │  00023540: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │  00023590: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ -000235e0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +000235e0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │  00023610: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ +00023630: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │  00023660: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │  00023680: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │ +00023690: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
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│ │ │ │ -000236d0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ +000236d0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │  00023700: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -00023720: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │ +00023720: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ +00023890: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ +00023bd0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │ -00023ea0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │ +00023ea0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ -00023fc0: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
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│ │ │ │ -00024000: 2064 696d 656e 7369 6f6e 2e20 2053 7562   dimension.  Sub
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│ │ │ │ +00023fc0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ +00023fe0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │ +00024000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024010: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
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│ │ │ │ -00024040: 6173 7420 6661 696c 6564 2c20 636f 6d70  ast failed, comp
│ │ │ │ -00024050: 7574 696e 6720 636f 6469 6d2e 2020 2020  uting codim.    
│ │ │ │ -00024060: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
│ │ │ │ +00024030: 6f6e 3a20 204c 6f6f 7020 7374 6570 2c20  on:  Loop step, 
│ │ │ │ +00024040: 6162 6f75 7420 746f 2063 6f6d 7075 7465  about to compute
│ │ │ │ +00024050: 2064 696d 656e 7369 6f6e 2e20 2053 7562   dimension.  Sub
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│ │ │ │  00024070: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
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│ │ │ │ -000240d0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -000240e0: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │ -000240f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024100: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │ +00024080: 6f6e 3a20 2069 7343 6f64 696d 4174 4c65  on:  isCodimAtLe
│ │ │ │ +00024090: 6173 7420 6661 696c 6564 2c20 636f 6d70  ast failed, comp
│ │ │ │ +000240a0: 7574 696e 6720 636f 6469 6d2e 2020 2020  uting codim.    
│ │ │ │ +000240b0: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
│ │ │ │ +000240c0: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
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│ │ │ │ +000240f0: 6e73 696f 6e20 636f 6d70 7574 6564 2c20  nsion computed, 
│ │ │ │ +00024100: 3d20 3120 2020 2020 2020 207c 0a7c 696e  = 1        |.|in
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│ │ │ │ -00024120: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -00024130: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │ +00024120: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +00024130: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
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│ │ │ │  00024c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024c90: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00024ca0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00024cb0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │ +00024cb0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │  00024cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00024d00: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │  00024d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024d30: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -00024d50: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ +00024d50: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │  00024d80: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -00024ea0: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │ +00024e90: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ +00024ee0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │ +00024f30: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ +00024f80: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00024f90: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
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│ │ │ │ +00024fd0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │  00025000: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ +00025020: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +00025030: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
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│ │ │ │ +00025070: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ +000250c0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +00025160: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │  00025190: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ +000251b0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +00025250: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ +000252a0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │ +000252f0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ -00025350: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
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│ │ │ │ -00025390: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +000253e0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +000253f0: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
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│ │ │ │ -00025460: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
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│ │ │ │ -00025490: 6162 6f75 7420 746f 2063 6f6d 7075 7465  about to compute
│ │ │ │ -000254a0: 2064 696d 656e 7369 6f6e 2e20 2053 7562   dimension.  Sub
│ │ │ │ -000254b0: 6d61 7472 6963 6573 2063 6f7c 0a7c 7265  matrices co|.|re
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│ │ │ │ +00025480: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +000254a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000254f0: 7574 696e 6720 636f 6469 6d2e 2020 2020  uting codim.    
│ │ │ │ -00025500: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
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│ │ │ │ +000254f0: 2064 696d 656e 7369 6f6e 2e20 2053 7562   dimension.  Sub
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│ │ │ │  00025510: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
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│ │ │ │ -00025570: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ -00025590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000255a0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │ +00025520: 6f6e 3a20 2069 7343 6f64 696d 4174 4c65  on:  isCodimAtLe
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│ │ │ │ +00025540: 7574 696e 6720 636f 6469 6d2e 2020 2020  uting codim.    
│ │ │ │ +00025550: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
│ │ │ │ +00025560: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
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│ │ │ │ +00025590: 6e73 696f 6e20 636f 6d70 7574 6564 2c20  nsion computed, 
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│ │ │ │ -000255c0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -000255d0: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │ +000255c0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │  000255f0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ +00025610: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00025620: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │  00025630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025640: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │  00025690: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ +000256b0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +000257a0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │ +000257f0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ +00025890: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +00025990: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
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│ │ │ │  000259d0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │  000259e0: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │  000259f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00025a30: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │  00025a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00025a70: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ +00025a70: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00025a80: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
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│ │ │ │ -00025ad0: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
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│ │ │ │ +00025b10: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │ +00025b60: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +00025bb0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ +00025c00: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +00025c50: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ -00025ca0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ +00025ca0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +00025cf0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ +00025d90: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │ -00025de0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │ +00025de0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +00025e30: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │  00025e60: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -00025e80: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ +00025e80: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00025e90: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
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│ │ │ │  00025eb0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -00025ed0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +00025ed0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │  00025f00: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -00025f20: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │ +00025f20: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │  00025f50: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -00025f70: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +00025f70: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │  00025fa0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -00025fc0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ +00025fc0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │  00025fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025ff0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00026000: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  00026010: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ +00026020: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
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│ │ │ │  00026040: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -00026060: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │ +00026060: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │  00026090: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -000260b0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -000260c0: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │ +000260b0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +000260c0: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
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│ │ │ │  000260e0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │  00026100: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │  00026110: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │  00026120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026130: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00026140: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00026150: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │ -00026ba0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ +00026ba0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │ -00026bf0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +00026bf0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │  00026c20: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ +00026c40: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00026c50: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │  00026c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026c70: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -00026ca0: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │ +00026c90: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │  00026cc0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ +00026ce0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │ +00026d30: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00026d40: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │  00026d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026d60: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -00026d80: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00026d90: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │ +00026d80: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +00026d90: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
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│ │ │ │ -00026dd0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +00026dd0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +00026de0: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
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│ │ │ │ -00026e30: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │ +00026e20: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00026e30: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
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│ │ │ │ -00026e70: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -00026e80: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │ +00026e70: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +00026e80: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
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│ │ │ │ +00026ec0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │  00026ef0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ +00026f10: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00026f20: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
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│ │ │ │ -00026f70: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │ +00026f70: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
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│ │ │ │ -00026fb0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ +00026fb0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +00026fc0: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
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│ │ │ │  00026fe0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │  00027000: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -00027010: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │ +00027010: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
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│ │ │ │  00027050: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -00027060: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │ +00027060: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
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│ │ │ │ -000270a0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -000270b0: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │ +000270a0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +000270b0: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
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│ │ │ │ -000270f0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -00027100: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │ +000270f0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +00027100: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
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│ │ │ │  00027120: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -00027150: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │ +00027150: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
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│ │ │ │ +00027320: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │ +000274b0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │ -00027670: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
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│ │ │ │ +000276f0: 6775 6c61 7220 6c6f 6375 7320 6469 6d65  gular locus dime
│ │ │ │ +00027700: 6e73 696f 6e20 636f 6d70 7574 6564 2c20  nsion computed, 
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│ │ │ │ +000279b0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ +00027a00: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00027a10: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
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│ │ │ │  00027aa0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ +00027af0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ +00027b50: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
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│ │ │ │  00027be0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │  00027bf0: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
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│ │ │ │ +00027c30: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +00027c40: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
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│ │ │ │ -00027c80: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ -00027cd0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ +00027cd0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +00027d20: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ +00027d70: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │  00027eb0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ -00027f00: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ +00027f00: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +00027f50: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ -00027fa0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +00027fa0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │ +00027ff0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +00028090: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │  000280c0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -000280e0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +000280e0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │  00028100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028110: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -00028130: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -00028140: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │ +00028130: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │  00028160: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │  00028180: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ -000281d0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +000281d0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +000281e0: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │  000281f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028200: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028210: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028220: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -00028230: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │ +00028220: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00028230: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │  00028240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028250: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028260: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  00028270: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │  00028280: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │  00028290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000282a0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  000282b0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  000282c0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -000282d0: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │ +000282d0: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │  000282e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000282f0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028300: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028310: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00028320: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │ +00028310: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +00028320: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │  00028330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028340: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028350: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028360: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -00028370: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │ +00028360: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +00028370: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │  00028380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028390: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  000283a0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  000283b0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -000283c0: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │ +000283c0: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │  000283d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000283e0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  000283f0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  00028400: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │  00028410: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │  00028420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028430: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028440: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028450: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00028460: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │ +00028450: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00028460: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │  00028470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028480: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028490: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -000284a0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -000284b0: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │ +000284a0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +000284b0: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │  000284c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000284d0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  000284e0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -000284f0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00028500: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │ +000284f0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +00028500: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │  00028510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028520: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028530: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028540: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -00028550: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │ +00028540: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +00028550: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │  00028560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028570: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028580: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  00028590: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │  000285a0: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │  000285b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000285c0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  000285d0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  000285e0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │  000285f0: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │  00028600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028610: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028620: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028630: 723a 2043 6820 2d2d 2075 7365 6420 362e  r: Ch -- used 6.
│ │ │ │ -00028640: 3539 3837 3173 2028 6370 7529 3b20 352e  59871s (cpu); 5.
│ │ │ │ -00028650: 3032 3137 3673 2028 7468 7265 6164 293b  02176s (thread);
│ │ │ │ -00028660: 2030 7320 2867 6329 2020 207c 0a7c 6f6f   0s (gc)   |.|oo
│ │ │ │ -00028670: 7369 6e67 2047 5265 764c 6578 536d 616c  sing GRevLexSmal
│ │ │ │ -00028680: 6c65 7374 5465 726d 2020 2020 2020 2020  lestTerm        
│ │ │ │ -00028690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000286a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000286b0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │ -000286c0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -000286d0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -000286e0: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │ +00028630: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +00028640: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │ +00028650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028660: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │ +00028670: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ +00028680: 723a 2043 6820 2d2d 2075 7365 6420 372e  r: Ch -- used 7.
│ │ │ │ +00028690: 3633 3030 3373 2028 6370 7529 3b20 352e  63003s (cpu); 5.
│ │ │ │ +000286a0: 3634 3839 3373 2028 7468 7265 6164 293b  64893s (thread);
│ │ │ │ +000286b0: 2030 7320 2867 6329 2020 207c 0a7c 6f6f   0s (gc)   |.|oo
│ │ │ │ +000286c0: 7369 6e67 2047 5265 764c 6578 536d 616c  sing GRevLexSmal
│ │ │ │ +000286d0: 6c65 7374 5465 726d 2020 2020 2020 2020  lestTerm        
│ │ │ │ +000286e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000286f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028700: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028710: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028720: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -00028730: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │ +00028720: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00028730: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │  00028740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028750: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028760: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028770: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -00028780: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │ +00028770: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +00028780: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │  00028790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000287a0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  000287b0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  000287c0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │  000287d0: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │  000287e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000287f0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028800: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028810: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -00028820: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │ +00028810: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00028820: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │  00028830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028840: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028850: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028860: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00028870: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │ +00028860: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +00028870: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │  00028880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028890: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  000288a0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -000288b0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -000288c0: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │ +000288b0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +000288c0: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │  000288d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000288e0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  000288f0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028900: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -00028910: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │ +00028900: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +00028910: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │  00028920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028930: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028940: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  00028950: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │  00028960: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │  00028970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028980: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028990: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -000289a0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -000289b0: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │ +000289a0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +000289b0: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │  000289c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000289d0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  000289e0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -000289f0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -00028a00: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │ +000289f0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +00028a00: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │  00028a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028a20: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028a30: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028a40: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00028a50: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │ +00028a40: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00028a50: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │  00028a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028a70: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028a80: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  00028a90: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00028aa0: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │ +00028aa0: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │  00028ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028ac0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028ad0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028ae0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -00028af0: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │ +00028ae0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +00028af0: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │  00028b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028b10: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028b20: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028b30: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00028b40: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │ +00028b30: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +00028b40: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │  00028b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028b60: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028b70: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  00028b80: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00028b90: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │ +00028b90: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │  00028ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028bb0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028bc0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028bd0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -00028be0: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │ +00028bd0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +00028be0: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │  00028bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028c00: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028c10: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028c20: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -00028c30: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │ +00028c20: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +00028c30: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │  00028c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028c50: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028c60: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028c70: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -00028c80: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │ +00028c70: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00028c80: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │  00028c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028ca0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028cb0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  00028cc0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -00028cd0: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │ +00028cd0: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │  00028ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028cf0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028d00: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028d10: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00028d20: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │ +00028d10: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +00028d20: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │  00028d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028d40: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028d50: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  00028d60: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00028d70: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │ +00028d70: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │  00028d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028d90: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028da0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  00028db0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │  00028dc0: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │  00028dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028de0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028df0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028e00: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -00028e10: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │ +00028e00: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +00028e10: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │  00028e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028e30: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
│ │ │ │ -00028e40: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
│ │ │ │ -00028e50: 6f6e 3a20 204c 6f6f 7020 7374 6570 2c20  on:  Loop step, 
│ │ │ │ -00028e60: 6162 6f75 7420 746f 2063 6f6d 7075 7465  about to compute
│ │ │ │ -00028e70: 2064 696d 656e 7369 6f6e 2e20 2053 7562   dimension.  Sub
│ │ │ │ -00028e80: 6d61 7472 6963 6573 2063 6f7c 0a7c 7265  matrices co|.|re
│ │ │ │ +00028e30: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │ +00028e40: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ +00028e50: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00028e60: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │ +00028e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028e80: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
│ │ │ │  00028e90: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
│ │ │ │ -00028ea0: 6f6e 3a20 2069 7343 6f64 696d 4174 4c65  on:  isCodimAtLe
│ │ │ │ -00028eb0: 6173 7420 6661 696c 6564 2c20 636f 6d70  ast failed, comp
│ │ │ │ -00028ec0: 7574 696e 6720 636f 6469 6d2e 2020 2020  uting codim.    
│ │ │ │ -00028ed0: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
│ │ │ │ +00028ea0: 6f6e 3a20 204c 6f6f 7020 7374 6570 2c20  on:  Loop step, 
│ │ │ │ +00028eb0: 6162 6f75 7420 746f 2063 6f6d 7075 7465  about to compute
│ │ │ │ +00028ec0: 2064 696d 656e 7369 6f6e 2e20 2053 7562   dimension.  Sub
│ │ │ │ +00028ed0: 6d61 7472 6963 6573 2063 6f7c 0a7c 7265  matrices co|.|re
│ │ │ │  00028ee0: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
│ │ │ │ -00028ef0: 6f6e 3a20 2070 6172 7469 616c 2073 696e  on:  partial sin
│ │ │ │ -00028f00: 6775 6c61 7220 6c6f 6375 7320 6469 6d65  gular locus dime
│ │ │ │ -00028f10: 6e73 696f 6e20 636f 6d70 7574 6564 2c20  nsion computed, 
│ │ │ │ -00028f20: 3d20 3120 2020 2020 2020 207c 0a7c 696e  = 1        |.|in
│ │ │ │ -00028f30: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028f40: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00028f50: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │ -00028f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028f70: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │ +00028ef0: 6f6e 3a20 2069 7343 6f64 696d 4174 4c65  on:  isCodimAtLe
│ │ │ │ +00028f00: 6173 7420 6661 696c 6564 2c20 636f 6d70  ast failed, comp
│ │ │ │ +00028f10: 7574 696e 6720 636f 6469 6d2e 2020 2020  uting codim.    
│ │ │ │ +00028f20: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
│ │ │ │ +00028f30: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
│ │ │ │ +00028f40: 6f6e 3a20 2070 6172 7469 616c 2073 696e  on:  partial sin
│ │ │ │ +00028f50: 6775 6c61 7220 6c6f 6375 7320 6469 6d65  gular locus dime
│ │ │ │ +00028f60: 6e73 696f 6e20 636f 6d70 7574 6564 2c20  nsion computed, 
│ │ │ │ +00028f70: 3d20 3120 2020 2020 2020 207c 0a7c 696e  = 1        |.|in
│ │ │ │  00028f80: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028f90: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -00028fa0: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │ +00028f90: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +00028fa0: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │  00028fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028fc0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028fd0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028fe0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -00028ff0: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │ +00028fe0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +00028ff0: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │  00029000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029010: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00029020: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  00029030: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │  00029040: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │  00029050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029060: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00029070: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  00029080: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │  00029090: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │  000290a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000290b0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  000290c0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -000290d0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -000290e0: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │ +000290d0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +000290e0: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │  000290f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029100: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00029110: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00029120: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00029130: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │ +00029120: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +00029130: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │  00029140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029150: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00029160: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00029170: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -00029180: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │ +00029170: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +00029180: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │  00029190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000291a0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  000291b0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  000291c0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │  000291d0: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │  000291e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000291f0: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
│ │ │ │ -00029200: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
│ │ │ │ -00029210: 6f6e 3a20 204c 6f6f 7020 7374 6570 2c20  on:  Loop step, 
│ │ │ │ -00029220: 6162 6f75 7420 746f 2063 6f6d 7075 7465  about to compute
│ │ │ │ -00029230: 2064 696d 656e 7369 6f6e 2e20 2053 7562   dimension.  Sub
│ │ │ │ -00029240: 6d61 7472 6963 6573 2063 6f7c 0a7c 7265  matrices co|.|re
│ │ │ │ +000291f0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │ +00029200: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ +00029210: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00029220: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │ +00029230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029240: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
│ │ │ │  00029250: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
│ │ │ │ -00029260: 6f6e 3a20 2069 7343 6f64 696d 4174 4c65  on:  isCodimAtLe
│ │ │ │ -00029270: 6173 7420 6661 696c 6564 2c20 636f 6d70  ast failed, comp
│ │ │ │ -00029280: 7574 696e 6720 636f 6469 6d2e 2020 2020  uting codim.    
│ │ │ │ -00029290: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
│ │ │ │ +00029260: 6f6e 3a20 204c 6f6f 7020 7374 6570 2c20  on:  Loop step, 
│ │ │ │ +00029270: 6162 6f75 7420 746f 2063 6f6d 7075 7465  about to compute
│ │ │ │ +00029280: 2064 696d 656e 7369 6f6e 2e20 2053 7562   dimension.  Sub
│ │ │ │ +00029290: 6d61 7472 6963 6573 2063 6f7c 0a7c 7265  matrices co|.|re
│ │ │ │  000292a0: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
│ │ │ │ -000292b0: 6f6e 3a20 2070 6172 7469 616c 2073 696e  on:  partial sin
│ │ │ │ -000292c0: 6775 6c61 7220 6c6f 6375 7320 6469 6d65  gular locus dime
│ │ │ │ -000292d0: 6e73 696f 6e20 636f 6d70 7574 6564 2c20  nsion computed, 
│ │ │ │ -000292e0: 3d20 3120 2020 2020 2020 207c 0a7c 7265  = 1        |.|re
│ │ │ │ +000292b0: 6f6e 3a20 2069 7343 6f64 696d 4174 4c65  on:  isCodimAtLe
│ │ │ │ +000292c0: 6173 7420 6661 696c 6564 2c20 636f 6d70  ast failed, comp
│ │ │ │ +000292d0: 7574 696e 6720 636f 6469 6d2e 2020 2020  uting codim.    
│ │ │ │ +000292e0: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
│ │ │ │  000292f0: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
│ │ │ │ -00029300: 6f6e 3a20 204c 6f6f 7020 636f 6d70 6c65  on:  Loop comple
│ │ │ │ -00029310: 7465 642c 2073 7562 6d61 7472 6963 6573  ted, submatrices
│ │ │ │ -00029320: 2063 6f6e 7369 6465 7265 6420 3d20 3332   considered = 32
│ │ │ │ -00029330: 382c 2061 6e64 2063 6f6d 707c 0a7c 2d2d  8, and comp|.|--
│ │ │ │ -00029340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 7273  -----------|.|rs
│ │ │ │ -00029390: 2c20 7765 2077 696c 6c20 636f 6d70 7574  , we will comput
│ │ │ │ -000293a0: 6520 7570 2074 6f20 3332 372e 3539 3920  e up to 327.599 
│ │ │ │ -000293b0: 6f66 2074 6865 6d2e 2020 2020 2020 2020  of them.        
│ │ │ │ -000293c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000293d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -000293e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000293f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029300: 6f6e 3a20 2070 6172 7469 616c 2073 696e  on:  partial sin
│ │ │ │ +00029310: 6775 6c61 7220 6c6f 6375 7320 6469 6d65  gular locus dime
│ │ │ │ +00029320: 6e73 696f 6e20 636f 6d70 7574 6564 2c20  nsion computed, 
│ │ │ │ +00029330: 3d20 3120 2020 2020 2020 207c 0a7c 7265  = 1        |.|re
│ │ │ │ +00029340: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
│ │ │ │ +00029350: 6f6e 3a20 204c 6f6f 7020 636f 6d70 6c65  on:  Loop comple
│ │ │ │ +00029360: 7465 642c 2073 7562 6d61 7472 6963 6573  ted, submatrices
│ │ │ │ +00029370: 2063 6f6e 7369 6465 7265 6420 3d20 3332   considered = 32
│ │ │ │ +00029380: 382c 2061 6e64 2063 6f6d 707c 0a7c 2d2d  8, and comp|.|--
│ │ │ │ +00029390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000293a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000293b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000293c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000293d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 7273  -----------|.|rs
│ │ │ │ +000293e0: 2c20 7765 2077 696c 6c20 636f 6d70 7574  , we will comput
│ │ │ │ +000293f0: 6520 7570 2074 6f20 3332 372e 3539 3920  e up to 327.599 
│ │ │ │ +00029400: 6f66 2074 6865 6d2e 2020 2020 2020 2020  of them.        
│ │ │ │  00029410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029420: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00029430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029470: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ @@ -10601,22 +10601,22 @@
│ │ │ │  00029680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000296a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000296b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000296c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000296d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000296e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000296f0: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ -00029700: 6964 6572 6564 3a20 392c 2061 6e64 2063  idered: 9, and c
│ │ │ │ -00029710: 6f6d 7075 7465 6420 3d20 3920 2020 2020  omputed = 9     
│ │ │ │ +000296f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00029700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029740: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00029750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029740: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ +00029750: 6964 6572 6564 3a20 392c 2061 6e64 2063  idered: 9, and c
│ │ │ │ +00029760: 6f6d 7075 7465 6420 3d20 3920 2020 2020  omputed = 9     
│ │ │ │  00029770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029790: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000297a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000297b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000297c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000297d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -10626,22 +10626,22 @@
│ │ │ │  00029810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029830: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00029840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029880: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ -00029890: 6964 6572 6564 3a20 3131 2c20 616e 6420  idered: 11, and 
│ │ │ │ -000298a0: 636f 6d70 7574 6564 203d 2031 3020 2020  computed = 10   
│ │ │ │ +00029880: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00029890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000298a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000298b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000298c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000298d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -000298e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000298f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000298d0: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ +000298e0: 6964 6572 6564 3a20 3131 2c20 616e 6420  idered: 11, and 
│ │ │ │ +000298f0: 636f 6d70 7574 6564 203d 2031 3020 2020  computed = 10   
│ │ │ │  00029900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029920: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00029930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -10661,22 +10661,22 @@
│ │ │ │  00029a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029a60: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00029a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029ab0: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ -00029ac0: 6964 6572 6564 3a20 3135 2c20 616e 6420  idered: 15, and 
│ │ │ │ -00029ad0: 636f 6d70 7574 6564 203d 2031 3320 2020  computed = 13   
│ │ │ │ +00029ab0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00029ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029b00: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00029b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029b00: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ +00029b10: 6964 6572 6564 3a20 3135 2c20 616e 6420  idered: 15, and 
│ │ │ │ +00029b20: 636f 6d70 7574 6564 203d 2031 3320 2020  computed = 13   
│ │ │ │  00029b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029b50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00029b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -10706,22 +10706,22 @@
│ │ │ │  00029d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029d30: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00029d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029d80: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ -00029d90: 6964 6572 6564 3a20 3231 2c20 616e 6420  idered: 21, and 
│ │ │ │ -00029da0: 636f 6d70 7574 6564 203d 2031 3820 2020  computed = 18   
│ │ │ │ +00029d80: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00029d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029dd0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00029de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029dd0: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ +00029de0: 6964 6572 6564 3a20 3231 2c20 616e 6420  idered: 21, and 
│ │ │ │ +00029df0: 636f 6d70 7574 6564 203d 2031 3820 2020  computed = 18   
│ │ │ │  00029e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029e20: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00029e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -10756,22 +10756,22 @@
│ │ │ │  0002a030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a050: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0002a060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a0a0: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ -0002a0b0: 6964 6572 6564 3a20 3238 2c20 616e 6420  idered: 28, and 
│ │ │ │ -0002a0c0: 636f 6d70 7574 6564 203d 2032 3320 2020  computed = 23   
│ │ │ │ +0002a0a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0002a0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a0f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002a100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a0f0: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ +0002a100: 6964 6572 6564 3a20 3238 2c20 616e 6420  idered: 28, and 
│ │ │ │ +0002a110: 636f 6d70 7574 6564 203d 2032 3320 2020  computed = 23   
│ │ │ │  0002a120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a140: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0002a150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -10816,22 +10816,22 @@
│ │ │ │  0002a3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a410: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0002a420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a460: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ -0002a470: 6964 6572 6564 3a20 3337 2c20 616e 6420  idered: 37, and 
│ │ │ │ -0002a480: 636f 6d70 7574 6564 203d 2033 3020 2020  computed = 30   
│ │ │ │ +0002a460: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0002a470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a4b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002a4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a4b0: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ +0002a4c0: 6964 6572 6564 3a20 3337 2c20 616e 6420  idered: 37, and 
│ │ │ │ +0002a4d0: 636f 6d70 7574 6564 203d 2033 3020 2020  computed = 30   
│ │ │ │  0002a4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a500: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0002a510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -10891,22 +10891,22 @@
│ │ │ │  0002a8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a8c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0002a8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a910: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ -0002a920: 6964 6572 6564 3a20 3439 2c20 616e 6420  idered: 49, and 
│ │ │ │ -0002a930: 636f 6d70 7574 6564 203d 2033 3620 2020  computed = 36   
│ │ │ │ +0002a910: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0002a920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a960: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002a970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a960: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ +0002a970: 6964 6572 6564 3a20 3439 2c20 616e 6420  idered: 49, and 
│ │ │ │ +0002a980: 636f 6d70 7574 6564 203d 2033 3620 2020  computed = 36   
│ │ │ │  0002a990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a9b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0002a9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -10981,22 +10981,22 @@
│ │ │ │  0002ae40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ae50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ae60: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0002ae70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ae80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ae90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002aea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aeb0: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ -0002aec0: 6964 6572 6564 3a20 3634 2c20 616e 6420  idered: 64, and 
│ │ │ │ -0002aed0: 636f 6d70 7574 6564 203d 2034 3420 2020  computed = 44   
│ │ │ │ +0002aeb0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0002aec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002aed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002aee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002aef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002af00: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002af10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002af20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002af00: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ +0002af10: 6964 6572 6564 3a20 3634 2c20 616e 6420  idered: 64, and 
│ │ │ │ +0002af20: 636f 6d70 7574 6564 203d 2034 3420 2020  computed = 44   
│ │ │ │  0002af30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002af40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002af50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0002af60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002af70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002af80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002af90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -11096,22 +11096,22 @@
│ │ │ │  0002b570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b590: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0002b5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b5e0: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ -0002b5f0: 6964 6572 6564 3a20 3834 2c20 616e 6420  idered: 84, and 
│ │ │ │ -0002b600: 636f 6d70 7574 6564 203d 2035 3620 2020  computed = 56   
│ │ │ │ +0002b5e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0002b5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b630: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002b640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b630: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ +0002b640: 6964 6572 6564 3a20 3834 2c20 616e 6420  idered: 84, and 
│ │ │ │ +0002b650: 636f 6d70 7574 6564 203d 2035 3620 2020  computed = 56   
│ │ │ │  0002b660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b680: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0002b690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -11241,22 +11241,22 @@
│ │ │ │  0002be80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002be90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bea0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0002beb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bef0: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ -0002bf00: 6964 6572 6564 3a20 3131 302c 2061 6e64  idered: 110, and
│ │ │ │ -0002bf10: 2063 6f6d 7075 7465 6420 3d20 3639 2020   computed = 69  
│ │ │ │ +0002bef0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0002bf00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bf10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bf20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bf30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bf40: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002bf50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bf60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bf40: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ +0002bf50: 6964 6572 6564 3a20 3131 302c 2061 6e64  idered: 110, and
│ │ │ │ +0002bf60: 2063 6f6d 7075 7465 6420 3d20 3639 2020   computed = 69  
│ │ │ │  0002bf70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bf80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bf90: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0002bfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bfc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -11426,22 +11426,22 @@
│ │ │ │  0002ca10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ca20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ca30: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0002ca40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ca50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ca60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ca70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ca80: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ -0002ca90: 6964 6572 6564 3a20 3134 342c 2061 6e64  idered: 144, and
│ │ │ │ -0002caa0: 2063 6f6d 7075 7465 6420 3d20 3833 2020   computed = 83  
│ │ │ │ +0002ca80: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0002ca90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002caa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cad0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002cae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002caf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cad0: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ +0002cae0: 6964 6572 6564 3a20 3134 342c 2061 6e64  idered: 144, and
│ │ │ │ +0002caf0: 2063 6f6d 7075 7465 6420 3d20 3833 2020   computed = 83  
│ │ │ │  0002cb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cb20: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0002cb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -11661,22 +11661,22 @@
│ │ │ │  0002d8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d8e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0002d8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d930: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ -0002d940: 6964 6572 6564 3a20 3138 382c 2061 6e64  idered: 188, and
│ │ │ │ -0002d950: 2063 6f6d 7075 7465 6420 3d20 3130 3520   computed = 105 
│ │ │ │ +0002d930: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0002d940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d980: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002d990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d980: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ +0002d990: 6964 6572 6564 3a20 3138 382c 2061 6e64  idered: 188, and
│ │ │ │ +0002d9a0: 2063 6f6d 7075 7465 6420 3d20 3130 3520   computed = 105 
│ │ │ │  0002d9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d9d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0002d9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002da00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002da10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -11961,22 +11961,22 @@
│ │ │ │  0002eb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002eb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002eba0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0002ebb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ebc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ebd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ebe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ebf0: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ -0002ec00: 6964 6572 6564 3a20 3234 352c 2061 6e64  idered: 245, and
│ │ │ │ -0002ec10: 2063 6f6d 7075 7465 6420 3d20 3133 3520   computed = 135 
│ │ │ │ +0002ebf0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0002ec00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ec10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ec20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ec30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ec40: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002ec50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ec60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ec40: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ +0002ec50: 6964 6572 6564 3a20 3234 352c 2061 6e64  idered: 245, and
│ │ │ │ +0002ec60: 2063 6f6d 7075 7465 6420 3d20 3133 3520   computed = 135 
│ │ │ │  0002ec70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ec80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ec90: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0002eca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ecb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ecc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ecd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -12351,22 +12351,22 @@
│ │ │ │  000303e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000303f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030400: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00030410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030450: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ -00030460: 6964 6572 6564 3a20 3331 392c 2061 6e64  idered: 319, and
│ │ │ │ -00030470: 2063 6f6d 7075 7465 6420 3d20 3137 3520   computed = 175 
│ │ │ │ +00030450: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00030460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000304a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -000304b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000304c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000304a0: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ +000304b0: 6964 6572 6564 3a20 3331 392c 2061 6e64  idered: 319, and
│ │ │ │ +000304c0: 2063 6f6d 7075 7465 6420 3d20 3137 3520   computed = 175 
│ │ │ │  000304d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000304e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000304f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00030500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -12411,357 +12411,357 @@
│ │ │ │  000307a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000307b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000307c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000307d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000307e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000307f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030810: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ -00030820: 6964 6572 6564 3a20 3332 382c 2061 6e64  idered: 328, and
│ │ │ │ -00030830: 2063 6f6d 7075 7465 6420 3d20 3138 3020   computed = 180 
│ │ │ │ +00030810: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00030820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030860: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00030870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030860: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ +00030870: 6964 6572 6564 3a20 3332 382c 2061 6e64  idered: 328, and
│ │ │ │ +00030880: 2063 6f6d 7075 7465 6420 3d20 3138 3020   computed = 180 
│ │ │ │  00030890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000308a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000308b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000308c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000308d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000308e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000308f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030900: 2020 2020 2020 2020 2020 207c 0a7c 7574             |.|ut
│ │ │ │ -00030910: 6564 203d 2031 3830 2e20 2073 696e 6775  ed = 180.  singu
│ │ │ │ -00030920: 6c61 7220 6c6f 6375 7320 6469 6d65 6e73  lar locus dimens
│ │ │ │ -00030930: 696f 6e20 6170 7065 6172 7320 746f 2062  ion appears to b
│ │ │ │ -00030940: 6520 3d20 3120 2020 2020 2020 2020 2020  e = 1           
│ │ │ │ -00030950: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -00030960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00030970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00030980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00030990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000309a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5468  -----------+..Th
│ │ │ │ -000309b0: 6520 6d61 7869 6d75 6d20 6e75 6d62 6572  e maximum number
│ │ │ │ -000309c0: 206f 6620 6d69 6e6f 7273 2063 6f6e 7369   of minors consi
│ │ │ │ -000309d0: 6465 7265 6420 6361 6e20 6265 2063 6f6e  dered can be con
│ │ │ │ -000309e0: 7472 6f6c 6c65 6420 6279 2074 6865 206f  trolled by the o
│ │ │ │ -000309f0: 7074 696f 6e0a 4d61 784d 696e 6f72 732e  ption.MaxMinors.
│ │ │ │ -00030a00: 2020 416c 7465 726e 6174 6976 656c 792c    Alternatively,
│ │ │ │ -00030a10: 2069 7420 6361 6e20 6265 2063 6f6e 7472   it can be contr
│ │ │ │ -00030a20: 6f6c 6c65 6420 696e 2061 206d 6f72 6520  olled in a more 
│ │ │ │ -00030a30: 7072 6563 6973 6520 7761 7920 6279 0a70  precise way by.p
│ │ │ │ -00030a40: 6173 7369 6e67 2061 2066 756e 6374 696f  assing a functio
│ │ │ │ -00030a50: 6e20 746f 2074 6865 206f 7074 696f 6e20  n to the option 
│ │ │ │ -00030a60: 4d61 784d 696e 6f72 732e 2020 5468 6973  MaxMinors.  This
│ │ │ │ -00030a70: 2066 756e 6374 696f 6e20 7368 6f75 6c64   function should
│ │ │ │ -00030a80: 2068 6176 6520 7477 6f0a 696e 7075 7473   have two.inputs
│ │ │ │ -00030a90: 3b20 7468 6520 6669 7273 7420 6973 206d  ; the first is m
│ │ │ │ -00030aa0: 696e 696d 756d 206e 756d 6265 7220 6f66  inimum number of
│ │ │ │ -00030ab0: 206d 696e 6f72 7320 6e65 6564 6564 2074   minors needed t
│ │ │ │ -00030ac0: 6f20 6465 7465 726d 696e 6520 7768 6574  o determine whet
│ │ │ │ -00030ad0: 6865 7220 7468 650a 7269 6e67 2069 7320  her the.ring is 
│ │ │ │ -00030ae0: 7265 6775 6c61 7220 696e 2063 6f64 696d  regular in codim
│ │ │ │ -00030af0: 656e 7369 6f6e 206e 2c20 616e 6420 7468  ension n, and th
│ │ │ │ -00030b00: 6520 7365 636f 6e64 2069 7320 7468 6520  e second is the 
│ │ │ │ -00030b10: 746f 7461 6c20 6e75 6d62 6572 206f 6620  total number of 
│ │ │ │ -00030b20: 6d69 6e6f 7273 0a61 7661 696c 6162 6c65  minors.available
│ │ │ │ -00030b30: 2069 6e20 7468 6520 4a61 636f 6269 616e   in the Jacobian
│ │ │ │ -00030b40: 2e20 5468 6520 6675 6e63 7469 6f6e 2072  . The function r
│ │ │ │ -00030b50: 6567 756c 6172 496e 436f 6469 6d65 6e73  egularInCodimens
│ │ │ │ -00030b60: 696f 6e20 646f 6573 206e 6f74 2072 6563  ion does not rec
│ │ │ │ -00030b70: 6f6d 7075 7465 0a64 6574 6572 6d69 6e61  ompute.determina
│ │ │ │ -00030b80: 6e74 732c 2073 6f20 4d61 784d 696e 6f72  nts, so MaxMinor
│ │ │ │ -00030b90: 7320 6f72 2069 7320 6f6e 6c79 2061 6e20  s or is only an 
│ │ │ │ -00030ba0: 7570 7065 7220 626f 756e 6420 6f6e 2074  upper bound on t
│ │ │ │ -00030bb0: 6865 206e 756d 6265 7220 6f66 206d 696e  he number of min
│ │ │ │ -00030bc0: 6f72 730a 636f 6d70 7574 6564 2e0a 0a2b  ors.computed...+
│ │ │ │ -00030bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00030be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00030bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00030c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00030c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00030c20: 6931 3720 3a20 7469 6d65 2072 6567 756c  i17 : time regul
│ │ │ │ -00030c30: 6172 496e 436f 6469 6d65 6e73 696f 6e28  arInCodimension(
│ │ │ │ -00030c40: 322c 2053 2c20 5665 7262 6f73 653d 3e74  2, S, Verbose=>t
│ │ │ │ -00030c50: 7275 652c 204d 6178 4d69 6e6f 7273 3d3e  rue, MaxMinors=>
│ │ │ │ -00030c60: 3330 2920 2020 2020 2020 2020 207c 0a7c  30)          |.|
│ │ │ │ -00030c70: 202d 2d20 7573 6564 2031 2e32 3930 3232   -- used 1.29022
│ │ │ │ -00030c80: 7320 2863 7075 293b 2030 2e39 3936 3437  s (cpu); 0.99647
│ │ │ │ -00030c90: 3673 2028 7468 7265 6164 293b 2030 7320  6s (thread); 0s 
│ │ │ │ -00030ca0: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │ -00030cb0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00030cc0: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ -00030cd0: 7369 6f6e 3a20 7269 6e67 2064 696d 656e  sion: ring dimen
│ │ │ │ -00030ce0: 7369 6f6e 203d 332c 2074 6865 7265 2061  sion =3, there a
│ │ │ │ -00030cf0: 7265 2031 3733 3235 2070 6f73 7369 626c  re 17325 possibl
│ │ │ │ -00030d00: 6520 3420 6279 2034 206d 696e 6f7c 0a7c  e 4 by 4 mino|.|
│ │ │ │ +00030900: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00030910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030950: 2020 2020 2020 2020 2020 207c 0a7c 7574             |.|ut
│ │ │ │ +00030960: 6564 203d 2031 3830 2e20 2073 696e 6775  ed = 180.  singu
│ │ │ │ +00030970: 6c61 7220 6c6f 6375 7320 6469 6d65 6e73  lar locus dimens
│ │ │ │ +00030980: 696f 6e20 6170 7065 6172 7320 746f 2062  ion appears to b
│ │ │ │ +00030990: 6520 3d20 3120 2020 2020 2020 2020 2020  e = 1           
│ │ │ │ +000309a0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +000309b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000309c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000309d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000309e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000309f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5468  -----------+..Th
│ │ │ │ +00030a00: 6520 6d61 7869 6d75 6d20 6e75 6d62 6572  e maximum number
│ │ │ │ +00030a10: 206f 6620 6d69 6e6f 7273 2063 6f6e 7369   of minors consi
│ │ │ │ +00030a20: 6465 7265 6420 6361 6e20 6265 2063 6f6e  dered can be con
│ │ │ │ +00030a30: 7472 6f6c 6c65 6420 6279 2074 6865 206f  trolled by the o
│ │ │ │ +00030a40: 7074 696f 6e0a 4d61 784d 696e 6f72 732e  ption.MaxMinors.
│ │ │ │ +00030a50: 2020 416c 7465 726e 6174 6976 656c 792c    Alternatively,
│ │ │ │ +00030a60: 2069 7420 6361 6e20 6265 2063 6f6e 7472   it can be contr
│ │ │ │ +00030a70: 6f6c 6c65 6420 696e 2061 206d 6f72 6520  olled in a more 
│ │ │ │ +00030a80: 7072 6563 6973 6520 7761 7920 6279 0a70  precise way by.p
│ │ │ │ +00030a90: 6173 7369 6e67 2061 2066 756e 6374 696f  assing a functio
│ │ │ │ +00030aa0: 6e20 746f 2074 6865 206f 7074 696f 6e20  n to the option 
│ │ │ │ +00030ab0: 4d61 784d 696e 6f72 732e 2020 5468 6973  MaxMinors.  This
│ │ │ │ +00030ac0: 2066 756e 6374 696f 6e20 7368 6f75 6c64   function should
│ │ │ │ +00030ad0: 2068 6176 6520 7477 6f0a 696e 7075 7473   have two.inputs
│ │ │ │ +00030ae0: 3b20 7468 6520 6669 7273 7420 6973 206d  ; the first is m
│ │ │ │ +00030af0: 696e 696d 756d 206e 756d 6265 7220 6f66  inimum number of
│ │ │ │ +00030b00: 206d 696e 6f72 7320 6e65 6564 6564 2074   minors needed t
│ │ │ │ +00030b10: 6f20 6465 7465 726d 696e 6520 7768 6574  o determine whet
│ │ │ │ +00030b20: 6865 7220 7468 650a 7269 6e67 2069 7320  her the.ring is 
│ │ │ │ +00030b30: 7265 6775 6c61 7220 696e 2063 6f64 696d  regular in codim
│ │ │ │ +00030b40: 656e 7369 6f6e 206e 2c20 616e 6420 7468  ension n, and th
│ │ │ │ +00030b50: 6520 7365 636f 6e64 2069 7320 7468 6520  e second is the 
│ │ │ │ +00030b60: 746f 7461 6c20 6e75 6d62 6572 206f 6620  total number of 
│ │ │ │ +00030b70: 6d69 6e6f 7273 0a61 7661 696c 6162 6c65  minors.available
│ │ │ │ +00030b80: 2069 6e20 7468 6520 4a61 636f 6269 616e   in the Jacobian
│ │ │ │ +00030b90: 2e20 5468 6520 6675 6e63 7469 6f6e 2072  . The function r
│ │ │ │ +00030ba0: 6567 756c 6172 496e 436f 6469 6d65 6e73  egularInCodimens
│ │ │ │ +00030bb0: 696f 6e20 646f 6573 206e 6f74 2072 6563  ion does not rec
│ │ │ │ +00030bc0: 6f6d 7075 7465 0a64 6574 6572 6d69 6e61  ompute.determina
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│ │ │ │ +00030be0: 7320 6f72 2069 7320 6f6e 6c79 2061 6e20  s or is only an 
│ │ │ │ +00030bf0: 7570 7065 7220 626f 756e 6420 6f6e 2074  upper bound on t
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│ │ │ │ +00030c10: 6f72 730a 636f 6d70 7574 6564 2e0a 0a2b  ors.computed...+
│ │ │ │ +00030c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00030c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00030c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00030c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00030c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00030c70: 6931 3720 3a20 7469 6d65 2072 6567 756c  i17 : time regul
│ │ │ │ +00030c80: 6172 496e 436f 6469 6d65 6e73 696f 6e28  arInCodimension(
│ │ │ │ +00030c90: 322c 2053 2c20 5665 7262 6f73 653d 3e74  2, S, Verbose=>t
│ │ │ │ +00030ca0: 7275 652c 204d 6178 4d69 6e6f 7273 3d3e  rue, MaxMinors=>
│ │ │ │ +00030cb0: 3330 2920 2020 2020 2020 2020 207c 0a7c  30)          |.|
│ │ │ │ +00030cc0: 202d 2d20 7573 6564 2031 2e35 3439 3635   -- used 1.54965
│ │ │ │ +00030cd0: 7320 2863 7075 293b 2031 2e31 3534 3331  s (cpu); 1.15431
│ │ │ │ +00030ce0: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ +00030cf0: 6763 2920 2020 2020 2020 2020 2020 2020  gc)             
│ │ │ │ +00030d00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00030d10: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ -00030d20: 7369 6f6e 3a20 4162 6f75 7420 746f 2065  sion: About to e
│ │ │ │ -00030d30: 6e74 6572 206c 6f6f 7020 2020 2020 2020  nter loop       
│ │ │ │ -00030d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030d50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00030d60: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ -00030d70: 6e6f 723a 2043 686f 6f73 696e 6720 4c65  nor: Choosing Le
│ │ │ │ -00030d80: 7853 6d61 6c6c 6573 7454 6572 6d20 2020  xSmallestTerm   
│ │ │ │ +00030d20: 7369 6f6e 3a20 7269 6e67 2064 696d 656e  sion: ring dimen
│ │ │ │ +00030d30: 7369 6f6e 203d 332c 2074 6865 7265 2061  sion =3, there a
│ │ │ │ +00030d40: 7265 2031 3733 3235 2070 6f73 7369 626c  re 17325 possibl
│ │ │ │ +00030d50: 6520 3420 6279 2034 206d 696e 6f7c 0a7c  e 4 by 4 mino|.|
│ │ │ │ +00030d60: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ +00030d70: 7369 6f6e 3a20 4162 6f75 7420 746f 2065  sion: About to e
│ │ │ │ +00030d80: 6e74 6572 206c 6f6f 7020 2020 2020 2020  nter loop       
│ │ │ │  00030d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030da0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00030db0: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │  00030dc0: 6e6f 723a 2043 686f 6f73 696e 6720 4c65  nor: Choosing Le
│ │ │ │  00030dd0: 7853 6d61 6c6c 6573 7454 6572 6d20 2020  xSmallestTerm   
│ │ │ │  00030de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030df0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00030e00: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ -00030e10: 6e6f 723a 2043 686f 6f73 696e 6720 4752  nor: Choosing GR
│ │ │ │ -00030e20: 6576 4c65 7853 6d61 6c6c 6573 7420 2020  evLexSmallest   
│ │ │ │ +00030e10: 6e6f 723a 2043 686f 6f73 696e 6720 4c65  nor: Choosing Le
│ │ │ │ +00030e20: 7853 6d61 6c6c 6573 7454 6572 6d20 2020  xSmallestTerm   
│ │ │ │  00030e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030e40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00030e50: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │  00030e60: 6e6f 723a 2043 686f 6f73 696e 6720 4752  nor: Choosing GR
│ │ │ │  00030e70: 6576 4c65 7853 6d61 6c6c 6573 7420 2020  evLexSmallest   
│ │ │ │  00030e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030e90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00030ea0: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ -00030eb0: 6e6f 723a 2043 686f 6f73 696e 6720 4c65  nor: Choosing Le
│ │ │ │ -00030ec0: 7853 6d61 6c6c 6573 7454 6572 6d20 2020  xSmallestTerm   
│ │ │ │ +00030eb0: 6e6f 723a 2043 686f 6f73 696e 6720 4752  nor: Choosing GR
│ │ │ │ +00030ec0: 6576 4c65 7853 6d61 6c6c 6573 7420 2020  evLexSmallest   
│ │ │ │  00030ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030ee0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00030ef0: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ -00030f00: 6e6f 723a 2043 686f 6f73 696e 6720 5261  nor: Choosing Ra
│ │ │ │ -00030f10: 6e64 6f6d 4e6f 6e5a 6572 6f20 2020 2020  ndomNonZero     
│ │ │ │ +00030f00: 6e6f 723a 2043 686f 6f73 696e 6720 4c65  nor: Choosing Le
│ │ │ │ +00030f10: 7853 6d61 6c6c 6573 7454 6572 6d20 2020  xSmallestTerm   
│ │ │ │  00030f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030f30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00030f40: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ -00030f50: 6e6f 723a 2043 686f 6f73 696e 6720 4c65  nor: Choosing Le
│ │ │ │ -00030f60: 7853 6d61 6c6c 6573 7420 2020 2020 2020  xSmallest       
│ │ │ │ +00030f50: 6e6f 723a 2043 686f 6f73 696e 6720 5261  nor: Choosing Ra
│ │ │ │ +00030f60: 6e64 6f6d 4e6f 6e5a 6572 6f20 2020 2020  ndomNonZero     
│ │ │ │  00030f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030f80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00030f90: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ -00030fa0: 6e6f 723a 2043 686f 6f73 696e 6720 5261  nor: Choosing Ra
│ │ │ │ -00030fb0: 6e64 6f6d 2020 2020 2020 2020 2020 2020  ndom            
│ │ │ │ +00030fa0: 6e6f 723a 2043 686f 6f73 696e 6720 4c65  nor: Choosing Le
│ │ │ │ +00030fb0: 7853 6d61 6c6c 6573 7420 2020 2020 2020  xSmallest       
│ │ │ │  00030fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030fd0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00030fe0: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │  00030ff0: 6e6f 723a 2043 686f 6f73 696e 6720 5261  nor: Choosing Ra
│ │ │ │ -00031000: 6e64 6f6d 4e6f 6e5a 6572 6f20 2020 2020  ndomNonZero     
│ │ │ │ +00031000: 6e64 6f6d 2020 2020 2020 2020 2020 2020  ndom            
│ │ │ │  00031010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031020: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00031030: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ -00031040: 7369 6f6e 3a20 204c 6f6f 7020 7374 6570  sion:  Loop step
│ │ │ │ -00031050: 2c20 6162 6f75 7420 746f 2063 6f6d 7075  , about to compu
│ │ │ │ -00031060: 7465 2064 696d 656e 7369 6f6e 2e20 2053  te dimension.  S
│ │ │ │ -00031070: 7562 6d61 7472 6963 6573 2063 6f7c 0a7c  ubmatrices co|.|
│ │ │ │ +00031030: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ +00031040: 6e6f 723a 2043 686f 6f73 696e 6720 5261  nor: Choosing Ra
│ │ │ │ +00031050: 6e64 6f6d 4e6f 6e5a 6572 6f20 2020 2020  ndomNonZero     
│ │ │ │ +00031060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031070: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00031080: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ -00031090: 7369 6f6e 3a20 2069 7343 6f64 696d 4174  sion:  isCodimAt
│ │ │ │ -000310a0: 4c65 6173 7420 6661 696c 6564 2c20 636f  Least failed, co
│ │ │ │ -000310b0: 6d70 7574 696e 6720 636f 6469 6d2e 2020  mputing codim.  
│ │ │ │ -000310c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031090: 7369 6f6e 3a20 204c 6f6f 7020 7374 6570  sion:  Loop step
│ │ │ │ +000310a0: 2c20 6162 6f75 7420 746f 2063 6f6d 7075  , about to compu
│ │ │ │ +000310b0: 7465 2064 696d 656e 7369 6f6e 2e20 2053  te dimension.  S
│ │ │ │ +000310c0: 7562 6d61 7472 6963 6573 2063 6f7c 0a7c  ubmatrices co|.|
│ │ │ │  000310d0: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ -000310e0: 7369 6f6e 3a20 2070 6172 7469 616c 2073  sion:  partial s
│ │ │ │ -000310f0: 696e 6775 6c61 7220 6c6f 6375 7320 6469  ingular locus di
│ │ │ │ -00031100: 6d65 6e73 696f 6e20 636f 6d70 7574 6564  mension computed
│ │ │ │ -00031110: 2c20 3d20 3220 2020 2020 2020 207c 0a7c  , = 2        |.|
│ │ │ │ -00031120: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ -00031130: 6e6f 723a 2043 686f 6f73 696e 6720 5261  nor: Choosing Ra
│ │ │ │ -00031140: 6e64 6f6d 2020 2020 2020 2020 2020 2020  ndom            
│ │ │ │ -00031150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031160: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000310e0: 7369 6f6e 3a20 2069 7343 6f64 696d 4174  sion:  isCodimAt
│ │ │ │ +000310f0: 4c65 6173 7420 6661 696c 6564 2c20 636f  Least failed, co
│ │ │ │ +00031100: 6d70 7574 696e 6720 636f 6469 6d2e 2020  mputing codim.  
│ │ │ │ +00031110: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031120: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ +00031130: 7369 6f6e 3a20 2070 6172 7469 616c 2073  sion:  partial s
│ │ │ │ +00031140: 696e 6775 6c61 7220 6c6f 6375 7320 6469  ingular locus di
│ │ │ │ +00031150: 6d65 6e73 696f 6e20 636f 6d70 7574 6564  mension computed
│ │ │ │ +00031160: 2c20 3d20 3220 2020 2020 2020 207c 0a7c  , = 2        |.|
│ │ │ │  00031170: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ -00031180: 6e6f 723a 2043 686f 6f73 696e 6720 4752  nor: Choosing GR
│ │ │ │ -00031190: 6576 4c65 7853 6d61 6c6c 6573 7420 2020  evLexSmallest   
│ │ │ │ +00031180: 6e6f 723a 2043 686f 6f73 696e 6720 5261  nor: Choosing Ra
│ │ │ │ +00031190: 6e64 6f6d 2020 2020 2020 2020 2020 2020  ndom            
│ │ │ │  000311a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000311b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000311c0: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ -000311d0: 7369 6f6e 3a20 204c 6f6f 7020 7374 6570  sion:  Loop step
│ │ │ │ -000311e0: 2c20 6162 6f75 7420 746f 2063 6f6d 7075  , about to compu
│ │ │ │ -000311f0: 7465 2064 696d 656e 7369 6f6e 2e20 2053  te dimension.  S
│ │ │ │ -00031200: 7562 6d61 7472 6963 6573 2063 6f7c 0a7c  ubmatrices co|.|
│ │ │ │ +000311c0: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ +000311d0: 6e6f 723a 2043 686f 6f73 696e 6720 4752  nor: Choosing GR
│ │ │ │ +000311e0: 6576 4c65 7853 6d61 6c6c 6573 7420 2020  evLexSmallest   
│ │ │ │ +000311f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031200: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00031210: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ -00031220: 7369 6f6e 3a20 2069 7343 6f64 696d 4174  sion:  isCodimAt
│ │ │ │ -00031230: 4c65 6173 7420 6661 696c 6564 2c20 636f  Least failed, co
│ │ │ │ -00031240: 6d70 7574 696e 6720 636f 6469 6d2e 2020  mputing codim.  
│ │ │ │ -00031250: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031220: 7369 6f6e 3a20 204c 6f6f 7020 7374 6570  sion:  Loop step
│ │ │ │ +00031230: 2c20 6162 6f75 7420 746f 2063 6f6d 7075  , about to compu
│ │ │ │ +00031240: 7465 2064 696d 656e 7369 6f6e 2e20 2053  te dimension.  S
│ │ │ │ +00031250: 7562 6d61 7472 6963 6573 2063 6f7c 0a7c  ubmatrices co|.|
│ │ │ │  00031260: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ -00031270: 7369 6f6e 3a20 2070 6172 7469 616c 2073  sion:  partial s
│ │ │ │ -00031280: 696e 6775 6c61 7220 6c6f 6375 7320 6469  ingular locus di
│ │ │ │ -00031290: 6d65 6e73 696f 6e20 636f 6d70 7574 6564  mension computed
│ │ │ │ -000312a0: 2c20 3d20 3220 2020 2020 2020 207c 0a7c  , = 2        |.|
│ │ │ │ -000312b0: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ -000312c0: 6e6f 723a 2043 686f 6f73 696e 6720 4c65  nor: Choosing Le
│ │ │ │ -000312d0: 7853 6d61 6c6c 6573 7454 6572 6d20 2020  xSmallestTerm   
│ │ │ │ -000312e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000312f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031270: 7369 6f6e 3a20 2069 7343 6f64 696d 4174  sion:  isCodimAt
│ │ │ │ +00031280: 4c65 6173 7420 6661 696c 6564 2c20 636f  Least failed, co
│ │ │ │ +00031290: 6d70 7574 696e 6720 636f 6469 6d2e 2020  mputing codim.  
│ │ │ │ +000312a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000312b0: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ +000312c0: 7369 6f6e 3a20 2070 6172 7469 616c 2073  sion:  partial s
│ │ │ │ +000312d0: 696e 6775 6c61 7220 6c6f 6375 7320 6469  ingular locus di
│ │ │ │ +000312e0: 6d65 6e73 696f 6e20 636f 6d70 7574 6564  mension computed
│ │ │ │ +000312f0: 2c20 3d20 3220 2020 2020 2020 207c 0a7c  , = 2        |.|
│ │ │ │  00031300: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │  00031310: 6e6f 723a 2043 686f 6f73 696e 6720 4c65  nor: Choosing Le
│ │ │ │ -00031320: 7853 6d61 6c6c 6573 7420 2020 2020 2020  xSmallest       
│ │ │ │ +00031320: 7853 6d61 6c6c 6573 7454 6572 6d20 2020  xSmallestTerm   
│ │ │ │  00031330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031340: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00031350: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │  00031360: 6e6f 723a 2043 686f 6f73 696e 6720 4c65  nor: Choosing Le
│ │ │ │  00031370: 7853 6d61 6c6c 6573 7420 2020 2020 2020  xSmallest       
│ │ │ │  00031380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031390: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  000313a0: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ -000313b0: 6e6f 723a 2043 686f 6f73 696e 6720 5261  nor: Choosing Ra
│ │ │ │ -000313c0: 6e64 6f6d 4e6f 6e5a 6572 6f20 2020 2020  ndomNonZero     
│ │ │ │ +000313b0: 6e6f 723a 2043 686f 6f73 696e 6720 4c65  nor: Choosing Le
│ │ │ │ +000313c0: 7853 6d61 6c6c 6573 7420 2020 2020 2020  xSmallest       
│ │ │ │  000313d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000313e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000313f0: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ -00031400: 7369 6f6e 3a20 204c 6f6f 7020 7374 6570  sion:  Loop step
│ │ │ │ -00031410: 2c20 6162 6f75 7420 746f 2063 6f6d 7075  , about to compu
│ │ │ │ -00031420: 7465 2064 696d 656e 7369 6f6e 2e20 2053  te dimension.  S
│ │ │ │ -00031430: 7562 6d61 7472 6963 6573 2063 6f7c 0a7c  ubmatrices co|.|
│ │ │ │ +000313f0: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ +00031400: 6e6f 723a 2043 686f 6f73 696e 6720 5261  nor: Choosing Ra
│ │ │ │ +00031410: 6e64 6f6d 4e6f 6e5a 6572 6f20 2020 2020  ndomNonZero     
│ │ │ │ +00031420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031430: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00031440: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ -00031450: 7369 6f6e 3a20 2069 7343 6f64 696d 4174  sion:  isCodimAt
│ │ │ │ -00031460: 4c65 6173 7420 6661 696c 6564 2c20 636f  Least failed, co
│ │ │ │ -00031470: 6d70 7574 696e 6720 636f 6469 6d2e 2020  mputing codim.  
│ │ │ │ -00031480: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031450: 7369 6f6e 3a20 204c 6f6f 7020 7374 6570  sion:  Loop step
│ │ │ │ +00031460: 2c20 6162 6f75 7420 746f 2063 6f6d 7075  , about to compu
│ │ │ │ +00031470: 7465 2064 696d 656e 7369 6f6e 2e20 2053  te dimension.  S
│ │ │ │ +00031480: 7562 6d61 7472 6963 6573 2063 6f7c 0a7c  ubmatrices co|.|
│ │ │ │  00031490: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ -000314a0: 7369 6f6e 3a20 2070 6172 7469 616c 2073  sion:  partial s
│ │ │ │ -000314b0: 696e 6775 6c61 7220 6c6f 6375 7320 6469  ingular locus di
│ │ │ │ -000314c0: 6d65 6e73 696f 6e20 636f 6d70 7574 6564  mension computed
│ │ │ │ -000314d0: 2c20 3d20 3120 2020 2020 2020 207c 0a7c  , = 1        |.|
│ │ │ │ -000314e0: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ -000314f0: 6e6f 723a 2043 686f 6f73 696e 6720 4752  nor: Choosing GR
│ │ │ │ -00031500: 6576 4c65 7853 6d61 6c6c 6573 7454 6572  evLexSmallestTer
│ │ │ │ -00031510: 6d20 2020 2020 2020 2020 2020 2020 2020  m               
│ │ │ │ -00031520: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000314a0: 7369 6f6e 3a20 2069 7343 6f64 696d 4174  sion:  isCodimAt
│ │ │ │ +000314b0: 4c65 6173 7420 6661 696c 6564 2c20 636f  Least failed, co
│ │ │ │ +000314c0: 6d70 7574 696e 6720 636f 6469 6d2e 2020  mputing codim.  
│ │ │ │ +000314d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000314e0: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ +000314f0: 7369 6f6e 3a20 2070 6172 7469 616c 2073  sion:  partial s
│ │ │ │ +00031500: 696e 6775 6c61 7220 6c6f 6375 7320 6469  ingular locus di
│ │ │ │ +00031510: 6d65 6e73 696f 6e20 636f 6d70 7574 6564  mension computed
│ │ │ │ +00031520: 2c20 3d20 3120 2020 2020 2020 207c 0a7c  , = 1        |.|
│ │ │ │  00031530: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ -00031540: 6e6f 723a 2043 686f 6f73 696e 6720 4c65  nor: Choosing Le
│ │ │ │ -00031550: 7853 6d61 6c6c 6573 7454 6572 6d20 2020  xSmallestTerm   
│ │ │ │ -00031560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031540: 6e6f 723a 2043 686f 6f73 696e 6720 4752  nor: Choosing GR
│ │ │ │ +00031550: 6576 4c65 7853 6d61 6c6c 6573 7454 6572  evLexSmallestTer
│ │ │ │ +00031560: 6d20 2020 2020 2020 2020 2020 2020 2020  m               
│ │ │ │  00031570: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00031580: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ -00031590: 6e6f 723a 2043 686f 6f73 696e 6720 5261  nor: Choosing Ra
│ │ │ │ -000315a0: 6e64 6f6d 4e6f 6e5a 6572 6f20 2020 2020  ndomNonZero     
│ │ │ │ +00031590: 6e6f 723a 2043 686f 6f73 696e 6720 4c65  nor: Choosing Le
│ │ │ │ +000315a0: 7853 6d61 6c6c 6573 7454 6572 6d20 2020  xSmallestTerm   
│ │ │ │  000315b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000315c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  000315d0: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ -000315e0: 6e6f 723a 2043 686f 6f73 696e 6720 4c65  nor: Choosing Le
│ │ │ │ -000315f0: 7853 6d61 6c6c 6573 7454 6572 6d20 2020  xSmallestTerm   
│ │ │ │ +000315e0: 6e6f 723a 2043 686f 6f73 696e 6720 5261  nor: Choosing Ra
│ │ │ │ +000315f0: 6e64 6f6d 4e6f 6e5a 6572 6f20 2020 2020  ndomNonZero     
│ │ │ │  00031600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031610: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00031620: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │  00031630: 6e6f 723a 2043 686f 6f73 696e 6720 4c65  nor: Choosing Le
│ │ │ │ -00031640: 7853 6d61 6c6c 6573 7420 2020 2020 2020  xSmallest       
│ │ │ │ +00031640: 7853 6d61 6c6c 6573 7454 6572 6d20 2020  xSmallestTerm   
│ │ │ │  00031650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031660: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00031670: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ -00031680: 6e6f 723a 2043 686f 6f73 696e 6720 5261  nor: Choosing Ra
│ │ │ │ -00031690: 6e64 6f6d 4e6f 6e5a 6572 6f20 2020 2020  ndomNonZero     
│ │ │ │ +00031680: 6e6f 723a 2043 686f 6f73 696e 6720 4c65  nor: Choosing Le
│ │ │ │ +00031690: 7853 6d61 6c6c 6573 7420 2020 2020 2020  xSmallest       
│ │ │ │  000316a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000316b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000316c0: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ -000316d0: 7369 6f6e 3a20 204c 6f6f 7020 7374 6570  sion:  Loop step
│ │ │ │ -000316e0: 2c20 6162 6f75 7420 746f 2063 6f6d 7075  , about to compu
│ │ │ │ -000316f0: 7465 2064 696d 656e 7369 6f6e 2e20 2053  te dimension.  S
│ │ │ │ -00031700: 7562 6d61 7472 6963 6573 2063 6f7c 0a7c  ubmatrices co|.|
│ │ │ │ +000316c0: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ +000316d0: 6e6f 723a 2043 686f 6f73 696e 6720 5261  nor: Choosing Ra
│ │ │ │ +000316e0: 6e64 6f6d 4e6f 6e5a 6572 6f20 2020 2020  ndomNonZero     
│ │ │ │ +000316f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031700: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00031710: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ -00031720: 7369 6f6e 3a20 2069 7343 6f64 696d 4174  sion:  isCodimAt
│ │ │ │ -00031730: 4c65 6173 7420 6661 696c 6564 2c20 636f  Least failed, co
│ │ │ │ -00031740: 6d70 7574 696e 6720 636f 6469 6d2e 2020  mputing codim.  
│ │ │ │ -00031750: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031720: 7369 6f6e 3a20 204c 6f6f 7020 7374 6570  sion:  Loop step
│ │ │ │ +00031730: 2c20 6162 6f75 7420 746f 2063 6f6d 7075  , about to compu
│ │ │ │ +00031740: 7465 2064 696d 656e 7369 6f6e 2e20 2053  te dimension.  S
│ │ │ │ +00031750: 7562 6d61 7472 6963 6573 2063 6f7c 0a7c  ubmatrices co|.|
│ │ │ │  00031760: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ -00031770: 7369 6f6e 3a20 2070 6172 7469 616c 2073  sion:  partial s
│ │ │ │ -00031780: 696e 6775 6c61 7220 6c6f 6375 7320 6469  ingular locus di
│ │ │ │ -00031790: 6d65 6e73 696f 6e20 636f 6d70 7574 6564  mension computed
│ │ │ │ -000317a0: 2c20 3d20 3120 2020 2020 2020 207c 0a7c  , = 1        |.|
│ │ │ │ -000317b0: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ -000317c0: 6e6f 723a 2043 686f 6f73 696e 6720 5261  nor: Choosing Ra
│ │ │ │ -000317d0: 6e64 6f6d 4e6f 6e5a 6572 6f20 2020 2020  ndomNonZero     
│ │ │ │ -000317e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000317f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031770: 7369 6f6e 3a20 2069 7343 6f64 696d 4174  sion:  isCodimAt
│ │ │ │ +00031780: 4c65 6173 7420 6661 696c 6564 2c20 636f  Least failed, co
│ │ │ │ +00031790: 6d70 7574 696e 6720 636f 6469 6d2e 2020  mputing codim.  
│ │ │ │ +000317a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000317b0: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ +000317c0: 7369 6f6e 3a20 2070 6172 7469 616c 2073  sion:  partial s
│ │ │ │ +000317d0: 696e 6775 6c61 7220 6c6f 6375 7320 6469  ingular locus di
│ │ │ │ +000317e0: 6d65 6e73 696f 6e20 636f 6d70 7574 6564  mension computed
│ │ │ │ +000317f0: 2c20 3d20 3120 2020 2020 2020 207c 0a7c  , = 1        |.|
│ │ │ │  00031800: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ -00031810: 6e6f 723a 2043 686f 6f73 696e 6720 4c65  nor: Choosing Le
│ │ │ │ -00031820: 7853 6d61 6c6c 6573 7420 2020 2020 2020  xSmallest       
│ │ │ │ +00031810: 6e6f 723a 2043 686f 6f73 696e 6720 5261  nor: Choosing Ra
│ │ │ │ +00031820: 6e64 6f6d 4e6f 6e5a 6572 6f20 2020 2020  ndomNonZero     
│ │ │ │  00031830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031840: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00031850: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ -00031860: 6e6f 723a 2043 686f 6f73 696e 6720 4752  nor: Choosing GR
│ │ │ │ -00031870: 6576 4c65 7853 6d61 6c6c 6573 7420 2020  evLexSmallest   
│ │ │ │ +00031860: 6e6f 723a 2043 686f 6f73 696e 6720 4c65  nor: Choosing Le
│ │ │ │ +00031870: 7853 6d61 6c6c 6573 7420 2020 2020 2020  xSmallest       
│ │ │ │  00031880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031890: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  000318a0: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │  000318b0: 6e6f 723a 2043 686f 6f73 696e 6720 4752  nor: Choosing GR
│ │ │ │ -000318c0: 6576 4c65 7853 6d61 6c6c 6573 7454 6572  evLexSmallestTer
│ │ │ │ -000318d0: 6d20 2020 2020 2020 2020 2020 2020 2020  m               
│ │ │ │ +000318c0: 6576 4c65 7853 6d61 6c6c 6573 7420 2020  evLexSmallest   
│ │ │ │ +000318d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000318e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  000318f0: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │  00031900: 6e6f 723a 2043 686f 6f73 696e 6720 4752  nor: Choosing GR
│ │ │ │  00031910: 6576 4c65 7853 6d61 6c6c 6573 7454 6572  evLexSmallestTer
│ │ │ │  00031920: 6d20 2020 2020 2020 2020 2020 2020 2020  m               
│ │ │ │  00031930: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00031940: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │  00031950: 6e6f 723a 2043 686f 6f73 696e 6720 4752  nor: Choosing GR
│ │ │ │ -00031960: 6576 4c65 7853 6d61 6c6c 6573 7420 2020  evLexSmallest   
│ │ │ │ -00031970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031960: 6576 4c65 7853 6d61 6c6c 6573 7454 6572  evLexSmallestTer
│ │ │ │ +00031970: 6d20 2020 2020 2020 2020 2020 2020 2020  m               
│ │ │ │  00031980: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00031990: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │  000319a0: 6e6f 723a 2043 686f 6f73 696e 6720 4752  nor: Choosing GR
│ │ │ │  000319b0: 6576 4c65 7853 6d61 6c6c 6573 7420 2020  evLexSmallest   
│ │ │ │  000319c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000319d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000319e0: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ -000319f0: 7369 6f6e 3a20 204c 6f6f 7020 7374 6570  sion:  Loop step
│ │ │ │ -00031a00: 2c20 6162 6f75 7420 746f 2063 6f6d 7075  , about to compu
│ │ │ │ -00031a10: 7465 2064 696d 656e 7369 6f6e 2e20 2053  te dimension.  S
│ │ │ │ -00031a20: 7562 6d61 7472 6963 6573 2063 6f7c 0a7c  ubmatrices co|.|
│ │ │ │ +000319e0: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ +000319f0: 6e6f 723a 2043 686f 6f73 696e 6720 4752  nor: Choosing GR
│ │ │ │ +00031a00: 6576 4c65 7853 6d61 6c6c 6573 7420 2020  evLexSmallest   
│ │ │ │ +00031a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031a20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00031a30: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ -00031a40: 7369 6f6e 3a20 2069 7343 6f64 696d 4174  sion:  isCodimAt
│ │ │ │ -00031a50: 4c65 6173 7420 6661 696c 6564 2c20 636f  Least failed, co
│ │ │ │ -00031a60: 6d70 7574 696e 6720 636f 6469 6d2e 2020  mputing codim.  
│ │ │ │ -00031a70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031a40: 7369 6f6e 3a20 204c 6f6f 7020 7374 6570  sion:  Loop step
│ │ │ │ +00031a50: 2c20 6162 6f75 7420 746f 2063 6f6d 7075  , about to compu
│ │ │ │ +00031a60: 7465 2064 696d 656e 7369 6f6e 2e20 2053  te dimension.  S
│ │ │ │ +00031a70: 7562 6d61 7472 6963 6573 2063 6f7c 0a7c  ubmatrices co|.|
│ │ │ │  00031a80: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ -00031a90: 7369 6f6e 3a20 2070 6172 7469 616c 2073  sion:  partial s
│ │ │ │ -00031aa0: 696e 6775 6c61 7220 6c6f 6375 7320 6469  ingular locus di
│ │ │ │ -00031ab0: 6d65 6e73 696f 6e20 636f 6d70 7574 6564  mension computed
│ │ │ │ -00031ac0: 2c20 3d20 3120 2020 2020 2020 207c 0a7c  , = 1        |.|
│ │ │ │ -00031ad0: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ -00031ae0: 6e6f 723a 2043 686f 6f73 696e 6720 4752  nor: Choosing GR
│ │ │ │ -00031af0: 6576 4c65 7853 6d61 6c6c 6573 7420 2020  evLexSmallest   
│ │ │ │ -00031b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031b10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031a90: 7369 6f6e 3a20 2069 7343 6f64 696d 4174  sion:  isCodimAt
│ │ │ │ +00031aa0: 4c65 6173 7420 6661 696c 6564 2c20 636f  Least failed, co
│ │ │ │ +00031ab0: 6d70 7574 696e 6720 636f 6469 6d2e 2020  mputing codim.  
│ │ │ │ +00031ac0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031ad0: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ +00031ae0: 7369 6f6e 3a20 2070 6172 7469 616c 2073  sion:  partial s
│ │ │ │ +00031af0: 696e 6775 6c61 7220 6c6f 6375 7320 6469  ingular locus di
│ │ │ │ +00031b00: 6d65 6e73 696f 6e20 636f 6d70 7574 6564  mension computed
│ │ │ │ +00031b10: 2c20 3d20 3120 2020 2020 2020 207c 0a7c  , = 1        |.|
│ │ │ │  00031b20: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │  00031b30: 6e6f 723a 2043 686f 6f73 696e 6720 4752  nor: Choosing GR
│ │ │ │ -00031b40: 6576 4c65 7853 6d61 6c6c 6573 7454 6572  evLexSmallestTer
│ │ │ │ -00031b50: 6d20 2020 2020 2020 2020 2020 2020 2020  m               
│ │ │ │ +00031b40: 6576 4c65 7853 6d61 6c6c 6573 7420 2020  evLexSmallest   
│ │ │ │ +00031b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031b60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00031b70: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ -00031b80: 7369 6f6e 3a20 204c 6f6f 7020 7374 6570  sion:  Loop step
│ │ │ │ -00031b90: 2c20 6162 6f75 7420 746f 2063 6f6d 7075  , about to compu
│ │ │ │ -00031ba0: 7465 2064 696d 656e 7369 6f6e 2e20 2053  te dimension.  S
│ │ │ │ -00031bb0: 7562 6d61 7472 6963 6573 2063 6f7c 0a7c  ubmatrices co|.|
│ │ │ │ +00031b70: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ +00031b80: 6e6f 723a 2043 686f 6f73 696e 6720 4752  nor: Choosing GR
│ │ │ │ +00031b90: 6576 4c65 7853 6d61 6c6c 6573 7454 6572  evLexSmallestTer
│ │ │ │ +00031ba0: 6d20 2020 2020 2020 2020 2020 2020 2020  m               
│ │ │ │ +00031bb0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00031bc0: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ -00031bd0: 7369 6f6e 3a20 2069 7343 6f64 696d 4174  sion:  isCodimAt
│ │ │ │ -00031be0: 4c65 6173 7420 6661 696c 6564 2c20 636f  Least failed, co
│ │ │ │ -00031bf0: 6d70 7574 696e 6720 636f 6469 6d2e 2020  mputing codim.  
│ │ │ │ -00031c00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031bd0: 7369 6f6e 3a20 204c 6f6f 7020 7374 6570  sion:  Loop step
│ │ │ │ +00031be0: 2c20 6162 6f75 7420 746f 2063 6f6d 7075  , about to compu
│ │ │ │ +00031bf0: 7465 2064 696d 656e 7369 6f6e 2e20 2053  te dimension.  S
│ │ │ │ +00031c00: 7562 6d61 7472 6963 6573 2063 6f7c 0a7c  ubmatrices co|.|
│ │ │ │  00031c10: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ -00031c20: 7369 6f6e 3a20 2070 6172 7469 616c 2073  sion:  partial s
│ │ │ │ -00031c30: 696e 6775 6c61 7220 6c6f 6375 7320 6469  ingular locus di
│ │ │ │ -00031c40: 6d65 6e73 696f 6e20 636f 6d70 7574 6564  mension computed
│ │ │ │ -00031c50: 2c20 3d20 3120 2020 2020 2020 207c 0a7c  , = 1        |.|
│ │ │ │ +00031c20: 7369 6f6e 3a20 2069 7343 6f64 696d 4174  sion:  isCodimAt
│ │ │ │ +00031c30: 4c65 6173 7420 6661 696c 6564 2c20 636f  Least failed, co
│ │ │ │ +00031c40: 6d70 7574 696e 6720 636f 6469 6d2e 2020  mputing codim.  
│ │ │ │ +00031c50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00031c60: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ -00031c70: 7369 6f6e 3a20 204c 6f6f 7020 636f 6d70  sion:  Loop comp
│ │ │ │ -00031c80: 6c65 7465 642c 2073 7562 6d61 7472 6963  leted, submatric
│ │ │ │ -00031c90: 6573 2063 6f6e 7369 6465 7265 6420 3d20  es considered = 
│ │ │ │ -00031ca0: 3330 2c20 616e 6420 636f 6d70 757c 0a7c  30, and compu|.|
│ │ │ │ -00031cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c  -------------|.|
│ │ │ │ -00031d00: 7273 2c20 7765 2077 696c 6c20 636f 6d70  rs, we will comp
│ │ │ │ -00031d10: 7574 6520 7570 2074 6f20 3330 206f 6620  ute up to 30 of 
│ │ │ │ -00031d20: 7468 656d 2e20 2020 2020 2020 2020 2020  them.           
│ │ │ │ -00031d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031d40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00031d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031c70: 7369 6f6e 3a20 2070 6172 7469 616c 2073  sion:  partial s
│ │ │ │ +00031c80: 696e 6775 6c61 7220 6c6f 6375 7320 6469  ingular locus di
│ │ │ │ +00031c90: 6d65 6e73 696f 6e20 636f 6d70 7574 6564  mension computed
│ │ │ │ +00031ca0: 2c20 3d20 3120 2020 2020 2020 207c 0a7c  , = 1        |.|
│ │ │ │ +00031cb0: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ +00031cc0: 7369 6f6e 3a20 204c 6f6f 7020 636f 6d70  sion:  Loop comp
│ │ │ │ +00031cd0: 6c65 7465 642c 2073 7562 6d61 7472 6963  leted, submatric
│ │ │ │ +00031ce0: 6573 2063 6f6e 7369 6465 7265 6420 3d20  es considered = 
│ │ │ │ +00031cf0: 3330 2c20 616e 6420 636f 6d70 757c 0a7c  30, and compu|.|
│ │ │ │ +00031d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00031d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00031d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00031d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00031d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c  -------------|.|
│ │ │ │ +00031d50: 7273 2c20 7765 2077 696c 6c20 636f 6d70  rs, we will comp
│ │ │ │ +00031d60: 7574 6520 7570 2074 6f20 3330 206f 6620  ute up to 30 of 
│ │ │ │ +00031d70: 7468 656d 2e20 2020 2020 2020 2020 2020  them.           
│ │ │ │  00031d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031d90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00031da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031de0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ @@ -12801,21 +12801,21 @@
│ │ │ │  00032000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032010: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00032020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032060: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00032070: 6e73 6964 6572 6564 3a20 392c 2061 6e64  nsidered: 9, and
│ │ │ │ -00032080: 2063 6f6d 7075 7465 6420 3d20 3720 2020   computed = 7   
│ │ │ │ +00032070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000320a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000320b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000320c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000320d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000320c0: 6e73 6964 6572 6564 3a20 392c 2061 6e64  nsidered: 9, and
│ │ │ │ +000320d0: 2063 6f6d 7075 7465 6420 3d20 3720 2020   computed = 7   
│ │ │ │  000320e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000320f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032100: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00032110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -12826,21 +12826,21 @@
│ │ │ │  00032190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000321a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  000321b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000321c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000321d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000321e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000321f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00032200: 6e73 6964 6572 6564 3a20 3131 2c20 616e  nsidered: 11, an
│ │ │ │ -00032210: 6420 636f 6d70 7574 6564 203d 2039 2020  d computed = 9  
│ │ │ │ +00032200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032240: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00032250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032250: 6e73 6964 6572 6564 3a20 3131 2c20 616e  nsidered: 11, an
│ │ │ │ +00032260: 6420 636f 6d70 7574 6564 203d 2039 2020  d computed = 9  
│ │ │ │  00032270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032290: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  000322a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000322b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000322c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000322d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -12861,21 +12861,21 @@
│ │ │ │  000323c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000323d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  000323e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000323f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032420: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00032430: 6e73 6964 6572 6564 3a20 3135 2c20 616e  nsidered: 15, an
│ │ │ │ -00032440: 6420 636f 6d70 7574 6564 203d 2031 3120  d computed = 11 
│ │ │ │ +00032430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032470: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00032480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032480: 6e73 6964 6572 6564 3a20 3135 2c20 616e  nsidered: 15, an
│ │ │ │ +00032490: 6420 636f 6d70 7574 6564 203d 2031 3120  d computed = 11 
│ │ │ │  000324a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000324b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000324c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  000324d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000324e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000324f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -12906,21 +12906,21 @@
│ │ │ │  00032690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000326a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  000326b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000326c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000326d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000326e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000326f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00032700: 6e73 6964 6572 6564 3a20 3231 2c20 616e  nsidered: 21, an
│ │ │ │ -00032710: 6420 636f 6d70 7574 6564 203d 2031 3620  d computed = 16 
│ │ │ │ +00032700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032740: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00032750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032750: 6e73 6964 6572 6564 3a20 3231 2c20 616e  nsidered: 21, an
│ │ │ │ +00032760: 6420 636f 6d70 7574 6564 203d 2031 3620  d computed = 16 
│ │ │ │  00032770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032790: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  000327a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000327b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000327c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000327d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -12956,21 +12956,21 @@
│ │ │ │  000329b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000329c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  000329d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000329e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000329f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032a10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00032a20: 6e73 6964 6572 6564 3a20 3238 2c20 616e  nsidered: 28, an
│ │ │ │ -00032a30: 6420 636f 6d70 7574 6564 203d 2032 3220  d computed = 22 
│ │ │ │ +00032a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032a60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00032a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032a70: 6e73 6964 6572 6564 3a20 3238 2c20 616e  nsidered: 28, an
│ │ │ │ +00032a80: 6420 636f 6d70 7574 6564 203d 2032 3220  d computed = 22 
│ │ │ │  00032a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032ab0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00032ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -12981,4040 +12981,4045 @@
│ │ │ │  00032b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032b50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00032b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032ba0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00032bb0: 6e73 6964 6572 6564 3a20 3330 2c20 616e  nsidered: 30, an
│ │ │ │ -00032bc0: 6420 636f 6d70 7574 6564 203d 2032 3320  d computed = 23 
│ │ │ │ +00032bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032bf0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00032c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032c00: 6e73 6964 6572 6564 3a20 3330 2c20 616e  nsidered: 30, an
│ │ │ │ +00032c10: 6420 636f 6d70 7574 6564 203d 2032 3320  d computed = 23 
│ │ │ │  00032c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032c40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00032c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032c90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00032ca0: 7465 6420 3d20 3233 2e20 2073 696e 6775  ted = 23.  singu
│ │ │ │ -00032cb0: 6c61 7220 6c6f 6375 7320 6469 6d65 6e73  lar locus dimens
│ │ │ │ -00032cc0: 696f 6e20 6170 7065 6172 7320 746f 2062  ion appears to b
│ │ │ │ -00032cd0: 6520 3d20 3120 2020 2020 2020 2020 2020  e = 1           
│ │ │ │ -00032ce0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00032cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00032d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00032d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00032d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00032d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ -00032d40: 5468 6973 2066 756e 6374 696f 6e20 6861  This function ha
│ │ │ │ -00032d50: 7320 6d61 6e79 206f 7074 696f 6e73 2077  s many options w
│ │ │ │ -00032d60: 6869 6368 2061 6c6c 6f77 2079 6f75 2074  hich allow you t
│ │ │ │ -00032d70: 6f20 6669 6e65 2074 756e 6520 7468 6520  o fine tune the 
│ │ │ │ -00032d80: 7374 7261 7465 6779 2075 7365 640a 746f  strategy used.to
│ │ │ │ -00032d90: 2066 696e 6420 696e 7465 7265 7374 696e   find interestin
│ │ │ │ -00032da0: 6720 6d69 6e6f 7273 2e20 596f 7520 6361  g minors. You ca
│ │ │ │ -00032db0: 6e20 7061 7373 2069 7420 6120 4861 7368  n pass it a Hash
│ │ │ │ -00032dc0: 5461 626c 6520 7370 6563 6966 7969 6e67  Table specifying
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│ │ │ │ -00032de0: 6120 7468 6520 6f70 7469 6f6e 2053 7472  a the option Str
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│ │ │ │ -00032e00: 6520 4c65 7853 6d61 6c6c 6573 743a 2053  e LexSmallest: S
│ │ │ │ -00032e10: 7472 6174 6567 7944 6566 6175 6c74 2c20  trategyDefault, 
│ │ │ │ -00032e20: 666f 7220 686f 7720 746f 0a63 6f6e 7374  for how to.const
│ │ │ │ -00032e30: 7275 6374 2074 6869 7320 4861 7368 5461  ruct this HashTa
│ │ │ │ -00032e40: 626c 652e 2054 6865 2064 6566 6175 6c74  ble. The default
│ │ │ │ -00032e50: 2073 7472 6174 6567 7920 6973 2053 7472   strategy is Str
│ │ │ │ -00032e60: 6174 6567 7944 6566 6175 6c74 2c20 7768  ategyDefault, wh
│ │ │ │ -00032e70: 6963 6820 7365 656d 730a 746f 2077 6f72  ich seems.to wor
│ │ │ │ -00032e80: 6b20 7765 6c6c 206f 6e20 7468 6520 6578  k well on the ex
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│ │ │ │ -00032eb0: 722c 2063 6175 7469 6f6e 206d 7573 7420  r, caution must 
│ │ │ │ -00032ec0: 6265 0a65 7865 7263 6973 6564 2c20 6265  be.exercised, be
│ │ │ │ -00032ed0: 6361 7573 652c 2065 7665 6e20 696e 2074  cause, even in t
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│ │ │ │ -00032ef0: 652c 2063 6572 7461 696e 2073 7472 6174  e, certain strat
│ │ │ │ -00032f00: 6567 6965 7320 776f 726b 2077 656c 6c0a  egies work well.
│ │ │ │ -00032f10: 7768 696c 6520 6f74 6865 7273 2064 6f20  while others do 
│ │ │ │ -00032f20: 6e6f 742e 2020 496e 2074 6865 2041 6265  not.  In the Abe
│ │ │ │ -00032f30: 6c69 616e 2073 7572 6661 6365 2065 7861  lian surface exa
│ │ │ │ -00032f40: 6d70 6c65 2c20 4c65 7853 6d61 6c6c 6573  mple, LexSmalles
│ │ │ │ -00032f50: 7420 776f 726b 7320 7665 7279 0a77 656c  t works very.wel
│ │ │ │ -00032f60: 6c2c 2077 6869 6c65 204c 6578 536d 616c  l, while LexSmal
│ │ │ │ -00032f70: 6c65 7374 5465 726d 2064 6f65 7320 6e6f  lestTerm does no
│ │ │ │ -00032f80: 7420 6576 656e 2074 7970 6963 616c 6c79  t even typically
│ │ │ │ -00032f90: 2063 6f72 7265 6374 6c79 2069 6465 6e74   correctly ident
│ │ │ │ -00032fa0: 6966 7920 7468 6520 7269 6e67 0a61 7320  ify the ring.as 
│ │ │ │ -00032fb0: 6e6f 6e73 696e 6775 6c61 7220 2874 6869  nonsingular (thi
│ │ │ │ -00032fc0: 7320 6973 2062 6563 6175 7365 2074 6865  s is because the
│ │ │ │ -00032fd0: 7265 2061 7265 2061 2073 6d61 6c6c 206e  re are a small n
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│ │ │ │ -00032ff0: 2077 6974 680a 6e6f 6e7a 6572 6f20 636f   with.nonzero co
│ │ │ │ -00033000: 6e73 7461 6e74 2074 6572 6d73 2c20 7768  nstant terms, wh
│ │ │ │ -00033010: 6963 6820 6172 6520 7365 6c65 6374 6564  ich are selected
│ │ │ │ -00033020: 2072 6570 6561 7465 646c 7929 2e20 486f   repeatedly). Ho
│ │ │ │ -00033030: 7765 7665 722c 2069 6e20 6f75 7220 6669  wever, in our fi
│ │ │ │ -00033040: 7273 740a 6578 616d 706c 652c 2074 6865  rst.example, the
│ │ │ │ -00033050: 204c 6578 536d 616c 6c65 7374 5465 726d   LexSmallestTerm
│ │ │ │ -00033060: 2069 7320 6d75 6368 2066 6173 7465 722c   is much faster,
│ │ │ │ -00033070: 2061 6e64 2052 616e 646f 6d20 646f 6573   and Random does
│ │ │ │ -00033080: 206e 6f74 2070 6572 666f 726d 2077 656c   not perform wel
│ │ │ │ -00033090: 6c0a 6174 2061 6c6c 2e0a 0a2b 2d2d 2d2d  l.at all...+----
│ │ │ │ -000330a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000330b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000330c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000330d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -000330e0: 3138 203a 2053 7472 6174 6567 7943 7572  18 : StrategyCur
│ │ │ │ -000330f0: 7265 6e74 2352 616e 646f 6d20 3d20 303b  rent#Random = 0;
│ │ │ │ -00033100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033110: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00033120: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00033130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033160: 2d2d 2b0a 7c69 3139 203a 2053 7472 6174  --+.|i19 : Strat
│ │ │ │ -00033170: 6567 7943 7572 7265 6e74 234c 6578 536d  egyCurrent#LexSm
│ │ │ │ -00033180: 616c 6c65 7374 203d 2031 3030 3b20 2020  allest = 100;   
│ │ │ │ -00033190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000331a0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -000331b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000331c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000331d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000331e0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3230 203a  --------+.|i20 :
│ │ │ │ -000331f0: 2053 7472 6174 6567 7943 7572 7265 6e74   StrategyCurrent
│ │ │ │ -00033200: 234c 6578 536d 616c 6c65 7374 5465 726d  #LexSmallestTerm
│ │ │ │ -00033210: 203d 2030 3b20 2020 2020 2020 2020 2020   = 0;           
│ │ │ │ -00033220: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -00033230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00033270: 7c69 3231 203a 2074 696d 6520 7265 6775  |i21 : time regu
│ │ │ │ -00033280: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ -00033290: 2832 2c20 522c 2053 7472 6174 6567 793d  (2, R, Strategy=
│ │ │ │ -000332a0: 3e53 7472 6174 6567 7943 7572 7265 6e74  >StrategyCurrent
│ │ │ │ -000332b0: 297c 0a7c 202d 2d20 7573 6564 2030 2e33  )|.| -- used 0.3
│ │ │ │ -000332c0: 3136 3532 7320 2863 7075 293b 2030 2e32  1652s (cpu); 0.2
│ │ │ │ -000332d0: 3231 3838 3973 2028 7468 7265 6164 293b  21889s (thread);
│ │ │ │ -000332e0: 2030 7320 2867 6329 2020 2020 2020 2020   0s (gc)        
│ │ │ │ -000332f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00033300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033330: 2020 2020 2020 207c 0a7c 6f32 3120 3d20         |.|o21 = 
│ │ │ │ -00033340: 7472 7565 2020 2020 2020 2020 2020 2020  true            
│ │ │ │ +00032ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032ce0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00032cf0: 7465 6420 3d20 3233 2e20 2073 696e 6775  ted = 23.  singu
│ │ │ │ +00032d00: 6c61 7220 6c6f 6375 7320 6469 6d65 6e73  lar locus dimens
│ │ │ │ +00032d10: 696f 6e20 6170 7065 6172 7320 746f 2062  ion appears to b
│ │ │ │ +00032d20: 6520 3d20 3120 2020 2020 2020 2020 2020  e = 1           
│ │ │ │ +00032d30: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00032d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00032d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00032d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00032d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00032d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ +00032d90: 5468 6973 2066 756e 6374 696f 6e20 6861  This function ha
│ │ │ │ +00032da0: 7320 6d61 6e79 206f 7074 696f 6e73 2077  s many options w
│ │ │ │ +00032db0: 6869 6368 2061 6c6c 6f77 2079 6f75 2074  hich allow you t
│ │ │ │ +00032dc0: 6f20 6669 6e65 2074 756e 6520 7468 6520  o fine tune the 
│ │ │ │ +00032dd0: 7374 7261 7465 6779 2075 7365 640a 746f  strategy used.to
│ │ │ │ +00032de0: 2066 696e 6420 696e 7465 7265 7374 696e   find interestin
│ │ │ │ +00032df0: 6720 6d69 6e6f 7273 2e20 596f 7520 6361  g minors. You ca
│ │ │ │ +00032e00: 6e20 7061 7373 2069 7420 6120 4861 7368  n pass it a Hash
│ │ │ │ +00032e10: 5461 626c 6520 7370 6563 6966 7969 6e67  Table specifying
│ │ │ │ +00032e20: 2074 6865 2073 7472 6174 6567 790a 7669   the strategy.vi
│ │ │ │ +00032e30: 6120 7468 6520 6f70 7469 6f6e 2053 7472  a the option Str
│ │ │ │ +00032e40: 6174 6567 792e 2020 5365 6520 2a6e 6f74  ategy.  See *not
│ │ │ │ +00032e50: 6520 4c65 7853 6d61 6c6c 6573 743a 2053  e LexSmallest: S
│ │ │ │ +00032e60: 7472 6174 6567 7944 6566 6175 6c74 2c20  trategyDefault, 
│ │ │ │ +00032e70: 666f 7220 686f 7720 746f 0a63 6f6e 7374  for how to.const
│ │ │ │ +00032e80: 7275 6374 2074 6869 7320 4861 7368 5461  ruct this HashTa
│ │ │ │ +00032e90: 626c 652e 2054 6865 2064 6566 6175 6c74  ble. The default
│ │ │ │ +00032ea0: 2073 7472 6174 6567 7920 6973 2053 7472   strategy is Str
│ │ │ │ +00032eb0: 6174 6567 7944 6566 6175 6c74 2c20 7768  ategyDefault, wh
│ │ │ │ +00032ec0: 6963 6820 7365 656d 730a 746f 2077 6f72  ich seems.to wor
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│ │ │ │ +00032ee0: 616d 706c 6573 2077 6520 6861 7665 2065  amples we have e
│ │ │ │ +00032ef0: 7870 6c6f 7265 642e 2020 486f 7765 7665  xplored.  Howeve
│ │ │ │ +00032f00: 722c 2063 6175 7469 6f6e 206d 7573 7420  r, caution must 
│ │ │ │ +00032f10: 6265 0a65 7865 7263 6973 6564 2c20 6265  be.exercised, be
│ │ │ │ +00032f20: 6361 7573 652c 2065 7665 6e20 696e 2074  cause, even in t
│ │ │ │ +00032f30: 6865 2065 7861 6d70 6c65 7320 6162 6f76  he examples abov
│ │ │ │ +00032f40: 652c 2063 6572 7461 696e 2073 7472 6174  e, certain strat
│ │ │ │ +00032f50: 6567 6965 7320 776f 726b 2077 656c 6c0a  egies work well.
│ │ │ │ +00032f60: 7768 696c 6520 6f74 6865 7273 2064 6f20  while others do 
│ │ │ │ +00032f70: 6e6f 742e 2020 496e 2074 6865 2041 6265  not.  In the Abe
│ │ │ │ +00032f80: 6c69 616e 2073 7572 6661 6365 2065 7861  lian surface exa
│ │ │ │ +00032f90: 6d70 6c65 2c20 4c65 7853 6d61 6c6c 6573  mple, LexSmalles
│ │ │ │ +00032fa0: 7420 776f 726b 7320 7665 7279 0a77 656c  t works very.wel
│ │ │ │ +00032fb0: 6c2c 2077 6869 6c65 204c 6578 536d 616c  l, while LexSmal
│ │ │ │ +00032fc0: 6c65 7374 5465 726d 2064 6f65 7320 6e6f  lestTerm does no
│ │ │ │ +00032fd0: 7420 6576 656e 2074 7970 6963 616c 6c79  t even typically
│ │ │ │ +00032fe0: 2063 6f72 7265 6374 6c79 2069 6465 6e74   correctly ident
│ │ │ │ +00032ff0: 6966 7920 7468 6520 7269 6e67 0a61 7320  ify the ring.as 
│ │ │ │ +00033000: 6e6f 6e73 696e 6775 6c61 7220 2874 6869  nonsingular (thi
│ │ │ │ +00033010: 7320 6973 2062 6563 6175 7365 2074 6865  s is because the
│ │ │ │ +00033020: 7265 2061 7265 2061 2073 6d61 6c6c 206e  re are a small n
│ │ │ │ +00033030: 756d 6265 7220 6f66 2065 6e74 7269 6573  umber of entries
│ │ │ │ +00033040: 2077 6974 680a 6e6f 6e7a 6572 6f20 636f   with.nonzero co
│ │ │ │ +00033050: 6e73 7461 6e74 2074 6572 6d73 2c20 7768  nstant terms, wh
│ │ │ │ +00033060: 6963 6820 6172 6520 7365 6c65 6374 6564  ich are selected
│ │ │ │ +00033070: 2072 6570 6561 7465 646c 7929 2e20 486f   repeatedly). Ho
│ │ │ │ +00033080: 7765 7665 722c 2069 6e20 6f75 7220 6669  wever, in our fi
│ │ │ │ +00033090: 7273 740a 6578 616d 706c 652c 2074 6865  rst.example, the
│ │ │ │ +000330a0: 204c 6578 536d 616c 6c65 7374 5465 726d   LexSmallestTerm
│ │ │ │ +000330b0: 2069 7320 6d75 6368 2066 6173 7465 722c   is much faster,
│ │ │ │ +000330c0: 2061 6e64 2052 616e 646f 6d20 646f 6573   and Random does
│ │ │ │ +000330d0: 206e 6f74 2070 6572 666f 726d 2077 656c   not perform wel
│ │ │ │ +000330e0: 6c0a 6174 2061 6c6c 2e0a 0a2b 2d2d 2d2d  l.at all...+----
│ │ │ │ +000330f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00033130: 3138 203a 2053 7472 6174 6567 7943 7572  18 : StrategyCur
│ │ │ │ +00033140: 7265 6e74 2352 616e 646f 6d20 3d20 303b  rent#Random = 0;
│ │ │ │ +00033150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033160: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00033170: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00033180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000331a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000331b0: 2d2d 2b0a 7c69 3139 203a 2053 7472 6174  --+.|i19 : Strat
│ │ │ │ +000331c0: 6567 7943 7572 7265 6e74 234c 6578 536d  egyCurrent#LexSm
│ │ │ │ +000331d0: 616c 6c65 7374 203d 2031 3030 3b20 2020  allest = 100;   
│ │ │ │ +000331e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000331f0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00033200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033230: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3230 203a  --------+.|i20 :
│ │ │ │ +00033240: 2053 7472 6174 6567 7943 7572 7265 6e74   StrategyCurrent
│ │ │ │ +00033250: 234c 6578 536d 616c 6c65 7374 5465 726d  #LexSmallestTerm
│ │ │ │ +00033260: 203d 2030 3b20 2020 2020 2020 2020 2020   = 0;           
│ │ │ │ +00033270: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00033280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000332a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000332b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000332c0: 7c69 3231 203a 2074 696d 6520 7265 6775  |i21 : time regu
│ │ │ │ +000332d0: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ +000332e0: 2832 2c20 522c 2053 7472 6174 6567 793d  (2, R, Strategy=
│ │ │ │ +000332f0: 3e53 7472 6174 6567 7943 7572 7265 6e74  >StrategyCurrent
│ │ │ │ +00033300: 297c 0a7c 202d 2d20 7573 6564 2030 2e33  )|.| -- used 0.3
│ │ │ │ +00033310: 3634 3535 3373 2028 6370 7529 3b20 302e  64553s (cpu); 0.
│ │ │ │ +00033320: 3233 3433 3735 7320 2874 6872 6561 6429  234375s (thread)
│ │ │ │ +00033330: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │ +00033340: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00033350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033370: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ -00033380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000333a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000333b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -000333c0: 6932 3220 3a20 7469 6d65 2072 6567 756c  i22 : time regul
│ │ │ │ -000333d0: 6172 496e 436f 6469 6d65 6e73 696f 6e28  arInCodimension(
│ │ │ │ -000333e0: 322c 2052 2c20 5374 7261 7465 6779 3d3e  2, R, Strategy=>
│ │ │ │ -000333f0: 5374 7261 7465 6779 4375 7272 656e 7429  StrategyCurrent)
│ │ │ │ -00033400: 7c0a 7c20 2d2d 2075 7365 6420 302e 3132  |.| -- used 0.12
│ │ │ │ -00033410: 3134 3132 7320 2863 7075 293b 2030 2e30  1412s (cpu); 0.0
│ │ │ │ -00033420: 3732 3931 3431 7320 2874 6872 6561 6429  729141s (thread)
│ │ │ │ -00033430: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │ -00033440: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00033450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033480: 2020 2020 2020 7c0a 7c6f 3232 203d 2074        |.|o22 = t
│ │ │ │ -00033490: 7275 6520 2020 2020 2020 2020 2020 2020  rue             
│ │ │ │ +00033370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033380: 2020 2020 2020 207c 0a7c 6f32 3120 3d20         |.|o21 = 
│ │ │ │ +00033390: 7472 7565 2020 2020 2020 2020 2020 2020  true            
│ │ │ │ +000333a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000333b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000333c0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +000333d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000333e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000333f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00033410: 6932 3220 3a20 7469 6d65 2072 6567 756c  i22 : time regul
│ │ │ │ +00033420: 6172 496e 436f 6469 6d65 6e73 696f 6e28  arInCodimension(
│ │ │ │ +00033430: 322c 2052 2c20 5374 7261 7465 6779 3d3e  2, R, Strategy=>
│ │ │ │ +00033440: 5374 7261 7465 6779 4375 7272 656e 7429  StrategyCurrent)
│ │ │ │ +00033450: 7c0a 7c20 2d2d 2075 7365 6420 302e 3134  |.| -- used 0.14
│ │ │ │ +00033460: 3836 3139 7320 2863 7075 293b 2030 2e30  8619s (cpu); 0.0
│ │ │ │ +00033470: 3836 3631 3739 7320 2874 6872 6561 6429  866179s (thread)
│ │ │ │ +00033480: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │ +00033490: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  000334a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000334b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000334c0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -000334d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000334e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000334f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00033510: 3233 203a 2074 696d 6520 7265 6775 6c61  23 : time regula
│ │ │ │ -00033520: 7249 6e43 6f64 696d 656e 7369 6f6e 2831  rInCodimension(1
│ │ │ │ -00033530: 2c20 532c 2053 7472 6174 6567 793d 3e53  , S, Strategy=>S
│ │ │ │ -00033540: 7472 6174 6567 7943 7572 7265 6e74 297c  trategyCurrent)|
│ │ │ │ -00033550: 0a7c 202d 2d20 7573 6564 2030 2e33 3630  .| -- used 0.360
│ │ │ │ -00033560: 3535 7320 2863 7075 293b 2030 2e32 3639  55s (cpu); 0.269
│ │ │ │ -00033570: 3031 3673 2028 7468 7265 6164 293b 2030  016s (thread); 0
│ │ │ │ -00033580: 7320 2867 6329 2020 2020 2020 2020 2020  s (gc)          
│ │ │ │ -00033590: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -000335a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000335b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000335c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000335d0: 2020 2020 207c 0a7c 6f32 3320 3d20 7472       |.|o23 = tr
│ │ │ │ -000335e0: 7565 2020 2020 2020 2020 2020 2020 2020  ue              
│ │ │ │ +000334c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000334d0: 2020 2020 2020 7c0a 7c6f 3232 203d 2074        |.|o22 = t
│ │ │ │ +000334e0: 7275 6520 2020 2020 2020 2020 2020 2020  rue             
│ │ │ │ +000334f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033510: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00033520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00033560: 3233 203a 2074 696d 6520 7265 6775 6c61  23 : time regula
│ │ │ │ +00033570: 7249 6e43 6f64 696d 656e 7369 6f6e 2831  rInCodimension(1
│ │ │ │ +00033580: 2c20 532c 2053 7472 6174 6567 793d 3e53  , S, Strategy=>S
│ │ │ │ +00033590: 7472 6174 6567 7943 7572 7265 6e74 297c  trategyCurrent)|
│ │ │ │ +000335a0: 0a7c 202d 2d20 7573 6564 2030 2e34 3733  .| -- used 0.473
│ │ │ │ +000335b0: 3131 3173 2028 6370 7529 3b20 302e 3334  111s (cpu); 0.34
│ │ │ │ +000335c0: 3131 3039 7320 2874 6872 6561 6429 3b20  1109s (thread); 
│ │ │ │ +000335d0: 3073 2028 6763 2920 2020 2020 2020 2020  0s (gc)         
│ │ │ │ +000335e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000335f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033610: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -00033620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ -00033660: 3420 3a20 7469 6d65 2072 6567 756c 6172  4 : time regular
│ │ │ │ -00033670: 496e 436f 6469 6d65 6e73 696f 6e28 312c  InCodimension(1,
│ │ │ │ -00033680: 2053 2c20 5374 7261 7465 6779 3d3e 5374   S, Strategy=>St
│ │ │ │ -00033690: 7261 7465 6779 4375 7272 656e 7429 7c0a  rategyCurrent)|.
│ │ │ │ -000336a0: 7c20 2d2d 2075 7365 6420 312e 3631 3534  | -- used 1.6154
│ │ │ │ -000336b0: 3773 2028 6370 7529 3b20 312e 3230 3539  7s (cpu); 1.2059
│ │ │ │ -000336c0: 3473 2028 7468 7265 6164 293b 2030 7320  4s (thread); 0s 
│ │ │ │ -000336d0: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │ -000336e0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000336f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033720: 2020 2020 7c0a 7c6f 3234 203d 2074 7275      |.|o24 = tru
│ │ │ │ -00033730: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │ +00033610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033620: 2020 2020 207c 0a7c 6f32 3320 3d20 7472       |.|o23 = tr
│ │ │ │ +00033630: 7565 2020 2020 2020 2020 2020 2020 2020  ue              
│ │ │ │ +00033640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033660: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00033670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000336a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +000336b0: 3420 3a20 7469 6d65 2072 6567 756c 6172  4 : time regular
│ │ │ │ +000336c0: 496e 436f 6469 6d65 6e73 696f 6e28 312c  InCodimension(1,
│ │ │ │ +000336d0: 2053 2c20 5374 7261 7465 6779 3d3e 5374   S, Strategy=>St
│ │ │ │ +000336e0: 7261 7465 6779 4375 7272 656e 7429 7c0a  rategyCurrent)|.
│ │ │ │ +000336f0: 7c20 2d2d 2075 7365 6420 322e 3136 3132  | -- used 2.1612
│ │ │ │ +00033700: 3273 2028 6370 7529 3b20 312e 3537 3632  2s (cpu); 1.5762
│ │ │ │ +00033710: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ +00033720: 6763 2920 2020 2020 2020 2020 2020 2020  gc)             
│ │ │ │ +00033730: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00033740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033760: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -00033770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000337a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3235  ----------+.|i25
│ │ │ │ -000337b0: 203a 2053 7472 6174 6567 7943 7572 7265   : StrategyCurre
│ │ │ │ -000337c0: 6e74 234c 6578 536d 616c 6c65 7374 203d  nt#LexSmallest =
│ │ │ │ -000337d0: 2030 3b20 2020 2020 2020 2020 2020 2020   0;             
│ │ │ │ -000337e0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -000337f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033830: 2b0a 7c69 3236 203a 2053 7472 6174 6567  +.|i26 : Strateg
│ │ │ │ -00033840: 7943 7572 7265 6e74 234c 6578 536d 616c  yCurrent#LexSmal
│ │ │ │ -00033850: 6c65 7374 5465 726d 203d 2031 3030 3b20  lestTerm = 100; 
│ │ │ │ -00033860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033870: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -00033880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000338a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000338b0: 2d2d 2d2d 2d2d 2b0a 7c69 3237 203a 2074  ------+.|i27 : t
│ │ │ │ -000338c0: 696d 6520 7265 6775 6c61 7249 6e43 6f64  ime regularInCod
│ │ │ │ -000338d0: 696d 656e 7369 6f6e 2832 2c20 522c 2053  imension(2, R, S
│ │ │ │ -000338e0: 7472 6174 6567 793d 3e53 7472 6174 6567  trategy=>Strateg
│ │ │ │ -000338f0: 7943 7572 7265 6e74 297c 0a7c 202d 2d20  yCurrent)|.| -- 
│ │ │ │ -00033900: 7573 6564 2032 2e31 3638 3339 7320 2863  used 2.16839s (c
│ │ │ │ -00033910: 7075 293b 2031 2e35 3836 3834 7320 2874  pu); 1.58684s (t
│ │ │ │ -00033920: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │ -00033930: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -00033940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00033980: 0a7c 6932 3820 3a20 7469 6d65 2072 6567  .|i28 : time reg
│ │ │ │ -00033990: 756c 6172 496e 436f 6469 6d65 6e73 696f  ularInCodimensio
│ │ │ │ -000339a0: 6e28 322c 2052 2c20 5374 7261 7465 6779  n(2, R, Strategy
│ │ │ │ -000339b0: 3d3e 5374 7261 7465 6779 4375 7272 656e  =>StrategyCurren
│ │ │ │ -000339c0: 7429 7c0a 7c20 2d2d 2075 7365 6420 322e  t)|.| -- used 2.
│ │ │ │ -000339d0: 3234 3937 3673 2028 6370 7529 3b20 312e  24976s (cpu); 1.
│ │ │ │ -000339e0: 3630 3638 3973 2028 7468 7265 6164 293b  60689s (thread);
│ │ │ │ -000339f0: 2030 7320 2867 6329 2020 2020 2020 2020   0s (gc)        
│ │ │ │ -00033a00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00033a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033a40: 2020 2020 2020 2020 7c0a 7c6f 3238 203d          |.|o28 =
│ │ │ │ -00033a50: 2074 7275 6520 2020 2020 2020 2020 2020   true           
│ │ │ │ +00033760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033770: 2020 2020 7c0a 7c6f 3234 203d 2074 7275      |.|o24 = tru
│ │ │ │ +00033780: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │ +00033790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000337a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000337b0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +000337c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000337d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000337e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000337f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3235  ----------+.|i25
│ │ │ │ +00033800: 203a 2053 7472 6174 6567 7943 7572 7265   : StrategyCurre
│ │ │ │ +00033810: 6e74 234c 6578 536d 616c 6c65 7374 203d  nt#LexSmallest =
│ │ │ │ +00033820: 2030 3b20 2020 2020 2020 2020 2020 2020   0;             
│ │ │ │ +00033830: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00033840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033880: 2b0a 7c69 3236 203a 2053 7472 6174 6567  +.|i26 : Strateg
│ │ │ │ +00033890: 7943 7572 7265 6e74 234c 6578 536d 616c  yCurrent#LexSmal
│ │ │ │ +000338a0: 6c65 7374 5465 726d 203d 2031 3030 3b20  lestTerm = 100; 
│ │ │ │ +000338b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000338c0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +000338d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000338e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000338f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033900: 2d2d 2d2d 2d2d 2b0a 7c69 3237 203a 2074  ------+.|i27 : t
│ │ │ │ +00033910: 696d 6520 7265 6775 6c61 7249 6e43 6f64  ime regularInCod
│ │ │ │ +00033920: 696d 656e 7369 6f6e 2832 2c20 522c 2053  imension(2, R, S
│ │ │ │ +00033930: 7472 6174 6567 793d 3e53 7472 6174 6567  trategy=>Strateg
│ │ │ │ +00033940: 7943 7572 7265 6e74 297c 0a7c 202d 2d20  yCurrent)|.| -- 
│ │ │ │ +00033950: 7573 6564 2032 2e36 3433 3573 2028 6370  used 2.6435s (cp
│ │ │ │ +00033960: 7529 3b20 312e 3832 3736 3673 2028 7468  u); 1.82766s (th
│ │ │ │ +00033970: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │ +00033980: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +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 2d2b  ---------------+
│ │ │ │ +000339d0: 0a7c 6932 3820 3a20 7469 6d65 2072 6567  .|i28 : time reg
│ │ │ │ +000339e0: 756c 6172 496e 436f 6469 6d65 6e73 696f  ularInCodimensio
│ │ │ │ +000339f0: 6e28 322c 2052 2c20 5374 7261 7465 6779  n(2, R, Strategy
│ │ │ │ +00033a00: 3d3e 5374 7261 7465 6779 4375 7272 656e  =>StrategyCurren
│ │ │ │ +00033a10: 7429 7c0a 7c20 2d2d 2075 7365 6420 322e  t)|.| -- used 2.
│ │ │ │ +00033a20: 3735 3038 3873 2028 6370 7529 3b20 312e  75088s (cpu); 1.
│ │ │ │ +00033a30: 3836 3538 3473 2028 7468 7265 6164 293b  86584s (thread);
│ │ │ │ +00033a40: 2030 7320 2867 6329 2020 2020 2020 2020   0s (gc)        
│ │ │ │ +00033a50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00033a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033a80: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -00033a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00033ad0: 7c69 3239 203a 2074 696d 6520 7265 6775  |i29 : time regu
│ │ │ │ -00033ae0: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ -00033af0: 2831 2c20 532c 2053 7472 6174 6567 793d  (1, S, Strategy=
│ │ │ │ -00033b00: 3e53 7472 6174 6567 7943 7572 7265 6e74  >StrategyCurrent
│ │ │ │ -00033b10: 297c 0a7c 202d 2d20 7573 6564 2030 2e34  )|.| -- used 0.4
│ │ │ │ -00033b20: 3139 3532 3873 2028 6370 7529 3b20 302e  19528s (cpu); 0.
│ │ │ │ -00033b30: 3333 3536 3738 7320 2874 6872 6561 6429  335678s (thread)
│ │ │ │ -00033b40: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │ -00033b50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00033b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033b90: 2020 2020 2020 207c 0a7c 6f32 3920 3d20         |.|o29 = 
│ │ │ │ -00033ba0: 7472 7565 2020 2020 2020 2020 2020 2020  true            
│ │ │ │ +00033a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033a90: 2020 2020 2020 2020 7c0a 7c6f 3238 203d          |.|o28 =
│ │ │ │ +00033aa0: 2074 7275 6520 2020 2020 2020 2020 2020   true           
│ │ │ │ +00033ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033ad0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00033ae0: 2d2d 2d2d 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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00033b20: 7c69 3239 203a 2074 696d 6520 7265 6775  |i29 : time regu
│ │ │ │ +00033b30: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ +00033b40: 2831 2c20 532c 2053 7472 6174 6567 793d  (1, S, Strategy=
│ │ │ │ +00033b50: 3e53 7472 6174 6567 7943 7572 7265 6e74  >StrategyCurrent
│ │ │ │ +00033b60: 297c 0a7c 202d 2d20 7573 6564 2030 2e35  )|.| -- used 0.5
│ │ │ │ +00033b70: 3237 3035 3473 2028 6370 7529 3b20 302e  27054s (cpu); 0.
│ │ │ │ +00033b80: 3339 3134 3234 7320 2874 6872 6561 6429  391424s (thread)
│ │ │ │ +00033b90: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │ +00033ba0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00033bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033bd0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ -00033be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00033c20: 6933 3020 3a20 7469 6d65 2072 6567 756c  i30 : time regul
│ │ │ │ -00033c30: 6172 496e 436f 6469 6d65 6e73 696f 6e28  arInCodimension(
│ │ │ │ -00033c40: 312c 2053 2c20 5374 7261 7465 6779 3d3e  1, S, Strategy=>
│ │ │ │ -00033c50: 5374 7261 7465 6779 4375 7272 656e 7429  StrategyCurrent)
│ │ │ │ -00033c60: 7c0a 7c20 2d2d 2075 7365 6420 302e 3638  |.| -- used 0.68
│ │ │ │ -00033c70: 3136 7320 2863 7075 293b 2030 2e35 3435  16s (cpu); 0.545
│ │ │ │ -00033c80: 3533 3473 2028 7468 7265 6164 293b 2030  534s (thread); 0
│ │ │ │ -00033c90: 7320 2867 6329 2020 2020 2020 2020 2020  s (gc)          
│ │ │ │ -00033ca0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00033cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033ce0: 2020 2020 2020 7c0a 7c6f 3330 203d 2074        |.|o30 = t
│ │ │ │ -00033cf0: 7275 6520 2020 2020 2020 2020 2020 2020  rue             
│ │ │ │ +00033bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033be0: 2020 2020 2020 207c 0a7c 6f32 3920 3d20         |.|o29 = 
│ │ │ │ +00033bf0: 7472 7565 2020 2020 2020 2020 2020 2020  true            
│ │ │ │ +00033c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033c20: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00033c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00033c70: 6933 3020 3a20 7469 6d65 2072 6567 756c  i30 : time regul
│ │ │ │ +00033c80: 6172 496e 436f 6469 6d65 6e73 696f 6e28  arInCodimension(
│ │ │ │ +00033c90: 312c 2053 2c20 5374 7261 7465 6779 3d3e  1, S, Strategy=>
│ │ │ │ +00033ca0: 5374 7261 7465 6779 4375 7272 656e 7429  StrategyCurrent)
│ │ │ │ +00033cb0: 7c0a 7c20 2d2d 2075 7365 6420 302e 3835  |.| -- used 0.85
│ │ │ │ +00033cc0: 3639 3538 7320 2863 7075 293b 2030 2e36  6958s (cpu); 0.6
│ │ │ │ +00033cd0: 3435 3837 3373 2028 7468 7265 6164 293b  45873s (thread);
│ │ │ │ +00033ce0: 2030 7320 2867 6329 2020 2020 2020 2020   0s (gc)        
│ │ │ │ +00033cf0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00033d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033d20: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -00033d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00033d70: 3331 203a 2074 696d 6520 7265 6775 6c61  31 : time regula
│ │ │ │ -00033d80: 7249 6e43 6f64 696d 656e 7369 6f6e 2831  rInCodimension(1
│ │ │ │ -00033d90: 2c20 532c 2053 7472 6174 6567 793d 3e53  , S, Strategy=>S
│ │ │ │ -00033da0: 7472 6174 6567 7952 616e 646f 6d29 207c  trategyRandom) |
│ │ │ │ -00033db0: 0a7c 202d 2d20 7573 6564 2030 2e39 3930  .| -- used 0.990
│ │ │ │ -00033dc0: 3137 3373 2028 6370 7529 3b20 302e 3831  173s (cpu); 0.81
│ │ │ │ -00033dd0: 3535 3538 7320 2874 6872 6561 6429 3b20  5558s (thread); 
│ │ │ │ -00033de0: 3073 2028 6763 2920 2020 2020 2020 2020  0s (gc)         
│ │ │ │ -00033df0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00033e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033e30: 2020 2020 207c 0a7c 6f33 3120 3d20 7472       |.|o31 = tr
│ │ │ │ -00033e40: 7565 2020 2020 2020 2020 2020 2020 2020  ue              
│ │ │ │ +00033d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033d30: 2020 2020 2020 7c0a 7c6f 3330 203d 2074        |.|o30 = t
│ │ │ │ +00033d40: 7275 6520 2020 2020 2020 2020 2020 2020  rue             
│ │ │ │ +00033d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033d70: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00033d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00033dc0: 3331 203a 2074 696d 6520 7265 6775 6c61  31 : time regula
│ │ │ │ +00033dd0: 7249 6e43 6f64 696d 656e 7369 6f6e 2831  rInCodimension(1
│ │ │ │ +00033de0: 2c20 532c 2053 7472 6174 6567 793d 3e53  , S, Strategy=>S
│ │ │ │ +00033df0: 7472 6174 6567 7952 616e 646f 6d29 207c  trategyRandom) |
│ │ │ │ +00033e00: 0a7c 202d 2d20 7573 6564 2031 2e31 3739  .| -- used 1.179
│ │ │ │ +00033e10: 3136 7320 2863 7075 293b 2030 2e39 3039  16s (cpu); 0.909
│ │ │ │ +00033e20: 3336 3173 2028 7468 7265 6164 293b 2030  361s (thread); 0
│ │ │ │ +00033e30: 7320 2867 6329 2020 2020 2020 2020 2020  s (gc)          
│ │ │ │ +00033e40: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00033e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033e70: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -00033e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933  -----------+.|i3
│ │ │ │ -00033ec0: 3220 3a20 7469 6d65 2072 6567 756c 6172  2 : time regular
│ │ │ │ -00033ed0: 496e 436f 6469 6d65 6e73 696f 6e28 312c  InCodimension(1,
│ │ │ │ -00033ee0: 2053 2c20 5374 7261 7465 6779 3d3e 5374   S, Strategy=>St
│ │ │ │ -00033ef0: 7261 7465 6779 5261 6e64 6f6d 2920 7c0a  rategyRandom) |.
│ │ │ │ -00033f00: 7c20 2d2d 2075 7365 6420 312e 3638 3130  | -- used 1.6810
│ │ │ │ -00033f10: 3373 2028 6370 7529 3b20 312e 3331 3931  3s (cpu); 1.3191
│ │ │ │ -00033f20: 3873 2028 7468 7265 6164 293b 2030 7320  8s (thread); 0s 
│ │ │ │ -00033f30: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │ -00033f40: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00033f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033f80: 2020 2020 7c0a 7c6f 3332 203d 2074 7275      |.|o32 = tru
│ │ │ │ -00033f90: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │ +00033e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033e80: 2020 2020 207c 0a7c 6f33 3120 3d20 7472       |.|o31 = tr
│ │ │ │ +00033e90: 7565 2020 2020 2020 2020 2020 2020 2020  ue              
│ │ │ │ +00033ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033ec0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00033ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933  -----------+.|i3
│ │ │ │ +00033f10: 3220 3a20 7469 6d65 2072 6567 756c 6172  2 : time regular
│ │ │ │ +00033f20: 496e 436f 6469 6d65 6e73 696f 6e28 312c  InCodimension(1,
│ │ │ │ +00033f30: 2053 2c20 5374 7261 7465 6779 3d3e 5374   S, Strategy=>St
│ │ │ │ +00033f40: 7261 7465 6779 5261 6e64 6f6d 2920 7c0a  rategyRandom) |.
│ │ │ │ +00033f50: 7c20 2d2d 2075 7365 6420 312e 3939 3037  | -- used 1.9907
│ │ │ │ +00033f60: 3673 2028 6370 7529 3b20 312e 3533 3937  6s (cpu); 1.5397
│ │ │ │ +00033f70: 3373 2028 7468 7265 6164 293b 2030 7320  3s (thread); 0s 
│ │ │ │ +00033f80: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │ +00033f90: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00033fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033fc0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -00033fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034000: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a54 6865  ----------+..The
│ │ │ │ -00034010: 206d 696e 696d 756d 206e 756d 6265 7220   minimum number 
│ │ │ │ -00034020: 6f66 206d 696e 6f72 7320 636f 6d70 7574  of minors comput
│ │ │ │ -00034030: 6564 2062 6566 6f72 6520 6368 6563 6b69  ed before checki
│ │ │ │ -00034040: 6e67 2074 6865 2063 6f64 696d 656e 7369  ng the codimensi
│ │ │ │ -00034050: 6f6e 2063 616e 2061 6c73 6f0a 6265 2063  on can also.be c
│ │ │ │ -00034060: 6f6e 7472 6f6c 6c65 6420 6279 2061 6e20  ontrolled by an 
│ │ │ │ -00034070: 6f70 7469 6f6e 204d 696e 4d69 6e6f 7273  option MinMinors
│ │ │ │ -00034080: 4675 6e63 7469 6f6e 2e20 2054 6869 7320  Function.  This 
│ │ │ │ -00034090: 6973 2073 686f 756c 6420 6265 2061 2066  is should be a f
│ │ │ │ -000340a0: 756e 6374 696f 6e20 6f66 0a61 2073 696e  unction of.a sin
│ │ │ │ -000340b0: 676c 6520 7661 7269 6162 6c65 2c20 7468  gle variable, th
│ │ │ │ -000340c0: 6520 6e75 6d62 6572 206f 6620 6d69 6e6f  e number of mino
│ │ │ │ -000340d0: 7273 2063 6f6d 7075 7465 642e 2020 4669  rs computed.  Fi
│ │ │ │ -000340e0: 6e61 6c6c 792c 2076 6961 2074 6865 206f  nally, via the o
│ │ │ │ -000340f0: 7074 696f 6e0a 436f 6469 6d43 6865 636b  ption.CodimCheck
│ │ │ │ -00034100: 4675 6e63 7469 6f6e 2c20 796f 7520 6361  Function, you ca
│ │ │ │ -00034110: 6e20 7061 7373 2074 6865 2072 6567 756c  n pass the regul
│ │ │ │ -00034120: 6172 496e 436f 6469 6d65 6e73 696f 6e20  arInCodimension 
│ │ │ │ -00034130: 6120 6675 6e63 7469 6f6e 2077 6869 6368  a function which
│ │ │ │ -00034140: 0a63 6f6e 7472 6f6c 7320 686f 7720 6672  .controls how fr
│ │ │ │ -00034150: 6571 7565 6e74 6c79 2074 6865 2063 6f64  equently the cod
│ │ │ │ -00034160: 696d 656e 7369 6f6e 206f 6620 7468 6520  imension of the 
│ │ │ │ -00034170: 7061 7274 6961 6c20 4a61 636f 6269 616e  partial Jacobian
│ │ │ │ -00034180: 2069 6465 616c 2069 730a 636f 6d70 7574   ideal is.comput
│ │ │ │ -00034190: 6564 2e20 2042 7920 6465 6661 756c 7420  ed.  By default 
│ │ │ │ -000341a0: 7468 6973 2069 7320 7468 6520 666c 6f6f  this is the floo
│ │ │ │ -000341b0: 7220 6f66 2031 2e33 5e6b 2e20 4669 6e61  r of 1.3^k. Fina
│ │ │ │ -000341c0: 6c6c 792c 2070 6173 7369 6e67 2074 6865  lly, passing the
│ │ │ │ -000341d0: 206f 7074 696f 6e0a 4d6f 6475 6c75 7320   option.Modulus 
│ │ │ │ -000341e0: 3d3e 2070 2077 696c 6c20 646f 2074 6865  => p will do the
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│ │ │ │ -000347d0: 6174 202a 6e6f 7465 0a46 6173 744d 696e  at *note.FastMin
│ │ │ │ -000347e0: 6f72 7353 7472 6174 6567 7954 7574 6f72  orsStrategyTutor
│ │ │ │ -000347f0: 6961 6c3a 2046 6173 744d 696e 6f72 7353  ial: FastMinorsS
│ │ │ │ -00034800: 7472 6174 6567 7954 7574 6f72 6961 6c2c  trategyTutorial,
│ │ │ │ -00034810: 2e0a 0a57 6520 6265 6769 6e20 7769 7468  ...We begin with
│ │ │ │ -00034820: 2074 6865 2066 6f6c 6c6f 7769 6e67 2069   the following i
│ │ │ │ -00034830: 6465 616c 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  deal...+--------
│ │ │ │ -00034840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034880: 2d2d 2d2d 2d2b 0a7c 6931 203a 2053 203d  -----+.|i1 : S =
│ │ │ │ -00034890: 205a 5a2f 3130 335b 785f 312e 2e78 5f39   ZZ/103[x_1..x_9
│ │ │ │ -000348a0: 5d3b 2020 2020 2020 2020 2020 2020 2020  ];              
│ │ │ │ -000348b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000348c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000348d0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -000348e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000348f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034920: 2d2d 2d2d 2d2b 0a7c 6932 203a 204a 203d  -----+.|i2 : J =
│ │ │ │ -00034930: 2069 6465 616c 2878 5f36 2a78 5f38 2d78   ideal(x_6*x_8-x
│ │ │ │ -00034940: 5f35 2a78 5f39 2c78 5f33 2a78 5f38 2d78  _5*x_9,x_3*x_8-x
│ │ │ │ -00034950: 5f32 2a78 5f39 2c78 5f36 2a78 5f37 2d78  _2*x_9,x_6*x_7-x
│ │ │ │ -00034960: 5f34 2a78 5f39 2c78 5f35 2a78 5f37 2d78  _4*x_9,x_5*x_7-x
│ │ │ │ -00034970: 5f34 2a78 5f7c 0a7c 2020 2020 2020 2020  _4*x_|.|        
│ │ │ │ -00034980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000349a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000349b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000349c0: 2020 2020 207c 0a7c 6f32 203a 2049 6465       |.|o2 : Ide
│ │ │ │ -000349d0: 616c 206f 6620 5320 2020 2020 2020 2020  al of S         
│ │ │ │ +000346d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000346e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000346f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00034700: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00034710: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00034720: 2a2a 2a2a 2a2a 0a0a 4465 7363 7269 7074  ******..Descript
│ │ │ │ +00034730: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ion.===========.
│ │ │ │ +00034740: 0a49 6e20 7468 6973 2074 7574 6f72 6961  .In this tutoria
│ │ │ │ +00034750: 6c20 7765 2065 7870 6c6f 7265 2074 6865  l we explore the
│ │ │ │ +00034760: 2064 6966 6665 7265 6e74 206f 7074 696f   different optio
│ │ │ │ +00034770: 6e73 206f 6620 5265 6775 6c61 7249 6e43  ns of RegularInC
│ │ │ │ +00034780: 6f64 696d 656e 7369 6f6e 2028 616e 640a  odimension (and.
│ │ │ │ +00034790: 7265 6c61 7465 6420 6675 6e63 7469 6f6e  related function
│ │ │ │ +000347a0: 7329 206f 6e20 736f 6d65 2063 6f6e 6520  s) on some cone 
│ │ │ │ +000347b0: 7369 6e67 756c 6172 6974 6965 732e 2020  singularities.  
│ │ │ │ +000347c0: 466f 7220 7468 6520 6d6f 7374 2070 6172  For the most par
│ │ │ │ +000347d0: 7420 7765 2077 696c 6c20 6e6f 740a 7461  t we will not.ta
│ │ │ │ +000347e0: 6c6b 2061 626f 7574 2074 6865 2053 7472  lk about the Str
│ │ │ │ +000347f0: 6174 6567 7920 6f70 7469 6f6e 2c20 7765  ategy option, we
│ │ │ │ +00034800: 2068 6176 6520 6120 7365 7061 7261 7465   have a separate
│ │ │ │ +00034810: 2074 7574 6f72 6961 6c20 666f 7220 7468   tutorial for th
│ │ │ │ +00034820: 6174 202a 6e6f 7465 0a46 6173 744d 696e  at *note.FastMin
│ │ │ │ +00034830: 6f72 7353 7472 6174 6567 7954 7574 6f72  orsStrategyTutor
│ │ │ │ +00034840: 6961 6c3a 2046 6173 744d 696e 6f72 7353  ial: FastMinorsS
│ │ │ │ +00034850: 7472 6174 6567 7954 7574 6f72 6961 6c2c  trategyTutorial,
│ │ │ │ +00034860: 2e0a 0a57 6520 6265 6769 6e20 7769 7468  ...We begin with
│ │ │ │ +00034870: 2074 6865 2066 6f6c 6c6f 7769 6e67 2069   the following i
│ │ │ │ +00034880: 6465 616c 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  deal...+--------
│ │ │ │ +00034890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000348a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000348b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000348c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000348d0: 2d2d 2d2d 2d2b 0a7c 6931 203a 2053 203d  -----+.|i1 : S =
│ │ │ │ +000348e0: 205a 5a2f 3130 335b 785f 312e 2e78 5f39   ZZ/103[x_1..x_9
│ │ │ │ +000348f0: 5d3b 2020 2020 2020 2020 2020 2020 2020  ];              
│ │ │ │ +00034900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034920: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00034930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034970: 2d2d 2d2d 2d2b 0a7c 6932 203a 204a 203d  -----+.|i2 : J =
│ │ │ │ +00034980: 2069 6465 616c 2878 5f36 2a78 5f38 2d78   ideal(x_6*x_8-x
│ │ │ │ +00034990: 5f35 2a78 5f39 2c78 5f33 2a78 5f38 2d78  _5*x_9,x_3*x_8-x
│ │ │ │ +000349a0: 5f32 2a78 5f39 2c78 5f36 2a78 5f37 2d78  _2*x_9,x_6*x_7-x
│ │ │ │ +000349b0: 5f34 2a78 5f39 2c78 5f35 2a78 5f37 2d78  _4*x_9,x_5*x_7-x
│ │ │ │ +000349c0: 5f34 2a78 5f7c 0a7c 2020 2020 2020 2020  _4*x_|.|        
│ │ │ │ +000349d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000349e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000349f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034a10: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ -00034a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034a60: 2d2d 2d2d 2d7c 0a7c 382c 785f 332a 785f  -----|.|8,x_3*x_
│ │ │ │ -00034a70: 372d 785f 312a 785f 392c 785f 322a 785f  7-x_1*x_9,x_2*x_
│ │ │ │ -00034a80: 372d 785f 312a 785f 382c 785f 332a 785f  7-x_1*x_8,x_3*x_
│ │ │ │ -00034a90: 352d 785f 322a 785f 362c 785f 332a 785f  5-x_2*x_6,x_3*x_
│ │ │ │ -00034aa0: 342d 785f 312a 785f 362c 785f 322a 785f  4-x_1*x_6,x_2*x_
│ │ │ │ -00034ab0: 342d 785f 317c 0a7c 2d2d 2d2d 2d2d 2d2d  4-x_1|.|--------
│ │ │ │ -00034ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034b00: 2d2d 2d2d 2d7c 0a7c 2a78 5f35 2c78 5f33  -----|.|*x_5,x_3
│ │ │ │ -00034b10: 5e33 2d78 5f36 5e33 2d78 5f39 5e33 2c78  ^3-x_6^3-x_9^3,x
│ │ │ │ -00034b20: 5f32 2a78 5f33 5e32 2d78 5f35 2a78 5f36  _2*x_3^2-x_5*x_6
│ │ │ │ -00034b30: 5e32 2d78 5f38 2a78 5f39 5e32 2c78 5f31  ^2-x_8*x_9^2,x_1
│ │ │ │ -00034b40: 2a78 5f33 5e32 2d78 5f34 2a78 5f36 5e32  *x_3^2-x_4*x_6^2
│ │ │ │ -00034b50: 2d78 5f37 2a7c 0a7c 2d2d 2d2d 2d2d 2d2d  -x_7*|.|--------
│ │ │ │ -00034b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034ba0: 2d2d 2d2d 2d7c 0a7c 785f 395e 322c 785f  -----|.|x_9^2,x_
│ │ │ │ -00034bb0: 325e 322a 785f 332d 785f 355e 322a 785f  2^2*x_3-x_5^2*x_
│ │ │ │ -00034bc0: 362d 785f 385e 322a 785f 392c 785f 312a  6-x_8^2*x_9,x_1*
│ │ │ │ -00034bd0: 785f 322a 785f 332d 785f 342a 785f 352a  x_2*x_3-x_4*x_5*
│ │ │ │ -00034be0: 785f 362d 785f 372a 785f 382a 785f 392c  x_6-x_7*x_8*x_9,
│ │ │ │ -00034bf0: 785f 315e 327c 0a7c 2d2d 2d2d 2d2d 2d2d  x_1^2|.|--------
│ │ │ │ -00034c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034c40: 2d2d 2d2d 2d7c 0a7c 2a78 5f33 2d78 5f34  -----|.|*x_3-x_4
│ │ │ │ -00034c50: 5e32 2a78 5f36 2d78 5f37 5e32 2a78 5f39  ^2*x_6-x_7^2*x_9
│ │ │ │ -00034c60: 2c78 5f32 5e33 2d78 5f35 5e33 2d78 5f38  ,x_2^3-x_5^3-x_8
│ │ │ │ -00034c70: 5e33 2c78 5f31 2a78 5f32 5e32 2d78 5f34  ^3,x_1*x_2^2-x_4
│ │ │ │ -00034c80: 2a78 5f35 5e32 2d78 5f37 2a78 5f38 5e32  *x_5^2-x_7*x_8^2
│ │ │ │ -00034c90: 2c78 5f31 5e7c 0a7c 2d2d 2d2d 2d2d 2d2d  ,x_1^|.|--------
│ │ │ │ -00034ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034ce0: 2d2d 2d2d 2d7c 0a7c 322a 785f 322d 785f  -----|.|2*x_2-x_
│ │ │ │ -00034cf0: 345e 322a 785f 352d 785f 375e 322a 785f  4^2*x_5-x_7^2*x_
│ │ │ │ -00034d00: 382c 785f 315e 332d 785f 345e 332d 785f  8,x_1^3-x_4^3-x_
│ │ │ │ -00034d10: 375e 3329 3b20 2020 2020 2020 2020 2020  7^3);           
│ │ │ │ -00034d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034d30: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -00034d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034d80: 2d2d 2d2d 2d2b 0a7c 6933 203a 2064 696d  -----+.|i3 : dim
│ │ │ │ -00034d90: 2028 532f 4a29 2020 2020 2020 2020 2020   (S/J)          
│ │ │ │ -00034da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034dd0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00034de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034a10: 2020 2020 207c 0a7c 6f32 203a 2049 6465       |.|o2 : Ide
│ │ │ │ +00034a20: 616c 206f 6620 5320 2020 2020 2020 2020  al of S         
│ │ │ │ +00034a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034a60: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +00034a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034ab0: 2d2d 2d2d 2d7c 0a7c 382c 785f 332a 785f  -----|.|8,x_3*x_
│ │ │ │ +00034ac0: 372d 785f 312a 785f 392c 785f 322a 785f  7-x_1*x_9,x_2*x_
│ │ │ │ +00034ad0: 372d 785f 312a 785f 382c 785f 332a 785f  7-x_1*x_8,x_3*x_
│ │ │ │ +00034ae0: 352d 785f 322a 785f 362c 785f 332a 785f  5-x_2*x_6,x_3*x_
│ │ │ │ +00034af0: 342d 785f 312a 785f 362c 785f 322a 785f  4-x_1*x_6,x_2*x_
│ │ │ │ +00034b00: 342d 785f 317c 0a7c 2d2d 2d2d 2d2d 2d2d  4-x_1|.|--------
│ │ │ │ +00034b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034b50: 2d2d 2d2d 2d7c 0a7c 2a78 5f35 2c78 5f33  -----|.|*x_5,x_3
│ │ │ │ +00034b60: 5e33 2d78 5f36 5e33 2d78 5f39 5e33 2c78  ^3-x_6^3-x_9^3,x
│ │ │ │ +00034b70: 5f32 2a78 5f33 5e32 2d78 5f35 2a78 5f36  _2*x_3^2-x_5*x_6
│ │ │ │ +00034b80: 5e32 2d78 5f38 2a78 5f39 5e32 2c78 5f31  ^2-x_8*x_9^2,x_1
│ │ │ │ +00034b90: 2a78 5f33 5e32 2d78 5f34 2a78 5f36 5e32  *x_3^2-x_4*x_6^2
│ │ │ │ +00034ba0: 2d78 5f37 2a7c 0a7c 2d2d 2d2d 2d2d 2d2d  -x_7*|.|--------
│ │ │ │ +00034bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034bf0: 2d2d 2d2d 2d7c 0a7c 785f 395e 322c 785f  -----|.|x_9^2,x_
│ │ │ │ +00034c00: 325e 322a 785f 332d 785f 355e 322a 785f  2^2*x_3-x_5^2*x_
│ │ │ │ +00034c10: 362d 785f 385e 322a 785f 392c 785f 312a  6-x_8^2*x_9,x_1*
│ │ │ │ +00034c20: 785f 322a 785f 332d 785f 342a 785f 352a  x_2*x_3-x_4*x_5*
│ │ │ │ +00034c30: 785f 362d 785f 372a 785f 382a 785f 392c  x_6-x_7*x_8*x_9,
│ │ │ │ +00034c40: 785f 315e 327c 0a7c 2d2d 2d2d 2d2d 2d2d  x_1^2|.|--------
│ │ │ │ +00034c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034c90: 2d2d 2d2d 2d7c 0a7c 2a78 5f33 2d78 5f34  -----|.|*x_3-x_4
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│ │ │ │ +00034cb0: 2c78 5f32 5e33 2d78 5f35 5e33 2d78 5f38  ,x_2^3-x_5^3-x_8
│ │ │ │ +00034cc0: 5e33 2c78 5f31 2a78 5f32 5e32 2d78 5f34  ^3,x_1*x_2^2-x_4
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│ │ │ │ +00034ce0: 2c78 5f31 5e7c 0a7c 2d2d 2d2d 2d2d 2d2d  ,x_1^|.|--------
│ │ │ │ +00034cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034d30: 2d2d 2d2d 2d7c 0a7c 322a 785f 322d 785f  -----|.|2*x_2-x_
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│ │ │ │ +00034d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00034d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00034de0: 2028 532f 4a29 2020 2020 2020 2020 2020   (S/J)          
│ │ │ │  00034df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034e20: 2020 2020 207c 0a7c 6f33 203d 2034 2020       |.|o3 = 4  
│ │ │ │ +00034e20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00034e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00034e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00034ef0: 2024 4524 2069 7320 616e 2065 6c6c 6970   $E$ is an ellip
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│ │ │ │ -00034f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00034fe0: 532f 4a29 2020 2020 2020 2020 2020 2020  S/J)            
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│ │ │ │ -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 207c 0a7c 6f34 203d 2074 7275 6520     |.|o4 = true 
│ │ │ │ -00035070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034e70: 2020 2020 207c 0a7c 6f33 203d 2034 2020       |.|o3 = 4  
│ │ │ │ +00034e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034ec0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00034ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034f10: 2d2d 2d2d 2d2b 0a0a 4974 2069 7320 7468  -----+..It is th
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│ │ │ │ +00034f40: 2024 4524 2069 7320 616e 2065 6c6c 6970   $E$ is an ellip
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│ │ │ │ +00034f70: 7769 7468 2061 2053 6567 7265 2065 6d62  with a Segre emb
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│ │ │ │ +00034ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00035030: 2f4a 2920 2020 2020 2020 2020 2020 2020  /J)             
│ │ │ │ +00035040: 207c 0a7c 202d 2d20 7573 6564 2031 2e30   |.| -- used 1.0
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│ │ │ │  00035080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035090: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -000350a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000350b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000350c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00035160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00035470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00035480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00035490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -000354f0: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64   |.|regularInCod
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│ │ │ │ -00035580: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00035090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000350a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000350b0: 0a7c 6f34 203d 2074 7275 6520 2020 2020  .|o4 = true     
│ │ │ │ +000350c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000350e0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +000350f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00035160: 3132 2e35 3636 3673 2028 6370 7529 3b20  12.5666s (cpu); 
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│ │ │ │ +00035190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000351a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000351b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000351c0: 2d2d 2b0a 0a57 6520 7472 7920 746f 2076  --+..We try to v
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│ │ │ │ +000354d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00035a00: 4368 6f6f 7369 6e67 2047 5265 764c 6578  Choosing GRevLex
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│ │ │ │ +00035b10: 6f6e 2063 6f6d 7075 7465 642c 203d 2033  on computed, = 3
│ │ │ │ +00035b20: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +00035b30: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +00035b40: 4368 6f6f 7369 6e67 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ +00035b50: 6c65 7374 2020 2020 2020 2020 2020 2020  lest            
│ │ │ │ +00035b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035b70: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +00035b80: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +00035b90: 4368 6f6f 7369 6e67 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ +00035ba0: 6c65 7374 5465 726d 2020 2020 2020 2020  lestTerm        
│ │ │ │  00035bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035bd0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ -00035be0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ -00035bf0: 6720 4752 6576 4c65 7853 6d61 6c6c 6573  g GRevLexSmalles
│ │ │ │ -00035c00: 7454 6572 6d20 2020 2020 2020 2020 2020  tTerm           
│ │ │ │ -00035c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035c20: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64   |.|regularInCod
│ │ │ │ -00035c30: 696d 656e 7369 6f6e 3a20 204c 6f6f 7020  imension:  Loop 
│ │ │ │ -00035c40: 7374 6570 2c20 6162 6f75 7420 746f 2063  step, about to c
│ │ │ │ -00035c50: 6f6d 7075 7465 2064 696d 656e 7369 6f6e  ompute dimension
│ │ │ │ -00035c60: 2e20 2053 7562 6d61 7472 6963 6573 2063  .  Submatrices c
│ │ │ │ -00035c70: 6f7c 0a7c 7265 6775 6c61 7249 6e43 6f64  o|.|regularInCod
│ │ │ │ -00035c80: 696d 656e 7369 6f6e 3a20 2069 7343 6f64  imension:  isCod
│ │ │ │ -00035c90: 696d 4174 4c65 6173 7420 6661 696c 6564  imAtLeast failed
│ │ │ │ -00035ca0: 2c20 636f 6d70 7574 696e 6720 636f 6469  , computing codi
│ │ │ │ -00035cb0: 6d2e 2020 2020 2020 2020 2020 2020 2020  m.              
│ │ │ │ -00035cc0: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64   |.|regularInCod
│ │ │ │ -00035cd0: 696d 656e 7369 6f6e 3a20 2070 6172 7469  imension:  parti
│ │ │ │ -00035ce0: 616c 2073 696e 6775 6c61 7220 6c6f 6375  al singular locu
│ │ │ │ -00035cf0: 7320 6469 6d65 6e73 696f 6e20 636f 6d70  s dimension comp
│ │ │ │ -00035d00: 7574 6564 2c20 3d20 3320 2020 2020 2020  uted, = 3       
│ │ │ │ -00035d10: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ -00035d20: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ -00035d30: 6720 5261 6e64 6f6d 4e6f 6e5a 6572 6f20  g RandomNonZero 
│ │ │ │ -00035d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035d60: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ -00035d70: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ -00035d80: 6720 5261 6e64 6f6d 4e6f 6e5a 6572 6f20  g RandomNonZero 
│ │ │ │ +00035bc0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +00035bd0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +00035be0: 4368 6f6f 7369 6e67 2052 616e 646f 6d20  Choosing Random 
│ │ │ │ +00035bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035c10: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +00035c20: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +00035c30: 4368 6f6f 7369 6e67 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ +00035c40: 536d 616c 6c65 7374 5465 726d 2020 2020  SmallestTerm    
│ │ │ │ +00035c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035c60: 2020 2020 2020 2020 7c0a 7c72 6567 756c          |.|regul
│ │ │ │ +00035c70: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +00035c80: 2020 4c6f 6f70 2073 7465 702c 2061 626f    Loop step, abo
│ │ │ │ +00035c90: 7574 2074 6f20 636f 6d70 7574 6520 6469  ut to compute di
│ │ │ │ +00035ca0: 6d65 6e73 696f 6e2e 2020 5375 626d 6174  mension.  Submat
│ │ │ │ +00035cb0: 7269 6365 7320 636f 7c0a 7c72 6567 756c  rices co|.|regul
│ │ │ │ +00035cc0: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +00035cd0: 2020 6973 436f 6469 6d41 744c 6561 7374    isCodimAtLeast
│ │ │ │ +00035ce0: 2066 6169 6c65 642c 2063 6f6d 7075 7469   failed, computi
│ │ │ │ +00035cf0: 6e67 2063 6f64 696d 2e20 2020 2020 2020  ng codim.       
│ │ │ │ +00035d00: 2020 2020 2020 2020 7c0a 7c72 6567 756c          |.|regul
│ │ │ │ +00035d10: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +00035d20: 2020 7061 7274 6961 6c20 7369 6e67 756c    partial singul
│ │ │ │ +00035d30: 6172 206c 6f63 7573 2064 696d 656e 7369  ar locus dimensi
│ │ │ │ +00035d40: 6f6e 2063 6f6d 7075 7465 642c 203d 2033  on computed, = 3
│ │ │ │ +00035d50: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +00035d60: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +00035d70: 4368 6f6f 7369 6e67 2052 616e 646f 6d4e  Choosing RandomN
│ │ │ │ +00035d80: 6f6e 5a65 726f 2020 2020 2020 2020 2020  onZero          
│ │ │ │  00035d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035db0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ -00035dc0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ -00035dd0: 6720 5261 6e64 6f6d 4e6f 6e5a 6572 6f20  g RandomNonZero 
│ │ │ │ +00035da0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +00035db0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +00035dc0: 4368 6f6f 7369 6e67 2052 616e 646f 6d4e  Choosing RandomN
│ │ │ │ +00035dd0: 6f6e 5a65 726f 2020 2020 2020 2020 2020  onZero          
│ │ │ │  00035de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035e00: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ -00035e10: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ -00035e20: 6720 4752 6576 4c65 7853 6d61 6c6c 6573  g GRevLexSmalles
│ │ │ │ -00035e30: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │ -00035e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035e50: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ -00035e60: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ -00035e70: 6720 4c65 7853 6d61 6c6c 6573 7454 6572  g LexSmallestTer
│ │ │ │ -00035e80: 6d20 2020 2020 2020 2020 2020 2020 2020  m               
│ │ │ │ -00035e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035ea0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ -00035eb0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ -00035ec0: 6720 5261 6e64 6f6d 2020 2020 2020 2020  g Random        
│ │ │ │ +00035df0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +00035e00: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +00035e10: 4368 6f6f 7369 6e67 2052 616e 646f 6d4e  Choosing RandomN
│ │ │ │ +00035e20: 6f6e 5a65 726f 2020 2020 2020 2020 2020  onZero          
│ │ │ │ +00035e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035e40: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +00035e50: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +00035e60: 4368 6f6f 7369 6e67 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ +00035e70: 536d 616c 6c65 7374 2020 2020 2020 2020  Smallest        
│ │ │ │ +00035e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035e90: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +00035ea0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +00035eb0: 4368 6f6f 7369 6e67 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ +00035ec0: 6c65 7374 5465 726d 2020 2020 2020 2020  lestTerm        
│ │ │ │  00035ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035ef0: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64   |.|regularInCod
│ │ │ │ -00035f00: 696d 656e 7369 6f6e 3a20 204c 6f6f 7020  imension:  Loop 
│ │ │ │ -00035f10: 7374 6570 2c20 6162 6f75 7420 746f 2063  step, about to c
│ │ │ │ -00035f20: 6f6d 7075 7465 2064 696d 656e 7369 6f6e  ompute dimension
│ │ │ │ -00035f30: 2e20 2053 7562 6d61 7472 6963 6573 2063  .  Submatrices c
│ │ │ │ -00035f40: 6f7c 0a7c 7265 6775 6c61 7249 6e43 6f64  o|.|regularInCod
│ │ │ │ -00035f50: 696d 656e 7369 6f6e 3a20 2069 7343 6f64  imension:  isCod
│ │ │ │ -00035f60: 696d 4174 4c65 6173 7420 6661 696c 6564  imAtLeast failed
│ │ │ │ -00035f70: 2c20 636f 6d70 7574 696e 6720 636f 6469  , computing codi
│ │ │ │ -00035f80: 6d2e 2020 2020 2020 2020 2020 2020 2020  m.              
│ │ │ │ -00035f90: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64   |.|regularInCod
│ │ │ │ -00035fa0: 696d 656e 7369 6f6e 3a20 2070 6172 7469  imension:  parti
│ │ │ │ -00035fb0: 616c 2073 696e 6775 6c61 7220 6c6f 6375  al singular locu
│ │ │ │ -00035fc0: 7320 6469 6d65 6e73 696f 6e20 636f 6d70  s dimension comp
│ │ │ │ -00035fd0: 7574 6564 2c20 3d20 3320 2020 2020 2020  uted, = 3       
│ │ │ │ -00035fe0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ -00035ff0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ -00036000: 6720 5261 6e64 6f6d 4e6f 6e5a 6572 6f20  g RandomNonZero 
│ │ │ │ -00036010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036030: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ -00036040: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ -00036050: 6720 4752 6576 4c65 7853 6d61 6c6c 6573  g GRevLexSmalles
│ │ │ │ -00036060: 7454 6572 6d20 2020 2020 2020 2020 2020  tTerm           
│ │ │ │ -00036070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036080: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ -00036090: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ -000360a0: 6720 5261 6e64 6f6d 2020 2020 2020 2020  g Random        
│ │ │ │ +00035ee0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +00035ef0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +00035f00: 4368 6f6f 7369 6e67 2052 616e 646f 6d20  Choosing Random 
│ │ │ │ +00035f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035f30: 2020 2020 2020 2020 7c0a 7c72 6567 756c          |.|regul
│ │ │ │ +00035f40: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +00035f50: 2020 4c6f 6f70 2073 7465 702c 2061 626f    Loop step, abo
│ │ │ │ +00035f60: 7574 2074 6f20 636f 6d70 7574 6520 6469  ut to compute di
│ │ │ │ +00035f70: 6d65 6e73 696f 6e2e 2020 5375 626d 6174  mension.  Submat
│ │ │ │ +00035f80: 7269 6365 7320 636f 7c0a 7c72 6567 756c  rices co|.|regul
│ │ │ │ +00035f90: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +00035fa0: 2020 6973 436f 6469 6d41 744c 6561 7374    isCodimAtLeast
│ │ │ │ +00035fb0: 2066 6169 6c65 642c 2063 6f6d 7075 7469   failed, computi
│ │ │ │ +00035fc0: 6e67 2063 6f64 696d 2e20 2020 2020 2020  ng codim.       
│ │ │ │ +00035fd0: 2020 2020 2020 2020 7c0a 7c72 6567 756c          |.|regul
│ │ │ │ +00035fe0: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +00035ff0: 2020 7061 7274 6961 6c20 7369 6e67 756c    partial singul
│ │ │ │ +00036000: 6172 206c 6f63 7573 2064 696d 656e 7369  ar locus dimensi
│ │ │ │ +00036010: 6f6e 2063 6f6d 7075 7465 642c 203d 2033  on computed, = 3
│ │ │ │ +00036020: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +00036030: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +00036040: 4368 6f6f 7369 6e67 2052 616e 646f 6d4e  Choosing RandomN
│ │ │ │ +00036050: 6f6e 5a65 726f 2020 2020 2020 2020 2020  onZero          
│ │ │ │ +00036060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036070: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +00036080: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +00036090: 4368 6f6f 7369 6e67 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ +000360a0: 536d 616c 6c65 7374 5465 726d 2020 2020  SmallestTerm    
│ │ │ │  000360b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000360c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000360d0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ -000360e0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ -000360f0: 6720 4c65 7853 6d61 6c6c 6573 7454 6572  g LexSmallestTer
│ │ │ │ -00036100: 6d20 2020 2020 2020 2020 2020 2020 2020  m               
│ │ │ │ -00036110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036120: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ -00036130: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ -00036140: 6720 4752 6576 4c65 7853 6d61 6c6c 6573  g GRevLexSmalles
│ │ │ │ -00036150: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │ -00036160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036170: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ -00036180: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ -00036190: 6720 4c65 7853 6d61 6c6c 6573 7420 2020  g LexSmallest   
│ │ │ │ +000360c0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +000360d0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +000360e0: 4368 6f6f 7369 6e67 2052 616e 646f 6d20  Choosing Random 
│ │ │ │ +000360f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036110: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +00036120: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +00036130: 4368 6f6f 7369 6e67 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ +00036140: 6c65 7374 5465 726d 2020 2020 2020 2020  lestTerm        
│ │ │ │ +00036150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036160: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +00036170: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +00036180: 4368 6f6f 7369 6e67 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ +00036190: 536d 616c 6c65 7374 2020 2020 2020 2020  Smallest        
│ │ │ │  000361a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000361b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000361c0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ -000361d0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ -000361e0: 6720 4c65 7853 6d61 6c6c 6573 7454 6572  g LexSmallestTer
│ │ │ │ -000361f0: 6d20 2020 2020 2020 2020 2020 2020 2020  m               
│ │ │ │ -00036200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036210: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64   |.|regularInCod
│ │ │ │ -00036220: 696d 656e 7369 6f6e 3a20 204c 6f6f 7020  imension:  Loop 
│ │ │ │ -00036230: 7374 6570 2c20 6162 6f75 7420 746f 2063  step, about to c
│ │ │ │ -00036240: 6f6d 7075 7465 2064 696d 656e 7369 6f6e  ompute dimension
│ │ │ │ -00036250: 2e20 2053 7562 6d61 7472 6963 6573 2063  .  Submatrices c
│ │ │ │ -00036260: 6f7c 0a7c 7265 6775 6c61 7249 6e43 6f64  o|.|regularInCod
│ │ │ │ -00036270: 696d 656e 7369 6f6e 3a20 2069 7343 6f64  imension:  isCod
│ │ │ │ -00036280: 696d 4174 4c65 6173 7420 6661 696c 6564  imAtLeast failed
│ │ │ │ -00036290: 2c20 636f 6d70 7574 696e 6720 636f 6469  , computing codi
│ │ │ │ -000362a0: 6d2e 2020 2020 2020 2020 2020 2020 2020  m.              
│ │ │ │ -000362b0: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64   |.|regularInCod
│ │ │ │ -000362c0: 696d 656e 7369 6f6e 3a20 2070 6172 7469  imension:  parti
│ │ │ │ -000362d0: 616c 2073 696e 6775 6c61 7220 6c6f 6375  al singular locu
│ │ │ │ -000362e0: 7320 6469 6d65 6e73 696f 6e20 636f 6d70  s dimension comp
│ │ │ │ -000362f0: 7574 6564 2c20 3d20 3320 2020 2020 2020  uted, = 3       
│ │ │ │ -00036300: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ -00036310: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ -00036320: 6720 4752 6576 4c65 7853 6d61 6c6c 6573  g GRevLexSmalles
│ │ │ │ -00036330: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │ -00036340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036350: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ -00036360: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ -00036370: 6720 4c65 7853 6d61 6c6c 6573 7420 2020  g LexSmallest   
│ │ │ │ +000361b0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +000361c0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +000361d0: 4368 6f6f 7369 6e67 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ +000361e0: 6c65 7374 2020 2020 2020 2020 2020 2020  lest            
│ │ │ │ +000361f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036200: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +00036210: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +00036220: 4368 6f6f 7369 6e67 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ +00036230: 6c65 7374 5465 726d 2020 2020 2020 2020  lestTerm        
│ │ │ │ +00036240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036250: 2020 2020 2020 2020 7c0a 7c72 6567 756c          |.|regul
│ │ │ │ +00036260: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +00036270: 2020 4c6f 6f70 2073 7465 702c 2061 626f    Loop step, abo
│ │ │ │ +00036280: 7574 2074 6f20 636f 6d70 7574 6520 6469  ut to compute di
│ │ │ │ +00036290: 6d65 6e73 696f 6e2e 2020 5375 626d 6174  mension.  Submat
│ │ │ │ +000362a0: 7269 6365 7320 636f 7c0a 7c72 6567 756c  rices co|.|regul
│ │ │ │ +000362b0: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +000362c0: 2020 6973 436f 6469 6d41 744c 6561 7374    isCodimAtLeast
│ │ │ │ +000362d0: 2066 6169 6c65 642c 2063 6f6d 7075 7469   failed, computi
│ │ │ │ +000362e0: 6e67 2063 6f64 696d 2e20 2020 2020 2020  ng codim.       
│ │ │ │ +000362f0: 2020 2020 2020 2020 7c0a 7c72 6567 756c          |.|regul
│ │ │ │ +00036300: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +00036310: 2020 7061 7274 6961 6c20 7369 6e67 756c    partial singul
│ │ │ │ +00036320: 6172 206c 6f63 7573 2064 696d 656e 7369  ar locus dimensi
│ │ │ │ +00036330: 6f6e 2063 6f6d 7075 7465 642c 203d 2033  on computed, = 3
│ │ │ │ +00036340: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +00036350: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +00036360: 4368 6f6f 7369 6e67 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ +00036370: 536d 616c 6c65 7374 2020 2020 2020 2020  Smallest        
│ │ │ │  00036380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000363a0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ -000363b0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ -000363c0: 6720 5261 6e64 6f6d 4e6f 6e5a 6572 6f20  g RandomNonZero 
│ │ │ │ +00036390: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +000363a0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +000363b0: 4368 6f6f 7369 6e67 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ +000363c0: 6c65 7374 2020 2020 2020 2020 2020 2020  lest            
│ │ │ │  000363d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000363e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000363f0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ -00036400: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ -00036410: 6720 5261 6e64 6f6d 2020 2020 2020 2020  g Random        
│ │ │ │ +000363e0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +000363f0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +00036400: 4368 6f6f 7369 6e67 2052 616e 646f 6d4e  Choosing RandomN
│ │ │ │ +00036410: 6f6e 5a65 726f 2020 2020 2020 2020 2020  onZero          
│ │ │ │  00036420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036440: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ -00036450: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ -00036460: 6720 5261 6e64 6f6d 4e6f 6e5a 6572 6f20  g RandomNonZero 
│ │ │ │ +00036430: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +00036440: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +00036450: 4368 6f6f 7369 6e67 2052 616e 646f 6d20  Choosing Random 
│ │ │ │ +00036460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036490: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ -000364a0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ -000364b0: 6720 4c65 7853 6d61 6c6c 6573 7454 6572  g LexSmallestTer
│ │ │ │ -000364c0: 6d20 2020 2020 2020 2020 2020 2020 2020  m               
│ │ │ │ -000364d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000364e0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ -000364f0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ -00036500: 6720 4c65 7853 6d61 6c6c 6573 7420 2020  g LexSmallest   
│ │ │ │ +00036480: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +00036490: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +000364a0: 4368 6f6f 7369 6e67 2052 616e 646f 6d4e  Choosing RandomN
│ │ │ │ +000364b0: 6f6e 5a65 726f 2020 2020 2020 2020 2020  onZero          
│ │ │ │ +000364c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000364d0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +000364e0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +000364f0: 4368 6f6f 7369 6e67 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ +00036500: 6c65 7374 5465 726d 2020 2020 2020 2020  lestTerm        
│ │ │ │  00036510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036530: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ -00036540: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ -00036550: 6720 4752 6576 4c65 7853 6d61 6c6c 6573  g GRevLexSmalles
│ │ │ │ -00036560: 7454 6572 6d20 2020 2020 2020 2020 2020  tTerm           
│ │ │ │ -00036570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036580: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ -00036590: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ -000365a0: 6720 4752 6576 4c65 7853 6d61 6c6c 6573  g GRevLexSmalles
│ │ │ │ -000365b0: 7454 6572 6d20 2020 2020 2020 2020 2020  tTerm           
│ │ │ │ -000365c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000365d0: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64   |.|regularInCod
│ │ │ │ -000365e0: 696d 656e 7369 6f6e 3a20 204c 6f6f 7020  imension:  Loop 
│ │ │ │ -000365f0: 7374 6570 2c20 6162 6f75 7420 746f 2063  step, about to c
│ │ │ │ -00036600: 6f6d 7075 7465 2064 696d 656e 7369 6f6e  ompute dimension
│ │ │ │ -00036610: 2e20 2053 7562 6d61 7472 6963 6573 2063  .  Submatrices c
│ │ │ │ -00036620: 6f7c 0a7c 7265 6775 6c61 7249 6e43 6f64  o|.|regularInCod
│ │ │ │ -00036630: 696d 656e 7369 6f6e 3a20 2069 7343 6f64  imension:  isCod
│ │ │ │ -00036640: 696d 4174 4c65 6173 7420 6661 696c 6564  imAtLeast failed
│ │ │ │ -00036650: 2c20 636f 6d70 7574 696e 6720 636f 6469  , computing codi
│ │ │ │ -00036660: 6d2e 2020 2020 2020 2020 2020 2020 2020  m.              
│ │ │ │ -00036670: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64   |.|regularInCod
│ │ │ │ -00036680: 696d 656e 7369 6f6e 3a20 2070 6172 7469  imension:  parti
│ │ │ │ -00036690: 616c 2073 696e 6775 6c61 7220 6c6f 6375  al singular locu
│ │ │ │ -000366a0: 7320 6469 6d65 6e73 696f 6e20 636f 6d70  s dimension comp
│ │ │ │ -000366b0: 7574 6564 2c20 3d20 3320 2020 2020 2020  uted, = 3       
│ │ │ │ -000366c0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ -000366d0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ -000366e0: 6720 4c65 7853 6d61 6c6c 6573 7420 2020  g LexSmallest   
│ │ │ │ -000366f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036710: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ -00036720: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ -00036730: 6720 4c65 7853 6d61 6c6c 6573 7454 6572  g LexSmallestTer
│ │ │ │ -00036740: 6d20 2020 2020 2020 2020 2020 2020 2020  m               
│ │ │ │ -00036750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036760: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ -00036770: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ -00036780: 6720 4c65 7853 6d61 6c6c 6573 7454 6572  g LexSmallestTer
│ │ │ │ -00036790: 6d20 2020 2020 2020 2020 2020 2020 2020  m               
│ │ │ │ -000367a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000367b0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ -000367c0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ -000367d0: 6720 5261 6e64 6f6d 2020 2020 2020 2020  g Random        
│ │ │ │ +00036520: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +00036530: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +00036540: 4368 6f6f 7369 6e67 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ +00036550: 6c65 7374 2020 2020 2020 2020 2020 2020  lest            
│ │ │ │ +00036560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036570: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +00036580: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +00036590: 4368 6f6f 7369 6e67 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ +000365a0: 536d 616c 6c65 7374 5465 726d 2020 2020  SmallestTerm    
│ │ │ │ +000365b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000365c0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +000365d0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +000365e0: 4368 6f6f 7369 6e67 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ +000365f0: 536d 616c 6c65 7374 5465 726d 2020 2020  SmallestTerm    
│ │ │ │ +00036600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036610: 2020 2020 2020 2020 7c0a 7c72 6567 756c          |.|regul
│ │ │ │ +00036620: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +00036630: 2020 4c6f 6f70 2073 7465 702c 2061 626f    Loop step, abo
│ │ │ │ +00036640: 7574 2074 6f20 636f 6d70 7574 6520 6469  ut to compute di
│ │ │ │ +00036650: 6d65 6e73 696f 6e2e 2020 5375 626d 6174  mension.  Submat
│ │ │ │ +00036660: 7269 6365 7320 636f 7c0a 7c72 6567 756c  rices co|.|regul
│ │ │ │ +00036670: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +00036680: 2020 6973 436f 6469 6d41 744c 6561 7374    isCodimAtLeast
│ │ │ │ +00036690: 2066 6169 6c65 642c 2063 6f6d 7075 7469   failed, computi
│ │ │ │ +000366a0: 6e67 2063 6f64 696d 2e20 2020 2020 2020  ng codim.       
│ │ │ │ +000366b0: 2020 2020 2020 2020 7c0a 7c72 6567 756c          |.|regul
│ │ │ │ +000366c0: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +000366d0: 2020 7061 7274 6961 6c20 7369 6e67 756c    partial singul
│ │ │ │ +000366e0: 6172 206c 6f63 7573 2064 696d 656e 7369  ar locus dimensi
│ │ │ │ +000366f0: 6f6e 2063 6f6d 7075 7465 642c 203d 2033  on computed, = 3
│ │ │ │ +00036700: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +00036710: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +00036720: 4368 6f6f 7369 6e67 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ +00036730: 6c65 7374 2020 2020 2020 2020 2020 2020  lest            
│ │ │ │ +00036740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036750: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +00036760: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +00036770: 4368 6f6f 7369 6e67 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ +00036780: 6c65 7374 5465 726d 2020 2020 2020 2020  lestTerm        
│ │ │ │ +00036790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000367a0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +000367b0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +000367c0: 4368 6f6f 7369 6e67 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ +000367d0: 6c65 7374 5465 726d 2020 2020 2020 2020  lestTerm        
│ │ │ │  000367e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000367f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036800: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ -00036810: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ -00036820: 6720 5261 6e64 6f6d 4e6f 6e5a 6572 6f20  g RandomNonZero 
│ │ │ │ +000367f0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +00036800: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +00036810: 4368 6f6f 7369 6e67 2052 616e 646f 6d20  Choosing Random 
│ │ │ │ +00036820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036850: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ -00036860: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ -00036870: 6720 4752 6576 4c65 7853 6d61 6c6c 6573  g GRevLexSmalles
│ │ │ │ -00036880: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │ -00036890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000368a0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ -000368b0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ -000368c0: 6720 5261 6e64 6f6d 2020 2020 2020 2020  g Random        
│ │ │ │ +00036840: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +00036850: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +00036860: 4368 6f6f 7369 6e67 2052 616e 646f 6d4e  Choosing RandomN
│ │ │ │ +00036870: 6f6e 5a65 726f 2020 2020 2020 2020 2020  onZero          
│ │ │ │ +00036880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036890: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +000368a0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +000368b0: 4368 6f6f 7369 6e67 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ +000368c0: 536d 616c 6c65 7374 2020 2020 2020 2020  Smallest        
│ │ │ │  000368d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000368e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000368f0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ -00036900: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ -00036910: 6720 4752 6576 4c65 7853 6d61 6c6c 6573  g GRevLexSmalles
│ │ │ │ -00036920: 7454 6572 6d20 2020 2020 2020 2020 2020  tTerm           
│ │ │ │ -00036930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036940: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ -00036950: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ -00036960: 6720 4c65 7853 6d61 6c6c 6573 7420 2020  g LexSmallest   
│ │ │ │ +000368e0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +000368f0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +00036900: 4368 6f6f 7369 6e67 2052 616e 646f 6d20  Choosing Random 
│ │ │ │ +00036910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036930: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +00036940: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +00036950: 4368 6f6f 7369 6e67 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ +00036960: 536d 616c 6c65 7374 5465 726d 2020 2020  SmallestTerm    
│ │ │ │  00036970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036990: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ -000369a0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ -000369b0: 6720 4c65 7853 6d61 6c6c 6573 7454 6572  g LexSmallestTer
│ │ │ │ -000369c0: 6d20 2020 2020 2020 2020 2020 2020 2020  m               
│ │ │ │ -000369d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000369e0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ -000369f0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ -00036a00: 6720 4c65 7853 6d61 6c6c 6573 7420 2020  g LexSmallest   
│ │ │ │ +00036980: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +00036990: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +000369a0: 4368 6f6f 7369 6e67 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ +000369b0: 6c65 7374 2020 2020 2020 2020 2020 2020  lest            
│ │ │ │ +000369c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000369d0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +000369e0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +000369f0: 4368 6f6f 7369 6e67 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ +00036a00: 6c65 7374 5465 726d 2020 2020 2020 2020  lestTerm        
│ │ │ │  00036a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036a30: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ -00036a40: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ -00036a50: 6720 5261 6e64 6f6d 2020 2020 2020 2020  g Random        
│ │ │ │ +00036a20: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +00036a30: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +00036a40: 4368 6f6f 7369 6e67 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ +00036a50: 6c65 7374 2020 2020 2020 2020 2020 2020  lest            
│ │ │ │  00036a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036a80: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64   |.|regularInCod
│ │ │ │ -00036a90: 696d 656e 7369 6f6e 3a20 204c 6f6f 7020  imension:  Loop 
│ │ │ │ -00036aa0: 7374 6570 2c20 6162 6f75 7420 746f 2063  step, about to c
│ │ │ │ -00036ab0: 6f6d 7075 7465 2064 696d 656e 7369 6f6e  ompute dimension
│ │ │ │ -00036ac0: 2e20 2053 7562 6d61 7472 6963 6573 2063  .  Submatrices c
│ │ │ │ -00036ad0: 6f7c 0a7c 7265 6775 6c61 7249 6e43 6f64  o|.|regularInCod
│ │ │ │ -00036ae0: 696d 656e 7369 6f6e 3a20 2073 696e 6775  imension:  singu
│ │ │ │ -00036af0: 6c61 724c 6f63 7573 2064 696d 656e 7369  larLocus dimensi
│ │ │ │ -00036b00: 6f6e 2076 6572 6966 6965 6420 6279 2069  on verified by i
│ │ │ │ -00036b10: 7343 6f64 696d 4174 4c65 6173 7420 2020  sCodimAtLeast   
│ │ │ │ -00036b20: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64   |.|regularInCod
│ │ │ │ -00036b30: 696d 656e 7369 6f6e 3a20 2070 6172 7469  imension:  parti
│ │ │ │ -00036b40: 616c 2073 696e 6775 6c61 7220 6c6f 6375  al singular locu
│ │ │ │ -00036b50: 7320 6469 6d65 6e73 696f 6e20 636f 6d70  s dimension comp
│ │ │ │ -00036b60: 7574 6564 2c20 3d20 3220 2020 2020 2020  uted, = 2       
│ │ │ │ -00036b70: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64   |.|regularInCod
│ │ │ │ -00036b80: 696d 656e 7369 6f6e 3a20 204c 6f6f 7020  imension:  Loop 
│ │ │ │ -00036b90: 636f 6d70 6c65 7465 642c 2073 7562 6d61  completed, subma
│ │ │ │ -00036ba0: 7472 6963 6573 2063 6f6e 7369 6465 7265  trices considere
│ │ │ │ -00036bb0: 6420 3d20 3439 2c20 616e 6420 636f 6d70  d = 49, and comp
│ │ │ │ -00036bc0: 757c 0a7c 6420 3d20 3339 2e20 2073 696e  u|.|d = 39.  sin
│ │ │ │ -00036bd0: 6775 6c61 7220 6c6f 6375 7320 6469 6d65  gular locus dime
│ │ │ │ -00036be0: 6e73 696f 6e20 6170 7065 6172 7320 746f  nsion appears to
│ │ │ │ -00036bf0: 2062 6520 3d20 3220 2020 2020 2020 2020   be = 2         
│ │ │ │ -00036c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036c10: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00036c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036a70: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +00036a80: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +00036a90: 4368 6f6f 7369 6e67 2052 616e 646f 6d20  Choosing Random 
│ │ │ │ +00036aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036ac0: 2020 2020 2020 2020 7c0a 7c72 6567 756c          |.|regul
│ │ │ │ +00036ad0: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +00036ae0: 2020 4c6f 6f70 2073 7465 702c 2061 626f    Loop step, abo
│ │ │ │ +00036af0: 7574 2074 6f20 636f 6d70 7574 6520 6469  ut to compute di
│ │ │ │ +00036b00: 6d65 6e73 696f 6e2e 2020 5375 626d 6174  mension.  Submat
│ │ │ │ +00036b10: 7269 6365 7320 636f 7c0a 7c72 6567 756c  rices co|.|regul
│ │ │ │ +00036b20: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +00036b30: 2020 7369 6e67 756c 6172 4c6f 6375 7320    singularLocus 
│ │ │ │ +00036b40: 6469 6d65 6e73 696f 6e20 7665 7269 6669  dimension verifi
│ │ │ │ +00036b50: 6564 2062 7920 6973 436f 6469 6d41 744c  ed by isCodimAtL
│ │ │ │ +00036b60: 6561 7374 2020 2020 7c0a 7c72 6567 756c  east    |.|regul
│ │ │ │ +00036b70: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +00036b80: 2020 7061 7274 6961 6c20 7369 6e67 756c    partial singul
│ │ │ │ +00036b90: 6172 206c 6f63 7573 2064 696d 656e 7369  ar locus dimensi
│ │ │ │ +00036ba0: 6f6e 2063 6f6d 7075 7465 642c 203d 2032  on computed, = 2
│ │ │ │ +00036bb0: 2020 2020 2020 2020 7c0a 7c72 6567 756c          |.|regul
│ │ │ │ +00036bc0: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +00036bd0: 2020 4c6f 6f70 2063 6f6d 706c 6574 6564    Loop completed
│ │ │ │ +00036be0: 2c20 7375 626d 6174 7269 6365 7320 636f  , submatrices co
│ │ │ │ +00036bf0: 6e73 6964 6572 6564 203d 2034 392c 2061  nsidered = 49, a
│ │ │ │ +00036c00: 6e64 2063 6f6d 7075 7c0a 7c64 203d 2033  nd compu|.|d = 3
│ │ │ │ +00036c10: 392e 2020 7369 6e67 756c 6172 206c 6f63  9.  singular loc
│ │ │ │ +00036c20: 7573 2064 696d 656e 7369 6f6e 2061 7070  us dimension app
│ │ │ │ +00036c30: 6561 7273 2074 6f20 6265 203d 2032 2020  ears to be = 2  
│ │ │ │  00036c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036c60: 207c 0a7c 6f36 203d 2074 7275 6520 2020   |.|o6 = true   
│ │ │ │ +00036c50: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00036c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036cb0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │ -00036cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036d00: 2d7c 0a7c 6e6f 7273 2c20 7765 2077 696c  -|.|nors, we wil
│ │ │ │ -00036d10: 6c20 636f 6d70 7574 6520 7570 2074 6f20  l compute up to 
│ │ │ │ -00036d20: 3435 322e 3930 3820 6f66 2074 6865 6d2e  452.908 of them.
│ │ │ │ -00036d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036d50: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00036d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036ca0: 2020 2020 2020 2020 7c0a 7c6f 3620 3d20          |.|o6 = 
│ │ │ │ +00036cb0: 7472 7565 2020 2020 2020 2020 2020 2020  true            
│ │ │ │ +00036cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036cf0: 2020 2020 2020 2020 7c0a 7c2d 2d2d 2d2d          |.|-----
│ │ │ │ +00036d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00036d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00036d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00036d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00036d40: 2d2d 2d2d 2d2d 2d2d 7c0a 7c6e 6f72 732c  --------|.|nors,
│ │ │ │ +00036d50: 2077 6520 7769 6c6c 2063 6f6d 7075 7465   we will compute
│ │ │ │ +00036d60: 2075 7020 746f 2034 3532 2e39 3038 206f   up to 452.908 o
│ │ │ │ +00036d70: 6620 7468 656d 2e20 2020 2020 2020 2020  f them.         
│ │ │ │  00036d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036da0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00036d90: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00036da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036df0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00036de0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00036df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036e40: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00036e30: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00036e40: 2020 2020 2020 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 2020 2020 2020 2020                  
│ │ │ │ -00036e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036e90: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00036e80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00036e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036ee0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00036ed0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00036ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036f30: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00036f20: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00036f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036f80: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00036f70: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00036f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036fd0: 207c 0a7c 6e73 6964 6572 6564 3a20 372c   |.|nsidered: 7,
│ │ │ │ -00036fe0: 2061 6e64 2063 6f6d 7075 7465 6420 3d20   and computed = 
│ │ │ │ -00036ff0: 3720 2020 2020 2020 2020 2020 2020 2020  7               
│ │ │ │ +00036fc0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00036fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037020: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00037030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037010: 2020 2020 2020 2020 7c0a 7c6e 7369 6465          |.|nside
│ │ │ │ +00037020: 7265 643a 2037 2c20 616e 6420 636f 6d70  red: 7, and comp
│ │ │ │ +00037030: 7574 6564 203d 2037 2020 2020 2020 2020  uted = 7        
│ │ │ │  00037040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037070: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037060: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000370a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000370b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000370c0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000370b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000370c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000370d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000370e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000370f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037110: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037100: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037160: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037150: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000371a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000371b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000371a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000371b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000371c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000371d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000371e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000371f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037200: 207c 0a7c 6e73 6964 6572 6564 3a20 3131   |.|nsidered: 11
│ │ │ │ -00037210: 2c20 616e 6420 636f 6d70 7574 6564 203d  , and computed =
│ │ │ │ -00037220: 2031 3020 2020 2020 2020 2020 2020 2020   10             
│ │ │ │ +000371f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037250: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00037260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037240: 2020 2020 2020 2020 7c0a 7c6e 7369 6465          |.|nside
│ │ │ │ +00037250: 7265 643a 2031 312c 2061 6e64 2063 6f6d  red: 11, and com
│ │ │ │ +00037260: 7075 7465 6420 3d20 3130 2020 2020 2020  puted = 10      
│ │ │ │  00037270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000372a0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037290: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000372a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000372b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000372c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000372d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000372e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000372f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000372e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000372f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037340: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037330: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037390: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037380: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000373a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000373b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000373c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000373d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000373e0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000373d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000373e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000373f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037430: 207c 0a7c 6e73 6964 6572 6564 3a20 3135   |.|nsidered: 15
│ │ │ │ -00037440: 2c20 616e 6420 636f 6d70 7574 6564 203d  , and computed =
│ │ │ │ -00037450: 2031 3320 2020 2020 2020 2020 2020 2020   13             
│ │ │ │ +00037420: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037480: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00037490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037470: 2020 2020 2020 2020 7c0a 7c6e 7369 6465          |.|nside
│ │ │ │ +00037480: 7265 643a 2031 352c 2061 6e64 2063 6f6d  red: 15, and com
│ │ │ │ +00037490: 7075 7465 6420 3d20 3133 2020 2020 2020  puted = 13      
│ │ │ │  000374a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000374b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000374c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000374d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000374c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000374d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000374e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000374f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037520: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037510: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037570: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037560: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000375a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000375b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000375c0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000375b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000375c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000375d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000375e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000375f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037610: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037600: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037660: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037650: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000376a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000376b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000376a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000376b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000376c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000376d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000376e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000376f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037700: 207c 0a7c 6e73 6964 6572 6564 3a20 3231   |.|nsidered: 21
│ │ │ │ -00037710: 2c20 616e 6420 636f 6d70 7574 6564 203d  , and computed =
│ │ │ │ -00037720: 2031 3820 2020 2020 2020 2020 2020 2020   18             
│ │ │ │ +000376f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037750: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00037760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037740: 2020 2020 2020 2020 7c0a 7c6e 7369 6465          |.|nside
│ │ │ │ +00037750: 7265 643a 2032 312c 2061 6e64 2063 6f6d  red: 21, and com
│ │ │ │ +00037760: 7075 7465 6420 3d20 3138 2020 2020 2020  puted = 18      
│ │ │ │  00037770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000377a0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037790: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000377a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000377b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000377c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000377d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000377e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000377f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000377e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000377f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037840: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037830: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037890: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037880: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000378a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000378b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000378c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000378d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000378e0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000378d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000378e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000378f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037930: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037920: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037980: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037970: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000379a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000379b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000379c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000379d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000379c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000379d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000379e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000379f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037a20: 207c 0a7c 6e73 6964 6572 6564 3a20 3238   |.|nsidered: 28
│ │ │ │ -00037a30: 2c20 616e 6420 636f 6d70 7574 6564 203d  , and computed =
│ │ │ │ -00037a40: 2032 3420 2020 2020 2020 2020 2020 2020   24             
│ │ │ │ +00037a10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037a70: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00037a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037a60: 2020 2020 2020 2020 7c0a 7c6e 7369 6465          |.|nside
│ │ │ │ +00037a70: 7265 643a 2032 382c 2061 6e64 2063 6f6d  red: 28, and com
│ │ │ │ +00037a80: 7075 7465 6420 3d20 3234 2020 2020 2020  puted = 24      
│ │ │ │  00037a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037ac0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037ab0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037b10: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037b00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037b60: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037b50: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037bb0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037ba0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037c00: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037bf0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037c50: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037c40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037ca0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037c90: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037cf0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037ce0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037d40: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037d30: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037d90: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037d80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037de0: 207c 0a7c 6e73 6964 6572 6564 3a20 3337   |.|nsidered: 37
│ │ │ │ -00037df0: 2c20 616e 6420 636f 6d70 7574 6564 203d  , and computed =
│ │ │ │ -00037e00: 2033 3020 2020 2020 2020 2020 2020 2020   30             
│ │ │ │ +00037dd0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037e30: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00037e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037e20: 2020 2020 2020 2020 7c0a 7c6e 7369 6465          |.|nside
│ │ │ │ +00037e30: 7265 643a 2033 372c 2061 6e64 2063 6f6d  red: 37, and com
│ │ │ │ +00037e40: 7075 7465 6420 3d20 3330 2020 2020 2020  puted = 30      
│ │ │ │  00037e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037e80: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037e70: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037ed0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037ec0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037f20: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037f10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037f70: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037f60: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037fc0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037fb0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00038000: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00038010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038060: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00038050: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00038060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000380a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000380b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000380a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000380b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000380c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000380d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000380e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000380f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038100: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000380f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00038100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038150: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00038140: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00038150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000381a0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00038190: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000381a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000381b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000381c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000381d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000381e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000381f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000381e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000381f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038240: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00038230: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00038240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038290: 207c 0a7c 6e73 6964 6572 6564 3a20 3439   |.|nsidered: 49
│ │ │ │ -000382a0: 2c20 616e 6420 636f 6d70 7574 6564 203d  , and computed =
│ │ │ │ -000382b0: 2033 3920 2020 2020 2020 2020 2020 2020   39             
│ │ │ │ +00038280: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00038290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000382a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000382b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000382c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000382d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000382e0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000382f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000382d0: 2020 2020 2020 2020 7c0a 7c6e 7369 6465          |.|nside
│ │ │ │ +000382e0: 7265 643a 2034 392c 2061 6e64 2063 6f6d  red: 49, and com
│ │ │ │ +000382f0: 7075 7465 6420 3d20 3339 2020 2020 2020  puted = 39      
│ │ │ │  00038300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038330: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00038320: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00038330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038380: 207c 0a7c 7465 202d 2d20 7573 6564 2031   |.|te -- used 1
│ │ │ │ -00038390: 2e34 3638 3131 7320 2863 7075 293b 2031  .46811s (cpu); 1
│ │ │ │ -000383a0: 2e30 3335 3538 7320 2874 6872 6561 6429  .03558s (thread)
│ │ │ │ -000383b0: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │ -000383c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000383d0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ -000383e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000383f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038420: 2d2b 0a0a 4d61 784d 696e 6f72 732e 2020  -+..MaxMinors.  
│ │ │ │ -00038430: 5468 6520 6669 7273 7420 6f75 7470 7574  The first output
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│ │ │ │ -000384e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000384f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00038390: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000383b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00038420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00038520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00038530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +000385d0: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │ +000385e0: 2020 2020 2020 2020 2020 2020 7c0a 7c72              |.|r
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│ │ │ │ +000386b0: 646f 6d20 2020 2020 2020 2020 2020 2020  dom             
│ │ │ │ +000386c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00038ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00038c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038c30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00038c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00038990: 656e 7369 6f6e 2063 6f6d 7075 7465 642c  ension computed,
│ │ │ │ +000389a0: 203d 2033 2020 2020 2020 2020 7c0a 7c69   = 3        |.|i
│ │ │ │ +000389b0: 6e74 6572 6e61 6c43 686f 6f73 654d 696e  nternalChooseMin
│ │ │ │ +000389c0: 6f72 3a20 4368 6f6f 7369 6e67 2052 616e  or: Choosing Ran
│ │ │ │ +000389d0: 646f 6d20 2020 2020 2020 2020 2020 2020  dom             
│ │ │ │ +000389e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000389f0: 2020 2020 2020 2020 2020 2020 7c0a 7c69              |.|i
│ │ │ │ +00038a00: 6e74 6572 6e61 6c43 686f 6f73 654d 696e  nternalChooseMin
│ │ │ │ +00038a10: 6f72 3a20 4368 6f6f 7369 6e67 2047 5265  or: Choosing GRe
│ │ │ │ +00038a20: 764c 6578 536d 616c 6c65 7374 2020 2020  vLexSmallest    
│ │ │ │ +00038a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038a40: 2020 2020 2020 2020 2020 2020 7c0a 7c69              |.|i
│ │ │ │ +00038a50: 6e74 6572 6e61 6c43 686f 6f73 654d 696e  nternalChooseMin
│ │ │ │ +00038a60: 6f72 3a20 4368 6f6f 7369 6e67 204c 6578  or: Choosing Lex
│ │ │ │ +00038a70: 536d 616c 6c65 7374 2020 2020 2020 2020  Smallest        
│ │ │ │ +00038a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038a90: 2020 2020 2020 2020 2020 2020 7c0a 7c72              |.|r
│ │ │ │ +00038aa0: 6567 756c 6172 496e 436f 6469 6d65 6e73  egularInCodimens
│ │ │ │ +00038ab0: 696f 6e3a 2020 4c6f 6f70 2073 7465 702c  ion:  Loop step,
│ │ │ │ +00038ac0: 2061 626f 7574 2074 6f20 636f 6d70 7574   about to comput
│ │ │ │ +00038ad0: 6520 6469 6d65 6e73 696f 6e2e 2020 5375  e dimension.  Su
│ │ │ │ +00038ae0: 626d 6174 7269 6365 7320 636f 7c0a 7c72  bmatrices co|.|r
│ │ │ │ +00038af0: 6567 756c 6172 496e 436f 6469 6d65 6e73  egularInCodimens
│ │ │ │ +00038b00: 696f 6e3a 2020 6973 436f 6469 6d41 744c  ion:  isCodimAtL
│ │ │ │ +00038b10: 6561 7374 2066 6169 6c65 642c 2063 6f6d  east failed, com
│ │ │ │ +00038b20: 7075 7469 6e67 2063 6f64 696d 2e20 2020  puting codim.   
│ │ │ │ +00038b30: 2020 2020 2020 2020 2020 2020 7c0a 7c72              |.|r
│ │ │ │ +00038b40: 6567 756c 6172 496e 436f 6469 6d65 6e73  egularInCodimens
│ │ │ │ +00038b50: 696f 6e3a 2020 7061 7274 6961 6c20 7369  ion:  partial si
│ │ │ │ +00038b60: 6e67 756c 6172 206c 6f63 7573 2064 696d  ngular locus dim
│ │ │ │ +00038b70: 656e 7369 6f6e 2063 6f6d 7075 7465 642c  ension computed,
│ │ │ │ +00038b80: 203d 2033 2020 2020 2020 2020 7c0a 7c72   = 3        |.|r
│ │ │ │ +00038b90: 6567 756c 6172 496e 436f 6469 6d65 6e73  egularInCodimens
│ │ │ │ +00038ba0: 696f 6e3a 2020 4c6f 6f70 2063 6f6d 706c  ion:  Loop compl
│ │ │ │ +00038bb0: 6574 6564 2c20 7375 626d 6174 7269 6365  eted, submatrice
│ │ │ │ +00038bc0: 7320 636f 6e73 6964 6572 6564 203d 2031  s considered = 1
│ │ │ │ +00038bd0: 302c 2061 6e64 2063 6f6d 7075 7c0a 7c2d  0, and compu|.|-
│ │ │ │ +00038be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00038bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00038c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00038c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00038c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c6e  ------------|.|n
│ │ │ │ +00038c30: 6f72 732c 2077 6520 7769 6c6c 2063 6f6d  ors, we will com
│ │ │ │ +00038c40: 7075 7465 2075 7020 746f 2031 3020 6f66  pute up to 10 of
│ │ │ │ +00038c50: 2074 6865 6d2e 2020 2020 2020 2020 2020   them.          
│ │ │ │  00038c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038c80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00038c70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00038c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00038cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038cd0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00038cc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00038cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00038d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038d20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00038d10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00038d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00038d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00038d60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00038d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00038db0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00038dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00038e00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00038e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00038e50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00038e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038eb0: 2020 2020 207c 0a7c 6e73 6964 6572 6564       |.|nsidered
│ │ │ │ -00038ec0: 3a20 372c 2061 6e64 2063 6f6d 7075 7465  : 7, and compute
│ │ │ │ -00038ed0: 6420 3d20 3720 2020 2020 2020 2020 2020  d = 7           
│ │ │ │ +00038ea0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00038eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038f00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00038f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038ef0: 2020 2020 2020 2020 2020 2020 7c0a 7c6e              |.|n
│ │ │ │ +00038f00: 7369 6465 7265 643a 2037 2c20 616e 6420  sidered: 7, and 
│ │ │ │ +00038f10: 636f 6d70 7574 6564 203d 2037 2020 2020  computed = 7    
│ │ │ │  00038f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00038f40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
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│ │ │ │ -00038fa0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00038f90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00038fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038ff0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00038fe0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00038ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039010: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00039030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039040: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00039030: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00039040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039090: 2020 2020 207c 0a7c 6e73 6964 6572 6564       |.|nsidered
│ │ │ │ -000390a0: 3a20 3130 2c20 616e 6420 636f 6d70 7574  : 10, and comput
│ │ │ │ -000390b0: 6564 203d 2031 3020 2020 2020 2020 2020  ed = 10         
│ │ │ │ +00039080: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00039090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000390a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000390b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000390c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000390d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000390e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000390f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000390d0: 2020 2020 2020 2020 2020 2020 7c0a 7c6e              |.|n
│ │ │ │ +000390e0: 7369 6465 7265 643a 2031 302c 2061 6e64  sidered: 10, and
│ │ │ │ +000390f0: 2063 6f6d 7075 7465 6420 3d20 3130 2020   computed = 10  
│ │ │ │  00039100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039130: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00039120: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00039130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039180: 2020 2020 207c 0a7c 7465 6420 3d20 3130       |.|ted = 10
│ │ │ │ -00039190: 2e20 2073 696e 6775 6c61 7220 6c6f 6375  .  singular locu
│ │ │ │ -000391a0: 7320 6469 6d65 6e73 696f 6e20 6170 7065  s dimension appe
│ │ │ │ -000391b0: 6172 7320 746f 2062 6520 3d20 3320 2020  ars to be = 3   
│ │ │ │ -000391c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000391d0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -000391e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000391f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00039200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00039210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00039220: 2d2d 2d2d 2d2b 0a0a 5468 6572 6520 6172  -----+..There ar
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│ │ │ │ -00039240: 7973 2074 6f20 636f 6e74 726f 6c20 7468  ys to control th
│ │ │ │ -00039250: 6520 4d61 784d 696e 6f72 7320 6f70 7469  e MaxMinors opti
│ │ │ │ -00039260: 6f6e 2c20 6275 7420 7468 6579 2077 696c  on, but they wil
│ │ │ │ -00039270: 6c20 6e6f 740a 6265 2064 6973 6375 7373  l not.be discuss
│ │ │ │ -00039280: 6564 2069 6e20 7468 6973 2074 7574 6f72  ed in this tutor
│ │ │ │ -00039290: 6961 6c2e 2020 5365 6520 2a6e 6f74 6520  ial.  See *note 
│ │ │ │ -000392a0: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ -000392b0: 7369 6f6e 3a0a 7265 6775 6c61 7249 6e43  sion:.regularInC
│ │ │ │ -000392c0: 6f64 696d 656e 7369 6f6e 2c2e 0a0a 5365  odimension,...Se
│ │ │ │ -000392d0: 6c65 6374 696e 6720 7375 626d 6174 7269  lecting submatri
│ │ │ │ -000392e0: 6365 7320 6f66 2074 6865 204a 6163 6f62  ces of the Jacob
│ │ │ │ -000392f0: 6961 6e2e 2020 5765 2061 6c73 6f20 7365  ian.  We also se
│ │ │ │ -00039300: 6520 6f75 7470 7574 206c 696b 653a 2060  e output like: `
│ │ │ │ -00039310: 6043 686f 6f73 696e 670a 4c65 7853 6d61  `Choosing.LexSma
│ │ │ │ -00039320: 6c6c 6573 7427 2720 6f72 2060 6043 686f  llest'' or ``Cho
│ │ │ │ -00039330: 6f73 696e 6720 5261 6e64 6f6d 2727 2e20  osing Random''. 
│ │ │ │ -00039340: 2054 6869 7320 6973 2073 6179 696e 6720   This is saying 
│ │ │ │ -00039350: 686f 7720 7765 2061 7265 2073 656c 6563  how we are selec
│ │ │ │ -00039360: 7469 6e67 2061 0a67 6976 656e 2073 7562  ting a.given sub
│ │ │ │ -00039370: 6d61 7472 6978 2e20 2046 6f72 2069 6e73  matrix.  For ins
│ │ │ │ -00039380: 7461 6e63 652c 2077 6520 6361 6e20 7275  tance, we can ru
│ │ │ │ -00039390: 6e3a 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  n:..+-----------
│ │ │ │ -000393a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000393b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000393c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000393d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000393e0: 2d2d 2b0a 7c69 3820 3a20 7469 6d65 2072  --+.|i8 : time r
│ │ │ │ -000393f0: 6567 756c 6172 496e 436f 6469 6d65 6e73  egularInCodimens
│ │ │ │ -00039400: 696f 6e28 312c 2053 2f4a 2c20 4d61 784d  ion(1, S/J, MaxM
│ │ │ │ -00039410: 696e 6f72 733d 3e31 302c 2053 7472 6174  inors=>10, Strat
│ │ │ │ -00039420: 6567 793d 3e53 7472 6174 6567 7952 616e  egy=>StrategyRan
│ │ │ │ -00039430: 646f 7c0a 7c20 2d2d 2075 7365 6420 302e  do|.| -- used 0.
│ │ │ │ -00039440: 3133 3836 3736 7320 2863 7075 293b 2030  138676s (cpu); 0
│ │ │ │ -00039450: 2e30 3938 3438 3173 2028 7468 7265 6164  .098481s (thread
│ │ │ │ -00039460: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │ -00039470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039480: 2020 7c0a 7c72 6567 756c 6172 496e 436f    |.|regularInCo
│ │ │ │ -00039490: 6469 6d65 6e73 696f 6e3a 2072 696e 6720  dimension: ring 
│ │ │ │ -000394a0: 6469 6d65 6e73 696f 6e20 3d34 2c20 7468  dimension =4, th
│ │ │ │ -000394b0: 6572 6520 6172 6520 3134 3635 3132 3820  ere are 1465128 
│ │ │ │ -000394c0: 706f 7373 6962 6c65 2035 2062 7920 3520  possible 5 by 5 
│ │ │ │ -000394d0: 6d69 7c0a 7c72 6567 756c 6172 496e 436f  mi|.|regularInCo
│ │ │ │ -000394e0: 6469 6d65 6e73 696f 6e3a 2041 626f 7574  dimension: About
│ │ │ │ -000394f0: 2074 6f20 656e 7465 7220 6c6f 6f70 2020   to enter loop  
│ │ │ │ -00039500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039520: 2020 7c0a 7c69 6e74 6572 6e61 6c43 686f    |.|internalCho
│ │ │ │ -00039530: 6f73 654d 696e 6f72 3a20 4368 6f6f 7369  oseMinor: Choosi
│ │ │ │ -00039540: 6e67 2052 616e 646f 6d20 2020 2020 2020  ng Random       
│ │ │ │ +00039170: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00039180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000391a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000391b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000391c0: 2020 2020 2020 2020 2020 2020 7c0a 7c74              |.|t
│ │ │ │ +000391d0: 6564 203d 2031 302e 2020 7369 6e67 756c  ed = 10.  singul
│ │ │ │ +000391e0: 6172 206c 6f63 7573 2064 696d 656e 7369  ar locus dimensi
│ │ │ │ +000391f0: 6f6e 2061 7070 6561 7273 2074 6f20 6265  on appears to be
│ │ │ │ +00039200: 203d 2033 2020 2020 2020 2020 2020 2020   = 3            
│ │ │ │ +00039210: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00039220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a54  ------------+..T
│ │ │ │ +00039270: 6865 7265 2061 7265 206f 7468 6572 2066  here are other f
│ │ │ │ +00039280: 696e 6572 2077 6179 7320 746f 2063 6f6e  iner ways to con
│ │ │ │ +00039290: 7472 6f6c 2074 6865 204d 6178 4d69 6e6f  trol the MaxMino
│ │ │ │ +000392a0: 7273 206f 7074 696f 6e2c 2062 7574 2074  rs option, but t
│ │ │ │ +000392b0: 6865 7920 7769 6c6c 206e 6f74 0a62 6520  hey will not.be 
│ │ │ │ +000392c0: 6469 7363 7573 7365 6420 696e 2074 6869  discussed in thi
│ │ │ │ +000392d0: 7320 7475 746f 7269 616c 2e20 2053 6565  s tutorial.  See
│ │ │ │ +000392e0: 202a 6e6f 7465 2072 6567 756c 6172 496e   *note regularIn
│ │ │ │ +000392f0: 436f 6469 6d65 6e73 696f 6e3a 0a72 6567  Codimension:.reg
│ │ │ │ +00039300: 756c 6172 496e 436f 6469 6d65 6e73 696f  ularInCodimensio
│ │ │ │ +00039310: 6e2c 2e0a 0a53 656c 6563 7469 6e67 2073  n,...Selecting s
│ │ │ │ +00039320: 7562 6d61 7472 6963 6573 206f 6620 7468  ubmatrices of th
│ │ │ │ +00039330: 6520 4a61 636f 6269 616e 2e20 2057 6520  e Jacobian.  We 
│ │ │ │ +00039340: 616c 736f 2073 6565 206f 7574 7075 7420  also see output 
│ │ │ │ +00039350: 6c69 6b65 3a20 6060 4368 6f6f 7369 6e67  like: ``Choosing
│ │ │ │ +00039360: 0a4c 6578 536d 616c 6c65 7374 2727 206f  .LexSmallest'' o
│ │ │ │ +00039370: 7220 6060 4368 6f6f 7369 6e67 2052 616e  r ``Choosing Ran
│ │ │ │ +00039380: 646f 6d27 272e 2020 5468 6973 2069 7320  dom''.  This is 
│ │ │ │ +00039390: 7361 7969 6e67 2068 6f77 2077 6520 6172  saying how we ar
│ │ │ │ +000393a0: 6520 7365 6c65 6374 696e 6720 610a 6769  e selecting a.gi
│ │ │ │ +000393b0: 7665 6e20 7375 626d 6174 7269 782e 2020  ven submatrix.  
│ │ │ │ +000393c0: 466f 7220 696e 7374 616e 6365 2c20 7765  For instance, we
│ │ │ │ +000393d0: 2063 616e 2072 756e 3a0a 0a2b 2d2d 2d2d   can run:..+----
│ │ │ │ +000393e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000393f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039420: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a  ---------+.|i8 :
│ │ │ │ +00039430: 2074 696d 6520 7265 6775 6c61 7249 6e43   time regularInC
│ │ │ │ +00039440: 6f64 696d 656e 7369 6f6e 2831 2c20 532f  odimension(1, S/
│ │ │ │ +00039450: 4a2c 204d 6178 4d69 6e6f 7273 3d3e 3130  J, MaxMinors=>10
│ │ │ │ +00039460: 2c20 5374 7261 7465 6779 3d3e 5374 7261  , Strategy=>Stra
│ │ │ │ +00039470: 7465 6779 5261 6e64 6f7c 0a7c 202d 2d20  tegyRando|.| -- 
│ │ │ │ +00039480: 7573 6564 2030 2e31 3839 3333 3473 2028  used 0.189334s (
│ │ │ │ +00039490: 6370 7529 3b20 302e 3132 3634 3638 7320  cpu); 0.126468s 
│ │ │ │ +000394a0: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │ +000394b0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +000394c0: 2020 2020 2020 2020 207c 0a7c 7265 6775           |.|regu
│ │ │ │ +000394d0: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ +000394e0: 3a20 7269 6e67 2064 696d 656e 7369 6f6e  : ring dimension
│ │ │ │ +000394f0: 203d 342c 2074 6865 7265 2061 7265 2031   =4, there are 1
│ │ │ │ +00039500: 3436 3531 3238 2070 6f73 7369 626c 6520  465128 possible 
│ │ │ │ +00039510: 3520 6279 2035 206d 697c 0a7c 7265 6775  5 by 5 mi|.|regu
│ │ │ │ +00039520: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ +00039530: 3a20 4162 6f75 7420 746f 2065 6e74 6572  : About to enter
│ │ │ │ +00039540: 206c 6f6f 7020 2020 2020 2020 2020 2020   loop           
│ │ │ │  00039550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039570: 2020 7c0a 7c69 6e74 6572 6e61 6c43 686f    |.|internalCho
│ │ │ │ -00039580: 6f73 654d 696e 6f72 3a20 4368 6f6f 7369  oseMinor: Choosi
│ │ │ │ -00039590: 6e67 2052 616e 646f 6d20 2020 2020 2020  ng Random       
│ │ │ │ +00039560: 2020 2020 2020 2020 207c 0a7c 696e 7465           |.|inte
│ │ │ │ +00039570: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a  rnalChooseMinor:
│ │ │ │ +00039580: 2043 686f 6f73 696e 6720 5261 6e64 6f6d   Choosing Random
│ │ │ │ +00039590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000395a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000395b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000395c0: 2020 7c0a 7c69 6e74 6572 6e61 6c43 686f    |.|internalCho
│ │ │ │ -000395d0: 6f73 654d 696e 6f72 3a20 4368 6f6f 7369  oseMinor: Choosi
│ │ │ │ -000395e0: 6e67 2052 616e 646f 6d20 2020 2020 2020  ng Random       
│ │ │ │ +000395b0: 2020 2020 2020 2020 207c 0a7c 696e 7465           |.|inte
│ │ │ │ +000395c0: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a  rnalChooseMinor:
│ │ │ │ +000395d0: 2043 686f 6f73 696e 6720 5261 6e64 6f6d   Choosing Random
│ │ │ │ +000395e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000395f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039610: 2020 7c0a 7c69 6e74 6572 6e61 6c43 686f    |.|internalCho
│ │ │ │ -00039620: 6f73 654d 696e 6f72 3a20 4368 6f6f 7369  oseMinor: Choosi
│ │ │ │ -00039630: 6e67 2052 616e 646f 6d20 2020 2020 2020  ng Random       
│ │ │ │ +00039600: 2020 2020 2020 2020 207c 0a7c 696e 7465           |.|inte
│ │ │ │ +00039610: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a  rnalChooseMinor:
│ │ │ │ +00039620: 2043 686f 6f73 696e 6720 5261 6e64 6f6d   Choosing Random
│ │ │ │ +00039630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039660: 2020 7c0a 7c69 6e74 6572 6e61 6c43 686f    |.|internalCho
│ │ │ │ -00039670: 6f73 654d 696e 6f72 3a20 4368 6f6f 7369  oseMinor: Choosi
│ │ │ │ -00039680: 6e67 2052 616e 646f 6d20 2020 2020 2020  ng Random       
│ │ │ │ +00039650: 2020 2020 2020 2020 207c 0a7c 696e 7465           |.|inte
│ │ │ │ +00039660: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a  rnalChooseMinor:
│ │ │ │ +00039670: 2043 686f 6f73 696e 6720 5261 6e64 6f6d   Choosing Random
│ │ │ │ +00039680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000396a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000396b0: 2020 7c0a 7c69 6e74 6572 6e61 6c43 686f    |.|internalCho
│ │ │ │ -000396c0: 6f73 654d 696e 6f72 3a20 4368 6f6f 7369  oseMinor: Choosi
│ │ │ │ -000396d0: 6e67 2052 616e 646f 6d20 2020 2020 2020  ng Random       
│ │ │ │ +000396a0: 2020 2020 2020 2020 207c 0a7c 696e 7465           |.|inte
│ │ │ │ +000396b0: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a  rnalChooseMinor:
│ │ │ │ +000396c0: 2043 686f 6f73 696e 6720 5261 6e64 6f6d   Choosing Random
│ │ │ │ +000396d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000396e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000396f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039700: 2020 7c0a 7c69 6e74 6572 6e61 6c43 686f    |.|internalCho
│ │ │ │ -00039710: 6f73 654d 696e 6f72 3a20 4368 6f6f 7369  oseMinor: Choosi
│ │ │ │ -00039720: 6e67 2052 616e 646f 6d20 2020 2020 2020  ng Random       
│ │ │ │ +000396f0: 2020 2020 2020 2020 207c 0a7c 696e 7465           |.|inte
│ │ │ │ +00039700: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a  rnalChooseMinor:
│ │ │ │ +00039710: 2043 686f 6f73 696e 6720 5261 6e64 6f6d   Choosing Random
│ │ │ │ +00039720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039750: 2020 7c0a 7c72 6567 756c 6172 496e 436f    |.|regularInCo
│ │ │ │ -00039760: 6469 6d65 6e73 696f 6e3a 2020 4c6f 6f70  dimension:  Loop
│ │ │ │ -00039770: 2073 7465 702c 2061 626f 7574 2074 6f20   step, about to 
│ │ │ │ -00039780: 636f 6d70 7574 6520 6469 6d65 6e73 696f  compute dimensio
│ │ │ │ -00039790: 6e2e 2020 5375 626d 6174 7269 6365 7320  n.  Submatrices 
│ │ │ │ -000397a0: 636f 7c0a 7c72 6567 756c 6172 496e 436f  co|.|regularInCo
│ │ │ │ -000397b0: 6469 6d65 6e73 696f 6e3a 2020 6973 436f  dimension:  isCo
│ │ │ │ -000397c0: 6469 6d41 744c 6561 7374 2066 6169 6c65  dimAtLeast faile
│ │ │ │ -000397d0: 642c 2063 6f6d 7075 7469 6e67 2063 6f64  d, computing cod
│ │ │ │ -000397e0: 696d 2e20 2020 2020 2020 2020 2020 2020  im.             
│ │ │ │ -000397f0: 2020 7c0a 7c72 6567 756c 6172 496e 436f    |.|regularInCo
│ │ │ │ -00039800: 6469 6d65 6e73 696f 6e3a 2020 7061 7274  dimension:  part
│ │ │ │ -00039810: 6961 6c20 7369 6e67 756c 6172 206c 6f63  ial singular loc
│ │ │ │ -00039820: 7573 2064 696d 656e 7369 6f6e 2063 6f6d  us dimension com
│ │ │ │ -00039830: 7075 7465 642c 203d 2033 2020 2020 2020  puted, = 3      
│ │ │ │ -00039840: 2020 7c0a 7c69 6e74 6572 6e61 6c43 686f    |.|internalCho
│ │ │ │ -00039850: 6f73 654d 696e 6f72 3a20 4368 6f6f 7369  oseMinor: Choosi
│ │ │ │ -00039860: 6e67 2052 616e 646f 6d20 2020 2020 2020  ng Random       
│ │ │ │ -00039870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039890: 2020 7c0a 7c69 6e74 6572 6e61 6c43 686f    |.|internalCho
│ │ │ │ -000398a0: 6f73 654d 696e 6f72 3a20 4368 6f6f 7369  oseMinor: Choosi
│ │ │ │ -000398b0: 6e67 2052 616e 646f 6d20 2020 2020 2020  ng Random       
│ │ │ │ +00039740: 2020 2020 2020 2020 207c 0a7c 696e 7465           |.|inte
│ │ │ │ +00039750: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a  rnalChooseMinor:
│ │ │ │ +00039760: 2043 686f 6f73 696e 6720 5261 6e64 6f6d   Choosing Random
│ │ │ │ +00039770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039790: 2020 2020 2020 2020 207c 0a7c 7265 6775           |.|regu
│ │ │ │ +000397a0: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ +000397b0: 3a20 204c 6f6f 7020 7374 6570 2c20 6162  :  Loop step, ab
│ │ │ │ +000397c0: 6f75 7420 746f 2063 6f6d 7075 7465 2064  out to compute d
│ │ │ │ +000397d0: 696d 656e 7369 6f6e 2e20 2053 7562 6d61  imension.  Subma
│ │ │ │ +000397e0: 7472 6963 6573 2063 6f7c 0a7c 7265 6775  trices co|.|regu
│ │ │ │ +000397f0: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ +00039800: 3a20 2069 7343 6f64 696d 4174 4c65 6173  :  isCodimAtLeas
│ │ │ │ +00039810: 7420 6661 696c 6564 2c20 636f 6d70 7574  t failed, comput
│ │ │ │ +00039820: 696e 6720 636f 6469 6d2e 2020 2020 2020  ing codim.      
│ │ │ │ +00039830: 2020 2020 2020 2020 207c 0a7c 7265 6775           |.|regu
│ │ │ │ +00039840: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ +00039850: 3a20 2070 6172 7469 616c 2073 696e 6775  :  partial singu
│ │ │ │ +00039860: 6c61 7220 6c6f 6375 7320 6469 6d65 6e73  lar locus dimens
│ │ │ │ +00039870: 696f 6e20 636f 6d70 7574 6564 2c20 3d20  ion computed, = 
│ │ │ │ +00039880: 3320 2020 2020 2020 207c 0a7c 696e 7465  3        |.|inte
│ │ │ │ +00039890: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a  rnalChooseMinor:
│ │ │ │ +000398a0: 2043 686f 6f73 696e 6720 5261 6e64 6f6d   Choosing Random
│ │ │ │ +000398b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000398c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000398d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000398e0: 2020 7c0a 7c69 6e74 6572 6e61 6c43 686f    |.|internalCho
│ │ │ │ -000398f0: 6f73 654d 696e 6f72 3a20 4368 6f6f 7369  oseMinor: Choosi
│ │ │ │ -00039900: 6e67 2052 616e 646f 6d20 2020 2020 2020  ng Random       
│ │ │ │ +000398d0: 2020 2020 2020 2020 207c 0a7c 696e 7465           |.|inte
│ │ │ │ +000398e0: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a  rnalChooseMinor:
│ │ │ │ +000398f0: 2043 686f 6f73 696e 6720 5261 6e64 6f6d   Choosing Random
│ │ │ │ +00039900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039930: 2020 7c0a 7c72 6567 756c 6172 496e 436f    |.|regularInCo
│ │ │ │ -00039940: 6469 6d65 6e73 696f 6e3a 2020 4c6f 6f70  dimension:  Loop
│ │ │ │ -00039950: 2073 7465 702c 2061 626f 7574 2074 6f20   step, about to 
│ │ │ │ -00039960: 636f 6d70 7574 6520 6469 6d65 6e73 696f  compute dimensio
│ │ │ │ -00039970: 6e2e 2020 5375 626d 6174 7269 6365 7320  n.  Submatrices 
│ │ │ │ -00039980: 636f 7c0a 7c72 6567 756c 6172 496e 436f  co|.|regularInCo
│ │ │ │ -00039990: 6469 6d65 6e73 696f 6e3a 2020 6973 436f  dimension:  isCo
│ │ │ │ -000399a0: 6469 6d41 744c 6561 7374 2066 6169 6c65  dimAtLeast faile
│ │ │ │ -000399b0: 642c 2063 6f6d 7075 7469 6e67 2063 6f64  d, computing cod
│ │ │ │ -000399c0: 696d 2e20 2020 2020 2020 2020 2020 2020  im.             
│ │ │ │ -000399d0: 2020 7c0a 7c72 6567 756c 6172 496e 436f    |.|regularInCo
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│ │ │ │ -000399f0: 6961 6c20 7369 6e67 756c 6172 206c 6f63  ial singular loc
│ │ │ │ -00039a00: 7573 2064 696d 656e 7369 6f6e 2063 6f6d  us dimension com
│ │ │ │ -00039a10: 7075 7465 642c 203d 2033 2020 2020 2020  puted, = 3      
│ │ │ │ -00039a20: 2020 7c0a 7c72 6567 756c 6172 496e 436f    |.|regularInCo
│ │ │ │ -00039a30: 6469 6d65 6e73 696f 6e3a 2020 4c6f 6f70  dimension:  Loop
│ │ │ │ -00039a40: 2063 6f6d 706c 6574 6564 2c20 7375 626d   completed, subm
│ │ │ │ -00039a50: 6174 7269 6365 7320 636f 6e73 6964 6572  atrices consider
│ │ │ │ -00039a60: 6564 203d 2031 302c 2061 6e64 2063 6f6d  ed = 10, and com
│ │ │ │ -00039a70: 7075 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d  pu|.|-----------
│ │ │ │ -00039a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00039a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00039aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00039ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00039ac0: 2d2d 7c0a 7c6d 2c20 5665 7262 6f73 653d  --|.|m, Verbose=
│ │ │ │ -00039ad0: 3e74 7275 6529 2020 2020 2020 2020 2020  >true)          
│ │ │ │ -00039ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039b10: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00039920: 2020 2020 2020 2020 207c 0a7c 696e 7465           |.|inte
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│ │ │ │ +00039940: 2043 686f 6f73 696e 6720 5261 6e64 6f6d   Choosing Random
│ │ │ │ +00039950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039970: 2020 2020 2020 2020 207c 0a7c 7265 6775           |.|regu
│ │ │ │ +00039980: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ +00039990: 3a20 204c 6f6f 7020 7374 6570 2c20 6162  :  Loop step, ab
│ │ │ │ +000399a0: 6f75 7420 746f 2063 6f6d 7075 7465 2064  out to compute d
│ │ │ │ +000399b0: 696d 656e 7369 6f6e 2e20 2053 7562 6d61  imension.  Subma
│ │ │ │ +000399c0: 7472 6963 6573 2063 6f7c 0a7c 7265 6775  trices co|.|regu
│ │ │ │ +000399d0: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ +000399e0: 3a20 2069 7343 6f64 696d 4174 4c65 6173  :  isCodimAtLeas
│ │ │ │ +000399f0: 7420 6661 696c 6564 2c20 636f 6d70 7574  t failed, comput
│ │ │ │ +00039a00: 696e 6720 636f 6469 6d2e 2020 2020 2020  ing codim.      
│ │ │ │ +00039a10: 2020 2020 2020 2020 207c 0a7c 7265 6775           |.|regu
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│ │ │ │ +00039a30: 3a20 2070 6172 7469 616c 2073 696e 6775  :  partial singu
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│ │ │ │ +00039a50: 696f 6e20 636f 6d70 7574 6564 2c20 3d20  ion computed, = 
│ │ │ │ +00039a60: 3320 2020 2020 2020 207c 0a7c 7265 6775  3        |.|regu
│ │ │ │ +00039a70: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ +00039a80: 3a20 204c 6f6f 7020 636f 6d70 6c65 7465  :  Loop complete
│ │ │ │ +00039a90: 642c 2073 7562 6d61 7472 6963 6573 2063  d, submatrices c
│ │ │ │ +00039aa0: 6f6e 7369 6465 7265 6420 3d20 3130 2c20  onsidered = 10, 
│ │ │ │ +00039ab0: 616e 6420 636f 6d70 757c 0a7c 2d2d 2d2d  and compu|.|----
│ │ │ │ +00039ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039b00: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 6d2c 2056  ---------|.|m, V
│ │ │ │ +00039b10: 6572 626f 7365 3d3e 7472 7565 2920 2020  erbose=>true)   
│ │ │ │  00039b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039b60: 2020 7c0a 7c6e 6f72 732c 2077 6520 7769    |.|nors, we wi
│ │ │ │ -00039b70: 6c6c 2063 6f6d 7075 7465 2075 7020 746f  ll compute up to
│ │ │ │ -00039b80: 2031 3020 6f66 2074 6865 6d2e 2020 2020   10 of them.    
│ │ │ │ +00039b50: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00039b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039bb0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00039bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039ba0: 2020 2020 2020 2020 207c 0a7c 6e6f 7273           |.|nors
│ │ │ │ +00039bb0: 2c20 7765 2077 696c 6c20 636f 6d70 7574  , we will comput
│ │ │ │ +00039bc0: 6520 7570 2074 6f20 3130 206f 6620 7468  e up to 10 of th
│ │ │ │ +00039bd0: 656d 2e20 2020 2020 2020 2020 2020 2020  em.             
│ │ │ │  00039be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039c00: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00039bf0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00039c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039c50: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00039c40: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00039c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039ca0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00039c90: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00039ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039cf0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00039ce0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00039cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039d40: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00039d30: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00039d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039d90: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00039d80: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00039d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039de0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00039dd0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00039de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039e30: 2020 7c0a 7c6e 7369 6465 7265 643a 2037    |.|nsidered: 7
│ │ │ │ -00039e40: 2c20 616e 6420 636f 6d70 7574 6564 203d  , and computed =
│ │ │ │ -00039e50: 2037 2020 2020 2020 2020 2020 2020 2020   7              
│ │ │ │ +00039e20: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00039e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039e80: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00039e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039e70: 2020 2020 2020 2020 207c 0a7c 6e73 6964           |.|nsid
│ │ │ │ +00039e80: 6572 6564 3a20 372c 2061 6e64 2063 6f6d  ered: 7, and com
│ │ │ │ +00039e90: 7075 7465 6420 3d20 3720 2020 2020 2020  puted = 7       
│ │ │ │  00039ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039ed0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00039ec0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00039ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039f20: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00039f10: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00039f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039f70: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00039f60: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00039f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039fc0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00039fb0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00039fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a010: 2020 7c0a 7c6e 7369 6465 7265 643a 2031    |.|nsidered: 1
│ │ │ │ -0003a020: 302c 2061 6e64 2063 6f6d 7075 7465 6420  0, and computed 
│ │ │ │ -0003a030: 3d20 3130 2020 2020 2020 2020 2020 2020  = 10            
│ │ │ │ +0003a000: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003a010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a060: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0003a070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a050: 2020 2020 2020 2020 207c 0a7c 6e73 6964           |.|nsid
│ │ │ │ +0003a060: 6572 6564 3a20 3130 2c20 616e 6420 636f  ered: 10, and co
│ │ │ │ +0003a070: 6d70 7574 6564 203d 2031 3020 2020 2020  mputed = 10     
│ │ │ │  0003a080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a0b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0003a0a0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003a0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a100: 2020 7c0a 7c74 6564 203d 2031 302e 2020    |.|ted = 10.  
│ │ │ │ -0003a110: 7369 6e67 756c 6172 206c 6f63 7573 2064  singular locus d
│ │ │ │ -0003a120: 696d 656e 7369 6f6e 2061 7070 6561 7273  imension appears
│ │ │ │ -0003a130: 2074 6f20 6265 203d 2033 2020 2020 2020   to be = 3      
│ │ │ │ -0003a140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a150: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ -0003a160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003a170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003a180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003a190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003a1a0: 2d2d 2b0a 0a61 6e64 206f 6e6c 7920 7261  --+..and only ra
│ │ │ │ -0003a1b0: 6e64 6f6d 2073 7562 6d61 7472 6963 6573  ndom submatrices
│ │ │ │ -0003a1c0: 2061 7265 2063 686f 7365 6e2e 2020 5765   are chosen.  We
│ │ │ │ -0003a1d0: 2064 6973 6375 7373 2073 7472 6174 6567   discuss strateg
│ │ │ │ -0003a1e0: 6965 7320 666f 7220 6368 6f6f 7369 6e67  ies for choosing
│ │ │ │ -0003a1f0: 0a73 7562 6d61 7472 6963 6573 206d 7563  .submatrices muc
│ │ │ │ -0003a200: 6820 6d6f 7265 2067 656e 6572 616c 6c79  h more generally
│ │ │ │ -0003a210: 2069 6e20 7468 6520 2a6e 6f74 6520 4661   in the *note Fa
│ │ │ │ -0003a220: 7374 4d69 6e6f 7273 5374 7261 7465 6779  stMinorsStrategy
│ │ │ │ -0003a230: 5475 746f 7269 616c 3a0a 4661 7374 4d69  Tutorial:.FastMi
│ │ │ │ -0003a240: 6e6f 7273 5374 7261 7465 6779 5475 746f  norsStrategyTuto
│ │ │ │ -0003a250: 7269 616c 2c2e 2052 6567 6172 646c 6573  rial,. Regardles
│ │ │ │ -0003a260: 732c 2061 6674 6572 2061 2063 6572 7461  s, after a certa
│ │ │ │ -0003a270: 696e 206e 756d 6265 7220 6f66 206d 696e  in number of min
│ │ │ │ -0003a280: 6f72 7320 6861 7665 0a62 6565 6e20 6c6f  ors have.been lo
│ │ │ │ -0003a290: 6f6b 6564 2061 742c 2077 6520 7365 6520  oked at, we see 
│ │ │ │ -0003a2a0: 6f75 7470 7574 206c 696e 6573 206c 696b  output lines lik
│ │ │ │ -0003a2b0: 653a 2020 6060 4c6f 6f70 2073 7465 702c  e:  ``Loop step,
│ │ │ │ -0003a2c0: 2061 626f 7574 2074 6f20 636f 6d70 7574   about to comput
│ │ │ │ -0003a2d0: 650a 6469 6d65 6e73 696f 6e2e 2020 5375  e.dimension.  Su
│ │ │ │ -0003a2e0: 626d 6174 7269 6365 7320 636f 6e73 6964  bmatrices consid
│ │ │ │ -0003a2f0: 6572 6564 3a20 372c 2061 6e64 2063 6f6d  ered: 7, and com
│ │ │ │ -0003a300: 7075 7465 6420 3d20 3727 272e 2020 5765  puted = 7''.  We
│ │ │ │ -0003a310: 206f 6e6c 7920 636f 6d70 7574 650a 6d69   only compute.mi
│ │ │ │ -0003a320: 6e6f 7273 2077 6520 6861 7665 6e27 7420  nors we haven't 
│ │ │ │ -0003a330: 636f 6e73 6964 6572 6564 2062 6566 6f72  considered befor
│ │ │ │ -0003a340: 652e 2020 536f 2061 7320 7765 2063 6f6d  e.  So as we com
│ │ │ │ -0003a350: 7075 7465 206d 6f72 6520 6d69 6e6f 7273  pute more minors
│ │ │ │ -0003a360: 2c20 7468 6572 6520 6361 6e0a 6265 2061  , there can.be a
│ │ │ │ -0003a370: 2064 6973 7469 6e63 7469 6f6e 2062 6574   distinction bet
│ │ │ │ -0003a380: 7765 656e 2063 6f6e 7369 6465 7265 6420  ween considered 
│ │ │ │ -0003a390: 616e 6420 636f 6d70 7574 6564 2e0a 0a43  and computed...C
│ │ │ │ -0003a3a0: 6f6d 7075 7469 6e67 206d 696e 6f72 7320  omputing minors 
│ │ │ │ -0003a3b0: 7673 2063 6f6e 7369 6465 7269 6e67 2074  vs considering t
│ │ │ │ -0003a3c0: 6865 2064 696d 656e 7369 6f6e 206f 6620  he dimension of 
│ │ │ │ -0003a3d0: 7768 6174 2068 6173 2062 6565 6e20 636f  what has been co
│ │ │ │ -0003a3e0: 6d70 7574 6564 2e0a 5065 7269 6f64 6963  mputed..Periodic
│ │ │ │ -0003a3f0: 616c 6c79 2077 6520 636f 6d70 7574 6520  ally we compute 
│ │ │ │ -0003a400: 7468 6520 636f 6469 6d65 6e73 696f 6e20  the codimension 
│ │ │ │ -0003a410: 6f66 2074 6865 2070 6172 7469 616c 2069  of the partial i
│ │ │ │ -0003a420: 6465 616c 206f 6620 6d69 6e6f 7273 2077  deal of minors w
│ │ │ │ -0003a430: 6520 6861 7665 0a63 6f6d 7075 7465 6420  e have.computed 
│ │ │ │ -0003a440: 736f 2066 6172 2e20 2054 6865 7265 2061  so far.  There a
│ │ │ │ -0003a450: 7265 2074 776f 206f 7074 696f 6e73 2074  re two options t
│ │ │ │ -0003a460: 6f20 636f 6e74 726f 6c20 7468 6973 2e20  o control this. 
│ │ │ │ -0003a470: 2046 6972 7374 2c20 7765 2063 616e 2074   First, we can t
│ │ │ │ -0003a480: 656c 6c0a 7468 6520 6675 6e63 7469 6f6e  ell.the function
│ │ │ │ -0003a490: 2077 6865 6e20 746f 2066 6972 7374 2063   when to first c
│ │ │ │ -0003a4a0: 6f6d 7075 7465 2074 6865 2064 696d 656e  ompute the dimen
│ │ │ │ -0003a4b0: 7369 6f6e 206f 6620 7468 6520 776f 726b  sion of the work
│ │ │ │ -0003a4c0: 696e 6720 7061 7274 6961 6c20 6964 6561  ing partial idea
│ │ │ │ -0003a4d0: 6c0a 6f66 206d 696e 6f72 732e 0a0a 2b2d  l.of minors...+-
│ │ │ │ -0003a4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003a4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003a500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003a510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003a520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0003a530: 3920 3a20 7469 6d65 2072 6567 756c 6172  9 : time regular
│ │ │ │ -0003a540: 496e 436f 6469 6d65 6e73 696f 6e28 312c  InCodimension(1,
│ │ │ │ -0003a550: 2053 2f4a 2c20 4d61 784d 696e 6f72 733d   S/J, MaxMinors=
│ │ │ │ -0003a560: 3e31 302c 204d 696e 4d69 6e6f 7273 4675  >10, MinMinorsFu
│ │ │ │ -0003a570: 6e63 7469 6f6e 203d 3e20 742d 7c0a 7c20  nction => t-|.| 
│ │ │ │ -0003a580: 2d2d 2075 7365 6420 302e 3536 3831 3132  -- used 0.568112
│ │ │ │ -0003a590: 7320 2863 7075 293b 2030 2e34 3434 3732  s (cpu); 0.44472
│ │ │ │ -0003a5a0: 3273 2028 7468 7265 6164 293b 2030 7320  2s (thread); 0s 
│ │ │ │ -0003a5b0: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │ -0003a5c0: 2020 2020 2020 2020 2020 2020 7c0a 7c72              |.|r
│ │ │ │ -0003a5d0: 6567 756c 6172 496e 436f 6469 6d65 6e73  egularInCodimens
│ │ │ │ -0003a5e0: 696f 6e3a 2072 696e 6720 6469 6d65 6e73  ion: ring dimens
│ │ │ │ -0003a5f0: 696f 6e20 3d34 2c20 7468 6572 6520 6172  ion =4, there ar
│ │ │ │ -0003a600: 6520 3134 3635 3132 3820 706f 7373 6962  e 1465128 possib
│ │ │ │ -0003a610: 6c65 2035 2062 7920 3520 6d69 7c0a 7c72  le 5 by 5 mi|.|r
│ │ │ │ -0003a620: 6567 756c 6172 496e 436f 6469 6d65 6e73  egularInCodimens
│ │ │ │ -0003a630: 696f 6e3a 2041 626f 7574 2074 6f20 656e  ion: About to en
│ │ │ │ -0003a640: 7465 7220 6c6f 6f70 2020 2020 2020 2020  ter loop        
│ │ │ │ -0003a650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a660: 2020 2020 2020 2020 2020 2020 7c0a 7c69              |.|i
│ │ │ │ -0003a670: 6e74 6572 6e61 6c43 686f 6f73 654d 696e  nternalChooseMin
│ │ │ │ -0003a680: 6f72 3a20 4368 6f6f 7369 6e67 2052 616e  or: Choosing Ran
│ │ │ │ -0003a690: 646f 6d4e 6f6e 5a65 726f 2020 2020 2020  domNonZero      
│ │ │ │ +0003a0f0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003a100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a140: 2020 2020 2020 2020 207c 0a7c 7465 6420           |.|ted 
│ │ │ │ +0003a150: 3d20 3130 2e20 2073 696e 6775 6c61 7220  = 10.  singular 
│ │ │ │ +0003a160: 6c6f 6375 7320 6469 6d65 6e73 696f 6e20  locus dimension 
│ │ │ │ +0003a170: 6170 7065 6172 7320 746f 2062 6520 3d20  appears to be = 
│ │ │ │ +0003a180: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ +0003a190: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0003a1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003a1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003a1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003a1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003a1e0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 616e 6420  ---------+..and 
│ │ │ │ +0003a1f0: 6f6e 6c79 2072 616e 646f 6d20 7375 626d  only random subm
│ │ │ │ +0003a200: 6174 7269 6365 7320 6172 6520 6368 6f73  atrices are chos
│ │ │ │ +0003a210: 656e 2e20 2057 6520 6469 7363 7573 7320  en.  We discuss 
│ │ │ │ +0003a220: 7374 7261 7465 6769 6573 2066 6f72 2063  strategies for c
│ │ │ │ +0003a230: 686f 6f73 696e 670a 7375 626d 6174 7269  hoosing.submatri
│ │ │ │ +0003a240: 6365 7320 6d75 6368 206d 6f72 6520 6765  ces much more ge
│ │ │ │ +0003a250: 6e65 7261 6c6c 7920 696e 2074 6865 202a  nerally in the *
│ │ │ │ +0003a260: 6e6f 7465 2046 6173 744d 696e 6f72 7353  note FastMinorsS
│ │ │ │ +0003a270: 7472 6174 6567 7954 7574 6f72 6961 6c3a  trategyTutorial:
│ │ │ │ +0003a280: 0a46 6173 744d 696e 6f72 7353 7472 6174  .FastMinorsStrat
│ │ │ │ +0003a290: 6567 7954 7574 6f72 6961 6c2c 2e20 5265  egyTutorial,. Re
│ │ │ │ +0003a2a0: 6761 7264 6c65 7373 2c20 6166 7465 7220  gardless, after 
│ │ │ │ +0003a2b0: 6120 6365 7274 6169 6e20 6e75 6d62 6572  a certain number
│ │ │ │ +0003a2c0: 206f 6620 6d69 6e6f 7273 2068 6176 650a   of minors have.
│ │ │ │ +0003a2d0: 6265 656e 206c 6f6f 6b65 6420 6174 2c20  been looked at, 
│ │ │ │ +0003a2e0: 7765 2073 6565 206f 7574 7075 7420 6c69  we see output li
│ │ │ │ +0003a2f0: 6e65 7320 6c69 6b65 3a20 2060 604c 6f6f  nes like:  ``Loo
│ │ │ │ +0003a300: 7020 7374 6570 2c20 6162 6f75 7420 746f  p step, about to
│ │ │ │ +0003a310: 2063 6f6d 7075 7465 0a64 696d 656e 7369   compute.dimensi
│ │ │ │ +0003a320: 6f6e 2e20 2053 7562 6d61 7472 6963 6573  on.  Submatrices
│ │ │ │ +0003a330: 2063 6f6e 7369 6465 7265 643a 2037 2c20   considered: 7, 
│ │ │ │ +0003a340: 616e 6420 636f 6d70 7574 6564 203d 2037  and computed = 7
│ │ │ │ +0003a350: 2727 2e20 2057 6520 6f6e 6c79 2063 6f6d  ''.  We only com
│ │ │ │ +0003a360: 7075 7465 0a6d 696e 6f72 7320 7765 2068  pute.minors we h
│ │ │ │ +0003a370: 6176 656e 2774 2063 6f6e 7369 6465 7265  aven't considere
│ │ │ │ +0003a380: 6420 6265 666f 7265 2e20 2053 6f20 6173  d before.  So as
│ │ │ │ +0003a390: 2077 6520 636f 6d70 7574 6520 6d6f 7265   we compute more
│ │ │ │ +0003a3a0: 206d 696e 6f72 732c 2074 6865 7265 2063   minors, there c
│ │ │ │ +0003a3b0: 616e 0a62 6520 6120 6469 7374 696e 6374  an.be a distinct
│ │ │ │ +0003a3c0: 696f 6e20 6265 7477 6565 6e20 636f 6e73  ion between cons
│ │ │ │ +0003a3d0: 6964 6572 6564 2061 6e64 2063 6f6d 7075  idered and compu
│ │ │ │ +0003a3e0: 7465 642e 0a0a 436f 6d70 7574 696e 6720  ted...Computing 
│ │ │ │ +0003a3f0: 6d69 6e6f 7273 2076 7320 636f 6e73 6964  minors vs consid
│ │ │ │ +0003a400: 6572 696e 6720 7468 6520 6469 6d65 6e73  ering the dimens
│ │ │ │ +0003a410: 696f 6e20 6f66 2077 6861 7420 6861 7320  ion of what has 
│ │ │ │ +0003a420: 6265 656e 2063 6f6d 7075 7465 642e 0a50  been computed..P
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│ │ │ │ +0003a480: 6d70 7574 6564 2073 6f20 6661 722e 2020  mputed so far.  
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│ │ │ │ +0003a540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003a550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003a560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003a570: 2d2d 2d2b 0a7c 6939 203a 2074 696d 6520  ---+.|i9 : time 
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│ │ │ │ +0003a590: 7369 6f6e 2831 2c20 532f 4a2c 204d 6178  sion(1, S/J, Max
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│ │ │ │ +0003a5b0: 696e 6f72 7346 756e 6374 696f 6e20 3d3e  inorsFunction =>
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│ │ │ │ +0003a5d0: 2e37 3337 3439 3473 2028 6370 7529 3b20  .737494s (cpu); 
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│ │ │ │ +0003a600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a610: 2020 207c 0a7c 7265 6775 6c61 7249 6e43     |.|regularInC
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│ │ │ │ +0003a660: 206d 697c 0a7c 7265 6775 6c61 7249 6e43   mi|.|regularInC
│ │ │ │ +0003a670: 6f64 696d 656e 7369 6f6e 3a20 4162 6f75  odimension: Abou
│ │ │ │ +0003a680: 7420 746f 2065 6e74 6572 206c 6f6f 7020  t to enter loop 
│ │ │ │ +0003a690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a6b0: 2020 2020 2020 2020 2020 2020 7c0a 7c69              |.|i
│ │ │ │ -0003a6c0: 6e74 6572 6e61 6c43 686f 6f73 654d 696e  nternalChooseMin
│ │ │ │ -0003a6d0: 6f72 3a20 4368 6f6f 7369 6e67 2052 616e  or: Choosing Ran
│ │ │ │ -0003a6e0: 646f 6d20 2020 2020 2020 2020 2020 2020  dom             
│ │ │ │ +0003a6b0: 2020 207c 0a7c 696e 7465 726e 616c 4368     |.|internalCh
│ │ │ │ +0003a6c0: 6f6f 7365 4d69 6e6f 723a 2043 686f 6f73  ooseMinor: Choos
│ │ │ │ +0003a6d0: 696e 6720 5261 6e64 6f6d 4e6f 6e5a 6572  ing RandomNonZer
│ │ │ │ +0003a6e0: 6f20 2020 2020 2020 2020 2020 2020 2020  o               
│ │ │ │  0003a6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a700: 2020 2020 2020 2020 2020 2020 7c0a 7c69              |.|i
│ │ │ │ -0003a710: 6e74 6572 6e61 6c43 686f 6f73 654d 696e  nternalChooseMin
│ │ │ │ -0003a720: 6f72 3a20 4368 6f6f 7369 6e67 204c 6578  or: Choosing Lex
│ │ │ │ -0003a730: 536d 616c 6c65 7374 2020 2020 2020 2020  Smallest        
│ │ │ │ +0003a700: 2020 207c 0a7c 696e 7465 726e 616c 4368     |.|internalCh
│ │ │ │ +0003a710: 6f6f 7365 4d69 6e6f 723a 2043 686f 6f73  ooseMinor: Choos
│ │ │ │ +0003a720: 696e 6720 5261 6e64 6f6d 2020 2020 2020  ing Random      
│ │ │ │ +0003a730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a750: 2020 2020 2020 2020 2020 2020 7c0a 7c72              |.|r
│ │ │ │ -0003a760: 6567 756c 6172 496e 436f 6469 6d65 6e73  egularInCodimens
│ │ │ │ -0003a770: 696f 6e3a 2020 4c6f 6f70 2073 7465 702c  ion:  Loop step,
│ │ │ │ -0003a780: 2061 626f 7574 2074 6f20 636f 6d70 7574   about to comput
│ │ │ │ -0003a790: 6520 6469 6d65 6e73 696f 6e2e 2020 5375  e dimension.  Su
│ │ │ │ -0003a7a0: 626d 6174 7269 6365 7320 636f 7c0a 7c72  bmatrices co|.|r
│ │ │ │ -0003a7b0: 6567 756c 6172 496e 436f 6469 6d65 6e73  egularInCodimens
│ │ │ │ -0003a7c0: 696f 6e3a 2020 6973 436f 6469 6d41 744c  ion:  isCodimAtL
│ │ │ │ -0003a7d0: 6561 7374 2066 6169 6c65 642c 2063 6f6d  east failed, com
│ │ │ │ -0003a7e0: 7075 7469 6e67 2063 6f64 696d 2e20 2020  puting codim.   
│ │ │ │ -0003a7f0: 2020 2020 2020 2020 2020 2020 7c0a 7c72              |.|r
│ │ │ │ -0003a800: 6567 756c 6172 496e 436f 6469 6d65 6e73  egularInCodimens
│ │ │ │ -0003a810: 696f 6e3a 2020 7061 7274 6961 6c20 7369  ion:  partial si
│ │ │ │ -0003a820: 6e67 756c 6172 206c 6f63 7573 2064 696d  ngular locus dim
│ │ │ │ -0003a830: 656e 7369 6f6e 2063 6f6d 7075 7465 642c  ension computed,
│ │ │ │ -0003a840: 203d 2033 2020 2020 2020 2020 7c0a 7c69   = 3        |.|i
│ │ │ │ -0003a850: 6e74 6572 6e61 6c43 686f 6f73 654d 696e  nternalChooseMin
│ │ │ │ -0003a860: 6f72 3a20 4368 6f6f 7369 6e67 204c 6578  or: Choosing Lex
│ │ │ │ -0003a870: 536d 616c 6c65 7374 5465 726d 2020 2020  SmallestTerm    
│ │ │ │ -0003a880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a890: 2020 2020 2020 2020 2020 2020 7c0a 7c69              |.|i
│ │ │ │ -0003a8a0: 6e74 6572 6e61 6c43 686f 6f73 654d 696e  nternalChooseMin
│ │ │ │ -0003a8b0: 6f72 3a20 4368 6f6f 7369 6e67 2047 5265  or: Choosing GRe
│ │ │ │ -0003a8c0: 764c 6578 536d 616c 6c65 7374 5465 726d  vLexSmallestTerm
│ │ │ │ +0003a750: 2020 207c 0a7c 696e 7465 726e 616c 4368     |.|internalCh
│ │ │ │ +0003a760: 6f6f 7365 4d69 6e6f 723a 2043 686f 6f73  ooseMinor: Choos
│ │ │ │ +0003a770: 696e 6720 4c65 7853 6d61 6c6c 6573 7420  ing LexSmallest 
│ │ │ │ +0003a780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a7a0: 2020 207c 0a7c 7265 6775 6c61 7249 6e43     |.|regularInC
│ │ │ │ +0003a7b0: 6f64 696d 656e 7369 6f6e 3a20 204c 6f6f  odimension:  Loo
│ │ │ │ +0003a7c0: 7020 7374 6570 2c20 6162 6f75 7420 746f  p step, about to
│ │ │ │ +0003a7d0: 2063 6f6d 7075 7465 2064 696d 656e 7369   compute dimensi
│ │ │ │ +0003a7e0: 6f6e 2e20 2053 7562 6d61 7472 6963 6573  on.  Submatrices
│ │ │ │ +0003a7f0: 2063 6f7c 0a7c 7265 6775 6c61 7249 6e43   co|.|regularInC
│ │ │ │ +0003a800: 6f64 696d 656e 7369 6f6e 3a20 2069 7343  odimension:  isC
│ │ │ │ +0003a810: 6f64 696d 4174 4c65 6173 7420 6661 696c  odimAtLeast fail
│ │ │ │ +0003a820: 6564 2c20 636f 6d70 7574 696e 6720 636f  ed, computing co
│ │ │ │ +0003a830: 6469 6d2e 2020 2020 2020 2020 2020 2020  dim.            
│ │ │ │ +0003a840: 2020 207c 0a7c 7265 6775 6c61 7249 6e43     |.|regularInC
│ │ │ │ +0003a850: 6f64 696d 656e 7369 6f6e 3a20 2070 6172  odimension:  par
│ │ │ │ +0003a860: 7469 616c 2073 696e 6775 6c61 7220 6c6f  tial singular lo
│ │ │ │ +0003a870: 6375 7320 6469 6d65 6e73 696f 6e20 636f  cus dimension co
│ │ │ │ +0003a880: 6d70 7574 6564 2c20 3d20 3320 2020 2020  mputed, = 3     
│ │ │ │ +0003a890: 2020 207c 0a7c 696e 7465 726e 616c 4368     |.|internalCh
│ │ │ │ +0003a8a0: 6f6f 7365 4d69 6e6f 723a 2043 686f 6f73  ooseMinor: Choos
│ │ │ │ +0003a8b0: 696e 6720 4c65 7853 6d61 6c6c 6573 7454  ing LexSmallestT
│ │ │ │ +0003a8c0: 6572 6d20 2020 2020 2020 2020 2020 2020  erm             
│ │ │ │  0003a8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a8e0: 2020 2020 2020 2020 2020 2020 7c0a 7c69              |.|i
│ │ │ │ -0003a8f0: 6e74 6572 6e61 6c43 686f 6f73 654d 696e  nternalChooseMin
│ │ │ │ -0003a900: 6f72 3a20 4368 6f6f 7369 6e67 2047 5265  or: Choosing GRe
│ │ │ │ -0003a910: 764c 6578 536d 616c 6c65 7374 2020 2020  vLexSmallest    
│ │ │ │ +0003a8e0: 2020 207c 0a7c 696e 7465 726e 616c 4368     |.|internalCh
│ │ │ │ +0003a8f0: 6f6f 7365 4d69 6e6f 723a 2043 686f 6f73  ooseMinor: Choos
│ │ │ │ +0003a900: 696e 6720 4752 6576 4c65 7853 6d61 6c6c  ing GRevLexSmall
│ │ │ │ +0003a910: 6573 7454 6572 6d20 2020 2020 2020 2020  estTerm         
│ │ │ │  0003a920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a930: 2020 2020 2020 2020 2020 2020 7c0a 7c72              |.|r
│ │ │ │ -0003a940: 6567 756c 6172 496e 436f 6469 6d65 6e73  egularInCodimens
│ │ │ │ -0003a950: 696f 6e3a 2020 4c6f 6f70 2073 7465 702c  ion:  Loop step,
│ │ │ │ -0003a960: 2061 626f 7574 2074 6f20 636f 6d70 7574   about to comput
│ │ │ │ -0003a970: 6520 6469 6d65 6e73 696f 6e2e 2020 5375  e dimension.  Su
│ │ │ │ -0003a980: 626d 6174 7269 6365 7320 636f 7c0a 7c72  bmatrices co|.|r
│ │ │ │ -0003a990: 6567 756c 6172 496e 436f 6469 6d65 6e73  egularInCodimens
│ │ │ │ -0003a9a0: 696f 6e3a 2020 6973 436f 6469 6d41 744c  ion:  isCodimAtL
│ │ │ │ -0003a9b0: 6561 7374 2066 6169 6c65 642c 2063 6f6d  east failed, com
│ │ │ │ -0003a9c0: 7075 7469 6e67 2063 6f64 696d 2e20 2020  puting codim.   
│ │ │ │ -0003a9d0: 2020 2020 2020 2020 2020 2020 7c0a 7c72              |.|r
│ │ │ │ -0003a9e0: 6567 756c 6172 496e 436f 6469 6d65 6e73  egularInCodimens
│ │ │ │ -0003a9f0: 696f 6e3a 2020 7061 7274 6961 6c20 7369  ion:  partial si
│ │ │ │ -0003aa00: 6e67 756c 6172 206c 6f63 7573 2064 696d  ngular locus dim
│ │ │ │ -0003aa10: 656e 7369 6f6e 2063 6f6d 7075 7465 642c  ension computed,
│ │ │ │ -0003aa20: 203d 2033 2020 2020 2020 2020 7c0a 7c69   = 3        |.|i
│ │ │ │ -0003aa30: 6e74 6572 6e61 6c43 686f 6f73 654d 696e  nternalChooseMin
│ │ │ │ -0003aa40: 6f72 3a20 4368 6f6f 7369 6e67 2047 5265  or: Choosing GRe
│ │ │ │ -0003aa50: 764c 6578 536d 616c 6c65 7374 5465 726d  vLexSmallestTerm
│ │ │ │ -0003aa60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003aa70: 2020 2020 2020 2020 2020 2020 7c0a 7c69              |.|i
│ │ │ │ -0003aa80: 6e74 6572 6e61 6c43 686f 6f73 654d 696e  nternalChooseMin
│ │ │ │ -0003aa90: 6f72 3a20 4368 6f6f 7369 6e67 204c 6578  or: Choosing Lex
│ │ │ │ -0003aaa0: 536d 616c 6c65 7374 5465 726d 2020 2020  SmallestTerm    
│ │ │ │ +0003a930: 2020 207c 0a7c 696e 7465 726e 616c 4368     |.|internalCh
│ │ │ │ +0003a940: 6f6f 7365 4d69 6e6f 723a 2043 686f 6f73  ooseMinor: Choos
│ │ │ │ +0003a950: 696e 6720 4752 6576 4c65 7853 6d61 6c6c  ing GRevLexSmall
│ │ │ │ +0003a960: 6573 7420 2020 2020 2020 2020 2020 2020  est             
│ │ │ │ +0003a970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a980: 2020 207c 0a7c 7265 6775 6c61 7249 6e43     |.|regularInC
│ │ │ │ +0003a990: 6f64 696d 656e 7369 6f6e 3a20 204c 6f6f  odimension:  Loo
│ │ │ │ +0003a9a0: 7020 7374 6570 2c20 6162 6f75 7420 746f  p step, about to
│ │ │ │ +0003a9b0: 2063 6f6d 7075 7465 2064 696d 656e 7369   compute dimensi
│ │ │ │ +0003a9c0: 6f6e 2e20 2053 7562 6d61 7472 6963 6573  on.  Submatrices
│ │ │ │ +0003a9d0: 2063 6f7c 0a7c 7265 6775 6c61 7249 6e43   co|.|regularInC
│ │ │ │ +0003a9e0: 6f64 696d 656e 7369 6f6e 3a20 2069 7343  odimension:  isC
│ │ │ │ +0003a9f0: 6f64 696d 4174 4c65 6173 7420 6661 696c  odimAtLeast fail
│ │ │ │ +0003aa00: 6564 2c20 636f 6d70 7574 696e 6720 636f  ed, computing co
│ │ │ │ +0003aa10: 6469 6d2e 2020 2020 2020 2020 2020 2020  dim.            
│ │ │ │ +0003aa20: 2020 207c 0a7c 7265 6775 6c61 7249 6e43     |.|regularInC
│ │ │ │ +0003aa30: 6f64 696d 656e 7369 6f6e 3a20 2070 6172  odimension:  par
│ │ │ │ +0003aa40: 7469 616c 2073 696e 6775 6c61 7220 6c6f  tial singular lo
│ │ │ │ +0003aa50: 6375 7320 6469 6d65 6e73 696f 6e20 636f  cus dimension co
│ │ │ │ +0003aa60: 6d70 7574 6564 2c20 3d20 3320 2020 2020  mputed, = 3     
│ │ │ │ +0003aa70: 2020 207c 0a7c 696e 7465 726e 616c 4368     |.|internalCh
│ │ │ │ +0003aa80: 6f6f 7365 4d69 6e6f 723a 2043 686f 6f73  ooseMinor: Choos
│ │ │ │ +0003aa90: 696e 6720 4752 6576 4c65 7853 6d61 6c6c  ing GRevLexSmall
│ │ │ │ +0003aaa0: 6573 7454 6572 6d20 2020 2020 2020 2020  estTerm         
│ │ │ │  0003aab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003aac0: 2020 2020 2020 2020 2020 2020 7c0a 7c72              |.|r
│ │ │ │ -0003aad0: 6567 756c 6172 496e 436f 6469 6d65 6e73  egularInCodimens
│ │ │ │ -0003aae0: 696f 6e3a 2020 4c6f 6f70 2073 7465 702c  ion:  Loop step,
│ │ │ │ -0003aaf0: 2061 626f 7574 2074 6f20 636f 6d70 7574   about to comput
│ │ │ │ -0003ab00: 6520 6469 6d65 6e73 696f 6e2e 2020 5375  e dimension.  Su
│ │ │ │ -0003ab10: 626d 6174 7269 6365 7320 636f 7c0a 7c72  bmatrices co|.|r
│ │ │ │ -0003ab20: 6567 756c 6172 496e 436f 6469 6d65 6e73  egularInCodimens
│ │ │ │ -0003ab30: 696f 6e3a 2020 6973 436f 6469 6d41 744c  ion:  isCodimAtL
│ │ │ │ -0003ab40: 6561 7374 2066 6169 6c65 642c 2063 6f6d  east failed, com
│ │ │ │ -0003ab50: 7075 7469 6e67 2063 6f64 696d 2e20 2020  puting codim.   
│ │ │ │ -0003ab60: 2020 2020 2020 2020 2020 2020 7c0a 7c72              |.|r
│ │ │ │ -0003ab70: 6567 756c 6172 496e 436f 6469 6d65 6e73  egularInCodimens
│ │ │ │ -0003ab80: 696f 6e3a 2020 7061 7274 6961 6c20 7369  ion:  partial si
│ │ │ │ -0003ab90: 6e67 756c 6172 206c 6f63 7573 2064 696d  ngular locus dim
│ │ │ │ -0003aba0: 656e 7369 6f6e 2063 6f6d 7075 7465 642c  ension computed,
│ │ │ │ -0003abb0: 203d 2033 2020 2020 2020 2020 7c0a 7c69   = 3        |.|i
│ │ │ │ -0003abc0: 6e74 6572 6e61 6c43 686f 6f73 654d 696e  nternalChooseMin
│ │ │ │ -0003abd0: 6f72 3a20 4368 6f6f 7369 6e67 204c 6578  or: Choosing Lex
│ │ │ │ -0003abe0: 536d 616c 6c65 7374 5465 726d 2020 2020  SmallestTerm    
│ │ │ │ -0003abf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ac00: 2020 2020 2020 2020 2020 2020 7c0a 7c69              |.|i
│ │ │ │ -0003ac10: 6e74 6572 6e61 6c43 686f 6f73 654d 696e  nternalChooseMin
│ │ │ │ -0003ac20: 6f72 3a20 4368 6f6f 7369 6e67 2047 5265  or: Choosing GRe
│ │ │ │ -0003ac30: 764c 6578 536d 616c 6c65 7374 5465 726d  vLexSmallestTerm
│ │ │ │ +0003aac0: 2020 207c 0a7c 696e 7465 726e 616c 4368     |.|internalCh
│ │ │ │ +0003aad0: 6f6f 7365 4d69 6e6f 723a 2043 686f 6f73  ooseMinor: Choos
│ │ │ │ +0003aae0: 696e 6720 4c65 7853 6d61 6c6c 6573 7454  ing LexSmallestT
│ │ │ │ +0003aaf0: 6572 6d20 2020 2020 2020 2020 2020 2020  erm             
│ │ │ │ +0003ab00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ab10: 2020 207c 0a7c 7265 6775 6c61 7249 6e43     |.|regularInC
│ │ │ │ +0003ab20: 6f64 696d 656e 7369 6f6e 3a20 204c 6f6f  odimension:  Loo
│ │ │ │ +0003ab30: 7020 7374 6570 2c20 6162 6f75 7420 746f  p step, about to
│ │ │ │ +0003ab40: 2063 6f6d 7075 7465 2064 696d 656e 7369   compute dimensi
│ │ │ │ +0003ab50: 6f6e 2e20 2053 7562 6d61 7472 6963 6573  on.  Submatrices
│ │ │ │ +0003ab60: 2063 6f7c 0a7c 7265 6775 6c61 7249 6e43   co|.|regularInC
│ │ │ │ +0003ab70: 6f64 696d 656e 7369 6f6e 3a20 2069 7343  odimension:  isC
│ │ │ │ +0003ab80: 6f64 696d 4174 4c65 6173 7420 6661 696c  odimAtLeast fail
│ │ │ │ +0003ab90: 6564 2c20 636f 6d70 7574 696e 6720 636f  ed, computing co
│ │ │ │ +0003aba0: 6469 6d2e 2020 2020 2020 2020 2020 2020  dim.            
│ │ │ │ +0003abb0: 2020 207c 0a7c 7265 6775 6c61 7249 6e43     |.|regularInC
│ │ │ │ +0003abc0: 6f64 696d 656e 7369 6f6e 3a20 2070 6172  odimension:  par
│ │ │ │ +0003abd0: 7469 616c 2073 696e 6775 6c61 7220 6c6f  tial singular lo
│ │ │ │ +0003abe0: 6375 7320 6469 6d65 6e73 696f 6e20 636f  cus dimension co
│ │ │ │ +0003abf0: 6d70 7574 6564 2c20 3d20 3320 2020 2020  mputed, = 3     
│ │ │ │ +0003ac00: 2020 207c 0a7c 696e 7465 726e 616c 4368     |.|internalCh
│ │ │ │ +0003ac10: 6f6f 7365 4d69 6e6f 723a 2043 686f 6f73  ooseMinor: Choos
│ │ │ │ +0003ac20: 696e 6720 4c65 7853 6d61 6c6c 6573 7454  ing LexSmallestT
│ │ │ │ +0003ac30: 6572 6d20 2020 2020 2020 2020 2020 2020  erm             
│ │ │ │  0003ac40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ac50: 2020 2020 2020 2020 2020 2020 7c0a 7c72              |.|r
│ │ │ │ -0003ac60: 6567 756c 6172 496e 436f 6469 6d65 6e73  egularInCodimens
│ │ │ │ -0003ac70: 696f 6e3a 2020 4c6f 6f70 2073 7465 702c  ion:  Loop step,
│ │ │ │ -0003ac80: 2061 626f 7574 2074 6f20 636f 6d70 7574   about to comput
│ │ │ │ -0003ac90: 6520 6469 6d65 6e73 696f 6e2e 2020 5375  e dimension.  Su
│ │ │ │ -0003aca0: 626d 6174 7269 6365 7320 636f 7c0a 7c72  bmatrices co|.|r
│ │ │ │ -0003acb0: 6567 756c 6172 496e 436f 6469 6d65 6e73  egularInCodimens
│ │ │ │ -0003acc0: 696f 6e3a 2020 6973 436f 6469 6d41 744c  ion:  isCodimAtL
│ │ │ │ -0003acd0: 6561 7374 2066 6169 6c65 642c 2063 6f6d  east failed, com
│ │ │ │ -0003ace0: 7075 7469 6e67 2063 6f64 696d 2e20 2020  puting codim.   
│ │ │ │ -0003acf0: 2020 2020 2020 2020 2020 2020 7c0a 7c72              |.|r
│ │ │ │ -0003ad00: 6567 756c 6172 496e 436f 6469 6d65 6e73  egularInCodimens
│ │ │ │ -0003ad10: 696f 6e3a 2020 7061 7274 6961 6c20 7369  ion:  partial si
│ │ │ │ -0003ad20: 6e67 756c 6172 206c 6f63 7573 2064 696d  ngular locus dim
│ │ │ │ -0003ad30: 656e 7369 6f6e 2063 6f6d 7075 7465 642c  ension computed,
│ │ │ │ -0003ad40: 203d 2033 2020 2020 2020 2020 7c0a 7c72   = 3        |.|r
│ │ │ │ -0003ad50: 6567 756c 6172 496e 436f 6469 6d65 6e73  egularInCodimens
│ │ │ │ -0003ad60: 696f 6e3a 2020 4c6f 6f70 2063 6f6d 706c  ion:  Loop compl
│ │ │ │ -0003ad70: 6574 6564 2c20 7375 626d 6174 7269 6365  eted, submatrice
│ │ │ │ -0003ad80: 7320 636f 6e73 6964 6572 6564 203d 2031  s considered = 1
│ │ │ │ -0003ad90: 302c 2061 6e64 2063 6f6d 7075 7c0a 7c2d  0, and compu|.|-
│ │ │ │ -0003ada0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003adb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003adc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003add0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003ade0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c3e  ------------|.|>
│ │ │ │ -0003adf0: 332c 2056 6572 626f 7365 3d3e 7472 7565  3, Verbose=>true
│ │ │ │ -0003ae00: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -0003ae10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ae20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ae30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003ae40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ac50: 2020 207c 0a7c 696e 7465 726e 616c 4368     |.|internalCh
│ │ │ │ +0003ac60: 6f6f 7365 4d69 6e6f 723a 2043 686f 6f73  ooseMinor: Choos
│ │ │ │ +0003ac70: 696e 6720 4752 6576 4c65 7853 6d61 6c6c  ing GRevLexSmall
│ │ │ │ +0003ac80: 6573 7454 6572 6d20 2020 2020 2020 2020  estTerm         
│ │ │ │ +0003ac90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003aca0: 2020 207c 0a7c 7265 6775 6c61 7249 6e43     |.|regularInC
│ │ │ │ +0003acb0: 6f64 696d 656e 7369 6f6e 3a20 204c 6f6f  odimension:  Loo
│ │ │ │ +0003acc0: 7020 7374 6570 2c20 6162 6f75 7420 746f  p step, about to
│ │ │ │ +0003acd0: 2063 6f6d 7075 7465 2064 696d 656e 7369   compute dimensi
│ │ │ │ +0003ace0: 6f6e 2e20 2053 7562 6d61 7472 6963 6573  on.  Submatrices
│ │ │ │ +0003acf0: 2063 6f7c 0a7c 7265 6775 6c61 7249 6e43   co|.|regularInC
│ │ │ │ +0003ad00: 6f64 696d 656e 7369 6f6e 3a20 2069 7343  odimension:  isC
│ │ │ │ +0003ad10: 6f64 696d 4174 4c65 6173 7420 6661 696c  odimAtLeast fail
│ │ │ │ +0003ad20: 6564 2c20 636f 6d70 7574 696e 6720 636f  ed, computing co
│ │ │ │ +0003ad30: 6469 6d2e 2020 2020 2020 2020 2020 2020  dim.            
│ │ │ │ +0003ad40: 2020 207c 0a7c 7265 6775 6c61 7249 6e43     |.|regularInC
│ │ │ │ +0003ad50: 6f64 696d 656e 7369 6f6e 3a20 2070 6172  odimension:  par
│ │ │ │ +0003ad60: 7469 616c 2073 696e 6775 6c61 7220 6c6f  tial singular lo
│ │ │ │ +0003ad70: 6375 7320 6469 6d65 6e73 696f 6e20 636f  cus dimension co
│ │ │ │ +0003ad80: 6d70 7574 6564 2c20 3d20 3320 2020 2020  mputed, = 3     
│ │ │ │ +0003ad90: 2020 207c 0a7c 7265 6775 6c61 7249 6e43     |.|regularInC
│ │ │ │ +0003ada0: 6f64 696d 656e 7369 6f6e 3a20 204c 6f6f  odimension:  Loo
│ │ │ │ +0003adb0: 7020 636f 6d70 6c65 7465 642c 2073 7562  p completed, sub
│ │ │ │ +0003adc0: 6d61 7472 6963 6573 2063 6f6e 7369 6465  matrices conside
│ │ │ │ +0003add0: 7265 6420 3d20 3130 2c20 616e 6420 636f  red = 10, and co
│ │ │ │ +0003ade0: 6d70 757c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d  mpu|.|----------
│ │ │ │ +0003adf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003ae00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003ae10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003ae20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003ae30: 2d2d 2d7c 0a7c 3e33 2c20 5665 7262 6f73  ---|.|>3, Verbos
│ │ │ │ +0003ae40: 653d 3e74 7275 6529 2020 2020 2020 2020  e=>true)        
│ │ │ │  0003ae50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ae60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ae70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ae80: 2020 2020 2020 2020 2020 2020 7c0a 7c6e              |.|n
│ │ │ │ -0003ae90: 6f72 732c 2077 6520 7769 6c6c 2063 6f6d  ors, we will com
│ │ │ │ -0003aea0: 7075 7465 2075 7020 746f 2031 3020 6f66  pute up to 10 of
│ │ │ │ -0003aeb0: 2074 6865 6d2e 2020 2020 2020 2020 2020   them.          
│ │ │ │ +0003ae80: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003ae90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003aea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003aeb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003aec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003aed0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003aee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003aef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003aed0: 2020 207c 0a7c 6e6f 7273 2c20 7765 2077     |.|nors, we w
│ │ │ │ +0003aee0: 696c 6c20 636f 6d70 7574 6520 7570 2074  ill compute up t
│ │ │ │ +0003aef0: 6f20 3130 206f 6620 7468 656d 2e20 2020  o 10 of them.   
│ │ │ │  0003af00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003af10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003af20: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003af20: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0003af30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003af40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003af50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003af60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003af70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003af70: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0003af80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003af90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003afa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003afb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003afc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003afc0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0003afd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003afe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003aff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b010: 2020 2020 2020 2020 2020 2020 7c0a 7c6e              |.|n
│ │ │ │ -0003b020: 7369 6465 7265 643a 2033 2c20 616e 6420  sidered: 3, and 
│ │ │ │ -0003b030: 636f 6d70 7574 6564 203d 2033 2020 2020  computed = 3    
│ │ │ │ +0003b010: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003b020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b060: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003b070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b060: 2020 207c 0a7c 6e73 6964 6572 6564 3a20     |.|nsidered: 
│ │ │ │ +0003b070: 332c 2061 6e64 2063 6f6d 7075 7465 6420  3, and computed 
│ │ │ │ +0003b080: 3d20 3320 2020 2020 2020 2020 2020 2020  = 3             
│ │ │ │  0003b090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b0b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003b0b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0003b0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b100: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003b100: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0003b110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b150: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003b150: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0003b160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b1a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003b1a0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0003b1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b1f0: 2020 2020 2020 2020 2020 2020 7c0a 7c6e              |.|n
│ │ │ │ -0003b200: 7369 6465 7265 643a 2036 2c20 616e 6420  sidered: 6, and 
│ │ │ │ -0003b210: 636f 6d70 7574 6564 203d 2036 2020 2020  computed = 6    
│ │ │ │ +0003b1f0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003b200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b240: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003b250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b240: 2020 207c 0a7c 6e73 6964 6572 6564 3a20     |.|nsidered: 
│ │ │ │ +0003b250: 362c 2061 6e64 2063 6f6d 7075 7465 6420  6, and computed 
│ │ │ │ +0003b260: 3d20 3620 2020 2020 2020 2020 2020 2020  = 6             
│ │ │ │  0003b270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b290: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003b290: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0003b2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b2e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003b2e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0003b2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b330: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003b330: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0003b340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b380: 2020 2020 2020 2020 2020 2020 7c0a 7c6e              |.|n
│ │ │ │ -0003b390: 7369 6465 7265 643a 2038 2c20 616e 6420  sidered: 8, and 
│ │ │ │ -0003b3a0: 636f 6d70 7574 6564 203d 2038 2020 2020  computed = 8    
│ │ │ │ +0003b380: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003b390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b3d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003b3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b3d0: 2020 207c 0a7c 6e73 6964 6572 6564 3a20     |.|nsidered: 
│ │ │ │ +0003b3e0: 382c 2061 6e64 2063 6f6d 7075 7465 6420  8, and computed 
│ │ │ │ +0003b3f0: 3d20 3820 2020 2020 2020 2020 2020 2020  = 8             
│ │ │ │  0003b400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b420: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003b420: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0003b430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b470: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003b470: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0003b480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b4c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003b4c0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0003b4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b510: 2020 2020 2020 2020 2020 2020 7c0a 7c6e              |.|n
│ │ │ │ -0003b520: 7369 6465 7265 643a 2031 302c 2061 6e64  sidered: 10, and
│ │ │ │ -0003b530: 2063 6f6d 7075 7465 6420 3d20 3130 2020   computed = 10  
│ │ │ │ +0003b510: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003b520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b560: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003b570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b560: 2020 207c 0a7c 6e73 6964 6572 6564 3a20     |.|nsidered: 
│ │ │ │ +0003b570: 3130 2c20 616e 6420 636f 6d70 7574 6564  10, and computed
│ │ │ │ +0003b580: 203d 2031 3020 2020 2020 2020 2020 2020   = 10           
│ │ │ │  0003b590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b5b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003b5b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0003b5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b600: 2020 2020 2020 2020 2020 2020 7c0a 7c74              |.|t
│ │ │ │ -0003b610: 6564 203d 2031 302e 2020 7369 6e67 756c  ed = 10.  singul
│ │ │ │ -0003b620: 6172 206c 6f63 7573 2064 696d 656e 7369  ar locus dimensi
│ │ │ │ -0003b630: 6f6e 2061 7070 6561 7273 2074 6f20 6265  on appears to be
│ │ │ │ -0003b640: 203d 2033 2020 2020 2020 2020 2020 2020   = 3            
│ │ │ │ -0003b650: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -0003b660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003b670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003b680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003b690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003b6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a4d  ------------+..M
│ │ │ │ -0003b6b0: 696e 4d69 6e6f 7273 4675 6e63 7469 6f6e  inMinorsFunction
│ │ │ │ -0003b6c0: 2e20 5765 2070 6173 7320 4d69 6e4d 696e  . We pass MinMin
│ │ │ │ -0003b6d0: 6f72 7346 756e 6374 696f 6e20 6120 6675  orsFunction a fu
│ │ │ │ -0003b6e0: 6e63 7469 6f6e 2077 6869 6368 2073 656e  nction which sen
│ │ │ │ -0003b6f0: 6473 2074 6865 206d 696e 696d 756d 0a6e  ds the minimum.n
│ │ │ │ -0003b700: 756d 6265 7220 6f66 206d 696e 6f72 7320  umber of minors 
│ │ │ │ -0003b710: 6e65 6564 6564 2074 6f20 7665 7269 6679  needed to verify
│ │ │ │ -0003b720: 2074 6861 7420 736f 6d65 7468 696e 6720   that something 
│ │ │ │ -0003b730: 6973 2072 6567 756c 6172 2069 6e20 636f  is regular in co
│ │ │ │ -0003b740: 6469 6d65 6e73 696f 6e20 246e 240a 2877  dimension $n$.(w
│ │ │ │ -0003b750: 6869 6368 2069 7320 616c 7761 7973 2024  hich is always $
│ │ │ │ -0003b760: 6e2b 3124 2920 746f 2074 6865 206e 756d  n+1$) to the num
│ │ │ │ -0003b770: 6265 7220 6f66 206d 696e 6f72 7320 746f  ber of minors to
│ │ │ │ -0003b780: 2063 6f6d 7075 7465 2062 6566 6f72 6520   compute before 
│ │ │ │ -0003b790: 636f 6d70 7574 696e 6720 7468 650a 6469  computing the.di
│ │ │ │ -0003b7a0: 6d65 6e73 696f 6e20 6f66 2074 6865 2070  mension of the p
│ │ │ │ -0003b7b0: 6172 7469 616c 2069 6465 616c 206f 6620  artial ideal of 
│ │ │ │ -0003b7c0: 6d69 6e6f 7273 2066 6f72 2074 6865 2066  minors for the f
│ │ │ │ -0003b7d0: 6972 7374 2074 696d 652e 2020 2059 6f75  irst time.   You
│ │ │ │ -0003b7e0: 2063 616e 2073 6565 2074 6861 740a 7468   can see that.th
│ │ │ │ -0003b7f0: 7265 6520 6d69 6e6f 7273 2077 6572 6520  ree minors were 
│ │ │ │ -0003b800: 636f 6d70 7574 6564 2069 6e20 7468 6520  computed in the 
│ │ │ │ -0003b810: 6162 6f76 6520 6578 616d 706c 6520 6265  above example be
│ │ │ │ -0003b820: 666f 7265 2077 6520 6174 7465 6d70 7420  fore we attempt 
│ │ │ │ -0003b830: 746f 2063 6f6d 7075 7465 0a63 6f64 696d  to compute.codim
│ │ │ │ -0003b840: 656e 7369 6f6e 2e0a 0a43 6f64 696d 4368  ension...CodimCh
│ │ │ │ -0003b850: 6563 6b46 756e 6374 696f 6e2e 2054 6865  eckFunction. The
│ │ │ │ -0003b860: 206f 7074 696f 6e20 436f 6469 6d43 6865   option CodimChe
│ │ │ │ -0003b870: 636b 4675 6e63 7469 6f6e 2063 6f6e 7472  ckFunction contr
│ │ │ │ -0003b880: 6f6c 7320 686f 7720 6672 6571 7565 6e74  ols how frequent
│ │ │ │ -0003b890: 6c79 2074 6865 0a64 696d 656e 7369 6f6e  ly the.dimension
│ │ │ │ -0003b8a0: 206f 6620 7468 6520 7061 7274 6961 6c20   of the partial 
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│ │ │ │ +0003b630: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0003b6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0003b6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0003b890: 436f 6469 6d43 6865 636b 4675 6e63 7469  CodimCheckFuncti
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│ │ │ │ +0003ba50: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..+-------------
│ │ │ │ +0003ba60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003ba70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003ba80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003ba90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003baa0: 2b0a 7c69 3130 203a 2074 696d 6520 7265  +.|i10 : time re
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│ │ │ │ +0003bae0: 6865 636b 4675 6e63 7469 6f6e 203d 3e20  heckFunction => 
│ │ │ │ +0003baf0: 7c0a 7c20 2d2d 2075 7365 6420 302e 3834  |.| -- used 0.84
│ │ │ │ +0003bb00: 3636 3432 7320 2863 7075 293b 2030 2e35  6642s (cpu); 0.5
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│ │ │ │ +0003bb20: 2030 7320 2867 6329 2020 2020 2020 2020   0s (gc)        
│ │ │ │ +0003bb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003bb40: 7c0a 7c72 6567 756c 6172 496e 436f 6469  |.|regularInCodi
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│ │ │ │ +0003bc10: 5465 726d 2020 2020 2020 2020 2020 2020  Term            
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│ │ │ │ +0003bc60: 5465 726d 2020 2020 2020 2020 2020 2020  Term            
│ │ │ │ +0003bc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003bc80: 7c0a 7c72 6567 756c 6172 496e 436f 6469  |.|regularInCodi
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│ │ │ │ +0003bcc0: 2020 5375 626d 6174 7269 6365 7320 636f    Submatrices co
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│ │ │ │ +0003bd20: 7c0a 7c72 6567 756c 6172 496e 436f 6469  |.|regularInCodi
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│ │ │ │ +0003bd40: 6c20 7369 6e67 756c 6172 206c 6f63 7573  l singular locus
│ │ │ │ +0003bd50: 2064 696d 656e 7369 6f6e 2063 6f6d 7075   dimension compu
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│ │ │ │ +0003bd70: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73  |.|internalChoos
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│ │ │ │ +0003bd90: 2047 5265 764c 6578 536d 616c 6c65 7374   GRevLexSmallest
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│ │ │ │  0003bdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003bdd0: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a  rnalChooseMinor:
│ │ │ │ -0003bde0: 2043 686f 6f73 696e 6720 5261 6e64 6f6d   Choosing Random
│ │ │ │ +0003bdc0: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73  |.|internalChoos
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│ │ │ │  0003be00: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003be70: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ -0003be80: 3a20 2069 7343 6f64 696d 4174 4c65 6173  :  isCodimAtLeas
│ │ │ │ -0003be90: 7420 6661 696c 6564 2c20 636f 6d70 7574  t failed, comput
│ │ │ │ -0003bea0: 696e 6720 636f 6469 6d2e 2020 2020 2020  ing codim.      
│ │ │ │ -0003beb0: 2020 2020 2020 2020 207c 0a7c 7265 6775           |.|regu
│ │ │ │ -0003bec0: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ -0003bed0: 3a20 2070 6172 7469 616c 2073 696e 6775  :  partial singu
│ │ │ │ -0003bee0: 6c61 7220 6c6f 6375 7320 6469 6d65 6e73  lar locus dimens
│ │ │ │ -0003bef0: 696f 6e20 636f 6d70 7574 6564 2c20 3d20  ion computed, = 
│ │ │ │ -0003bf00: 3420 2020 2020 2020 207c 0a7c 696e 7465  4        |.|inte
│ │ │ │ -0003bf10: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a  rnalChooseMinor:
│ │ │ │ -0003bf20: 2043 686f 6f73 696e 6720 4752 6576 4c65   Choosing GRevLe
│ │ │ │ -0003bf30: 7853 6d61 6c6c 6573 7454 6572 6d20 2020  xSmallestTerm   
│ │ │ │ -0003bf40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bf50: 2020 2020 2020 2020 207c 0a7c 696e 7465           |.|inte
│ │ │ │ -0003bf60: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a  rnalChooseMinor:
│ │ │ │ -0003bf70: 2043 686f 6f73 696e 6720 4c65 7853 6d61   Choosing LexSma
│ │ │ │ -0003bf80: 6c6c 6573 7420 2020 2020 2020 2020 2020  llest           
│ │ │ │ +0003be10: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73  |.|internalChoos
│ │ │ │ +0003be20: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67  eMinor: Choosing
│ │ │ │ +0003be30: 2052 616e 646f 6d20 2020 2020 2020 2020   Random         
│ │ │ │ +0003be40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003be50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003be60: 7c0a 7c72 6567 756c 6172 496e 436f 6469  |.|regularInCodi
│ │ │ │ +0003be70: 6d65 6e73 696f 6e3a 2020 4c6f 6f70 2073  mension:  Loop s
│ │ │ │ +0003be80: 7465 702c 2061 626f 7574 2074 6f20 636f  tep, about to co
│ │ │ │ +0003be90: 6d70 7574 6520 6469 6d65 6e73 696f 6e2e  mpute dimension.
│ │ │ │ +0003bea0: 2020 5375 626d 6174 7269 6365 7320 636f    Submatrices co
│ │ │ │ +0003beb0: 7c0a 7c72 6567 756c 6172 496e 436f 6469  |.|regularInCodi
│ │ │ │ +0003bec0: 6d65 6e73 696f 6e3a 2020 6973 436f 6469  mension:  isCodi
│ │ │ │ +0003bed0: 6d41 744c 6561 7374 2066 6169 6c65 642c  mAtLeast failed,
│ │ │ │ +0003bee0: 2063 6f6d 7075 7469 6e67 2063 6f64 696d   computing codim
│ │ │ │ +0003bef0: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ +0003bf00: 7c0a 7c72 6567 756c 6172 496e 436f 6469  |.|regularInCodi
│ │ │ │ +0003bf10: 6d65 6e73 696f 6e3a 2020 7061 7274 6961  mension:  partia
│ │ │ │ +0003bf20: 6c20 7369 6e67 756c 6172 206c 6f63 7573  l singular locus
│ │ │ │ +0003bf30: 2064 696d 656e 7369 6f6e 2063 6f6d 7075   dimension compu
│ │ │ │ +0003bf40: 7465 642c 203d 2034 2020 2020 2020 2020  ted, = 4        
│ │ │ │ +0003bf50: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73  |.|internalChoos
│ │ │ │ +0003bf60: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67  eMinor: Choosing
│ │ │ │ +0003bf70: 2047 5265 764c 6578 536d 616c 6c65 7374   GRevLexSmallest
│ │ │ │ +0003bf80: 5465 726d 2020 2020 2020 2020 2020 2020  Term            
│ │ │ │  0003bf90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bfa0: 2020 2020 2020 2020 207c 0a7c 696e 7465           |.|inte
│ │ │ │ -0003bfb0: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a  rnalChooseMinor:
│ │ │ │ -0003bfc0: 2043 686f 6f73 696e 6720 5261 6e64 6f6d   Choosing Random
│ │ │ │ -0003bfd0: 4e6f 6e5a 6572 6f20 2020 2020 2020 2020  NonZero         
│ │ │ │ +0003bfa0: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73  |.|internalChoos
│ │ │ │ +0003bfb0: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67  eMinor: Choosing
│ │ │ │ +0003bfc0: 204c 6578 536d 616c 6c65 7374 2020 2020   LexSmallest    
│ │ │ │ +0003bfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bff0: 2020 2020 2020 2020 207c 0a7c 696e 7465           |.|inte
│ │ │ │ -0003c000: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a  rnalChooseMinor:
│ │ │ │ -0003c010: 2043 686f 6f73 696e 6720 4752 6576 4c65   Choosing GRevLe
│ │ │ │ -0003c020: 7853 6d61 6c6c 6573 7420 2020 2020 2020  xSmallest       
│ │ │ │ +0003bff0: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73  |.|internalChoos
│ │ │ │ +0003c000: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67  eMinor: Choosing
│ │ │ │ +0003c010: 2052 616e 646f 6d4e 6f6e 5a65 726f 2020   RandomNonZero  
│ │ │ │ +0003c020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c040: 2020 2020 2020 2020 207c 0a7c 696e 7465           |.|inte
│ │ │ │ -0003c050: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a  rnalChooseMinor:
│ │ │ │ -0003c060: 2043 686f 6f73 696e 6720 4752 6576 4c65   Choosing GRevLe
│ │ │ │ -0003c070: 7853 6d61 6c6c 6573 7420 2020 2020 2020  xSmallest       
│ │ │ │ +0003c040: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73  |.|internalChoos
│ │ │ │ +0003c050: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67  eMinor: Choosing
│ │ │ │ +0003c060: 2047 5265 764c 6578 536d 616c 6c65 7374   GRevLexSmallest
│ │ │ │ +0003c070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c090: 2020 2020 2020 2020 207c 0a7c 7265 6775           |.|regu
│ │ │ │ -0003c0a0: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ -0003c0b0: 3a20 204c 6f6f 7020 7374 6570 2c20 6162  :  Loop step, ab
│ │ │ │ -0003c0c0: 6f75 7420 746f 2063 6f6d 7075 7465 2064  out to compute d
│ │ │ │ -0003c0d0: 696d 656e 7369 6f6e 2e20 2053 7562 6d61  imension.  Subma
│ │ │ │ -0003c0e0: 7472 6963 6573 2063 6f7c 0a7c 7265 6775  trices co|.|regu
│ │ │ │ -0003c0f0: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ -0003c100: 3a20 2069 7343 6f64 696d 4174 4c65 6173  :  isCodimAtLeas
│ │ │ │ -0003c110: 7420 6661 696c 6564 2c20 636f 6d70 7574  t failed, comput
│ │ │ │ -0003c120: 696e 6720 636f 6469 6d2e 2020 2020 2020  ing codim.      
│ │ │ │ -0003c130: 2020 2020 2020 2020 207c 0a7c 7265 6775           |.|regu
│ │ │ │ -0003c140: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ -0003c150: 3a20 2070 6172 7469 616c 2073 696e 6775  :  partial singu
│ │ │ │ -0003c160: 6c61 7220 6c6f 6375 7320 6469 6d65 6e73  lar locus dimens
│ │ │ │ -0003c170: 696f 6e20 636f 6d70 7574 6564 2c20 3d20  ion computed, = 
│ │ │ │ -0003c180: 3420 2020 2020 2020 207c 0a7c 696e 7465  4        |.|inte
│ │ │ │ -0003c190: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a  rnalChooseMinor:
│ │ │ │ -0003c1a0: 2043 686f 6f73 696e 6720 4c65 7853 6d61   Choosing LexSma
│ │ │ │ -0003c1b0: 6c6c 6573 7454 6572 6d20 2020 2020 2020  llestTerm       
│ │ │ │ -0003c1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c1d0: 2020 2020 2020 2020 207c 0a7c 696e 7465           |.|inte
│ │ │ │ -0003c1e0: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a  rnalChooseMinor:
│ │ │ │ -0003c1f0: 2043 686f 6f73 696e 6720 4752 6576 4c65   Choosing GRevLe
│ │ │ │ -0003c200: 7853 6d61 6c6c 6573 7454 6572 6d20 2020  xSmallestTerm   
│ │ │ │ +0003c090: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73  |.|internalChoos
│ │ │ │ +0003c0a0: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67  eMinor: Choosing
│ │ │ │ +0003c0b0: 2047 5265 764c 6578 536d 616c 6c65 7374   GRevLexSmallest
│ │ │ │ +0003c0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c0e0: 7c0a 7c72 6567 756c 6172 496e 436f 6469  |.|regularInCodi
│ │ │ │ +0003c0f0: 6d65 6e73 696f 6e3a 2020 4c6f 6f70 2073  mension:  Loop s
│ │ │ │ +0003c100: 7465 702c 2061 626f 7574 2074 6f20 636f  tep, about to co
│ │ │ │ +0003c110: 6d70 7574 6520 6469 6d65 6e73 696f 6e2e  mpute dimension.
│ │ │ │ +0003c120: 2020 5375 626d 6174 7269 6365 7320 636f    Submatrices co
│ │ │ │ +0003c130: 7c0a 7c72 6567 756c 6172 496e 436f 6469  |.|regularInCodi
│ │ │ │ +0003c140: 6d65 6e73 696f 6e3a 2020 6973 436f 6469  mension:  isCodi
│ │ │ │ +0003c150: 6d41 744c 6561 7374 2066 6169 6c65 642c  mAtLeast failed,
│ │ │ │ +0003c160: 2063 6f6d 7075 7469 6e67 2063 6f64 696d   computing codim
│ │ │ │ +0003c170: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ +0003c180: 7c0a 7c72 6567 756c 6172 496e 436f 6469  |.|regularInCodi
│ │ │ │ +0003c190: 6d65 6e73 696f 6e3a 2020 7061 7274 6961  mension:  partia
│ │ │ │ +0003c1a0: 6c20 7369 6e67 756c 6172 206c 6f63 7573  l singular locus
│ │ │ │ +0003c1b0: 2064 696d 656e 7369 6f6e 2063 6f6d 7075   dimension compu
│ │ │ │ +0003c1c0: 7465 642c 203d 2034 2020 2020 2020 2020  ted, = 4        
│ │ │ │ +0003c1d0: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73  |.|internalChoos
│ │ │ │ +0003c1e0: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67  eMinor: Choosing
│ │ │ │ +0003c1f0: 204c 6578 536d 616c 6c65 7374 5465 726d   LexSmallestTerm
│ │ │ │ +0003c200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c220: 2020 2020 2020 2020 207c 0a7c 696e 7465           |.|inte
│ │ │ │ -0003c230: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a  rnalChooseMinor:
│ │ │ │ -0003c240: 2043 686f 6f73 696e 6720 4c65 7853 6d61   Choosing LexSma
│ │ │ │ -0003c250: 6c6c 6573 7420 2020 2020 2020 2020 2020  llest           
│ │ │ │ +0003c220: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73  |.|internalChoos
│ │ │ │ +0003c230: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67  eMinor: Choosing
│ │ │ │ +0003c240: 2047 5265 764c 6578 536d 616c 6c65 7374   GRevLexSmallest
│ │ │ │ +0003c250: 5465 726d 2020 2020 2020 2020 2020 2020  Term            
│ │ │ │  0003c260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c270: 2020 2020 2020 2020 207c 0a7c 696e 7465           |.|inte
│ │ │ │ -0003c280: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a  rnalChooseMinor:
│ │ │ │ -0003c290: 2043 686f 6f73 696e 6720 4752 6576 4c65   Choosing GRevLe
│ │ │ │ -0003c2a0: 7853 6d61 6c6c 6573 7420 2020 2020 2020  xSmallest       
│ │ │ │ +0003c270: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73  |.|internalChoos
│ │ │ │ +0003c280: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67  eMinor: Choosing
│ │ │ │ +0003c290: 204c 6578 536d 616c 6c65 7374 2020 2020   LexSmallest    
│ │ │ │ +0003c2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c2c0: 2020 2020 2020 2020 207c 0a7c 696e 7465           |.|inte
│ │ │ │ -0003c2d0: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a  rnalChooseMinor:
│ │ │ │ -0003c2e0: 2043 686f 6f73 696e 6720 5261 6e64 6f6d   Choosing Random
│ │ │ │ -0003c2f0: 4e6f 6e5a 6572 6f20 2020 2020 2020 2020  NonZero         
│ │ │ │ +0003c2c0: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73  |.|internalChoos
│ │ │ │ +0003c2d0: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67  eMinor: Choosing
│ │ │ │ +0003c2e0: 2047 5265 764c 6578 536d 616c 6c65 7374   GRevLexSmallest
│ │ │ │ +0003c2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c310: 2020 2020 2020 2020 207c 0a7c 7265 6775           |.|regu
│ │ │ │ -0003c320: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ -0003c330: 3a20 204c 6f6f 7020 7374 6570 2c20 6162  :  Loop step, ab
│ │ │ │ -0003c340: 6f75 7420 746f 2063 6f6d 7075 7465 2064  out to compute d
│ │ │ │ -0003c350: 696d 656e 7369 6f6e 2e20 2053 7562 6d61  imension.  Subma
│ │ │ │ -0003c360: 7472 6963 6573 2063 6f7c 0a7c 7265 6775  trices co|.|regu
│ │ │ │ -0003c370: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ -0003c380: 3a20 2069 7343 6f64 696d 4174 4c65 6173  :  isCodimAtLeas
│ │ │ │ -0003c390: 7420 6661 696c 6564 2c20 636f 6d70 7574  t failed, comput
│ │ │ │ -0003c3a0: 696e 6720 636f 6469 6d2e 2020 2020 2020  ing codim.      
│ │ │ │ -0003c3b0: 2020 2020 2020 2020 207c 0a7c 7265 6775           |.|regu
│ │ │ │ -0003c3c0: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ -0003c3d0: 3a20 2070 6172 7469 616c 2073 696e 6775  :  partial singu
│ │ │ │ -0003c3e0: 6c61 7220 6c6f 6375 7320 6469 6d65 6e73  lar locus dimens
│ │ │ │ -0003c3f0: 696f 6e20 636f 6d70 7574 6564 2c20 3d20  ion computed, = 
│ │ │ │ -0003c400: 3320 2020 2020 2020 207c 0a7c 696e 7465  3        |.|inte
│ │ │ │ -0003c410: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a  rnalChooseMinor:
│ │ │ │ -0003c420: 2043 686f 6f73 696e 6720 4c65 7853 6d61   Choosing LexSma
│ │ │ │ -0003c430: 6c6c 6573 7420 2020 2020 2020 2020 2020  llest           
│ │ │ │ -0003c440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c450: 2020 2020 2020 2020 207c 0a7c 696e 7465           |.|inte
│ │ │ │ -0003c460: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a  rnalChooseMinor:
│ │ │ │ -0003c470: 2043 686f 6f73 696e 6720 4c65 7853 6d61   Choosing LexSma
│ │ │ │ -0003c480: 6c6c 6573 7454 6572 6d20 2020 2020 2020  llestTerm       
│ │ │ │ +0003c310: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73  |.|internalChoos
│ │ │ │ +0003c320: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67  eMinor: Choosing
│ │ │ │ +0003c330: 2052 616e 646f 6d4e 6f6e 5a65 726f 2020   RandomNonZero  
│ │ │ │ +0003c340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c360: 7c0a 7c72 6567 756c 6172 496e 436f 6469  |.|regularInCodi
│ │ │ │ +0003c370: 6d65 6e73 696f 6e3a 2020 4c6f 6f70 2073  mension:  Loop s
│ │ │ │ +0003c380: 7465 702c 2061 626f 7574 2074 6f20 636f  tep, about to co
│ │ │ │ +0003c390: 6d70 7574 6520 6469 6d65 6e73 696f 6e2e  mpute dimension.
│ │ │ │ +0003c3a0: 2020 5375 626d 6174 7269 6365 7320 636f    Submatrices co
│ │ │ │ +0003c3b0: 7c0a 7c72 6567 756c 6172 496e 436f 6469  |.|regularInCodi
│ │ │ │ +0003c3c0: 6d65 6e73 696f 6e3a 2020 6973 436f 6469  mension:  isCodi
│ │ │ │ +0003c3d0: 6d41 744c 6561 7374 2066 6169 6c65 642c  mAtLeast failed,
│ │ │ │ +0003c3e0: 2063 6f6d 7075 7469 6e67 2063 6f64 696d   computing codim
│ │ │ │ +0003c3f0: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ +0003c400: 7c0a 7c72 6567 756c 6172 496e 436f 6469  |.|regularInCodi
│ │ │ │ +0003c410: 6d65 6e73 696f 6e3a 2020 7061 7274 6961  mension:  partia
│ │ │ │ +0003c420: 6c20 7369 6e67 756c 6172 206c 6f63 7573  l singular locus
│ │ │ │ +0003c430: 2064 696d 656e 7369 6f6e 2063 6f6d 7075   dimension compu
│ │ │ │ +0003c440: 7465 642c 203d 2033 2020 2020 2020 2020  ted, = 3        
│ │ │ │ +0003c450: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73  |.|internalChoos
│ │ │ │ +0003c460: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67  eMinor: Choosing
│ │ │ │ +0003c470: 204c 6578 536d 616c 6c65 7374 2020 2020   LexSmallest    
│ │ │ │ +0003c480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c4a0: 2020 2020 2020 2020 207c 0a7c 696e 7465           |.|inte
│ │ │ │ -0003c4b0: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a  rnalChooseMinor:
│ │ │ │ -0003c4c0: 2043 686f 6f73 696e 6720 5261 6e64 6f6d   Choosing Random
│ │ │ │ -0003c4d0: 4e6f 6e5a 6572 6f20 2020 2020 2020 2020  NonZero         
│ │ │ │ +0003c4a0: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73  |.|internalChoos
│ │ │ │ +0003c4b0: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67  eMinor: Choosing
│ │ │ │ +0003c4c0: 204c 6578 536d 616c 6c65 7374 5465 726d   LexSmallestTerm
│ │ │ │ +0003c4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c4f0: 2020 2020 2020 2020 207c 0a7c 696e 7465           |.|inte
│ │ │ │ -0003c500: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a  rnalChooseMinor:
│ │ │ │ -0003c510: 2043 686f 6f73 696e 6720 4752 6576 4c65   Choosing GRevLe
│ │ │ │ -0003c520: 7853 6d61 6c6c 6573 7454 6572 6d20 2020  xSmallestTerm   
│ │ │ │ +0003c4f0: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73  |.|internalChoos
│ │ │ │ +0003c500: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67  eMinor: Choosing
│ │ │ │ +0003c510: 2052 616e 646f 6d4e 6f6e 5a65 726f 2020   RandomNonZero  
│ │ │ │ +0003c520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c540: 2020 2020 2020 2020 207c 0a7c 696e 7465           |.|inte
│ │ │ │ -0003c550: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a  rnalChooseMinor:
│ │ │ │ -0003c560: 2043 686f 6f73 696e 6720 4752 6576 4c65   Choosing GRevLe
│ │ │ │ -0003c570: 7853 6d61 6c6c 6573 7454 6572 6d20 2020  xSmallestTerm   
│ │ │ │ +0003c540: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73  |.|internalChoos
│ │ │ │ +0003c550: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67  eMinor: Choosing
│ │ │ │ +0003c560: 2047 5265 764c 6578 536d 616c 6c65 7374   GRevLexSmallest
│ │ │ │ +0003c570: 5465 726d 2020 2020 2020 2020 2020 2020  Term            
│ │ │ │  0003c580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c590: 2020 2020 2020 2020 207c 0a7c 7265 6775           |.|regu
│ │ │ │ -0003c5a0: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ -0003c5b0: 3a20 204c 6f6f 7020 7374 6570 2c20 6162  :  Loop step, ab
│ │ │ │ -0003c5c0: 6f75 7420 746f 2063 6f6d 7075 7465 2064  out to compute d
│ │ │ │ -0003c5d0: 696d 656e 7369 6f6e 2e20 2053 7562 6d61  imension.  Subma
│ │ │ │ -0003c5e0: 7472 6963 6573 2063 6f7c 0a7c 7265 6775  trices co|.|regu
│ │ │ │ -0003c5f0: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ -0003c600: 3a20 2069 7343 6f64 696d 4174 4c65 6173  :  isCodimAtLeas
│ │ │ │ -0003c610: 7420 6661 696c 6564 2c20 636f 6d70 7574  t failed, comput
│ │ │ │ -0003c620: 696e 6720 636f 6469 6d2e 2020 2020 2020  ing codim.      
│ │ │ │ -0003c630: 2020 2020 2020 2020 207c 0a7c 7265 6775           |.|regu
│ │ │ │ -0003c640: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ -0003c650: 3a20 2070 6172 7469 616c 2073 696e 6775  :  partial singu
│ │ │ │ -0003c660: 6c61 7220 6c6f 6375 7320 6469 6d65 6e73  lar locus dimens
│ │ │ │ -0003c670: 696f 6e20 636f 6d70 7574 6564 2c20 3d20  ion computed, = 
│ │ │ │ -0003c680: 3320 2020 2020 2020 207c 0a7c 696e 7465  3        |.|inte
│ │ │ │ -0003c690: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a  rnalChooseMinor:
│ │ │ │ -0003c6a0: 2043 686f 6f73 696e 6720 4c65 7853 6d61   Choosing LexSma
│ │ │ │ -0003c6b0: 6c6c 6573 7420 2020 2020 2020 2020 2020  llest           
│ │ │ │ -0003c6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c6d0: 2020 2020 2020 2020 207c 0a7c 696e 7465           |.|inte
│ │ │ │ -0003c6e0: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a  rnalChooseMinor:
│ │ │ │ -0003c6f0: 2043 686f 6f73 696e 6720 5261 6e64 6f6d   Choosing Random
│ │ │ │ -0003c700: 4e6f 6e5a 6572 6f20 2020 2020 2020 2020  NonZero         
│ │ │ │ +0003c590: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73  |.|internalChoos
│ │ │ │ +0003c5a0: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67  eMinor: Choosing
│ │ │ │ +0003c5b0: 2047 5265 764c 6578 536d 616c 6c65 7374   GRevLexSmallest
│ │ │ │ +0003c5c0: 5465 726d 2020 2020 2020 2020 2020 2020  Term            
│ │ │ │ +0003c5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c5e0: 7c0a 7c72 6567 756c 6172 496e 436f 6469  |.|regularInCodi
│ │ │ │ +0003c5f0: 6d65 6e73 696f 6e3a 2020 4c6f 6f70 2073  mension:  Loop s
│ │ │ │ +0003c600: 7465 702c 2061 626f 7574 2074 6f20 636f  tep, about to co
│ │ │ │ +0003c610: 6d70 7574 6520 6469 6d65 6e73 696f 6e2e  mpute dimension.
│ │ │ │ +0003c620: 2020 5375 626d 6174 7269 6365 7320 636f    Submatrices co
│ │ │ │ +0003c630: 7c0a 7c72 6567 756c 6172 496e 436f 6469  |.|regularInCodi
│ │ │ │ +0003c640: 6d65 6e73 696f 6e3a 2020 6973 436f 6469  mension:  isCodi
│ │ │ │ +0003c650: 6d41 744c 6561 7374 2066 6169 6c65 642c  mAtLeast failed,
│ │ │ │ +0003c660: 2063 6f6d 7075 7469 6e67 2063 6f64 696d   computing codim
│ │ │ │ +0003c670: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ +0003c680: 7c0a 7c72 6567 756c 6172 496e 436f 6469  |.|regularInCodi
│ │ │ │ +0003c690: 6d65 6e73 696f 6e3a 2020 7061 7274 6961  mension:  partia
│ │ │ │ +0003c6a0: 6c20 7369 6e67 756c 6172 206c 6f63 7573  l singular locus
│ │ │ │ +0003c6b0: 2064 696d 656e 7369 6f6e 2063 6f6d 7075   dimension compu
│ │ │ │ +0003c6c0: 7465 642c 203d 2033 2020 2020 2020 2020  ted, = 3        
│ │ │ │ +0003c6d0: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73  |.|internalChoos
│ │ │ │ +0003c6e0: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67  eMinor: Choosing
│ │ │ │ +0003c6f0: 204c 6578 536d 616c 6c65 7374 2020 2020   LexSmallest    
│ │ │ │ +0003c700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c720: 2020 2020 2020 2020 207c 0a7c 696e 7465           |.|inte
│ │ │ │ -0003c730: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a  rnalChooseMinor:
│ │ │ │ -0003c740: 2043 686f 6f73 696e 6720 4c65 7853 6d61   Choosing LexSma
│ │ │ │ -0003c750: 6c6c 6573 7454 6572 6d20 2020 2020 2020  llestTerm       
│ │ │ │ +0003c720: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73  |.|internalChoos
│ │ │ │ +0003c730: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67  eMinor: Choosing
│ │ │ │ +0003c740: 2052 616e 646f 6d4e 6f6e 5a65 726f 2020   RandomNonZero  
│ │ │ │ +0003c750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c770: 2020 2020 2020 2020 207c 0a7c 696e 7465           |.|inte
│ │ │ │ -0003c780: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a  rnalChooseMinor:
│ │ │ │ -0003c790: 2043 686f 6f73 696e 6720 4752 6576 4c65   Choosing GRevLe
│ │ │ │ -0003c7a0: 7853 6d61 6c6c 6573 7420 2020 2020 2020  xSmallest       
│ │ │ │ +0003c770: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73  |.|internalChoos
│ │ │ │ +0003c780: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67  eMinor: Choosing
│ │ │ │ +0003c790: 204c 6578 536d 616c 6c65 7374 5465 726d   LexSmallestTerm
│ │ │ │ +0003c7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c7c0: 2020 2020 2020 2020 207c 0a7c 696e 7465           |.|inte
│ │ │ │ -0003c7d0: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a  rnalChooseMinor:
│ │ │ │ -0003c7e0: 2043 686f 6f73 696e 6720 4752 6576 4c65   Choosing GRevLe
│ │ │ │ -0003c7f0: 7853 6d61 6c6c 6573 7454 6572 6d20 2020  xSmallestTerm   
│ │ │ │ +0003c7c0: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73  |.|internalChoos
│ │ │ │ +0003c7d0: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67  eMinor: Choosing
│ │ │ │ +0003c7e0: 2047 5265 764c 6578 536d 616c 6c65 7374   GRevLexSmallest
│ │ │ │ +0003c7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c810: 2020 2020 2020 2020 207c 0a7c 7265 6775           |.|regu
│ │ │ │ -0003c820: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ -0003c830: 3a20 204c 6f6f 7020 7374 6570 2c20 6162  :  Loop step, ab
│ │ │ │ -0003c840: 6f75 7420 746f 2063 6f6d 7075 7465 2064  out to compute d
│ │ │ │ -0003c850: 696d 656e 7369 6f6e 2e20 2053 7562 6d61  imension.  Subma
│ │ │ │ -0003c860: 7472 6963 6573 2063 6f7c 0a7c 7265 6775  trices co|.|regu
│ │ │ │ -0003c870: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ -0003c880: 3a20 2069 7343 6f64 696d 4174 4c65 6173  :  isCodimAtLeas
│ │ │ │ -0003c890: 7420 6661 696c 6564 2c20 636f 6d70 7574  t failed, comput
│ │ │ │ -0003c8a0: 696e 6720 636f 6469 6d2e 2020 2020 2020  ing codim.      
│ │ │ │ -0003c8b0: 2020 2020 2020 2020 207c 0a7c 7265 6775           |.|regu
│ │ │ │ -0003c8c0: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ -0003c8d0: 3a20 2070 6172 7469 616c 2073 696e 6775  :  partial singu
│ │ │ │ -0003c8e0: 6c61 7220 6c6f 6375 7320 6469 6d65 6e73  lar locus dimens
│ │ │ │ -0003c8f0: 696f 6e20 636f 6d70 7574 6564 2c20 3d20  ion computed, = 
│ │ │ │ -0003c900: 3320 2020 2020 2020 207c 0a7c 7265 6775  3        |.|regu
│ │ │ │ -0003c910: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ -0003c920: 3a20 204c 6f6f 7020 636f 6d70 6c65 7465  :  Loop complete
│ │ │ │ -0003c930: 642c 2073 7562 6d61 7472 6963 6573 2063  d, submatrices c
│ │ │ │ -0003c940: 6f6e 7369 6465 7265 6420 3d20 3235 2c20  onsidered = 25, 
│ │ │ │ -0003c950: 616e 6420 636f 6d70 757c 0a7c 2d2d 2d2d  and compu|.|----
│ │ │ │ -0003c960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003c970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003c980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003c990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003c9a0: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 742d 3e74  ---------|.|t->t
│ │ │ │ -0003c9b0: 2f35 2c20 4d69 6e4d 696e 6f72 7346 756e  /5, MinMinorsFun
│ │ │ │ -0003c9c0: 6374 696f 6e20 3d3e 2074 2d3e 322c 2056  ction => t->2, V
│ │ │ │ -0003c9d0: 6572 626f 7365 3d3e 7472 7565 2920 2020  erbose=>true)   
│ │ │ │ -0003c9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c9f0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0003ca00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ca10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ca20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c810: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73  |.|internalChoos
│ │ │ │ +0003c820: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67  eMinor: Choosing
│ │ │ │ +0003c830: 2047 5265 764c 6578 536d 616c 6c65 7374   GRevLexSmallest
│ │ │ │ +0003c840: 5465 726d 2020 2020 2020 2020 2020 2020  Term            
│ │ │ │ +0003c850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c860: 7c0a 7c72 6567 756c 6172 496e 436f 6469  |.|regularInCodi
│ │ │ │ +0003c870: 6d65 6e73 696f 6e3a 2020 4c6f 6f70 2073  mension:  Loop s
│ │ │ │ +0003c880: 7465 702c 2061 626f 7574 2074 6f20 636f  tep, about to co
│ │ │ │ +0003c890: 6d70 7574 6520 6469 6d65 6e73 696f 6e2e  mpute dimension.
│ │ │ │ +0003c8a0: 2020 5375 626d 6174 7269 6365 7320 636f    Submatrices co
│ │ │ │ +0003c8b0: 7c0a 7c72 6567 756c 6172 496e 436f 6469  |.|regularInCodi
│ │ │ │ +0003c8c0: 6d65 6e73 696f 6e3a 2020 6973 436f 6469  mension:  isCodi
│ │ │ │ +0003c8d0: 6d41 744c 6561 7374 2066 6169 6c65 642c  mAtLeast failed,
│ │ │ │ +0003c8e0: 2063 6f6d 7075 7469 6e67 2063 6f64 696d   computing codim
│ │ │ │ +0003c8f0: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ +0003c900: 7c0a 7c72 6567 756c 6172 496e 436f 6469  |.|regularInCodi
│ │ │ │ +0003c910: 6d65 6e73 696f 6e3a 2020 7061 7274 6961  mension:  partia
│ │ │ │ +0003c920: 6c20 7369 6e67 756c 6172 206c 6f63 7573  l singular locus
│ │ │ │ +0003c930: 2064 696d 656e 7369 6f6e 2063 6f6d 7075   dimension compu
│ │ │ │ +0003c940: 7465 642c 203d 2033 2020 2020 2020 2020  ted, = 3        
│ │ │ │ +0003c950: 7c0a 7c72 6567 756c 6172 496e 436f 6469  |.|regularInCodi
│ │ │ │ +0003c960: 6d65 6e73 696f 6e3a 2020 4c6f 6f70 2063  mension:  Loop c
│ │ │ │ +0003c970: 6f6d 706c 6574 6564 2c20 7375 626d 6174  ompleted, submat
│ │ │ │ +0003c980: 7269 6365 7320 636f 6e73 6964 6572 6564  rices considered
│ │ │ │ +0003c990: 203d 2032 352c 2061 6e64 2063 6f6d 7075   = 25, and compu
│ │ │ │ +0003c9a0: 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.|-------------
│ │ │ │ +0003c9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003c9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003c9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003c9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003c9f0: 7c0a 7c74 2d3e 742f 352c 204d 696e 4d69  |.|t->t/5, MinMi
│ │ │ │ +0003ca00: 6e6f 7273 4675 6e63 7469 6f6e 203d 3e20  norsFunction => 
│ │ │ │ +0003ca10: 742d 3e32 2c20 5665 7262 6f73 653d 3e74  t->2, Verbose=>t
│ │ │ │ +0003ca20: 7275 6529 2020 2020 2020 2020 2020 2020  rue)            
│ │ │ │  0003ca30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ca40: 2020 2020 2020 2020 207c 0a7c 6e6f 7273           |.|nors
│ │ │ │ -0003ca50: 2c20 7765 2077 696c 6c20 636f 6d70 7574  , we will comput
│ │ │ │ -0003ca60: 6520 7570 2074 6f20 3235 206f 6620 7468  e up to 25 of th
│ │ │ │ -0003ca70: 656d 2e20 2020 2020 2020 2020 2020 2020  em.             
│ │ │ │ +0003ca40: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0003ca50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ca60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ca70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ca80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ca90: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0003caa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ca90: 7c0a 7c6e 6f72 732c 2077 6520 7769 6c6c  |.|nors, we will
│ │ │ │ +0003caa0: 2063 6f6d 7075 7465 2075 7020 746f 2032   compute up to 2
│ │ │ │ +0003cab0: 3520 6f66 2074 6865 6d2e 2020 2020 2020  5 of them.      
│ │ │ │  0003cac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cae0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003cae0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0003caf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cb30: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003cb30: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0003cb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cb80: 2020 2020 2020 2020 207c 0a7c 6e73 6964           |.|nsid
│ │ │ │ -0003cb90: 6572 6564 3a20 322c 2061 6e64 2063 6f6d  ered: 2, and com
│ │ │ │ -0003cba0: 7075 7465 6420 3d20 3220 2020 2020 2020  puted = 2       
│ │ │ │ +0003cb80: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0003cb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003cba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cbd0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0003cbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003cbd0: 7c0a 7c6e 7369 6465 7265 643a 2032 2c20  |.|nsidered: 2, 
│ │ │ │ +0003cbe0: 616e 6420 636f 6d70 7574 6564 203d 2032  and computed = 2
│ │ │ │  0003cbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cc20: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003cc20: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0003cc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cc70: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003cc70: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0003cc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ccb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ccc0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003ccc0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0003ccd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ccf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cd10: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003cd10: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0003cd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cd60: 2020 2020 2020 2020 207c 0a7c 6e73 6964           |.|nsid
│ │ │ │ -0003cd70: 6572 6564 3a20 352c 2061 6e64 2063 6f6d  ered: 5, and com
│ │ │ │ -0003cd80: 7075 7465 6420 3d20 3520 2020 2020 2020  puted = 5       
│ │ │ │ +0003cd60: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0003cd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003cd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cdb0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0003cdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003cdb0: 7c0a 7c6e 7369 6465 7265 643a 2035 2c20  |.|nsidered: 5, 
│ │ │ │ +0003cdc0: 616e 6420 636f 6d70 7574 6564 203d 2035  and computed = 5
│ │ │ │  0003cdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ce00: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003ce00: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0003ce10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ce20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ce30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ce40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ce50: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003ce50: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0003ce60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ce70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ce80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ce90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cea0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003cea0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0003ceb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ced0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cef0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003cef0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0003cf00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cf10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cf20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cf30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cf40: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003cf40: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0003cf50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cf60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cf70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cf80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cf90: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003cf90: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0003cfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cfc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cfe0: 2020 2020 2020 2020 207c 0a7c 6e73 6964           |.|nsid
│ │ │ │ -0003cff0: 6572 6564 3a20 3130 2c20 616e 6420 636f  ered: 10, and co
│ │ │ │ -0003d000: 6d70 7574 6564 203d 2031 3020 2020 2020  mputed = 10     
│ │ │ │ +0003cfe0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0003cff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d030: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0003d040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d030: 7c0a 7c6e 7369 6465 7265 643a 2031 302c  |.|nsidered: 10,
│ │ │ │ +0003d040: 2061 6e64 2063 6f6d 7075 7465 6420 3d20   and computed = 
│ │ │ │ +0003d050: 3130 2020 2020 2020 2020 2020 2020 2020  10              
│ │ │ │  0003d060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d080: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003d080: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0003d090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d0d0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003d0d0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0003d0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d120: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003d120: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0003d130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d170: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003d170: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0003d180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d1c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003d1c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0003d1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d210: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003d210: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0003d220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d260: 2020 2020 2020 2020 207c 0a7c 6e73 6964           |.|nsid
│ │ │ │ -0003d270: 6572 6564 3a20 3135 2c20 616e 6420 636f  ered: 15, and co
│ │ │ │ -0003d280: 6d70 7574 6564 203d 2031 3520 2020 2020  mputed = 15     
│ │ │ │ +0003d260: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0003d270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d2b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0003d2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d2b0: 7c0a 7c6e 7369 6465 7265 643a 2031 352c  |.|nsidered: 15,
│ │ │ │ +0003d2c0: 2061 6e64 2063 6f6d 7075 7465 6420 3d20   and computed = 
│ │ │ │ +0003d2d0: 3135 2020 2020 2020 2020 2020 2020 2020  15              
│ │ │ │  0003d2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d300: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003d300: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0003d310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d350: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003d350: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0003d360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d3a0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003d3a0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0003d3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d3f0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003d3f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0003d400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d440: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003d440: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0003d450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d490: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003d490: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0003d4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d4e0: 2020 2020 2020 2020 207c 0a7c 6e73 6964           |.|nsid
│ │ │ │ -0003d4f0: 6572 6564 3a20 3230 2c20 616e 6420 636f  ered: 20, and co
│ │ │ │ -0003d500: 6d70 7574 6564 203d 2031 3820 2020 2020  mputed = 18     
│ │ │ │ +0003d4e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0003d4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d530: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0003d540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d530: 7c0a 7c6e 7369 6465 7265 643a 2032 302c  |.|nsidered: 20,
│ │ │ │ +0003d540: 2061 6e64 2063 6f6d 7075 7465 6420 3d20   and computed = 
│ │ │ │ +0003d550: 3138 2020 2020 2020 2020 2020 2020 2020  18              
│ │ │ │  0003d560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d580: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003d580: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0003d590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d5d0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003d5d0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0003d5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d620: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003d620: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0003d630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d670: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003d670: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0003d680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d6c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003d6c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0003d6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d710: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003d710: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0003d720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d760: 2020 2020 2020 2020 207c 0a7c 6e73 6964           |.|nsid
│ │ │ │ -0003d770: 6572 6564 3a20 3235 2c20 616e 6420 636f  ered: 25, and co
│ │ │ │ -0003d780: 6d70 7574 6564 203d 2032 3320 2020 2020  mputed = 23     
│ │ │ │ +0003d760: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0003d770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d7b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0003d7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d7b0: 7c0a 7c6e 7369 6465 7265 643a 2032 352c  |.|nsidered: 25,
│ │ │ │ +0003d7c0: 2061 6e64 2063 6f6d 7075 7465 6420 3d20   and computed = 
│ │ │ │ +0003d7d0: 3233 2020 2020 2020 2020 2020 2020 2020  23              
│ │ │ │  0003d7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d800: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003d800: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0003d810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d850: 2020 2020 2020 2020 207c 0a7c 7465 6420           |.|ted 
│ │ │ │ -0003d860: 3d20 3233 2e20 2073 696e 6775 6c61 7220  = 23.  singular 
│ │ │ │ -0003d870: 6c6f 6375 7320 6469 6d65 6e73 696f 6e20  locus dimension 
│ │ │ │ -0003d880: 6170 7065 6172 7320 746f 2062 6520 3d20  appears to be = 
│ │ │ │ -0003d890: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ -0003d8a0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -0003d8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003d8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003d8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003d8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003d8f0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 6973 436f  ---------+..isCo
│ │ │ │ -0003d900: 6469 6d41 744c 6561 7374 2061 6e64 2064  dimAtLeast and d
│ │ │ │ -0003d910: 696d 2e20 2057 6520 7365 6520 7468 6520  im.  We see the 
│ │ │ │ -0003d920: 6c69 6e65 7320 6162 6f75 7420 7468 6520  lines about the 
│ │ │ │ -0003d930: 6060 6973 436f 6469 6d41 744c 6561 7374  ``isCodimAtLeast
│ │ │ │ -0003d940: 2066 6169 6c65 6427 272e 0a54 6869 7320   failed''..This 
│ │ │ │ -0003d950: 6d65 616e 7320 7468 6174 2069 7343 6f64  means that isCod
│ │ │ │ -0003d960: 696d 4174 4c65 6173 7420 7761 7320 6e6f  imAtLeast was no
│ │ │ │ -0003d970: 7420 656e 6f75 6768 206f 6e20 6974 7320  t enough on its 
│ │ │ │ -0003d980: 6f77 6e20 746f 2076 6572 6966 7920 7468  own to verify th
│ │ │ │ -0003d990: 6174 206f 7572 0a72 696e 6720 6973 2072  at our.ring is r
│ │ │ │ -0003d9a0: 6567 756c 6172 2069 6e20 636f 6469 6d65  egular in codime
│ │ │ │ -0003d9b0: 6e73 696f 6e20 312e 2020 4166 7465 7220  nsion 1.  After 
│ │ │ │ -0003d9c0: 7468 6973 2c20 6060 7061 7274 6961 6c20  this, ``partial 
│ │ │ │ -0003d9d0: 7369 6e67 756c 6172 206c 6f63 7573 0a64  singular locus.d
│ │ │ │ -0003d9e0: 696d 656e 7369 6f6e 2063 6f6d 7075 7465  imension compute
│ │ │ │ -0003d9f0: 6427 2720 696e 6469 6361 7465 7320 7765  d'' indicates we
│ │ │ │ -0003da00: 2064 6964 2061 2063 6f6d 706c 6574 6520   did a complete 
│ │ │ │ -0003da10: 6469 6d65 6e73 696f 6e20 636f 6d70 7574  dimension comput
│ │ │ │ -0003da20: 6174 696f 6e20 6f66 2074 6865 0a70 6172  ation of the.par
│ │ │ │ -0003da30: 7469 616c 2069 6465 616c 2064 6566 696e  tial ideal defin
│ │ │ │ -0003da40: 696e 6720 7468 6520 7369 6e67 756c 6172  ing the singular
│ │ │ │ -0003da50: 206c 6f63 7573 2e20 2048 6f77 2069 7343   locus.  How isC
│ │ │ │ -0003da60: 6f64 696d 4174 4c65 6173 7420 6973 2063  odimAtLeast is c
│ │ │ │ -0003da70: 616c 6c65 6420 6361 6e20 6265 0a63 6f6e  alled can be.con
│ │ │ │ -0003da80: 7472 6f6c 6c65 6420 7669 6120 7468 6520  trolled via the 
│ │ │ │ -0003da90: 6f70 7469 6f6e 7320 5350 6169 7273 4675  options SPairsFu
│ │ │ │ -0003daa0: 6e63 7469 6f6e 2061 6e64 2050 6169 724c  nction and PairL
│ │ │ │ -0003dab0: 696d 6974 2c20 7768 6963 6820 6172 6520  imit, which are 
│ │ │ │ -0003dac0: 7369 6d70 6c79 0a70 6173 7365 6420 746f  simply.passed to
│ │ │ │ -0003dad0: 202a 6e6f 7465 2069 7343 6f64 696d 4174   *note isCodimAt
│ │ │ │ -0003dae0: 4c65 6173 743a 2069 7343 6f64 696d 4174  Least: isCodimAt
│ │ │ │ -0003daf0: 4c65 6173 742c 2e20 2059 6f75 2063 616e  Least,.  You can
│ │ │ │ -0003db00: 2066 6f72 6365 2074 6865 2066 756e 6374   force the funct
│ │ │ │ -0003db10: 696f 6e20 746f 0a6f 6e6c 7920 7573 6520  ion to.only use 
│ │ │ │ -0003db20: 6973 436f 6469 6d41 744c 6561 7374 2061  isCodimAtLeast a
│ │ │ │ -0003db30: 6e64 206e 6f74 2063 616c 6c20 6469 6d65  nd not call dime
│ │ │ │ -0003db40: 6e73 696f 6e20 6279 2073 6574 7469 6e67  nsion by setting
│ │ │ │ -0003db50: 2055 7365 4f6e 6c79 4661 7374 436f 6469   UseOnlyFastCodi
│ │ │ │ -0003db60: 6d20 3d3e 0a74 7275 652e 0a0a 2b2d 2d2d  m =>.true...+---
│ │ │ │ -0003db70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003db80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003db90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003dba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003dbb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131  ----------+.|i11
│ │ │ │ -0003dbc0: 203a 2074 696d 6520 7265 6775 6c61 7249   : time regularI
│ │ │ │ -0003dbd0: 6e43 6f64 696d 656e 7369 6f6e 2831 2c20  nCodimension(1, 
│ │ │ │ -0003dbe0: 532f 4a2c 204d 6178 4d69 6e6f 7273 3d3e  S/J, MaxMinors=>
│ │ │ │ -0003dbf0: 3235 2c20 5573 654f 6e6c 7946 6173 7443  25, UseOnlyFastC
│ │ │ │ -0003dc00: 6f64 696d 203d 3e20 7472 7c0a 7c20 2d2d  odim => tr|.| --
│ │ │ │ -0003dc10: 2075 7365 6420 302e 3436 3734 3534 7320   used 0.467454s 
│ │ │ │ -0003dc20: 2863 7075 293b 2030 2e33 3233 3438 3373  (cpu); 0.323483s
│ │ │ │ -0003dc30: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │ -0003dc40: 6329 2020 2020 2020 2020 2020 2020 2020  c)              
│ │ │ │ -0003dc50: 2020 2020 2020 2020 2020 7c0a 7c72 6567            |.|reg
│ │ │ │ -0003dc60: 756c 6172 496e 436f 6469 6d65 6e73 696f  ularInCodimensio
│ │ │ │ -0003dc70: 6e3a 2072 696e 6720 6469 6d65 6e73 696f  n: ring dimensio
│ │ │ │ -0003dc80: 6e20 3d34 2c20 7468 6572 6520 6172 6520  n =4, there are 
│ │ │ │ -0003dc90: 3134 3635 3132 3820 706f 7373 6962 6c65  1465128 possible
│ │ │ │ -0003dca0: 2035 2062 7920 3520 6d69 7c0a 7c72 6567   5 by 5 mi|.|reg
│ │ │ │ -0003dcb0: 756c 6172 496e 436f 6469 6d65 6e73 696f  ularInCodimensio
│ │ │ │ -0003dcc0: 6e3a 2041 626f 7574 2074 6f20 656e 7465  n: About to ente
│ │ │ │ -0003dcd0: 7220 6c6f 6f70 2020 2020 2020 2020 2020  r loop          
│ │ │ │ -0003dce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dcf0: 2020 2020 2020 2020 2020 7c0a 7c69 6e74            |.|int
│ │ │ │ -0003dd00: 6572 6e61 6c43 686f 6f73 654d 696e 6f72  ernalChooseMinor
│ │ │ │ -0003dd10: 3a20 4368 6f6f 7369 6e67 2047 5265 764c  : Choosing GRevL
│ │ │ │ -0003dd20: 6578 536d 616c 6c65 7374 2020 2020 2020  exSmallest      
│ │ │ │ +0003d850: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0003d860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d8a0: 7c0a 7c74 6564 203d 2032 332e 2020 7369  |.|ted = 23.  si
│ │ │ │ +0003d8b0: 6e67 756c 6172 206c 6f63 7573 2064 696d  ngular locus dim
│ │ │ │ +0003d8c0: 656e 7369 6f6e 2061 7070 6561 7273 2074  ension appears t
│ │ │ │ +0003d8d0: 6f20 6265 203d 2033 2020 2020 2020 2020  o be = 3        
│ │ │ │ +0003d8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d8f0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0003d900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003d910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003d920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003d930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0003dbe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003dbf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0003dc80: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │ +0003dc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003dca0: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64   |.|regularInCod
│ │ │ │ +0003dcb0: 696d 656e 7369 6f6e 3a20 7269 6e67 2064  imension: ring d
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│ │ │ │ +0003dd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003dd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dd40: 2020 2020 2020 2020 2020 7c0a 7c69 6e74            |.|int
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│ │ │ │ -0003dd60: 3a20 4368 6f6f 7369 6e67 204c 6578 536d  : Choosing LexSm
│ │ │ │ -0003dd70: 616c 6c65 7374 2020 2020 2020 2020 2020  allest          
│ │ │ │ +0003dd40: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
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│ │ │ │ +0003dd60: 6720 4752 6576 4c65 7853 6d61 6c6c 6573  g GRevLexSmalles
│ │ │ │ +0003dd70: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │  0003dd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dd90: 2020 2020 2020 2020 2020 7c0a 7c69 6e74            |.|int
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│ │ │ │ -0003ddb0: 3a20 4368 6f6f 7369 6e67 2052 616e 646f  : Choosing Rando
│ │ │ │ -0003ddc0: 6d4e 6f6e 5a65 726f 2020 2020 2020 2020  mNonZero        
│ │ │ │ +0003dd90: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
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│ │ │ │ +0003ddc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ddd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dde0: 2020 2020 2020 2020 2020 7c0a 7c69 6e74            |.|int
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│ │ │ │ -0003de00: 3a20 4368 6f6f 7369 6e67 2052 616e 646f  : Choosing Rando
│ │ │ │ -0003de10: 6d4e 6f6e 5a65 726f 2020 2020 2020 2020  mNonZero        
│ │ │ │ +0003dde0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ +0003ddf0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ +0003de00: 6720 5261 6e64 6f6d 4e6f 6e5a 6572 6f20  g RandomNonZero 
│ │ │ │ +0003de10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003de20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003de30: 2020 2020 2020 2020 2020 7c0a 7c69 6e74            |.|int
│ │ │ │ -0003de40: 6572 6e61 6c43 686f 6f73 654d 696e 6f72  ernalChooseMinor
│ │ │ │ -0003de50: 3a20 4368 6f6f 7369 6e67 2047 5265 764c  : Choosing GRevL
│ │ │ │ -0003de60: 6578 536d 616c 6c65 7374 5465 726d 2020  exSmallestTerm  
│ │ │ │ +0003de30: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ +0003de40: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ +0003de50: 6720 5261 6e64 6f6d 4e6f 6e5a 6572 6f20  g RandomNonZero 
│ │ │ │ +0003de60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003de70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003de80: 2020 2020 2020 2020 2020 7c0a 7c69 6e74            |.|int
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│ │ │ │ -0003dea0: 3a20 4368 6f6f 7369 6e67 204c 6578 536d  : Choosing LexSm
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│ │ │ │ +0003de80: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
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│ │ │ │ +0003deb0: 7454 6572 6d20 2020 2020 2020 2020 2020  tTerm           
│ │ │ │  0003dec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ded0: 2020 2020 2020 2020 2020 7c0a 7c69 6e74            |.|int
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│ │ │ │ -0003def0: 3a20 4368 6f6f 7369 6e67 2052 616e 646f  : Choosing Rando
│ │ │ │ -0003df00: 6d20 2020 2020 2020 2020 2020 2020 2020  m               
│ │ │ │ +0003ded0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
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│ │ │ │ +0003df00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003df10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003df20: 2020 2020 2020 2020 2020 7c0a 7c72 6567            |.|reg
│ │ │ │ -0003df30: 756c 6172 496e 436f 6469 6d65 6e73 696f  ularInCodimensio
│ │ │ │ -0003df40: 6e3a 2020 4c6f 6f70 2073 7465 702c 2061  n:  Loop step, a
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│ │ │ │ -0003df60: 6469 6d65 6e73 696f 6e2e 2020 5375 626d  dimension.  Subm
│ │ │ │ -0003df70: 6174 7269 6365 7320 636f 7c0a 7c72 6567  atrices co|.|reg
│ │ │ │ -0003df80: 756c 6172 496e 436f 6469 6d65 6e73 696f  ularInCodimensio
│ │ │ │ -0003df90: 6e3a 2020 7061 7274 6961 6c20 7369 6e67  n:  partial sing
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│ │ │ │ -0003dfb0: 7369 6f6e 2063 6f6d 7075 7465 642c 203d  sion computed, =
│ │ │ │ -0003dfc0: 2034 2020 2020 2020 2020 7c0a 7c69 6e74   4        |.|int
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│ │ │ │ -0003dfe0: 3a20 4368 6f6f 7369 6e67 204c 6578 536d  : Choosing LexSm
│ │ │ │ -0003dff0: 616c 6c65 7374 2020 2020 2020 2020 2020  allest          
│ │ │ │ -0003e000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e010: 2020 2020 2020 2020 2020 7c0a 7c69 6e74            |.|int
│ │ │ │ -0003e020: 6572 6e61 6c43 686f 6f73 654d 696e 6f72  ernalChooseMinor
│ │ │ │ -0003e030: 3a20 4368 6f6f 7369 6e67 2047 5265 764c  : Choosing GRevL
│ │ │ │ -0003e040: 6578 536d 616c 6c65 7374 5465 726d 2020  exSmallestTerm  
│ │ │ │ +0003df20: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ +0003df30: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ +0003df40: 6720 5261 6e64 6f6d 2020 2020 2020 2020  g Random        
│ │ │ │ +0003df50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003df60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003df70: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64   |.|regularInCod
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│ │ │ │ +0003dfa0: 6f6d 7075 7465 2064 696d 656e 7369 6f6e  ompute dimension
│ │ │ │ +0003dfb0: 2e20 2053 7562 6d61 7472 6963 6573 2063  .  Submatrices c
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│ │ │ │ +0003dfd0: 696d 656e 7369 6f6e 3a20 2070 6172 7469  imension:  parti
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│ │ │ │ +0003e000: 7574 6564 2c20 3d20 3420 2020 2020 2020  uted, = 4       
│ │ │ │ +0003e010: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
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│ │ │ │ +0003e030: 6720 4c65 7853 6d61 6c6c 6573 7420 2020  g LexSmallest   
│ │ │ │ +0003e040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e060: 2020 2020 2020 2020 2020 7c0a 7c69 6e74            |.|int
│ │ │ │ -0003e070: 6572 6e61 6c43 686f 6f73 654d 696e 6f72  ernalChooseMinor
│ │ │ │ -0003e080: 3a20 4368 6f6f 7369 6e67 2047 5265 764c  : Choosing GRevL
│ │ │ │ -0003e090: 6578 536d 616c 6c65 7374 2020 2020 2020  exSmallest      
│ │ │ │ +0003e060: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ +0003e070: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ +0003e080: 6720 4752 6576 4c65 7853 6d61 6c6c 6573  g GRevLexSmalles
│ │ │ │ +0003e090: 7454 6572 6d20 2020 2020 2020 2020 2020  tTerm           
│ │ │ │  0003e0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e0b0: 2020 2020 2020 2020 2020 7c0a 7c69 6e74            |.|int
│ │ │ │ -0003e0c0: 6572 6e61 6c43 686f 6f73 654d 696e 6f72  ernalChooseMinor
│ │ │ │ -0003e0d0: 3a20 4368 6f6f 7369 6e67 2047 5265 764c  : Choosing GRevL
│ │ │ │ -0003e0e0: 6578 536d 616c 6c65 7374 2020 2020 2020  exSmallest      
│ │ │ │ +0003e0b0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
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│ │ │ │ +0003e0d0: 6720 4752 6576 4c65 7853 6d61 6c6c 6573  g GRevLexSmalles
│ │ │ │ +0003e0e0: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │  0003e0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e100: 2020 2020 2020 2020 2020 7c0a 7c72 6567            |.|reg
│ │ │ │ -0003e110: 756c 6172 496e 436f 6469 6d65 6e73 696f  ularInCodimensio
│ │ │ │ -0003e120: 6e3a 2020 4c6f 6f70 2073 7465 702c 2061  n:  Loop step, a
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│ │ │ │ -0003e140: 6469 6d65 6e73 696f 6e2e 2020 5375 626d  dimension.  Subm
│ │ │ │ -0003e150: 6174 7269 6365 7320 636f 7c0a 7c72 6567  atrices co|.|reg
│ │ │ │ -0003e160: 756c 6172 496e 436f 6469 6d65 6e73 696f  ularInCodimensio
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│ │ │ │ -0003e180: 756c 6172 206c 6f63 7573 2064 696d 656e  ular locus dimen
│ │ │ │ -0003e190: 7369 6f6e 2063 6f6d 7075 7465 642c 203d  sion computed, =
│ │ │ │ -0003e1a0: 2034 2020 2020 2020 2020 7c0a 7c69 6e74   4        |.|int
│ │ │ │ -0003e1b0: 6572 6e61 6c43 686f 6f73 654d 696e 6f72  ernalChooseMinor
│ │ │ │ -0003e1c0: 3a20 4368 6f6f 7369 6e67 204c 6578 536d  : Choosing LexSm
│ │ │ │ -0003e1d0: 616c 6c65 7374 5465 726d 2020 2020 2020  allestTerm      
│ │ │ │ -0003e1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003e210: 3a20 4368 6f6f 7369 6e67 204c 6578 536d  : Choosing LexSm
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│ │ │ │ +0003e100: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
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│ │ │ │ +0003e130: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │ +0003e140: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0003e180: 6f6d 7075 7465 2064 696d 656e 7369 6f6e  ompute dimension
│ │ │ │ +0003e190: 2e20 2053 7562 6d61 7472 6963 6573 2063  .  Submatrices c
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│ │ │ │ +0003e1c0: 616c 2073 696e 6775 6c61 7220 6c6f 6375  al singular locu
│ │ │ │ +0003e1d0: 7320 6469 6d65 6e73 696f 6e20 636f 6d70  s dimension comp
│ │ │ │ +0003e1e0: 7574 6564 2c20 3d20 3420 2020 2020 2020  uted, = 4       
│ │ │ │ +0003e1f0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ +0003e200: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
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│ │ │ │ +0003e220: 6d20 2020 2020 2020 2020 2020 2020 2020  m               
│ │ │ │  0003e230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e240: 2020 2020 2020 2020 2020 7c0a 7c69 6e74            |.|int
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│ │ │ │ -0003e260: 3a20 4368 6f6f 7369 6e67 2047 5265 764c  : Choosing GRevL
│ │ │ │ -0003e270: 6578 536d 616c 6c65 7374 5465 726d 2020  exSmallestTerm  
│ │ │ │ +0003e240: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ +0003e250: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ +0003e260: 6720 4c65 7853 6d61 6c6c 6573 7420 2020  g LexSmallest   
│ │ │ │ +0003e270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e290: 2020 2020 2020 2020 2020 7c0a 7c69 6e74            |.|int
│ │ │ │ -0003e2a0: 6572 6e61 6c43 686f 6f73 654d 696e 6f72  ernalChooseMinor
│ │ │ │ -0003e2b0: 3a20 4368 6f6f 7369 6e67 204c 6578 536d  : Choosing LexSm
│ │ │ │ -0003e2c0: 616c 6c65 7374 5465 726d 2020 2020 2020  allestTerm      
│ │ │ │ +0003e290: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ +0003e2a0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ +0003e2b0: 6720 4752 6576 4c65 7853 6d61 6c6c 6573  g GRevLexSmalles
│ │ │ │ +0003e2c0: 7454 6572 6d20 2020 2020 2020 2020 2020  tTerm           
│ │ │ │  0003e2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e2e0: 2020 2020 2020 2020 2020 7c0a 7c72 6567            |.|reg
│ │ │ │ -0003e2f0: 756c 6172 496e 436f 6469 6d65 6e73 696f  ularInCodimensio
│ │ │ │ -0003e300: 6e3a 2020 4c6f 6f70 2073 7465 702c 2061  n:  Loop step, a
│ │ │ │ -0003e310: 626f 7574 2074 6f20 636f 6d70 7574 6520  bout to compute 
│ │ │ │ -0003e320: 6469 6d65 6e73 696f 6e2e 2020 5375 626d  dimension.  Subm
│ │ │ │ -0003e330: 6174 7269 6365 7320 636f 7c0a 7c72 6567  atrices co|.|reg
│ │ │ │ -0003e340: 756c 6172 496e 436f 6469 6d65 6e73 696f  ularInCodimensio
│ │ │ │ -0003e350: 6e3a 2020 7369 6e67 756c 6172 4c6f 6375  n:  singularLocu
│ │ │ │ -0003e360: 7320 6469 6d65 6e73 696f 6e20 7665 7269  s dimension veri
│ │ │ │ -0003e370: 6669 6564 2062 7920 6973 436f 6469 6d41  fied by isCodimA
│ │ │ │ -0003e380: 744c 6561 7374 2020 2020 7c0a 7c72 6567  tLeast    |.|reg
│ │ │ │ -0003e390: 756c 6172 496e 436f 6469 6d65 6e73 696f  ularInCodimensio
│ │ │ │ -0003e3a0: 6e3a 2020 7061 7274 6961 6c20 7369 6e67  n:  partial sing
│ │ │ │ -0003e3b0: 756c 6172 206c 6f63 7573 2064 696d 656e  ular locus dimen
│ │ │ │ -0003e3c0: 7369 6f6e 2063 6f6d 7075 7465 642c 203d  sion computed, =
│ │ │ │ -0003e3d0: 2032 2020 2020 2020 2020 7c0a 7c72 6567   2        |.|reg
│ │ │ │ -0003e3e0: 756c 6172 496e 436f 6469 6d65 6e73 696f  ularInCodimensio
│ │ │ │ -0003e3f0: 6e3a 2020 4c6f 6f70 2063 6f6d 706c 6574  n:  Loop complet
│ │ │ │ -0003e400: 6564 2c20 7375 626d 6174 7269 6365 7320  ed, submatrices 
│ │ │ │ -0003e410: 636f 6e73 6964 6572 6564 203d 2031 352c  considered = 15,
│ │ │ │ -0003e420: 2061 6e64 2063 6f6d 7075 7c0a 7c20 2020   and compu|.|   
│ │ │ │ -0003e430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e470: 2020 2020 2020 2020 2020 7c0a 7c6f 3131            |.|o11
│ │ │ │ -0003e480: 203d 2074 7275 6520 2020 2020 2020 2020   = true         
│ │ │ │ +0003e2e0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ +0003e2f0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ +0003e300: 6720 4c65 7853 6d61 6c6c 6573 7454 6572  g LexSmallestTer
│ │ │ │ +0003e310: 6d20 2020 2020 2020 2020 2020 2020 2020  m               
│ │ │ │ +0003e320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e330: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64   |.|regularInCod
│ │ │ │ +0003e340: 696d 656e 7369 6f6e 3a20 204c 6f6f 7020  imension:  Loop 
│ │ │ │ +0003e350: 7374 6570 2c20 6162 6f75 7420 746f 2063  step, about to c
│ │ │ │ +0003e360: 6f6d 7075 7465 2064 696d 656e 7369 6f6e  ompute dimension
│ │ │ │ +0003e370: 2e20 2053 7562 6d61 7472 6963 6573 2063  .  Submatrices c
│ │ │ │ +0003e380: 6f7c 0a7c 7265 6775 6c61 7249 6e43 6f64  o|.|regularInCod
│ │ │ │ +0003e390: 696d 656e 7369 6f6e 3a20 2073 696e 6775  imension:  singu
│ │ │ │ +0003e3a0: 6c61 724c 6f63 7573 2064 696d 656e 7369  larLocus dimensi
│ │ │ │ +0003e3b0: 6f6e 2076 6572 6966 6965 6420 6279 2069  on verified by i
│ │ │ │ +0003e3c0: 7343 6f64 696d 4174 4c65 6173 7420 2020  sCodimAtLeast   
│ │ │ │ +0003e3d0: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64   |.|regularInCod
│ │ │ │ +0003e3e0: 696d 656e 7369 6f6e 3a20 2070 6172 7469  imension:  parti
│ │ │ │ +0003e3f0: 616c 2073 696e 6775 6c61 7220 6c6f 6375  al singular locu
│ │ │ │ +0003e400: 7320 6469 6d65 6e73 696f 6e20 636f 6d70  s dimension comp
│ │ │ │ +0003e410: 7574 6564 2c20 3d20 3220 2020 2020 2020  uted, = 2       
│ │ │ │ +0003e420: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64   |.|regularInCod
│ │ │ │ +0003e430: 696d 656e 7369 6f6e 3a20 204c 6f6f 7020  imension:  Loop 
│ │ │ │ +0003e440: 636f 6d70 6c65 7465 642c 2073 7562 6d61  completed, subma
│ │ │ │ +0003e450: 7472 6963 6573 2063 6f6e 7369 6465 7265  trices considere
│ │ │ │ +0003e460: 6420 3d20 3135 2c20 616e 6420 636f 6d70  d = 15, and comp
│ │ │ │ +0003e470: 757c 0a7c 2020 2020 2020 2020 2020 2020  u|.|            
│ │ │ │ +0003e480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e4c0: 2020 2020 2020 2020 2020 7c0a 7c2d 2d2d            |.|---
│ │ │ │ -0003e4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e510: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c75 652c  ----------|.|ue,
│ │ │ │ -0003e520: 2056 6572 626f 7365 3d3e 7472 7565 2920   Verbose=>true) 
│ │ │ │ -0003e530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e560: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0003e570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e4c0: 207c 0a7c 6f31 3120 3d20 7472 7565 2020   |.|o11 = true  
│ │ │ │ +0003e4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e510: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │ +0003e520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003e530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003e540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003e550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003e560: 2d7c 0a7c 7565 2c20 5665 7262 6f73 653d  -|.|ue, Verbose=
│ │ │ │ +0003e570: 3e74 7275 6529 2020 2020 2020 2020 2020  >true)          
│ │ │ │  0003e580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e5b0: 2020 2020 2020 2020 2020 7c0a 7c6e 6f72            |.|nor
│ │ │ │ -0003e5c0: 732c 2077 6520 7769 6c6c 2063 6f6d 7075  s, we will compu
│ │ │ │ -0003e5d0: 7465 2075 7020 746f 2032 3520 6f66 2074  te up to 25 of t
│ │ │ │ -0003e5e0: 6865 6d2e 2020 2020 2020 2020 2020 2020  hem.            
│ │ │ │ +0003e5b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0003e5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e600: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0003e610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e600: 207c 0a7c 6e6f 7273 2c20 7765 2077 696c   |.|nors, we wil
│ │ │ │ +0003e610: 6c20 636f 6d70 7574 6520 7570 2074 6f20  l compute up to 
│ │ │ │ +0003e620: 3235 206f 6620 7468 656d 2e20 2020 2020  25 of them.     
│ │ │ │  0003e630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e650: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0003e650: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003e660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e6a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0003e6a0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003e6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e6f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0003e6f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003e700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e740: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0003e740: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003e750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e790: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0003e790: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003e7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e7e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0003e7e0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003e7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e830: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0003e830: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003e840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e880: 2020 2020 2020 2020 2020 7c0a 7c6e 7369            |.|nsi
│ │ │ │ -0003e890: 6465 7265 643a 2037 2c20 616e 6420 636f  dered: 7, and co
│ │ │ │ -0003e8a0: 6d70 7574 6564 203d 2037 2020 2020 2020  mputed = 7      
│ │ │ │ +0003e880: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0003e890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e8d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0003e8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e8d0: 207c 0a7c 6e73 6964 6572 6564 3a20 372c   |.|nsidered: 7,
│ │ │ │ +0003e8e0: 2061 6e64 2063 6f6d 7075 7465 6420 3d20   and computed = 
│ │ │ │ +0003e8f0: 3720 2020 2020 2020 2020 2020 2020 2020  7               
│ │ │ │  0003e900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e920: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0003e920: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003e930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e970: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0003e970: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003e980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e9c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0003e9c0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003e9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ea00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ea10: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0003ea10: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003ea20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ea30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ea40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ea50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ea60: 2020 2020 2020 2020 2020 7c0a 7c6e 7369            |.|nsi
│ │ │ │ -0003ea70: 6465 7265 643a 2031 312c 2061 6e64 2063  dered: 11, and c
│ │ │ │ -0003ea80: 6f6d 7075 7465 6420 3d20 3130 2020 2020  omputed = 10    
│ │ │ │ +0003ea60: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0003ea70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ea80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ea90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003eaa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003eab0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0003eac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ead0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003eab0: 207c 0a7c 6e73 6964 6572 6564 3a20 3131   |.|nsidered: 11
│ │ │ │ +0003eac0: 2c20 616e 6420 636f 6d70 7574 6564 203d  , and computed =
│ │ │ │ +0003ead0: 2031 3020 2020 2020 2020 2020 2020 2020   10             
│ │ │ │  0003eae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003eaf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003eb00: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0003eb00: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003eb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003eb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003eb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003eb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003eb50: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0003eb50: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003eb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003eb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003eb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003eb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003eba0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0003eba0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003ebb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ebc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ebd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ebe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ebf0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0003ebf0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003ec00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ec10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ec20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ec30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ec40: 2020 2020 2020 2020 2020 7c0a 7c6e 7369            |.|nsi
│ │ │ │ -0003ec50: 6465 7265 643a 2031 352c 2061 6e64 2063  dered: 15, and c
│ │ │ │ -0003ec60: 6f6d 7075 7465 6420 3d20 3133 2020 2020  omputed = 13    
│ │ │ │ +0003ec40: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0003ec50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ec60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ec70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ec80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ec90: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0003eca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ecb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ec90: 207c 0a7c 6e73 6964 6572 6564 3a20 3135   |.|nsidered: 15
│ │ │ │ +0003eca0: 2c20 616e 6420 636f 6d70 7574 6564 203d  , and computed =
│ │ │ │ +0003ecb0: 2031 3320 2020 2020 2020 2020 2020 2020   13             
│ │ │ │  0003ecc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ecd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ece0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0003ece0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  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 2020 2020 2020 7c0a 7c74 6564            |.|ted
│ │ │ │ -0003ed40: 203d 2031 332e 2020 7369 6e67 756c 6172   = 13.  singular
│ │ │ │ -0003ed50: 206c 6f63 7573 2064 696d 656e 7369 6f6e   locus dimension
│ │ │ │ -0003ed60: 2061 7070 6561 7273 2074 6f20 6265 203d   appears to be =
│ │ │ │ -0003ed70: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -0003ed80: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ -0003ed90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003eda0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003edb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003edc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003edd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a54 6869  ----------+..Thi
│ │ │ │ -0003ede0: 7320 6361 6e20 6265 2075 7365 6675 6c20  s can be useful 
│ │ │ │ -0003edf0: 6966 2074 6865 2066 756e 6374 696f 6e20  if the function 
│ │ │ │ -0003ee00: 6973 2068 616e 6769 6e67 2077 6865 6e20  is hanging when 
│ │ │ │ -0003ee10: 7472 7969 6e67 2074 6f20 636f 6d70 7574  trying to comput
│ │ │ │ -0003ee20: 6520 7468 650a 6469 6d65 6e73 696f 6e2c  e the.dimension,
│ │ │ │ -0003ee30: 2062 7574 2079 6f75 206d 6179 2077 6973   but you may wis
│ │ │ │ -0003ee40: 6820 696e 6372 6561 7365 2050 6169 724c  h increase PairL
│ │ │ │ -0003ee50: 696d 6974 2e0a 0a53 756d 6d61 7279 2e20  imit...Summary. 
│ │ │ │ -0003ee60: 2049 6620 796f 7520 6578 7065 6374 2074   If you expect t
│ │ │ │ -0003ee70: 6861 7420 6669 6e64 696e 6720 6120 7375  hat finding a su
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│ │ │ │ -0003f3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003f3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0003f520: 646f 6d69 7a65 642c 206d 6f6e 6f6d 6961  domized, monomia
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│ │ │ │ -0003f540: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0003f550: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0003f560: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0003f570: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0003ed30: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0003ed40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ed50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ed60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ed70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ed80: 207c 0a7c 7465 6420 3d20 3133 2e20 2073   |.|ted = 13.  s
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│ │ │ │ +0003edc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003edd0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +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  ----------------
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│ │ │ │ +0003f3b0: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +0003f3c0: 5379 6d62 6f6c 2c2e 0a0a 2d2d 2d2d 2d2d  Symbol,...------
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│ │ │ │ +0003f3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003f3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003f400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0003f480: 2f46 6173 744d 696e 6f72 732e 0a6d 323a  /FastMinors..m2:
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│ │ │ │ +0003f4c0: 6c79 6e6f 6d69 616c 5269 6e67 2c20 4e65  lynomialRing, Ne
│ │ │ │ +0003f4d0: 7874 3a20 5374 7261 7465 6779 4465 6661  xt: StrategyDefa
│ │ │ │ +0003f4e0: 756c 742c 2050 7265 763a 2052 6567 756c  ult, Prev: Regul
│ │ │ │ +0003f4f0: 6172 496e 436f 6469 6d65 6e73 696f 6e54  arInCodimensionT
│ │ │ │ +0003f500: 7574 6f72 6961 6c2c 2055 703a 2054 6f70  utorial, Up: Top
│ │ │ │ +0003f510: 0a0a 7265 6f72 6465 7250 6f6c 796e 6f6d  ..reorderPolynom
│ │ │ │ +0003f520: 6961 6c52 696e 6720 2d2d 2070 726f 6475  ialRing -- produ
│ │ │ │ +0003f530: 6365 7320 616e 2069 736f 6d6f 7270 6869  ces an isomorphi
│ │ │ │ +0003f540: 6320 706f 6c79 6e6f 6d69 616c 2072 696e  c polynomial rin
│ │ │ │ +0003f550: 6720 7769 7468 2061 2064 6966 6665 7265  g with a differe
│ │ │ │ +0003f560: 6e74 2c20 7261 6e64 6f6d 697a 6564 2c20  nt, randomized, 
│ │ │ │ +0003f570: 6d6f 6e6f 6d69 616c 206f 7264 6572 0a2a  monomial order.*
│ │ │ │  0003f580: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0003f590: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0003f5a0: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a  ****..  * Usage:
│ │ │ │ -0003f5b0: 200a 2020 2020 2020 2020 5231 203d 2072   .        R1 = r
│ │ │ │ -0003f5c0: 656f 7264 6572 506f 6c79 6e6f 6d69 616c  eorderPolynomial
│ │ │ │ -0003f5d0: 5269 6e67 286f 7264 6572 5479 7065 2c20  Ring(orderType, 
│ │ │ │ -0003f5e0: 5229 0a20 202a 2049 6e70 7574 733a 0a20  R).  * Inputs:. 
│ │ │ │ -0003f5f0: 2020 2020 202a 2052 2c20 6120 2a6e 6f74       * R, a *not
│ │ │ │ -0003f600: 6520 7269 6e67 3a20 284d 6163 6175 6c61  e ring: (Macaula
│ │ │ │ -0003f610: 7932 446f 6329 5269 6e67 2c2c 2061 2070  y2Doc)Ring,, a p
│ │ │ │ -0003f620: 6f6c 796e 6f6d 6961 6c20 7269 6e67 0a20  olynomial ring. 
│ │ │ │ -0003f630: 2020 2020 202a 206f 7264 6572 5479 7065       * orderType
│ │ │ │ -0003f640: 2c20 6120 2a6e 6f74 6520 7379 6d62 6f6c  , a *note symbol
│ │ │ │ -0003f650: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ -0003f660: 5379 6d62 6f6c 2c2c 2061 2076 616c 6964  Symbol,, a valid
│ │ │ │ -0003f670: 206d 6f6e 6f6d 6961 6c0a 2020 2020 2020   monomial.      
│ │ │ │ -0003f680: 2020 6f72 6465 722c 2073 7563 6820 6173    order, such as
│ │ │ │ -0003f690: 2047 5265 764c 6578 0a20 202a 204f 7574   GRevLex.  * Out
│ │ │ │ -0003f6a0: 7075 7473 3a0a 2020 2020 2020 2a20 532c  puts:.      * S,
│ │ │ │ -0003f6b0: 2061 202a 6e6f 7465 2072 696e 673a 2028   a *note ring: (
│ │ │ │ -0003f6c0: 4d61 6361 756c 6179 3244 6f63 2952 696e  Macaulay2Doc)Rin
│ │ │ │ -0003f6d0: 672c 2c20 6120 706f 6c79 6e6f 6d69 616c  g,, a polynomial
│ │ │ │ -0003f6e0: 2072 696e 6720 7769 7468 2061 206e 6577   ring with a new
│ │ │ │ -0003f6f0: 0a20 2020 2020 2020 2072 616e 646f 6d20  .        random 
│ │ │ │ -0003f700: 6d6f 6e6f 6d69 616c 206f 7264 6572 0a0a  monomial order..
│ │ │ │ -0003f710: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
│ │ │ │ -0003f720: 3d3d 3d3d 3d3d 3d0a 0a54 6869 7320 6675  =======..This fu
│ │ │ │ -0003f730: 6e63 7469 6f6e 2074 616b 6573 2061 2070  nction takes a p
│ │ │ │ -0003f740: 6f6c 796e 6f6d 6961 6c20 7269 6e67 2061  olynomial ring a
│ │ │ │ -0003f750: 6e64 2070 726f 6475 6365 7320 6120 6e65  nd produces a ne
│ │ │ │ -0003f760: 7720 706f 6c79 6e6f 6d69 616c 2072 696e  w polynomial rin
│ │ │ │ -0003f770: 6720 7769 7468 0a4d 6f6e 6f6d 6961 6c4f  g with.MonomialO
│ │ │ │ -0003f780: 7264 6572 206f 6620 7479 7065 206f 7264  rder of type ord
│ │ │ │ -0003f790: 6572 5479 7065 2e20 5468 6520 6f72 6465  erType. The orde
│ │ │ │ -0003f7a0: 7220 6f66 2074 6865 2076 6172 6961 626c  r of the variabl
│ │ │ │ -0003f7b0: 6573 2069 7320 7261 6e64 6f6d 697a 6564  es is randomized
│ │ │ │ -0003f7c0: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...+------------
│ │ │ │ -0003f7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003f7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0003f7f0: 7c69 3120 3a20 5220 3d20 5151 5b78 2c79  |i1 : R = QQ[x,y
│ │ │ │ -0003f800: 2c7a 2c77 5d3b 2020 2020 2020 2020 2020  ,z,w];          
│ │ │ │ -0003f810: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0003f5a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0003f5b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0003f5c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0003f5d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0003f5e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a  ***********..  *
│ │ │ │ +0003f5f0: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ +0003f600: 2052 3120 3d20 7265 6f72 6465 7250 6f6c   R1 = reorderPol
│ │ │ │ +0003f610: 796e 6f6d 6961 6c52 696e 6728 6f72 6465  ynomialRing(orde
│ │ │ │ +0003f620: 7254 7970 652c 2052 290a 2020 2a20 496e  rType, R).  * In
│ │ │ │ +0003f630: 7075 7473 3a0a 2020 2020 2020 2a20 522c  puts:.      * R,
│ │ │ │ +0003f640: 2061 202a 6e6f 7465 2072 696e 673a 2028   a *note ring: (
│ │ │ │ +0003f650: 4d61 6361 756c 6179 3244 6f63 2952 696e  Macaulay2Doc)Rin
│ │ │ │ +0003f660: 672c 2c20 6120 706f 6c79 6e6f 6d69 616c  g,, a polynomial
│ │ │ │ +0003f670: 2072 696e 670a 2020 2020 2020 2a20 6f72   ring.      * or
│ │ │ │ +0003f680: 6465 7254 7970 652c 2061 202a 6e6f 7465  derType, a *note
│ │ │ │ +0003f690: 2073 796d 626f 6c3a 2028 4d61 6361 756c   symbol: (Macaul
│ │ │ │ +0003f6a0: 6179 3244 6f63 2953 796d 626f 6c2c 2c20  ay2Doc)Symbol,, 
│ │ │ │ +0003f6b0: 6120 7661 6c69 6420 6d6f 6e6f 6d69 616c  a valid monomial
│ │ │ │ +0003f6c0: 0a20 2020 2020 2020 206f 7264 6572 2c20  .        order, 
│ │ │ │ +0003f6d0: 7375 6368 2061 7320 4752 6576 4c65 780a  such as GRevLex.
│ │ │ │ +0003f6e0: 2020 2a20 4f75 7470 7574 733a 0a20 2020    * Outputs:.   
│ │ │ │ +0003f6f0: 2020 202a 2053 2c20 6120 2a6e 6f74 6520     * S, a *note 
│ │ │ │ +0003f700: 7269 6e67 3a20 284d 6163 6175 6c61 7932  ring: (Macaulay2
│ │ │ │ +0003f710: 446f 6329 5269 6e67 2c2c 2061 2070 6f6c  Doc)Ring,, a pol
│ │ │ │ +0003f720: 796e 6f6d 6961 6c20 7269 6e67 2077 6974  ynomial ring wit
│ │ │ │ +0003f730: 6820 6120 6e65 770a 2020 2020 2020 2020  h a new.        
│ │ │ │ +0003f740: 7261 6e64 6f6d 206d 6f6e 6f6d 6961 6c20  random monomial 
│ │ │ │ +0003f750: 6f72 6465 720a 0a44 6573 6372 6970 7469  order..Descripti
│ │ │ │ +0003f760: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  on.===========..
│ │ │ │ +0003f770: 5468 6973 2066 756e 6374 696f 6e20 7461  This function ta
│ │ │ │ +0003f780: 6b65 7320 6120 706f 6c79 6e6f 6d69 616c  kes a polynomial
│ │ │ │ +0003f790: 2072 696e 6720 616e 6420 7072 6f64 7563   ring and produc
│ │ │ │ +0003f7a0: 6573 2061 206e 6577 2070 6f6c 796e 6f6d  es a new polynom
│ │ │ │ +0003f7b0: 6961 6c20 7269 6e67 2077 6974 680a 4d6f  ial ring with.Mo
│ │ │ │ +0003f7c0: 6e6f 6d69 616c 4f72 6465 7220 6f66 2074  nomialOrder of t
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│ │ │ │ +0003f7f0: 7661 7269 6162 6c65 7320 6973 2072 616e  variables is ran
│ │ │ │ +0003f800: 646f 6d69 7a65 642e 0a0a 2b2d 2d2d 2d2d  domized...+-----
│ │ │ │ +0003f810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003f820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003f830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003f840: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20  --------+.|i2 : 
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│ │ │ │ -0003f870: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0003f880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f8a0: 2020 7c0a 7c6f 3220 3d20 7472 7565 2020    |.|o2 = true  
│ │ │ │ -0003f8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f8c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0003f8d0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -0003f8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003f8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
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│ │ │ │ -0003f930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f950: 2020 2020 2020 7c0a 7c6f 3320 3d20 5151        |.|o3 = QQ
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│ │ │ │ -0003f970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f980: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003f990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f9b0: 7c0a 7c6f 3320 3a20 506f 6c79 6e6f 6d69  |.|o3 : Polynomi
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│ │ │ │ -0003f9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003f9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003fa00: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420  ----------+.|i4 
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│ │ │ │ -0003fa20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fa30: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0003fa40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fa50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fa60: 2020 2020 7c0a 7c6f 3420 3d20 7472 7565      |.|o4 = true
│ │ │ │ -0003fa70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fa80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fa90: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ -0003faa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003fab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0003fac0: 7c69 3520 3a20 7920 3e20 7a20 2020 2020  |i5 : y > z     
│ │ │ │ -0003fad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fae0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0003faf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fb10: 2020 2020 2020 2020 7c0a 7c6f 3520 3d20          |.|o5 = 
│ │ │ │ -0003fb20: 6661 6c73 6520 2020 2020 2020 2020 2020  false           
│ │ │ │ -0003fb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003fb50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003fb60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003fb70: 2d2d 2b0a 7c69 3620 3a20 7a20 3e20 7720  --+.|i6 : z > w 
│ │ │ │ -0003fb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fb90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0003fba0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0003fbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fbc0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003fbd0: 3620 3d20 7472 7565 2020 2020 2020 2020  6 = true        
│ │ │ │ -0003fbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003fc00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003fc10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0003fc50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0003fc60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ -0003fc70: 2020 2a20 2272 656f 7264 6572 506f 6c79    * "reorderPoly
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│ │ │ │ -0003fcb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
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│ │ │ │ -0003fce0: 6961 6c52 696e 673a 2072 656f 7264 6572  ialRing: reorder
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│ │ │ │ -0003fd20: 6f70 7469 6f6e 733a 2028 4d61 6361 756c  options: (Macaul
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│ │ │ │ -0003fd40: 6374 696f 6e57 6974 684f 7074 696f 6e73  ctionWithOptions
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│ │ │ │ -0003fd60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003fd70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003fd80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003fd90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003fda0: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520  ---..The source 
│ │ │ │ -0003fdb0: 6f66 2074 6869 7320 646f 6375 6d65 6e74  of this document
│ │ │ │ -0003fdc0: 2069 7320 696e 0a2f 6275 696c 642f 7265   is in./build/re
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│ │ │ │ -0003fde0: 6d61 6361 756c 6179 322d 312e 3235 2e31  macaulay2-1.25.1
│ │ │ │ -0003fdf0: 312b 6473 2f4d 322f 4d61 6361 756c 6179  1+ds/M2/Macaulay
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│ │ │ │ -0003feb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0003fec0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ -000400a0: 2046 6f72 2061 206d 6f72 6520 6465 7461   For a more deta
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│ │ │ │ -000400e0: 2046 6173 744d 696e 6f72 7353 7472 6174   FastMinorsStrat
│ │ │ │ -000400f0: 6567 7954 7574 6f72 6961 6c3a 0a46 6173  egyTutorial:.Fas
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│ │ │ │ -00040180: 4865 7572 6973 7469 6320 7375 626d 6174  Heuristic submat
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│ │ │ │ -000402d0: 5468 6174 2070 6f69 6e74 2069 730a 2020  That point is.  
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│ │ │ │ -00040570: 2020 2a20 5374 7261 7465 6779 506f 696e    * StrategyPoin
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│ │ │ │ -00040890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000408a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00040940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -000409a0: 672d 625e 322a 692d 612a 632a 692c 645e  g-b^2*i-a*c*i,d^
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│ │ │ │ -000409d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000409e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00040a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00040450: 2020 2073 7562 6d61 7472 6978 2c20 6f72     submatrix, or
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│ │ │ │ +00040480: 6f6c 756d 6e73 206f 7220 726f 7773 2e0a  olumns or rows..
│ │ │ │ +00040490: 5468 6572 6520 7765 2068 6967 686c 6967  There we highlig
│ │ │ │ +000404a0: 6874 2066 6976 6520 7072 652d 7072 6f67  ht five pre-prog
│ │ │ │ +000404b0: 7261 6d6d 6564 2073 7472 6174 6567 6965  rammed strategie
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│ │ │ │ +000404d0: 6520 7573 6572 2e0a 2020 2a20 5374 7261  e user..  * Stra
│ │ │ │ +000404e0: 7465 6779 4465 6661 756c 743a 2074 6869  tegyDefault: thi
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│ │ │ │ +00040550: 7562 6d61 7472 6963 6573 2e0a 2020 2a20  ubmatrices..  * 
│ │ │ │ +00040560: 5374 7261 7465 6779 4465 6661 756c 744e  StrategyDefaultN
│ │ │ │ +00040570: 6f6e 5261 6e64 6f6d 3a20 7468 6973 2075  onRandom: this u
│ │ │ │ +00040580: 7365 7320 6120 6d69 7820 6f66 2068 6575  ses a mix of heu
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│ │ │ │ +000405a0: 616e 646f 6d0a 2020 2020 7375 626d 6174  andom.    submat
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│ │ │ │ +000405c0: 6567 7950 6f69 6e74 733a 2074 6869 7320  egyPoints: this 
│ │ │ │ +000405d0: 6f6e 6c79 2075 7365 7320 7261 7469 6f6e  only uses ration
│ │ │ │ +000405e0: 616c 202f 2067 656f 6d65 7472 6963 2070  al / geometric p
│ │ │ │ +000405f0: 6f69 6e74 7320 746f 2066 696e 640a 2020  oints to find.  
│ │ │ │ +00040600: 2020 7375 626d 6174 7269 6365 732e 0a20    submatrices.. 
│ │ │ │ +00040610: 202a 2053 7472 6174 6567 7944 6566 6175   * StrategyDefau
│ │ │ │ +00040620: 6c74 5769 7468 506f 696e 7473 3a20 7468  ltWithPoints: th
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│ │ │ │ +00040650: 7375 626d 6174 7269 6365 730a 2020 2020  submatrices.    
│ │ │ │ +00040660: 6368 6f73 656e 2077 6974 6820 7261 7469  chosen with rati
│ │ │ │ +00040670: 6f6e 616c 2061 6e64 2067 656f 6d65 7472  onal and geometr
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│ │ │ │ +00040be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040bf0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
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│ │ │ │ +00040c90: 2020 7c0a 7c20 2d2d 2031 2e34 3730 3131    |.| -- 1.47011
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│ │ │ │ -00040ce0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
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│ │ │ │ +00040ce0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
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│ │ │ │  00040d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00040d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040d30: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
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│ │ │ │ -00040d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a49 6e20  -----------+.In 
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│ │ │ │ -00040dc0: 2061 7665 7261 6765 2074 696d 6520 746f   average time to
│ │ │ │ -00040dd0: 2063 6f6d 706c 6574 696f 6e0a 6f66 2065   completion.of e
│ │ │ │ -00040de0: 6163 6820 6f66 2074 6865 2061 626f 7665  ach of the above
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│ │ │ │ -00040e10: 5374 7261 7465 6779 4465 6661 756c 743a  StrategyDefault:
│ │ │ │ -00040e20: 2031 2e36 3520 7365 636f 6e64 730a 2020   1.65 seconds.  
│ │ │ │ -00040e30: 2a20 5374 7261 7465 6779 5261 6e64 6f6d  * StrategyRandom
│ │ │ │ -00040e40: 3a20 382e 3332 2073 6563 6f6e 6473 0a20  : 8.32 seconds. 
│ │ │ │ -00040e50: 202a 2053 7472 6174 6567 7944 6566 6175   * StrategyDefau
│ │ │ │ -00040e60: 6c74 4e6f 6e52 616e 646f 6d3a 2030 2e39  ltNonRandom: 0.9
│ │ │ │ -00040e70: 3920 7365 636f 6e64 730a 2020 2a20 5374  9 seconds.  * St
│ │ │ │ -00040e80: 7261 7465 6779 506f 696e 7473 3a20 332e  rategyPoints: 3.
│ │ │ │ -00040e90: 3237 2073 6563 6f6e 6473 0a20 202a 2053  27 seconds.  * S
│ │ │ │ -00040ea0: 7472 6174 6567 7944 6566 6175 6c74 5769  trategyDefaultWi
│ │ │ │ -00040eb0: 7468 506f 696e 7473 3a20 332e 3337 0a52  thPoints: 3.37.R
│ │ │ │ -00040ec0: 6f75 6768 6c79 2073 7065 616b 696e 672c  oughly speaking,
│ │ │ │ -00040ed0: 2068 6575 7269 7374 6963 7320 7465 6e64   heuristics tend
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│ │ │ │ -00040f30: 7220 7369 6e63 6520 7468 6579 2063 6f6e  r since they con
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│ │ │ │ -00040fa0: 616e 7461 6765 7320 6173 2074 7970 6963  antages as typic
│ │ │ │ -00040fb0: 616c 6c79 2066 6577 6572 2073 7469 6c6c  ally fewer still
│ │ │ │ -00040fc0: 206d 696e 6f72 7320 6172 6520 6e65 6564   minors are need
│ │ │ │ -00040fd0: 6564 2028 6865 6e63 6520 6966 2063 6f6d  ed (hence if com
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│ │ │ │ -00040ff0: 2073 6c6f 7720 5374 7261 7465 6779 506f   slow StrategyPo
│ │ │ │ -00041000: 696e 7473 2069 7320 6120 676f 6f64 2063  ints is a good c
│ │ │ │ -00041010: 686f 6963 6529 2e20 2048 6f77 6576 6572  hoice).  However
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│ │ │ │ -00041e30: 2c20 616e 6420 5261 6e64 6f6d 4e6f 6e5a  , and RandomNonZ
│ │ │ │ -00041e40: 6572 6f20 6561 6368 0a20 202a 2053 7472  ero each.  * Str
│ │ │ │ -00041e50: 6174 6567 7944 6566 6175 6c74 4e6f 6e52  ategyDefaultNonR
│ │ │ │ -00041e60: 616e 646f 6d3a 2032 3525 206f 6620 7468  andom: 25% of th
│ │ │ │ -00041e70: 6520 6d61 7472 6963 6573 2061 7265 204c  e matrices are L
│ │ │ │ -00041e80: 6578 536d 616c 6c65 7374 2c0a 2020 2020  exSmallest,.    
│ │ │ │ -00041e90: 4c65 7853 6d61 6c6c 6573 7454 6572 6d2c  LexSmallestTerm,
│ │ │ │ -00041ea0: 2047 5265 764c 6578 536d 616c 6c65 7374   GRevLexSmallest
│ │ │ │ -00041eb0: 2061 6e64 2c20 4752 6576 4c65 784c 6172   and, GRevLexLar
│ │ │ │ -00041ec0: 6765 7374 2065 6163 680a 2020 2a20 5374  gest each.  * St
│ │ │ │ -00041ed0: 7261 7465 6779 4c65 7853 6d61 6c6c 6573  rategyLexSmalles
│ │ │ │ -00041ee0: 743a 2035 3025 206f 6620 7468 6520 6d61  t: 50% of the ma
│ │ │ │ -00041ef0: 7472 6963 6573 2061 7265 204c 6578 536d  trices are LexSm
│ │ │ │ -00041f00: 616c 6c65 7374 2061 6e64 2035 3025 2061  allest and 50% a
│ │ │ │ -00041f10: 7265 0a20 2020 204c 6578 536d 616c 6c65  re.    LexSmalle
│ │ │ │ -00041f20: 7374 5465 726d 0a20 202a 2053 7472 6174  stTerm.  * Strat
│ │ │ │ -00041f30: 6567 7947 5265 764c 6578 536d 616c 6c65  egyGRevLexSmalle
│ │ │ │ -00041f40: 7374 3a20 3530 2520 6f66 2074 6865 206d  st: 50% of the m
│ │ │ │ -00041f50: 6174 7269 6365 7320 6172 6520 4752 6576  atrices are GRev
│ │ │ │ -00041f60: 4c65 7853 6d61 6c6c 6573 7420 616e 6420  LexSmallest and 
│ │ │ │ -00041f70: 3530 250a 2020 2020 6172 6520 4752 6576  50%.    are GRev
│ │ │ │ -00041f80: 4c65 784c 6172 6765 7374 0a20 202a 2053  LexLargest.  * S
│ │ │ │ -00041f90: 7472 6174 6567 7952 616e 646f 6d3a 2063  trategyRandom: c
│ │ │ │ -00041fa0: 686f 6f73 6573 2031 3030 2520 7261 6e64  hooses 100% rand
│ │ │ │ -00041fb0: 6f6d 2073 7562 6d61 7472 6963 6573 2e0a  om submatrices..
│ │ │ │ -00041fc0: 2020 2a20 5374 7261 7465 6779 506f 696e    * StrategyPoin
│ │ │ │ -00041fd0: 7473 3a20 6368 6f6f 7365 2061 6c6c 2073  ts: choose all s
│ │ │ │ -00041fe0: 7562 6d61 7472 6963 6573 2076 6961 2050  ubmatrices via P
│ │ │ │ -00041ff0: 6f69 6e74 732e 0a20 202a 2053 7472 6174  oints..  * Strat
│ │ │ │ -00042000: 6567 7944 6566 6175 6c74 5769 7468 506f  egyDefaultWithPo
│ │ │ │ -00042010: 696e 7473 3a20 6c69 6b65 2053 7472 6174  ints: like Strat
│ │ │ │ -00042020: 6567 7944 6566 6175 6c74 2062 7574 2072  egyDefault but r
│ │ │ │ -00042030: 6570 6c61 6365 7320 7468 6520 5261 6e64  eplaces the Rand
│ │ │ │ -00042040: 6f6d 2061 6e64 0a20 2020 2052 616e 646f  om and.    Rando
│ │ │ │ -00042050: 6d4e 6f6e 5a65 726f 2073 7562 6d61 7472  mNonZero submatr
│ │ │ │ -00042060: 6963 6573 2061 7320 7769 7468 206d 6174  ices as with mat
│ │ │ │ -00042070: 7269 6365 7320 6368 6f73 656e 2061 7320  rices chosen as 
│ │ │ │ -00042080: 696e 2050 6f69 6e74 732e 0a41 6464 6974  in Points..Addit
│ │ │ │ -00042090: 696f 6e61 6c6c 792c 2061 204d 7574 6162  ionally, a Mutab
│ │ │ │ -000420a0: 6c65 4861 7368 5461 626c 6520 6e61 6d65  leHashTable name
│ │ │ │ -000420b0: 6420 5374 7261 7465 6779 4375 7272 656e  d StrategyCurren
│ │ │ │ -000420c0: 7420 6973 2061 6c73 6f20 6578 706f 7274  t is also export
│ │ │ │ -000420d0: 6564 2e20 2049 740a 6265 6769 6e73 2061  ed.  It.begins a
│ │ │ │ -000420e0: 7320 7468 6520 6465 6661 756c 7420 7374  s the default st
│ │ │ │ -000420f0: 7261 7465 6779 2c20 6275 7420 7468 6520  rategy, but the 
│ │ │ │ -00042100: 7573 6572 2063 616e 206d 6f64 6966 7920  user can modify 
│ │ │ │ -00042110: 6974 2e0a 0a55 7369 6e67 2061 2073 696e  it...Using a sin
│ │ │ │ -00042120: 676c 6520 6865 7572 6973 7469 6320 2041  gle heuristic  A
│ │ │ │ -00042130: 6c74 6572 6e61 7469 7665 6c79 2c20 6966  lternatively, if
│ │ │ │ -00042140: 2074 6865 2075 7365 7220 6f6e 6c79 2077   the user only w
│ │ │ │ -00042150: 616e 7473 2074 6f20 7573 6520 7361 790a  ants to use say.
│ │ │ │ -00042160: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │ -00042170: 7468 6579 2063 616e 2073 6574 2c20 5374  they can set, St
│ │ │ │ -00042180: 7261 7465 6779 2074 6f20 706f 696e 7420  rategy to point 
│ │ │ │ -00042190: 746f 2074 6861 7420 7379 6d62 6f6c 2c20  to that symbol, 
│ │ │ │ -000421a0: 696e 7374 6561 6420 6f66 2061 0a63 7265  instead of a.cre
│ │ │ │ -000421b0: 6174 696e 6720 6120 6375 7374 6f6d 2073  ating a custom s
│ │ │ │ -000421c0: 7472 6174 6567 7920 4861 7368 5461 626c  trategy HashTabl
│ │ │ │ -000421d0: 652e 2020 466f 7220 6578 616d 706c 653a  e.  For example:
│ │ │ │ -000421e0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -000421f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00042200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00042210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00042220: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20  --------+.|i4 : 
│ │ │ │ -00042230: 656c 6170 7365 6454 696d 6520 7265 6775  elapsedTime regu
│ │ │ │ -00042240: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ -00042250: 2831 2c20 542c 2053 7472 6174 6567 793d  (1, T, Strategy=
│ │ │ │ -00042260: 3e4c 6578 536d 616c 6c65 7374 5465 726d  >LexSmallestTerm
│ │ │ │ -00042270: 297c 0a7c 202d 2d20 2e39 3832 3939 3473  )|.| -- .982994s
│ │ │ │ -00042280: 2065 6c61 7073 6564 2020 2020 2020 2020   elapsed        
│ │ │ │ -00042290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000422a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000422b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -000422c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040d30: 2020 7c0a 7c6f 3220 3d20 7472 7565 2020    |.|o2 = true  
│ │ │ │ +00040d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040d80: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00040d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00040da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00040db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00040dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00040dd0: 2d2d 2b0a 496e 2074 6869 7320 7061 7274  --+.In this part
│ │ │ │ +00040de0: 6963 756c 6172 2065 7861 6d70 6c65 2c20  icular example, 
│ │ │ │ +00040df0: 6f6e 206f 6e65 206d 6163 6869 6e65 2c20  on one machine, 
│ │ │ │ +00040e00: 7765 206c 6973 7420 6176 6572 6167 6520  we list average 
│ │ │ │ +00040e10: 7469 6d65 2074 6f20 636f 6d70 6c65 7469  time to completi
│ │ │ │ +00040e20: 6f6e 0a6f 6620 6561 6368 206f 6620 7468  on.of each of th
│ │ │ │ +00040e30: 6520 6162 6f76 6520 7374 7261 7465 6769  e above strategi
│ │ │ │ +00040e40: 6573 2061 6674 6572 2031 3030 2072 756e  es after 100 run
│ │ │ │ +00040e50: 732e 0a20 202a 2053 7472 6174 6567 7944  s..  * StrategyD
│ │ │ │ +00040e60: 6566 6175 6c74 3a20 312e 3635 2073 6563  efault: 1.65 sec
│ │ │ │ +00040e70: 6f6e 6473 0a20 202a 2053 7472 6174 6567  onds.  * Strateg
│ │ │ │ +00040e80: 7952 616e 646f 6d3a 2038 2e33 3220 7365  yRandom: 8.32 se
│ │ │ │ +00040e90: 636f 6e64 730a 2020 2a20 5374 7261 7465  conds.  * Strate
│ │ │ │ +00040ea0: 6779 4465 6661 756c 744e 6f6e 5261 6e64  gyDefaultNonRand
│ │ │ │ +00040eb0: 6f6d 3a20 302e 3939 2073 6563 6f6e 6473  om: 0.99 seconds
│ │ │ │ +00040ec0: 0a20 202a 2053 7472 6174 6567 7950 6f69  .  * StrategyPoi
│ │ │ │ +00040ed0: 6e74 733a 2033 2e32 3720 7365 636f 6e64  nts: 3.27 second
│ │ │ │ +00040ee0: 730a 2020 2a20 5374 7261 7465 6779 4465  s.  * StrategyDe
│ │ │ │ +00040ef0: 6661 756c 7457 6974 6850 6f69 6e74 733a  faultWithPoints:
│ │ │ │ +00040f00: 2033 2e33 370a 526f 7567 686c 7920 7370   3.37.Roughly sp
│ │ │ │ +00040f10: 6561 6b69 6e67 2c20 6865 7572 6973 7469  eaking, heuristi
│ │ │ │ +00040f20: 6373 2074 656e 6420 746f 2070 726f 7669  cs tend to provi
│ │ │ │ +00040f30: 6465 206d 6f72 6520 696e 666f 726d 6174  de more informat
│ │ │ │ +00040f40: 696f 6e20 7468 616e 2072 616e 646f 6d0a  ion than random.
│ │ │ │ +00040f50: 7375 626d 6174 7269 6365 7320 616e 6420  submatrices and 
│ │ │ │ +00040f60: 736f 2074 6865 7920 776f 726b 206d 7563  so they work muc
│ │ │ │ +00040f70: 6820 6661 7374 6572 2073 696e 6365 2074  h faster since t
│ │ │ │ +00040f80: 6865 7920 636f 6e73 6964 6572 2066 6172  hey consider far
│ │ │ │ +00040f90: 2066 6577 6572 0a73 7562 6d61 7472 6963   fewer.submatric
│ │ │ │ +00040fa0: 6573 2e20 2046 7265 7175 656e 746c 7920  es.  Frequently 
│ │ │ │ +00040fb0: 616c 736f 2c20 636f 6d70 7574 696e 6720  also, computing 
│ │ │ │ +00040fc0: 7261 6e64 6f6d 206f 7220 7261 7469 6f6e  random or ration
│ │ │ │ +00040fd0: 616c 2070 6f69 6e74 7320 646f 6573 2068  al points does h
│ │ │ │ +00040fe0: 6176 650a 6164 7661 6e74 6167 6573 2061  ave.advantages a
│ │ │ │ +00040ff0: 7320 7479 7069 6361 6c6c 7920 6665 7765  s typically fewe
│ │ │ │ +00041000: 7220 7374 696c 6c20 6d69 6e6f 7273 2061  r still minors a
│ │ │ │ +00041010: 7265 206e 6565 6465 6420 2868 656e 6365  re needed (hence
│ │ │ │ +00041020: 2069 6620 636f 6d70 7574 696e 670a 6d69   if computing.mi
│ │ │ │ +00041030: 6e6f 7273 2069 7320 736c 6f77 2053 7472  nors is slow Str
│ │ │ │ +00041040: 6174 6567 7950 6f69 6e74 7320 6973 2061  ategyPoints is a
│ │ │ │ +00041050: 2067 6f6f 6420 6368 6f69 6365 292e 2020   good choice).  
│ │ │ │ +00041060: 486f 7765 7665 722c 2073 6f6d 6574 696d  However, sometim
│ │ │ │ +00041070: 6573 2074 6861 740a 6e6f 6e2d 7472 6976  es that.non-triv
│ │ │ │ +00041080: 6961 6c20 706f 696e 7420 636f 6d70 7574  ial point comput
│ │ │ │ +00041090: 6174 696f 6e20 7769 6c6c 2062 6563 6f6d  ation will becom
│ │ │ │ +000410a0: 6520 7374 7563 6b20 2869 6e20 7468 6520  e stuck (in the 
│ │ │ │ +000410b0: 6162 6f76 6520 6578 616d 706c 652c 2074  above example, t
│ │ │ │ +000410c0: 6865 0a6d 6564 6961 6e20 7469 6d65 2066  he.median time f
│ │ │ │ +000410d0: 6f72 2053 7472 6174 6567 7950 6f69 6e74  or StrategyPoint
│ │ │ │ +000410e0: 7320 616e 6420 5374 7261 7465 6779 4465  s and StrategyDe
│ │ │ │ +000410f0: 6661 756c 7457 6974 6850 6f69 6e74 7320  faultWithPoints 
│ │ │ │ +00041100: 7761 7320 636c 6f73 6520 746f 2031 2e35  was close to 1.5
│ │ │ │ +00041110: 0a73 6563 6f6e 6473 2c20 6275 7420 6120  .seconds, but a 
│ │ │ │ +00041120: 636f 7570 6c65 2072 756e 7320 696e 2065  couple runs in e
│ │ │ │ +00041130: 6163 6820 6361 7365 2077 6572 6520 6f72  ach case were or
│ │ │ │ +00041140: 6465 7273 206f 6620 6d61 676e 6974 7564  ders of magnitud
│ │ │ │ +00041150: 6520 736c 6f77 6572 292e 0a0a 4375 7374  e slower)...Cust
│ │ │ │ +00041160: 6f6d 2053 7472 6174 6567 6965 730a 5468  om Strategies.Th
│ │ │ │ +00041170: 6520 7573 6572 2063 616e 2063 7265 6174  e user can creat
│ │ │ │ +00041180: 6520 7468 6569 7220 6f77 6e20 7374 7261  e their own stra
│ │ │ │ +00041190: 7465 6769 6573 2061 7320 7765 6c6c 2c20  tegies as well, 
│ │ │ │ +000411a0: 6173 2077 6520 6e6f 7720 6578 706c 6169  as we now explai
│ │ │ │ +000411b0: 6e2e 2020 496e 0a70 6172 7469 6375 6c61  n.  In.particula
│ │ │ │ +000411c0: 722c 2074 6865 2075 7365 7220 6361 6e20  r, the user can 
│ │ │ │ +000411d0: 6576 656e 2063 7573 746f 6d69 7a65 2074  even customize t
│ │ │ │ +000411e0: 6865 2068 6575 7269 7374 6963 7320 7573  he heuristics us
│ │ │ │ +000411f0: 6564 2e20 2053 6565 2062 656c 6f77 2066  ed.  See below f
│ │ │ │ +00041200: 6f72 2068 6f77 0a74 6f20 6561 7369 6c79  or how.to easily
│ │ │ │ +00041210: 2075 7365 206f 6e6c 7920 6120 7369 6e67   use only a sing
│ │ │ │ +00041220: 6c65 2068 6575 7269 7374 6963 2e20 546f  le heuristic. To
│ │ │ │ +00041230: 2063 7573 746f 6d20 7374 7261 7465 6779   custom strategy
│ │ │ │ +00041240: 2069 7320 7370 6563 6966 6965 6420 6279   is specified by
│ │ │ │ +00041250: 2061 0a48 6173 6854 6162 6c65 2077 6869   a.HashTable whi
│ │ │ │ +00041260: 6368 206d 7573 7420 6861 7665 2074 6865  ch must have the
│ │ │ │ +00041270: 2066 6f6c 6c6f 7769 6e67 206b 6579 732e   following keys.
│ │ │ │ +00041280: 0a20 202a 2047 5265 764c 6578 4c61 7267  .  * GRevLexLarg
│ │ │ │ +00041290: 6573 743a 2074 7279 2074 6f20 6669 6e64  est: try to find
│ │ │ │ +000412a0: 2073 7562 6d61 7472 6963 6573 2077 6865   submatrices whe
│ │ │ │ +000412b0: 7265 2065 6163 6820 726f 7720 616e 6420  re each row and 
│ │ │ │ +000412c0: 636f 6c75 6d6e 2068 6173 2061 0a20 2020  column has a.   
│ │ │ │ +000412d0: 206c 6172 6765 2065 6e74 7279 2077 6974   large entry wit
│ │ │ │ +000412e0: 6820 7265 7370 6563 7420 746f 2061 2072  h respect to a r
│ │ │ │ +000412f0: 616e 646f 6d20 4752 6576 4c65 786f 7264  andom GRevLexord
│ │ │ │ +00041300: 6572 2e0a 2020 2a20 4752 6576 4c65 7853  er..  * GRevLexS
│ │ │ │ +00041310: 6d61 6c6c 6573 743a 2074 7279 2074 6f20  mallest: try to 
│ │ │ │ +00041320: 6669 6e64 2073 7562 6d61 7472 6963 6573  find submatrices
│ │ │ │ +00041330: 2077 6865 7265 2065 6163 6820 726f 7720   where each row 
│ │ │ │ +00041340: 616e 6420 636f 6c75 6d6e 2068 6173 2061  and column has a
│ │ │ │ +00041350: 0a20 2020 2073 6d61 6c6c 2065 6e74 7279  .    small entry
│ │ │ │ +00041360: 2077 6974 6820 7265 7370 6563 7420 746f   with respect to
│ │ │ │ +00041370: 2061 2072 616e 646f 6d20 4752 6576 4c65   a random GRevLe
│ │ │ │ +00041380: 786f 7264 6572 2e0a 2020 2a20 4752 6576  xorder..  * GRev
│ │ │ │ +00041390: 4c65 7853 6d61 6c6c 6573 7454 6572 6d3a  LexSmallestTerm:
│ │ │ │ +000413a0: 2066 696e 6420 7375 626d 6174 7269 6365   find submatrice
│ │ │ │ +000413b0: 7320 7768 6572 6520 6561 6368 2072 6f77  s where each row
│ │ │ │ +000413c0: 2061 6e64 2063 6f6c 756d 6e20 6861 7320   and column has 
│ │ │ │ +000413d0: 616e 0a20 2020 2065 6e74 7279 2077 6974  an.    entry wit
│ │ │ │ +000413e0: 6820 6120 736d 616c 6c20 7465 726d 2077  h a small term w
│ │ │ │ +000413f0: 6974 6820 7265 7370 6563 7420 746f 2061  ith respect to a
│ │ │ │ +00041400: 2072 616e 646f 6d20 4752 6576 4c65 786f   random GRevLexo
│ │ │ │ +00041410: 7264 6572 2e0a 2020 2a20 4c65 784c 6172  rder..  * LexLar
│ │ │ │ +00041420: 6765 7374 3a20 7472 7920 746f 2066 696e  gest: try to fin
│ │ │ │ +00041430: 6420 7375 626d 6174 7269 6365 7320 7768  d submatrices wh
│ │ │ │ +00041440: 6572 6520 6561 6368 2072 6f77 2061 6e64  ere each row and
│ │ │ │ +00041450: 2063 6f6c 756d 6e20 6861 7320 6120 6c61   column has a la
│ │ │ │ +00041460: 7267 650a 2020 2020 656e 7472 7920 7769  rge.    entry wi
│ │ │ │ +00041470: 7468 2072 6573 7065 6374 2074 6f20 6120  th respect to a 
│ │ │ │ +00041480: 7261 6e64 6f6d 204c 6578 6f72 6465 722e  random Lexorder.
│ │ │ │ +00041490: 0a20 202a 204c 6578 536d 616c 6c65 7374  .  * LexSmallest
│ │ │ │ +000414a0: 3a20 7472 7920 746f 2066 696e 6420 7375  : try to find su
│ │ │ │ +000414b0: 626d 6174 7269 6365 7320 7768 6572 6520  bmatrices where 
│ │ │ │ +000414c0: 6561 6368 2072 6f77 2061 6e64 2063 6f6c  each row and col
│ │ │ │ +000414d0: 756d 6e20 6861 7320 6120 736d 616c 6c0a  umn has a small.
│ │ │ │ +000414e0: 2020 2020 656e 7472 7920 7769 7468 2072      entry with r
│ │ │ │ +000414f0: 6573 7065 6374 2074 6f20 6120 7261 6e64  espect to a rand
│ │ │ │ +00041500: 6f6d 204c 6578 6f72 6465 722e 0a20 202a  om Lexorder..  *
│ │ │ │ +00041510: 204c 6578 536d 616c 6c65 7374 5465 726d   LexSmallestTerm
│ │ │ │ +00041520: 3a20 6669 6e64 2073 7562 6d61 7472 6963  : find submatric
│ │ │ │ +00041530: 6573 2077 6865 7265 2065 6163 6820 726f  es where each ro
│ │ │ │ +00041540: 7720 616e 6420 636f 6c75 6d6e 2068 6173  w and column has
│ │ │ │ +00041550: 2061 6e20 656e 7472 790a 2020 2020 7769   an entry.    wi
│ │ │ │ +00041560: 7468 2061 2073 6d61 6c6c 2074 6572 6d20  th a small term 
│ │ │ │ +00041570: 7769 7468 2072 6573 7065 6374 2074 6f20  with respect to 
│ │ │ │ +00041580: 6120 7261 6e64 6f6d 204c 6578 6f72 6465  a random Lexorde
│ │ │ │ +00041590: 722e 0a20 202a 2052 616e 646f 6d3a 2066  r..  * Random: f
│ │ │ │ +000415a0: 696e 6420 7261 6e64 6f6d 2073 7562 6d61  ind random subma
│ │ │ │ +000415b0: 7472 6963 6573 200a 2020 2a20 5261 6e64  trices .  * Rand
│ │ │ │ +000415c0: 6f6d 4e6f 6e7a 6572 6f3a 2066 696e 6420  omNonzero: find 
│ │ │ │ +000415d0: 7261 6e64 6f6d 2073 7562 6d61 7472 6963  random submatric
│ │ │ │ +000415e0: 6573 2074 6861 7420 6861 7665 206e 6f6e  es that have non
│ │ │ │ +000415f0: 7a65 726f 2072 6f77 7320 616e 6420 636f  zero rows and co
│ │ │ │ +00041600: 6c75 6d6e 730a 2020 2a20 506f 696e 7473  lumns.  * Points
│ │ │ │ +00041610: 3a20 6669 6e64 2073 7562 6d61 7472 6963  : find submatric
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│ │ │ │ +00042800: 6162 6c65 0a                             able.
│ │ ├── ./usr/share/info/FiniteFittingIdeals.info.gz
│ │ │ ├── FiniteFittingIdeals.info
│ │ │ │ @@ -1017,17 +1017,17 @@
│ │ │ │  00003f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00003f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  00003fa0: 0a7c 6931 3520 3a20 7469 6d65 2049 3d63  .|i15 : time I=c
│ │ │ │  00003fb0: 6f31 4669 7474 696e 6728 4b33 2920 2020  o1Fitting(K3)   
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│ │ │ │  00003fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003fe0: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d            |.| --
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│ │ │ │ +00003ff0: 2075 7365 6420 302e 3030 3333 3136 3635   used 0.00331665
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│ │ │ │ +00004010: 3337 3373 2028 7468 7265 6164 293b 2030  373s (thread); 0
│ │ │ │  00004020: 7320 2867 6329 2020 2020 2020 2020 2020  s (gc)          
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│ │ │ │  00004050: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00004070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004080: 7c0a 7c6f 3135 203d 2069 6465 616c 2028  |.|o15 = ideal (
│ │ │ │ @@ -1055,17 +1055,17 @@
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│ │ │ │  00004230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004240: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e    |.| -- used 0.
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│ │ │ │ -00004260: 2030 2e30 3036 3330 3538 3873 2028 7468   0.00630588s (th
│ │ │ │ -00004270: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
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│ │ │ │  00004280: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00004290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000042a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000042b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000042c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000042d0: 2020 2020 2020 2020 7c0a 7c6f 3136 203a          |.|o16 :
│ │ │ │  000042e0: 2049 6465 616c 206f 6620 5220 2020 2020   Ideal of R
│ │ ├── ./usr/share/info/ForeignFunctions.info.gz
│ │ │ ├── ForeignFunctions.info
│ │ │ │ @@ -3504,16 +3504,16 @@
│ │ │ │  0000daf0: 2078 203d 206d 616c 6c6f 6320 3820 2020   x = malloc 8   
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│ │ │ │  0000db10: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0000db60: 7c0a 7c6f 3137 203d 2030 7837 6663 3038  |.|o17 = 0x7fc08
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│ │ │ │  0000db90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
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│ │ │ │  0000dbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000dbd0: 2020 2020 2020 2020 7c0a 7c6f 3137 203a          |.|o17 :
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│ │ │ │  0000e430: 7065 6e20 6120 7368 6172 6564 206c 6962  pen a shared lib
│ │ │ │ @@ -5393,29 +5393,29 @@
│ │ │ │  00015100: 6520 706f 696e 7465 722e 0a0a 2b2d 2d2d  e pointer...+---
│ │ │ │  00015110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │  00015130: 6931 203a 2070 7472 203d 2061 6464 7265  i1 : ptr = addre
│ │ │ │  00015140: 7373 2069 6e74 2030 2020 2020 2020 2020  ss int 0        
│ │ │ │  00015150: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00015160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015170: 2020 207c 0a7c 6f31 203d 2030 7837 6666     |.|o1 = 0x7ff
│ │ │ │ -00015180: 3961 3432 3335 6532 3020 2020 2020 2020  9a4235e20       
│ │ │ │ +00015170: 2020 207c 0a7c 6f31 203d 2030 7837 6566     |.|o1 = 0x7ef
│ │ │ │ +00015180: 6566 3532 6231 3730 3020 2020 2020 2020  ef52b1700       
│ │ │ │  00015190: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000151a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000151b0: 2020 2020 2020 2020 207c 0a7c 6f31 203a           |.|o1 :
│ │ │ │  000151c0: 2050 6f69 6e74 6572 2020 2020 2020 2020   Pointer        
│ │ │ │  000151d0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │  000151e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000151f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  00015200: 0a7c 6932 203a 2076 6f69 6473 7461 7220  .|i2 : voidstar 
│ │ │ │  00015210: 7074 7220 2020 2020 2020 2020 2020 2020  ptr             
│ │ │ │  00015220: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00015230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015240: 2020 2020 207c 0a7c 6f32 203d 2030 7837       |.|o2 = 0x7
│ │ │ │ -00015250: 6666 3961 3432 3335 6532 3020 2020 2020  ff9a4235e20     
│ │ │ │ +00015250: 6566 6566 3532 6231 3730 3020 2020 2020  efef52b1700     
│ │ │ │  00015260: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  00015270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015280: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │  00015290: 203a 2046 6f72 6569 676e 4f62 6a65 6374   : ForeignObject
│ │ │ │  000152a0: 206f 6620 7479 7065 2076 6f69 642a 7c0a   of type void*|.
│ │ │ │  000152b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  000152c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ @@ -5496,16 +5496,16 @@
│ │ │ │  00015770: 7970 6520 696e 7433 327c 0a2b 2d2d 2d2d  ype int32|.+----
│ │ │ │  00015780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │  000157a0: 3220 3a20 7074 7220 3d20 6164 6472 6573  2 : ptr = addres
│ │ │ │  000157b0: 7320 7820 2020 2020 2020 2020 2020 207c  s x            |
│ │ │ │  000157c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000157d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000157e0: 2020 7c0a 7c6f 3220 3d20 3078 3766 6639    |.|o2 = 0x7ff9
│ │ │ │ -000157f0: 6131 3936 3834 3830 2020 2020 2020 2020  a1968480        
│ │ │ │ +000157e0: 2020 7c0a 7c6f 3220 3d20 3078 3765 6665    |.|o2 = 0x7efe
│ │ │ │ +000157f0: 6632 3964 6663 3130 2020 2020 2020 2020  f29dfc10        
│ │ │ │  00015800: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00015810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015820: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20          |.|o2 : 
│ │ │ │  00015830: 506f 696e 7465 7220 2020 2020 2020 2020  Pointer         
│ │ │ │  00015840: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  00015850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ @@ -5646,16 +5646,16 @@
│ │ │ │  000160d0: 696e 7465 7273 2e0a 0a2b 2d2d 2d2d 2d2d  inters...+------
│ │ │ │  000160e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000160f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │  00016100: 203a 2070 7472 203d 2076 6f69 6473 7461   : ptr = voidsta
│ │ │ │  00016110: 7220 6164 6472 6573 7320 696e 7420 357c  r address int 5|
│ │ │ │  00016120: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00016130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016140: 2020 207c 0a7c 6f31 203d 2030 7837 6666     |.|o1 = 0x7ff
│ │ │ │ -00016150: 3961 3139 3638 3166 3020 2020 2020 2020  9a19681f0       
│ │ │ │ +00016140: 2020 207c 0a7c 6f31 203d 2030 7837 6566     |.|o1 = 0x7ef
│ │ │ │ +00016150: 6566 3261 3263 6338 3020 2020 2020 2020  ef2a2cc80       
│ │ │ │  00016160: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00016170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016180: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │  00016190: 203a 2046 6f72 6569 676e 4f62 6a65 6374   : ForeignObject
│ │ │ │  000161a0: 206f 6620 7479 7065 2076 6f69 642a 207c   of type void* |
│ │ │ │  000161b0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  000161c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ @@ -5744,15 +5744,15 @@
│ │ │ │  000166f0: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...+------------
│ │ │ │  00016700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016710: 2d2d 2d2d 2b0a 7c69 3120 3a20 7074 7220  ----+.|i1 : ptr 
│ │ │ │  00016720: 3d20 6765 744d 656d 6f72 7920 3820 2020  = getMemory 8   
│ │ │ │  00016730: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00016740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016750: 2020 2020 2020 2020 2020 7c0a 7c6f 3120            |.|o1 
│ │ │ │ -00016760: 3d20 3078 3766 6639 6162 6131 3931 3630  = 0x7ff9aba19160
│ │ │ │ +00016760: 3d20 3078 3765 6666 3031 3762 3934 3930  = 0x7eff017b9490
│ │ │ │  00016770: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00016780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000167a0: 7c0a 7c6f 3120 3a20 466f 7265 6967 6e4f  |.|o1 : ForeignO
│ │ │ │  000167b0: 626a 6563 7420 6f66 2074 7970 6520 766f  bject of type vo
│ │ │ │  000167c0: 6964 2a7c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  id*|.+----------
│ │ │ │  000167d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ @@ -5767,16 +5767,16 @@
│ │ │ │  00016860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  00016880: 7c69 3220 3a20 7074 7220 3d20 6765 744d  |i2 : ptr = getM
│ │ │ │  00016890: 656d 6f72 7928 382c 2041 746f 6d69 6320  emory(8, Atomic 
│ │ │ │  000168a0: 3d3e 2074 7275 6529 7c0a 7c20 2020 2020  => true)|.|     
│ │ │ │  000168b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000168c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000168d0: 2020 7c0a 7c6f 3220 3d20 3078 3766 6639    |.|o2 = 0x7ff9
│ │ │ │ -000168e0: 6131 3931 6530 6330 2020 2020 2020 2020  a191e0c0        
│ │ │ │ +000168d0: 2020 7c0a 7c6f 3220 3d20 3078 3765 6665    |.|o2 = 0x7efe
│ │ │ │ +000168e0: 6632 3964 6633 3430 2020 2020 2020 2020  f29df340        
│ │ │ │  000168f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00016900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016920: 2020 2020 2020 7c0a 7c6f 3220 3a20 466f        |.|o2 : Fo
│ │ │ │  00016930: 7265 6967 6e4f 626a 6563 7420 6f66 2074  reignObject of t
│ │ │ │  00016940: 7970 6520 766f 6964 2a20 2020 2020 2020  ype void*       
│ │ │ │  00016950: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ @@ -5795,16 +5795,16 @@
│ │ │ │  00016a20: 6361 6c6c 792e 0a0a 2b2d 2d2d 2d2d 2d2d  cally...+-------
│ │ │ │  00016a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016a40: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a  ---------+.|i3 :
│ │ │ │  00016a50: 2070 7472 203d 2067 6574 4d65 6d6f 7279   ptr = getMemory
│ │ │ │  00016a60: 2069 6e74 2020 2020 2020 2020 7c0a 7c20   int        |.| 
│ │ │ │  00016a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016a80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016a90: 0a7c 6f33 203d 2030 7837 6666 3961 3139  .|o3 = 0x7ff9a19
│ │ │ │ -00016aa0: 3638 6664 3020 2020 2020 2020 2020 2020  68fd0           
│ │ │ │ +00016a90: 0a7c 6f33 203d 2030 7837 6566 6566 3239  .|o3 = 0x7efef29
│ │ │ │ +00016aa0: 6466 3233 3020 2020 2020 2020 2020 2020  df230           
│ │ │ │  00016ab0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00016ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016ad0: 2020 2020 207c 0a7c 6f33 203a 2046 6f72       |.|o3 : For
│ │ │ │  00016ae0: 6569 676e 4f62 6a65 6374 206f 6620 7479  eignObject of ty
│ │ │ │  00016af0: 7065 2076 6f69 642a 7c0a 2b2d 2d2d 2d2d  pe void*|.+-----
│ │ │ │  00016b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5365  -----------+..Se
│ │ │ │ @@ -5965,18 +5965,18 @@
│ │ │ │  000174c0: 7320 696e 7420 312c 2061 6464 7265 7373  s int 1, address
│ │ │ │  000174d0: 2069 6e74 2032 7d20 2020 2020 2020 2020   int 2}         
│ │ │ │  000174e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000174f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017520: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00017530: 0a7c 6f33 203d 207b 3078 3766 6639 6131  .|o3 = {0x7ff9a1
│ │ │ │ -00017540: 3962 3236 3430 2c20 3078 3766 6639 6131  9b2640, 0x7ff9a1
│ │ │ │ -00017550: 3962 3235 6530 2c20 3078 3766 6639 6131  9b25e0, 0x7ff9a1
│ │ │ │ -00017560: 3962 3235 6430 7d20 2020 2020 2020 2020  9b25d0}         
│ │ │ │ +00017530: 0a7c 6f33 203d 207b 3078 3765 6665 6632  .|o3 = {0x7efef2
│ │ │ │ +00017540: 6132 6331 3330 2c20 3078 3765 6665 6632  a2c130, 0x7efef2
│ │ │ │ +00017550: 6132 6331 3230 2c20 3078 3765 6665 6632  a2c120, 0x7efef2
│ │ │ │ +00017560: 6132 6330 6630 7d20 2020 2020 2020 2020  a2c0f0}         
│ │ │ │  00017570: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00017580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000175a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000175b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000175c0: 2020 2020 2020 2020 207c 0a7c 6f33 203a           |.|o3 :
│ │ │ │  000175d0: 2046 6f72 6569 676e 4f62 6a65 6374 206f   ForeignObject o
│ │ │ │ @@ -6431,17 +6431,17 @@
│ │ │ │  000191e0: 7373 2069 6e74 2030 2c20 6164 6472 6573  ss int 0, addres
│ │ │ │  000191f0: 7320 696e 7420 312c 2061 6464 7265 7373  s int 1, address
│ │ │ │  00019200: 2069 6e74 2032 7d7c 0a7c 2020 2020 2020   int 2}|.|      
│ │ │ │  00019210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019240: 2020 2020 2020 2020 207c 0a7c 6f32 203d           |.|o2 =
│ │ │ │ -00019250: 207b 3078 3766 6639 6131 3962 3233 6430   {0x7ff9a19b23d0
│ │ │ │ -00019260: 2c20 3078 3766 6639 6131 3962 3233 6330  , 0x7ff9a19b23c0
│ │ │ │ -00019270: 2c20 3078 3766 6639 6131 3962 3233 6230  , 0x7ff9a19b23b0
│ │ │ │ +00019250: 207b 3078 3765 6665 6632 6137 3566 3530   {0x7efef2a75f50
│ │ │ │ +00019260: 2c20 3078 3765 6665 6632 6137 3566 3430  , 0x7efef2a75f40
│ │ │ │ +00019270: 2c20 3078 3765 6665 6632 6137 3566 3230  , 0x7efef2a75f20
│ │ │ │  00019280: 7d20 2020 2020 2020 2020 207c 0a7c 2020  }          |.|  
│ │ │ │  00019290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000192a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000192b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000192c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  000192d0: 6f32 203a 2046 6f72 6569 676e 4f62 6a65  o2 : ForeignObje
│ │ │ │  000192e0: 6374 206f 6620 7479 7065 2076 6f69 642a  ct of type void*
│ │ │ │ @@ -7909,15 +7909,15 @@
│ │ │ │  0001ee40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ee50: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  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 2020 2020 7c0a                |.
│ │ │ │  0001eea0: 7c6f 3220 3d20 4861 7368 5461 626c 657b  |o2 = HashTable{
│ │ │ │ -0001eeb0: 2262 6172 2220 3d3e 2036 2e39 3531 3837  "bar" => 6.95187
│ │ │ │ +0001eeb0: 2262 6172 2220 3d3e 2036 2e38 3938 3631  "bar" => 6.89861
│ │ │ │  0001eec0: 652d 3331 307d 2020 2020 2020 2020 2020  e-310}          
│ │ │ │  0001eed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001eee0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001eef0: 2020 2020 2020 2020 2020 2020 2020 2266                "f
│ │ │ │  0001ef00: 6f6f 2220 3d3e 2032 3720 2020 2020 2020  oo" => 27       
│ │ │ │  0001ef10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ef20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -8160,26 +8160,26 @@
│ │ │ │  0001fdf0: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 7065  ------+.|i2 : pe
│ │ │ │  0001fe00: 656b 2078 2020 2020 2020 2020 2020 2020  ek x            
│ │ │ │  0001fe10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  0001fe20: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0001fe30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fe40: 2020 2020 2020 7c0a 7c6f 3220 3d20 696e        |.|o2 = in
│ │ │ │  0001fe50: 7433 327b 4164 6472 6573 7320 3d3e 2030  t32{Address => 0
│ │ │ │ -0001fe60: 7837 6666 3961 3139 3638 6561 307d 7c0a  x7ff9a1968ea0}|.
│ │ │ │ +0001fe60: 7837 6566 6566 3239 6466 3661 307d 7c0a  x7efef29df6a0}|.
│ │ │ │  0001fe70: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  0001fe80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001fe90: 2d2d 2d2d 2d2d 2b0a 0a54 6f20 6765 7420  ------+..To get 
│ │ │ │  0001fea0: 7468 6973 2c20 7573 6520 2a6e 6f74 6520  this, use *note 
│ │ │ │  0001feb0: 6164 6472 6573 733a 2061 6464 7265 7373  address: address
│ │ │ │  0001fec0: 2c2e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  ,...+-----------
│ │ │ │  0001fed0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │  0001fee0: 6164 6472 6573 7320 7820 2020 2020 7c0a  address x     |.
│ │ │ │  0001fef0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0001ff00: 2020 2020 7c0a 7c6f 3320 3d20 3078 3766      |.|o3 = 0x7f
│ │ │ │ -0001ff10: 6639 6131 3936 3865 6130 7c0a 7c20 2020  f9a1968ea0|.|   
│ │ │ │ +0001ff00: 2020 2020 7c0a 7c6f 3320 3d20 3078 3765      |.|o3 = 0x7e
│ │ │ │ +0001ff10: 6665 6632 3964 6636 6130 7c0a 7c20 2020  fef29df6a0|.|   
│ │ │ │  0001ff20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ff30: 7c0a 7c6f 3320 3a20 506f 696e 7465 7220  |.|o3 : Pointer 
│ │ │ │  0001ff40: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │  0001ff50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a55  ------------+..U
│ │ │ │  0001ff60: 7365 202a 6e6f 7465 2063 6c61 7373 3a20  se *note class: 
│ │ │ │  0001ff70: 284d 6163 6175 6c61 7932 446f 6329 636c  (Macaulay2Doc)cl
│ │ │ │  0001ff80: 6173 732c 2074 6f20 6465 7465 726d 696e  ass, to determin
│ │ │ │ @@ -8881,29 +8881,29 @@
│ │ │ │  00022b00: 626a 6563 7473 2e0a 0a2b 2d2d 2d2d 2d2d  bjects...+------
│ │ │ │  00022b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520  ----------+.|i5 
│ │ │ │  00022b30: 3a20 7820 3d20 766f 6964 7374 6172 2061  : x = voidstar a
│ │ │ │  00022b40: 6464 7265 7373 2069 6e74 2035 207c 0a7c  ddress int 5 |.|
│ │ │ │  00022b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022b70: 7c0a 7c6f 3520 3d20 3078 3766 6639 6131  |.|o5 = 0x7ff9a1
│ │ │ │ -00022b80: 3936 3838 3630 2020 2020 2020 2020 2020  968860          
│ │ │ │ +00022b70: 7c0a 7c6f 3520 3d20 3078 3765 6665 6632  |.|o5 = 0x7efef2
│ │ │ │ +00022b80: 6132 6363 6530 2020 2020 2020 2020 2020  a2cce0          
│ │ │ │  00022b90: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00022ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022bb0: 2020 2020 2020 7c0a 7c6f 3520 3a20 466f        |.|o5 : Fo
│ │ │ │  00022bc0: 7265 6967 6e4f 626a 6563 7420 6f66 2074  reignObject of t
│ │ │ │  00022bd0: 7970 6520 766f 6964 2a7c 0a2b 2d2d 2d2d  ype void*|.+----
│ │ │ │  00022be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │  00022c00: 3620 3a20 7661 6c75 6520 7820 2020 2020  6 : value x     
│ │ │ │  00022c10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00022c20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00022c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022c40: 2020 7c0a 7c6f 3620 3d20 3078 3766 6639    |.|o6 = 0x7ff9
│ │ │ │ -00022c50: 6131 3936 3838 3630 2020 2020 2020 2020  a1968860        
│ │ │ │ +00022c40: 2020 7c0a 7c6f 3620 3d20 3078 3765 6665    |.|o6 = 0x7efe
│ │ │ │ +00022c50: 6632 6132 6363 6530 2020 2020 2020 2020  f2a2cce0        
│ │ │ │  00022c60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00022c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022c80: 2020 2020 2020 2020 7c0a 7c6f 3620 3a20          |.|o6 : 
│ │ │ │  00022c90: 506f 696e 7465 7220 2020 2020 2020 2020  Pointer         
│ │ │ │  00022ca0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  00022cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ @@ -9430,50 +9430,50 @@
│ │ │ │  00024d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024d60: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2063  -------+.|i5 : c
│ │ │ │  00024d70: 6f6c 6c65 6374 4761 7262 6167 6528 2920  ollectGarbage() 
│ │ │ │  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 207c 0a7c               |.|
│ │ │ │  00024db0: 6672 6565 696e 6720 6d65 6d6f 7279 2061  freeing memory a
│ │ │ │ -00024dc0: 7420 3078 3766 6639 3934 3037 6639 3030  t 0x7ff99407f900
│ │ │ │ +00024dc0: 7420 3078 3765 6665 6463 3037 6639 3030  t 0x7efedc07f900
│ │ │ │  00024dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024df0: 2020 207c 0a7c 6672 6565 696e 6720 6d65     |.|freeing me
│ │ │ │ -00024e00: 6d6f 7279 2061 7420 3078 3766 6639 3934  mory at 0x7ff994
│ │ │ │ -00024e10: 3037 6639 3230 2020 2020 2020 2020 2020  07f920          
│ │ │ │ +00024e00: 6d6f 7279 2061 7420 3078 3765 6665 6463  mory at 0x7efedc
│ │ │ │ +00024e10: 3037 6638 6530 2020 2020 2020 2020 2020  07f8e0          
│ │ │ │  00024e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024e30: 2020 2020 2020 2020 207c 0a7c 6672 6565           |.|free
│ │ │ │  00024e40: 696e 6720 6d65 6d6f 7279 2061 7420 3078  ing memory at 0x
│ │ │ │ -00024e50: 3766 6639 3934 3037 6638 6130 2020 2020  7ff99407f8a0    
│ │ │ │ +00024e50: 3765 6665 6463 3037 6638 6330 2020 2020  7efedc07f8c0    
│ │ │ │  00024e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024e70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00024e80: 0a7c 6672 6565 696e 6720 6d65 6d6f 7279  .|freeing memory
│ │ │ │ -00024e90: 2061 7420 3078 3766 6639 3934 3037 6631   at 0x7ff99407f1
│ │ │ │ -00024ea0: 6330 2020 2020 2020 2020 2020 2020 2020  c0              
│ │ │ │ +00024e90: 2061 7420 3078 3765 6665 6463 3037 6638   at 0x7efedc07f8
│ │ │ │ +00024ea0: 6130 2020 2020 2020 2020 2020 2020 2020  a0              
│ │ │ │  00024eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024ec0: 2020 2020 207c 0a7c 6672 6565 696e 6720       |.|freeing 
│ │ │ │ -00024ed0: 6d65 6d6f 7279 2061 7420 3078 3766 6639  memory at 0x7ff9
│ │ │ │ -00024ee0: 3934 3037 6631 6130 2020 2020 2020 2020  9407f1a0        
│ │ │ │ +00024ed0: 6d65 6d6f 7279 2061 7420 3078 3765 6665  memory at 0x7efe
│ │ │ │ +00024ee0: 6463 3037 6638 3830 2020 2020 2020 2020  dc07f880        
│ │ │ │  00024ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024f00: 2020 2020 2020 2020 2020 207c 0a7c 6672             |.|fr
│ │ │ │  00024f10: 6565 696e 6720 6d65 6d6f 7279 2061 7420  eeing memory at 
│ │ │ │ -00024f20: 3078 3766 6639 3934 3037 6638 3630 2020  0x7ff99407f860  
│ │ │ │ +00024f20: 3078 3765 6665 6463 3037 6638 3630 2020  0x7efedc07f860  
│ │ │ │  00024f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024f50: 207c 0a7c 6672 6565 696e 6720 6d65 6d6f   |.|freeing memo
│ │ │ │ -00024f60: 7279 2061 7420 3078 3766 6639 3934 3037  ry at 0x7ff99407
│ │ │ │ -00024f70: 6638 3830 2020 2020 2020 2020 2020 2020  f880            
│ │ │ │ +00024f60: 7279 2061 7420 3078 3765 6665 6463 3037  ry at 0x7efedc07
│ │ │ │ +00024f70: 6631 6130 2020 2020 2020 2020 2020 2020  f1a0            
│ │ │ │  00024f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024f90: 2020 2020 2020 207c 0a7c 6672 6565 696e         |.|freein
│ │ │ │ -00024fa0: 6720 6d65 6d6f 7279 2061 7420 3078 3766  g memory at 0x7f
│ │ │ │ -00024fb0: 6639 3934 3037 6638 6330 2020 2020 2020  f99407f8c0      
│ │ │ │ +00024fa0: 6720 6d65 6d6f 7279 2061 7420 3078 3765  g memory at 0x7e
│ │ │ │ +00024fb0: 6665 6463 3037 6639 3230 2020 2020 2020  fedc07f920      
│ │ │ │  00024fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024fd0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00024fe0: 6672 6565 696e 6720 6d65 6d6f 7279 2061  freeing memory a
│ │ │ │ -00024ff0: 7420 3078 3766 6639 3934 3037 6638 6530  t 0x7ff99407f8e0
│ │ │ │ +00024ff0: 7420 3078 3765 6665 6463 3037 6631 6330  t 0x7efedc07f1c0
│ │ │ │  00025000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025020: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  00025030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025060: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520  ---------+..See 
│ │ │ │ @@ -9546,50 +9546,50 @@
│ │ │ │  00025490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000254a0: 2d2d 2d2b 0a7c 6932 203a 2070 6565 6b20  ---+.|i2 : peek 
│ │ │ │  000254b0: 7820 2020 2020 2020 2020 2020 2020 2020  x               
│ │ │ │  000254c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000254d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000254e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000254f0: 2020 207c 0a7c 6f32 203d 2069 6e74 3332     |.|o2 = int32
│ │ │ │ -00025500: 7b41 6464 7265 7373 203d 3e20 3078 3766  {Address => 0x7f
│ │ │ │ -00025510: 6639 6131 3931 6535 3230 7d7c 0a2b 2d2d  f9a191e520}|.+--
│ │ │ │ +00025500: 7b41 6464 7265 7373 203d 3e20 3078 3765  {Address => 0x7e
│ │ │ │ +00025510: 6665 6632 3964 6637 6430 7d7c 0a2b 2d2d  fef29df7d0}|.+--
│ │ │ │  00025520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025540: 2d2d 2d2b 0a0a 5468 6573 6520 706f 696e  ---+..These poin
│ │ │ │  00025550: 7465 7273 2063 616e 2062 6520 6163 6365  ters can be acce
│ │ │ │  00025560: 7373 6564 2075 7369 6e67 202a 6e6f 7465  ssed using *note
│ │ │ │  00025570: 2061 6464 7265 7373 3a20 6164 6472 6573   address: addres
│ │ │ │  00025580: 732c 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  s,...+----------
│ │ │ │  00025590: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320  ----------+.|i3 
│ │ │ │  000255a0: 3a20 7074 7220 3d20 6164 6472 6573 7320  : ptr = address 
│ │ │ │  000255b0: 787c 0a7c 2020 2020 2020 2020 2020 2020  x|.|            
│ │ │ │  000255c0: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20          |.|o3 = 
│ │ │ │ -000255d0: 3078 3766 6639 6131 3931 6535 3230 207c  0x7ff9a191e520 |
│ │ │ │ +000255d0: 3078 3765 6665 6632 3964 6637 6430 207c  0x7efef29df7d0 |
│ │ │ │  000255e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000255f0: 2020 2020 2020 7c0a 7c6f 3320 3a20 506f        |.|o3 : Po
│ │ │ │  00025600: 696e 7465 7220 2020 2020 2020 207c 0a2b  inter        |.+
│ │ │ │  00025610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025620: 2d2d 2d2d 2b0a 0a53 696d 706c 6520 6172  ----+..Simple ar
│ │ │ │  00025630: 6974 686d 6574 6963 2063 616e 2062 6520  ithmetic can be 
│ │ │ │  00025640: 7065 7266 6f72 6d65 6420 6f6e 2070 6f69  performed on poi
│ │ │ │  00025650: 6e74 6572 732e 0a0a 2b2d 2d2d 2d2d 2d2d  nters...+-------
│ │ │ │  00025660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │  00025670: 3420 3a20 7074 7220 2b20 3520 2020 2020  4 : ptr + 5     
│ │ │ │  00025680: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00025690: 2020 2020 2020 2020 7c0a 7c6f 3420 3d20          |.|o4 = 
│ │ │ │ -000256a0: 3078 3766 6639 6131 3931 6535 3235 7c0a  0x7ff9a191e525|.
│ │ │ │ +000256a0: 3078 3765 6665 6632 3964 6637 6435 7c0a  0x7efef29df7d5|.
│ │ │ │  000256b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000256c0: 2020 2020 7c0a 7c6f 3420 3a20 506f 696e      |.|o4 : Poin
│ │ │ │  000256d0: 7465 7220 2020 2020 2020 7c0a 2b2d 2d2d  ter       |.+---
│ │ │ │  000256e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000256f0: 2b0a 7c69 3520 3a20 7074 7220 2d20 3320  +.|i5 : ptr - 3 
│ │ │ │  00025700: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00025710: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00025720: 3520 3d20 3078 3766 6639 6131 3931 6535  5 = 0x7ff9a191e5
│ │ │ │ -00025730: 3164 7c0a 7c20 2020 2020 2020 2020 2020  1d|.|           
│ │ │ │ +00025720: 3520 3d20 3078 3765 6665 6632 3964 6637  5 = 0x7efef29df7
│ │ │ │ +00025730: 6364 7c0a 7c20 2020 2020 2020 2020 2020  cd|.|           
│ │ │ │  00025740: 2020 2020 2020 2020 7c0a 7c6f 3520 3a20          |.|o5 : 
│ │ │ │  00025750: 506f 696e 7465 7220 2020 2020 2020 7c0a  Pointer       |.
│ │ │ │  00025760: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00025770: 2d2d 2d2d 2b0a 2a20 4d65 6e75 3a0a 0a2a  ----+.* Menu:..*
│ │ │ │  00025780: 206e 756c 6c50 6f69 6e74 6572 3a3a 2020   nullPointer::  
│ │ │ │  00025790: 2020 2020 2020 2020 2020 2020 2020 2074                 t
│ │ │ │  000257a0: 6865 206e 756c 6c20 706f 696e 7465 720a  he null pointer.
│ │ │ │ @@ -9758,15 +9758,15 @@
│ │ │ │  000261d0: 740a 7573 6564 2062 7920 6c69 6266 6669  t.used by libffi
│ │ │ │  000261e0: 2074 6f20 6964 656e 7469 6679 2074 6865   to identify the
│ │ │ │  000261f0: 2074 7970 652e 0a0a 2b2d 2d2d 2d2d 2d2d   type...+-------
│ │ │ │  00026200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │  00026210: 3120 3a20 6164 6472 6573 7320 696e 7420  1 : address int 
│ │ │ │  00026220: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00026230: 2020 2020 2020 2020 7c0a 7c6f 3120 3d20          |.|o1 = 
│ │ │ │ -00026240: 3078 3536 3362 3034 3137 3062 3430 7c0a  0x563b04170b40|.
│ │ │ │ +00026240: 3078 3535 6266 6664 3464 3662 3430 7c0a  0x55bffd4d6b40|.
│ │ │ │  00026250: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00026260: 2020 2020 7c0a 7c6f 3120 3a20 506f 696e      |.|o1 : Poin
│ │ │ │  00026270: 7465 7220 2020 2020 2020 7c0a 2b2d 2d2d  ter       |.+---
│ │ │ │  00026280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026290: 2b0a 0a49 6620 7820 6973 2061 2066 6f72  +..If x is a for
│ │ │ │  000262a0: 6569 676e 206f 626a 6563 742c 2074 6865  eign object, the
│ │ │ │  000262b0: 6e20 7468 6973 2072 6574 7572 6e73 2074  n this returns t
│ │ │ │ @@ -9775,16 +9775,16 @@
│ │ │ │  000262e0: 6861 7665 7320 6c69 6b65 2074 6865 2026  haves like the &
│ │ │ │  000262f0: 2022 6164 6472 6573 732d 6f66 2220 6f70   "address-of" op
│ │ │ │  00026300: 6572 6174 6f72 2069 6e20 432e 0a0a 2b2d  erator in C...+-
│ │ │ │  00026310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026320: 2d2d 2b0a 7c69 3220 3a20 6164 6472 6573  --+.|i2 : addres
│ │ │ │  00026330: 7320 696e 7420 3520 7c0a 7c20 2020 2020  s int 5 |.|     
│ │ │ │  00026340: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00026350: 7c6f 3220 3d20 3078 3766 6639 6131 3931  |o2 = 0x7ff9a191
│ │ │ │ -00026360: 6536 3330 7c0a 7c20 2020 2020 2020 2020  e630|.|         
│ │ │ │ +00026350: 7c6f 3220 3d20 3078 3765 6665 6632 3964  |o2 = 0x7efef29d
│ │ │ │ +00026360: 6661 3630 7c0a 7c20 2020 2020 2020 2020  fa60|.|         
│ │ │ │  00026370: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │  00026380: 3a20 506f 696e 7465 7220 2020 2020 2020  : Pointer       
│ │ │ │  00026390: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  000263a0: 2d2d 2d2d 2d2d 2b0a 0a57 6179 7320 746f  ------+..Ways to
│ │ │ │  000263b0: 2075 7365 2061 6464 7265 7373 3a0a 3d3d   use address:.==
│ │ │ │  000263c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │  000263d0: 3d3d 0a0a 2020 2a20 2261 6464 7265 7373  ==..  * "address
│ │ │ │ @@ -9866,16 +9866,16 @@
│ │ │ │  00026890: 7970 6520 696e 7433 327c 0a2b 2d2d 2d2d  ype int32|.+----
│ │ │ │  000268a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000268b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │  000268c0: 3220 3a20 7074 7220 3d20 6164 6472 6573  2 : ptr = addres
│ │ │ │  000268d0: 7320 7820 2020 2020 2020 2020 2020 207c  s x            |
│ │ │ │  000268e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000268f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026900: 2020 7c0a 7c6f 3220 3d20 3078 3766 6639    |.|o2 = 0x7ff9
│ │ │ │ -00026910: 6131 3936 3835 6430 2020 2020 2020 2020  a19685d0        
│ │ │ │ +00026900: 2020 7c0a 7c6f 3220 3d20 3078 3765 6665    |.|o2 = 0x7efe
│ │ │ │ +00026910: 6632 3964 6638 3330 2020 2020 2020 2020  f29df830        
│ │ │ │  00026920: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00026930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026940: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20          |.|o2 : 
│ │ │ │  00026950: 506f 696e 7465 7220 2020 2020 2020 2020  Pointer         
│ │ │ │  00026960: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  00026970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ ├── ./usr/share/info/FourTiTwo.info.gz
│ │ │ ├── FourTiTwo.info
│ │ │ │ @@ -838,25 +838,25 @@
│ │ │ │  00003450: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00003460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00003470: 2d2d 2d2d 2b0a 7c69 3220 3a20 7320 3d20  ----+.|i2 : s = 
│ │ │ │  00003480: 7465 6d70 6f72 6172 7946 696c 654e 616d  temporaryFileNam
│ │ │ │  00003490: 6528 2920 2020 2020 207c 0a7c 2020 2020  e()      |.|    
│ │ │ │  000034a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000034b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000034c0: 7c6f 3220 3d20 2f74 6d70 2f4d 322d 3135  |o2 = /tmp/M2-15
│ │ │ │ -000034d0: 3935 342d 302f 3020 2020 2020 2020 2020  954-0/0         
│ │ │ │ +000034c0: 7c6f 3220 3d20 2f74 6d70 2f4d 322d 3230  |o2 = /tmp/M2-20
│ │ │ │ +000034d0: 3935 352d 302f 3020 2020 2020 2020 2020  955-0/0         
│ │ │ │  000034e0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  000034f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00003500: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │  00003510: 4620 3d20 6f70 656e 4f75 7428 7329 2020  F = openOut(s)  
│ │ │ │  00003520: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00003530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003550: 2020 7c0a 7c6f 3320 3d20 2f74 6d70 2f4d    |.|o3 = /tmp/M
│ │ │ │ -00003560: 322d 3135 3935 342d 302f 3020 2020 2020  2-15954-0/0     
│ │ │ │ +00003560: 322d 3230 3935 352d 302f 3020 2020 2020  2-20955-0/0     
│ │ │ │  00003570: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00003580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003590: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │  000035a0: 3320 3a20 4669 6c65 2020 2020 2020 2020  3 : File        
│ │ │ │  000035b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000035c0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  000035d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ @@ -866,15 +866,15 @@
│ │ │ │  00003610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00003620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00003630: 2b0a 7c69 3520 3a20 636c 6f73 6528 4629  +.|i5 : close(F)
│ │ │ │  00003640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003650: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00003660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003670: 2020 2020 2020 2020 2020 7c0a 7c6f 3520            |.|o5 
│ │ │ │ -00003680: 3d20 2f74 6d70 2f4d 322d 3135 3935 342d  = /tmp/M2-15954-
│ │ │ │ +00003680: 3d20 2f74 6d70 2f4d 322d 3230 3935 352d  = /tmp/M2-20955-
│ │ │ │  00003690: 302f 3020 2020 2020 2020 2020 2020 207c  0/0            |
│ │ │ │  000036a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000036b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000036c0: 2020 2020 7c0a 7c6f 3520 3a20 4669 6c65      |.|o5 : File
│ │ │ │  000036d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000036e0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │  000036f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ ├── ./usr/share/info/FrobeniusThresholds.info.gz
│ │ │ ├── FrobeniusThresholds.info
│ │ │ │ @@ -2692,16 +2692,16 @@
│ │ │ │  0000a830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a850: 2d2d 2d2d 2d2b 0a7c 6932 3720 3a20 7469  -----+.|i27 : ti
│ │ │ │  0000a860: 6d65 206e 756d 6572 6963 2066 7074 2866  me numeric fpt(f
│ │ │ │  0000a870: 2c20 4465 7074 684f 6653 6561 7263 6820  , DepthOfSearch 
│ │ │ │  0000a880: 3d3e 2033 2c20 4669 6e61 6c41 7474 656d  => 3, FinalAttem
│ │ │ │  0000a890: 7074 203d 3e20 7472 7565 297c 0a7c 202d  pt => true)|.| -
│ │ │ │ -0000a8a0: 2d20 7573 6564 2032 2e32 3936 3837 7320  - used 2.29687s 
│ │ │ │ -0000a8b0: 2863 7075 293b 2031 2e33 3632 3173 2028  (cpu); 1.3621s (
│ │ │ │ +0000a8a0: 2d20 7573 6564 2032 2e35 3939 3973 2028  - used 2.5999s (
│ │ │ │ +0000a8b0: 6370 7529 3b20 312e 3435 3538 3573 2028  cpu); 1.45585s (
│ │ │ │  0000a8c0: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 0s (gc)
│ │ │ │  0000a8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a8e0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0000a8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a920: 2020 2020 2020 207c 0a7c 6f32 3720 3d20         |.|o27 = 
│ │ │ │ @@ -2723,17 +2723,17 @@
│ │ │ │  0000aa20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000aa30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  0000aa40: 0a7c 6932 3820 3a20 7469 6d65 2066 7074  .|i28 : time fpt
│ │ │ │  0000aa50: 2866 2c20 4465 7074 684f 6653 6561 7263  (f, DepthOfSearc
│ │ │ │  0000aa60: 6820 3d3e 2033 2c20 4174 7465 6d70 7473  h => 3, Attempts
│ │ │ │  0000aa70: 203d 3e20 3729 2020 2020 2020 2020 2020   => 7)          
│ │ │ │  0000aa80: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ -0000aa90: 2031 2e32 3936 3831 7320 2863 7075 293b   1.29681s (cpu);
│ │ │ │ -0000aaa0: 2030 2e37 3933 3939 7320 2874 6872 6561   0.79399s (threa
│ │ │ │ -0000aab0: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │ +0000aa90: 2031 2e34 3939 3838 7320 2863 7075 293b   1.49988s (cpu);
│ │ │ │ +0000aaa0: 2030 2e38 3738 3536 3373 2028 7468 7265   0.878563s (thre
│ │ │ │ +0000aab0: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │  0000aac0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0000aad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000aae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000aaf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ab00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ab10: 207c 0a7c 2020 2020 2020 3120 2020 2020   |.|      1     
│ │ │ │  0000ab20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -2762,16 +2762,16 @@
│ │ │ │  0000ac90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000aca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000acb0: 2d2d 2d2d 2d2b 0a7c 6932 3920 3a20 7469  -----+.|i29 : ti
│ │ │ │  0000acc0: 6d65 2066 7074 2866 2c20 4465 7074 684f  me fpt(f, DepthO
│ │ │ │  0000acd0: 6653 6561 7263 6820 3d3e 2034 2920 2020  fSearch => 4)   
│ │ │ │  0000ace0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000acf0: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ -0000ad00: 2d20 7573 6564 2031 2e31 3036 3833 7320  - used 1.10683s 
│ │ │ │ -0000ad10: 2863 7075 293b 2030 2e37 3139 3637 3173  (cpu); 0.719671s
│ │ │ │ +0000ad00: 2d20 7573 6564 2031 2e32 3034 3033 7320  - used 1.20403s 
│ │ │ │ +0000ad10: 2863 7075 293b 2030 2e37 3131 3636 3473  (cpu); 0.711664s
│ │ │ │  0000ad20: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │  0000ad30: 6329 2020 2020 2020 2020 2020 2020 2020  c)              
│ │ │ │  0000ad40: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0000ad50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ad60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ad70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ad80: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ @@ -3670,18 +3670,18 @@
│ │ │ │  0000e550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000e560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000e570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000e580: 2b0a 7c69 3135 203a 2074 696d 6520 6672  +.|i15 : time fr
│ │ │ │  0000e590: 6f62 656e 6975 734e 7528 332c 2066 2920  obeniusNu(3, f) 
│ │ │ │  0000e5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000e5b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0000e5c0: 7c20 2d2d 2075 7365 6420 302e 3030 3430  | -- used 0.0040
│ │ │ │ -0000e5d0: 3038 3235 7320 2863 7075 293b 2030 2e30  0825s (cpu); 0.0
│ │ │ │ -0000e5e0: 3034 3139 3432 3773 2028 7468 7265 6164  0419427s (thread
│ │ │ │ -0000e5f0: 293b 2030 7320 2867 6329 2020 7c0a 7c20  ); 0s (gc)  |.| 
│ │ │ │ +0000e5c0: 7c20 2d2d 2075 7365 6420 302e 3030 3339  | -- used 0.0039
│ │ │ │ +0000e5d0: 3630 3137 7320 2863 7075 293b 2030 2e30  6017s (cpu); 0.0
│ │ │ │ +0000e5e0: 3035 3331 3931 7320 2874 6872 6561 6429  053191s (thread)
│ │ │ │ +0000e5f0: 3b20 3073 2028 6763 2920 2020 7c0a 7c20  ; 0s (gc)   |.| 
│ │ │ │  0000e600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000e610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000e620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000e630: 2020 2020 2020 2020 2020 7c0a 7c6f 3135            |.|o15
│ │ │ │  0000e640: 203d 2033 3735 3620 2020 2020 2020 2020   = 3756         
│ │ │ │  0000e650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000e660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -3690,16 +3690,16 @@
│ │ │ │  0000e690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000e6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000e6b0: 2d2d 2d2d 2d2d 2b0a 7c69 3136 203a 2074  ------+.|i16 : t
│ │ │ │  0000e6c0: 696d 6520 6672 6f62 656e 6975 734e 7528  ime frobeniusNu(
│ │ │ │  0000e6d0: 332c 2066 2c20 5573 6553 7065 6369 616c  3, f, UseSpecial
│ │ │ │  0000e6e0: 416c 676f 7269 7468 6d73 203d 3e20 6661  Algorithms => fa
│ │ │ │  0000e6f0: 6c73 6529 7c0a 7c20 2d2d 2075 7365 6420  lse)|.| -- used 
│ │ │ │ -0000e700: 302e 3338 3230 3837 7320 2863 7075 293b  0.382087s (cpu);
│ │ │ │ -0000e710: 2030 2e32 3531 3130 3273 2028 7468 7265   0.251102s (thre
│ │ │ │ +0000e700: 302e 3431 3135 3238 7320 2863 7075 293b  0.411528s (cpu);
│ │ │ │ +0000e710: 2030 2e32 3836 3839 3173 2028 7468 7265   0.286891s (thre
│ │ │ │  0000e720: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │  0000e730: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0000e740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000e750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000e760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000e770: 7c0a 7c6f 3136 203d 2033 3735 3620 2020  |.|o16 = 3756   
│ │ │ │  0000e780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -3805,17 +3805,17 @@
│ │ │ │  0000edc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000edd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000ede0: 2b0a 7c69 3139 203a 2074 696d 6520 6672  +.|i19 : time fr
│ │ │ │  0000edf0: 6f62 656e 6975 734e 7528 342c 2066 2920  obeniusNu(4, f) 
│ │ │ │  0000ee00: 2d2d 2043 6f6e 7461 696e 6d65 6e74 5465  -- ContainmentTe
│ │ │ │  0000ee10: 7374 2069 7320 7365 7420 746f 2046 726f  st is set to Fro
│ │ │ │  0000ee20: 6265 6e69 7573 526f 6f74 2c20 6279 2020  beniusRoot, by  
│ │ │ │ -0000ee30: 7c0a 7c20 2d2d 2075 7365 6420 302e 3237  |.| -- used 0.27
│ │ │ │ -0000ee40: 3039 3739 7320 2863 7075 293b 2030 2e31  0979s (cpu); 0.1
│ │ │ │ -0000ee50: 3834 3236 3973 2028 7468 7265 6164 293b  84269s (thread);
│ │ │ │ +0000ee30: 7c0a 7c20 2d2d 2075 7365 6420 302e 3435  |.| -- used 0.45
│ │ │ │ +0000ee40: 3534 3639 7320 2863 7075 293b 2030 2e32  5469s (cpu); 0.2
│ │ │ │ +0000ee50: 3731 3832 3473 2028 7468 7265 6164 293b  71824s (thread);
│ │ │ │  0000ee60: 2030 7320 2867 6329 2020 2020 2020 2020   0s (gc)        
│ │ │ │  0000ee70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ee80: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0000ee90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000eea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000eeb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000eec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -3840,17 +3840,17 @@
│ │ │ │  0000eff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000f000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000f010: 2b0a 7c69 3230 203a 2074 696d 6520 6672  +.|i20 : time fr
│ │ │ │  0000f020: 6f62 656e 6975 734e 7528 342c 2066 2c20  obeniusNu(4, f, 
│ │ │ │  0000f030: 436f 6e74 6169 6e6d 656e 7454 6573 7420  ContainmentTest 
│ │ │ │  0000f040: 3d3e 2053 7461 6e64 6172 6450 6f77 6572  => StandardPower
│ │ │ │  0000f050: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -0000f060: 7c0a 7c20 2d2d 2075 7365 6420 312e 3531  |.| -- used 1.51
│ │ │ │ -0000f070: 3637 3773 2028 6370 7529 3b20 312e 3133  677s (cpu); 1.13
│ │ │ │ -0000f080: 3234 3773 2028 7468 7265 6164 293b 2030  247s (thread); 0
│ │ │ │ +0000f060: 7c0a 7c20 2d2d 2075 7365 6420 312e 3433  |.| -- used 1.43
│ │ │ │ +0000f070: 3635 3173 2028 6370 7529 3b20 312e 3233  651s (cpu); 1.23
│ │ │ │ +0000f080: 3633 3573 2028 7468 7265 6164 293b 2030  635s (thread); 0
│ │ │ │  0000f090: 7320 2867 6329 2020 2020 2020 2020 2020  s (gc)          
│ │ │ │  0000f0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f0b0: 7c0a 7c20 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -4015,17 +4015,17 @@
│ │ │ │  0000fae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000faf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  0000fb00: 0a7c 6932 3720 3a20 7469 6d65 2066 726f  .|i27 : time fro
│ │ │ │  0000fb10: 6265 6e69 7573 4e75 2835 2c20 6629 202d  beniusNu(5, f) -
│ │ │ │  0000fb20: 2d20 7573 6573 2062 696e 6172 7920 7365  - uses binary se
│ │ │ │  0000fb30: 6172 6368 2028 6465 6661 756c 7429 2020  arch (default)  
│ │ │ │  0000fb40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000fb50: 0a7c 202d 2d20 7573 6564 2031 2e30 3537  .| -- used 1.057
│ │ │ │ -0000fb60: 3535 7320 2863 7075 293b 2030 2e36 3935  55s (cpu); 0.695
│ │ │ │ -0000fb70: 3335 3473 2028 7468 7265 6164 293b 2030  354s (thread); 0
│ │ │ │ +0000fb50: 0a7c 202d 2d20 7573 6564 2031 2e31 3636  .| -- used 1.166
│ │ │ │ +0000fb60: 3831 7320 2863 7075 293b 2030 2e37 3232  81s (cpu); 0.722
│ │ │ │ +0000fb70: 3533 3873 2028 7468 7265 6164 293b 2030  538s (thread); 0
│ │ │ │  0000fb80: 7320 2867 6329 2020 2020 2020 2020 2020  s (gc)          
│ │ │ │  0000fb90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000fba0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000fbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fbe0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ @@ -4040,18 +4040,18 @@
│ │ │ │  0000fc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000fc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  0000fc90: 0a7c 6932 3820 3a20 7469 6d65 2066 726f  .|i28 : time fro
│ │ │ │  0000fca0: 6265 6e69 7573 4e75 2835 2c20 662c 2053  beniusNu(5, f, S
│ │ │ │  0000fcb0: 6561 7263 6820 3d3e 204c 696e 6561 7229  earch => Linear)
│ │ │ │  0000fcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fcd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000fce0: 0a7c 202d 2d20 7573 6564 2031 2e35 3135  .| -- used 1.515
│ │ │ │ -0000fcf0: 3732 7320 2863 7075 293b 2030 2e39 3531  72s (cpu); 0.951
│ │ │ │ -0000fd00: 3931 3573 2028 7468 7265 6164 293b 2030  915s (thread); 0
│ │ │ │ -0000fd10: 7320 2867 6329 2020 2020 2020 2020 2020  s (gc)          
│ │ │ │ +0000fce0: 0a7c 202d 2d20 7573 6564 2031 2e37 3137  .| -- used 1.717
│ │ │ │ +0000fcf0: 3173 2028 6370 7529 3b20 312e 3033 3636  1s (cpu); 1.0366
│ │ │ │ +0000fd00: 3473 2028 7468 7265 6164 293b 2030 7320  4s (thread); 0s 
│ │ │ │ +0000fd10: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │  0000fd20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000fd30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000fd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fd70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000fd80: 0a7c 6f32 3820 3d20 3131 3234 2020 2020  .|o28 = 1124    
│ │ │ │ @@ -4085,18 +4085,18 @@
│ │ │ │  0000ff40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000ff50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  0000ff60: 0a7c 6933 3020 3a20 7469 6d65 2066 726f  .|i30 : time fro
│ │ │ │  0000ff70: 6265 6e69 7573 4e75 2832 2c20 4d2c 204d  beniusNu(2, M, M
│ │ │ │  0000ff80: 5e32 2920 2d2d 2075 7365 7320 6269 6e61  ^2) -- uses bina
│ │ │ │  0000ff90: 7279 2073 6561 7263 6820 2864 6566 6175  ry search (defau
│ │ │ │  0000ffa0: 6c74 2920 2020 2020 2020 2020 2020 207c  lt)            |
│ │ │ │ -0000ffb0: 0a7c 202d 2d20 7573 6564 2031 2e37 3236  .| -- used 1.726
│ │ │ │ -0000ffc0: 3433 7320 2863 7075 293b 2031 2e33 3835  43s (cpu); 1.385
│ │ │ │ -0000ffd0: 3832 7320 2874 6872 6561 6429 3b20 3073  82s (thread); 0s
│ │ │ │ -0000ffe0: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │ +0000ffb0: 0a7c 202d 2d20 7573 6564 2031 2e37 3332  .| -- used 1.732
│ │ │ │ +0000ffc0: 3434 7320 2863 7075 293b 2031 2e34 3338  44s (cpu); 1.438
│ │ │ │ +0000ffd0: 3673 2028 7468 7265 6164 293b 2030 7320  6s (thread); 0s 
│ │ │ │ +0000ffe0: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │  0000fff0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00010000: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00010010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010040: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00010050: 0a7c 6f33 3020 3d20 3937 2020 2020 2020  .|o30 = 97      
│ │ │ │ @@ -4110,17 +4110,17 @@
│ │ │ │  000100d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000100e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  000100f0: 0a7c 6933 3120 3a20 7469 6d65 2066 726f  .|i31 : time fro
│ │ │ │  00010100: 6265 6e69 7573 4e75 2832 2c20 4d2c 204d  beniusNu(2, M, M
│ │ │ │  00010110: 5e32 2c20 5365 6172 6368 203d 3e20 4c69  ^2, Search => Li
│ │ │ │  00010120: 6e65 6172 2920 2d2d 2062 7574 206c 696e  near) -- but lin
│ │ │ │  00010130: 6561 7220 7365 6172 6368 2067 6574 737c  ear search gets|
│ │ │ │ -00010140: 0a7c 202d 2d20 7573 6564 2030 2e36 3133  .| -- used 0.613
│ │ │ │ -00010150: 3132 3473 2028 6370 7529 3b20 302e 3531  124s (cpu); 0.51
│ │ │ │ -00010160: 3536 3035 7320 2874 6872 6561 6429 3b20  5605s (thread); 
│ │ │ │ +00010140: 0a7c 202d 2d20 7573 6564 2030 2e35 3535  .| -- used 0.555
│ │ │ │ +00010150: 3235 3173 2028 6370 7529 3b20 302e 3439  251s (cpu); 0.49
│ │ │ │ +00010160: 3331 3234 7320 2874 6872 6561 6429 3b20  3124s (thread); 
│ │ │ │  00010170: 3073 2028 6763 2920 2020 2020 2020 2020  0s (gc)         
│ │ │ │  00010180: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00010190: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000101a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000101b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000101c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000101d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ ├── ./usr/share/info/GKMVarieties.info.gz
│ │ │ ├── GKMVarieties.info
│ │ │ │ @@ -17563,16 +17563,16 @@
│ │ │ │  000449a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000449b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000449c0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3720  ---------+.|i27 
│ │ │ │  000449d0: 3a20 7469 6d65 2043 203d 206f 7262 6974  : time C = orbit
│ │ │ │  000449e0: 436c 6f73 7572 6528 582c 4d61 7429 2020  Closure(X,Mat)  
│ │ │ │  000449f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044a00: 2020 2020 2020 2020 207c 0a7c 202d 2d20           |.| -- 
│ │ │ │ -00044a10: 7573 6564 2030 2e35 3933 3831 7320 2863  used 0.59381s (c
│ │ │ │ -00044a20: 7075 293b 2030 2e33 3437 3038 3673 2028  pu); 0.347086s (
│ │ │ │ +00044a10: 7573 6564 2031 2e33 3134 3034 7320 2863  used 1.31404s (c
│ │ │ │ +00044a20: 7075 293b 2030 2e34 3030 3636 3973 2028  pu); 0.400669s (
│ │ │ │  00044a30: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 0s (gc)
│ │ │ │  00044a40: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00044a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044a80: 2020 2020 2020 2020 207c 0a7c 6f32 3720           |.|o27 
│ │ │ │  00044a90: 3d20 616e 2022 6571 7569 7661 7269 616e  = an "equivarian
│ │ │ │ @@ -17591,16 +17591,16 @@
│ │ │ │  00044b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00044b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00044b80: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3820  ---------+.|i28 
│ │ │ │  00044b90: 3a20 7469 6d65 2043 203d 206f 7262 6974  : time C = orbit
│ │ │ │  00044ba0: 436c 6f73 7572 6528 582c 4d61 742c 2052  Closure(X,Mat, R
│ │ │ │  00044bb0: 5245 464d 6574 686f 6420 3d3e 2074 7275  REFMethod => tru
│ │ │ │  00044bc0: 6529 2020 2020 2020 207c 0a7c 202d 2d20  e)       |.| -- 
│ │ │ │ -00044bd0: 7573 6564 2031 2e38 3034 3734 7320 2863  used 1.80474s (c
│ │ │ │ -00044be0: 7075 293b 2031 2e30 3337 3039 7320 2874  pu); 1.03709s (t
│ │ │ │ +00044bd0: 7573 6564 2032 2e37 3835 3737 7320 2863  used 2.78577s (c
│ │ │ │ +00044be0: 7075 293b 2031 2e30 3131 3732 7320 2874  pu); 1.01172s (t
│ │ │ │  00044bf0: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │  00044c00: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00044c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044c40: 2020 2020 2020 2020 207c 0a7c 6f32 3820           |.|o28 
│ │ │ │  00044c50: 3d20 616e 2022 6571 7569 7661 7269 616e  = an "equivarian
│ │ ├── ./usr/share/info/Graphs.info.gz
│ │ │ ├── Graphs.info
│ │ │ │ @@ -18078,15 +18078,15 @@
│ │ │ │  000469d0: 7973 2048 2020 2020 2020 2020 2020 2020  ys H            
│ │ │ │  000469e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000469f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00046a00: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00046a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046a30: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20          |.|o3 = 
│ │ │ │ -00046a40: 7b6d 6170 2c20 6469 6772 6170 682c 206e  {map, digraph, n
│ │ │ │ +00046a40: 7b64 6967 7261 7068 2c20 6d61 702c 206e  {digraph, map, n
│ │ │ │  00046a50: 6577 4469 6772 6170 687d 2020 2020 2020  ewDigraph}      
│ │ │ │  00046a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046a70: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00046a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046aa0: 2020 2020 2020 2020 2020 7c0a 7c6f 3320            |.|o3 
│ │ │ │  00046ab0: 3a20 4c69 7374 2020 2020 2020 2020 2020  : List
│ │ ├── ./usr/share/info/GroebnerWalk.info.gz
│ │ │ ├── GroebnerWalk.info
│ │ │ │ @@ -207,16 +207,16 @@
│ │ │ │  00000ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00000cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00000d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  00000d10: 7c69 3520 3a20 656c 6170 7365 6454 696d  |i5 : elapsedTim
│ │ │ │  00000d20: 6520 6762 2049 3220 2020 2020 2020 2020  e gb I2         
│ │ │ │  00000d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00000d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00000d50: 2020 2020 2020 7c0a 7c20 2d2d 2032 2e38        |.| -- 2.8
│ │ │ │ -00000d60: 3136 3034 7320 656c 6170 7365 6420 2020  1604s elapsed   
│ │ │ │ +00000d50: 2020 2020 2020 7c0a 7c20 2d2d 2032 2e31        |.| -- 2.1
│ │ │ │ +00000d60: 3334 3173 2065 6c61 7073 6564 2020 2020  341s elapsed    
│ │ │ │  00000d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00000d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00000d90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  00000da0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00000db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00000dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00000dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -249,15 +249,15 @@
│ │ │ │  00000f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00000f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00000fa0: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 2065  -------+.|i6 : e
│ │ │ │  00000fb0: 6c61 7073 6564 5469 6d65 2067 726f 6562  lapsedTime groeb
│ │ │ │  00000fc0: 6e65 7257 616c 6b28 6762 2049 312c 2052  nerWalk(gb I1, R
│ │ │ │  00000fd0: 3229 2020 2020 2020 2020 2020 2020 2020  2)              
│ │ │ │  00000fe0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00000ff0: 7c20 2d2d 2032 2e34 3537 3436 7320 656c  | -- 2.45746s el
│ │ │ │ +00000ff0: 7c20 2d2d 2031 2e37 3736 3131 7320 656c  | -- 1.77611s el
│ │ │ │  00001000: 6170 7365 6420 2020 2020 2020 2020 2020  apsed           
│ │ │ │  00001010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001030: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00001040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001060: 2020 2020 2020 2020 2020 2020 2020 2020
│ │ ├── ./usr/share/info/Hadamard.info.gz
│ │ │ ├── Hadamard.info
│ │ │ │ @@ -580,1432 +580,1442 @@
│ │ │ │  00002430: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00002440: 2020 2020 2020 2020 2031 2020 2020 2020           1      
│ │ │ │  00002450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002480: 2020 2020 7c0a 7c6f 3220 3d20 7b50 6f69      |.|o2 = {Poi
│ │ │ │  00002490: 6e74 7b31 2c20 302c 202d 7d2c 2050 6f69  nt{1, 0, -}, Poi
│ │ │ │ -000024a0: 6e74 7b31 2c20 382c 2036 347d 2c20 506f  nt{1, 8, 64}, Po
│ │ │ │ -000024b0: 696e 747b 312c 2034 2c20 387d 2c20 506f  int{1, 4, 8}, Po
│ │ │ │ -000024c0: 696e 747b 312c 2030 2c20 3136 7d2c 2020  int{1, 0, 16},  
│ │ │ │ +000024a0: 6e74 7b31 2c20 302c 2032 7d2c 2050 6f69  nt{1, 0, 2}, Poi
│ │ │ │ +000024b0: 6e74 7b31 2c20 322c 2031 7d2c 2050 6f69  nt{1, 2, 1}, Poi
│ │ │ │ +000024c0: 6e74 7b31 2c20 302c 2034 7d2c 2020 2020  nt{1, 0, 4},    
│ │ │ │  000024d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000024e0: 2020 2020 2020 2020 2034 2020 2020 2020           4      
│ │ │ │ +000024e0: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │  000024f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002520: 2020 2020 7c0a 7c20 2020 2020 2d2d 2d2d      |.|     ----
│ │ │ │  00002530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002570: 2d2d 2d2d 7c0a 7c20 2020 2020 2020 2020  ----|.|         
│ │ │ │ -00002580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00002590: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ +00002580: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ +00002590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000025a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000025b0: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ +000025b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000025c0: 2020 2020 7c0a 7c20 2020 2020 506f 696e      |.|     Poin
│ │ │ │ -000025d0: 747b 312c 2030 2c20 317d 2c20 506f 696e  t{1, 0, 1}, Poin
│ │ │ │ -000025e0: 747b 312c 2031 2c20 2d7d 2c20 506f 696e  t{1, 1, -}, Poin
│ │ │ │ -000025f0: 747b 312c 2030 2c20 327d 2c20 506f 696e  t{1, 0, 2}, Poin
│ │ │ │ -00002600: 747b 312c 2030 2c20 2d7d 2c20 506f 696e  t{1, 0, -}, Poin
│ │ │ │ -00002610: 747b 312c 7c0a 7c20 2020 2020 2020 2020  t{1,|.|         
│ │ │ │ -00002620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00002630: 2020 2020 2020 2020 3820 2020 2020 2020          8       
│ │ │ │ +000025d0: 747b 312c 2030 2c20 2d7d 2c20 506f 696e  t{1, 0, -}, Poin
│ │ │ │ +000025e0: 747b 312c 2038 2c20 3634 7d2c 2050 6f69  t{1, 8, 64}, Poi
│ │ │ │ +000025f0: 6e74 7b31 2c20 342c 2038 7d2c 2050 6f69  nt{1, 4, 8}, Poi
│ │ │ │ +00002600: 6e74 7b31 2c20 302c 2031 367d 2c20 2020  nt{1, 0, 16},   
│ │ │ │ +00002610: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00002620: 2020 2020 2020 2020 3420 2020 2020 2020          4       
│ │ │ │ +00002630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00002650: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +00002650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002660: 2020 2020 7c0a 7c20 2020 2020 2d2d 2d2d      |.|     ----
│ │ │ │  00002670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000026a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000026b0: 2d2d 2d2d 7c0a 7c20 2020 2020 322c 2031  ----|.|     2, 1
│ │ │ │ -000026c0: 7d2c 2050 6f69 6e74 7b31 2c20 302c 2034  }, Point{1, 0, 4
│ │ │ │ -000026d0: 7d7d 2020 2020 2020 2020 2020 2020 2020  }}              
│ │ │ │ +000026b0: 2d2d 2d2d 7c0a 7c20 2020 2020 2020 2020  ----|.|         
│ │ │ │ +000026c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000026d0: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │  000026e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000026f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00002700: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00002710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00002720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00002700: 2020 2020 7c0a 7c20 2020 2020 506f 696e      |.|     Poin
│ │ │ │ +00002710: 747b 312c 2030 2c20 317d 2c20 506f 696e  t{1, 0, 1}, Poin
│ │ │ │ +00002720: 747b 312c 2031 2c20 2d7d 7d20 2020 2020  t{1, 1, -}}     
│ │ │ │  00002730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00002750: 2020 2020 7c0a 7c6f 3220 3a20 4c69 7374      |.|o2 : List
│ │ │ │ +00002750: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00002760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00002770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00002770: 2020 2020 2020 2020 3820 2020 2020 2020          8       
│ │ │ │  00002780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000027a0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -000027b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000027c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000027d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000027e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000027f0: 2d2d 2d2d 2b0a 0a57 6179 7320 746f 2075  ----+..Ways to u
│ │ │ │ -00002800: 7365 2074 6869 7320 6d65 7468 6f64 3a0a  se this method:.
│ │ │ │ -00002810: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00002820: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ -00002830: 6f74 6520 6861 6461 6d61 7264 506f 7765  ote hadamardPowe
│ │ │ │ -00002840: 7228 4c69 7374 2c5a 5a29 3a20 6861 6461  r(List,ZZ): hada
│ │ │ │ -00002850: 6d61 7264 506f 7765 725f 6c70 4c69 7374  mardPower_lpList
│ │ │ │ -00002860: 5f63 6d5a 5a5f 7270 2c20 2d2d 2063 6f6d  _cmZZ_rp, -- com
│ │ │ │ -00002870: 7075 7465 7320 7468 650a 2020 2020 2472  putes the.    $r
│ │ │ │ -00002880: 242d 7468 2048 6164 6d61 7264 2070 6f77  $-th Hadmard pow
│ │ │ │ -00002890: 6572 7320 6f66 2061 2073 6574 2070 6f69  ers of a set poi
│ │ │ │ -000028a0: 6e74 730a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  nts.------------
│ │ │ │ -000028b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000028c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000028d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000028e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000028f0: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520  ---..The source 
│ │ │ │ -00002900: 6f66 2074 6869 7320 646f 6375 6d65 6e74  of this document
│ │ │ │ -00002910: 2069 7320 696e 0a2f 6275 696c 642f 7265   is in./build/re
│ │ │ │ -00002920: 7072 6f64 7563 6962 6c65 2d70 6174 682f  producible-path/
│ │ │ │ -00002930: 6d61 6361 756c 6179 322d 312e 3235 2e31  macaulay2-1.25.1
│ │ │ │ -00002940: 312b 6473 2f4d 322f 4d61 6361 756c 6179  1+ds/M2/Macaulay
│ │ │ │ -00002950: 322f 7061 636b 6167 6573 2f48 6164 616d  2/packages/Hadam
│ │ │ │ -00002960: 6172 642e 6d32 0a3a 3339 383a 302e 0a1f  ard.m2.:398:0...
│ │ │ │ -00002970: 0a46 696c 653a 2048 6164 616d 6172 642e  .File: Hadamard.
│ │ │ │ -00002980: 696e 666f 2c20 4e6f 6465 3a20 6861 6461  info, Node: hada
│ │ │ │ -00002990: 6d61 7264 5072 6f64 7563 742c 204e 6578  mardProduct, Nex
│ │ │ │ -000029a0: 743a 2068 6164 616d 6172 6450 726f 6475  t: hadamardProdu
│ │ │ │ -000029b0: 6374 5f6c 7049 6465 616c 5f63 6d49 6465  ct_lpIdeal_cmIde
│ │ │ │ -000029c0: 616c 5f72 702c 2050 7265 763a 2068 6164  al_rp, Prev: had
│ │ │ │ -000029d0: 616d 6172 6450 6f77 6572 5f6c 704c 6973  amardPower_lpLis
│ │ │ │ -000029e0: 745f 636d 5a5a 5f72 702c 2055 703a 2054  t_cmZZ_rp, Up: T
│ │ │ │ -000029f0: 6f70 0a0a 6861 6461 6d61 7264 5072 6f64  op..hadamardProd
│ │ │ │ -00002a00: 7563 7420 2d2d 2063 6f6d 7075 7465 7320  uct -- computes 
│ │ │ │ -00002a10: 7468 6520 4861 6461 6d61 7264 2070 726f  the Hadamard pro
│ │ │ │ -00002a20: 6475 6374 206f 6620 7661 7269 6574 6965  duct of varietie
│ │ │ │ -00002a30: 730a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  s.**************
│ │ │ │ -00002a40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00002a50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00002a60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a  ***************.
│ │ │ │ -00002a70: 0a57 6179 7320 746f 2075 7365 2068 6164  .Ways to use had
│ │ │ │ -00002a80: 616d 6172 6450 726f 6475 6374 3a0a 3d3d  amardProduct:.==
│ │ │ │ -00002a90: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00002aa0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ -00002ab0: 2a6e 6f74 6520 6861 6461 6d61 7264 5072  *note hadamardPr
│ │ │ │ -00002ac0: 6f64 7563 7428 4964 6561 6c2c 4964 6561  oduct(Ideal,Idea
│ │ │ │ -00002ad0: 6c29 3a20 6861 6461 6d61 7264 5072 6f64  l): hadamardProd
│ │ │ │ -00002ae0: 7563 745f 6c70 4964 6561 6c5f 636d 4964  uct_lpIdeal_cmId
│ │ │ │ -00002af0: 6561 6c5f 7270 2c20 2d2d 0a20 2020 2048  eal_rp, --.    H
│ │ │ │ -00002b00: 6164 616d 6172 6420 7072 6f64 7563 7420  adamard product 
│ │ │ │ -00002b10: 6f66 2074 776f 2068 6f6d 6f67 656e 656f  of two homogeneo
│ │ │ │ -00002b20: 7573 2069 6465 616c 730a 2020 2a20 2a6e  us ideals.  * *n
│ │ │ │ -00002b30: 6f74 6520 6861 6461 6d61 7264 5072 6f64  ote hadamardProd
│ │ │ │ -00002b40: 7563 7428 4c69 7374 293a 2068 6164 616d  uct(List): hadam
│ │ │ │ -00002b50: 6172 6450 726f 6475 6374 5f6c 704c 6973  ardProduct_lpLis
│ │ │ │ -00002b60: 745f 7270 2c20 2d2d 2048 6164 616d 6172  t_rp, -- Hadamar
│ │ │ │ -00002b70: 6420 7072 6f64 7563 740a 2020 2020 6f66  d product.    of
│ │ │ │ -00002b80: 2061 206c 6973 7420 6f66 2068 6f6d 6f67   a list of homog
│ │ │ │ -00002b90: 656e 656f 7573 2069 6465 616c 732c 206f  eneous ideals, o
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│ │ │ │ -00002bd0: 6164 616d 6172 6450 726f 6475 6374 5f6c  adamardProduct_l
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│ │ │ │ -00002bf0: 202d 2d0a 2020 2020 4861 6461 6d61 7264   --.    Hadamard
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│ │ │ │ -00002c40: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563  =====..The objec
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│ │ │ │ -00002c60: 5072 6f64 7563 743a 2068 6164 616d 6172  Product: hadamar
│ │ │ │ -00002c70: 6450 726f 6475 6374 2c20 6973 2061 202a  dProduct, is a *
│ │ │ │ -00002c80: 6e6f 7465 206d 6574 686f 6420 6675 6e63  note method func
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│ │ │ │ -00002ca0: 446f 6329 4d65 7468 6f64 4675 6e63 7469  Doc)MethodFuncti
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│ │ │ │ -00002ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00002cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00002e20: 2d2d 2048 6164 616d 6172 6420 7072 6f64  -- Hadamard prod
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│ │ │ │ -00002e50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00002e60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00002e70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00002e80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00002e90: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 4675  ********..  * Fu
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│ │ │ │ -00002ee0: 2020 2020 6861 6461 6d61 7264 5072 6f64      hadamardProd
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│ │ │ │ -000031b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -000031f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +000027a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000027b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000027c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000027d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000027e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000027f0: 2020 2020 7c0a 7c6f 3220 3a20 4c69 7374      |.|o2 : List
│ │ │ │ +00002800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00002810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00002820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00002830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00002840: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00002850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00002860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00002870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00002880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00002890: 2d2d 2d2d 2b0a 0a57 6179 7320 746f 2075  ----+..Ways to u
│ │ │ │ +000028a0: 7365 2074 6869 7320 6d65 7468 6f64 3a0a  se this method:.
│ │ │ │ +000028b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +000028c0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ +000028d0: 6f74 6520 6861 6461 6d61 7264 506f 7765  ote hadamardPowe
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│ │ │ │ +000028f0: 6d61 7264 506f 7765 725f 6c70 4c69 7374  mardPower_lpList
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│ │ │ │ +00002910: 7075 7465 7320 7468 650a 2020 2020 2472  putes the.    $r
│ │ │ │ +00002920: 242d 7468 2048 6164 6d61 7264 2070 6f77  $-th Hadmard pow
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│ │ │ │ +00002950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00002960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00002970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00002980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00002990: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520  ---..The source 
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│ │ │ │ +000029b0: 2069 7320 696e 0a2f 6275 696c 642f 7265   is in./build/re
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│ │ │ │ +000029d0: 6d61 6361 756c 6179 322d 312e 3235 2e31  macaulay2-1.25.1
│ │ │ │ +000029e0: 312b 6473 2f4d 322f 4d61 6361 756c 6179  1+ds/M2/Macaulay
│ │ │ │ +000029f0: 322f 7061 636b 6167 6573 2f48 6164 616d  2/packages/Hadam
│ │ │ │ +00002a00: 6172 642e 6d32 0a3a 3339 383a 302e 0a1f  ard.m2.:398:0...
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│ │ │ │ +00002a50: 6374 5f6c 7049 6465 616c 5f63 6d49 6465  ct_lpIdeal_cmIde
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│ │ │ │ +00002a70: 616d 6172 6450 6f77 6572 5f6c 704c 6973  amardPower_lpLis
│ │ │ │ +00002a80: 745f 636d 5a5a 5f72 702c 2055 703a 2054  t_cmZZ_rp, Up: T
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│ │ │ │ +00002aa0: 7563 7420 2d2d 2063 6f6d 7075 7465 7320  uct -- computes 
│ │ │ │ +00002ab0: 7468 6520 4861 6461 6d61 7264 2070 726f  the Hadamard pro
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│ │ │ │ +00002ad0: 730a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  s.**************
│ │ │ │ +00002ae0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00002af0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00002b00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a  ***************.
│ │ │ │ +00002b10: 0a57 6179 7320 746f 2075 7365 2068 6164  .Ways to use had
│ │ │ │ +00002b20: 616d 6172 6450 726f 6475 6374 3a0a 3d3d  amardProduct:.==
│ │ │ │ +00002b30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00002b40: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ +00002b50: 2a6e 6f74 6520 6861 6461 6d61 7264 5072  *note hadamardPr
│ │ │ │ +00002b60: 6f64 7563 7428 4964 6561 6c2c 4964 6561  oduct(Ideal,Idea
│ │ │ │ +00002b70: 6c29 3a20 6861 6461 6d61 7264 5072 6f64  l): hadamardProd
│ │ │ │ +00002b80: 7563 745f 6c70 4964 6561 6c5f 636d 4964  uct_lpIdeal_cmId
│ │ │ │ +00002b90: 6561 6c5f 7270 2c20 2d2d 0a20 2020 2048  eal_rp, --.    H
│ │ │ │ +00002ba0: 6164 616d 6172 6420 7072 6f64 7563 7420  adamard product 
│ │ │ │ +00002bb0: 6f66 2074 776f 2068 6f6d 6f67 656e 656f  of two homogeneo
│ │ │ │ +00002bc0: 7573 2069 6465 616c 730a 2020 2a20 2a6e  us ideals.  * *n
│ │ │ │ +00002bd0: 6f74 6520 6861 6461 6d61 7264 5072 6f64  ote hadamardProd
│ │ │ │ +00002be0: 7563 7428 4c69 7374 293a 2068 6164 616d  uct(List): hadam
│ │ │ │ +00002bf0: 6172 6450 726f 6475 6374 5f6c 704c 6973  ardProduct_lpLis
│ │ │ │ +00002c00: 745f 7270 2c20 2d2d 2048 6164 616d 6172  t_rp, -- Hadamar
│ │ │ │ +00002c10: 6420 7072 6f64 7563 740a 2020 2020 6f66  d product.    of
│ │ │ │ +00002c20: 2061 206c 6973 7420 6f66 2068 6f6d 6f67   a list of homog
│ │ │ │ +00002c30: 656e 656f 7573 2069 6465 616c 732c 206f  eneous ideals, o
│ │ │ │ +00002c40: 7220 706f 696e 7473 0a20 202a 202a 6e6f  r points.  * *no
│ │ │ │ +00002c50: 7465 2068 6164 616d 6172 6450 726f 6475  te hadamardProdu
│ │ │ │ +00002c60: 6374 284c 6973 742c 4c69 7374 293a 2068  ct(List,List): h
│ │ │ │ +00002c70: 6164 616d 6172 6450 726f 6475 6374 5f6c  adamardProduct_l
│ │ │ │ +00002c80: 704c 6973 745f 636d 4c69 7374 5f72 702c  pList_cmList_rp,
│ │ │ │ +00002c90: 202d 2d0a 2020 2020 4861 6461 6d61 7264   --.    Hadamard
│ │ │ │ +00002ca0: 2070 726f 6475 6374 206f 6620 7477 6f20   product of two 
│ │ │ │ +00002cb0: 7365 7473 206f 6620 706f 696e 7473 0a0a  sets of points..
│ │ │ │ +00002cc0: 466f 7220 7468 6520 7072 6f67 7261 6d6d  For the programm
│ │ │ │ +00002cd0: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  er.=============
│ │ │ │ +00002ce0: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563  =====..The objec
│ │ │ │ +00002cf0: 7420 2a6e 6f74 6520 6861 6461 6d61 7264  t *note hadamard
│ │ │ │ +00002d00: 5072 6f64 7563 743a 2068 6164 616d 6172  Product: hadamar
│ │ │ │ +00002d10: 6450 726f 6475 6374 2c20 6973 2061 202a  dProduct, is a *
│ │ │ │ +00002d20: 6e6f 7465 206d 6574 686f 6420 6675 6e63  note method func
│ │ │ │ +00002d30: 7469 6f6e 3a0a 284d 6163 6175 6c61 7932  tion:.(Macaulay2
│ │ │ │ +00002d40: 446f 6329 4d65 7468 6f64 4675 6e63 7469  Doc)MethodFuncti
│ │ │ │ +00002d50: 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d  on,...----------
│ │ │ │ +00002d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00002d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00002d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00002d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00002da0: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263  -----..The sourc
│ │ │ │ +00002db0: 6520 6f66 2074 6869 7320 646f 6375 6d65  e of this docume
│ │ │ │ +00002dc0: 6e74 2069 7320 696e 0a2f 6275 696c 642f  nt is in./build/
│ │ │ │ +00002dd0: 7265 7072 6f64 7563 6962 6c65 2d70 6174  reproducible-pat
│ │ │ │ +00002de0: 682f 6d61 6361 756c 6179 322d 312e 3235  h/macaulay2-1.25
│ │ │ │ +00002df0: 2e31 312b 6473 2f4d 322f 4d61 6361 756c  .11+ds/M2/Macaul
│ │ │ │ +00002e00: 6179 322f 7061 636b 6167 6573 2f48 6164  ay2/packages/Had
│ │ │ │ +00002e10: 616d 6172 642e 6d32 0a3a 3235 313a 302e  amard.m2.:251:0.
│ │ │ │ +00002e20: 0a1f 0a46 696c 653a 2048 6164 616d 6172  ...File: Hadamar
│ │ │ │ +00002e30: 642e 696e 666f 2c20 4e6f 6465 3a20 6861  d.info, Node: ha
│ │ │ │ +00002e40: 6461 6d61 7264 5072 6f64 7563 745f 6c70  damardProduct_lp
│ │ │ │ +00002e50: 4964 6561 6c5f 636d 4964 6561 6c5f 7270  Ideal_cmIdeal_rp
│ │ │ │ +00002e60: 2c20 4e65 7874 3a20 6861 6461 6d61 7264  , Next: hadamard
│ │ │ │ +00002e70: 5072 6f64 7563 745f 6c70 4c69 7374 5f72  Product_lpList_r
│ │ │ │ +00002e80: 702c 2050 7265 763a 2068 6164 616d 6172  p, Prev: hadamar
│ │ │ │ +00002e90: 6450 726f 6475 6374 2c20 5570 3a20 546f  dProduct, Up: To
│ │ │ │ +00002ea0: 700a 0a68 6164 616d 6172 6450 726f 6475  p..hadamardProdu
│ │ │ │ +00002eb0: 6374 2849 6465 616c 2c49 6465 616c 2920  ct(Ideal,Ideal) 
│ │ │ │ +00002ec0: 2d2d 2048 6164 616d 6172 6420 7072 6f64  -- Hadamard prod
│ │ │ │ +00002ed0: 7563 7420 6f66 2074 776f 2068 6f6d 6f67  uct of two homog
│ │ │ │ +00002ee0: 656e 656f 7573 2069 6465 616c 730a 2a2a  eneous ideals.**
│ │ │ │ +00002ef0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00002f00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00002f10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00002f20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00002f30: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 4675  ********..  * Fu
│ │ │ │ +00002f40: 6e63 7469 6f6e 3a20 2a6e 6f74 6520 6861  nction: *note ha
│ │ │ │ +00002f50: 6461 6d61 7264 5072 6f64 7563 743a 2068  damardProduct: h
│ │ │ │ +00002f60: 6164 616d 6172 6450 726f 6475 6374 2c0a  adamardProduct,.
│ │ │ │ +00002f70: 2020 2a20 5573 6167 653a 200a 2020 2020    * Usage: .    
│ │ │ │ +00002f80: 2020 2020 6861 6461 6d61 7264 5072 6f64      hadamardProd
│ │ │ │ +00002f90: 7563 7428 492c 4a29 0a20 202a 2049 6e70  uct(I,J).  * Inp
│ │ │ │ +00002fa0: 7574 733a 0a20 2020 2020 202a 2049 2c20  uts:.      * I, 
│ │ │ │ +00002fb0: 616e 202a 6e6f 7465 2069 6465 616c 3a20  an *note ideal: 
│ │ │ │ +00002fc0: 284d 6163 6175 6c61 7932 446f 6329 4964  (Macaulay2Doc)Id
│ │ │ │ +00002fd0: 6561 6c2c 2c20 2868 6f6d 6f67 656e 656f  eal,, (homogeneo
│ │ │ │ +00002fe0: 7573 290a 2020 2020 2020 2a20 4a2c 2061  us).      * J, a
│ │ │ │ +00002ff0: 6e20 2a6e 6f74 6520 6964 6561 6c3a 2028  n *note ideal: (
│ │ │ │ +00003000: 4d61 6361 756c 6179 3244 6f63 2949 6465  Macaulay2Doc)Ide
│ │ │ │ +00003010: 616c 2c2c 2028 686f 6d6f 6765 6e65 6f75  al,, (homogeneou
│ │ │ │ +00003020: 7329 0a20 202a 204f 7574 7075 7473 3a0a  s).  * Outputs:.
│ │ │ │ +00003030: 2020 2020 2020 2a20 616e 202a 6e6f 7465        * an *note
│ │ │ │ +00003040: 2069 6465 616c 3a20 284d 6163 6175 6c61   ideal: (Macaula
│ │ │ │ +00003050: 7932 446f 6329 4964 6561 6c2c 2c20 0a0a  y2Doc)Ideal,, ..
│ │ │ │ +00003060: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
│ │ │ │ +00003070: 3d3d 3d3d 3d3d 3d0a 0a47 6976 656e 2074  =======..Given t
│ │ │ │ +00003080: 776f 2070 726f 6a65 6374 6976 6520 7375  wo projective su
│ │ │ │ +00003090: 6276 6172 6965 7469 6573 2024 5824 2061  bvarieties $X$ a
│ │ │ │ +000030a0: 6e64 2024 5924 2c20 7468 6569 7220 4861  nd $Y$, their Ha
│ │ │ │ +000030b0: 6461 6d61 7264 2070 726f 6475 6374 2069  damard product i
│ │ │ │ +000030c0: 730a 6465 6669 6e65 6420 6173 2074 6865  s.defined as the
│ │ │ │ +000030d0: 205a 6172 6973 6b69 2063 6c6f 7375 7265   Zariski closure
│ │ │ │ +000030e0: 206f 6620 7468 6520 7365 7420 6f66 2028   of the set of (
│ │ │ │ +000030f0: 7765 6c6c 2d64 6566 696e 6564 2920 656e  well-defined) en
│ │ │ │ +00003100: 7472 7977 6973 6520 7072 6f64 7563 7473  trywise products
│ │ │ │ +00003110: 0a6f 6620 7061 6972 7320 6f66 2070 6f69  .of pairs of poi
│ │ │ │ +00003120: 6e74 7320 696e 2074 6865 2063 6172 7465  nts in the carte
│ │ │ │ +00003130: 7369 616e 2070 726f 6475 6374 2024 5820  sian product $X 
│ │ │ │ +00003140: 5c74 696d 6573 2059 242e 2054 6869 7320  \times Y$. This 
│ │ │ │ +00003150: 6361 6e20 616c 736f 2062 650a 7265 6761  can also be.rega
│ │ │ │ +00003160: 7264 6564 2061 7320 7468 6520 696d 6167  rded as the imag
│ │ │ │ +00003170: 6520 6f66 2074 6865 2053 6567 7265 2070  e of the Segre p
│ │ │ │ +00003180: 726f 6475 6374 206f 6620 2458 205c 7469  roduct of $X \ti
│ │ │ │ +00003190: 6d65 7320 5924 2076 6961 2074 6865 206c  mes Y$ via the l
│ │ │ │ +000031a0: 696e 6561 720a 7072 6f6a 6563 7469 6f6e  inear.projection
│ │ │ │ +000031b0: 206f 6e20 7468 6520 247a 5f7b 6969 7d24   on the $z_{ii}$
│ │ │ │ +000031c0: 2063 6f6f 7264 696e 6174 6573 2e20 5468   coordinates. Th
│ │ │ │ +000031d0: 6520 6c61 7474 6572 2069 7320 7468 6520  e latter is the 
│ │ │ │ +000031e0: 7761 7920 7468 6520 6675 6e63 7469 6f6e  way the function
│ │ │ │ +000031f0: 2069 730a 696d 706c 656d 656e 7465 642e   is.implemented.
│ │ │ │ +00003200: 0a0a 436f 6e73 6964 6572 2066 6f72 2065  ..Consider for e
│ │ │ │ +00003210: 7861 6d70 6c65 2074 6865 2065 6e74 7279  xample the entry
│ │ │ │ +00003220: 7769 7365 2070 726f 6475 6374 206f 6620  wise product of 
│ │ │ │ +00003230: 7477 6f20 706f 696e 7473 2e0a 0a2b 2d2d  two points...+--
│ │ │ │  00003240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00003250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003260: 2d2d 2d2b 0a7c 6933 203a 2071 203d 2070  ---+.|i3 : q = p
│ │ │ │ -00003270: 6f69 6e74 207b 312c 2d31 2c2d 312c 2d31  oint {1,-1,-1,-1
│ │ │ │ -00003280: 7d3b 2020 2020 2020 2020 207c 0a2b 2d2d  };         |.+--
│ │ │ │ +00003260: 2d2d 2d2b 0a7c 6931 203a 2053 203d 2051  ---+.|i1 : S = Q
│ │ │ │ +00003270: 515b 782c 792c 7a2c 745d 3b20 2020 2020  Q[x,y,z,t];     
│ │ │ │ +00003280: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  00003290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000032a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000032b0: 2d2d 2d2b 0a7c 6934 203a 2069 6465 616c  ---+.|i4 : ideal
│ │ │ │ -000032c0: 4f66 5072 6f6a 6563 7469 7665 506f 696e  OfProjectivePoin
│ │ │ │ -000032d0: 7473 287b 702a 717d 2c53 297c 0a7c 2020  ts({p*q},S)|.|  
│ │ │ │ -000032e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000032f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003300: 2020 207c 0a7c 6f34 203d 2069 6465 616c     |.|o4 = ideal
│ │ │ │ -00003310: 2028 327a 202d 2074 2c20 3279 202d 2074   (2z - t, 2y - t
│ │ │ │ -00003320: 2c20 3278 202b 2074 2920 207c 0a7c 2020  , 2x + t)  |.|  
│ │ │ │ -00003330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003350: 2020 207c 0a7c 6f34 203a 2049 6465 616c     |.|o4 : Ideal
│ │ │ │ -00003360: 206f 6620 5320 2020 2020 2020 2020 2020   of S           
│ │ │ │ -00003370: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -00003380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000033a0: 2d2d 2d2b 0a0a 5468 6973 2063 616e 2062  ---+..This can b
│ │ │ │ -000033b0: 6520 636f 6d70 7574 6564 2061 6c73 6f20  e computed also 
│ │ │ │ -000033c0: 6672 6f6d 2074 6865 6972 2064 6566 696e  from their defin
│ │ │ │ -000033d0: 696e 6720 6964 6561 6c73 2061 7320 6578  ing ideals as ex
│ │ │ │ -000033e0: 706c 6169 6e65 642e 0a0a 2b2d 2d2d 2d2d  plained...+-----
│ │ │ │ -000033f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00003410: 7c69 3520 3a20 4950 203d 2069 6465 616c  |i5 : IP = ideal
│ │ │ │ -00003420: 2878 2d79 2c78 2d7a 2c32 2a78 2d74 2920  (x-y,x-z,2*x-t) 
│ │ │ │ -00003430: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00003440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003450: 2020 2020 2020 2020 2020 7c0a 7c6f 3520            |.|o5 
│ │ │ │ -00003460: 3d20 6964 6561 6c20 2878 202d 2079 2c20  = ideal (x - y, 
│ │ │ │ -00003470: 7820 2d20 7a2c 2032 7820 2d20 7429 2020  x - z, 2x - t)  
│ │ │ │ -00003480: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00003490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000034a0: 2020 2020 2020 7c0a 7c6f 3520 3a20 4964        |.|o5 : Id
│ │ │ │ -000034b0: 6561 6c20 6f66 2053 2020 2020 2020 2020  eal of S        
│ │ │ │ -000034c0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -000034d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000034e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000034f0: 2d2d 2b0a 7c69 3620 3a20 4951 203d 2069  --+.|i6 : IQ = i
│ │ │ │ -00003500: 6465 616c 2878 2b79 2c78 2b7a 2c78 2b74  deal(x+y,x+z,x+t
│ │ │ │ -00003510: 2920 2020 2020 2020 7c0a 7c20 2020 2020  )       |.|     
│ │ │ │ -00003520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003530: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00003540: 7c6f 3620 3d20 6964 6561 6c20 2878 202b  |o6 = ideal (x +
│ │ │ │ -00003550: 2079 2c20 7820 2b20 7a2c 2078 202b 2074   y, x + z, x + t
│ │ │ │ -00003560: 2920 2020 7c0a 7c20 2020 2020 2020 2020  )   |.|         
│ │ │ │ -00003570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003580: 2020 2020 2020 2020 2020 7c0a 7c6f 3620            |.|o6 
│ │ │ │ -00003590: 3a20 4964 6561 6c20 6f66 2053 2020 2020  : Ideal of S    
│ │ │ │ -000035a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000035b0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ -000035c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000035d0: 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20 6861  ------+.|i7 : ha
│ │ │ │ -000035e0: 6461 6d61 7264 5072 6f64 7563 7428 4950  damardProduct(IP
│ │ │ │ -000035f0: 2c49 5129 2020 2020 2020 2020 7c0a 7c20  ,IQ)        |.| 
│ │ │ │ -00003600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000032b0: 2d2d 2d2b 0a7c 6932 203a 2070 203d 2070  ---+.|i2 : p = p
│ │ │ │ +000032c0: 6f69 6e74 207b 312c 312c 312c 327d 3b20  oint {1,1,1,2}; 
│ │ │ │ +000032d0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +000032e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000032f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003300: 2d2d 2d2b 0a7c 6933 203a 2071 203d 2070  ---+.|i3 : q = p
│ │ │ │ +00003310: 6f69 6e74 207b 312c 2d31 2c2d 312c 2d31  oint {1,-1,-1,-1
│ │ │ │ +00003320: 7d3b 2020 2020 2020 2020 207c 0a2b 2d2d  };         |.+--
│ │ │ │ +00003330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003350: 2d2d 2d2b 0a7c 6934 203a 2069 6465 616c  ---+.|i4 : ideal
│ │ │ │ +00003360: 4f66 5072 6f6a 6563 7469 7665 506f 696e  OfProjectivePoin
│ │ │ │ +00003370: 7473 287b 702a 717d 2c53 297c 0a7c 2020  ts({p*q},S)|.|  
│ │ │ │ +00003380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00003390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000033a0: 2020 207c 0a7c 6f34 203d 2069 6465 616c     |.|o4 = ideal
│ │ │ │ +000033b0: 2028 327a 202d 2074 2c20 3279 202d 2074   (2z - t, 2y - t
│ │ │ │ +000033c0: 2c20 3278 202b 2074 2920 207c 0a7c 2020  , 2x + t)  |.|  
│ │ │ │ +000033d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000033e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000033f0: 2020 207c 0a7c 6f34 203a 2049 6465 616c     |.|o4 : Ideal
│ │ │ │ +00003400: 206f 6620 5320 2020 2020 2020 2020 2020   of S           
│ │ │ │ +00003410: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00003420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003440: 2d2d 2d2b 0a0a 5468 6973 2063 616e 2062  ---+..This can b
│ │ │ │ +00003450: 6520 636f 6d70 7574 6564 2061 6c73 6f20  e computed also 
│ │ │ │ +00003460: 6672 6f6d 2074 6865 6972 2064 6566 696e  from their defin
│ │ │ │ +00003470: 696e 6720 6964 6561 6c73 2061 7320 6578  ing ideals as ex
│ │ │ │ +00003480: 706c 6169 6e65 642e 0a0a 2b2d 2d2d 2d2d  plained...+-----
│ │ │ │ +00003490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000034a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000034b0: 7c69 3520 3a20 4950 203d 2069 6465 616c  |i5 : IP = ideal
│ │ │ │ +000034c0: 2878 2d79 2c78 2d7a 2c32 2a78 2d74 2920  (x-y,x-z,2*x-t) 
│ │ │ │ +000034d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000034e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000034f0: 2020 2020 2020 2020 2020 7c0a 7c6f 3520            |.|o5 
│ │ │ │ +00003500: 3d20 6964 6561 6c20 2878 202d 2079 2c20  = ideal (x - y, 
│ │ │ │ +00003510: 7820 2d20 7a2c 2032 7820 2d20 7429 2020  x - z, 2x - t)  
│ │ │ │ +00003520: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00003530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00003540: 2020 2020 2020 7c0a 7c6f 3520 3a20 4964        |.|o5 : Id
│ │ │ │ +00003550: 6561 6c20 6f66 2053 2020 2020 2020 2020  eal of S        
│ │ │ │ +00003560: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00003570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003590: 2d2d 2b0a 7c69 3620 3a20 4951 203d 2069  --+.|i6 : IQ = i
│ │ │ │ +000035a0: 6465 616c 2878 2b79 2c78 2b7a 2c78 2b74  deal(x+y,x+z,x+t
│ │ │ │ +000035b0: 2920 2020 2020 2020 7c0a 7c20 2020 2020  )       |.|     
│ │ │ │ +000035c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000035d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000035e0: 7c6f 3620 3d20 6964 6561 6c20 2878 202b  |o6 = ideal (x +
│ │ │ │ +000035f0: 2079 2c20 7820 2b20 7a2c 2078 202b 2074   y, x + z, x + t
│ │ │ │ +00003600: 2920 2020 7c0a 7c20 2020 2020 2020 2020  )   |.|         
│ │ │ │  00003610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003620: 2020 7c0a 7c6f 3720 3d20 6964 6561 6c20    |.|o7 = ideal 
│ │ │ │ -00003630: 2832 7a20 2d20 742c 2032 7920 2d20 742c  (2z - t, 2y - t,
│ │ │ │ -00003640: 2032 7820 2b20 7429 7c0a 7c20 2020 2020   2x + t)|.|     
│ │ │ │ -00003650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003660: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00003670: 7c6f 3720 3a20 4964 6561 6c20 6f66 2053  |o7 : Ideal of S
│ │ │ │ -00003680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003690: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -000036a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000036b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a57 6520  ----------+..We 
│ │ │ │ -000036c0: 6361 6e20 616c 736f 2063 6f6e 7369 6465  can also conside
│ │ │ │ -000036d0: 7220 4861 6461 6d61 7264 2070 726f 6475  r Hadamard produ
│ │ │ │ -000036e0: 6374 206f 6620 6869 6768 6572 2064 696d  ct of higher dim
│ │ │ │ -000036f0: 656e 7369 6f6e 616c 2076 6172 6965 7469  ensional varieti
│ │ │ │ -00003700: 6573 2e20 466f 720a 6578 616d 706c 652c  es. For.example,
│ │ │ │ -00003710: 2074 6865 2048 6164 616d 6172 6420 7072   the Hadamard pr
│ │ │ │ -00003720: 6f64 7563 7420 6f66 2074 776f 206c 696e  oduct of two lin
│ │ │ │ -00003730: 6573 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  es...+----------
│ │ │ │ +00003620: 2020 2020 2020 2020 2020 7c0a 7c6f 3620            |.|o6 
│ │ │ │ +00003630: 3a20 4964 6561 6c20 6f66 2053 2020 2020  : Ideal of S    
│ │ │ │ +00003640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00003650: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00003660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003670: 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20 6861  ------+.|i7 : ha
│ │ │ │ +00003680: 6461 6d61 7264 5072 6f64 7563 7428 4950  damardProduct(IP
│ │ │ │ +00003690: 2c49 5129 2020 2020 2020 2020 7c0a 7c20  ,IQ)        |.| 
│ │ │ │ +000036a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000036b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000036c0: 2020 7c0a 7c6f 3720 3d20 6964 6561 6c20    |.|o7 = ideal 
│ │ │ │ +000036d0: 2832 7a20 2d20 742c 2032 7920 2d20 742c  (2z - t, 2y - t,
│ │ │ │ +000036e0: 2032 7820 2b20 7429 7c0a 7c20 2020 2020   2x + t)|.|     
│ │ │ │ +000036f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00003700: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00003710: 7c6f 3720 3a20 4964 6561 6c20 6f66 2053  |o7 : Ideal of S
│ │ │ │ +00003720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00003730: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  00003740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003780: 2d2d 2d2b 0a7c 6938 203a 2049 203d 2069  ---+.|i8 : I = i
│ │ │ │ -00003790: 6465 616c 2872 616e 646f 6d28 312c 5329  deal(random(1,S)
│ │ │ │ -000037a0: 2c72 616e 646f 6d28 312c 5329 293b 2020  ,random(1,S));  
│ │ │ │ -000037b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000037c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000037d0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -000037e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000037f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003820: 2020 207c 0a7c 6f38 203a 2049 6465 616c     |.|o8 : Ideal
│ │ │ │ -00003830: 206f 6620 5320 2020 2020 2020 2020 2020   of S           
│ │ │ │ -00003840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00003750: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a57 6520  ----------+..We 
│ │ │ │ +00003760: 6361 6e20 616c 736f 2063 6f6e 7369 6465  can also conside
│ │ │ │ +00003770: 7220 4861 6461 6d61 7264 2070 726f 6475  r Hadamard produ
│ │ │ │ +00003780: 6374 206f 6620 6869 6768 6572 2064 696d  ct of higher dim
│ │ │ │ +00003790: 656e 7369 6f6e 616c 2076 6172 6965 7469  ensional varieti
│ │ │ │ +000037a0: 6573 2e20 466f 720a 6578 616d 706c 652c  es. For.example,
│ │ │ │ +000037b0: 2074 6865 2048 6164 616d 6172 6420 7072   the Hadamard pr
│ │ │ │ +000037c0: 6f64 7563 7420 6f66 2074 776f 206c 696e  oduct of two lin
│ │ │ │ +000037d0: 6573 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  es...+----------
│ │ │ │ +000037e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000037f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003820: 2d2d 2d2b 0a7c 6938 203a 2049 203d 2069  ---+.|i8 : I = i
│ │ │ │ +00003830: 6465 616c 2872 616e 646f 6d28 312c 5329  deal(random(1,S)
│ │ │ │ +00003840: 2c72 616e 646f 6d28 312c 5329 293b 2020  ,random(1,S));  
│ │ │ │  00003850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003870: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -00003880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000038a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000038b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000038c0: 2d2d 2d2b 0a7c 6939 203a 204a 203d 2069  ---+.|i9 : J = i
│ │ │ │ -000038d0: 6465 616c 2872 616e 646f 6d28 312c 5329  deal(random(1,S)
│ │ │ │ -000038e0: 2c72 616e 646f 6d28 312c 5329 293b 2020  ,random(1,S));  
│ │ │ │ +00003870: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00003880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00003890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000038a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000038b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000038c0: 2020 207c 0a7c 6f38 203a 2049 6465 616c     |.|o8 : Ideal
│ │ │ │ +000038d0: 206f 6620 5320 2020 2020 2020 2020 2020   of S           
│ │ │ │ +000038e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000038f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003910: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00003920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003960: 2020 207c 0a7c 6f39 203a 2049 6465 616c     |.|o9 : Ideal
│ │ │ │ -00003970: 206f 6620 5320 2020 2020 2020 2020 2020   of S           
│ │ │ │ -00003980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00003910: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00003920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003960: 2d2d 2d2b 0a7c 6939 203a 204a 203d 2069  ---+.|i9 : J = i
│ │ │ │ +00003970: 6465 616c 2872 616e 646f 6d28 312c 5329  deal(random(1,S)
│ │ │ │ +00003980: 2c72 616e 646f 6d28 312c 5329 293b 2020  ,random(1,S));  
│ │ │ │  00003990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000039a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000039b0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -000039c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000039d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000039e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000039f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003a00: 2d2d 2d2b 0a7c 6931 3020 3a20 6861 6461  ---+.|i10 : hada
│ │ │ │ -00003a10: 6d61 7264 5072 6f64 7563 7428 492c 4a29  mardProduct(I,J)
│ │ │ │ +000039b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000039c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000039d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000039e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000039f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00003a00: 2020 207c 0a7c 6f39 203a 2049 6465 616c     |.|o9 : Ideal
│ │ │ │ +00003a10: 206f 6620 5320 2020 2020 2020 2020 2020   of S           
│ │ │ │  00003a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003a50: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00003a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003aa0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00003ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003ac0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00003a50: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00003a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003aa0: 2d2d 2d2b 0a7c 6931 3020 3a20 6861 6461  ---+.|i10 : hada
│ │ │ │ +00003ab0: 6d61 7264 5072 6f64 7563 7428 492c 4a29  mardProduct(I,J)
│ │ │ │ +00003ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003ae0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -00003af0: 2020 207c 0a7c 6f31 3020 3d20 6964 6561     |.|o10 = idea
│ │ │ │ -00003b00: 6c28 3136 3430 3636 3634 3132 3030 3078  l(1640666412000x
│ │ │ │ -00003b10: 2020 2d20 3132 3739 3834 3138 3831 3530    - 127984188150
│ │ │ │ -00003b20: 782a 7920 2d20 3435 3435 3236 3334 3530  x*y - 4545263450
│ │ │ │ -00003b30: 7920 202b 2020 2020 2020 2020 2020 2020  y  +            
│ │ │ │ -00003b40: 2020 207c 0a7c 2020 2020 2020 2d2d 2d2d     |.|      ----
│ │ │ │ -00003b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003b90: 2d2d 2d7c 0a7c 2020 2020 2020 2020 2020  ---|.|          
│ │ │ │ -00003ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003bc0: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -00003bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003be0: 2020 207c 0a7c 2020 2020 2020 3338 3630     |.|      3860
│ │ │ │ -00003bf0: 3239 3838 3432 3430 782a 7a20 2b20 3130  29884240x*z + 10
│ │ │ │ -00003c00: 3938 3231 3134 3332 3579 2a7a 202d 2031  982114325y*z - 1
│ │ │ │ -00003c10: 3336 3630 3532 3638 387a 2020 2b20 3639  366052688z  + 69
│ │ │ │ -00003c20: 3438 3239 3932 3535 3632 3578 2a74 202d  48299255625x*t -
│ │ │ │ -00003c30: 2020 207c 0a7c 2020 2020 2020 2d2d 2d2d     |.|      ----
│ │ │ │ -00003c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003c80: 2d2d 2d7c 0a7c 2020 2020 2020 2020 2020  ---|.|          
│ │ │ │ -00003c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003cb0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -00003cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003cd0: 2020 207c 0a7c 2020 2020 2020 3637 3636     |.|      6766
│ │ │ │ -00003ce0: 3331 3536 3536 3735 792a 7420 2b20 3637  31565675y*t + 67
│ │ │ │ -00003cf0: 3432 3131 3331 3234 357a 2a74 202b 2031  421131245z*t + 1
│ │ │ │ -00003d00: 3531 3439 3235 3639 3633 3735 7420 2920  514925696375t ) 
│ │ │ │ -00003d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003d20: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00003ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00003af0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00003b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00003b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00003b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00003b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00003b40: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00003b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00003b60: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00003b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00003b80: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +00003b90: 2020 207c 0a7c 6f31 3020 3d20 6964 6561     |.|o10 = idea
│ │ │ │ +00003ba0: 6c28 3136 3430 3636 3634 3132 3030 3078  l(1640666412000x
│ │ │ │ +00003bb0: 2020 2d20 3132 3739 3834 3138 3831 3530    - 127984188150
│ │ │ │ +00003bc0: 782a 7920 2d20 3435 3435 3236 3334 3530  x*y - 4545263450
│ │ │ │ +00003bd0: 7920 202b 2020 2020 2020 2020 2020 2020  y  +            
│ │ │ │ +00003be0: 2020 207c 0a7c 2020 2020 2020 2d2d 2d2d     |.|      ----
│ │ │ │ +00003bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003c30: 2d2d 2d7c 0a7c 2020 2020 2020 2020 2020  ---|.|          
│ │ │ │ +00003c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00003c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00003c60: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +00003c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00003c80: 2020 207c 0a7c 2020 2020 2020 3338 3630     |.|      3860
│ │ │ │ +00003c90: 3239 3838 3432 3430 782a 7a20 2b20 3130  29884240x*z + 10
│ │ │ │ +00003ca0: 3938 3231 3134 3332 3579 2a7a 202d 2031  982114325y*z - 1
│ │ │ │ +00003cb0: 3336 3630 3532 3638 387a 2020 2b20 3639  366052688z  + 69
│ │ │ │ +00003cc0: 3438 3239 3932 3535 3632 3578 2a74 202d  48299255625x*t -
│ │ │ │ +00003cd0: 2020 207c 0a7c 2020 2020 2020 2d2d 2d2d     |.|      ----
│ │ │ │ +00003ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003d20: 2d2d 2d7c 0a7c 2020 2020 2020 2020 2020  ---|.|          
│ │ │ │  00003d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00003d50: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │  00003d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003d70: 2020 207c 0a7c 6f31 3020 3a20 4964 6561     |.|o10 : Idea
│ │ │ │ -00003d80: 6c20 6f66 2053 2020 2020 2020 2020 2020  l of S          
│ │ │ │ -00003d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00003d70: 2020 207c 0a7c 2020 2020 2020 3637 3636     |.|      6766
│ │ │ │ +00003d80: 3331 3536 3536 3735 792a 7420 2b20 3637  31565675y*t + 67
│ │ │ │ +00003d90: 3432 3131 3331 3234 357a 2a74 202b 2031  421131245z*t + 1
│ │ │ │ +00003da0: 3531 3439 3235 3639 3633 3735 7420 2920  514925696375t ) 
│ │ │ │  00003db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003dc0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -00003dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003e10: 2d2d 2d2b 0a0a 5761 7973 2074 6f20 7573  ---+..Ways to us
│ │ │ │ -00003e20: 6520 7468 6973 206d 6574 686f 643a 0a3d  e this method:.=
│ │ │ │ -00003e30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00003e40: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f  =======..  * *no
│ │ │ │ -00003e50: 7465 2068 6164 616d 6172 6450 726f 6475  te hadamardProdu
│ │ │ │ -00003e60: 6374 2849 6465 616c 2c49 6465 616c 293a  ct(Ideal,Ideal):
│ │ │ │ -00003e70: 2068 6164 616d 6172 6450 726f 6475 6374   hadamardProduct
│ │ │ │ -00003e80: 5f6c 7049 6465 616c 5f63 6d49 6465 616c  _lpIdeal_cmIdeal
│ │ │ │ -00003e90: 5f72 702c 202d 2d0a 2020 2020 4861 6461  _rp, --.    Hada
│ │ │ │ -00003ea0: 6d61 7264 2070 726f 6475 6374 206f 6620  mard product of 
│ │ │ │ -00003eb0: 7477 6f20 686f 6d6f 6765 6e65 6f75 7320  two homogeneous 
│ │ │ │ -00003ec0: 6964 6561 6c73 0a2d 2d2d 2d2d 2d2d 2d2d  ideals.---------
│ │ │ │ -00003ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003f10: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572  ------..The sour
│ │ │ │ -00003f20: 6365 206f 6620 7468 6973 2064 6f63 756d  ce of this docum
│ │ │ │ -00003f30: 656e 7420 6973 2069 6e0a 2f62 7569 6c64  ent is in./build
│ │ │ │ -00003f40: 2f72 6570 726f 6475 6369 626c 652d 7061  /reproducible-pa
│ │ │ │ -00003f50: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32  th/macaulay2-1.2
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│ │ │ │ -00004090: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000040a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000040b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000040c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a  **************..
│ │ │ │ -000040d0: 2020 2a20 4675 6e63 7469 6f6e 3a20 2a6e    * Function: *n
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│ │ │ │ -000041e0: 2a6e 6f74 6520 6964 6561 6c3a 2028 4d61  *note ideal: (Ma
│ │ │ │ -000041f0: 6361 756c 6179 3244 6f63 2949 6465 616c  caulay2Doc)Ideal
│ │ │ │ -00004200: 2c2c 200a 2020 2020 2020 2a20 616e 2069  ,, .      * an i
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│ │ │ │ -00004230: 2050 6f69 6e74 2c2c 200a 0a44 6573 6372   Point,, ..Descr
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│ │ │ │ -00004250: 3d3d 0a0a 5468 6520 4861 6461 6d61 7264  ==..The Hadamard
│ │ │ │ -00004260: 2070 726f 6475 6374 206f 6620 6120 6c69   product of a li
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│ │ │ │ -000042a0: 6174 6976 656c 7920 7468 6520 6269 6e61  atively the bina
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│ │ │ │ -00004320: 696e 7420 5f73 7420 506f 696e 742c 2e0a  int _st Point,..
│ │ │ │ -00004330: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00004340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00004350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00004360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00004370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00004380: 0a7c 6931 203a 2053 203d 2051 515b 782c  .|i1 : S = QQ[x,
│ │ │ │ -00004390: 792c 7a2c 745d 3b20 2020 2020 2020 2020  y,z,t];         
│ │ │ │ -000043a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000043b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000043c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00003dc0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00003dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00003de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00003df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00003e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00003e20: 6c20 6f66 2053 2020 2020 2020 2020 2020  l of S          
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│ │ │ │ +00003e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00003e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00003e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003eb0: 2d2d 2d2b 0a0a 5761 7973 2074 6f20 7573  ---+..Ways to us
│ │ │ │ +00003ec0: 6520 7468 6973 206d 6574 686f 643a 0a3d  e this method:.=
│ │ │ │ +00003ed0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ +00003f10: 2068 6164 616d 6172 6450 726f 6475 6374   hadamardProduct
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│ │ │ │ +00003f30: 5f72 702c 202d 2d0a 2020 2020 4861 6461  _rp, --.    Hada
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│ │ │ │ +00003f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003fb0: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572  ------..The sour
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│ │ │ │ +00004000: 352e 3131 2b64 732f 4d32 2f4d 6163 6175  5.11+ds/M2/Macau
│ │ │ │ +00004010: 6c61 7932 2f70 6163 6b61 6765 732f 4861  lay2/packages/Ha
│ │ │ │ +00004020: 6461 6d61 7264 2e6d 320a 3a32 3933 3a30  damard.m2.:293:0
│ │ │ │ +00004030: 2e0a 1f0a 4669 6c65 3a20 4861 6461 6d61  ....File: Hadama
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│ │ │ │ +00004050: 6164 616d 6172 6450 726f 6475 6374 5f6c  adamardProduct_l
│ │ │ │ +00004060: 704c 6973 745f 7270 2c20 4e65 7874 3a20  pList_rp, Next: 
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│ │ │ │ +000040a0: 5072 6f64 7563 745f 6c70 4964 6561 6c5f  Product_lpIdeal_
│ │ │ │ +000040b0: 636d 4964 6561 6c5f 7270 2c20 5570 3a20  cmIdeal_rp, Up: 
│ │ │ │ +000040c0: 546f 700a 0a68 6164 616d 6172 6450 726f  Top..hadamardPro
│ │ │ │ +000040d0: 6475 6374 284c 6973 7429 202d 2d20 4861  duct(List) -- Ha
│ │ │ │ +000040e0: 6461 6d61 7264 2070 726f 6475 6374 206f  damard product o
│ │ │ │ +000040f0: 6620 6120 6c69 7374 206f 6620 686f 6d6f  f a list of homo
│ │ │ │ +00004100: 6765 6e65 6f75 7320 6964 6561 6c73 2c20  geneous ideals, 
│ │ │ │ +00004110: 6f72 2070 6f69 6e74 730a 2a2a 2a2a 2a2a  or points.******
│ │ │ │ +00004120: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00004130: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00004140: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00004150: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00004160: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a  **************..
│ │ │ │ +00004170: 2020 2a20 4675 6e63 7469 6f6e 3a20 2a6e    * Function: *n
│ │ │ │ +00004180: 6f74 6520 6861 6461 6d61 7264 5072 6f64  ote hadamardProd
│ │ │ │ +00004190: 7563 743a 2068 6164 616d 6172 6450 726f  uct: hadamardPro
│ │ │ │ +000041a0: 6475 6374 2c0a 2020 2a20 5573 6167 653a  duct,.  * Usage:
│ │ │ │ +000041b0: 200a 2020 2020 2020 2020 6861 6461 6d61   .        hadama
│ │ │ │ +000041c0: 7264 5072 6f64 7563 7428 4c29 0a20 202a  rdProduct(L).  *
│ │ │ │ +000041d0: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ +000041e0: 204c 2c20 6120 2a6e 6f74 6520 6c69 7374   L, a *note list
│ │ │ │ +000041f0: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +00004200: 4c69 7374 2c2c 206f 6620 2a6e 6f74 6520  List,, of *note 
│ │ │ │ +00004210: 2868 6f6d 6f67 656e 656f 7573 2920 6964  (homogeneous) id
│ │ │ │ +00004220: 6561 6c73 3a0a 2020 2020 2020 2020 284d  eals:.        (M
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│ │ │ │ +00004240: 6c2c 206f 7220 2a6e 6f74 6520 2870 726f  l, or *note (pro
│ │ │ │ +00004250: 6a65 6374 6976 6529 2070 6f69 6e74 733a  jective) points:
│ │ │ │ +00004260: 2050 6f69 6e74 2c0a 2020 2a20 4f75 7470   Point,.  * Outp
│ │ │ │ +00004270: 7574 733a 0a20 2020 2020 202a 2061 6e20  uts:.      * an 
│ │ │ │ +00004280: 2a6e 6f74 6520 6964 6561 6c3a 2028 4d61  *note ideal: (Ma
│ │ │ │ +00004290: 6361 756c 6179 3244 6f63 2949 6465 616c  caulay2Doc)Ideal
│ │ │ │ +000042a0: 2c2c 200a 2020 2020 2020 2a20 616e 2069  ,, .      * an i
│ │ │ │ +000042b0: 6e73 7461 6e63 6520 6f66 2074 6865 2074  nstance of the t
│ │ │ │ +000042c0: 7970 6520 2a6e 6f74 6520 506f 696e 743a  ype *note Point:
│ │ │ │ +000042d0: 2050 6f69 6e74 2c2c 200a 0a44 6573 6372   Point,, ..Descr
│ │ │ │ +000042e0: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d  iption.=========
│ │ │ │ +000042f0: 3d3d 0a0a 5468 6520 4861 6461 6d61 7264  ==..The Hadamard
│ │ │ │ +00004300: 2070 726f 6475 6374 206f 6620 6120 6c69   product of a li
│ │ │ │ +00004310: 7374 206f 6620 6964 6561 6c73 206f 7220  st of ideals or 
│ │ │ │ +00004320: 706f 696e 7473 2063 6f6e 7374 7275 6374  points construct
│ │ │ │ +00004330: 6564 2062 7920 7573 696e 670a 6974 6572  ed by using.iter
│ │ │ │ +00004340: 6174 6976 656c 7920 7468 6520 6269 6e61  atively the bina
│ │ │ │ +00004350: 7279 2066 756e 6374 696f 6e20 2a6e 6f74  ry function *not
│ │ │ │ +00004360: 6520 6861 6461 6d61 7264 5072 6f64 7563  e hadamardProduc
│ │ │ │ +00004370: 7428 4964 6561 6c2c 4964 6561 6c29 3a0a  t(Ideal,Ideal):.
│ │ │ │ +00004380: 6861 6461 6d61 7264 5072 6f64 7563 745f  hadamardProduct_
│ │ │ │ +00004390: 6c70 4964 6561 6c5f 636d 4964 6561 6c5f  lpIdeal_cmIdeal_
│ │ │ │ +000043a0: 7270 2c2c 206f 7220 202a 6e6f 7465 2050  rp,, or  *note P
│ │ │ │ +000043b0: 6f69 6e74 202a 2050 6f69 6e74 3a20 506f  oint * Point: Po
│ │ │ │ +000043c0: 696e 7420 5f73 7420 506f 696e 742c 2e0a  int _st Point,..
│ │ │ │  000043d0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  000043e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000043f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00004400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00004410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00004420: 0a7c 6932 203a 2049 203d 2069 6465 616c  .|i2 : I = ideal
│ │ │ │ -00004430: 2872 616e 646f 6d28 312c 5329 2c72 616e  (random(1,S),ran
│ │ │ │ -00004440: 646f 6d28 312c 5329 293b 2020 2020 2020  dom(1,S));      
│ │ │ │ +00004420: 0a7c 6931 203a 2053 203d 2051 515b 782c  .|i1 : S = QQ[x,
│ │ │ │ +00004430: 792c 7a2c 745d 3b20 2020 2020 2020 2020  y,z,t];         
│ │ │ │ +00004440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004460: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00004470: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00004480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00004490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000044a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000044b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000044c0: 0a7c 6f32 203a 2049 6465 616c 206f 6620  .|o2 : Ideal of 
│ │ │ │ -000044d0: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │ -000044e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004470: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00004480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00004490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000044a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000044b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +000044c0: 0a7c 6932 203a 2049 203d 2069 6465 616c  .|i2 : I = ideal
│ │ │ │ +000044d0: 2872 616e 646f 6d28 312c 5329 2c72 616e  (random(1,S),ran
│ │ │ │ +000044e0: 646f 6d28 312c 5329 293b 2020 2020 2020  dom(1,S));      
│ │ │ │  000044f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004500: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00004510: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00004520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00004530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00004540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00004550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00004560: 0a7c 6933 203a 204a 203d 2069 6465 616c  .|i3 : J = ideal
│ │ │ │ -00004570: 2872 616e 646f 6d28 312c 5329 2c72 616e  (random(1,S),ran
│ │ │ │ -00004580: 646f 6d28 312c 5329 293b 2020 2020 2020  dom(1,S));      
│ │ │ │ +00004510: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00004520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004550: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00004560: 0a7c 6f32 203a 2049 6465 616c 206f 6620  .|o2 : Ideal of 
│ │ │ │ +00004570: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │ +00004580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000045a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000045b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000045c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000045d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000045e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000045f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00004600: 0a7c 6f33 203a 2049 6465 616c 206f 6620  .|o3 : Ideal of 
│ │ │ │ -00004610: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │ -00004620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000045b0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000045c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000045d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000045e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000045f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00004600: 0a7c 6933 203a 204a 203d 2069 6465 616c  .|i3 : J = ideal
│ │ │ │ +00004610: 2872 616e 646f 6d28 312c 5329 2c72 616e  (random(1,S),ran
│ │ │ │ +00004620: 646f 6d28 312c 5329 293b 2020 2020 2020  dom(1,S));      
│ │ │ │  00004630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004640: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00004650: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00004660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00004670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00004680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00004690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000046a0: 0a7c 6934 203a 204c 203d 207b 492c 4a7d  .|i4 : L = {I,J}
│ │ │ │ -000046b0: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ +00004650: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00004660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004690: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000046a0: 0a7c 6f33 203a 2049 6465 616c 206f 6620  .|o3 : Ideal of 
│ │ │ │ +000046b0: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │  000046c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000046d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000046e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  000046f0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00004700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00004710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00004720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00004730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00004740: 0a7c 6935 203a 2068 6164 616d 6172 6450  .|i5 : hadamardP
│ │ │ │ -00004750: 726f 6475 6374 284c 2920 2020 2020 2020  roduct(L)       
│ │ │ │ +00004740: 0a7c 6934 203a 204c 203d 207b 492c 4a7d  .|i4 : L = {I,J}
│ │ │ │ +00004750: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │  00004760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004780: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00004790: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000047a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000047b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000047c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000047d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000047e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000047f0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +00004790: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000047a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000047b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000047c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000047d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +000047e0: 0a7c 6935 203a 2068 6164 616d 6172 6450  .|i5 : hadamardP
│ │ │ │ +000047f0: 726f 6475 6374 284c 2920 2020 2020 2020  roduct(L)       
│ │ │ │  00004800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00004810: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +00004810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004820: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00004830: 0a7c 6f35 203d 2069 6465 616c 2831 3634  .|o5 = ideal(164
│ │ │ │ -00004840: 3036 3636 3431 3230 3030 7820 202d 2031  0666412000x  - 1
│ │ │ │ -00004850: 3237 3938 3431 3838 3135 3078 2a79 202d  27984188150x*y -
│ │ │ │ -00004860: 2034 3534 3532 3633 3435 3079 2020 2b20   4545263450y  + 
│ │ │ │ -00004870: 3338 3630 3239 3838 3432 3430 782a 7a7c  386029884240x*z|
│ │ │ │ -00004880: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d  .|     ---------
│ │ │ │ -00004890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000048a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000048b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000048c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -000048d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000048e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000048f0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -00004900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00004910: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00004920: 0a7c 2020 2020 202b 2031 3039 3832 3131  .|     + 1098211
│ │ │ │ -00004930: 3433 3235 792a 7a20 2d20 3133 3636 3035  4325y*z - 136605
│ │ │ │ -00004940: 3236 3838 7a20 202b 2036 3934 3832 3939  2688z  + 6948299
│ │ │ │ -00004950: 3235 3536 3235 782a 7420 2d20 3637 3636  255625x*t - 6766
│ │ │ │ -00004960: 3331 3536 3536 3735 792a 7420 2b20 207c  31565675y*t +  |
│ │ │ │ -00004970: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d  .|     ---------
│ │ │ │ -00004980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00004990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000049a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000049b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -000049c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000049d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000049e0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -000049f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00004a00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00004a10: 0a7c 2020 2020 2036 3734 3231 3133 3132  .|     674211312
│ │ │ │ -00004a20: 3435 7a2a 7420 2b20 3135 3134 3932 3536  45z*t + 15149256
│ │ │ │ -00004a30: 3936 3337 3574 2029 2020 2020 2020 2020  96375t )        
│ │ │ │ -00004a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00004a50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00004830: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00004840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004870: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00004880: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00004890: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +000048a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000048b0: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +000048c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000048d0: 0a7c 6f35 203d 2069 6465 616c 2831 3634  .|o5 = ideal(164
│ │ │ │ +000048e0: 3036 3636 3431 3230 3030 7820 202d 2031  0666412000x  - 1
│ │ │ │ +000048f0: 3237 3938 3431 3838 3135 3078 2a79 202d  27984188150x*y -
│ │ │ │ +00004900: 2034 3534 3532 3633 3435 3079 2020 2b20   4545263450y  + 
│ │ │ │ +00004910: 3338 3630 3239 3838 3432 3430 782a 7a7c  386029884240x*z|
│ │ │ │ +00004920: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d  .|     ---------
│ │ │ │ +00004930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00004940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00004950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00004960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00004970: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00004980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004990: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +000049a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000049b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000049c0: 0a7c 2020 2020 202b 2031 3039 3832 3131  .|     + 1098211
│ │ │ │ +000049d0: 3433 3235 792a 7a20 2d20 3133 3636 3035  4325y*z - 136605
│ │ │ │ +000049e0: 3236 3838 7a20 202b 2036 3934 3832 3939  2688z  + 6948299
│ │ │ │ +000049f0: 3235 3536 3235 782a 7420 2d20 3637 3636  255625x*t - 6766
│ │ │ │ +00004a00: 3331 3536 3536 3735 792a 7420 2b20 207c  31565675y*t +  |
│ │ │ │ +00004a10: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d  .|     ---------
│ │ │ │ +00004a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00004a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00004a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00004a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │  00004a60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00004a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00004a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004a80: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │  00004a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004aa0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00004ab0: 0a7c 6f35 203a 2049 6465 616c 206f 6620  .|o5 : Ideal of 
│ │ │ │ -00004ac0: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │ -00004ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004ab0: 0a7c 2020 2020 2036 3734 3231 3133 3132  .|     674211312
│ │ │ │ +00004ac0: 3435 7a2a 7420 2b20 3135 3134 3932 3536  45z*t + 15149256
│ │ │ │ +00004ad0: 3936 3337 3574 2029 2020 2020 2020 2020  96375t )        
│ │ │ │  00004ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004af0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00004b00: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00004b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00004b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00004b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00004b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00004b50: 0a7c 6936 203a 2050 203d 2070 6f69 6e74  .|i6 : P = point
│ │ │ │ -00004b60: 5c7b 7b31 2c32 2c33 7d2c 7b2d 312c 312c  \{{1,2,3},{-1,1,
│ │ │ │ -00004b70: 317d 2c7b 312c 312f 322c 2d31 2f33 7d7d  1},{1,1/2,-1/3}}
│ │ │ │ +00004b00: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00004b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004b40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00004b50: 0a7c 6f35 203a 2049 6465 616c 206f 6620  .|o5 : Ideal of 
│ │ │ │ +00004b60: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │ +00004b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004b90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00004ba0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00004bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00004bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00004bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00004be0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00004bf0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00004c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00004c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00004c20: 2020 3120 2020 2031 2020 2020 2020 2020    1    1        
│ │ │ │ +00004ba0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00004bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00004bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00004bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00004be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00004bf0: 0a7c 6936 203a 2050 203d 2070 6f69 6e74  .|i6 : P = point
│ │ │ │ +00004c00: 5c7b 7b31 2c32 2c33 7d2c 7b2d 312c 312c  \{{1,2,3},{-1,1,
│ │ │ │ +00004c10: 317d 2c7b 312c 312f 322c 2d31 2f33 7d7d  1},{1,1/2,-1/3}}
│ │ │ │ +00004c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004c30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00004c40: 0a7c 6f36 203d 207b 506f 696e 747b 312c  .|o6 = {Point{1,
│ │ │ │ -00004c50: 2032 2c20 337d 2c20 506f 696e 747b 2d31   2, 3}, Point{-1
│ │ │ │ -00004c60: 2c20 312c 2031 7d2c 2050 6f69 6e74 7b31  , 1, 1}, Point{1
│ │ │ │ -00004c70: 2c20 2d2c 202d 202d 7d7d 2020 2020 2020  , -, - -}}      
│ │ │ │ +00004c40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00004c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004c80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00004c90: 0a7c 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 3220 2020 2033 2020 2020 2020 2020    2    3        
│ │ │ │ +00004cc0: 2020 3120 2020 2031 2020 2020 2020 2020    1    1        
│ │ │ │  00004cd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00004ce0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00004cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00004d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00004d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004ce0: 0a7c 6f36 203d 207b 506f 696e 747b 312c  .|o6 = {Point{1,
│ │ │ │ +00004cf0: 2032 2c20 337d 2c20 506f 696e 747b 2d31   2, 3}, Point{-1
│ │ │ │ +00004d00: 2c20 312c 2031 7d2c 2050 6f69 6e74 7b31  , 1, 1}, Point{1
│ │ │ │ +00004d10: 2c20 2d2c 202d 202d 7d7d 2020 2020 2020  , -, - -}}      
│ │ │ │  00004d20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00004d30: 0a7c 6f36 203a 204c 6973 7420 2020 2020  .|o6 : List     
│ │ │ │ +00004d30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00004d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00004d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004d60: 2020 3220 2020 2033 2020 2020 2020 2020    2    3        
│ │ │ │  00004d70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00004d80: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00004d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00004da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00004db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00004dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00004dd0: 0a7c 6937 203a 2068 6164 616d 6172 6450  .|i7 : hadamardP
│ │ │ │ -00004de0: 726f 6475 6374 2850 2920 2020 2020 2020  roduct(P)       
│ │ │ │ +00004d80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00004d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004dc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00004dd0: 0a7c 6f36 203a 204c 6973 7420 2020 2020  .|o6 : List     
│ │ │ │ +00004de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004e10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00004e20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00004e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00004e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00004e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00004e60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00004e70: 0a7c 6f37 203d 2050 6f69 6e74 7b2d 312c  .|o7 = Point{-1,
│ │ │ │ -00004e80: 2031 2c20 2d31 7d20 2020 2020 2020 2020   1, -1}         
│ │ │ │ +00004e20: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00004e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00004e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00004e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00004e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00004e70: 0a7c 6937 203a 2068 6164 616d 6172 6450  .|i7 : hadamardP
│ │ │ │ +00004e80: 726f 6475 6374 2850 2920 2020 2020 2020  roduct(P)       
│ │ │ │  00004e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004eb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00004ec0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00004ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004f00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00004f10: 0a7c 6f37 203a 2050 6f69 6e74 2020 2020  .|o7 : Point    
│ │ │ │ -00004f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004f10: 0a7c 6f37 203d 2050 6f69 6e74 7b2d 312c  .|o7 = Point{-1,
│ │ │ │ +00004f20: 2031 2c20 2d31 7d20 2020 2020 2020 2020   1, -1}         
│ │ │ │  00004f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004f50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00004f60: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00004f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00004f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00004f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00004fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00004fb0: 0a0a 5761 7973 2074 6f20 7573 6520 7468  ..Ways to use th
│ │ │ │ -00004fc0: 6973 206d 6574 686f 643a 0a3d 3d3d 3d3d  is method:.=====
│ │ │ │ -00004fd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00004fe0: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2068  ===..  * *note h
│ │ │ │ -00004ff0: 6164 616d 6172 6450 726f 6475 6374 284c  adamardProduct(L
│ │ │ │ -00005000: 6973 7429 3a20 6861 6461 6d61 7264 5072  ist): hadamardPr
│ │ │ │ -00005010: 6f64 7563 745f 6c70 4c69 7374 5f72 702c  oduct_lpList_rp,
│ │ │ │ -00005020: 202d 2d20 4861 6461 6d61 7264 2070 726f   -- Hadamard pro
│ │ │ │ -00005030: 6475 6374 0a20 2020 206f 6620 6120 6c69  duct.    of a li
│ │ │ │ -00005040: 7374 206f 6620 686f 6d6f 6765 6e65 6f75  st of homogeneou
│ │ │ │ -00005050: 7320 6964 6561 6c73 2c20 6f72 2070 6f69  s ideals, or poi
│ │ │ │ -00005060: 6e74 730a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  nts.------------
│ │ │ │ -00005070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000050a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000050b0: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520  ---..The source 
│ │ │ │ -000050c0: 6f66 2074 6869 7320 646f 6375 6d65 6e74  of this document
│ │ │ │ -000050d0: 2069 7320 696e 0a2f 6275 696c 642f 7265   is in./build/re
│ │ │ │ -000050e0: 7072 6f64 7563 6962 6c65 2d70 6174 682f  producible-path/
│ │ │ │ -000050f0: 6d61 6361 756c 6179 322d 312e 3235 2e31  macaulay2-1.25.1
│ │ │ │ -00005100: 312b 6473 2f4d 322f 4d61 6361 756c 6179  1+ds/M2/Macaulay
│ │ │ │ -00005110: 322f 7061 636b 6167 6573 2f48 6164 616d  2/packages/Hadam
│ │ │ │ -00005120: 6172 642e 6d32 0a3a 3332 303a 302e 0a1f  ard.m2.:320:0...
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│ │ │ │ -00005140: 696e 666f 2c20 4e6f 6465 3a20 6861 6461  info, Node: hada
│ │ │ │ -00005150: 6d61 7264 5072 6f64 7563 745f 6c70 4c69  mardProduct_lpLi
│ │ │ │ -00005160: 7374 5f63 6d4c 6973 745f 7270 2c20 4e65  st_cmList_rp, Ne
│ │ │ │ -00005170: 7874 3a20 6964 6561 6c4f 6650 726f 6a65  xt: idealOfProje
│ │ │ │ -00005180: 6374 6976 6550 6f69 6e74 732c 2050 7265  ctivePoints, Pre
│ │ │ │ -00005190: 763a 2068 6164 616d 6172 6450 726f 6475  v: hadamardProdu
│ │ │ │ -000051a0: 6374 5f6c 704c 6973 745f 7270 2c20 5570  ct_lpList_rp, Up
│ │ │ │ -000051b0: 3a20 546f 700a 0a68 6164 616d 6172 6450  : Top..hadamardP
│ │ │ │ -000051c0: 726f 6475 6374 284c 6973 742c 4c69 7374  roduct(List,List
│ │ │ │ -000051d0: 2920 2d2d 2048 6164 616d 6172 6420 7072  ) -- Hadamard pr
│ │ │ │ -000051e0: 6f64 7563 7420 6f66 2074 776f 2073 6574  oduct of two set
│ │ │ │ -000051f0: 7320 6f66 2070 6f69 6e74 730a 2a2a 2a2a  s of points.****
│ │ │ │ -00005200: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00005210: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00005220: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00005230: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00005240: 0a0a 2020 2a20 4675 6e63 7469 6f6e 3a20  ..  * Function: 
│ │ │ │ -00005250: 2a6e 6f74 6520 6861 6461 6d61 7264 5072  *note hadamardPr
│ │ │ │ -00005260: 6f64 7563 743a 2068 6164 616d 6172 6450  oduct: hadamardP
│ │ │ │ -00005270: 726f 6475 6374 2c0a 2020 2a20 5573 6167  roduct,.  * Usag
│ │ │ │ -00005280: 653a 200a 2020 2020 2020 2020 6861 6461  e: .        hada
│ │ │ │ -00005290: 6d61 7264 5072 6f64 7563 7428 4c2c 4d29  mardProduct(L,M)
│ │ │ │ -000052a0: 0a20 202a 2049 6e70 7574 733a 0a20 2020  .  * Inputs:.   
│ │ │ │ -000052b0: 2020 202a 204c 2c20 6120 2a6e 6f74 6520     * L, a *note 
│ │ │ │ -000052c0: 6c69 7374 3a20 284d 6163 6175 6c61 7932  list: (Macaulay2
│ │ │ │ -000052d0: 446f 6329 4c69 7374 2c2c 206f 6620 2a6e  Doc)List,, of *n
│ │ │ │ -000052e0: 6f74 6520 506f 696e 743a 2050 6f69 6e74  ote Point: Point
│ │ │ │ -000052f0: 2c0a 2020 2020 2020 2a20 4d2c 2061 202a  ,.      * M, a *
│ │ │ │ -00005300: 6e6f 7465 206c 6973 743a 2028 4d61 6361  note list: (Maca
│ │ │ │ -00005310: 756c 6179 3244 6f63 294c 6973 742c 2c20  ulay2Doc)List,, 
│ │ │ │ -00005320: 6f66 202a 6e6f 7465 2050 6f69 6e74 3a20  of *note Point: 
│ │ │ │ -00005330: 506f 696e 742c 0a20 202a 204f 7574 7075  Point,.  * Outpu
│ │ │ │ -00005340: 7473 3a0a 2020 2020 2020 2a20 6120 2a6e  ts:.      * a *n
│ │ │ │ -00005350: 6f74 6520 6c69 7374 3a20 284d 6163 6175  ote list: (Macau
│ │ │ │ -00005360: 6c61 7932 446f 6329 4c69 7374 2c2c 206f  lay2Doc)List,, o
│ │ │ │ -00005370: 6620 2a6e 6f74 6520 506f 696e 743a 2050  f *note Point: P
│ │ │ │ -00005380: 6f69 6e74 2c0a 0a44 6573 6372 6970 7469  oint,..Descripti
│ │ │ │ -00005390: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  on.===========..
│ │ │ │ -000053a0: 4769 7665 6e20 7477 6f20 7365 7473 206f  Given two sets o
│ │ │ │ -000053b0: 6620 706f 696e 7473 2024 4c24 2061 6e64  f points $L$ and
│ │ │ │ -000053c0: 2024 4d24 2072 6574 7572 6e73 2074 6865   $M$ returns the
│ │ │ │ -000053d0: 206c 6973 7420 6f66 2028 7765 6c6c 2d64   list of (well-d
│ │ │ │ -000053e0: 6566 696e 6564 290a 656e 7472 7977 6973  efined).entrywis
│ │ │ │ -000053f0: 6520 6d75 6c74 6970 6c69 6361 7469 6f6e  e multiplication
│ │ │ │ -00005400: 206f 6620 7061 6972 7320 6f66 2070 6f69   of pairs of poi
│ │ │ │ -00005410: 6e74 7320 696e 2074 6865 2063 6172 7465  nts in the carte
│ │ │ │ -00005420: 7369 616e 2070 726f 6475 6374 2024 4c5c  sian product $L\
│ │ │ │ -00005430: 7469 6d65 730a 4d24 2e0a 0a2b 2d2d 2d2d  times.M$...+----
│ │ │ │ -00005440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005460: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20  --------+.|i1 : 
│ │ │ │ -00005470: 4c20 3d20 7b70 6f69 6e74 7b30 2c31 7d2c  L = {point{0,1},
│ │ │ │ -00005480: 2070 6f69 6e74 7b31 2c32 7d7d 3b20 2020   point{1,2}};   
│ │ │ │ -00005490: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -000054a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000054b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000054c0: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 4d20  ------+.|i2 : M 
│ │ │ │ -000054d0: 3d20 7b70 6f69 6e74 7b31 2c30 7d2c 2070  = {point{1,0}, p
│ │ │ │ -000054e0: 6f69 6e74 7b32 2c32 7d7d 3b20 2020 2020  oint{2,2}};     
│ │ │ │ -000054f0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -00005500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005520: 2d2d 2d2d 2b0a 7c69 3320 3a20 6861 6461  ----+.|i3 : hada
│ │ │ │ -00005530: 6d61 7264 5072 6f64 7563 7428 4c2c 4d29  mardProduct(L,M)
│ │ │ │ -00005540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005550: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00005560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005580: 2020 7c0a 7c6f 3320 3d20 7b50 6f69 6e74    |.|o3 = {Point
│ │ │ │ -00005590: 7b31 2c20 307d 2c20 506f 696e 747b 302c  {1, 0}, Point{0,
│ │ │ │ -000055a0: 2032 7d2c 2050 6f69 6e74 7b32 2c20 347d   2}, Point{2, 4}
│ │ │ │ -000055b0: 7d7c 0a7c 2020 2020 2020 2020 2020 2020  }|.|            
│ │ │ │ -000055c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000055d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000055e0: 7c0a 7c6f 3320 3a20 4c69 7374 2020 2020  |.|o3 : List    
│ │ │ │ -000055f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005600: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00005610: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00005620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00005640: 0a57 6179 7320 746f 2075 7365 2074 6869  .Ways to use thi
│ │ │ │ -00005650: 7320 6d65 7468 6f64 3a0a 3d3d 3d3d 3d3d  s method:.======
│ │ │ │ -00005660: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00005670: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6861  ==..  * *note ha
│ │ │ │ -00005680: 6461 6d61 7264 5072 6f64 7563 7428 4c69  damardProduct(Li
│ │ │ │ -00005690: 7374 2c4c 6973 7429 3a20 6861 6461 6d61  st,List): hadama
│ │ │ │ -000056a0: 7264 5072 6f64 7563 745f 6c70 4c69 7374  rdProduct_lpList
│ │ │ │ -000056b0: 5f63 6d4c 6973 745f 7270 2c20 2d2d 0a20  _cmList_rp, --. 
│ │ │ │ -000056c0: 2020 2048 6164 616d 6172 6420 7072 6f64     Hadamard prod
│ │ │ │ -000056d0: 7563 7420 6f66 2074 776f 2073 6574 7320  uct of two sets 
│ │ │ │ -000056e0: 6f66 2070 6f69 6e74 730a 2d2d 2d2d 2d2d  of points.------
│ │ │ │ -000056f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005730: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073  ---------..The s
│ │ │ │ -00005740: 6f75 7263 6520 6f66 2074 6869 7320 646f  ource of this do
│ │ │ │ -00005750: 6375 6d65 6e74 2069 7320 696e 0a2f 6275  cument is in./bu
│ │ │ │ -00005760: 696c 642f 7265 7072 6f64 7563 6962 6c65  ild/reproducible
│ │ │ │ -00005770: 2d70 6174 682f 6d61 6361 756c 6179 322d  -path/macaulay2-
│ │ │ │ -00005780: 312e 3235 2e31 312b 6473 2f4d 322f 4d61  1.25.11+ds/M2/Ma
│ │ │ │ -00005790: 6361 756c 6179 322f 7061 636b 6167 6573  caulay2/packages
│ │ │ │ -000057a0: 2f48 6164 616d 6172 642e 6d32 0a3a 3334  /Hadamard.m2.:34
│ │ │ │ -000057b0: 353a 302e 0a1f 0a46 696c 653a 2048 6164  5:0....File: Had
│ │ │ │ -000057c0: 616d 6172 642e 696e 666f 2c20 4e6f 6465  amard.info, Node
│ │ │ │ -000057d0: 3a20 6964 6561 6c4f 6650 726f 6a65 6374  : idealOfProject
│ │ │ │ -000057e0: 6976 6550 6f69 6e74 732c 204e 6578 743a  ivePoints, Next:
│ │ │ │ -000057f0: 2050 6f69 6e74 2c20 5072 6576 3a20 6861   Point, Prev: ha
│ │ │ │ -00005800: 6461 6d61 7264 5072 6f64 7563 745f 6c70  damardProduct_lp
│ │ │ │ -00005810: 4c69 7374 5f63 6d4c 6973 745f 7270 2c20  List_cmList_rp, 
│ │ │ │ -00005820: 5570 3a20 546f 700a 0a69 6465 616c 4f66  Up: Top..idealOf
│ │ │ │ -00005830: 5072 6f6a 6563 7469 7665 506f 696e 7473  ProjectivePoints
│ │ │ │ -00005840: 202d 2d20 636f 6d70 7574 6573 2074 6865   -- computes the
│ │ │ │ -00005850: 2069 6465 616c 206f 6620 7365 7420 6f66   ideal of set of
│ │ │ │ -00005860: 2070 6f69 6e74 730a 2a2a 2a2a 2a2a 2a2a   points.********
│ │ │ │ -00005870: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00005880: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00005890: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000058a0: 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 6167  ******..  * Usag
│ │ │ │ -000058b0: 653a 200a 2020 2020 2020 2020 6964 6561  e: .        idea
│ │ │ │ -000058c0: 6c4f 6650 726f 6a65 6374 6976 6550 6f69  lOfProjectivePoi
│ │ │ │ -000058d0: 6e74 7328 4c2c 5329 0a20 202a 2049 6e70  nts(L,S).  * Inp
│ │ │ │ -000058e0: 7574 733a 0a20 2020 2020 202a 204c 2c20  uts:.      * L, 
│ │ │ │ -000058f0: 6120 2a6e 6f74 6520 6c69 7374 3a20 284d  a *note list: (M
│ │ │ │ -00005900: 6163 6175 6c61 7932 446f 6329 4c69 7374  acaulay2Doc)List
│ │ │ │ -00005910: 2c2c 206f 6620 2a6e 6f74 6520 2870 726f  ,, of *note (pro
│ │ │ │ -00005920: 6a65 6374 6976 6529 2070 6f69 6e74 733a  jective) points:
│ │ │ │ -00005930: 0a20 2020 2020 2020 2050 6f69 6e74 2c0a  .        Point,.
│ │ │ │ -00005940: 2020 2020 2020 2a20 532c 2061 202a 6e6f        * S, a *no
│ │ │ │ -00005950: 7465 2072 696e 673a 2028 4d61 6361 756c  te ring: (Macaul
│ │ │ │ -00005960: 6179 3244 6f63 2952 696e 672c 2c20 0a20  ay2Doc)Ring,, . 
│ │ │ │ -00005970: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ -00005980: 2020 2a20 492c 2061 6e20 2a6e 6f74 6520    * I, an *note 
│ │ │ │ -00005990: 6964 6561 6c3a 2028 4d61 6361 756c 6179  ideal: (Macaulay
│ │ │ │ -000059a0: 3244 6f63 2949 6465 616c 2c2c 200a 0a44  2Doc)Ideal,, ..D
│ │ │ │ -000059b0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ -000059c0: 3d3d 3d3d 3d3d 0a0a 4769 7665 6e20 6120  ======..Given a 
│ │ │ │ -000059d0: 7365 7420 706f 696e 7473 2024 5824 2c20  set points $X$, 
│ │ │ │ -000059e0: 6974 2072 6574 7572 6e73 2074 6865 2064  it returns the d
│ │ │ │ -000059f0: 6566 696e 696e 6720 6964 6561 6c20 6f66  efining ideal of
│ │ │ │ -00005a00: 2024 4928 5829 240a 0a2b 2d2d 2d2d 2d2d   $I(X)$..+------
│ │ │ │ -00005a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005a50: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2053  -------+.|i1 : S
│ │ │ │ -00005a60: 203d 2051 515b 782c 792c 7a5d 2020 2020   = QQ[x,y,z]    
│ │ │ │ -00005a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005aa0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00005ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005af0: 2020 2020 2020 207c 0a7c 6f31 203d 2053         |.|o1 = S
│ │ │ │ -00005b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004f60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00004f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004fa0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00004fb0: 0a7c 6f37 203a 2050 6f69 6e74 2020 2020  .|o7 : Point    
│ │ │ │ +00004fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004ff0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00005000: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00005010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00005020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00005030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00005040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00005050: 0a0a 5761 7973 2074 6f20 7573 6520 7468  ..Ways to use th
│ │ │ │ +00005060: 6973 206d 6574 686f 643a 0a3d 3d3d 3d3d  is method:.=====
│ │ │ │ +00005070: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00005080: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2068  ===..  * *note h
│ │ │ │ +00005090: 6164 616d 6172 6450 726f 6475 6374 284c  adamardProduct(L
│ │ │ │ +000050a0: 6973 7429 3a20 6861 6461 6d61 7264 5072  ist): hadamardPr
│ │ │ │ +000050b0: 6f64 7563 745f 6c70 4c69 7374 5f72 702c  oduct_lpList_rp,
│ │ │ │ +000050c0: 202d 2d20 4861 6461 6d61 7264 2070 726f   -- Hadamard pro
│ │ │ │ +000050d0: 6475 6374 0a20 2020 206f 6620 6120 6c69  duct.    of a li
│ │ │ │ +000050e0: 7374 206f 6620 686f 6d6f 6765 6e65 6f75  st of homogeneou
│ │ │ │ +000050f0: 7320 6964 6561 6c73 2c20 6f72 2070 6f69  s ideals, or poi
│ │ │ │ +00005100: 6e74 730a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  nts.------------
│ │ │ │ +00005110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00005120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00005130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00005140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00005150: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520  ---..The source 
│ │ │ │ +00005160: 6f66 2074 6869 7320 646f 6375 6d65 6e74  of this document
│ │ │ │ +00005170: 2069 7320 696e 0a2f 6275 696c 642f 7265   is in./build/re
│ │ │ │ +00005180: 7072 6f64 7563 6962 6c65 2d70 6174 682f  producible-path/
│ │ │ │ +00005190: 6d61 6361 756c 6179 322d 312e 3235 2e31  macaulay2-1.25.1
│ │ │ │ +000051a0: 312b 6473 2f4d 322f 4d61 6361 756c 6179  1+ds/M2/Macaulay
│ │ │ │ +000051b0: 322f 7061 636b 6167 6573 2f48 6164 616d  2/packages/Hadam
│ │ │ │ +000051c0: 6172 642e 6d32 0a3a 3332 303a 302e 0a1f  ard.m2.:320:0...
│ │ │ │ +000051d0: 0a46 696c 653a 2048 6164 616d 6172 642e  .File: Hadamard.
│ │ │ │ +000051e0: 696e 666f 2c20 4e6f 6465 3a20 6861 6461  info, Node: hada
│ │ │ │ +000051f0: 6d61 7264 5072 6f64 7563 745f 6c70 4c69  mardProduct_lpLi
│ │ │ │ +00005200: 7374 5f63 6d4c 6973 745f 7270 2c20 4e65  st_cmList_rp, Ne
│ │ │ │ +00005210: 7874 3a20 6964 6561 6c4f 6650 726f 6a65  xt: idealOfProje
│ │ │ │ +00005220: 6374 6976 6550 6f69 6e74 732c 2050 7265  ctivePoints, Pre
│ │ │ │ +00005230: 763a 2068 6164 616d 6172 6450 726f 6475  v: hadamardProdu
│ │ │ │ +00005240: 6374 5f6c 704c 6973 745f 7270 2c20 5570  ct_lpList_rp, Up
│ │ │ │ +00005250: 3a20 546f 700a 0a68 6164 616d 6172 6450  : Top..hadamardP
│ │ │ │ +00005260: 726f 6475 6374 284c 6973 742c 4c69 7374  roduct(List,List
│ │ │ │ +00005270: 2920 2d2d 2048 6164 616d 6172 6420 7072  ) -- Hadamard pr
│ │ │ │ +00005280: 6f64 7563 7420 6f66 2074 776f 2073 6574  oduct of two set
│ │ │ │ +00005290: 7320 6f66 2070 6f69 6e74 730a 2a2a 2a2a  s of points.****
│ │ │ │ +000052a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000052b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000052c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000052d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000052e0: 0a0a 2020 2a20 4675 6e63 7469 6f6e 3a20  ..  * Function: 
│ │ │ │ +000052f0: 2a6e 6f74 6520 6861 6461 6d61 7264 5072  *note hadamardPr
│ │ │ │ +00005300: 6f64 7563 743a 2068 6164 616d 6172 6450  oduct: hadamardP
│ │ │ │ +00005310: 726f 6475 6374 2c0a 2020 2a20 5573 6167  roduct,.  * Usag
│ │ │ │ +00005320: 653a 200a 2020 2020 2020 2020 6861 6461  e: .        hada
│ │ │ │ +00005330: 6d61 7264 5072 6f64 7563 7428 4c2c 4d29  mardProduct(L,M)
│ │ │ │ +00005340: 0a20 202a 2049 6e70 7574 733a 0a20 2020  .  * Inputs:.   
│ │ │ │ +00005350: 2020 202a 204c 2c20 6120 2a6e 6f74 6520     * L, a *note 
│ │ │ │ +00005360: 6c69 7374 3a20 284d 6163 6175 6c61 7932  list: (Macaulay2
│ │ │ │ +00005370: 446f 6329 4c69 7374 2c2c 206f 6620 2a6e  Doc)List,, of *n
│ │ │ │ +00005380: 6f74 6520 506f 696e 743a 2050 6f69 6e74  ote Point: Point
│ │ │ │ +00005390: 2c0a 2020 2020 2020 2a20 4d2c 2061 202a  ,.      * M, a *
│ │ │ │ +000053a0: 6e6f 7465 206c 6973 743a 2028 4d61 6361  note list: (Maca
│ │ │ │ +000053b0: 756c 6179 3244 6f63 294c 6973 742c 2c20  ulay2Doc)List,, 
│ │ │ │ +000053c0: 6f66 202a 6e6f 7465 2050 6f69 6e74 3a20  of *note Point: 
│ │ │ │ +000053d0: 506f 696e 742c 0a20 202a 204f 7574 7075  Point,.  * Outpu
│ │ │ │ +000053e0: 7473 3a0a 2020 2020 2020 2a20 6120 2a6e  ts:.      * a *n
│ │ │ │ +000053f0: 6f74 6520 6c69 7374 3a20 284d 6163 6175  ote list: (Macau
│ │ │ │ +00005400: 6c61 7932 446f 6329 4c69 7374 2c2c 206f  lay2Doc)List,, o
│ │ │ │ +00005410: 6620 2a6e 6f74 6520 506f 696e 743a 2050  f *note Point: P
│ │ │ │ +00005420: 6f69 6e74 2c0a 0a44 6573 6372 6970 7469  oint,..Descripti
│ │ │ │ +00005430: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  on.===========..
│ │ │ │ +00005440: 4769 7665 6e20 7477 6f20 7365 7473 206f  Given two sets o
│ │ │ │ +00005450: 6620 706f 696e 7473 2024 4c24 2061 6e64  f points $L$ and
│ │ │ │ +00005460: 2024 4d24 2072 6574 7572 6e73 2074 6865   $M$ returns the
│ │ │ │ +00005470: 206c 6973 7420 6f66 2028 7765 6c6c 2d64   list of (well-d
│ │ │ │ +00005480: 6566 696e 6564 290a 656e 7472 7977 6973  efined).entrywis
│ │ │ │ +00005490: 6520 6d75 6c74 6970 6c69 6361 7469 6f6e  e multiplication
│ │ │ │ +000054a0: 206f 6620 7061 6972 7320 6f66 2070 6f69   of pairs of poi
│ │ │ │ +000054b0: 6e74 7320 696e 2074 6865 2063 6172 7465  nts in the carte
│ │ │ │ +000054c0: 7369 616e 2070 726f 6475 6374 2024 4c5c  sian product $L\
│ │ │ │ +000054d0: 7469 6d65 730a 4d24 2e0a 0a2b 2d2d 2d2d  times.M$...+----
│ │ │ │ +000054e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000054f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00005500: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20  --------+.|i1 : 
│ │ │ │ +00005510: 4c20 3d20 7b70 6f69 6e74 7b30 2c31 7d2c  L = {point{0,1},
│ │ │ │ +00005520: 2070 6f69 6e74 7b31 2c32 7d7d 3b20 2020   point{1,2}};   
│ │ │ │ +00005530: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00005540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00005550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00005560: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 4d20  ------+.|i2 : M 
│ │ │ │ +00005570: 3d20 7b70 6f69 6e74 7b31 2c30 7d2c 2070  = {point{1,0}, p
│ │ │ │ +00005580: 6f69 6e74 7b32 2c32 7d7d 3b20 2020 2020  oint{2,2}};     
│ │ │ │ +00005590: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +000055a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000055b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000055c0: 2d2d 2d2d 2b0a 7c69 3320 3a20 6861 6461  ----+.|i3 : hada
│ │ │ │ +000055d0: 6d61 7264 5072 6f64 7563 7428 4c2c 4d29  mardProduct(L,M)
│ │ │ │ +000055e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000055f0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00005600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00005610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00005620: 2020 7c0a 7c6f 3320 3d20 7b50 6f69 6e74    |.|o3 = {Point
│ │ │ │ +00005630: 7b32 2c20 347d 2c20 506f 696e 747b 312c  {2, 4}, Point{1,
│ │ │ │ +00005640: 2030 7d2c 2050 6f69 6e74 7b30 2c20 327d   0}, Point{0, 2}
│ │ │ │ +00005650: 7d7c 0a7c 2020 2020 2020 2020 2020 2020  }|.|            
│ │ │ │ +00005660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00005670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00005680: 7c0a 7c6f 3320 3a20 4c69 7374 2020 2020  |.|o3 : List    
│ │ │ │ +00005690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000056a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000056b0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000056c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000056d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000056e0: 0a57 6179 7320 746f 2075 7365 2074 6869  .Ways to use thi
│ │ │ │ +000056f0: 7320 6d65 7468 6f64 3a0a 3d3d 3d3d 3d3d  s method:.======
│ │ │ │ +00005700: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00005710: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6861  ==..  * *note ha
│ │ │ │ +00005720: 6461 6d61 7264 5072 6f64 7563 7428 4c69  damardProduct(Li
│ │ │ │ +00005730: 7374 2c4c 6973 7429 3a20 6861 6461 6d61  st,List): hadama
│ │ │ │ +00005740: 7264 5072 6f64 7563 745f 6c70 4c69 7374  rdProduct_lpList
│ │ │ │ +00005750: 5f63 6d4c 6973 745f 7270 2c20 2d2d 0a20  _cmList_rp, --. 
│ │ │ │ +00005760: 2020 2048 6164 616d 6172 6420 7072 6f64     Hadamard prod
│ │ │ │ +00005770: 7563 7420 6f66 2074 776f 2073 6574 7320  uct of two sets 
│ │ │ │ +00005780: 6f66 2070 6f69 6e74 730a 2d2d 2d2d 2d2d  of points.------
│ │ │ │ +00005790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000057a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000057b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000057c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000057d0: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073  ---------..The s
│ │ │ │ +000057e0: 6f75 7263 6520 6f66 2074 6869 7320 646f  ource of this do
│ │ │ │ +000057f0: 6375 6d65 6e74 2069 7320 696e 0a2f 6275  cument is in./bu
│ │ │ │ +00005800: 696c 642f 7265 7072 6f64 7563 6962 6c65  ild/reproducible
│ │ │ │ +00005810: 2d70 6174 682f 6d61 6361 756c 6179 322d  -path/macaulay2-
│ │ │ │ +00005820: 312e 3235 2e31 312b 6473 2f4d 322f 4d61  1.25.11+ds/M2/Ma
│ │ │ │ +00005830: 6361 756c 6179 322f 7061 636b 6167 6573  caulay2/packages
│ │ │ │ +00005840: 2f48 6164 616d 6172 642e 6d32 0a3a 3334  /Hadamard.m2.:34
│ │ │ │ +00005850: 353a 302e 0a1f 0a46 696c 653a 2048 6164  5:0....File: Had
│ │ │ │ +00005860: 616d 6172 642e 696e 666f 2c20 4e6f 6465  amard.info, Node
│ │ │ │ +00005870: 3a20 6964 6561 6c4f 6650 726f 6a65 6374  : idealOfProject
│ │ │ │ +00005880: 6976 6550 6f69 6e74 732c 204e 6578 743a  ivePoints, Next:
│ │ │ │ +00005890: 2050 6f69 6e74 2c20 5072 6576 3a20 6861   Point, Prev: ha
│ │ │ │ +000058a0: 6461 6d61 7264 5072 6f64 7563 745f 6c70  damardProduct_lp
│ │ │ │ +000058b0: 4c69 7374 5f63 6d4c 6973 745f 7270 2c20  List_cmList_rp, 
│ │ │ │ +000058c0: 5570 3a20 546f 700a 0a69 6465 616c 4f66  Up: Top..idealOf
│ │ │ │ +000058d0: 5072 6f6a 6563 7469 7665 506f 696e 7473  ProjectivePoints
│ │ │ │ +000058e0: 202d 2d20 636f 6d70 7574 6573 2074 6865   -- computes the
│ │ │ │ +000058f0: 2069 6465 616c 206f 6620 7365 7420 6f66   ideal of set of
│ │ │ │ +00005900: 2070 6f69 6e74 730a 2a2a 2a2a 2a2a 2a2a   points.********
│ │ │ │ +00005910: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00005920: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00005930: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00005940: 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 6167  ******..  * Usag
│ │ │ │ +00005950: 653a 200a 2020 2020 2020 2020 6964 6561  e: .        idea
│ │ │ │ +00005960: 6c4f 6650 726f 6a65 6374 6976 6550 6f69  lOfProjectivePoi
│ │ │ │ +00005970: 6e74 7328 4c2c 5329 0a20 202a 2049 6e70  nts(L,S).  * Inp
│ │ │ │ +00005980: 7574 733a 0a20 2020 2020 202a 204c 2c20  uts:.      * L, 
│ │ │ │ +00005990: 6120 2a6e 6f74 6520 6c69 7374 3a20 284d  a *note list: (M
│ │ │ │ +000059a0: 6163 6175 6c61 7932 446f 6329 4c69 7374  acaulay2Doc)List
│ │ │ │ +000059b0: 2c2c 206f 6620 2a6e 6f74 6520 2870 726f  ,, of *note (pro
│ │ │ │ +000059c0: 6a65 6374 6976 6529 2070 6f69 6e74 733a  jective) points:
│ │ │ │ +000059d0: 0a20 2020 2020 2020 2050 6f69 6e74 2c0a  .        Point,.
│ │ │ │ +000059e0: 2020 2020 2020 2a20 532c 2061 202a 6e6f        * S, a *no
│ │ │ │ +000059f0: 7465 2072 696e 673a 2028 4d61 6361 756c  te ring: (Macaul
│ │ │ │ +00005a00: 6179 3244 6f63 2952 696e 672c 2c20 0a20  ay2Doc)Ring,, . 
│ │ │ │ +00005a10: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ +00005a20: 2020 2a20 492c 2061 6e20 2a6e 6f74 6520    * I, an *note 
│ │ │ │ +00005a30: 6964 6561 6c3a 2028 4d61 6361 756c 6179  ideal: (Macaulay
│ │ │ │ +00005a40: 3244 6f63 2949 6465 616c 2c2c 200a 0a44  2Doc)Ideal,, ..D
│ │ │ │ +00005a50: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ +00005a60: 3d3d 3d3d 3d3d 0a0a 4769 7665 6e20 6120  ======..Given a 
│ │ │ │ +00005a70: 7365 7420 706f 696e 7473 2024 5824 2c20  set points $X$, 
│ │ │ │ +00005a80: 6974 2072 6574 7572 6e73 2074 6865 2064  it returns the d
│ │ │ │ +00005a90: 6566 696e 696e 6720 6964 6561 6c20 6f66  efining ideal of
│ │ │ │ +00005aa0: 2024 4928 5829 240a 0a2b 2d2d 2d2d 2d2d   $I(X)$..+------
│ │ │ │ +00005ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00005ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00005ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00005ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00005af0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2053  -------+.|i1 : S
│ │ │ │ +00005b00: 203d 2051 515b 782c 792c 7a5d 2020 2020   = QQ[x,y,z]    
│ │ │ │  00005b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005b40: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00005b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005b90: 2020 2020 2020 207c 0a7c 6f31 203a 2050         |.|o1 : P
│ │ │ │ -00005ba0: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020  olynomialRing   
│ │ │ │ +00005b90: 2020 2020 2020 207c 0a7c 6f31 203d 2053         |.|o1 = S
│ │ │ │ +00005ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005be0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -00005bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005c30: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2058  -------+.|i2 : X
│ │ │ │ -00005c40: 203d 207b 706f 696e 747b 312c 312c 307d   = {point{1,1,0}
│ │ │ │ -00005c50: 2c70 6f69 6e74 7b30 2c31 2c31 7d2c 706f  ,point{0,1,1},po
│ │ │ │ -00005c60: 696e 747b 312c 322c 2d31 7d7d 2020 2020  int{1,2,-1}}    
│ │ │ │ +00005be0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00005bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00005c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00005c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00005c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00005c30: 2020 2020 2020 207c 0a7c 6f31 203a 2050         |.|o1 : P
│ │ │ │ +00005c40: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020  olynomialRing   
│ │ │ │ +00005c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00005c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005c80: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00005c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005cd0: 2020 2020 2020 207c 0a7c 6f32 203d 207b         |.|o2 = {
│ │ │ │ -00005ce0: 506f 696e 747b 312c 2031 2c20 307d 2c20  Point{1, 1, 0}, 
│ │ │ │ -00005cf0: 506f 696e 747b 302c 2031 2c20 317d 2c20  Point{0, 1, 1}, 
│ │ │ │ -00005d00: 506f 696e 747b 312c 2032 2c20 2d31 7d7d  Point{1, 2, -1}}
│ │ │ │ +00005c80: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00005c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00005ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00005cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00005cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00005cd0: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2058  -------+.|i2 : X
│ │ │ │ +00005ce0: 203d 207b 706f 696e 747b 312c 312c 307d   = {point{1,1,0}
│ │ │ │ +00005cf0: 2c70 6f69 6e74 7b30 2c31 2c31 7d2c 706f  ,point{0,1,1},po
│ │ │ │ +00005d00: 696e 747b 312c 322c 2d31 7d7d 2020 2020  int{1,2,-1}}    
│ │ │ │  00005d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005d20: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00005d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005d70: 2020 2020 2020 207c 0a7c 6f32 203a 204c         |.|o2 : L
│ │ │ │ -00005d80: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ -00005d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00005d70: 2020 2020 2020 207c 0a7c 6f32 203d 207b         |.|o2 = {
│ │ │ │ +00005d80: 506f 696e 747b 312c 2031 2c20 307d 2c20  Point{1, 1, 0}, 
│ │ │ │ +00005d90: 506f 696e 747b 302c 2031 2c20 317d 2c20  Point{0, 1, 1}, 
│ │ │ │ +00005da0: 506f 696e 747b 312c 2032 2c20 2d31 7d7d  Point{1, 2, -1}}
│ │ │ │  00005db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005dc0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -00005dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005e10: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2049  -------+.|i3 : I
│ │ │ │ -00005e20: 203d 2069 6465 616c 4f66 5072 6f6a 6563   = idealOfProjec
│ │ │ │ -00005e30: 7469 7665 506f 696e 7473 2858 2c53 2920  tivePoints(X,S) 
│ │ │ │ +00005dc0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00005dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00005de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00005df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00005e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00005e10: 2020 2020 2020 207c 0a7c 6f32 203a 204c         |.|o2 : L
│ │ │ │ +00005e20: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ +00005e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005e60: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00005e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005eb0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00005ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005ed0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -00005ee0: 3220 2020 2020 2020 2020 2020 3220 2020  2           2   
│ │ │ │ -00005ef0: 2032 2020 2020 2032 2020 2020 2020 2020   2     2        
│ │ │ │ -00005f00: 2020 2020 3220 207c 0a7c 6f33 203d 2069      2  |.|o3 = i
│ │ │ │ -00005f10: 6465 616c 2028 3378 2a7a 202d 2079 2a7a  deal (3x*z - y*z
│ │ │ │ -00005f20: 202b 207a 202c 2033 782a 7920 2d20 3379   + z , 3x*y - 3y
│ │ │ │ -00005f30: 2020 2d20 792a 7a20 2b20 347a 202c 2033    - y*z + 4z , 3
│ │ │ │ -00005f40: 7820 202d 2033 7920 202d 2032 792a 7a20  x  - 3y  - 2y*z 
│ │ │ │ -00005f50: 2b20 357a 202c 207c 0a7c 2020 2020 202d  + 5z , |.|     -
│ │ │ │ -00005f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005fa0: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2020  -------|.|      
│ │ │ │ -00005fb0: 3220 2020 2020 2020 3220 2020 2020 3320  2       2     3 
│ │ │ │ -00005fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005ff0: 2020 2020 2020 207c 0a7c 2020 2020 2079         |.|     y
│ │ │ │ -00006000: 207a 202b 2079 2a7a 2020 2d20 327a 2029   z + y*z  - 2z )
│ │ │ │ -00006010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00006020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00006030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00006040: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00006050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00005e60: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00005e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00005e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00005e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00005ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00005eb0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2049  -------+.|i3 : I
│ │ │ │ +00005ec0: 203d 2069 6465 616c 4f66 5072 6f6a 6563   = idealOfProjec
│ │ │ │ +00005ed0: 7469 7665 506f 696e 7473 2858 2c53 2920  tivePoints(X,S) 
│ │ │ │ +00005ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00005ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00005f00: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00005f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00005f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00005f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00005f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00005f50: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00005f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00005f70: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +00005f80: 3220 2020 2020 2020 2020 2020 3220 2020  2           2   
│ │ │ │ +00005f90: 2032 2020 2020 2032 2020 2020 2020 2020   2     2        
│ │ │ │ +00005fa0: 2020 2020 3220 207c 0a7c 6f33 203d 2069      2  |.|o3 = i
│ │ │ │ +00005fb0: 6465 616c 2028 3378 2a7a 202d 2079 2a7a  deal (3x*z - y*z
│ │ │ │ +00005fc0: 202b 207a 202c 2033 782a 7920 2d20 3379   + z , 3x*y - 3y
│ │ │ │ +00005fd0: 2020 2d20 792a 7a20 2b20 347a 202c 2033    - y*z + 4z , 3
│ │ │ │ +00005fe0: 7820 202d 2033 7920 202d 2032 792a 7a20  x  - 3y  - 2y*z 
│ │ │ │ +00005ff0: 2b20 357a 202c 207c 0a7c 2020 2020 202d  + 5z , |.|     -
│ │ │ │ +00006000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006040: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2020  -------|.|      
│ │ │ │ +00006050: 3220 2020 2020 2020 3220 2020 2020 3320  2       2     3 
│ │ │ │  00006060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00006090: 2020 2020 2020 207c 0a7c 6f33 203a 2049         |.|o3 : I
│ │ │ │ -000060a0: 6465 616c 206f 6620 5320 2020 2020 2020  deal of S       
│ │ │ │ +00006090: 2020 2020 2020 207c 0a7c 2020 2020 2079         |.|     y
│ │ │ │ +000060a0: 207a 202b 2079 2a7a 2020 2d20 327a 2029   z + y*z  - 2z )
│ │ │ │  000060b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000060c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000060d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000060e0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -000060f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00006100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00006110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00006120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00006130: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2049  -------+.|i4 : I
│ │ │ │ -00006140: 3220 3d20 6861 6461 6d61 7264 506f 7765  2 = hadamardPowe
│ │ │ │ -00006150: 7228 492c 3229 2020 2020 2020 2020 2020  r(I,2)          
│ │ │ │ +000060e0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000060f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006130: 2020 2020 2020 207c 0a7c 6f33 203a 2049         |.|o3 : I
│ │ │ │ +00006140: 6465 616c 206f 6620 5320 2020 2020 2020  deal of S       
│ │ │ │ +00006150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00006180: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00006190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000061a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000061b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000061c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000061d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000061e0: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ -000061f0: 2032 2020 2020 2020 3220 2020 2020 3320   2      2     3 
│ │ │ │ -00006200: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -00006210: 2032 2020 2020 2020 2032 2020 2020 3320   2       2    3 
│ │ │ │ -00006220: 2020 2020 3220 207c 0a7c 6f34 203d 2069      2  |.|o4 = i
│ │ │ │ -00006230: 6465 616c 2028 7920 7a20 2d20 3138 782a  deal (y z - 18x*
│ │ │ │ -00006240: 7a20 202b 2079 2a7a 2020 2d20 327a 202c  z  + y*z  - 2z ,
│ │ │ │ -00006250: 2078 2a79 2a7a 202d 2034 782a 7a20 2c20   x*y*z - 4x*z , 
│ │ │ │ -00006260: 7820 7a20 2d20 782a 7a20 2c20 3278 2020  x z - x*z , 2x  
│ │ │ │ -00006270: 2d20 3378 2079 207c 0a7c 2020 2020 202d  - 3x y |.|     -
│ │ │ │ -00006280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00006290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000062a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000062b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000062c0: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2020  -------|.|      
│ │ │ │ -000062d0: 2020 2020 3220 2020 2020 2020 3220 2020      2       2   
│ │ │ │ -000062e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000062f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00006300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00006310: 2020 2020 2020 207c 0a7c 2020 2020 202b         |.|     +
│ │ │ │ -00006320: 2078 2a79 2020 2d20 3678 2a7a 2029 2020   x*y  - 6x*z )  
│ │ │ │ -00006330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00006340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00006350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00006360: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00006370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006180: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00006190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000061a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000061b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000061c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000061d0: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2049  -------+.|i4 : I
│ │ │ │ +000061e0: 3220 3d20 6861 6461 6d61 7264 506f 7765  2 = hadamardPowe
│ │ │ │ +000061f0: 7228 492c 3229 2020 2020 2020 2020 2020  r(I,2)          
│ │ │ │ +00006200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006220: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00006230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006270: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00006280: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +00006290: 2032 2020 2020 2020 3220 2020 2020 3320   2      2     3 
│ │ │ │ +000062a0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +000062b0: 2032 2020 2020 2020 2032 2020 2020 3320   2       2    3 
│ │ │ │ +000062c0: 2020 2020 3220 207c 0a7c 6f34 203d 2069      2  |.|o4 = i
│ │ │ │ +000062d0: 6465 616c 2028 7920 7a20 2d20 3138 782a  deal (y z - 18x*
│ │ │ │ +000062e0: 7a20 202b 2079 2a7a 2020 2d20 327a 202c  z  + y*z  - 2z ,
│ │ │ │ +000062f0: 2078 2a79 2a7a 202d 2034 782a 7a20 2c20   x*y*z - 4x*z , 
│ │ │ │ +00006300: 7820 7a20 2d20 782a 7a20 2c20 3278 2020  x z - x*z , 2x  
│ │ │ │ +00006310: 2d20 3378 2079 207c 0a7c 2020 2020 202d  - 3x y |.|     -
│ │ │ │ +00006320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006360: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2020  -------|.|      
│ │ │ │ +00006370: 2020 2020 3220 2020 2020 2020 3220 2020      2       2   
│ │ │ │  00006380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000063a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000063b0: 2020 2020 2020 207c 0a7c 6f34 203a 2049         |.|o4 : I
│ │ │ │ -000063c0: 6465 616c 206f 6620 5320 2020 2020 2020  deal of S       
│ │ │ │ +000063b0: 2020 2020 2020 207c 0a7c 2020 2020 202b         |.|     +
│ │ │ │ +000063c0: 2078 2a79 2020 2d20 3678 2a7a 2029 2020   x*y  - 6x*z )  
│ │ │ │  000063d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000063e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000063f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00006400: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -00006410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00006420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00006430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00006440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00006450: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2058  -------+.|i5 : X
│ │ │ │ -00006460: 3220 3d20 6861 6461 6d61 7264 506f 7765  2 = hadamardPowe
│ │ │ │ -00006470: 7228 582c 3229 2020 2020 2020 2020 2020  r(X,2)          
│ │ │ │ +00006400: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00006410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006450: 2020 2020 2020 207c 0a7c 6f34 203a 2049         |.|o4 : I
│ │ │ │ +00006460: 6465 616c 206f 6620 5320 2020 2020 2020  deal of S       
│ │ │ │ +00006470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000064a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000064b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000064c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000064d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000064e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00006540: 2020 2020 2020 207c 0a7c 2020 2020 202d         |.|     -
│ │ │ │ -00006550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00006560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00006570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00006580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -000065a0: 6f69 6e74 7b31 2c20 312c 2030 7d2c 2050  oint{1, 1, 0}, P
│ │ │ │ -000065b0: 6f69 6e74 7b31 2c20 322c 2030 7d7d 2020  oint{1, 2, 0}}  
│ │ │ │ -000065c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000065d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000065e0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000065f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00006610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00006620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00006630: 2020 2020 2020 207c 0a7c 6f35 203a 204c         |.|o5 : L
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│ │ │ │ -00006650: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000064f0: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2058  -------+.|i5 : X
│ │ │ │ +00006500: 3220 3d20 6861 6461 6d61 7264 506f 7765  2 = hadamardPowe
│ │ │ │ +00006510: 7228 582c 3229 2020 2020 2020 2020 2020  r(X,2)          
│ │ │ │ +00006520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006540: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00006550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006590: 2020 2020 2020 207c 0a7c 6f35 203d 207b         |.|o5 = {
│ │ │ │ +000065a0: 506f 696e 747b 312c 2034 2c20 317d 2c20  Point{1, 4, 1}, 
│ │ │ │ +000065b0: 506f 696e 747b 302c 2032 2c20 2d31 7d2c  Point{0, 2, -1},
│ │ │ │ +000065c0: 2050 6f69 6e74 7b30 2c20 312c 2030 7d2c   Point{0, 1, 0},
│ │ │ │ +000065d0: 2050 6f69 6e74 7b30 2c20 312c 2031 7d2c   Point{0, 1, 1},
│ │ │ │ +000065e0: 2020 2020 2020 207c 0a7c 2020 2020 202d         |.|     -
│ │ │ │ +000065f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006630: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2050  -------|.|     P
│ │ │ │ +00006640: 6f69 6e74 7b31 2c20 312c 2030 7d2c 2050  oint{1, 1, 0}, P
│ │ │ │ +00006650: 6f69 6e74 7b31 2c20 322c 2030 7d7d 2020  oint{1, 2, 0}}  
│ │ │ │  00006660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00006680: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -00006690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000066a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000066b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000066c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000066d0: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 2049  -------+.|i6 : I
│ │ │ │ -000066e0: 3220 3d3d 2069 6465 616c 4f66 5072 6f6a  2 == idealOfProj
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│ │ │ │ -00006700: 5329 2020 2020 2020 2020 2020 2020 2020  S)              
│ │ │ │ +00006680: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00006690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000066a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000066b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000066c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000066d0: 2020 2020 2020 207c 0a7c 6f35 203a 204c         |.|o5 : L
│ │ │ │ +000066e0: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ +000066f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00006720: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00006730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00006740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00006750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00006760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00006770: 2020 2020 2020 207c 0a7c 6f36 203d 2074         |.|o6 = t
│ │ │ │ -00006780: 7275 6520 2020 2020 2020 2020 2020 2020  rue             
│ │ │ │ -00006790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000067a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006720: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00006730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006770: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 2049  -------+.|i6 : I
│ │ │ │ +00006780: 3220 3d3d 2069 6465 616c 4f66 5072 6f6a  2 == idealOfProj
│ │ │ │ +00006790: 6563 7469 7665 506f 696e 7473 2858 322c  ectivePoints(X2,
│ │ │ │ +000067a0: 5329 2020 2020 2020 2020 2020 2020 2020  S)              
│ │ │ │  000067b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000067c0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -000067d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000067e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000067f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00006800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00006810: 2d2d 2d2d 2d2d 2d2b 0a0a 5761 7973 2074  -------+..Ways t
│ │ │ │ -00006820: 6f20 7573 6520 6964 6561 6c4f 6650 726f  o use idealOfPro
│ │ │ │ -00006830: 6a65 6374 6976 6550 6f69 6e74 733a 0a3d  jectivePoints:.=
│ │ │ │ -00006840: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00006850: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ -000068a0: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  r.==============
│ │ │ │ -000068b0: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374  ====..The object
│ │ │ │ -000068c0: 202a 6e6f 7465 2069 6465 616c 4f66 5072   *note idealOfPr
│ │ │ │ -000068d0: 6f6a 6563 7469 7665 506f 696e 7473 3a20  ojectivePoints: 
│ │ │ │ -000068e0: 6964 6561 6c4f 6650 726f 6a65 6374 6976  idealOfProjectiv
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│ │ │ │ -00006940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00006950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00006960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00006970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00006980: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365  ----..The source
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│ │ │ │ -000069a0: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72  t is in./build/r
│ │ │ │ -000069b0: 6570 726f 6475 6369 626c 652d 7061 7468  eproducible-path
│ │ │ │ -000069c0: 2f6d 6163 6175 6c61 7932 2d31 2e32 352e  /macaulay2-1.25.
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│ │ │ │ -00006a40: 6f6a 6563 7469 7665 506f 696e 7473 2c20  ojectivePoints, 
│ │ │ │ -00006a50: 5570 3a20 546f 700a 0a50 6f69 6e74 202d  Up: Top..Point -
│ │ │ │ -00006a60: 2d20 6120 6e65 7720 7479 7065 2066 6f72  - a new type for
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│ │ │ │ -00006a90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00006aa0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00006ab0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a  **************..
│ │ │ │ -00006ac0: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
│ │ │ │ -00006ad0: 3d3d 3d3d 3d3d 3d0a 0a41 2070 6f69 6e74  =======..A point
│ │ │ │ -00006ae0: 2069 6e20 7072 6f6a 6563 7469 7665 2073   in projective s
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│ │ │ │ -00006b30: 2c2e 2041 6e20 656c 656d 656e 7420 6f66  ,. An element of
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│ │ │ │ -00006b70: 4261 7369 634c 6973 742c 2e0a 0a4d 6574  BasicList,...Met
│ │ │ │ -00006b80: 686f 6473 2074 6861 7420 7573 6520 616e  hods that use an
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│ │ │ │ -00006ba0: 2050 6f69 6e74 3a0a 3d3d 3d3d 3d3d 3d3d   Point:.========
│ │ │ │ -00006bb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00006bc0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00006bd0: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 506f  ==..  * *note Po
│ │ │ │ -00006be0: 696e 7420 2a20 506f 696e 743a 2050 6f69  int * Point: Poi
│ │ │ │ -00006bf0: 6e74 205f 7374 2050 6f69 6e74 2c20 2d2d  nt _st Point, --
│ │ │ │ -00006c00: 2065 6e74 7279 7769 7365 2070 726f 6475   entrywise produ
│ │ │ │ -00006c10: 6374 206f 6620 7477 6f0a 2020 2020 7072  ct of two.    pr
│ │ │ │ -00006c20: 6f6a 6563 7469 7665 2070 6f69 6e74 730a  ojective points.
│ │ │ │ -00006c30: 2020 2a20 2a6e 6f74 6520 506f 696e 7420    * *note Point 
│ │ │ │ -00006c40: 3d3d 2050 6f69 6e74 3a20 506f 696e 7420  == Point: Point 
│ │ │ │ -00006c50: 3d3d 2050 6f69 6e74 2c20 2d2d 2063 6865  == Point, -- che
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│ │ │ │ -00006c70: 776f 2070 726f 6a65 6374 6976 650a 2020  wo projective.  
│ │ │ │ -00006c80: 2020 706f 696e 7473 0a0a 466f 7220 7468    points..For th
│ │ │ │ -00006c90: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d  e programmer.===
│ │ │ │ -00006ca0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ -00006cb0: 0a54 6865 206f 626a 6563 7420 2a6e 6f74  .The object *not
│ │ │ │ -00006cc0: 6520 506f 696e 743a 2050 6f69 6e74 2c20  e Point: Point, 
│ │ │ │ -00006cd0: 6973 2061 202a 6e6f 7465 2074 7970 653a  is a *note type:
│ │ │ │ -00006ce0: 2028 4d61 6361 756c 6179 3244 6f63 2954   (Macaulay2Doc)T
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│ │ │ │ -00006d10: 6520 4261 7369 634c 6973 743a 2028 4d61  e BasicList: (Ma
│ │ │ │ -00006d20: 6361 756c 6179 3244 6f63 2942 6173 6963  caulay2Doc)Basic
│ │ │ │ -00006d30: 4c69 7374 2c20 3c20 2a6e 6f74 6520 5468  List, < *note Th
│ │ │ │ -00006d40: 696e 673a 0a28 4d61 6361 756c 6179 3244  ing:.(Macaulay2D
│ │ │ │ -00006d50: 6f63 2954 6869 6e67 2c2e 0a0a 2d2d 2d2d  oc)Thing,...----
│ │ │ │ -00006d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00006d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00006da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865  -----------..The
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│ │ │ │ -00006e00: 4d61 6361 756c 6179 322f 7061 636b 6167  Macaulay2/packag
│ │ │ │ -00006e10: 6573 2f48 6164 616d 6172 642e 6d32 0a3a  es/Hadamard.m2.:
│ │ │ │ -00006e20: 3138 333a 302e 0a1f 0a46 696c 653a 2048  183:0....File: H
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│ │ │ │ -00006e60: 2c20 5072 6576 3a20 506f 696e 742c 2055  , Prev: Point, U
│ │ │ │ -00006e70: 703a 2054 6f70 0a0a 706f 696e 7420 2d2d  p: Top..point --
│ │ │ │ -00006e80: 2063 6f6e 7374 7275 6374 7320 6120 7072   constructs a pr
│ │ │ │ -00006e90: 6f6a 6563 7469 7665 2070 6f69 6e74 2066  ojective point f
│ │ │ │ -00006ea0: 726f 6d20 7468 6520 6c69 7374 2028 6f72  rom the list (or
│ │ │ │ -00006eb0: 2061 7272 6179 2920 6f66 2063 6f6f 7264   array) of coord
│ │ │ │ -00006ec0: 696e 6174 6573 2e0a 2a2a 2a2a 2a2a 2a2a  inates..********
│ │ │ │ -00006ed0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00006ee0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00006ef0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00006f00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00006f10: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 7361  *******..  * Usa
│ │ │ │ -00006f20: 6765 3a20 0a20 2020 2020 2020 2070 6f69  ge: .        poi
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│ │ │ │ -00006f40: 3a0a 2020 2020 2020 2a20 4c2c 2061 202a  :.      * L, a *
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│ │ │ │ -000070e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00007110: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
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│ │ │ │ -00007230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00007240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00007250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54  -------------..T
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│ │ │ │ -000072c0: 6167 6573 2f48 6164 616d 6172 642e 6d32  ages/Hadamard.m2
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│ │ │ │ -00007300: 506f 696e 742c 204e 6578 743a 2050 6f69  Point, Next: Poi
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│ │ │ │ -00007370: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00007380: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00007390: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000073a0: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ -000073b0: 4f70 6572 6174 6f72 3a20 2a6e 6f74 6520  Operator: *note 
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│ │ │ │ -00007430: 742c 2c20 0a20 2020 2020 202a 2071 2c20  t,, .      * q, 
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│ │ │ │ -000074a0: 2050 6f69 6e74 3a20 506f 696e 742c 2c20   Point: Point,, 
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│ │ │ │ -000074d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000074e0: 2d2d 2d2d 2b0a 7c69 3120 3a20 7020 3d20  ----+.|i1 : p = 
│ │ │ │ -000074f0: 706f 696e 7420 7b31 2c32 2c33 7d3b 207c  point {1,2,3}; |
│ │ │ │ -00007500: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00007510: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220  ----------+.|i2 
│ │ │ │ -00007520: 3a20 7120 3d20 706f 696e 7420 7b2d 312c  : q = point {-1,
│ │ │ │ -00007530: 322c 357d 3b7c 0a2b 2d2d 2d2d 2d2d 2d2d  2,5};|.+--------
│ │ │ │ -00007540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00007550: 2b0a 7c69 3320 3a20 7020 2a20 7120 2020  +.|i3 : p * q   
│ │ │ │ -00007560: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00007570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00007580: 2020 2020 2020 7c0a 7c6f 3320 3d20 506f        |.|o3 = Po
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│ │ │ │ -000075a0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000075b0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -000075c0: 3320 3a20 506f 696e 7420 2020 2020 2020  3 : Point       
│ │ │ │ -000075d0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +000067c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000067d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000067e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000067f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006810: 2020 2020 2020 207c 0a7c 6f36 203d 2074         |.|o6 = t
│ │ │ │ +00006820: 7275 6520 2020 2020 2020 2020 2020 2020  rue             
│ │ │ │ +00006830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006860: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00006870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000068a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000068b0: 2d2d 2d2d 2d2d 2d2b 0a0a 5761 7973 2074  -------+..Ways t
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│ │ │ │ +000068e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +000068f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ +00006980: 6964 6561 6c4f 6650 726f 6a65 6374 6976  idealOfProjectiv
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│ │ │ │ +000069e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000069f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006a20: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365  ----..The source
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│ │ │ │ +00006b30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00006b40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00006b50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a  **************..
│ │ │ │ +00006b60: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
│ │ │ │ +00006b70: 3d3d 3d3d 3d3d 3d0a 0a41 2070 6f69 6e74  =======..A point
│ │ │ │ +00006b80: 2069 6e20 7072 6f6a 6563 7469 7665 2073   in projective s
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│ │ │ │ +00006c40: 2050 6f69 6e74 3a0a 3d3d 3d3d 3d3d 3d3d   Point:.========
│ │ │ │ +00006c50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00006c60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ +00006cd0: 2020 2a20 2a6e 6f74 6520 506f 696e 7420    * *note Point 
│ │ │ │ +00006ce0: 3d3d 2050 6f69 6e74 3a20 506f 696e 7420  == Point: Point 
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│ │ │ │ +00006dd0: 4c69 7374 2c20 3c20 2a6e 6f74 6520 5468  List, < *note Th
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│ │ │ │ +00006df0: 6f63 2954 6869 6e67 2c2e 0a0a 2d2d 2d2d  oc)Thing,...----
│ │ │ │ +00006e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865  -----------..The
│ │ │ │ +00006e50: 2073 6f75 7263 6520 6f66 2074 6869 7320   source of this 
│ │ │ │ +00006e60: 646f 6375 6d65 6e74 2069 7320 696e 0a2f  document is in./
│ │ │ │ +00006e70: 6275 696c 642f 7265 7072 6f64 7563 6962  build/reproducib
│ │ │ │ +00006e80: 6c65 2d70 6174 682f 6d61 6361 756c 6179  le-path/macaulay
│ │ │ │ +00006e90: 322d 312e 3235 2e31 312b 6473 2f4d 322f  2-1.25.11+ds/M2/
│ │ │ │ +00006ea0: 4d61 6361 756c 6179 322f 7061 636b 6167  Macaulay2/packag
│ │ │ │ +00006eb0: 6573 2f48 6164 616d 6172 642e 6d32 0a3a  es/Hadamard.m2.:
│ │ │ │ +00006ec0: 3138 333a 302e 0a1f 0a46 696c 653a 2048  183:0....File: H
│ │ │ │ +00006ed0: 6164 616d 6172 642e 696e 666f 2c20 4e6f  adamard.info, No
│ │ │ │ +00006ee0: 6465 3a20 706f 696e 742c 204e 6578 743a  de: point, Next:
│ │ │ │ +00006ef0: 2050 6f69 6e74 205f 7374 2050 6f69 6e74   Point _st Point
│ │ │ │ +00006f00: 2c20 5072 6576 3a20 506f 696e 742c 2055  , Prev: Point, U
│ │ │ │ +00006f10: 703a 2054 6f70 0a0a 706f 696e 7420 2d2d  p: Top..point --
│ │ │ │ +00006f20: 2063 6f6e 7374 7275 6374 7320 6120 7072   constructs a pr
│ │ │ │ +00006f30: 6f6a 6563 7469 7665 2070 6f69 6e74 2066  ojective point f
│ │ │ │ +00006f40: 726f 6d20 7468 6520 6c69 7374 2028 6f72  rom the list (or
│ │ │ │ +00006f50: 2061 7272 6179 2920 6f66 2063 6f6f 7264   array) of coord
│ │ │ │ +00006f60: 696e 6174 6573 2e0a 2a2a 2a2a 2a2a 2a2a  inates..********
│ │ │ │ +00006f70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00006f80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00006f90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00006fa0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00006fb0: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 7361  *******..  * Usa
│ │ │ │ +00006fc0: 6765 3a20 0a20 2020 2020 2020 2070 6f69  ge: .        poi
│ │ │ │ +00006fd0: 6e74 284c 290a 2020 2a20 496e 7075 7473  nt(L).  * Inputs
│ │ │ │ +00006fe0: 3a0a 2020 2020 2020 2a20 4c2c 2061 202a  :.      * L, a *
│ │ │ │ +00006ff0: 6e6f 7465 206c 6973 743a 2028 4d61 6361  note list: (Maca
│ │ │ │ +00007000: 756c 6179 3244 6f63 294c 6973 742c 2c20  ulay2Doc)List,, 
│ │ │ │ +00007010: 6f72 202a 6e6f 7465 2061 7272 6179 3a0a  or *note array:.
│ │ │ │ +00007020: 2020 2020 2020 2020 284d 6163 6175 6c61          (Macaula
│ │ │ │ +00007030: 7932 446f 6329 4172 7261 792c 206f 7220  y2Doc)Array, or 
│ │ │ │ +00007040: 2a6e 6f74 6520 7669 7369 626c 6520 6c69  *note visible li
│ │ │ │ +00007050: 7374 3a20 284d 6163 6175 6c61 7932 446f  st: (Macaulay2Do
│ │ │ │ +00007060: 6329 5669 7369 626c 654c 6973 742c 0a20  c)VisibleList,. 
│ │ │ │ +00007070: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ +00007080: 2020 2a20 616e 2069 6e73 7461 6e63 6520    * an instance 
│ │ │ │ +00007090: 6f66 2074 6865 2074 7970 6520 2a6e 6f74  of the type *not
│ │ │ │ +000070a0: 6520 506f 696e 743a 2050 6f69 6e74 2c2c  e Point: Point,,
│ │ │ │ +000070b0: 200a 0a44 6573 6372 6970 7469 6f6e 0a3d   ..Description.=
│ │ │ │ +000070c0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2b2d 2d2d  ==========..+---
│ │ │ │ +000070d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000070e0: 2b0a 7c69 3120 3a20 706f 696e 7420 7b31  +.|i1 : point {1
│ │ │ │ +000070f0: 2c32 2c33 7d20 7c0a 7c20 2020 2020 2020  ,2,3} |.|       
│ │ │ │ +00007100: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00007110: 3120 3d20 506f 696e 747b 312c 2032 2c20  1 = Point{1, 2, 
│ │ │ │ +00007120: 337d 7c0a 7c20 2020 2020 2020 2020 2020  3}|.|           
│ │ │ │ +00007130: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20          |.|o1 : 
│ │ │ │ +00007140: 506f 696e 7420 2020 2020 2020 2020 7c0a  Point         |.
│ │ │ │ +00007150: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00007160: 2d2d 2d2d 2b0a 7c69 3220 3a20 706f 696e  ----+.|i2 : poin
│ │ │ │ +00007170: 7420 5b31 2c34 2c36 5d20 7c0a 7c20 2020  t [1,4,6] |.|   
│ │ │ │ +00007180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00007190: 7c0a 7c6f 3220 3d20 506f 696e 747b 312c  |.|o2 = Point{1,
│ │ │ │ +000071a0: 2034 2c20 367d 7c0a 7c20 2020 2020 2020   4, 6}|.|       
│ │ │ │ +000071b0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +000071c0: 3220 3a20 506f 696e 7420 2020 2020 2020  2 : Point       
│ │ │ │ +000071d0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +000071e0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a57 6179 7320  --------+..Ways 
│ │ │ │ +000071f0: 746f 2075 7365 2070 6f69 6e74 3a0a 3d3d  to use point:.==
│ │ │ │ +00007200: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00007210: 0a0a 2020 2a20 2270 6f69 6e74 2856 6973  ..  * "point(Vis
│ │ │ │ +00007220: 6962 6c65 4c69 7374 2922 0a0a 466f 7220  ibleList)"..For 
│ │ │ │ +00007230: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d  the programmer.=
│ │ │ │ +00007240: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00007250: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e  =..The object *n
│ │ │ │ +00007260: 6f74 6520 706f 696e 743a 2070 6f69 6e74  ote point: point
│ │ │ │ +00007270: 2c20 6973 2061 202a 6e6f 7465 206d 6574  , is a *note met
│ │ │ │ +00007280: 686f 6420 6675 6e63 7469 6f6e 3a0a 284d  hod function:.(M
│ │ │ │ +00007290: 6163 6175 6c61 7932 446f 6329 4d65 7468  acaulay2Doc)Meth
│ │ │ │ +000072a0: 6f64 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d  odFunction,...--
│ │ │ │ +000072b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000072c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000072d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000072e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000072f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54  -------------..T
│ │ │ │ +00007300: 6865 2073 6f75 7263 6520 6f66 2074 6869  he source of thi
│ │ │ │ +00007310: 7320 646f 6375 6d65 6e74 2069 7320 696e  s document is in
│ │ │ │ +00007320: 0a2f 6275 696c 642f 7265 7072 6f64 7563  ./build/reproduc
│ │ │ │ +00007330: 6962 6c65 2d70 6174 682f 6d61 6361 756c  ible-path/macaul
│ │ │ │ +00007340: 6179 322d 312e 3235 2e31 312b 6473 2f4d  ay2-1.25.11+ds/M
│ │ │ │ +00007350: 322f 4d61 6361 756c 6179 322f 7061 636b  2/Macaulay2/pack
│ │ │ │ +00007360: 6167 6573 2f48 6164 616d 6172 642e 6d32  ages/Hadamard.m2
│ │ │ │ +00007370: 0a3a 3234 343a 302e 0a1f 0a46 696c 653a  .:244:0....File:
│ │ │ │ +00007380: 2048 6164 616d 6172 642e 696e 666f 2c20   Hadamard.info, 
│ │ │ │ +00007390: 4e6f 6465 3a20 506f 696e 7420 5f73 7420  Node: Point _st 
│ │ │ │ +000073a0: 506f 696e 742c 204e 6578 743a 2050 6f69  Point, Next: Poi
│ │ │ │ +000073b0: 6e74 203d 3d20 506f 696e 742c 2050 7265  nt == Point, Pre
│ │ │ │ +000073c0: 763a 2070 6f69 6e74 2c20 5570 3a20 546f  v: point, Up: To
│ │ │ │ +000073d0: 700a 0a50 6f69 6e74 202a 2050 6f69 6e74  p..Point * Point
│ │ │ │ +000073e0: 202d 2d20 656e 7472 7977 6973 6520 7072   -- entrywise pr
│ │ │ │ +000073f0: 6f64 7563 7420 6f66 2074 776f 2070 726f  oduct of two pro
│ │ │ │ +00007400: 6a65 6374 6976 6520 706f 696e 7473 0a2a  jective points.*
│ │ │ │ +00007410: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00007420: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00007430: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00007440: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ +00007450: 4f70 6572 6174 6f72 3a20 2a6e 6f74 6520  Operator: *note 
│ │ │ │ +00007460: 2a3a 2028 4d61 6361 756c 6179 3244 6f63  *: (Macaulay2Doc
│ │ │ │ +00007470: 295f 7374 2c0a 2020 2a20 5573 6167 653a  )_st,.  * Usage:
│ │ │ │ +00007480: 200a 2020 2020 2020 2020 7020 2a20 710a   .        p * q.
│ │ │ │ +00007490: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ +000074a0: 2020 2a20 702c 2061 6e20 696e 7374 616e    * p, an instan
│ │ │ │ +000074b0: 6365 206f 6620 7468 6520 7479 7065 202a  ce of the type *
│ │ │ │ +000074c0: 6e6f 7465 2050 6f69 6e74 3a20 506f 696e  note Point: Poin
│ │ │ │ +000074d0: 742c 2c20 0a20 2020 2020 202a 2071 2c20  t,, .      * q, 
│ │ │ │ +000074e0: 616e 2069 6e73 7461 6e63 6520 6f66 2074  an instance of t
│ │ │ │ +000074f0: 6865 2074 7970 6520 2a6e 6f74 6520 506f  he type *note Po
│ │ │ │ +00007500: 696e 743a 2050 6f69 6e74 2c2c 200a 2020  int: Point,, .  
│ │ │ │ +00007510: 2a20 4f75 7470 7574 733a 0a20 2020 2020  * Outputs:.     
│ │ │ │ +00007520: 202a 2061 6e20 696e 7374 616e 6365 206f   * an instance o
│ │ │ │ +00007530: 6620 7468 6520 7479 7065 202a 6e6f 7465  f the type *note
│ │ │ │ +00007540: 2050 6f69 6e74 3a20 506f 696e 742c 2c20   Point: Point,, 
│ │ │ │ +00007550: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d  ..Description.==
│ │ │ │ +00007560: 3d3d 3d3d 3d3d 3d3d 3d0a 0a2b 2d2d 2d2d  =========..+----
│ │ │ │ +00007570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00007580: 2d2d 2d2d 2b0a 7c69 3120 3a20 7020 3d20  ----+.|i1 : p = 
│ │ │ │ +00007590: 706f 696e 7420 7b31 2c32 2c33 7d3b 207c  point {1,2,3}; |
│ │ │ │ +000075a0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000075b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220  ----------+.|i2 
│ │ │ │ +000075c0: 3a20 7120 3d20 706f 696e 7420 7b2d 312c  : q = point {-1,
│ │ │ │ +000075d0: 322c 357d 3b7c 0a2b 2d2d 2d2d 2d2d 2d2d  2,5};|.+--------
│ │ │ │  000075e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000075f0: 2d2d 2b0a 0a4e 6f74 6520 7468 6174 2074  --+..Note that t
│ │ │ │ -00007600: 6869 7320 6f70 6572 6174 696f 6e20 6973  his operation is
│ │ │ │ -00007610: 206e 6f74 2061 6c77 6179 7320 7765 6c6c   not always well
│ │ │ │ -00007620: 2d64 6566 696e 6564 2069 6e20 7072 6f6a  -defined in proj
│ │ │ │ -00007630: 6563 7469 7665 2073 7061 6365 2c20 652e  ective space, e.
│ │ │ │ -00007640: 672e 2c0a 7468 6520 4861 6461 6d61 7264  g.,.the Hadamard
│ │ │ │ -00007650: 2070 726f 6475 6374 206f 6620 7468 6520   product of the 
│ │ │ │ -00007660: 706f 696e 7473 2024 5b31 3a30 5d24 2061  points $[1:0]$ a
│ │ │ │ -00007670: 6e64 2024 5b30 3a31 5d24 2069 7320 6e6f  nd $[0:1]$ is no
│ │ │ │ -00007680: 7420 7765 6c6c 2d64 6566 696e 6564 0a0a  t well-defined..
│ │ │ │ -00007690: 5761 7973 2074 6f20 7573 6520 7468 6973  Ways to use this
│ │ │ │ -000076a0: 206d 6574 686f 643a 0a3d 3d3d 3d3d 3d3d   method:.=======
│ │ │ │ -000076b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000076c0: 3d0a 0a20 202a 202a 6e6f 7465 2050 6f69  =..  * *note Poi
│ │ │ │ -000076d0: 6e74 202a 2050 6f69 6e74 3a20 506f 696e  nt * Point: Poin
│ │ │ │ -000076e0: 7420 5f73 7420 506f 696e 742c 202d 2d20  t _st Point, -- 
│ │ │ │ -000076f0: 656e 7472 7977 6973 6520 7072 6f64 7563  entrywise produc
│ │ │ │ -00007700: 7420 6f66 2074 776f 0a20 2020 2070 726f  t of two.    pro
│ │ │ │ -00007710: 6a65 6374 6976 6520 706f 696e 7473 0a2d  jective points.-
│ │ │ │ -00007720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00007730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00007740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00007750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00007760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a  --------------..
│ │ │ │ -00007770: 5468 6520 736f 7572 6365 206f 6620 7468  The source of th
│ │ │ │ -00007780: 6973 2064 6f63 756d 656e 7420 6973 2069  is document is i
│ │ │ │ -00007790: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475  n./build/reprodu
│ │ │ │ -000077a0: 6369 626c 652d 7061 7468 2f6d 6163 6175  cible-path/macau
│ │ │ │ -000077b0: 6c61 7932 2d31 2e32 352e 3131 2b64 732f  lay2-1.25.11+ds/
│ │ │ │ -000077c0: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163  M2/Macaulay2/pac
│ │ │ │ -000077d0: 6b61 6765 732f 4861 6461 6d61 7264 2e6d  kages/Hadamard.m
│ │ │ │ -000077e0: 320a 3a32 3035 3a30 2e0a 1f0a 4669 6c65  2.:205:0....File
│ │ │ │ -000077f0: 3a20 4861 6461 6d61 7264 2e69 6e66 6f2c  : Hadamard.info,
│ │ │ │ -00007800: 204e 6f64 653a 2050 6f69 6e74 203d 3d20   Node: Point == 
│ │ │ │ -00007810: 506f 696e 742c 2050 7265 763a 2050 6f69  Point, Prev: Poi
│ │ │ │ -00007820: 6e74 205f 7374 2050 6f69 6e74 2c20 5570  nt _st Point, Up
│ │ │ │ -00007830: 3a20 546f 700a 0a50 6f69 6e74 203d 3d20  : Top..Point == 
│ │ │ │ -00007840: 506f 696e 7420 2d2d 2063 6865 636b 2065  Point -- check e
│ │ │ │ -00007850: 7175 616c 6974 7920 6f66 2074 776f 2070  quality of two p
│ │ │ │ -00007860: 726f 6a65 6374 6976 6520 706f 696e 7473  rojective points
│ │ │ │ -00007870: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  .***************
│ │ │ │ -00007880: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00007890: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000078a0: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ -000078b0: 4f70 6572 6174 6f72 3a20 2a6e 6f74 6520  Operator: *note 
│ │ │ │ -000078c0: 3d3d 3a20 284d 6163 6175 6c61 7932 446f  ==: (Macaulay2Do
│ │ │ │ -000078d0: 6329 3d3d 2c0a 2020 2a20 5573 6167 653a  c)==,.  * Usage:
│ │ │ │ -000078e0: 200a 2020 2020 2020 2020 7020 3d3d 2071   .        p == q
│ │ │ │ -000078f0: 0a20 202a 2049 6e70 7574 733a 0a20 2020  .  * Inputs:.   
│ │ │ │ -00007900: 2020 202a 2070 2c20 616e 2069 6e73 7461     * p, an insta
│ │ │ │ -00007910: 6e63 6520 6f66 2074 6865 2074 7970 6520  nce of the type 
│ │ │ │ -00007920: 2a6e 6f74 6520 506f 696e 743a 2050 6f69  *note Point: Poi
│ │ │ │ -00007930: 6e74 2c2c 200a 2020 2020 2020 2a20 712c  nt,, .      * q,
│ │ │ │ -00007940: 2061 6e20 696e 7374 616e 6365 206f 6620   an instance of 
│ │ │ │ -00007950: 7468 6520 7479 7065 202a 6e6f 7465 2050  the type *note P
│ │ │ │ -00007960: 6f69 6e74 3a20 506f 696e 742c 2c20 0a20  oint: Point,, . 
│ │ │ │ -00007970: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ -00007980: 2020 2a20 6120 2a6e 6f74 6520 426f 6f6c    * a *note Bool
│ │ │ │ -00007990: 6561 6e20 7661 6c75 653a 2028 4d61 6361  ean value: (Maca
│ │ │ │ -000079a0: 756c 6179 3244 6f63 2942 6f6f 6c65 616e  ulay2Doc)Boolean
│ │ │ │ -000079b0: 2c2c 200a 0a44 6573 6372 6970 7469 6f6e  ,, ..Description
│ │ │ │ -000079c0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2b2d  .===========..+-
│ │ │ │ -000079d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000079e0: 2d2d 2d2d 2b0a 7c69 3120 3a20 7020 3d20  ----+.|i1 : p = 
│ │ │ │ -000079f0: 706f 696e 7420 7b31 2c31 7d3b 7c0a 2b2d  point {1,1};|.+-
│ │ │ │ -00007a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00007a10: 2d2d 2d2d 2b0a 7c69 3220 3a20 7120 3d20  ----+.|i2 : q = 
│ │ │ │ -00007a20: 706f 696e 7420 7b32 2c32 7d3b 7c0a 2b2d  point {2,2};|.+-
│ │ │ │ -00007a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00007a40: 2d2d 2d2d 2b0a 7c69 3320 3a20 7020 3d3d  ----+.|i3 : p ==
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│ │ │ │ -00007a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00007ad0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
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│ │ │ │ +00007610: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00007640: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00007650: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00007660: 3320 3a20 506f 696e 7420 2020 2020 2020  3 : Point       
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│ │ │ │ +00007680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00007690: 2d2d 2b0a 0a4e 6f74 6520 7468 6174 2074  --+..Note that t
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│ │ │ │ +00007730: 5761 7973 2074 6f20 7573 6520 7468 6973  Ways to use this
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│ │ │ │ +00007750: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00007760: 3d0a 0a20 202a 202a 6e6f 7465 2050 6f69  =..  * *note Poi
│ │ │ │ +00007770: 6e74 202a 2050 6f69 6e74 3a20 506f 696e  nt * Point: Poin
│ │ │ │ +00007780: 7420 5f73 7420 506f 696e 742c 202d 2d20  t _st Point, -- 
│ │ │ │ +00007790: 656e 7472 7977 6973 6520 7072 6f64 7563  entrywise produc
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│ │ │ │ +000077b0: 6a65 6374 6976 6520 706f 696e 7473 0a2d  jective points.-
│ │ │ │ +000077c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000077d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000077e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000077f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00007800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a  --------------..
│ │ │ │ +00007810: 5468 6520 736f 7572 6365 206f 6620 7468  The source of th
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│ │ │ │ +00007870: 6b61 6765 732f 4861 6461 6d61 7264 2e6d  kages/Hadamard.m
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│ │ │ │ +000078b0: 506f 696e 742c 2050 7265 763a 2050 6f69  Point, Prev: Poi
│ │ │ │ +000078c0: 6e74 205f 7374 2050 6f69 6e74 2c20 5570  nt _st Point, Up
│ │ │ │ +000078d0: 3a20 546f 700a 0a50 6f69 6e74 203d 3d20  : Top..Point == 
│ │ │ │ +000078e0: 506f 696e 7420 2d2d 2063 6865 636b 2065  Point -- check e
│ │ │ │ +000078f0: 7175 616c 6974 7920 6f66 2074 776f 2070  quality of two p
│ │ │ │ +00007900: 726f 6a65 6374 6976 6520 706f 696e 7473  rojective points
│ │ │ │ +00007910: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  .***************
│ │ │ │ +00007920: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00007930: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00007940: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ +00007950: 4f70 6572 6174 6f72 3a20 2a6e 6f74 6520  Operator: *note 
│ │ │ │ +00007960: 3d3d 3a20 284d 6163 6175 6c61 7932 446f  ==: (Macaulay2Do
│ │ │ │ +00007970: 6329 3d3d 2c0a 2020 2a20 5573 6167 653a  c)==,.  * Usage:
│ │ │ │ +00007980: 200a 2020 2020 2020 2020 7020 3d3d 2071   .        p == q
│ │ │ │ +00007990: 0a20 202a 2049 6e70 7574 733a 0a20 2020  .  * Inputs:.   
│ │ │ │ +000079a0: 2020 202a 2070 2c20 616e 2069 6e73 7461     * p, an insta
│ │ │ │ +000079b0: 6e63 6520 6f66 2074 6865 2074 7970 6520  nce of the type 
│ │ │ │ +000079c0: 2a6e 6f74 6

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